%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU839^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n111.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:16 EDT 2014

% Result   : Timeout 300.09s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU839^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n111.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:36:06 CDT 2014
% % CPUTime  : 300.09 
% 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 0xeb6440>, <kernel.Type object at 0xff2878>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (S:(a->Prop)) (T:(a->Prop)), (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))) of role conjecture named cGAZING_THM26_pme
% Conjecture to prove = (forall (S:(a->Prop)) (T:(a->Prop)), (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (S:(a->Prop)) (T:(a->Prop)), (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))']
% Parameter a:Type.
% Trying to prove (forall (S:(a->Prop)) (T:(a->Prop)), (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (x:a)=> ((or (S x)) (T x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (x:a)=> ((or (T x)) (S x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x01:(P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P0 ((or (S x)) (T x)))
% Found x01:(P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P0 ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found x01:(P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P0 ((or (T x)) (S x)))
% Found x01:(P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P0 ((or (T x)) (S x)))
% Found x:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (S Xz)) (T Xz))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found x:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (S Xz)) (T Xz))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (T x)) (S x))))
% Found x:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (S Xz)) (T Xz))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (T x)) (S x))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (x:a)=> ((or (S x)) (T x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (x:a)=> ((or (T x)) (S x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found x:(P0 b)
% Instantiate: b:=(fun (Xz:a)=> ((or (S Xz)) (T Xz))):(a->Prop)
% Found (fun (x:(P0 b))=> x) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of ((P0 b)->(P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of (P b)
% Found x3:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x3:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x3) as proof of (P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x3:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x3) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (x:a)=> ((or (S x)) (T x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (T Xz)) (S Xz))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (x:a)=> ((or (S x)) (T x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (T Xz)) (S Xz))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (S x)) (T x))))
% Found x:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (T Xz)) (S Xz))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (S x)) (T x))))
% Found x0:(P ((or (S x)) (T x)))
% Instantiate: b:=((or (S x)) (T x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found x0:(P ((or (S x)) (T x)))
% Instantiate: b:=((or (S x)) (T x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (x:a)=> ((or (T x)) (S x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P2 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P2 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P2 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P1 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P2 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x3:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x3:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x3) as proof of (P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x3:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x3) as proof of (P0 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (x:a)=> ((or (T x)) (S x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found x01:(P1 ((or (S x)) (T x)))
% Found (fun (x01:(P1 ((or (S x)) (T x))))=> x01) as proof of (P1 ((or (S x)) (T x)))
% Found (fun (x01:(P1 ((or (S x)) (T x))))=> x01) as proof of (P2 ((or (S x)) (T x)))
% Found x01:(P1 ((or (S x)) (T x)))
% Found (fun (x01:(P1 ((or (S x)) (T x))))=> x01) as proof of (P1 ((or (S x)) (T x)))
% Found (fun (x01:(P1 ((or (S x)) (T x))))=> x01) as proof of (P2 ((or (S x)) (T x)))
% Found x01:(P1 ((or (S x)) (T x)))
% Found (fun (x01:(P1 ((or (S x)) (T x))))=> x01) as proof of (P1 ((or (S x)) (T x)))
% Found (fun (x01:(P1 ((or (S x)) (T x))))=> x01) as proof of (P2 ((or (S x)) (T x)))
% Found x01:(P1 ((or (S x)) (T x)))
% Found (fun (x01:(P1 ((or (S x)) (T x))))=> x01) as proof of (P1 ((or (S x)) (T x)))
% Found (fun (x01:(P1 ((or (S x)) (T x))))=> x01) as proof of (P2 ((or (S x)) (T x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (x:a)=> ((or (T x)) (S x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b0)
