TSTP Solution File: SEU845^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU845^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n187.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:16 EDT 2014
% Result : Timeout 300.01s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU845^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n187.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:36:11 CDT 2014
% % CPUTime : 300.01
% 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 0x12dd4d0>, <kernel.Type object at 0x12dd7a0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))))) of role conjecture named cGAZING_THM12_pme
% Conjecture to prove = (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))))']
% Parameter a:Type.
% Trying to prove (forall (S:(a->Prop)) (T:(a->Prop)), ((forall (Xx:a), ((S Xx)->(T Xx)))->(((eq (a->Prop)) S) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% 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 x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) 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) 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) (fun (x:a)=> ((and (T x)) (((and (T x)) ((S x)->False))->False))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% 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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) (fun (x:a)=> ((and (T x)) (not ((and (T x)) (not (S x)))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 x0:(P S)
% Instantiate: b:=S:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) 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 x0:(P S)
% Instantiate: b:=S:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% 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 x0:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x0 as proof of (P0 f)
% 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)) ((and (T x1)) (not ((and (T x1)) (not (S x1))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (T x1)) (not ((and (T x1)) (not (S x1))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (T x1)) (not ((and (T x1)) (not (S x1))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (T x1)) (not ((and (T x1)) (not (S x1))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (T x)) (not ((and (T x)) (not (S x)))))))
% Found x0:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x0 as proof of (P0 f)
% 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)) ((and (T x1)) (not ((and (T x1)) (not (S x1))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (T x1)) (not ((and (T x1)) (not (S x1))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (T x1)) (not ((and (T x1)) (not (S x1))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (T x1)) (not ((and (T x1)) (not (S x1))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (T x)) (not ((and (T x)) (not (S x)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1: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) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) (fun (x:a)=> ((and (T x)) (((and (T x)) ((S x)->False))->False))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found x0:(P0 b)
% Instantiate: b:=S:(a->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 S)
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 S))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) 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) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% 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 x0:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Instantiate: b:=(fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))):(a->Prop)
% Found x0 as proof of (P0 b)
% 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 (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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% 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) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) (fun (x:a)=> ((and (T x)) (not ((and (T x)) (not (S x)))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found x0:(P0 b)
% Instantiate: b:=S:(a->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 S)
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 S))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x0:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Instantiate: b:=(fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))):(a->Prop)
% Found x0 as proof of (P0 b)
% 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 x0:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))):(a->Prop)
% Found x0 as proof of (P0 f)
% 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)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found x0:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))):(a->Prop)
% Found x0 as proof of (P0 f)
% 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)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found x1:(P (S x0))
% Instantiate: b:=(S x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found x1:(P (S x0))
% Instantiate: b:=(S x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) 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 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) (fun (x:a)=> ((and (T x)) (((and (T x)) ((S x)->False))->False))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) 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 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) (fun (x:a)=> ((and (T x)) (((and (T x)) ((S x)->False))->False))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) 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 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) (fun (x:a)=> ((and (T x)) (((and (T x)) ((S x)->False))->False))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found x0:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Instantiate: f:=(fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))):(a->Prop)
% Found x0 as proof of (P0 f)
% 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)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found x0:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Instantiate: f:=(fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))):(a->Prop)
% Found x0 as proof of (P0 f)
% 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)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (S x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found x1:(P (S x0))
% Instantiate: b:=(S x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found x1:(P (S x0))
% Instantiate: b:=(S x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% 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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) (fun (x:a)=> ((and (T x)) (not ((and (T x)) (not (S x)))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) 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 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) (fun (x:a)=> ((and (T x)) (not ((and (T x)) (not (S x)))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) 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 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) (fun (x:a)=> ((and (T x)) (not ((and (T x)) (not (S x)))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x10:(P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P2 (S x0))
% Found x10:(P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P2 (S x0))
% Found x10:(P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P2 (S x0))
% Found x10:(P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P2 (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) (fun (x:a)=> ((and (T x)) (((and (T x)) ((S x)->False))->False))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b0)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found x1:(P0 b)
% Instantiate: b:=(S x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (S x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (S x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found x1:(P0 b)
% Instantiate: b:=(S x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (S x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (S x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S 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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S 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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found x1:(P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Instantiate: b:=((and (T x0)) (((and (T x0)) ((S x0)->False))->False)):Prop
