TSTP Solution File: SEU913^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU913^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n095.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:22 EDT 2014
% Result : Timeout 300.04s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU913^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n095.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:42:26 CDT 2014
% % CPUTime : 300.04
% 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 0x1370320>, <kernel.Type object at 0x1370680>) of role type named c_type
% Using role type
% Declaring c:Type
% FOF formula (<kernel.Constant object at 0x13e1ab8>, <kernel.Type object at 0x1370488>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x1370200>, <kernel.Type object at 0x1370098>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1370680>, <kernel.DependentProduct object at 0x13dfef0>) of role type named g
% Using role type
% Declaring g:((c->Prop)->(b->Prop))
% FOF formula (<kernel.Constant object at 0x1370488>, <kernel.DependentProduct object at 0x13dfd88>) of role type named f
% Using role type
% Declaring f:((c->Prop)->(a->Prop))
% FOF formula (<kernel.Constant object at 0x1370200>, <kernel.DependentProduct object at 0x13dfab8>) of role type named h
% Using role type
% Declaring h:((c->Prop)->(((a->Prop)->((b->Prop)->Prop))->Prop))
% FOF formula (<kernel.Constant object at 0x1370680>, <kernel.DependentProduct object at 0x13dfbd8>) of role type named cC
% Using role type
% Declaring cC:((c->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x1370440>, <kernel.DependentProduct object at 0x13dfdd0>) of role type named cB
% Using role type
% Declaring cB:((b->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x1370680>, <kernel.DependentProduct object at 0x13dfd40>) of role type named cA
% Using role type
% Declaring cA:((a->Prop)->Prop)
% FOF formula (((and ((and ((and ((and ((and ((and ((and (forall (Xx:(c->Prop)), ((cC Xx)->(cA (f Xx))))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:a), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(cB (g Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:b), ((Xe Xx0)->((g Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((h Xx) Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv))))))))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and ((h Xx) Xr)) ((Xr S) (fun (Xy:b)=> False)))))) (S Xx0)))))) (f Xx)))))) (forall (Xx:(c->Prop)), ((cC Xx)->(((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and ((h Xx) Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) (g Xx)))))->(forall (Xx:(c->Prop)), ((cC Xx)->(((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))))) of role conjecture named cDOMTHM15_pme
% Conjecture to prove = (((and ((and ((and ((and ((and ((and ((and (forall (Xx:(c->Prop)), ((cC Xx)->(cA (f Xx))))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:a), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(cB (g Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:b), ((Xe Xx0)->((g Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((h Xx) Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv))))))))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and ((h Xx) Xr)) ((Xr S) (fun (Xy:b)=> False)))))) (S Xx0)))))) (f Xx)))))) (forall (Xx:(c->Prop)), ((cC Xx)->(((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and ((h Xx) Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) (g Xx)))))->(forall (Xx:(c->Prop)), ((cC Xx)->(((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))))):Prop
% Parameter c_DUMMY:c.
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['(((and ((and ((and ((and ((and ((and ((and (forall (Xx:(c->Prop)), ((cC Xx)->(cA (f Xx))))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:a), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(cB (g Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:b), ((Xe Xx0)->((g Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((h Xx) Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv))))))))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and ((h Xx) Xr)) ((Xr S) (fun (Xy:b)=> False)))))) (S Xx0)))))) (f Xx)))))) (forall (Xx:(c->Prop)), ((cC Xx)->(((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and ((h Xx) Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) (g Xx)))))->(forall (Xx:(c->Prop)), ((cC Xx)->(((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))))))']
% Parameter c:Type.
% Parameter b:Type.
% Parameter a:Type.
% Parameter g:((c->Prop)->(b->Prop)).
% Parameter f:((c->Prop)->(a->Prop)).
% Parameter h:((c->Prop)->(((a->Prop)->((b->Prop)->Prop))->Prop)).
% Parameter cC:((c->Prop)->Prop).
% Parameter cB:((b->Prop)->Prop).
% Parameter cA:((a->Prop)->Prop).
