TSTP Solution File: ALG294^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : ALG294^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n092.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 13:18:23 EDT 2014
% Result : Timeout 300.06s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : ALG294^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n092.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 09:10:21 CDT 2014
% % CPUTime : 300.06
% 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 0x1da3ef0>, <kernel.Type object at 0x1d88cf8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1c71c68>, <kernel.DependentProduct object at 0x1d88fc8>) of role type named cR
% Using role type
% Declaring cR:(a->a)
% FOF formula (<kernel.Constant object at 0x1da3ef0>, <kernel.DependentProduct object at 0x1d88d40>) of role type named cL
% Using role type
% Declaring cL:(a->a)
% FOF formula (<kernel.Constant object at 0x1da3ef0>, <kernel.DependentProduct object at 0x1d88dd0>) of role type named cPSI
% Using role type
% Declaring cPSI:((a->Prop)->(a->Prop))
% FOF formula (<kernel.Constant object at 0x1d88fc8>, <kernel.DependentProduct object at 0x1d88ef0>) of role type named cP
% Using role type
% Declaring cP:(a->(a->a))
% FOF formula (<kernel.Constant object at 0x1d88f38>, <kernel.Constant object at 0x1d88ef0>) of role type named cZ
% Using role type
% Declaring cZ:a
% FOF formula (<kernel.Constant object at 0x1d88d40>, <kernel.DependentProduct object at 0x1d88dd0>) of role type named cPHI
% Using role type
% Declaring cPHI:((a->Prop)->(a->Prop))
% FOF formula (((and ((and ((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X Xt)) (forall (Xu:a), ((X Xu)->(X (cL Xu)))))))->(X cZ))))) (forall (X:(a->Prop)) (Xz:a), ((iff ((cPHI X) Xz)) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_14:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_14))))))->(X Xx_14)))) ((cPHI (fun (Xy:a)=> (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xy)))))))) Xz)))))))) (forall (X:(a->Prop)) (Xz:a), ((iff ((cPSI X) Xz)) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_15:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_15))))))->(X Xx_15)))) ((cPSI (fun (Xy:a)=> (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xy)))))))) Xz)))))))->(((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) of role conjecture named cPU_X2310B_pme
% Conjecture to prove = (((and ((and ((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X Xt)) (forall (Xu:a), ((X Xu)->(X (cL Xu)))))))->(X cZ))))) (forall (X:(a->Prop)) (Xz:a), ((iff ((cPHI X) Xz)) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_14:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_14))))))->(X Xx_14)))) ((cPHI (fun (Xy:a)=> (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xy)))))))) Xz)))))))) (forall (X:(a->Prop)) (Xz:a), ((iff ((cPSI X) Xz)) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_15:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_15))))))->(X Xx_15)))) ((cPSI (fun (Xy:a)=> (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xy)))))))) Xz)))))))->(((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))):Prop
% We need to prove ['(((and ((and ((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X Xt)) (forall (Xu:a), ((X Xu)->(X (cL Xu)))))))->(X cZ))))) (forall (X:(a->Prop)) (Xz:a), ((iff ((cPHI X) Xz)) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_14:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_14))))))->(X Xx_14)))) ((cPHI (fun (Xy:a)=> (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xy)))))))) Xz)))))))) (forall (X:(a->Prop)) (Xz:a), ((iff ((cPSI X) Xz)) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_15:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_15))))))->(X Xx_15)))) ((cPSI (fun (Xy:a)=> (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xy)))))))) Xz)))))))->(((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))']
% Parameter a:Type.
% Parameter cR:(a->a).
% Parameter cL:(a->a).
% Parameter cPSI:((a->Prop)->(a->Prop)).
% Parameter cP:(a->(a->a)).
% Parameter cZ:a.
% Parameter cPHI:((a->Prop)->(a->Prop)).
