TSTP Solution File: SET596^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SET596^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n110.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 13:30:47 EDT 2014
% Result : Timeout 300.09s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SET596^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n110.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 10:13:56 CDT 2014
% % CPUTime : 300.09
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x19e3878>, <kernel.Type object at 0x19e3830>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), (((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))->(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> False)))) of role conjecture named cBOOL_PROP_55_pme
% Conjecture to prove = (forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), (((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))->(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> False)))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), (((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))->(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> False))))']
% Parameter a:Type.
% Trying to prove (forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), (((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False)))->(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> False))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (x:a)=> ((and (X x)) (Z x)))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z 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_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (x:a)=> ((and (X x)) (Z x)))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found x1:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z 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)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (x:a)=> ((and (X x)) (Z x)))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (x:a)=> ((and (X x)) (Z x)))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P ((and (X x0)) (Z x0)))->(P ((and (X x0)) (Z x0))))
% Found (eq_ref00 P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found ((eq_ref0 ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% 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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (x:a)=> ((and (X x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((and (X x0)) (Z x0)))->(P ((and (X x0)) (Z x0))))
% Found (eq_ref00 P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found ((eq_ref0 ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (x:a)=> ((and (X x)) (Z x)))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (x:a)=> ((and (X x)) (Z x)))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% 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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (x:a)=> ((and (X x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% 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)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z 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:=(x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (x:a)=> ((and (X x)) (Z x)))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found x10:=(x1 (fun (x3:(a->Prop))=> (P ((and (X x2)) (Z x2))))):((P ((and (X x2)) (Z x2)))->(P ((and (X x2)) (Z x2))))
% Found (x1 (fun (x3:(a->Prop))=> (P ((and (X x2)) (Z x2))))) as proof of (P0 ((and (X x2)) (Z x2)))
% Found (x1 (fun (x3:(a->Prop))=> (P ((and (X x2)) (Z x2))))) as proof of (P0 ((and (X x2)) (Z x2)))
% Found x10:=(x1 (fun (x3:(a->Prop))=> (P ((and (X x2)) (Z x2))))):((P ((and (X x2)) (Z x2)))->(P ((and (X x2)) (Z x2))))
% Found (x1 (fun (x3:(a->Prop))=> (P ((and (X x2)) (Z x2))))) as proof of (P0 ((and (X x2)) (Z x2)))
% Found (x1 (fun (x3:(a->Prop))=> (P ((and (X x2)) (Z x2))))) as proof of (P0 ((and (X x2)) (Z x2)))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% 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)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z 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 eq_ref00:=(eq_ref0 ((and (X x2)) (Z x2))):(((eq Prop) ((and (X x2)) (Z x2))) ((and (X x2)) (Z x2)))
% Found (eq_ref0 ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) 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 x2)) (Z x2))):(((eq Prop) ((and (X x2)) (Z x2))) ((and (X x2)) (Z x2)))
% Found (eq_ref0 ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) 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:=(x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P False)->(P False))
% Found (eq_ref00 P) as proof of (P0 False)
% Found ((eq_ref0 False) P) as proof of (P0 False)
% Found (((eq_ref Prop) False) P) as proof of (P0 False)
% Found (((eq_ref Prop) False) P) as proof of (P0 False)
% Found eq_ref000:=(eq_ref00 P):((P False)->(P False))
% Found (eq_ref00 P) as proof of (P0 False)
% Found ((eq_ref0 False) P) as proof of (P0 False)
% Found (((eq_ref Prop) False) P) as proof of (P0 False)
% Found (((eq_ref Prop) False) P) as proof of (P0 False)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z 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)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found x1:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Found x1 as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x1 as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z 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)=> ((and (X Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% 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)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z 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)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z 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)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found x20:=(x2 (fun (x3:(a->Prop))=> (P ((and (X x0)) (Z x0))))):((P ((and (X x0)) (Z x0)))->(P ((and (X x0)) (Z x0))))