% Found x02:(P ((or (S x)) (T x)))
% Found (fun (x02:(P ((or (S x)) (T x))))=> x02) as proof of (P ((or (S x)) (T x)))
% Found (fun (x02:(P ((or (S x)) (T x))))=> x02) as proof of (P0 ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found x02:(P ((or (S x)) (T x)))
% Found (fun (x02:(P ((or (S x)) (T x))))=> x02) as proof of (P ((or (S x)) (T x)))
% Found (fun (x02:(P ((or (S x)) (T x))))=> x02) as proof of (P0 ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found x0:(P0 b)
% Instantiate: b:=((or (S x)) (T x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 ((or (S x)) (T x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 ((or (S x)) (T x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found x0:(P0 b)
% Instantiate: b:=((or (S x)) (T x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 ((or (S x)) (T x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 ((or (S x)) (T x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (x:a)=> ((or (T x)) (S x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found x0:(P ((or (T x)) (S x)))
% Instantiate: b:=((or (T x)) (S x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found x0:(P ((or (T x)) (S x)))
% Instantiate: b:=((or (T x)) (S x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (x:a)=> ((or (S x)) (T x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x0:(P ((or (T x)) (S x)))
% Instantiate: b:=((or (T x)) (S x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found x0:(P ((or (T x)) (S x)))
% Instantiate: b:=((or (T x)) (S x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P2 ((or (T x)) (S x)))
% Found x01:(P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P2 ((or (T x)) (S x)))
% Found x01:(P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P2 ((or (T x)) (S x)))
% Found x01:(P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P2 ((or (T x)) (S x)))
% Found x01:(P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P2 ((or (T x)) (S x)))
% Found x01:(P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P2 ((or (T x)) (S x)))
% Found x01:(P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P2 ((or (T x)) (S x)))
% Found x01:(P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P1 ((or (T x)) (S x)))
% Found (fun (x01:(P1 ((or (T x)) (S x))))=> x01) as proof of (P2 ((or (T x)) (S x)))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found x02:(P ((or (T x)) (S x)))
% Found (fun (x02:(P ((or (T x)) (S x))))=> x02) as proof of (P ((or (T x)) (S x)))
% Found (fun (x02:(P ((or (T x)) (S x))))=> x02) as proof of (P0 ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found x02:(P ((or (T x)) (S x)))
% Found (fun (x02:(P ((or (T x)) (S x))))=> x02) as proof of (P ((or (T x)) (S x)))
% Found (fun (x02:(P ((or (T x)) (S x))))=> x02) as proof of (P0 ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found x02:(P ((or (T x)) (S x)))
% Found (fun (x02:(P ((or (T x)) (S x))))=> x02) as proof of (P ((or (T x)) (S x)))
% Found (fun (x02:(P ((or (T x)) (S x))))=> x02) as proof of (P0 ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found x02:(P ((or (T x)) (S x)))
% Found (fun (x02:(P ((or (T x)) (S x))))=> x02) as proof of (P ((or (T x)) (S x)))
% Found (fun (x02:(P ((or (T x)) (S x))))=> x02) as proof of (P0 ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found x0:(P0 b)
% Instantiate: b:=((or (T x)) (S x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 ((or (T x)) (S x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 ((or (T x)) (S x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found x0:(P0 b)
% Instantiate: b:=((or (T x)) (S x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 ((or (T x)) (S x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 ((or (T x)) (S x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (x:a)=> ((or (T x)) (S x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (T x))
% Found x0:(P ((or (S x)) (T x)))
% Instantiate: b:=((or (S x)) (T x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found x0:(P ((or (S x)) (T x)))
% Instantiate: b:=((or (S x)) (T x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x01:(P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P0 ((or (S x)) (T x)))
% Found x01:(P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P0 ((or (S x)) (T x)))
% Found x01:(P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P0 ((or (S x)) (T x)))
% Found x01:(P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P0 ((or (S x)) (T x)))
% Found x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P0 ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P0 ((or (T x)) (S x)))
% Found x01:(P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P0 ((or (T x)) (S x)))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x02:(P ((or (S x)) (T x)))
% Found (fun (x02:(P ((or (S x)) (T x))))=> x02) as proof of (P ((or (S x)) (T x)))
% Found (fun (x02:(P ((or (S x)) (T x))))=> x02) as proof of (P0 ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found x02:(P ((or (S x)) (T x)))
% Found (fun (x02:(P ((or (S x)) (T x))))=> x02) as proof of (P ((or (S x)) (T x)))
