TSTP Solution File: SEU856^5 by cocATP---0.2.0
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%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU856^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n118.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:17 EDT 2014
% Result : Timeout 300.09s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU856^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n118.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:37:16 CDT 2014
% % CPUTime : 300.09
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1872b00>, <kernel.Type object at 0x1872e60>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (S:(a->Prop)) (T:(a->Prop)), ((iff (((eq (a->Prop)) S) T)) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) (fun (Xx:a)=> False)))) of role conjecture named cGAZING_THM46_pme
% Conjecture to prove = (forall (S:(a->Prop)) (T:(a->Prop)), ((iff (((eq (a->Prop)) S) T)) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) (fun (Xx:a)=> False)))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (S:(a->Prop)) (T:(a->Prop)), ((iff (((eq (a->Prop)) S) T)) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) (fun (Xx:a)=> False))))']
% Parameter a:Type.
% Trying to prove (forall (S:(a->Prop)) (T:(a->Prop)), ((iff (((eq (a->Prop)) S) T)) (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) (fun (Xx:a)=> False))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))->(P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) (fun (x:a)=> ((or ((and (T x)) (not (S x)))) ((and (S x)) (not (T x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))->(P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))->(P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))->(P (fun (x:a)=> ((or ((and (T x)) ((S x)->False))) ((and (S x)) ((T x)->False))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))->(P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))->(P (fun (x:a)=> ((or ((and (T x)) (not (S x)))) ((and (S x)) (not (T x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) (fun (x:a)=> ((or ((and (T x)) ((S x)->False))) ((and (S x)) ((T x)->False)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T 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) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))->(P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) (fun (x:a)=> ((or ((and (S x)) (not (S x)))) ((and (S x)) (not (S x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))->(P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))->(P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))->(P (fun (x:a)=> ((or ((and (T x)) ((S x)->False))) ((and (S x)) ((T x)->False))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))->(P (fun (x:a)=> ((or ((and (T x)) ((S x)->False))) ((and (S x)) ((T x)->False))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))->(P (fun (x:a)=> ((or ((and (T x)) (not (S x)))) ((and (S x)) (not (T x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))->(P (fun (x:a)=> ((or ((and (T x)) (not (S x)))) ((and (S x)) (not (T x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))->(P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))->(P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S 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 (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eq_ref00:=(eq_ref0 T):(((eq (a->Prop)) T) T)
% Found (eq_ref0 T) as proof of (((eq (a->Prop)) T) b)
% Found ((eq_ref (a->Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found ((eq_ref (a->Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found ((eq_ref (a->Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eq_ref00:=(eq_ref0 T):(((eq (a->Prop)) T) T)
% Found (eq_ref0 T) as proof of (((eq (a->Prop)) T) b)
% Found ((eq_ref (a->Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found ((eq_ref (a->Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found ((eq_ref (a->Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) 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 (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))->(P (fun (x:a)=> ((or ((and (S x)) (not (S x)))) ((and (S x)) (not (S x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P T))):((P T)->(P T))
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found x0:=(x (fun (x0:(a->Prop))=> (P T))):((P T)->(P T))
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found eq_ref00:=(eq_ref0 ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))):(((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found (eq_ref0 ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))):(((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found (eq_ref0 ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (S x0)))):((P (S x0))->(P (S x0)))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (S x0)))):((P (S x0))->(P (S x0)))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))->(P (fun (x:a)=> ((or ((and (S x)) (not (S x)))) ((and (S x)) (not (S x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))->(P (fun (x:a)=> ((or ((and (S x)) (not (S x)))) ((and (S x)) (not (S x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P T))):((P T)->(P T))
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found x0:=(x (fun (x0:(a->Prop))=> (P T))):((P T)->(P T))
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found eq_ref00:=(eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))):(((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found eq_ref00:=(eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))):(((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) 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 (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eq_ref00:=(eq_ref0 T):(((eq (a->Prop)) T) T)
% Found (eq_ref0 T) as proof of (((eq (a->Prop)) T) b)
% Found ((eq_ref (a->Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found ((eq_ref (a->Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found ((eq_ref (a->Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (S x0)))):((P (S x0))->(P (S x0)))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))->(P ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))->(P ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (S x0)))):((P (S x0))->(P (S x0)))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> 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 (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eta_expansion000:=(eta_expansion00 T):(((eq (a->Prop)) T) (fun (x:a)=> (T x)))
% Found (eta_expansion00 T) as proof of (((eq (a->Prop)) T) b)
% Found ((eta_expansion0 Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) 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 (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) 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 (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P T))):((P T)->(P T))
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found x0:=(x (fun (x0:(a->Prop))=> (P T))):((P T)->(P T))
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found eq_ref00:=(eq_ref0 ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))):(((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found (eq_ref0 ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))):(((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found (eq_ref0 ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) as proof of (((eq Prop) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Instantiate: b:=(fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x0:(P S)
