TSTP Solution File: PUZ109^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : PUZ109^5 : TPTP v6.1.0. Bugfixed v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n109.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 13:28:59 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 : PUZ109^5 : TPTP v6.1.0. Bugfixed v5.2.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n109.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 08:20:46 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 0x274e7e8>, <kernel.Constant object at 0x274e560>) of role type named c1_type
% Using role type
% Declaring c1:fofType
% FOF formula (<kernel.Constant object at 0x261c248>, <kernel.DependentProduct object at 0x274e758>) of role type named s_type
% Using role type
% Declaring s:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0x274ec20>, <kernel.DependentProduct object at 0x274e7e8>) of role type named cCKB6_BLACK_type
% Using role type
% Declaring cCKB6_BLACK:(fofType->(fofType->Prop))
% FOF formula (((eq (fofType->(fofType->Prop))) cCKB6_BLACK) (fun (Xu:fofType) (Xv:fofType)=> (forall (Xw:(fofType->(fofType->Prop))), (((and ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))->((Xw Xu) Xv))))) of role definition named cCKB6_BLACK_def
% A new definition: (((eq (fofType->(fofType->Prop))) cCKB6_BLACK) (fun (Xu:fofType) (Xv:fofType)=> (forall (Xw:(fofType->(fofType->Prop))), (((and ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))->((Xw Xu) Xv)))))
% Defined: cCKB6_BLACK:=(fun (Xu:fofType) (Xv:fofType)=> (forall (Xw:(fofType->(fofType->Prop))), (((and ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))->((Xw Xu) Xv))))
% FOF formula (forall (Xj:fofType) (Xk:fofType), (((cCKB6_BLACK Xj) Xk)->((cCKB6_BLACK (s Xj)) (s Xk)))) of role conjecture named cCKB6_L9000
% Conjecture to prove = (forall (Xj:fofType) (Xk:fofType), (((cCKB6_BLACK Xj) Xk)->((cCKB6_BLACK (s Xj)) (s Xk)))):Prop
% We need to prove ['(forall (Xj:fofType) (Xk:fofType), (((cCKB6_BLACK Xj) Xk)->((cCKB6_BLACK (s Xj)) (s Xk))))']
% Parameter fofType:Type.
% Parameter c1:fofType.
% Parameter s:(fofType->fofType).
% Definition cCKB6_BLACK:=(fun (Xu:fofType) (Xv:fofType)=> (forall (Xw:(fofType->(fofType->Prop))), (((and ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))->((Xw Xu) Xv)))):(fofType->(fofType->Prop)).
% Trying to prove (forall (Xj:fofType) (Xk:fofType), (((cCKB6_BLACK Xj) Xk)->((cCKB6_BLACK (s Xj)) (s Xk))))
% Found x200:=(x20 (s Xk0)):(((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (x20 (s Xk0)) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found ((x2 (s Xj0)) (s Xk0)) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (Xk0:fofType)=> ((x2 (s Xj0)) (s Xk0))) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (Xj0:fofType) (Xk0:fofType)=> ((x2 (s Xj0)) (s Xk0))) as proof of (forall (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found (fun (Xj0:fofType) (Xk0:fofType)=> ((x2 (s Xj0)) (s Xk0))) as proof of (forall (Xj0:fofType) (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found x300:=(x30 (s Xk0)):(((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (x30 (s Xk0)) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found ((x3 (s Xj0)) (s Xk0)) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (Xk0:fofType)=> ((x3 (s Xj0)) (s Xk0))) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (Xj0:fofType) (Xk0:fofType)=> ((x3 (s Xj0)) (s Xk0))) as proof of (forall (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found (fun (Xj0:fofType) (Xk0:fofType)=> ((x3 (s Xj0)) (s Xk0))) as proof of (forall (Xj0:fofType) (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found x4000:=(x400 x2):((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (x400 x2) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found ((x40 (s Xk0)) x2) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (((x4 (s Xj0)) (s Xk0)) x2) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (fun (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)) as proof of ((forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)) as proof of (((Xw c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found (and_rect00 (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2))) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found ((and_rect0 ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2))) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (((fun (P:Type) (x3:(((Xw c1) c1)->((forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))) P) x3) x0)) ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2))) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (fun (x2:((Xw (s Xj0)) (s Xk0)))=> (((fun (P:Type) (x3:(((Xw c1) c1)->((forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))) P) x3) x0)) ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)))) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (fun (Xk0:fofType) (x2:((Xw (s Xj0)) (s