TSTP Solution File: COM024^5 by cocATP---0.2.0

%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : COM024^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n190.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 13:20:11 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  : COM024^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n190.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 07:43:31 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 0x1c094d0>, <kernel.DependentProduct object at 0x1c07998>) of role type named cTM
% Using role type
% Declaring cTM:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0x1e527e8>, <kernel.DependentProduct object at 0x1c07998>) of role type named cTH
% Using role type
% Declaring cTH:((fofType->fofType)->fofType)
% FOF formula ((forall (G:(fofType->fofType)), (((eq (fofType->fofType)) (cTM (cTH G))) G))->(forall (F:(fofType->fofType)), ((ex fofType) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))))) of role conjecture named cTHM9
% Conjecture to prove = ((forall (G:(fofType->fofType)), (((eq (fofType->fofType)) (cTM (cTH G))) G))->(forall (F:(fofType->fofType)), ((ex fofType) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((forall (G:(fofType->fofType)), (((eq (fofType->fofType)) (cTM (cTH G))) G))->(forall (F:(fofType->fofType)), ((ex fofType) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N))))))']
% Parameter fofType:Type.
% Parameter cTM:(fofType->(fofType->fofType)).
% Parameter cTH:((fofType->fofType)->fofType).
% Trying to prove ((forall (G:(fofType->fofType)), (((eq (fofType->fofType)) (cTM (cTH G))) G))->(forall (F:(fofType->fofType)), ((ex fofType) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N))))))
% Found eta_expansion000:=(eta_expansion00 (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))):(((eq (fofType->Prop)) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))) (fun (x:fofType)=> (((eq (fofType->fofType)) (cTM (F x))) (cTM x))))
% Found (eta_expansion00 (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))) as proof of (((eq (fofType->Prop)) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))) b)
% Found ((eta_expansion0 Prop) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))) as proof of (((eq (fofType->Prop)) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))) b)
% Found (((eta_expansion fofType) Prop) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))) as proof of (((eq (fofType->Prop)) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))) b)
% Found (((eta_expansion fofType) Prop) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))) as proof of (((eq (fofType->Prop)) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))) b)
% Found (((eta_expansion fofType) Prop) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))) as proof of (((eq (fofType->Prop)) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N)))) 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->fofType)) (cTM (F x0))) (cTM x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->fofType)) (cTM (F x0))) (cTM x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->fofType)) (cTM (F x0))) (cTM x0)))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->fofType)) (cTM (F x0))) (cTM x0)))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((eq (fofType->fofType)) (cTM (F x))) (cTM 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->fofType)) (cTM (F x0))) (cTM x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->fofType)) (cTM (F x0))) (cTM x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->fofType)) (cTM (F x0))) (cTM x0)))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->fofType)) (cTM (F x0))) (cTM x0)))
% Found (fun (x0:fofType)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (((eq (fofType->fofType)) (cTM (F x))) (cTM x))))
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found eta_expansion000:=(eta_expansion00 (cTM (F x0))):(((eq (fofType->fofType)) (cTM (F x0))) (fun (x:fofType)=> ((cTM (F x0)) x)))
% Found (eta_expansion00 (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found ((eta_expansion0 fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found (((eta_expansion fofType) fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found (((eta_expansion fofType) fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found (((eta_expansion fofType) fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM (F x0)) x1))->(P ((cTM (F x0)) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM (F x0)) x1))->(P ((cTM (F x0)) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found eq_ref00:=(eq_ref0 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (cTM x0))
