TSTP Solution File: SET173^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SET173^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n117.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 13:30:19 EDT 2014
% Result : Timeout 300.04s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SET173^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n117.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 09:16:06 CDT 2014
% % CPUTime : 300.04
% 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 0x276ef80>, <kernel.Type object at 0x276ed88>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) X)) of role conjecture named cBOOL_PROP_68_pme
% Conjecture to prove = (forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) X)):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) X))']
% Parameter a:Type.
% Trying to prove (forall (X:(a->Prop)) (Y:(a->Prop)), (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) X))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found x2:(P X)
% Found (fun (x2:(P X))=> x2) as proof of (P X)
% Found (fun (x2:(P X))=> x2) as proof of (P0 X)
% Found x2:(P X)
% Found (fun (x2:(P X))=> x2) as proof of (P X)
% Found (fun (x2:(P X))=> x2) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found x01:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found x01:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))):(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)) (X x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (X x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (X x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (X x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))):(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)) (X x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (X x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (X x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (X x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found x:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (x:a)=> ((and (X x)) ((or (X x)) (Y x)))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found x:(P X)
% Instantiate: f:=X:(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)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) ((or (X x)) (Y x)))))
% Found x:(P X)
% Instantiate: f:=X:(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)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) ((or (X x)) (Y x)))))
% Found x0:(P ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=((and (X x)) ((or (X x)) (Y x))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found x0:(P ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=((and (X x)) ((or (X x)) (Y x))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found x2:(P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P2 X)
% Found x2:(P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P2 X)
% Found x3:(P X)
% Found (fun (x3:(P X))=> x3) as proof of (P X)
% Found (fun (x3:(P X))=> x3) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% 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) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x2:(P X)
% Found (fun (x2:(P X))=> x2) as proof of (P X)
% Found (fun (x2:(P X))=> x2) as proof of (P0 X)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y 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 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y 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 x2:(P X)
% Found (fun (x2:(P X))=> x2) as proof of (P X)
% Found (fun (x2:(P X))=> x2) as proof of (P0 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 X)
% Found (fun (x2:(P X))=> x2) as proof of (P X)
% Found (fun (x2:(P X))=> x2) as proof of (P0 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 eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y 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 ((and (X x)) ((or (X x)) (Y x))))
% Found (fun (x01:(P ((and (X x)) ((or (X x)) (Y x)))))=> x01) as proof of (P ((and (X x)) ((or (X x)) (Y x))))
% Found (fun (x01:(P ((and (X x)) ((or (X x)) (Y x)))))=> x01) as proof of (P0 ((and (X x)) ((or (X x)) (Y x))))
% Found x01:(P ((and (X x)) ((or (X x)) (Y x))))
% Found (fun (x01:(P ((and (X x)) ((or (X x)) (Y x)))))=> x01) as proof of (P ((and (X x)) ((or (X x)) (Y x))))
% Found (fun (x01:(P ((and (X x)) ((or (X x)) (Y x)))))=> x01) as proof of (P0 ((and (X x)) ((or (X x)) (Y x))))
% Found x01:(P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P2 (X x))
% Found x01:(P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P2 (X x))
% Found x01:(P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P2 (X x))
% Found x01:(P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P2 (X x))
% Found x01:(P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P2 (X x))
% Found x01:(P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P1 (X x))
% Found (fun (x01:(P1 (X x)))=> x01) as proof of (P2 (X x))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) X)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) X)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) X)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) X)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found x02:(P (X x))
% Found (fun (x02:(P (X x)))=> x02) as proof of (P (X x))
% Found (fun (x02:(P (X x)))=> x02) as proof of (P0 (X x))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found x02:(P (X x))
% Found (fun (x02:(P (X x)))=> x02) as proof of (P (X x))
% Found (fun (x02:(P (X x)))=> x02) as proof of (P0 (X x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found x02:(P (X x))
% Found (fun (x02:(P (X x)))=> x02) as proof of (P (X x))
% Found (fun (x02:(P (X x)))=> x02) as proof of (P0 (X x))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found x02:(P (X x))
% Found (fun (x02:(P (X x)))=> x02) as proof of (P (X x))
% Found (fun (x02:(P (X x)))=> x02) as proof of (P0 (X x))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found x0:(P0 b)
% Instantiate: b:=(X x):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (X x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (X x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found x0:(P0 b)
% Instantiate: b:=(X x):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (X x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (X x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found x0:(P ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=((and (X x)) ((or (X x)) (Y x))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found x0:(P ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=((and (X x)) ((or (X x)) (Y x))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found x2:(P X)
% Found (fun (x2:(P X))=> x2) as proof of (P X)
% Found (fun (x2:(P X))=> x2) as proof of (P0 X)
% Found x2:(P X)
% Found (fun (x2:(P X))=> x2) as proof of (P X)
% Found (fun (x2:(P X))=> x2) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X 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 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X 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 X)
% Found (fun (x2:(P X))=> x2) as proof of (P X)
% Found (fun (x2:(P X))=> x2) as proof of (P0 X)
% Found x2:(P X)
% Found (fun (x2:(P X))=> x2) as proof of (P X)
% Found (fun (x2:(P X))=> x2) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((or (X x)) (Y x)))
% Found x0:(P ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=(fun (x1:a)=> ((and (X x1)) ((or (X x1)) (Y x1)))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found x0:(P ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=(fun (x1:a)=> ((and (X x1)) ((or (X x1)) (Y x1)))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) 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 b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y 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) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y 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) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y 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) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X 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 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found x01:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found x01:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X 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:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found x01:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X 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) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X 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 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found x01:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found x01:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found x01:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found x01:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found x01:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found x01:(P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P (X x))
