TSTP Solution File: SET617^5 by cocATP---0.2.0
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%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SET617^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n184.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 13:30:51 EDT 2014
% Result : Timeout 300.05s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SET617^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n184.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 10:22:46 CDT 2014
% % CPUTime : 300.05
% 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 0x1046ea8>, <kernel.Type object at 0x10465a8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (X:(a->Prop)), ((and (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) X)) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X))) of role conjecture named cBOOL_PROP_92_pme
% Conjecture to prove = (forall (X:(a->Prop)), ((and (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) X)) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:(a->Prop)), ((and (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) X)) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X)))']
% Parameter a:Type.
% Trying to prove (forall (X:(a->Prop)), ((and (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) X)) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) 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 eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) (fun (x:a)=> ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) 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 eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) (fun (x:a)=> ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) 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 eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) (fun (x:a)=> ((or ((and False) (not (X x)))) ((and (X x)) (not False)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) 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 eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) (fun (x:a)=> ((or ((and (X x)) (not False))) ((and False) (not (X x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X)):(((eq Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X)) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X))
% Found (eq_ref0 (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X)) as proof of (((eq Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X)) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X)) as proof of (((eq Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X)) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X)) as proof of (((eq Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X)) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X)) as proof of (((eq Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X)) b)
% Found eq_ref00:=(eq_ref0 (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X)):(((eq Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X)) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X))
% Found (eq_ref0 (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X)) as proof of (((eq Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X)) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X)) as proof of (((eq Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X)) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X)) as proof of (((eq Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X)) b)
% Found ((eq_ref Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X)) as proof of (((eq Prop) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% 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_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_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% 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 (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 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 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X 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 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) 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 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) 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 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X 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 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 x:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Instantiate: b:=(fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))):(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 (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Instantiate: b:=(fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))):(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 (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Instantiate: b:=(fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))):(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 (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Instantiate: b:=(fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))):(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 (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))):(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))):(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))):(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) 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 (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) (fun (x:a)=> ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) 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 (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 (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 (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 (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 (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))):(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))):(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))):(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))):(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) (fun (x:a)=> ((or ((and (X x)) (not False))) ((and False) (not (X x))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) 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 eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) (fun (x:a)=> ((or ((and False) (not (X x)))) ((and (X x)) (not False)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (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 (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 (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 (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 Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% 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) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% 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) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% 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) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% 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) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% 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) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% 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) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% 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) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% 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) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (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 (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) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (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) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% 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) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% 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) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% 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) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (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) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (X x)) (not False))) ((and False) (not (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) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) (fun (x:a)=> ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) 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 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 (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) (fun (x:a)=> ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) 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 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 X)
% Instantiate: b:=X:(a->Prop)
% Found x as proof of (P0 b)
% Found x:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x as proof of (P0 b)
% 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 (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) (fun (x:a)=> ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) 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 (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) (fun (x:a)=> ((or ((and False) (not (X x)))) ((and (X x)) (not False)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) 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 (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) (fun (x:a)=> ((or ((and (X x)) (not False))) ((and False) (not (X x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b0)
% Found x:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x as proof of (P0 b)
% Found x:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% 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 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 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 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) (fun (x:a)=> ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) 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 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) (fun (x:a)=> ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))) b)
% Found x:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% 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 x0:(P ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Instantiate: b:=((or ((and (X x)) (False->False))) ((and False) ((X x)->False))):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Instantiate: b:=((or ((and False) ((X x)->False))) ((and (X x)) (False->False))):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Instantiate: b:=((or ((and False) ((X x)->False))) ((and (X x)) (False->False))):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Instantiate: b:=((or ((and (X x)) (False->False))) ((and False) ((X x)->False))):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)) 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 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 eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) (fun (x:a)=> ((or ((and False) (not (X x)))) ((and (X x)) (not False)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) 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 eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) (fun (x:a)=> ((or ((and (X x)) (not False))) ((and False) (not (X x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))) b)