% Found (eq_ref0 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref000:=(eq_ref00 P):((P ((cTM x0) x1))->(P ((cTM x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM x0) x1))->(P ((cTM x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (N:fofType)=> (((eq (fofType->fofType)) (cTM (F N))) (cTM N))))
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found x1:(P (cTM (F x0)))
% Instantiate: b:=(cTM (F x0)):(fofType->fofType)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (cTM x0))
% Found (eq_ref0 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found eta_expansion000:=(eta_expansion00 (cTM (F x0))):(((eq (fofType->fofType)) (cTM (F x0))) (fun (x:fofType)=> ((cTM (F x0)) x)))
% Found (eta_expansion00 (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found ((eta_expansion0 fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found (((eta_expansion fofType) fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found (((eta_expansion fofType) fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found (((eta_expansion fofType) fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found x1:(P (cTM (F x0)))
% Instantiate: f:=(cTM (F x0)):(fofType->fofType)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq fofType) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq fofType) (f x2)) ((cTM x0) x2))
% Found ((eq_ref fofType) (f x2)) as proof of (((eq fofType) (f x2)) ((cTM x0) x2))
% Found ((eq_ref fofType) (f x2)) as proof of (((eq fofType) (f x2)) ((cTM x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref fofType) (f x2))) as proof of (((eq fofType) (f x2)) ((cTM x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref fofType) (f x2))) as proof of (forall (x:fofType), (((eq fofType) (f x)) ((cTM x0) x)))
% Found x1:(P (cTM (F x0)))
% Instantiate: f:=(cTM (F x0)):(fofType->fofType)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq fofType) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq fofType) (f x2)) ((cTM x0) x2))
% Found ((eq_ref fofType) (f x2)) as proof of (((eq fofType) (f x2)) ((cTM x0) x2))
% Found ((eq_ref fofType) (f x2)) as proof of (((eq fofType) (f x2)) ((cTM x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref fofType) (f x2))) as proof of (((eq fofType) (f x2)) ((cTM x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref fofType) (f x2))) as proof of (forall (x:fofType), (((eq fofType) (f x)) ((cTM x0) x)))
% Found x2:=(x (cTM x0)):(((eq (fofType->fofType)) (cTM (cTH (cTM x0)))) (cTM x0))
% Instantiate: b:=(cTM x0):(fofType->fofType)
% Found x2 as proof of (((eq (fofType->fofType)) (cTM (cTH (cTM x0)))) b)
% Found x1:(P (cTM (F x0)))
% Instantiate: b:=(cTM (F x0)):(fofType->fofType)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found eq_ref00:=(eq_ref0 (cTM (F x0))):(((eq (fofType->fofType)) (cTM (F x0))) (cTM (F x0)))
% Found (eq_ref0 (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found ((eq_ref (fofType->fofType)) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found ((eq_ref (fofType->fofType)) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found ((eq_ref (fofType->fofType)) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM x0))
% Found eq_ref00:=(eq_ref0 (cTM (F x0))):(((eq (fofType->fofType)) (cTM (F x0))) (cTM (F x0)))
% Found (eq_ref0 (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found ((eq_ref (fofType->fofType)) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found ((eq_ref (fofType->fofType)) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found ((eq_ref (fofType->fofType)) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (fofType->fofType)) b) as proof of (P b)
% Found ((eq_ref (fofType->fofType)) b) as proof of (P b)
% Found ((eq_ref (fofType->fofType)) b) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (fun (x:fofType)=> ((cTM x0) x)))
% Found (eta_expansion00 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eta_expansion0 fofType) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion fofType) fofType) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion fofType) fofType) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion fofType) fofType) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM (F x0)) x1))->(P ((cTM (F x0)) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM (F x0)) x1))->(P ((cTM (F x0)) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found x1:(P (cTM x0))
% Instantiate: b:=(cTM x0):(fofType->fofType)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (cTM (F x0))):(((eq (fofType->fofType)) (cTM (F x0))) (cTM (F x0)))
