%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SET587^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n106.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:30:46 EDT 2014

% Result   : Timeout 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SET587^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n106.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 10:12:51 CDT 2014
% % CPUTime  : 300.02 
% 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 0x2859710>, <kernel.Type object at 0x28595f0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (X:(a->Prop)) (Y:(a->Prop)), ((iff (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> False))) (forall (Xx:a), ((X Xx)->(Y Xx))))) of role conjecture named cBOOL_PROP_45_pme
% Conjecture to prove = (forall (X:(a->Prop)) (Y:(a->Prop)), ((iff (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> False))) (forall (Xx:a), ((X Xx)->(Y Xx))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:(a->Prop)) (Y:(a->Prop)), ((iff (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> False))) (forall (Xx:a), ((X Xx)->(Y Xx)))))']
% Parameter a:Type.
% Trying to prove (forall (X:(a->Prop)) (Y:(a->Prop)), ((iff (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> False))) (forall (Xx:a), ((X Xx)->(Y Xx)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (x:a)=> ((and (X x)) (not (Y x)))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (x:a)=> ((and (X x)) ((Y x)->False))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x10:(P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found x10:(P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 f)
% 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 eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (x:a)=> ((and (X x)) ((Y x)->False))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% 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 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x10:(P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found x10:(P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (x:a)=> ((and (X x)) ((Y x)->False))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x0:(P0 b)
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (x:a)=> ((and (X x)) (not (Y x)))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 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 Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x0:(P0 b)
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x10:(P False)
% Found (fun (x10:(P False))=> x10) as proof of (P False)
% Found (fun (x10:(P False))=> x10) as proof of (P0 False)
% Found x10:(P False)
% Found (fun (x10:(P False))=> x10) as proof of (P False)
% Found (fun (x10:(P False))=> x10) as proof of (P0 False)
% 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 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (x:a)=> ((and (X x)) ((Y x)->False))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x10:(P False)
% Found (fun (x10:(P False))=> x10) as proof of (P False)
% Found (fun (x10:(P False))=> x10) as proof of (P0 False)
% Found x10:(P False)
% Found (fun (x10:(P False))=> x10) as proof of (P False)
% Found (fun (x10:(P False))=> x10) as proof of (P0 False)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (x:a)=> ((and (X x)) ((Y x)->False))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x0:(P0 b)
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (x:a)=> ((and (X x)) ((Y x)->False))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (x:a)=> ((and (X x)) (not (Y x)))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x0:(P0 b)
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x1:(P ((and (X x0)) ((Y x0)->False)))
% Instantiate: b:=((and (X x0)) ((Y x0)->False)):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P ((and (X x0)) ((Y x0)->False)))
% Instantiate: b:=((and (X x0)) ((Y x0)->False)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found x10:(P False)
% Found (fun (x10:(P False))=> x10) as proof of (P False)
% Found (fun (x10:(P False))=> x10) as proof of (P0 False)
% Found x10:(P False)
% Found (fun (x10:(P False))=> x10) as proof of (P False)
% Found (fun (x10:(P False))=> x10) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x1:(P ((and (X x0)) (not (Y x0))))
% Instantiate: b:=((and (X x0)) (not (Y x0))):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P ((and (X x0)) (not (Y x0))))
% Instantiate: b:=((and (X x0)) (not (Y x0))):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (x:a)=> ((and (X x)) ((Y x)->False))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (x:a)=> ((and (X x)) ((Y x)->False))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x10:(P False)
% Found (fun (x10:(P False))=> x10) as proof of (P False)
% Found (fun (x10:(P False))=> x10) as proof of (P0 False)
% Found x10:(P False)
% Found (fun (x10:(P False))=> x10) as proof of (P False)
% Found (fun (x10:(P False))=> x10) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (x:a)=> ((and (X x)) (not (Y x)))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (x:a)=> ((and (X x)) (not (Y x)))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (x:a)=> ((and (X x)) ((Y x)->False))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found x1:(P ((and (X x0)) ((Y x0)->False)))
% Instantiate: b:=((and (X x0)) ((Y x0)->False)):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P ((and (X x0)) ((Y x0)->False)))
% Instantiate: b:=((and (X x0)) ((Y x0)->False)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (not (Y x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (not (Y x)))))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x10:(P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x10:(P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found x1:(P ((and (X x0)) (not (Y x0))))
% Instantiate: b:=((and (X x0)) (not (Y x0))):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P ((and (X x0)) (not (Y x0))))
% Instantiate: b:=((and (X x0)) (not (Y x0))):Prop
% Found x1 as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found x10:(P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P2 ((and (X x0)) (not (Y x0))))
% Found x10:(P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P2 ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x10:(P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P2 ((and (X x0)) (not (Y x0))))
% Found x10:(P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P2 ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x1:(P0 b)
% Instantiate: b:=((and (X x0)) (not (Y x0))):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((and (X x0)) (not (Y x0)))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x1:(P0 b)
% Instantiate: b:=((and (X x0)) (not (Y x0))):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((and (X x0)) (not (Y x0)))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found x1:(P False)
% Instantiate: b:=False:Prop
% Found x1 as proof of (P0 b)
% Found x1:(P False)
% Instantiate: b:=False:Prop
% Found x1 as proof of (P0 b)
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x1:(P False)
% Instantiate: b:=False:Prop
% Found x1 as proof of (P0 b)
% Found x1:(P False)
% Instantiate: b:=False:Prop
% Found x1 as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x1:(P False)
% Instantiate: b:=False:Prop
% Found x1 as proof of (P0 b)
% Found x1:(P False)
% Instantiate: b:=False:Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (not (Y x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (not (Y x1))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (not (Y x)))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P1 (fun (Xx:a)=> False))
% Found (fun (x00:(P1 (fun (Xx:a)=> False)))=> x00) as proof of (P2 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x10:(P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x10:(P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x1:(P False)
% Instantiate: b:=False:Prop
% Found x1 as proof of (P0 b)
% Found x1:(P False)
% Instantiate: b:=False:Prop
% Found x1 as proof of (P0 b)
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found x00:(P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P (fun (Xx:a)=> False))
% Found (fun (x00:(P (fun (Xx:a)=> False)))=> x00) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (x:a)=> ((and (X x)) (not (Y x)))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) b)
% Found x10:(P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P2 ((and (X x0)) (not (Y x0))))
% Found x10:(P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P2 ((and (X x0)) (not (Y x0))))
% Found x10:(P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P2 ((and (X x0)) (not (Y x0))))
% Found x10:(P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P1 ((and (X x0)) (not (Y x0))))
% Found (fun (x10:(P1 ((and (X x0)) (not (Y x0)))))=> x10) as proof of (P2 ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x1:(P0 b)
% Instantiate: b:=((and (X x0)) (not (Y x0))):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((and (X x0)) (not (Y x0)))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found x1:(P0 b)
% Instantiate: b:=((and (X x0)) (not (Y x0))):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 ((and (X x0)) (not (Y x0))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 ((and (X x0)) (not (Y x0)))))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->F
% EOF
%------------------------------------------------------------------------------