Xk0)))=> (((fun (P:Type) (x3:(((Xw c1) c1)->((forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))) P) x3) x0)) ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)))) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (Xj0:fofType) (Xk0:fofType) (x2:((Xw (s Xj0)) (s Xk0)))=> (((fun (P:Type) (x3:(((Xw c1) c1)->((forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))) P) x3) x0)) ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)))) as proof of (forall (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found (fun (Xj0:fofType) (Xk0:fofType) (x2:((Xw (s Xj0)) (s Xk0)))=> (((fun (P:Type) (x3:(((Xw c1) c1)->((forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))) P) x3) x0)) ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)))) as proof of (forall (Xj0:fofType) (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found x200:=(x20 (s Xk0)):(((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (x20 (s Xk0)) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found ((x2 (s Xj0)) (s Xk0)) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (Xk0:fofType)=> ((x2 (s Xj0)) (s Xk0))) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (Xj0:fofType) (Xk0:fofType)=> ((x2 (s Xj0)) (s Xk0))) as proof of (forall (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found (fun (Xj0:fofType) (Xk0:fofType)=> ((x2 (s Xj0)) (s Xk0))) as proof of (forall (Xj0:fofType) (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found x300:=(x30 (s Xk0)):(((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (x30 (s Xk0)) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found ((x3 (s Xj0)) (s Xk0)) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (Xk0:fofType)=> ((x3 (s Xj0)) (s Xk0))) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (Xj0:fofType) (Xk0:fofType)=> ((x3 (s Xj0)) (s Xk0))) as proof of (forall (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found (fun (Xj0:fofType) (Xk0:fofType)=> ((x3 (s Xj0)) (s Xk0))) as proof of (forall (Xj0:fofType) (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x40:=(x4 (s (s Xj1))):(forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (x4 (s (s Xj1))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (fun (Xj1:fofType)=> (x4 (s (s Xj1)))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (fun (Xj1:fofType)=> (x4 (s (s Xj1)))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s (s Xj))) Xk)->((and ((Xw (s (s (s (s Xj))))) Xk)) ((Xw (s (s (s Xj)))) (s Xk)))))
% Found x40:=(x4 (s (s Xj1))):(forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (x4 (s (s Xj1))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (fun (Xj1:fofType)=> (x4 (s (s Xj1)))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (fun (Xj1:fofType)=> (x4 (s (s Xj1)))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s (s Xj))) Xk)->((and ((Xw (s (s (s (s Xj))))) Xk)) ((Xw (s (s (s Xj)))) (s Xk)))))
% Found x20:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))
% Found x20 as proof of (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))
% Found x20:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))
% Found x20 as proof of (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))
% Found x400:=(x40 (s Xk1)):(((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (x40 (s Xk1)) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found ((x4 (s Xj1)) (s Xk1)) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (Xj1:fofType) (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (forall (Xk:fofType), (((Xw (s Xj1)) (s Xk))->((and ((Xw (s (s (s Xj1)))) (s Xk))) ((Xw (s (s Xj1))) (s (s Xk))))))
% Found (fun (Xj1:fofType) (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s Xj)) (s Xk))->((and ((Xw (s (s (s Xj)))) (s Xk))) ((Xw (s (s Xj))) (s (s Xk))))))
% Found x400:=(x40 (s Xk1)):(((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (x40 (s Xk1)) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found ((x4 (s Xj1)) (s Xk1)) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (Xj1:fofType) (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (forall (Xk:fofType), (((Xw (s Xj1)) (s Xk))->((and ((Xw (s (s (s Xj1)))) (s Xk))) ((Xw (s (s Xj1))) (s (s Xk))))))
% Found (fun (Xj1:fofType) (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s Xj)) (s Xk))->((and ((Xw (s (s (s Xj)))) (s Xk))) ((Xw (s (s Xj))) (s (s Xk))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x4000:=(x400 x2):((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (x400 x2) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found ((x40 (s Xk0)) x2) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (((x4 (s Xj0)) (s Xk0)) x2) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (fun (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)) as proof of ((forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)) as proof of (((Xw c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found (and_rect00 (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2))) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found ((and_rect0 ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2))) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (((fun (P:Type) (x3:(((Xw c1) c1)->((forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))) P) x3) x0)) ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2))) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (fun (x2:((Xw (s Xj0)) (s Xk0)))=> (((fun (P:Type) (x3:(((Xw c1) c1)->((forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))) P) x3) x0)) ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)))) as proof of ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))