% Found (eq_ref0 (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found ((eq_ref (fofType->fofType)) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found ((eq_ref (fofType->fofType)) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found ((eq_ref (fofType->fofType)) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((cTM (F x0)) x1))->(P ((cTM (F x0)) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM (F x0)) x1))->(P ((cTM (F x0)) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found eq_ref00:=(eq_ref0 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (cTM x0))
% Found (eq_ref0 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found eq_ref00:=(eq_ref0 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (cTM x0))
% Found (eq_ref0 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found x1:(P (cTM (F x0)))
% Instantiate: a:=(F x0):fofType
% Found x1 as proof of (P0 (cTM a))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found x1:(P (cTM (F x0)))
% Instantiate: f:=(cTM (F x0)):(fofType->fofType)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq fofType) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM x0))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM x0))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM x0))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM x0))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (forall (x:fofType), (((eq fofType) (f x)) ((cTM (cTH (cTM x0))) x)))
% Found x1:(P (cTM (F x0)))
% Instantiate: f:=(cTM (F x0)):(fofType->fofType)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq fofType) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM x0))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM x0))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM x0))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM x0))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (forall (x:fofType), (((eq fofType) (f x)) ((cTM (cTH (cTM x0))) x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->fofType)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->fofType)) b0) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b0) as proof of (((eq (fofType->fofType)) b0) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b0) as proof of (((eq (fofType->fofType)) b0) (cTM x0))
% Found ((eq_ref (fofType->fofType)) b0) as proof of (((eq (fofType->fofType)) b0) (cTM x0))
% Found eta_expansion000:=(eta_expansion00 (cTM (F x0))):(((eq (fofType->fofType)) (cTM (F x0))) (fun (x:fofType)=> ((cTM (F x0)) x)))
% Found (eta_expansion00 (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b0)
% Found ((eta_expansion0 fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b0)
% Found (((eta_expansion fofType) fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b0)
% Found (((eta_expansion fofType) fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b0)
% Found (((eta_expansion fofType) fofType) (cTM (F x0))) as proof of (((eq (fofType->fofType)) (cTM (F x0))) b0)
% Found x2:(P ((cTM (F x0)) x1))
% Instantiate: b:=((cTM (F x0)) x1):fofType
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found x2:(P ((cTM (F x0)) x1))
% Instantiate: b:=((cTM (F x0)) x1):fofType
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found x1:(P (F x0))
% Instantiate: b:=(F x0):fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found x1:(P (F x0))
% Instantiate: b:=(F x0):fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 (cTM x0))->(P1 (cTM x0)))
% Found (eq_ref00 P1) as proof of (P2 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (cTM x0))->(P1 (cTM x0)))
% Found (eq_ref00 P1) as proof of (P2 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (cTM x0))->(P1 (cTM x0)))
% Found (eq_ref00 P1) as proof of (P2 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (cTM x0))->(P1 (cTM x0)))
% Found (eq_ref00 P1) as proof of (P2 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found x1:(P (cTM x0))
% Instantiate: f:=(cTM x0):(fofType->fofType)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq fofType) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq fofType) (f x2)) ((cTM (F x0)) x2))
% Found ((eq_ref fofType) (f x2)) as proof of (((eq fofType) (f x2)) ((cTM (F x0)) x2))
% Found ((eq_ref fofType) (f x2)) as proof of (((eq fofType) (f x2)) ((cTM (F x0)) x2))
% Found (fun (x2:fofType)=> ((eq_ref fofType) (f x2))) as proof of (((eq fofType) (f x2)) ((cTM (F x0)) x2))
% Found (fun (x2:fofType)=> ((eq_ref fofType) (f x2))) as proof of (forall (x:fofType), (((eq fofType) (f x)) ((cTM (F x0)) x)))
% Found x1:(P (cTM x0))
% Instantiate: f:=(cTM x0):(fofType->fofType)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq fofType) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq fofType) (f x2)) ((cTM (F x0)) x2))