% Found (x2 (fun (x3:(a->Prop))=> (P ((and (X x0)) (Z x0))))) as proof of (P0 ((and (X x0)) (Z x0)))
% Found (x2 (fun (x3:(a->Prop))=> (P ((and (X x0)) (Z x0))))) as proof of (P0 ((and (X x0)) (Z x0)))
% Found x20:=(x2 (fun (x3:(a->Prop))=> (P ((and (X x0)) (Z x0))))):((P ((and (X x0)) (Z x0)))->(P ((and (X x0)) (Z x0))))
% Found (x2 (fun (x3:(a->Prop))=> (P ((and (X x0)) (Z x0))))) as proof of (P0 ((and (X x0)) (Z x0)))
% Found (x2 (fun (x3:(a->Prop))=> (P ((and (X x0)) (Z x0))))) as proof of (P0 ((and (X x0)) (Z x0)))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(P (fun (x:a)=> ((and (Y x)) (Z x)))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(P (fun (x:a)=> ((and (Y x)) (Z x)))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (X x2)) (Z x2)))->(P ((and (X x2)) (Z x2))))
% Found (eq_ref00 P) as proof of (P0 ((and (X x2)) (Z x2)))
% Found ((eq_ref0 ((and (X x2)) (Z x2))) P) as proof of (P0 ((and (X x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (X x2)) (Z x2))) P) as proof of (P0 ((and (X x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (X x2)) (Z x2))) P) as proof of (P0 ((and (X x2)) (Z x2)))
% Found eq_ref000:=(eq_ref00 P):((P ((and (X x2)) (Z x2)))->(P ((and (X x2)) (Z x2))))
% Found (eq_ref00 P) as proof of (P0 ((and (X x2)) (Z x2)))
% Found ((eq_ref0 ((and (X x2)) (Z x2))) P) as proof of (P0 ((and (X x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (X x2)) (Z x2))) P) as proof of (P0 ((and (X x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (X x2)) (Z x2))) P) as proof of (P0 ((and (X x2)) (Z x2)))
% Found x1:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Found x1 as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x1 as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z 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)=> ((and (X Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 ((and (X x2)) (Z x2))):(((eq Prop) ((and (X x2)) (Z x2))) ((and (X x2)) (Z x2)))
% Found (eq_ref0 ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 ((and (X x2)) (Z x2))):(((eq Prop) ((and (X x2)) (Z x2))) ((and (X x2)) (Z x2)))
% Found (eq_ref0 ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (X x2)) (Z x2))) as proof of (((eq Prop) ((and (X x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(P (fun (x:a)=> ((and (Y x)) (Z x)))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z 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)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Z 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 x0:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) 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)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x10:=(x1 (fun (x3:(a->Prop))=> (P False))):((P False)->(P False))
% Found (x1 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found (x1 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found x10:=(x1 (fun (x3:(a->Prop))=> (P False))):((P False)->(P False))
% Found (x1 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found (x1 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found eq_ref000:=(eq_ref00 P):((P ((and (X x1)) (Z x1)))->(P ((and (X x1)) (Z x1))))
% Found (eq_ref00 P) as proof of (P0 ((and (X x1)) (Z x1)))
% Found ((eq_ref0 ((and (X x1)) (Z x1))) P) as proof of (P0 ((and (X x1)) (Z x1)))
% Found (((eq_ref Prop) ((and (X x1)) (Z x1))) P) as proof of (P0 ((and (X x1)) (Z x1)))
% Found (((eq_ref Prop) ((and (X x1)) (Z x1))) P) as proof of (P0 ((and (X x1)) (Z x1)))
% Found eq_ref000:=(eq_ref00 P):((P ((and (X x1)) (Z x1)))->(P ((and (X x1)) (Z x1))))
% Found (eq_ref00 P) as proof of (P0 ((and (X x1)) (Z x1)))
% Found ((eq_ref0 ((and (X x1)) (Z x1))) P) as proof of (P0 ((and (X x1)) (Z x1)))
% Found (((eq_ref Prop) ((and (X x1)) (Z x1))) P) as proof of (P0 ((and (X x1)) (Z x1)))
% Found (((eq_ref Prop) ((and (X x1)) (Z x1))) P) as proof of (P0 ((and (X x1)) (Z x1)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% 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 x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% 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 x1)) (Z x1))):(((eq Prop) ((and (X x1)) (Z x1))) ((and (X x1)) (Z x1)))
% Found (eq_ref0 ((and (X x1)) (Z x1))) as proof of (((eq Prop) ((and (X x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (X x1)) (Z x1))) as proof of (((eq Prop) ((and (X x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (X x1)) (Z x1))) as proof of (((eq Prop) ((and (X x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (X x1)) (Z x1))) as proof of (((eq Prop) ((and (X x1)) (Z x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x1)) (Z x1)))
% Found eq_ref00:=(eq_ref0 ((and (X x1)) (Z x1))):(((eq Prop) ((and (X x1)) (Z x1))) ((and (X x1)) (Z x1)))
% Found (eq_ref0 ((and (X x1)) (Z x1))) as proof of (((eq Prop) ((and (X x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (X x1)) (Z x1))) as proof of (((eq Prop) ((and (X x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (X x1)) (Z x1))) as proof of (((eq Prop) ((and (X x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (X x1)) (Z x1))) as proof of (((eq Prop) ((and (X x1)) (Z x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x1)) (Z x1)))
% 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 x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% 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 x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z 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_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z 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 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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (x:a)=> ((and (X x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found x10:=(x1 (fun (x3:(a->Prop))=> (P False))):((P False)->(P False))