% Found (fun (Xk0:fofType) (x2:((Xw (s Xj0)) (s Xk0)))=> (((fun (P:Type) (x3:(((Xw c1) c1)->((forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))) P) x3) x0)) ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)))) as proof of (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0)))))
% Found (fun (Xj0:fofType) (Xk0:fofType) (x2:((Xw (s Xj0)) (s Xk0)))=> (((fun (P:Type) (x3:(((Xw c1) c1)->((forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))) P) x3) x0)) ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)))) as proof of (forall (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found (fun (Xj0:fofType) (Xk0:fofType) (x2:((Xw (s Xj0)) (s Xk0)))=> (((fun (P:Type) (x3:(((Xw c1) c1)->((forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj0:fofType) (Xk0:fofType), (((Xw Xj0) Xk0)->((and ((Xw (s (s Xj0))) Xk0)) ((Xw (s Xj0)) (s Xk0)))))) P) x3) x0)) ((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))) (fun (x3:((Xw c1) c1)) (x4:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))=> (((x4 (s Xj0)) (s Xk0)) x2)))) as proof of (forall (Xj0:fofType) (Xk0:fofType), (((Xw (s Xj0)) (s Xk0))->((and ((Xw (s (s (s Xj0)))) (s Xk0))) ((Xw (s (s Xj0))) (s (s Xk0))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x5000:=(x500 x3):((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (x500 x3) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found ((x50 Xk1) x3) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (((x5 (s (s Xj1))) Xk1) x3) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (fun (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)) as proof of ((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1))))
% Found (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)) as proof of (((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))))
% Found (and_rect00 (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3))) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found ((and_rect0 ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3))) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3))) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (fun (x3:((Xw (s (s Xj1))) Xk1))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)))) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (fun (Xk1:fofType) (x3:((Xw (s (s Xj1))) Xk1))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)))) as proof of (((Xw (s (s Xj1))) Xk1)->((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1))))
% Found (fun (Xj1:fofType) (Xk1:fofType) (x3:((Xw (s (s Xj1))) Xk1))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (fun (Xj1:fofType) (Xk1:fofType) (x3:((Xw (s (s Xj1))) Xk1))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s (s Xj))) Xk)->((and ((Xw (s (s (s (s Xj))))) Xk)) ((Xw (s (s (s Xj)))) (s Xk)))))
% Found x5000:=(x500 x3):((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (x500 x3) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found ((x50 (s Xk1)) x3) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (((x5 (s Xj1)) (s Xk1)) x3) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (fun (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)) as proof of ((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)) as proof of (((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))))
% Found (and_rect00 (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3))) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found ((and_rect0 ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3))) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3))) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (fun (x3:((Xw (s Xj1)) (s Xk1)))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)))) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (fun (Xk1:fofType) (x3:((Xw (s Xj1)) (s Xk1)))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)))) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (Xj1:fofType) (Xk1:fofType) (x3:((Xw (s Xj1)) (s Xk1)))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)))) as proof of (forall (Xk:fofType), (((Xw (s Xj1)) (s Xk))->((and ((Xw (s (s (s Xj1)))) (s Xk))) ((Xw (s (s Xj1))) (s (s Xk))))))