% Instantiate: b:=S:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 T):(((eq (a->Prop)) T) (fun (x:a)=> (T x)))
% Found (eta_expansion00 T) as proof of (((eq (a->Prop)) T) b)
% Found ((eta_expansion0 Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (S x0)))):((P (S x0))->(P (S x0)))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (S x0)))):((P (S x0))->(P (S x0)))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P T))):((P T)->(P T))
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found x0:=(x (fun (x0:(a->Prop))=> (P T))):((P T)->(P T))
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found (x (fun (x0:(a->Prop))=> (P T))) as proof of (P0 T)
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))):(((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))):(((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))):(((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))):(((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))):(((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))):(((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Instantiate: b:=(fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P S)
% Instantiate: b:=S:(a->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 T):(((eq (a->Prop)) T) (fun (x:a)=> (T x)))
% Found (eta_expansion00 T) as proof of (((eq (a->Prop)) T) b)
% Found ((eta_expansion0 Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found x0:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))):(((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found eq_ref00:=(eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))):(((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found ((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) as proof of (((eq Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (T x0))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))->(P ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found ((eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))->(P ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found ((eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))->(P ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found ((eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))->(P ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found ((eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))->(P ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found ((eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))->(P ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found ((eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))->(P ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))->(P ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found (((eq_ref Prop) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0))))) P) as proof of (P0 ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (S x0)))):((P (S x0))->(P (S x0)))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found (x (fun (x1:(a->Prop))=> (P (S x0)))) as proof of (P0 (S x0))
% Found eq_ref000:=(eq_ref00 P):((P (S x0))->(P (S x0)))
% Found (eq_ref00 P) as proof of (P0 (S x0))
% Found ((eq_ref0 (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found (((eq_ref Prop) (S x0)) P) as proof of (P0 (S x0))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) (fun (x:a)=> ((or ((and (T x)) ((S x)->False))) ((and (S x)) ((T x)->False)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T 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) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (T x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (T x)))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))->(P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) (fun (x:a)=> ((or ((and (T x)) ((S x)->False))) ((and (S x)) ((T x)->False)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T 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) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x0 as proof of (P0 f)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> 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 (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) 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 (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) (fun (x:a)=> ((or ((and (T x)) (not (S x)))) ((and (S x)) (not (T x))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Instantiate: b:=(fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P S)
% Instantiate: b:=S:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (T x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (T x)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 T):(((eq (a->Prop)) T) (fun (x:a)=> (T x)))
% Found (eta_expansion00 T) as proof of (((eq (a->Prop)) T) b)
% Found ((eta_expansion0 Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))->(P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) (fun (x:a)=> ((or ((and (T x)) (not (S x)))) ((and (S x)) (not (T x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) ((S x0)->False))) ((and (S x0)) ((T x0)->False))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found eq_ref00:=(eq_ref0 (T x0)):(((eq Prop) (T x0)) (T x0))
% Found (eq_ref0 (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found eq_ref00:=(eq_ref0 (T x0)):(((eq Prop) (T x0)) (T x0))
% Found (eq_ref0 (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found eq_ref00:=(eq_ref0 (T x0)):(((eq Prop) (T x0)) (T x0))
% Found (eq_ref0 (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found eq_ref00:=(eq_ref0 (T x0)):(((eq Prop) (T x0)) (T x0))
% Found (eq_ref0 (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P False))):((P False)->(P False))
% Found (x (fun (x1:(a->Prop))=> (P False))) as proof of (P0 False)
% Found (x (fun (x1:(a->Prop))=> (P False))) as proof of (P0 False)
% Found x1:=(x (fun (x1:(a->Prop))=> (P False))):((P False)->(P False))
% Found (x (fun (x1:(a->Prop))=> (P False))) as proof of (P0 False)
% Found (x (fun (x1:(a->Prop))=> (P False))) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 T):(((eq (a->Prop)) T) (fun (x:a)=> (T x)))
% Found (eta_expansion00 T) as proof of (((eq (a->Prop)) T) b)
% Found ((eta_expansion0 Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found eq_ref000:=(eq_ref00 P):((P (T x0))->(P (T x0)))
% Found (eq_ref00 P) as proof of (P0 (T x0))
% Found ((eq_ref0 (T x0)) P) as proof of (P0 (T x0))
% Found (((eq_ref Prop) (T x0)) P) as proof of (P0 (T x0))
% Found (((eq_ref Prop) (T x0)) P) as proof of (P0 (T x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (T x0)))):((P (T x0))->(P (T x0)))
% Found (x (fun (x1:(a->Prop))=> (P (T x0)))) as proof of (P0 (T x0))
% Found (x (fun (x1:(a->Prop))=> (P (T x0)))) as proof of (P0 (T x0))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Instantiate: b:=(fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P S)
% Instantiate: b:=S:(a->Prop)
% Found x0 as proof of (P0 b)
% Found x1:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Instantiate: b:=(fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))):(a->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x0 as proof of (P0 f)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 T):(((eq (a->Prop)) T) (fun (x:a)=> (T x)))