% Found ((eq_ref fofType) (f x2)) as proof of (((eq fofType) (f x2)) ((cTM (F x0)) x2))
% Found ((eq_ref fofType) (f x2)) as proof of (((eq fofType) (f x2)) ((cTM (F x0)) x2))
% Found (fun (x2:fofType)=> ((eq_ref fofType) (f x2))) as proof of (((eq fofType) (f x2)) ((cTM (F x0)) x2))
% Found (fun (x2:fofType)=> ((eq_ref fofType) (f x2))) as proof of (forall (x:fofType), (((eq fofType) (f x)) ((cTM (F x0)) x)))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM (F x0)) x1))->(P1 ((cTM (F x0)) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM (F x0)) x1))->(P1 ((cTM (F x0)) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM (F x0)) x1))->(P1 ((cTM (F x0)) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM (F x0)) x1))->(P1 ((cTM (F x0)) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P1) as proof of (P2 ((cTM (F x0)) x1))
% Found x1:(P (cTM (F x0)))
% Instantiate: a:=(F x0):fofType
% Found x1 as proof of (P0 (cTM a))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) (cTH (cTM x0)))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (cTH (cTM x0)))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (cTH (cTM x0)))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (cTH (cTM x0)))
% Found eq_ref000:=(eq_ref00 P1):((P1 (cTM x0))->(P1 (cTM x0)))
% Found (eq_ref00 P1) as proof of (P2 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (cTM x0))->(P1 (cTM x0)))
% Found (eq_ref00 P1) as proof of (P2 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found eq_ref000:=(eq_ref00 P1):((P1 (cTM x0))->(P1 (cTM x0)))
% Found (eq_ref00 P1) as proof of (P2 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (cTM x0))->(P1 (cTM x0)))
% Found (eq_ref00 P1) as proof of (P2 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P1) as proof of (P2 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (fun (x:fofType)=> ((cTM x0) x)))
% Found (eta_expansion_dep00 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (fun (x:fofType)=> ((cTM x0) x)))
% Found (eta_expansion_dep00 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (fun (x:fofType)=> ((cTM x0) x)))
% Found (eta_expansion_dep00 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->fofType)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b) as proof of (((eq (fofType->fofType)) b) (cTM (F x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (fun (x:fofType)=> ((cTM x0) x)))
% Found (eta_expansion_dep00 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found eq_ref000:=(eq_ref00 P):((P (cTM (F x0)))->(P (cTM (F x0))))
% Found (eq_ref00 P) as proof of (P0 (cTM (F x0)))
% Found ((eq_ref0 (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found (((eq_ref (fofType->fofType)) (cTM (F x0))) P) as proof of (P0 (cTM (F x0)))
% Found x1:(P (cTM (F x0)))
% Instantiate: b:=(cTM (F x0)):(fofType->fofType)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cTM (cTH (cTM (cTH (cTM x0)))))):(((eq (fofType->fofType)) (cTM (cTH (cTM (cTH (cTM x0)))))) (fun (x:fofType)=> ((cTM (cTH (cTM (cTH (cTM x0))))) x)))
% Found (eta_expansion_dep00 (cTM (cTH (cTM (cTH (cTM x0)))))) as proof of (((eq (fofType->fofType)) (cTM (cTH (cTM (cTH (cTM x0)))))) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> fofType)) (cTM (cTH (cTM (cTH (cTM x0)))))) as proof of (((eq (fofType->fofType)) (cTM (cTH (cTM (cTH (cTM x0)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) (cTM (cTH (cTM (cTH (cTM x0)))))) as proof of (((eq (fofType->fofType)) (cTM (cTH (cTM (cTH (cTM x0)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) (cTM (cTH (cTM (cTH (cTM x0)))))) as proof of (((eq (fofType->fofType)) (cTM (cTH (cTM (cTH (cTM x0)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> fofType)) (cTM (cTH (cTM (cTH (cTM x0)))))) as proof of (((eq (fofType->fofType)) (cTM (cTH (cTM (cTH (cTM x0)))))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((cTM (F x0)) x1))->(P ((cTM (F x0)) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM (F x0)) x1))->(P ((cTM (F x0)) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM x0) x1))->(P ((cTM x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM x0) x1))->(P ((cTM x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM x0) x1))->(P ((cTM x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM x0) x1))->(P ((cTM x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM x0) x1))->(P ((cTM x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM x0) x1))->(P ((cTM x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found ((eq_ref fofType) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref00:=(eq_ref0 (b x1)):(((eq fofType) (b x1)) (b x1))
% Found (eq_ref0 (b x1)) as proof of (P b)