% Found (fun (Xj1:fofType) (Xk1:fofType) (x3:((Xw (s Xj1)) (s Xk1)))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s Xj)) (s Xk))->((and ((Xw (s (s (s Xj)))) (s Xk))) ((Xw (s (s Xj))) (s (s Xk))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x30:=(x3 (s (s Xj00))):(forall (Xk00:fofType), (((Xw0 (s (s Xj00))) Xk00)->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))))
% Found (x3 (s (s Xj00))) as proof of (forall (Xk00:fofType), (((Xw0 (s (s Xj00))) Xk00)->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))))
% Found (fun (Xj00:fofType)=> (x3 (s (s Xj00)))) as proof of (forall (Xk00:fofType), (((Xw0 (s (s Xj00))) Xk00)->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))))
% Found (fun (Xj00:fofType)=> (x3 (s (s Xj00)))) as proof of (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 (s (s Xj00))) Xk00)->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))))
% Found x30:=(x3 (s (s Xj00))):(forall (Xk00:fofType), (((Xw0 (s (s Xj00))) Xk00)->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))))
% Found (x3 (s (s Xj00))) as proof of (forall (Xk00:fofType), (((Xw0 (s (s Xj00))) Xk00)->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))))
% Found (fun (Xj00:fofType)=> (x3 (s (s Xj00)))) as proof of (forall (Xk00:fofType), (((Xw0 (s (s Xj00))) Xk00)->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))))
% Found (fun (Xj00:fofType)=> (x3 (s (s Xj00)))) as proof of (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 (s (s Xj00))) Xk00)->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))))
% Found x40:=(x4 (s (s Xj1))):(forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (x4 (s (s Xj1))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (fun (Xj1:fofType)=> (x4 (s (s Xj1)))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (fun (Xj1:fofType)=> (x4 (s (s Xj1)))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s (s Xj))) Xk)->((and ((Xw (s (s (s (s Xj))))) Xk)) ((Xw (s (s (s Xj)))) (s Xk)))))
% Found x40:=(x4 (s (s Xj1))):(forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (x4 (s (s Xj1))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (fun (Xj1:fofType)=> (x4 (s (s Xj1)))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (fun (Xj1:fofType)=> (x4 (s (s Xj1)))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s (s Xj))) Xk)->((and ((Xw (s (s (s (s Xj))))) Xk)) ((Xw (s (s (s Xj)))) (s Xk)))))
% Found x20:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))
% Found x20 as proof of (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))
% Found x20:(forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))
% Found x20 as proof of (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00)))))
% Found x300:=(x30 (s Xk00)):(((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00)))))
% Found (x30 (s Xk00)) as proof of (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00)))))
% Found ((x3 (s Xj00)) (s Xk00)) as proof of (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00)))))
% Found (fun (Xk00:fofType)=> ((x3 (s Xj00)) (s Xk00))) as proof of (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00)))))
% Found (fun (Xj00:fofType) (Xk00:fofType)=> ((x3 (s Xj00)) (s Xk00))) as proof of (forall (Xk00:fofType), (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))))
% Found (fun (Xj00:fofType) (Xk00:fofType)=> ((x3 (s Xj00)) (s Xk00))) as proof of (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))))
% Found x300:=(x30 (s Xk00)):(((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00)))))
% Found (x30 (s Xk00)) as proof of (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00)))))
% Found ((x3 (s Xj00)) (s Xk00)) as proof of (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00)))))
% Found (fun (Xk00:fofType)=> ((x3 (s Xj00)) (s Xk00))) as proof of (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00)))))
% Found (fun (Xj00:fofType) (Xk00:fofType)=> ((x3 (s Xj00)) (s Xk00))) as proof of (forall (Xk00:fofType), (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))))
% Found (fun (Xj00:fofType) (Xk00:fofType)=> ((x3 (s Xj00)) (s Xk00))) as proof of (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))))
% Found x400:=(x40 (s Xk1)):(((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (x40 (s Xk1)) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found ((x4 (s Xj1)) (s Xk1)) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (Xj1:fofType) (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (forall (Xk:fofType), (((Xw (s Xj1)) (s Xk))->((and ((Xw (s (s (s Xj1)))) (s Xk))) ((Xw (s (s Xj1))) (s (s Xk))))))
% Found (fun (Xj1:fofType) (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s Xj)) (s Xk))->((and ((Xw (s (s (s Xj)))) (s Xk))) ((Xw (s (s Xj))) (s (s Xk))))))
% Found x400:=(x40 (s Xk1)):(((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (x40 (s Xk1)) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found ((x4 (s Xj1)) (s Xk1)) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (Xj1:fofType) (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (forall (Xk:fofType), (((Xw (s Xj1)) (s Xk))->((and ((Xw (s (s (s Xj1)))) (s Xk))) ((Xw (s (s Xj1))) (s (s Xk))))))