% Found x2:(P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found x:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x as proof of (P0 f)
% 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 (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% 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_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% 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 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 (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found x0:(P ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Instantiate: b:=((or ((and (X x)) (not False))) ((and False) (not (X x)))):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Instantiate: b:=((or ((and False) (not (X x)))) ((and (X x)) (not False))):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Instantiate: b:=((or ((and (X x)) (not False))) ((and False) (not (X x)))):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Instantiate: b:=((or ((and False) (not (X x)))) ((and (X x)) (not False))):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)) 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 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 x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found x2:(P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found x2:(P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found x2:(P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and (X x0)) (not False))) ((and False) (not (X x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and (X x0)) (not False))) ((and False) (not (X x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and (X x0)) (not False))) ((and False) (not (X x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or ((and (X x0)) (not False))) ((and False) (not (X x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((and (X x)) (not False))) ((and False) (not (X x))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and False) (not (X x0)))) ((and (X x0)) (not False))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and False) (not (X x0)))) ((and (X x0)) (not False))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and False) (not (X x0)))) ((and (X x0)) (not False))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or ((and False) (not (X x0)))) ((and (X x0)) (not False))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and False) (not (X x0)))) ((and (X x0)) (not False))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and False) (not (X x0)))) ((and (X x0)) (not False))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and False) (not (X x0)))) ((and (X x0)) (not False))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or ((and False) (not (X x0)))) ((and (X x0)) (not False))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and (X x0)) (not False))) ((and False) (not (X x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and (X x0)) (not False))) ((and False) (not (X x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and (X x0)) (not False))) ((and False) (not (X x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or ((and (X x0)) (not False))) ((and False) (not (X x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((and (X x)) (not False))) ((and False) (not (X 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% 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 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found x2:(P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz)))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found x2:(P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% 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 (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% 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 (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% 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 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 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 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 eq_ref00:=(eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X 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 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) 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 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) 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 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X 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 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) 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 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X 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 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) 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 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X 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 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X 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 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) 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 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% 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 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% 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 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 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 eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% 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 eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% 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 eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% 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 eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (False->False))) ((and False) ((X Xz)->False)))))
% 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False)))))
% 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 (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 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) ((X Xz)->False))) ((and (X Xz)) (False->False))))) X))
% 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 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 x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 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 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False))))) X))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% 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_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% 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 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 (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and False) (not (X Xz)))) ((and (X Xz)) (not False)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (X Xz)) (not False))) ((and False) (not (X Xz))))))
% 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 (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 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 x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))):(((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False))))
% Found (eq_ref0 ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) as proof of (((eq Prop) ((or ((and (X x)) (False->False))) ((and False) ((X x)->False)))) 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 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))):(((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False))))
% Found (eq_ref0 ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) b)
% Found ((eq_ref Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) as proof of (((eq Prop) ((or ((and False) ((X x)->False))) ((and (X x)) (False->False)))) 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 (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 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 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 (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) 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_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 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 (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 as proof of (P0 b)
% Found x0:(P (X x))
% Instantiate: b:=(X x):Prop
% Found x0 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 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 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 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 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 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 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 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 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 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (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 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b0)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (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 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b0)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X 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 ((or ((and (X x)) (not False))) ((and False) (not (X x))))):(((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) ((or ((and (X x)) (not False))) ((and False) (not (X x)))))
% Found (eq_ref0 ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) b)
% Found ((eq_ref Prop) ((or ((and (X x)) (not False))) ((and False) (not (X x))))) as proof of (((eq Prop) ((or ((and (X x)) (not False))) ((and False) (not (X 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 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))):(((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) ((or ((and False) (not (X x)))) ((and (X x)) (not False))))
% Found (eq_ref0 ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) b)
% Found ((eq_ref Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))) as proof of (((eq Prop) ((or ((and False) (not (X x)))) ((and (X x)) (not False)))
% EOF
%------------------------------------------------------------------------------