% Found (fun (x02:(P ((or (S x)) (T x))))=> x02) as proof of (P0 ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (S x))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x01:(P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P0 ((or (T x)) (S x)))
% Found x01:(P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P0 ((or (T x)) (S x)))
% Found x01:(P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P0 ((or (T x)) (S x)))
% Found x01:(P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P0 ((or (T x)) (S x)))
% Found x01:(P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P0 ((or (T x)) (S x)))
% Found x01:(P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P ((or (T x)) (S x)))
% Found (fun (x01:(P ((or (T x)) (S x))))=> x01) as proof of (P0 ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b0)
% Found x01:(P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P0 ((or (S x)) (T x)))
% Found x01:(P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P0 ((or (S x)) (T x)))
% Found x01:(P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P0 ((or (S x)) (T x)))
% Found x01:(P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P ((or (S x)) (T x)))
% Found (fun (x01:(P ((or (S x)) (T x))))=> x01) as proof of (P0 ((or (S x)) (T x)))
% Found x:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (S Xz)) (T Xz))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found x:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (S Xz)) (T Xz))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (T x)) (S x))))
% Found x:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (S Xz)) (T Xz))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or (T x0)) (S x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (T x)) (S x))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (x:a)=> ((or (S x)) (T x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or (S x)) (T x))):(((eq Prop) ((or (S x)) (T x))) ((or (S x)) (T x)))
% Found (eq_ref0 ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found ((eq_ref Prop) ((or (S x)) (T x))) as proof of (((eq Prop) ((or (S x)) (T x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (T x)) (S x)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) (fun (x:a)=> ((or (T x)) (S x))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (T Xz)) (S Xz)))) b)
% Found x:(P0 b)
% Instantiate: b:=(fun (Xz:a)=> ((or (S Xz)) (T Xz))):(a->Prop)
% Found (fun (x:(P0 b))=> x) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of ((P0 b)->(P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))
% Found (fun (P0:((a->Prop)->Prop)) (x:(P0 b))=> x) as proof of (P b)
% Found x3:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x3:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x3) as proof of (P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (fun (x3:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz)))))=> x3) as proof of (P0 (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (x:a)=> ((or (S x)) (T x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found x:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Instantiate: b:=(fun (Xz:a)=> ((or (T Xz)) (S Xz))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Found (eq_ref0 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b)
% Found x:(P (fun (Xz:a)=> ((or (S Xz)) (T Xz))))
% Instantiate: a0:=(fun (Xz:a)=> ((or (S Xz)) (T Xz))):(a->Prop)
% Found x as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))):(((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) (fun (x:a)=> ((or (S x)) (T x))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or (S Xz)) (T Xz)))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or (S x)) (T x)))
% Found x:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (T Xz)) (S Xz))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (S x)) (T x))))
% Found x:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (T Xz)) (S Xz))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (S x)) (T x))))
% Found x:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (T Xz)) (S Xz))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (S x)) (T x))))
% Found x:(P (fun (Xz:a)=> ((or (T Xz)) (S Xz))))
% Instantiate: f:=(fun (Xz:a)=> ((or (T Xz)) (S Xz))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or (S x0)) (T x0)))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or (S x)) (T x))))
% Found x0:(P ((or (S x)) (T x)))
% Instantiate: b:=((or (S x)) (T x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found x0:(P ((or (S x)) (T x)))
% Instantiate: b:=((or (S x)) (T x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found x0:(P ((or (S x)) (T x)))
% Instantiate: b:=((or (S x)) (T x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref0 ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found ((eq_ref Prop) ((or (T x)) (S x))) as proof of (((eq Prop) ((or (T x)) (S x))) b)
% Found x0:(P ((or (S x)) (T x)))
% Instantiate: b:=((or (S x)) (T x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or (T x)) (S x))):(((eq Prop) ((or (T x)) (S x))) ((or (T x)) (S x)))
% Found (eq_ref
% EOF
%------------------------------------------------------------------------------