% Found (fun (Xj1:fofType) (Xk1:fofType)=> ((x4 (s Xj1)) (s Xk1))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s Xj)) (s Xk))->((and ((Xw (s (s (s Xj)))) (s Xk))) ((Xw (s (s Xj))) (s (s Xk))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00:((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x00 as proof of ((and ((Xw c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw Xj00) Xk00)->((and ((Xw (s (s Xj00))) Xk00)) ((Xw (s Xj00)) (s Xk00))))))
% Found x5:((and ((and ((cCKB6_BLACK (s (s (s c1)))) Xk2)) ((cCKB6_BLACK (s (s c1))) c1))) ((and ((cCKB6_BLACK (s (s c1))) c1)) ((cCKB6_BLACK (s c1)) (s c1))))
% Found x5 as proof of ((and ((and ((cCKB6_BLACK (s (s (s c1)))) Xk2)) ((cCKB6_BLACK (s (s c1))) c1))) ((and ((cCKB6_BLACK (s (s c1))) c1)) ((cCKB6_BLACK (s c1)) (s c1))))
% Found x3000:=(x300 x20):((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))
% Found (x300 x20) as proof of ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))
% Found ((x30 Xk00) x20) as proof of ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))
% Found (((x3 (s (s Xj00))) Xk00) x20) as proof of ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))
% Found (fun (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s (s Xj00))) Xk00) x20)) as proof of ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))
% Found (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s (s Xj00))) Xk00) x20)) as proof of ((forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000)))))->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00))))
% Found (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s (s Xj00))) Xk00) x20)) as proof of (((Xw0 c1) c1)->((forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000)))))->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))))
% Found (and_rect00 (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s (s Xj00))) Xk00) x20))) as proof of ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))
% Found ((and_rect0 ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s (s Xj00))) Xk00) x20))) as proof of ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))
% Found (((fun (P:Type) (x2:(((Xw0 c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))->P)))=> (((((and_rect ((Xw0 c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))) P) x2) x00)) ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s (s Xj00))) Xk00) x20))) as proof of ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))
% Found (fun (x20:((Xw0 (s (s Xj00))) Xk00))=> (((fun (P:Type) (x2:(((Xw0 c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))->P)))=> (((((and_rect ((Xw0 c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))) P) x2) x00)) ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s (s Xj00))) Xk00) x20)))) as proof of ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))
% Found (fun (Xk00:fofType) (x20:((Xw0 (s (s Xj00))) Xk00))=> (((fun (P:Type) (x2:(((Xw0 c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))->P)))=> (((((and_rect ((Xw0 c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))) P) x2) x00)) ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s (s Xj00))) Xk00) x20)))) as proof of (((Xw0 (s (s Xj00))) Xk00)->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00))))
% Found (fun (Xj00:fofType) (Xk00:fofType) (x20:((Xw0 (s (s Xj00))) Xk00))=> (((fun (P:Type) (x2:(((Xw0 c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))->P)))=> (((((and_rect ((Xw0 c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))) P) x2) x00)) ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s (s Xj00))) Xk00) x20)))) as proof of (forall (Xk00:fofType), (((Xw0 (s (s Xj00))) Xk00)->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))))
% Found (fun (Xj00:fofType) (Xk00:fofType) (x20:((Xw0 (s (s Xj00))) Xk00))=> (((fun (P:Type) (x2:(((Xw0 c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))->P)))=> (((((and_rect ((Xw0 c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))) P) x2) x00)) ((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s (s Xj00))) Xk00) x20)))) as proof of (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 (s (s Xj00))) Xk00)->((and ((Xw0 (s (s (s (s Xj00))))) Xk00)) ((Xw0 (s (s (s Xj00)))) (s Xk00)))))
% Found x3000:=(x300 x20):((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))
% Found (x300 x20) as proof of ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))
% Found ((x30 (s Xk00)) x20) as proof of ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))
% Found (((x3 (s Xj00)) (s Xk00)) x20) as proof of ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))
% Found (fun (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s Xj00)) (s Xk00)) x20)) as proof of ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))