% Found ((eq_ref fofType) (b x1)) as proof of (P b)
% Found ((eq_ref fofType) (b x1)) as proof of (P b)
% Found ((eq_ref fofType) (b x1)) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (fun (x:fofType)=> ((cTM x0) x)))
% Found (eta_expansion_dep00 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found eq_ref00:=(eq_ref0 (b x1)):(((eq fofType) (b x1)) (b x1))
% Found (eq_ref0 (b x1)) as proof of (P b)
% Found ((eq_ref fofType) (b x1)) as proof of (P b)
% Found ((eq_ref fofType) (b x1)) as proof of (P b)
% Found ((eq_ref fofType) (b x1)) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (fun (x:fofType)=> ((cTM x0) x)))
% Found (eta_expansion_dep00 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found x1:(P x0)
% Instantiate: x0:=(F b):fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found x1:(P (cTM x0))
% Instantiate: a:=x0:fofType
% Found x1 as proof of (P0 (cTM a))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) (F x0))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (F x0))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (F x0))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (F x0))
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found x1:(P (cTM x0))
% Instantiate: f:=(cTM x0):(fofType->fofType)
% Found x1 as proof of (P0 f)
% Found x1:(P (cTM x0))
% Instantiate: f:=(cTM x0):(fofType->fofType)
% Found x1 as proof of (P0 f)
% Found x2:(P ((cTM x0) x1))
% Instantiate: b:=((cTM x0) x1):fofType
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found x2:(P ((cTM x0) x1))
% Instantiate: b:=((cTM x0) x1):fofType
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq fofType) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (F x0)))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (F x0)))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (F x0)))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (F x0)))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (forall (x:fofType), (((eq fofType) (f x)) ((cTM (cTH (cTM (F x0)))) x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq fofType) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (F x0)))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (F x0)))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (F x0)))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (F x0)))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (forall (x:fofType), (((eq fofType) (f x)) ((cTM (cTH (cTM (F x0)))) x)))
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found (((eq_ref fofType) x0) P) as proof of (P0 x0)
% Found x1:(P x0)
% Instantiate: x0:=(F b):fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found x1:(P x0)
% Instantiate: b:=x0:fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->fofType)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->fofType)) b0) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b0) as proof of (((eq (fofType->fofType)) b0) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b0) as proof of (((eq (fofType->fofType)) b0) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b0) as proof of (((eq (fofType->fofType)) b0) (cTM (F x0)))
% Found eq_ref00:=(eq_ref0 (cTM x0)):(((eq (fofType->fofType)) (cTM x0)) (cTM x0))
% Found (eq_ref0 (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b0)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b0)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b0)
% Found ((eq_ref (fofType->fofType)) (cTM x0)) as proof of (((eq (fofType->fofType)) (cTM x0)) b0)
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found x2:(P ((cTM x0) x1))
% Instantiate: b:=((cTM x0) x1):fofType
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found x2:(P ((cTM x0) x1))
% Instantiate: b:=((cTM x0) x1):fofType
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found x1:(P x0)
% Instantiate: b:=x0:fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref00:=(eq_ref0 ((cTM x0) x1)):(((eq fofType) ((cTM x0) x1)) ((cTM x0) x1))
% Found (eq_ref0 ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found ((eq_ref fofType) ((cTM x0) x1)) as proof of (((eq fofType) ((cTM x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found x1:(P x0)
% Instantiate: b:=x0:fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found x1:(P (cTM (F x0)))
% Instantiate: f:=(cTM (F x0)):(fofType->fofType)
% Found x1 as proof of (P0 f)
% Found x1:(P (cTM (F x0)))