% Found (eta_expansion00 T) as proof of (((eq (a->Prop)) T) b)
% Found ((eta_expansion0 Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion a) Prop) T) as proof of (((eq (a->Prop)) T) 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 eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b0)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))):(((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))):(((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found ((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) as proof of (((eq Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))):(((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))):(((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))):(((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found ((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) as proof of (((eq Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))):(((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found ((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) as proof of (((eq Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found eq_ref00:=(eq_ref0 (T x0)):(((eq Prop) (T x0)) (T x0))
% Found (eq_ref0 (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found eq_ref00:=(eq_ref0 (T x0)):(((eq Prop) (T x0)) (T x0))
% Found (eq_ref0 (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 (T x0)):(((eq Prop) (T x0)) (T x0))
% Found (eq_ref0 (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found eq_ref00:=(eq_ref0 (T x0)):(((eq Prop) (T x0)) (T x0))
% Found (eq_ref0 (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found ((eq_ref Prop) (T x0)) as proof of (((eq Prop) (T x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x0))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (T x0)) (not (S x0)))) ((and (S x0)) (not (T x0)))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> 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 (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 T):(((eq (a->Prop)) T) (fun (x:a)=> (T x)))
% Found (eta_expansion_dep00 T) as proof of (((eq (a->Prop)) T) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) T) as proof of (((eq (a->Prop)) T) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))->(P ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found ((eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))->(P ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found ((eq_ref0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found (((eq_ref Prop) ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00))))) P) as proof of (P0 ((or ((and (S x00)) (not (S x00)))) ((and (S x00)) (not (S x00)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (T x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (T x)))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))->(P ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found ((eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))->(P (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) (fun (x:a)=> ((or ((and (T x)) ((S x)->False))) ((and (S x)) ((T x)->False)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) ((S Xz)->False))) ((and (S Xz)) ((T Xz)->False))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))->(P ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found ((eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))->(P ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found ((eq_ref0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found (((eq_ref Prop) ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1))))) P) as proof of (P0 ((or ((and (S x1)) (not (S x1)))) ((and (S x1)) (not (S x1)))))
% Found eq_ref000:=(eq_ref00 P):((P ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))->(P ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))))
% Found (eq_ref00 P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found ((eq_ref0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found (((eq_ref Prop) ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0))))) P) as proof of (P0 ((or ((and (S x0)) (not (S x0)))) ((and (S x0)) (not (S x0)))))
% Found x1:=(x (fun (x1:(a->Prop))=> (P False))):((P False)->(P False))
% Found (x (fun (x1:(a->Prop))=> (P False))) as proof of (P0 False)
% Found (x (fun (x1:(a->Prop))=> (P False))) as proof of (P0 False)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (T x0)))):((P (T x0))->(P (T x0)))
% Found (x (fun (x1:(a->Prop))=> (P (T x0)))) as proof of (P0 (T x0))
% Found (x (fun (x1:(a->Prop))=> (P (T x0)))) as proof of (P0 (T x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P False))):((P False)->(P False))
% Found (x (fun (x1:(a->Prop))=> (P False))) as proof of (P0 False)
% Found (x (fun (x1:(a->Prop))=> (P False))) as proof of (P0 False)
% Found eq_ref000:=(eq_ref00 P):((P (T x0))->(P (T x0)))
% Found (eq_ref00 P) as proof of (P0 (T x0))
% Found ((eq_ref0 (T x0)) P) as proof of (P0 (T x0))
% Found (((eq_ref Prop) (T x0)) P) as proof of (P0 (T x0))
% Found (((eq_ref Prop) (T x0)) P) as proof of (P0 (T x0))
% Found x0:(P T)
% Instantiate: b:=T:(a->Prop)
% Found x0 as proof of (P0 b)
% Found x1:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))):(a->Prop)
% Found x1 as proof of (P0 f)
% Found x1:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))):(a->Prop)
% Found x1 as proof of (P0 f)
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) 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 eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b0)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found x0:=(x (fun (x0:(a->Prop))=> (P S))):((P S)->(P S))
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found (x (fun (x0:(a->Prop))=> (P S))) as proof of (P0 S)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) T)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) (fun (x:a)=> ((or ((and (S x)) (not (S x)))) ((and (S x)) (not (S x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) False)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) False)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) False)
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) False)
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) False)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) False)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) False)
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) False)
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))->(P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) (fun (x:a)=> ((or ((and (S x)) (not (S x)))) ((and (S x)) (not (S x))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (T x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (T x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (T x)))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))->(P (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (T Xz)) (not (S Xz)))) ((and (S Xz)) (not (T Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x0:(P T)
% Instantiate: b:=T:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))->(P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))->(P (fun (x:a)=> ((or ((and (S x)) (not (S x)))) ((and (S x)) (not (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz))))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not (S Xz)))) ((and (S Xz)) (not (S Xz)))))) P) as proof of (P0 (fun (Xz:a)=>
% EOF
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