% Found (x1 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found (x1 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found x10:=(x1 (fun (x3:(a->Prop))=> (P False))):((P False)->(P False))
% Found (x1 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found (x1 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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 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 x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% 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 x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% 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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (x:a)=> ((and (X x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z 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)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z 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)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z 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 x2:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x2 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 x1:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (x:a)=> ((and (X x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (P b)
% Found ((eta_expansion0 Prop) b) as proof of (P b)
% Found (((eta_expansion a) Prop) b) as proof of (P b)
% Found (((eta_expansion a) Prop) b) as proof of (P b)
% Found (((eta_expansion a) Prop) b) as proof of (P 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_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Z 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 x2:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) False)
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) False)
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x20:=(x2 (fun (x3:(a->Prop))=> (P False))):((P False)->(P False))
% Found (x2 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found (x2 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found x20:=(x2 (fun (x3:(a->Prop))=> (P False))):((P False)->(P False))
% Found (x2 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found (x2 (fun (x3:(a->Prop))=> (P False))) as proof of (P0 False)
% Found eq_ref000:=(eq_ref00 P):((P ((and (Y x2)) (Z x2)))->(P ((and (Y x2)) (Z x2))))
% Found (eq_ref00 P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found ((eq_ref0 ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Y x2)) (Z x2)))->(P ((and (Y x2)) (Z x2))))
% Found (eq_ref00 P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found ((eq_ref0 ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Y x2)) (Z x2)))->(P ((and (Y x2)) (Z x2))))
% Found (eq_ref00 P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found ((eq_ref0 ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Y x2)) (Z x2)))->(P ((and (Y x2)) (Z x2))))
% Found (eq_ref00 P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found ((eq_ref0 ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) 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)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z 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 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% 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)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z 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)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Z x0)))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) False)
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) False)
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) False)
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x1:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (x:a)=> ((and (X x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found eta_expansion0000:=(eta_expansion000 P):((P b)->(P (fun (x:a)=> (b x))))
% Found (eta_expansion000 P) as proof of (P0 b)
% Found ((eta_expansion00 b) P) as proof of (P0 b)
% Found (((eta_expansion0 Prop) b) P) as proof of (P0 b)
% Found ((((eta_expansion a) Prop) b) P) as proof of (P0 b)
% Found ((((eta_expansion a) Prop) b) P) as proof of (P0 b)
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x20:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x20 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((and (Y x2)) (Z x2)))->(P ((and (Y x2)) (Z x2))))
% Found (eq_ref00 P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found ((eq_ref0 ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Y x2)) (Z x2)))->(P ((and (Y x2)) (Z x2))))
% Found (eq_ref00 P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found ((eq_ref0 ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found (((eq_ref Prop) ((and (Y x2)) (Z x2))) P) as proof of (P0 ((and (Y x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x2)) (Z x2)))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x1:(P ((and (X x0)) (Z x0)))
% Instantiate: b:=((and (X x0)) (Z 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 x1:(P ((and (X x0)) (Z x0)))
% Instantiate: b:=((and (X x0)) (Z 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 eta_expansion0000:=(eta_expansion000 P):((P b)->(P (fun (x:a)=> (b x))))
% Found (eta_expansion000 P) as proof of (P0 b)
% Found ((eta_expansion00 b) P) as proof of (P0 b)
% Found (((eta_expansion0 Prop) b) P) as proof of (P0 b)
% Found ((((eta_expansion a) Prop) b) P) as proof of (P0 b)
% Found ((((eta_expansion a) Prop) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (a->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (a->Prop)) b) P) as proof of (P0 b)