% Instantiate: f:=(cTM (F x0)):(fofType->fofType)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq fofType) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (cTH (cTM x0))))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (cTH (cTM x0))))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (cTH (cTM x0))))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (cTH (cTM x0))))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (forall (x:fofType), (((eq fofType) (f x)) ((cTM (cTH (cTM (cTH (cTM x0))))) x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq fofType) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (cTH (cTM x0))))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (cTH (cTM x0))))) x3))
% Found ((eq_ref fofType) (f x3)) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (cTH (cTM x0))))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (((eq fofType) (f x3)) ((cTM (cTH (cTM (cTH (cTM x0))))) x3))
% Found (fun (x3:fofType)=> ((eq_ref fofType) (f x3))) as proof of (forall (x:fofType), (((eq fofType) (f x)) ((cTM (cTH (cTM (cTH (cTM x0))))) x)))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b0)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b0)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b0)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((cTM x0) x1))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((cTM x0) x1))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((cTM x0) x1))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((cTM x0) x1))
% Found eq_ref00:=(eq_ref0 ((cTM (F x0)) x1)):(((eq fofType) ((cTM (F x0)) x1)) ((cTM (F x0)) x1))
% Found (eq_ref0 ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b0)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b0)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b0)
% Found ((eq_ref fofType) ((cTM (F x0)) x1)) as proof of (((eq fofType) ((cTM (F x0)) x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) ((cTM x0) x1))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((cTM x0) x1))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((cTM x0) x1))
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) ((cTM x0) x1))
% Found x1:(P (cTM x0))
% Instantiate: b:=(cTM x0):(fofType->fofType)
% Found x1 as proof of (P0 b)
% Found x2:=(x (cTM (F x0))):(((eq (fofType->fofType)) (cTM (cTH (cTM (F x0))))) (cTM (F x0)))
% Instantiate: b:=(cTM (F x0)):(fofType->fofType)
% Found x2 as proof of (((eq (fofType->fofType)) (cTM (cTH (cTM (F x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found x1:(P x0)
% Instantiate: b:=x0:fofType
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (F x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq fofType) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) b)
% Found ((eq_ref fofType) x0) as proof of (((eq fofType) x0) 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 x1:(P (cTM x0))
% Instantiate: a:=x0:fofType
% Found x1 as proof of (P0 (cTM a))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) (cTH (cTM (F x0))))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (cTH (cTM (F x0))))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (cTH (cTM (F x0))))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (cTH (cTM (F x0))))
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b0)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b0)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b0)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) x0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) x0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) x0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) x0)
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM x0) x1))->(P1 ((cTM x0) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM x0) x1))->(P1 ((cTM x0) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM x0) x1))->(P1 ((cTM x0) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM x0) x1))->(P1 ((cTM x0) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM x0) x1))->(P1 ((cTM x0) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM x0) x1))->(P1 ((cTM x0) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM x0) x1))->(P1 ((cTM x0) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((cTM x0) x1))->(P1 ((cTM x0) x1)))
% Found (eq_ref00 P1) as proof of (P2 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P1) as proof of (P2 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->fofType)) b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->fofType)) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->fofType)) b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->fofType)) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found eq_ref000:=(eq_ref00 P1):((P1 x0)->(P1 x0))
% Found (eq_ref00 P1) as proof of (P2 x0)
% Found ((eq_ref0 x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found eq_ref000:=(eq_ref00 P1):((P1 x0)->(P1 x0))
% Found (eq_ref00 P1) as proof of (P2 x0)
