TSTP Solution File: SEV150^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV150^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n112.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 13:33:48 EDT 2014
% Result : Timeout 300.03s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV150^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n112.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:14:06 CDT 2014
% % CPUTime : 300.03
% 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 0x1ec9518>, <kernel.Type object at 0x1ec9830>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))) of role conjecture named cTHM251A_pme
% Conjecture to prove = (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))))']
% Parameter a:Type.
% Trying to prove (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))), (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->(a->Prop))) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->(a->Prop))) b) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found eq_ref00:=(eq_ref0 (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))):(((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (eq_ref0 (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found ((eq_ref (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P0 (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P0 (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P0 (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P0 (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->(a->Prop))) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->(a->Prop))) b) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->(a->Prop))) b) as proof of (((eq (a->(a->Prop))) b) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xp) Xw)) ((S Xp) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))):(((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))) (fun (x:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (eta_expansion00 (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found ((eta_expansion0 (a->Prop)) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found (((eta_expansion a) (a->Prop)) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->(a->Prop))) (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))):(((eq (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) (fun (x0:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 x0)))))
% Found (eta_expansion_dep00 (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found eq_ref00:=(eq_ref0 (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))):(((eq (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (eq_ref0 (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) as proof of (((eq (a->Prop)) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P0 (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P0 (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x01) as proof of (P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x01) as proof of (P0 (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x01) as proof of (P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x01) as proof of (P0 (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x01) as proof of (P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x01) as proof of (P0 (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x01) as proof of (P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x01:(P (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq))))))=> x01) as proof of (P0 (fun (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found eq_ref00:=(eq_ref0 (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 y)))):(((eq Prop) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 y)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 y))))
% Found (eq_ref0 (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 y)))) as proof of (((eq Prop) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 y)))) b)
% Found ((eq_ref Prop) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 y)))) as proof of (((eq Prop) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 y)))) b)
% Found ((eq_ref Prop) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 y)))) as proof of (((eq Prop) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 y)))) b)
% Found ((eq_ref Prop) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 y)))) as proof of (((eq Prop) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or ((R x) Xw)) ((S x) Xw))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq0 Xv))))->(Xq0 y)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 y))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S x) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 y))))
% Found x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq)))))
% Found (fun (x2:(P (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((S Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((S Xu) Xv))->(Xq1 Xv))))->(Xq1 Xw))))->(Xq0 Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((or (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((R Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))) (forall (Xq1:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq1 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq1 Xu0)) ((S Xu0) Xv0))->(Xq1 Xv0))))->(Xq1 Xv)))))->(Xq0 Xv))))->(Xq0 Xq))))))=> x2) as proof of (P0 (fun (Xp:a) (Xq:a)=> (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq1:(a->Prop)), (((and (forall (Xw0:a), (((R Xp) Xw0)->(Xq1 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq1 Xu)) ((R Xu) Xv))->(Xq1 Xv))))-
% EOF
%------------------------------------------------------------------------------