% Found x20:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x20 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (Y x2)) (Z x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (Y x2)) (Z x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found x20:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x20 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (Y x2)) (Z x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (Y x2)) (Z x2)))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found x2:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z 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_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)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z 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)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z 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)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Z 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 x0:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (fun (Xx:a)=> False))->(P1 (fun (Xx:a)=> False)))
% Found (eq_ref00 P1) as proof of (P2 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P1):((P1 (fun (Xx:a)=> False))->(P1 (fun (Xx:a)=> False)))
% Found (eq_ref00 P1) as proof of (P2 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P1):((P1 (fun (Xx:a)=> False))->(P1 (fun (Xx:a)=> False)))
% Found (eq_ref00 P1) as proof of (P2 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P1):((P1 (fun (Xx:a)=> False))->(P1 (fun (Xx:a)=> False)))
% Found (eq_ref00 P1) as proof of (P2 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P1) as proof of (P2 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z 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 (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((and (X x0)) (Z x0)))->(P1 ((and (X x0)) (Z x0))))
% Found (eq_ref00 P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found ((eq_ref0 ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (x:a)=> ((and (X x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z 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 (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((and (X x0)) (Z x0)))->(P1 ((and (X x0)) (Z x0))))
% Found (eq_ref00 P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found ((eq_ref0 ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((and (X x0)) (Z x0)))->(P1 ((and (X x0)) (Z x0))))
% Found (eq_ref00 P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found ((eq_ref0 ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found eq_ref000:=(eq_ref00 P1):((P1 ((and (X x0)) (Z x0)))->(P1 ((and (X x0)) (Z x0))))
% Found (eq_ref00 P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found ((eq_ref0 ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P1) as proof of (P2 ((and (X x0)) (Z x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z 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)=> ((and (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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_ref000:=(eq_ref00 P):((P ((and (X x0)) (Z x0)))->(P ((and (X x0)) (Z x0))))
% Found (eq_ref00 P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found ((eq_ref0 ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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_ref000:=(eq_ref00 P):((P ((and (X x0)) (Z x0)))->(P ((and (X x0)) (Z x0))))
% Found (eq_ref00 P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found ((eq_ref0 ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found (((eq_ref Prop) ((and (X x0)) (Z x0))) P) as proof of (P0 ((and (X x0)) (Z x0)))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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)) (Z x0))):(((eq Prop) ((and (X x0)) (Z x0))) ((and (X x0)) (Z x0)))
% Found (eq_ref0 ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Z x0))) as proof of (((eq Prop) ((and (X x0)) (Z 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:(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> False))
% Instantiate: b0:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x1 as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b0)
% 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)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Z x))))
% 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 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Z Xx)))) b0)
% 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)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Z x))))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Y x1)) (Z x1)))->(P ((and (Y x1)) (Z x1))))
% Found (eq_ref00 P) as proof of (P0 ((and (Y x1)) (Z x1)))
% Found ((eq_ref0 ((and (Y x1)) (Z x1))) P) as proof of (P0 ((and (Y x1)) (Z x1)))
% Found (((eq_ref Prop) ((and (Y x1)) (Z x1))) P) as proof of (P0 ((and (Y x1)) (Z x1)))
% Found (((eq_ref Prop) ((and (Y x1)) (Z x1))) P) as proof of (P0 ((and (Y x1)) (Z x1)))
% Found eq_ref000:=(eq_ref00 P):((P ((and (Y x1)) (Z x1)))->(P ((and (Y x1)) (Z x1))))
% Found (eq_ref00 P) as proof of (P0 ((and (Y x1)) (Z x1)))
% Found ((eq_ref0 ((and (Y x1)) (Z x1))) P) as proof of (P0 ((and (Y x1)) (Z x1)))
% Found (((eq_ref Prop) ((and (Y x1)) (Z x1))) P) as proof of (P0 ((and (Y x1)) (Z x1)))
% Found (((eq_ref Prop) ((and (Y x1)) (Z x1))) P) as proof of (P0 ((and (Y x1)) (Z x1)))
% Found eq_ref00:=(eq_ref0 ((and (Y x1)) (Z x1))):(((eq Prop) ((and (Y x1)) (Z x1))) ((and (Y x1)) (Z x1)))
% Found (eq_ref0 ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found eq_ref00:=(eq_ref0 ((and (Y x1)) (Z x1))):(((eq Prop) ((and (Y x1)) (Z x1))) ((and (Y x1)) (Z x1)))
% Found (eq_ref0 ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found eq_ref00:=(eq_ref0 ((and (Y x1)) (Z x1))):(((eq Prop) ((and (Y x1)) (Z x1))) ((and (Y x1)) (Z x1)))
% Found (eq_ref0 ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found eq_ref00:=(eq_ref0 ((and (Y x1)) (Z x1))):(((eq Prop) ((and (Y x1)) (Z x1))) ((and (Y x1)) (Z x1)))
% Found (eq_ref0 ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found ((eq_ref Prop) ((and (Y x1)) (Z x1))) as proof of (((eq Prop) ((and (Y x1)) (Z x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x1)) (Z x1)))
% 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 P
% EOF
%------------------------------------------------------------------------------