TSTP Solution File: SEV166^5 by cocATP---0.2.0

%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV166^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n097.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 13:33:49 EDT 2014

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

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV166^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n097.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 08:18:21 CDT 2014
% % CPUTime  : 300.09 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x14c4cf8>, <kernel.Type object at 0x14c47a0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), ((iff (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) ((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv)))) of role conjecture named cTHM182_pme
% Conjecture to prove = (forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), ((iff (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) ((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv)))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), ((iff (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) ((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv))))']
% Parameter a:Type.
% Trying to prove (forall (Xx:a) (Xy:a) (Xu:a) (Xv:a), ((iff (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) ((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv))))
% Found x100:=(x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))):((P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy)))->(P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xv))))
% Found (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))) as proof of ((P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy)))->(P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xv))))
% Found (fun (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))) as proof of ((P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy)))->(P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xv))))
% Found (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))) as proof of (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xv)))
% Found (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))) as proof of ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xv))))
% Found (x00 (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))))) as proof of ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv))))
% Found ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))))) as proof of ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv))))
% Found (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))))) as proof of ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv))))
% Found (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))))) as proof of ((((eq a) Xx) Xu)->((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))))
% Found (and_rect00 (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))))))) as proof of (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))
% Found ((and_rect0 (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))))))) as proof of (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))
% Found (((fun (P:Type) (x0:((((eq a) Xx) Xu)->((((eq a) Xy) Xv)->P)))=> (((((and_rect (((eq a) Xx) Xu)) (((eq a) Xy) Xv)) P) x0) x)) (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))))))) as proof of (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))
% Found (fun (x:((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv)))=> (((fun (P:Type) (x0:((((eq a) Xx) Xu)->((((eq a) Xy) Xv)->P)))=> (((((and_rect (((eq a) Xx) Xu)) (((eq a) Xy) Xv)) P) x0) x)) (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))))))) as proof of (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))
% Found (fun (x:((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv)))=> (((fun (P:Type) (x0:((((eq a) Xx) Xu)->((((eq a) Xy) Xv)->P)))=> (((((and_rect (((eq a) Xx) Xu)) (((eq a) Xy) Xv)) P) x0) x)) (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))))))) as proof of (((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv))->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv))))
% Found x100:=(x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))):((P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy)))->(P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xv))))
% Found (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))) as proof of ((P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy)))->(P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xv))))
% Found (fun (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))) as proof of ((P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy)))->(P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xv))))
% Found (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))) as proof of (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xv)))
% Found (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))) as proof of ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xv))))
% Found (x00 (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))))) as proof of ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv))))
% Found ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))))) as proof of ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv))))
% Found (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))))) as proof of ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv))))