% Found (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s Xj00)) (s Xk00)) x20)) as proof of ((forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000)))))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00)))))
% Found (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s Xj00)) (s Xk00)) x20)) as proof of (((Xw0 c1) c1)->((forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000)))))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))))
% Found (and_rect00 (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s Xj00)) (s Xk00)) x20))) as proof of ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))
% Found ((and_rect0 ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s Xj00)) (s Xk00)) x20))) as proof of ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))
% Found (((fun (P:Type) (x2:(((Xw0 c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))->P)))=> (((((and_rect ((Xw0 c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))) P) x2) x00)) ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s Xj00)) (s Xk00)) x20))) as proof of ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))
% Found (fun (x20:((Xw0 (s Xj00)) (s Xk00)))=> (((fun (P:Type) (x2:(((Xw0 c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))->P)))=> (((((and_rect ((Xw0 c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))) P) x2) x00)) ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s Xj00)) (s Xk00)) x20)))) as proof of ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))
% Found (fun (Xk00:fofType) (x20:((Xw0 (s Xj00)) (s Xk00)))=> (((fun (P:Type) (x2:(((Xw0 c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))->P)))=> (((((and_rect ((Xw0 c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))) P) x2) x00)) ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s Xj00)) (s Xk00)) x20)))) as proof of (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00)))))
% Found (fun (Xj00:fofType) (Xk00:fofType) (x20:((Xw0 (s Xj00)) (s Xk00)))=> (((fun (P:Type) (x2:(((Xw0 c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))->P)))=> (((((and_rect ((Xw0 c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))) P) x2) x00)) ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s Xj00)) (s Xk00)) x20)))) as proof of (forall (Xk00:fofType), (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))))
% Found (fun (Xj00:fofType) (Xk00:fofType) (x20:((Xw0 (s Xj00)) (s Xk00)))=> (((fun (P:Type) (x2:(((Xw0 c1) c1)->((forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))->P)))=> (((((and_rect ((Xw0 c1) c1)) (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 Xj00) Xk00)->((and ((Xw0 (s (s Xj00))) Xk00)) ((Xw0 (s Xj00)) (s Xk00)))))) P) x2) x00)) ((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))) (fun (x2:((Xw0 c1) c1)) (x3:(forall (Xj000:fofType) (Xk000:fofType), (((Xw0 Xj000) Xk000)->((and ((Xw0 (s (s Xj000))) Xk000)) ((Xw0 (s Xj000)) (s Xk000))))))=> (((x3 (s Xj00)) (s Xk00)) x20)))) as proof of (forall (Xj00:fofType) (Xk00:fofType), (((Xw0 (s Xj00)) (s Xk00))->((and ((Xw0 (s (s (s Xj00)))) (s Xk00))) ((Xw0 (s (s Xj00))) (s (s Xk00))))))
% Found x400:(forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400 as proof of (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400:(forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400 as proof of (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400:(forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400 as proof of (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400:(forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400 as proof of (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x5000:=(x500 x3):((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (x500 x3) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found ((x50 (s Xk1)) x3) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (((x5 (s Xj1)) (s Xk1)) x3) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (fun (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)) as proof of ((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)) as proof of (((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))))
% Found (and_rect00 (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3))) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found ((and_rect0 ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3))) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3))) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (fun (x3:((Xw (s Xj1)) (s Xk1)))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)))) as proof of ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))
% Found (fun (Xk1:fofType) (x3:((Xw (s Xj1)) (s Xk1)))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)))) as proof of (((Xw (s Xj1)) (s Xk1))->((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1)))))