% Trying to prove (((and ((and ((and ((and ((and ((and ((and (((eq a) (cL cZ)) cZ)) (((eq a) (cR cZ)) cZ))) (forall (Xx:a) (Xy:a), (((eq a) (cL ((cP Xx) Xy))) Xx)))) (forall (Xx:a) (Xy:a), (((eq a) (cR ((cP Xx) Xy))) Xy)))) (forall (Xt:a), ((iff (not (((eq a) Xt) cZ))) (((eq a) Xt) ((cP (cL Xt)) (cR Xt))))))) (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> ((and (X Xt)) (forall (Xu:a), ((X Xu)->(X (cL Xu)))))))->(X cZ))))) (forall (X:(a->Prop)) (Xz:a), ((iff ((cPHI X) Xz)) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_14:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_14))))))->(X Xx_14)))) ((cPHI (fun (Xy:a)=> (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xy)))))))) Xz)))))))) (forall (X:(a->Prop)) (Xz:a), ((iff ((cPSI X) Xz)) ((ex a) (fun (Xx:a)=> ((and (forall (Xx_15:a), ((forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xx_15))))))->(X Xx_15)))) ((cPSI (fun (Xy:a)=> (forall (X0:(a->Prop)), (((and (X0 Xx)) (forall (Xz0:a), ((X0 Xz0)->(X0 (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X0 Xv)) (((eq a) (cR Xv)) Xy)))))))) Xz)))))))->(((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))->(P (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((eq_ref0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))->(P (fun (x:a)=> ((or (((eq a) x) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((eta_expansion00 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (((eta_expansion0 Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))->(P (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((eq_ref0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))):(((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) (fun (x:a)=> ((or (((eq a) x) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (eta_expansion00 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))->(P (fun (x:a)=> ((or (((eq a) x) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((eta_expansion00 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (((eta_expansion0 Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found eq_ref00:=(eq_ref0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))):(((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (eq_ref0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) as proof of (((eq (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))->(P (fun (x:a)=> ((or (((eq a) x) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((eta_expansion00 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (((eta_expansion0 Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((((eta_expansion a) Prop) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))->(P (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((eq_ref0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (((eq_ref (a->Prop)) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) (fun (x:a)=> ((or ((or ((or (((eq a) x) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) x) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))->(P (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))->(P (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found ((eq_ref0 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 (fun (Xz:a)=> ((or ((or ((or (((eq a) Xz) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) Xz) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xz) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))->(P ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (eq_ref00 P) as proof of (P0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found ((eq_ref0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (((eq_ref Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (((eq_ref Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found eq_ref000:=(eq_ref00 P):((P ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))->(P ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found (eq_ref00 P) as proof of (P0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found ((eq_ref0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (((eq_ref Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (((eq_ref Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) P) as proof of (P0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found eq_ref00:=(eq_ref0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))):(((eq Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (eq_ref0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found ((eq_ref Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found ((eq_ref Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found ((eq_ref Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((or ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) x0) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((or ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) x0) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((or ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) x0) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((or ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) x0) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))
% Found eq_ref00:=(eq_ref0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))):(((eq Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))
% Found (eq_ref0 ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found ((eq_ref Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found ((eq_ref Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found ((eq_ref Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) as proof of (((eq Prop) ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((or ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) x0) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((or ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) x0) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((or ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) x0) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((or ((or (((eq a) x0) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))) (((eq a) x0) cZ))) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x0) ((cP Xx) Xy))) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz0:a), ((X Xz0)->(X (cL Xz0)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))->(P (fun (x:a)=> ((or (((eq a) x) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) x) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy))))))))))
% Found ((eta_expansion_dep00 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)))))))))) P) as proof of (P0 (fun (Xu:a)=> ((or (((eq a) Xu) cZ)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((and (((eq a) Xu) ((cP Xx) Xy))) ((or ((cPHI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (Xz:a), ((X Xz)->(X (cL Xz)))))->((ex a) (fun (Xv:a)=> ((and (X Xv)) (((eq a) (cR Xv)) Xy0)))))))) Xy)) ((cPSI (fun (Xy0:a)=> (forall (X:(a->Prop)), (((and (X Xx)) (forall (
% EOF
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