% Found ((eq_ref0 x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found eq_ref000:=(eq_ref00 P1):((P1 x0)->(P1 x0))
% Found (eq_ref00 P1) as proof of (P2 x0)
% Found ((eq_ref0 x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found eq_ref000:=(eq_ref00 P1):((P1 x0)->(P1 x0))
% Found (eq_ref00 P1) as proof of (P2 x0)
% Found ((eq_ref0 x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found eq_ref000:=(eq_ref00 P1):((P1 x0)->(P1 x0))
% Found (eq_ref00 P1) as proof of (P2 x0)
% Found ((eq_ref0 x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found eq_ref000:=(eq_ref00 P1):((P1 x0)->(P1 x0))
% Found (eq_ref00 P1) as proof of (P2 x0)
% Found ((eq_ref0 x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->fofType)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->fofType)) b0) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b0) as proof of (((eq (fofType->fofType)) b0) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b0) as proof of (((eq (fofType->fofType)) b0) (cTM (F x0)))
% Found ((eq_ref (fofType->fofType)) b0) as proof of (((eq (fofType->fofType)) b0) (cTM (F x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->fofType)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->fofType)) b) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> fofType)) b) as proof of (((eq (fofType->fofType)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) b) as proof of (((eq (fofType->fofType)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) b) as proof of (((eq (fofType->fofType)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> fofType)) b) as proof of (((eq (fofType->fofType)) b) b0)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref fofType) b) P) as proof of (P0 b)
% Found (((eq_ref fofType) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref fofType) b) P) as proof of (P0 b)
% Found (((eq_ref fofType) b) P) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (F x0)):(((eq fofType) (F x0)) (F x0))
% Found (eq_ref0 (F x0)) as proof of (((eq fofType) (F x0)) b0)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b0)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b0)
% Found ((eq_ref fofType) (F x0)) as proof of (((eq fofType) (F x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) x0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) x0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) x0)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) x0)
% Found eq_ref000:=(eq_ref00 P):((P ((cTM (F x0)) x1))->(P ((cTM (F x0)) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM (F x0)) x1))->(P ((cTM (F x0)) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM (F x0)) x1))
% Found ((eq_ref0 ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found (((eq_ref fofType) ((cTM (F x0)) x1)) P) as proof of (P0 ((cTM (F x0)) x1))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref fofType) b) P) as proof of (P0 b)
% Found (((eq_ref fofType) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 x0)->(P1 x0))
% Found (eq_ref00 P1) as proof of (P2 x0)
% Found ((eq_ref0 x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found eq_ref000:=(eq_ref00 P1):((P1 x0)->(P1 x0))
% Found (eq_ref00 P1) as proof of (P2 x0)
% Found ((eq_ref0 x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found (((eq_ref fofType) x0) P1) as proof of (P2 x0)
% Found eq_ref000:=(eq_ref00 P):((P (cTM x0))->(P (cTM x0)))
% Found (eq_ref00 P) as proof of (P0 (cTM x0))
% Found ((eq_ref0 (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found (((eq_ref (fofType->fofType)) (cTM x0)) P) as proof of (P0 (cTM x0))
% Found x1:(P (cTM (F x0)))
% Instantiate: a:=(F x0):fofType
% Found x1 as proof of (P0 (cTM a))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) (cTH (cTM (cTH (cTM x0)))))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (cTH (cTM (cTH (cTM x0)))))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (cTH (cTM (cTH (cTM x0)))))
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) (cTH (cTM (cTH (cTM x0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM x0) x1))->(P ((cTM x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P ((cTM x0) x1))->(P ((cTM x0) x1)))
% Found (eq_ref00 P) as proof of (P0 ((cTM x0) x1))
% Found ((eq_ref0 ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found (((eq_ref fofType) ((cTM x0) x1)) P) as proof of (P0 ((cTM x0) x1))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref fofType) b) P) as proof of (P0 b)
% Found (((eq_ref fofType) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P ((cTM x0) x1))->(P ((cTM x0) x1)))
% Found (eq_ref00 P)
% EOF
%------------------------------------------------------------------------------