% Found (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))))) as proof of ((((eq a) Xx) Xu)->((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))))
% Found (and_rect00 (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))))))) as proof of (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))
% Found ((and_rect0 (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))))))) as proof of (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))
% Found (((fun (P:Type) (x0:((((eq a) Xx) Xu)->((((eq a) Xy) Xv)->P)))=> (((((and_rect (((eq a) Xx) Xu)) (((eq a) Xy) Xv)) P) x0) x)) (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1))))))))) as proof of (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))
% Found (fun (x:((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv)))=> (((fun (P:Type) (x0:((((eq a) Xx) Xu)->((((eq a) Xy) Xv)->P)))=> (((((and_rect (((eq a) Xx) Xu)) (((eq a) Xy) Xv)) P) x0) x)) (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))))))) as proof of (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))
% Found (fun (x:((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv)))=> (((fun (P:Type) (x0:((((eq a) Xx) Xu)->((((eq a) Xy) Xv)->P)))=> (((((and_rect (((eq a) Xx) Xu)) (((eq a) Xy) Xv)) P) x0) x)) (((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))) (fun (x0:(((eq a) Xx) Xu))=> ((x0 (fun (x2:a)=> ((((eq a) Xy) Xv)->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg x2) Xv)))))) (fun (x10:(((eq a) Xy) Xv)) (P:(((a->(a->a))->a)->Prop))=> (x10 (fun (x1:a)=> (P (fun (Xg:(a->(a->a)))=> ((Xg Xx) x1)))))))))) as proof of (((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv))->(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv))))
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 (((eq a) Xy) Xv)):(((eq Prop) (((eq a) Xy) Xv)) (((eq a) Xy) Xv))
% Found (eq_ref0 (((eq a) Xy) Xv)) as proof of (((eq Prop) (((eq a) Xy) Xv)) b)
% Found ((eq_ref Prop) (((eq a) Xy) Xv)) as proof of (((eq Prop) (((eq a) Xy) Xv)) b)
% Found ((eq_ref Prop) (((eq a) Xy) Xv)) as proof of (((eq Prop) (((eq a) Xy) Xv)) b)
% Found ((eq_ref Prop) (((eq a) Xy) Xv)) as proof of (((eq Prop) (((eq a) Xy) Xv)) b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found eq_ref00:=(eq_ref0 (((eq a) Xy) Xv)):(((eq Prop) (((eq a) Xy) Xv)) (((eq a) Xy) Xv))
% Found (eq_ref0 (((eq a) Xy) Xv)) as proof of (((eq Prop) (((eq a) Xy) Xv)) b)
% Found ((eq_ref Prop) (((eq a) Xy) Xv)) as proof of (((eq Prop) (((eq a) Xy) Xv)) b)
% Found ((eq_ref Prop) (((eq a) Xy) Xv)) as proof of (((eq Prop) (((eq a) Xy) Xv)) b)
% Found ((eq_ref Prop) (((eq a) Xy) Xv)) as proof of (((eq Prop) (((eq a) Xy) Xv)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found x0:(P Xx)
% Instantiate: b:=Xx:a
% Found x0 as proof of (P0 b)
% Found x0:(P Xy)
% Instantiate: b:=Xy:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found x0:(P Xy)
% Instantiate: b:=Xy:a
% Found x0 as proof of (P0 b)
% Found x0:(P Xx)
% Instantiate: b:=Xx:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found x0:(P Xu)
% Instantiate: b:=Xu:a
% Found x0 as proof of (P0 b)
% Found x0:(P Xv)
% Instantiate: b:=Xv:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:(P Xv)
% Instantiate: b:=Xv:a
% Found x0 as proof of (P0 b)
% Found x0:(P Xu)
% Instantiate: b:=Xu:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xu))):((P Xu)->(P Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xv))):((P Xv)->(P Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) Xy) Xv))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Xy) Xv))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Xy) Xv))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Xy) Xv))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xv))):((P Xv)->(P Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xu))):((P Xu)->(P Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xu))):((P Xu)->(P Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of b
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) Xy) Xv))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Xy) Xv))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Xy) Xv))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Xy) Xv))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found x0:(P Xx)
% Instantiate: b:=Xx:a
% Found x0 as proof of (P0 b)
% Found x0:(P Xy)
% Instantiate: b:=Xy:a
% Found x0 as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xv))):((P Xv)->(P Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found x0:(P Xx)
% Instantiate: b:=Xx:a
% Found x0 as proof of (P0 b)
% Found x0:(P Xy)
% Instantiate: b:=Xy:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found x0:(P Xy)
% Instantiate: a0:=Xy:a
% Found x0 as proof of (P0 a0)
% Found x0:(P Xx)
% Instantiate: a0:=Xx:a
% Found x0 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found x0:(P Xu)
% Instantiate: b:=Xu:a
% Found x0 as proof of (P0 b)
% Found x0:(P Xv)
% Instantiate: b:=Xv:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x0:(P Xx)
% Instantiate: a0:=Xx:a
% Found x0 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found x0:(P Xy)
% Instantiate: a0:=Xy:a
% Found x0 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xv)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xu)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found x0:(P Xu)
% Instantiate: b:=Xu:a
% Found x0 as proof of (P0 b)
% Found x0:(P Xv)