% Found (fun (x01:(P (X x)))=> x01) as proof of (P0 (X x))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y 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 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y 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 (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) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X 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) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X 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 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b0)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y 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) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X 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 x01:(P ((and (X x)) ((or (X x)) (Y x))))
% Found (fun (x01:(P ((and (X x)) ((or (X x)) (Y x)))))=> x01) as proof of (P ((and (X x)) ((or (X x)) (Y x))))
% Found (fun (x01:(P ((and (X x)) ((or (X x)) (Y x)))))=> x01) as proof of (P0 ((and (X x)) ((or (X x)) (Y x))))
% Found x01:(P ((and (X x)) ((or (X x)) (Y x))))
% Found (fun (x01:(P ((and (X x)) ((or (X x)) (Y x)))))=> x01) as proof of (P ((and (X x)) ((or (X x)) (Y x))))
% Found (fun (x01:(P ((and (X x)) ((or (X x)) (Y x)))))=> x01) as proof of (P0 ((and (X x)) ((or (X x)) (Y 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 x:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))):(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)) (X x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (X x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (X x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (X x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))):(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)) (X x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (X x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (X x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (X x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found x:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Instantiate: a0:=(fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))):(a->Prop)
% Found x as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% 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) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y 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) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x)) ((or (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (x:a)=> ((and (X x)) ((or (X x)) (Y x)))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% 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 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b0)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X 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 x:(P X)
% Instantiate: f:=X:(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)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) ((or (X x)) (Y x)))))
% Found x:(P X)
% Instantiate: f:=X:(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)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) ((or (X x)) (Y x)))))
% Found x:(P X)
% Instantiate: f:=X:(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)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) ((or (X x)) (Y x)))))
% Found x:(P X)
% Instantiate: f:=X:(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)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) ((or (X x)) (Y x)))))
% Found x0:(P ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=((and (X x)) ((or (X x)) (Y x))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found x0:(P ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=((and (X x)) ((or (X x)) (Y x))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found x0:(P ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=((and (X x)) ((or (X x)) (Y x))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found x0:(P ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=((and (X x)) ((or (X x)) (Y x))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x:(P1 X)
% Instantiate: b:=X:(a->Prop)
% Found x as proof of (P2 b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (x:a)=> ((and (X x)) ((or (X x)) (Y x)))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (fun (x2:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))))=> x2) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Instantiate: b0:=(fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))):(a->Prop)
% Found x as proof of (P0 b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 ((and (X x)) ((or (X x)) (Y x)))):(((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) ((and (X x)) ((or (X x)) (Y x))))
% Found (eq_ref0 ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found ((eq_ref Prop) ((and (X x)) ((or (X x)) (Y x)))) as proof of (((eq Prop) ((and (X x)) ((or (X x)) (Y x)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (x:a)=> ((and (X x)) ((or (X x)) (Y x)))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x2:(P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P2 X)
% Found x2:(P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P2 X)
% Found x2:(P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P2 X)
% Found x2:(P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P1 X)
% Found (fun (x2:(P1 X))=> x2) as proof of (P2 X)
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))):(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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))):(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)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x:(P (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found x as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (x:a)=> ((and (X x)) ((or (X x)) (Y x)))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found x3:(P X)
% Found (fun (x3:(P X))=> x3) as proof of (P X)
% Found (fun (x3:(P X))=> x3) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found x3:(P X)
% Found (fun (x3:(P X))=> x3) as proof of (P X)
% Found (fun (x3:(P X))=> x3) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found x:(P X)
% Instantiate: a0:=X:(a->Prop)
% Found x as proof of (P0 a0)
% 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 (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found x:(P1 X)
% Instantiate: f:=X:(a->Prop)
% Found x as proof of (P2 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)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) ((or (X x)) (Y x)))))
% Found x:(P1 X)
% Instantiate: f:=X:(a->Prop)
% Found x as proof of (P2 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)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (X x0)) ((or (X x0)) (Y x0))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) ((or (X x)) (Y 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 eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b0)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) (fun (x:a)=> ((and (X x)) ((or (X x)) (Y x)))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((or (X Xx)) (Y Xx))))) b0)
% Found x00:(P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P2 X)
% Found x0:(P1 ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=((and (X x)) ((or (X x)) (Y x))):Prop
% Found x0 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found x0:(P1 ((and (X x)) ((or (X x)) (Y x))))
% Instantiate: b:=((and (X x)) ((or (X x)) (Y x))):Prop
% Found x0 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 (X x)):(((eq Prop) (X x)) (X x))
% Found (eq_ref0 (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found ((eq_ref Prop) (X x)) as proof of (((eq Prop) (X x)) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b0)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% 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) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% 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) Pro
% EOF
%------------------------------------------------------------------------------