% Found x1 as proof of (P0 b)
% 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 x1:(P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Instantiate: b:=((and (T x0)) (((and (T x0)) ((S x0)->False))->False)):Prop
% Found x1 as proof of (P0 b)
% 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 eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) S)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) S)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) S)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) S)
% Found x10:(P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P2 (S x0))
% Found x10:(P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P2 (S x0))
% Found x10:(P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P2 (S x0))
% Found x10:(P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P1 (S x0))
% Found (fun (x10:(P1 (S x0)))=> x10) as proof of (P2 (S 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)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (((and (T x0)) ((S x0)->False))->False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% 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 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) (fun (x:a)=> ((and (T x)) (not ((and (T x)) (not (S x)))))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b0)
% Found x00:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found x1:(P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Instantiate: b:=((and (T x0)) (((and (T x0)) ((S x0)->False))->False)):Prop
% Found x1 as proof of (P0 b)
% 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 x1:(P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Instantiate: b:=((and (T x0)) (((and (T x0)) ((S x0)->False))->False)):Prop
% Found x1 as proof of (P0 b)
% 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 x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found x1:(P0 b)
% Instantiate: b:=(S x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (S x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (S x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found x1:(P0 b)
% Instantiate: b:=(S x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (S x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (S x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found x1:(P ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Instantiate: b:=((and (T x0)) (not ((and (T x0)) (not (S x0))))):Prop
% Found x1 as proof of (P0 b)
% 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 x1:(P ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Instantiate: b:=((and (T x0)) (not ((and (T x0)) (not (S x0))))):Prop
% Found x1 as proof of (P0 b)
% 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 (S x0)):(((eq Prop) (S x0)) (S x0))
% Found (eq_ref0 (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S 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)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% 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 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% 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)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b 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)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found ((eq_ref Prop) (S x0)) as proof of (((eq Prop) (S x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% 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 eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b)
% Found x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found x1:(P ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Instantiate: b:=((and (T x0)) (not ((and (T x0)) (not (S x0))))):Prop
% Found x1 as proof of (P0 b)
% 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 x1:(P ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Instantiate: b:=((and (T x0)) (not ((and (T x0)) (not (S x0))))):Prop
% Found x1 as proof of (P0 b)
% 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 x00:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found x00:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found x00:(P S)
% Found (fun (x00:(P S))=> x00) as proof of (P S)
% Found (fun (x00:(P S))=> x00) as proof of (P0 S)
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (not ((and (T x0)) (not (S x0)))))
% 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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T 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) (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 x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% 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 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) (fun (x:a)=> ((and (T x)) (((and (T x)) ((S x)->False))->False))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) (fun (x:a)=> ((and (T x)) (((and (T x)) ((S x)->False))->False))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (((and (T Xx)) ((S Xx)->False))->False)))) b)
% Found x1:(P (S x0))
% Instantiate: b:=(S x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found x1:(P (S x0))
% Instantiate: b:=(S x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b)
% 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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx)))))))
% 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) (fun (x:a)=> ((and (T x)) (not ((and (T x)) (not (S x)))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) (fun (x:a)=> ((and (T x)) (not ((and (T x)) (not (S x)))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (T Xx)) (not ((and (T Xx)) (not (S Xx))))))) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (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 (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 x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S 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) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found x1:(P (S x0))
% Instantiate: b:=(S x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found x1:(P (S x0))
% Instantiate: b:=(S x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))):(((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (eq_ref0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) as proof of (((eq Prop) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x0))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% 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 (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 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found x10:(P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P (S x0))
% Found (fun (x10:(P (S x0)))=> x10) as proof of (P0 (S x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (T x0)) (((and (T x0)) ((S x0)->False))->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 x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (fun (x10:(P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))))=> x10) as proof of (P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (fun (x10:(P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))))=> x10) as proof of (P0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found x10:(P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (fun (x10:(P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))))=> x10) as proof of (P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found (fun (x10:(P ((and (T x0)) (((and (T x0)) ((S x0)->False))->False))))=> x10) as proof of (P0 ((and (T x0)) (((and (T x0)) ((S x0)->False))->False)))
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found eq_ref00:=(eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))):(((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) ((and (T x0)) (not ((and (T x0)) (not (S x0))))))
% Found (eq_ref0 ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as proof of (((eq Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) b0)
% Found ((eq_ref Prop) ((and (T x0)) (not ((and (T x0)) (not (S x0)))))) as pro
% EOF
%------------------------------------------------------------------------------