% Instantiate: b:=Xv:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found x0:(P1 Xu)
% Instantiate: b:=Xu:a
% Found x0 as proof of (P2 b)
% Found x0:(P1 Xv)
% Instantiate: b:=Xv:a
% Found x0 as proof of (P2 b)
% Found x0:(P1 Xv)
% Instantiate: b:=Xv:a
% Found x0 as proof of (P2 b)
% Found x0:(P1 Xu)
% Instantiate: b:=Xu:a
% Found x0 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x0:(P Xv)
% Instantiate: b:=Xv:a
% Found x0 as proof of (P0 b)
% Found x0:(P Xu)
% Instantiate: b:=Xu:a
% Found x0 as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xu)->(P1 Xu))
% Found (eq_ref00 P1) as proof of (P2 Xu)
% Found ((eq_ref0 Xu) P1) as proof of (P2 Xu)
% Found (((eq_ref a) Xu) P1) as proof of (P2 Xu)
% Found (((eq_ref a) Xu) P1) as proof of (P2 Xu)
% Found x1:=(x (fun (x1:((a->(a->a))->a))=> (P1 Xv))):((P1 Xv)->(P1 Xv))
% Found (x (fun (x1:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found (x (fun (x1:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found x0:(P b)
% Found x0 as proof of (P0 Xx)
% Found x0:(P b)
% Found x0 as proof of (P0 Xy)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xv))):((P Xv)->(P Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xu))):((P Xu)->(P Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xv))):((P Xv)->(P Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xu))):((P Xu)->(P Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found x0:(P b)
% Instantiate: b0:=b:a
% Found x0 as proof of (P0 b0)
% Found x0:(P b)
% Instantiate: b0:=b:a
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found x0:(P Xu)
% Instantiate: a0:=Xu:a
% Found x0 as proof of (P0 a0)
% Found x0:(P Xv)
% Instantiate: a0:=Xv:a
% Found x0 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xv)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xv)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xv)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xu)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xu)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xu)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P0 Xv))):((P0 Xv)->(P0 Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xv))) as proof of (P1 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xv))) as proof of (P1 Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P0 Xu))):((P0 Xu)->(P0 Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xu))) as proof of (P1 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xu))) as proof of (P1 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of a0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found x0:(P1 Xu)
% Instantiate: b:=Xu:a
% Found x0 as proof of (P2 b)
% Found x0:(P1 Xu)
% Instantiate: b:=Xu:a
% Found x0 as proof of (P2 b)
% Found x0:(P1 Xv)
% Instantiate: b:=Xv:a
% Found x0 as proof of (P2 b)
% Found x0:(P1 Xv)
% Instantiate: b:=Xv:a
% Found x0 as proof of (P2 b)
% Found x0:(P Xu)
% Instantiate: b:=Xu:a
% Found x0 as proof of (P0 b)
% Found x0:(P Xv)
% Instantiate: b:=Xv:a
% Found x0 as proof of (P0 b)
% Found x:(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))
% Instantiate: b:=(forall (P:(((a->(a->a))->a)->Prop)), ((P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy)))->(P (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv))))):Prop
% Found x as proof of a0
% Found x0:(P b)
% Found x0 as proof of (P0 Xy)
% Found x0:(P b)
% Found x0 as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x1:=(x (fun (x1:((a->(a->a))->a))=> (P1 Xv))):((P1 Xv)->(P1 Xv))
% Found (x (fun (x1:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found (x (fun (x1:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found x1:=(x (fun (x1:((a->(a->a))->a))=> (P1 Xu))):((P1 Xu)->(P1 Xu))
% Found (x (fun (x1:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found (x (fun (x1:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xv))):((P Xv)->(P Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xv))):((P Xv)->(P Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xu))):((P Xu)->(P Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xu))):((P Xu)->(P Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 Xu))):((P1 Xu)->(P1 Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 Xv))):((P1 Xv)->(P1 Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 Xu))):((P1 Xu)->(P1 Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 Xv))):((P1 Xv)->(P1 Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found x0:(P b)
% Instantiate: b0:=b:a
% Found x0 as proof of (P0 b0)
% Found x0:(P b)
% Instantiate: b0:=b:a
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found x0:(P Xu)
% Instantiate: a0:=Xu:a
% Found x0 as proof of (P0 a0)
% Found x0:(P Xv)
% Instantiate: a0:=Xv:a
% Found x0 as proof of (P0 a0)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xv))):((P Xv)->(P Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found eq_ref000:=(eq_ref00 P):((P Xu)->(P Xu))
% Found (eq_ref00 P) as proof of (P0 Xu)
% Found ((eq_ref0 Xu) P) as proof of (P0 Xu)
% Found (((eq_ref a) Xu) P) as proof of (P0 Xu)
% Found (((eq_ref a) Xu) P) as proof of (P0 Xu)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b0))):((P b0)->(P b0))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b0))):((P b0)->(P b0))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (P Xy)
% Found ((eq_ref a) Xy) as proof of (P Xy)