% Trying to prove (((and ((and ((and ((and ((and ((and ((and (forall (Xx:(c->Prop)), ((cC Xx)->(cA (f Xx))))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:a), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(cB (g Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:b), ((Xe Xx0)->((g Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((h Xx) Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv))))))))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((h Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(((eq (a->Prop)) (fun (Xx0:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and ((h Xx) Xr)) ((Xr S) (fun (Xy:b)=> False)))))) (S Xx0)))))) (f Xx)))))) (forall (Xx:(c->Prop)), ((cC Xx)->(((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and ((h Xx) Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) (g Xx)))))->(forall (Xx:(c->Prop)), ((cC Xx)->(((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))))))
% Found eq_ref000:=(eq_ref00 P):((P (h Xx))->(P (h Xx)))
% Found (eq_ref00 P) as proof of (P0 (h Xx))
% Found ((eq_ref0 (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found eq_ref000:=(eq_ref00 P):((P (h Xx))->(P (h Xx)))
% Found (eq_ref00 P) as proof of (P0 (h Xx))
% Found ((eq_ref0 (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found eta_expansion000:=(eta_expansion00 (h Xx)):(((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) (fun (x:((a->Prop)->((b->Prop)->Prop)))=> ((h Xx) x)))
% Found (eta_expansion00 (h Xx)) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) b0)
% Found ((eta_expansion0 Prop) (h Xx)) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) b0)
% Found (((eta_expansion ((a->Prop)->((b->Prop)->Prop))) Prop) (h Xx)) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) b0)
% Found (((eta_expansion ((a->Prop)->((b->Prop)->Prop))) Prop) (h Xx)) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) b0)
% Found (((eta_expansion ((a->Prop)->((b->Prop)->Prop))) Prop) (h Xx)) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (x:((a->Prop)->((b->Prop)->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found ((eta_expansion_dep0 (fun (x2:((a->Prop)->((b->Prop)->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found (((eta_expansion_dep ((a->Prop)->((b->Prop)->Prop))) (fun (x2:((a->Prop)->((b->Prop)->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found (((eta_expansion_dep ((a->Prop)->((b->Prop)->Prop))) (fun (x2:((a->Prop)->((b->Prop)->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found (((eta_expansion_dep ((a->Prop)->((b->Prop)->Prop))) (fun (x2:((a->Prop)->((b->Prop)->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found eq_ref000:=(eq_ref00 P):((P (h Xx))->(P (h Xx)))
% Found (eq_ref00 P) as proof of (P0 (h Xx))
% Found ((eq_ref0 (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found eq_ref000:=(eq_ref00 P):((P (h Xx))->(P (h Xx)))
% Found (eq_ref00 P) as proof of (P0 (h Xx))
% Found ((eq_ref0 (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found eq_ref00:=(eq_ref0 (h Xx)):(((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) (h Xx))
% Found (eq_ref0 (h Xx)) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) b0)
% Found ((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) b0)
% Found ((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) b0)
% Found ((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (x:((a->Prop)->((b->Prop)->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found (((eta_expansion ((a->Prop)->((b->Prop)->Prop))) Prop) b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found (((eta_expansion ((a->Prop)->((b->Prop)->Prop))) Prop) b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found (((eta_expansion ((a->Prop)->((b->Prop)->Prop))) Prop) b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found eq_ref000:=(eq_ref00 P):((P (h Xx))->(P (h Xx)))
% Found (eq_ref00 P) as proof of (P0 (h Xx))
% Found ((eq_ref0 (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found eq_ref000:=(eq_ref00 P):((P (h Xx))->(P (h Xx)))
% Found (eq_ref00 P) as proof of (P0 (h Xx))
% Found ((eq_ref0 (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (h Xx)) P) as proof of (P0 (h Xx))
% Found eq_ref00:=(eq_ref0 b0):(((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (h Xx))
% Found ((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (h Xx))
% Found ((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (h Xx))
% Found ((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) b0) (h Xx))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))):(((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) (fun (x:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((x Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found (eta_expansion_dep00 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) b0)
% Found ((eta_expansion_dep0 (fun (x2:((a->Prop)->((b->Prop)->Prop)))=> Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) b0)
% Found (((eta_expansion_dep ((a->Prop)->((b->Prop)->Prop))) (fun (x2:((a->Prop)->((b->Prop)->Prop)))=> Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) b0)
% Found (((eta_expansion_dep ((a->Prop)->((b->Prop)->Prop))) (fun (x2:((a->Prop)->((b->Prop)->Prop)))=> Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) b0)
% Found (((eta_expansion_dep ((a->Prop)->((b->Prop)->Prop))) (fun (x2:((a->Prop)->((b->Prop)->Prop)))=> Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) as proof of (((eq (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))->(P (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found ((eq_ref0 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) P) as proof of (P0 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) P) as proof of (P0 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) P) as proof of (P0 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))->(P (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found ((eq_ref0 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) P) as proof of (P0 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) P) as proof of (P0 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found (((eq_ref (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe))))))))))) P) as proof of (P0 (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->((f Xx) Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))))
% Found eq_ref000:=(eq_ref00 P):((P ((h Xx) x1))->(P ((h Xx) x1)))
% Found (eq_ref00 P) as proof of (P0 ((h Xx) x1))
% Found ((eq_ref0 ((h Xx) x1)) P) as proof of (P0 ((h Xx) x1))
% Found (((eq_ref Prop) ((h Xx) x1)) P) as proof of (P0 (
% EOF
%------------------------------------------------------------------------------