% Found (fun (Xj1:fofType) (Xk1:fofType) (x3:((Xw (s Xj1)) (s Xk1)))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)))) as proof of (forall (Xk:fofType), (((Xw (s Xj1)) (s Xk))->((and ((Xw (s (s (s Xj1)))) (s Xk))) ((Xw (s (s Xj1))) (s (s Xk))))))
% Found (fun (Xj1:fofType) (Xk1:fofType) (x3:((Xw (s Xj1)) (s Xk1)))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s Xj1)))) (s Xk1))) ((Xw (s (s Xj1))) (s (s Xk1))))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s Xj1)) (s Xk1)) x3)))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s Xj)) (s Xk))->((and ((Xw (s (s (s Xj)))) (s Xk))) ((Xw (s (s Xj))) (s (s Xk))))))
% Found x5000:=(x500 x3):((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (x500 x3) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found ((x50 Xk1) x3) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (((x5 (s (s Xj1))) Xk1) x3) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (fun (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)) as proof of ((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1))))
% Found (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)) as proof of (((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))))
% Found (and_rect00 (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3))) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found ((and_rect0 ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3))) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3))) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (fun (x3:((Xw (s (s Xj1))) Xk1))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)))) as proof of ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))
% Found (fun (Xk1:fofType) (x3:((Xw (s (s Xj1))) Xk1))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)))) as proof of (((Xw (s (s Xj1))) Xk1)->((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1))))
% Found (fun (Xj1:fofType) (Xk1:fofType) (x3:((Xw (s (s Xj1))) Xk1))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj1))) Xk)->((and ((Xw (s (s (s (s Xj1))))) Xk)) ((Xw (s (s (s Xj1)))) (s Xk)))))
% Found (fun (Xj1:fofType) (Xk1:fofType) (x3:((Xw (s (s Xj1))) Xk1))=> (((fun (P:Type) (x4:(((Xw c1) c1)->((forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))->P)))=> (((((and_rect ((Xw c1) c1)) (forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk)))))) P) x4) x2)) ((and ((Xw (s (s (s (s Xj1))))) Xk1)) ((Xw (s (s (s Xj1)))) (s Xk1)))) (fun (x4:((Xw c1) c1)) (x5:(forall (Xj:fofType) (Xk:fofType), (((Xw Xj) Xk)->((and ((Xw (s (s Xj))) Xk)) ((Xw (s Xj)) (s Xk))))))=> (((x5 (s (s Xj1))) Xk1) x3)))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s (s Xj))) Xk)->((and ((Xw (s (s (s (s Xj))))) Xk)) ((Xw (s (s (s Xj)))) (s Xk)))))
% Found x400:(forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400 as proof of (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400:(forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400 as proof of (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400:(forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400 as proof of (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400:(forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x400 as proof of (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10)))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200:((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x200 as proof of ((and ((Xw c1) c1)) (forall (Xj10:fofType) (Xk10:fofType), (((Xw Xj10) Xk10)->((and ((Xw (s (s Xj10))) Xk10)) ((Xw (s Xj10)) (s Xk10))))))
% Found x50:=(x5 (s (s Xj2))):(forall (Xk:fofType), (((Xw (s (s Xj2))) Xk)->((and ((Xw (s (s (s (s Xj2))))) Xk)) ((Xw (s (s (s Xj2)))) (s Xk)))))
% Found (x5 (s (s Xj2))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj2))) Xk)->((and ((Xw (s (s (s (s Xj2))))) Xk)) ((Xw (s (s (s Xj2)))) (s Xk)))))
% Found (fun (Xj2:fofType)=> (x5 (s (s Xj2)))) as proof of (forall (Xk:fofType), (((Xw (s (s Xj2))) Xk)->((and ((Xw (s (s (s (s Xj2))))) Xk)) ((Xw (s (s (s Xj2)))) (s Xk)))))
% Found (fun (Xj2:fofType)=> (x5 (s (s Xj2)))) as proof of (forall (Xj:fofType) (Xk:fofType), (((Xw (s (s Xj))) Xk)->((and ((Xw (s (s (s (s Xj))))) Xk)) ((Xw (s (s (s Xj)))) (s Xk)))))
% Found x6:((and (((fun (x700:fofType) (x60:fofType)=> ((and ((and ((cCKB6_BLACK (s (s (s x700)))) Xk3)) ((cCKB6_BLACK (s (s x700))) x60))) ((and ((cCKB6_BLACK (s (s x700))) x60)) ((cCKB6_BLACK (s x700)) (s x60))))) (s (s Xj2))) Xk2)) ((and ((and ((cCKB6_BLACK (s (s (s (s Xj2))))) Xk3)) ((cCKB6_BLACK (s (s (s Xj2)))) (s Xk2)))) ((and ((cCKB6_BLACK (s (s (s Xj2)))) (s Xk2))) ((cCKB6_BLACK (s (s Xj2))) (s (s Xk2))))))
% Found x6 as proof of ((and (((fun (x700:fofType) (x60:fofType)=> ((and ((and ((cCKB6_BLACK (s (s (s x700)))) Xk3)) ((cCKB6_BLACK (s (s x700))) x60))) ((and ((cCKB6_BLACK (s (s x700))) x60)) ((cCKB6_BLACK (s x700)) (s x60))))) (s (s Xj2))) Xk2)) ((and
% EOF
%------------------------------------------------------------------------------