% Found ((eq_ref a) Xy) as proof of (P Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xu)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xu)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xu)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xu)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (P Xx)
% Found ((eq_ref a) Xx) as proof of (P Xx)
% Found ((eq_ref a) Xx) as proof of (P Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xv)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xv)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xv)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xv)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found x0:(P Xv)
% Instantiate: b0:=Xv:a
% Found x0 as proof of (P0 b0)
% Found x0:(P Xu)
% Instantiate: b0:=Xu:a
% Found x0 as proof of (P0 b0)
% Found x0:(P Xu)
% Instantiate: b0:=Xu:a
% Found x0 as proof of (P0 b0)
% Found x0:(P b)
% Instantiate: b0:=b:a
% Found x0 as proof of (P0 b0)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P0 Xv))):((P0 Xv)->(P0 Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xv))) as proof of (P1 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xv))) as proof of (P1 Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P0 Xu))):((P0 Xu)->(P0 Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xu))) as proof of (P1 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xu))) as proof of (P1 Xu)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found iff_sym0:=(iff_sym ((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv))):(forall (B:Prop), (((iff ((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv))) B)->((iff B) ((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv)))))
% Instantiate: a0:=(forall (B:Prop), (((iff ((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv))) B)->((iff B) ((and (((eq a) Xx) Xu)) (((eq a) Xy) Xv))))):Prop
% Found iff_sym0 as proof of a0
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found x:(((eq ((a->(a->a))->a)) (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy))) (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv)))
% Instantiate: b:=(forall (P:(((a->(a->a))->a)->Prop)), ((P (fun (Xg:(a->(a->a)))=> ((Xg Xx) Xy)))->(P (fun (Xg:(a->(a->a)))=> ((Xg Xu) Xv))))):Prop
% Found x as proof of a0
% Found x0:(P b)
% Found x0 as proof of (P0 Xv)
% Found x0:(P Xu)
% Found x0 as proof of (P0 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 Xu))):((P1 Xu)->(P1 Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 Xv))):((P1 Xv)->(P1 Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 Xu))):((P1 Xu)->(P1 Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 Xv))):((P1 Xv)->(P1 Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P0 Xx))):((P0 Xx)->(P0 Xx))
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xx))) as proof of (P1 Xx)
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xx))) as proof of (P1 Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P0 Xx))):((P0 Xx)->(P0 Xx))
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xx))) as proof of (P1 Xx)
% Found (x (fun (x0:((a->(a->a))->a))=> (P0 Xx))) as proof of (P1 Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 Xv))):((P1 Xv)->(P1 Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xv))) as proof of (P2 Xv)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 Xu))):((P1 Xu)->(P1 Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 Xu))) as proof of (P2 Xu)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xu)->(P1 Xu))
% Found (eq_ref00 P1) as proof of (P2 Xu)
% Found ((eq_ref0 Xu) P1) as proof of (P2 Xu)
% Found (((eq_ref a) Xu) P1) as proof of (P2 Xu)
% Found (((eq_ref a) Xu) P1) as proof of (P2 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xu))):((P Xu)->(P Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b0))):((P b0)->(P b0))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P b0))):((P b0)->(P b0))
% Found (x (fun (x0:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found (x (fun (x0:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b1)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b1)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b1)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b1)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b1)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b1)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b1)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xu))):((P Xu)->(P Xu))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xu))) as proof of (P0 Xu)
% Found x0:=(x (fun (x0:((a->(a->a))->a))=> (P Xv))):((P Xv)->(P Xv))
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found (x (fun (x0:((a->(a->a))->a))=> (P Xv))) as proof of (P0 Xv)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (P Xy)
% Found ((eq_ref a) Xy) as proof of (P Xy)
% Found ((eq_ref a) Xy) as proof of (P Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xv)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (P Xx)
% Found ((eq_ref a) Xx) as proof of (P Xx)
% Found ((eq_ref a) Xx) as proof of (P Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xu)
% Found eq_ref00:=(eq_ref0 Xv):(((eq a) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found ((eq_ref a) Xv) as proof of (((eq a) Xv) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq a) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found ((eq_ref a) Xu) as proof of (((eq a) Xu) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proo
% EOF
%------------------------------------------------------------------------------