TSTP Solution File: PUZ124^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : PUZ124^5 : TPTP v6.1.0. Bugfixed v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n103.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 13:29:01 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 : PUZ124^5 : TPTP v6.1.0. Bugfixed v5.2.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n103.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 08:25:31 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 0x1c25cf8>, <kernel.Constant object at 0x1c25830>) of role type named c1_type
% Using role type
% Declaring c1:fofType
% FOF formula (<kernel.Constant object at 0x1e31a28>, <kernel.Single object at 0x1c25d88>) of role type named c2_type
% Using role type
% Declaring c2:fofType
% FOF formula (<kernel.Constant object at 0x1c25950>, <kernel.Single object at 0x1c256c8>) of role type named c3_type
% Using role type
% Declaring c3:fofType
% FOF formula (<kernel.Constant object at 0x1c25cf8>, <kernel.Single object at 0x1c259e0>) of role type named c4_type
% Using role type
% Declaring c4:fofType
% FOF formula (<kernel.Constant object at 0x1c25638>, <kernel.DependentProduct object at 0x1c25950>) of role type named g_type
% Using role type
% Declaring g:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0x1c25680>, <kernel.DependentProduct object at 0x1c254d0>) of role type named s_type
% Using role type
% Declaring s:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0x1c25f38>, <kernel.DependentProduct object at 0x1c25638>) of role type named cCKB6_BLACK_type
% Using role type
% Declaring cCKB6_BLACK:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x1c25488>, <kernel.DependentProduct object at 0x1c25680>) of role type named cCKB6_H_type
% Using role type
% Declaring cCKB6_H:(fofType->(fofType->(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 (((eq (fofType->(fofType->(fofType->(fofType->Prop))))) cCKB6_H) (fun (Xx:fofType) (Xy:fofType) (Xu:fofType) (Xv:fofType)=> ((and ((cCKB6_BLACK Xx) Xy)) ((or ((or ((or ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c1)) (((eq fofType) Xu) (s (s (s Xx)))))) (((eq fofType) Xv) (s Xy)))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c2)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) (s (s Xy)))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c3)) (((eq fofType) Xu) (s Xx)))) (((eq fofType) Xv) (s Xy))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c4)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) Xy)))))) of role definition named cCKB6_H_def
% A new definition: (((eq (fofType->(fofType->(fofType->(fofType->Prop))))) cCKB6_H) (fun (Xx:fofType) (Xy:fofType) (Xu:fofType) (Xv:fofType)=> ((and ((cCKB6_BLACK Xx) Xy)) ((or ((or ((or ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c1)) (((eq fofType) Xu) (s (s (s Xx)))))) (((eq fofType) Xv) (s Xy)))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c2)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) (s (s Xy)))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c3)) (((eq fofType) Xu) (s Xx)))) (((eq fofType) Xv) (s Xy))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c4)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) Xy))))))
% Defined: cCKB6_H:=(fun (Xx:fofType) (Xy:fofType) (Xu:fofType) (Xv:fofType)=> ((and ((cCKB6_BLACK Xx) Xy)) ((or ((or ((or ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c1)) (((eq fofType) Xu) (s (s (s Xx)))))) (((eq fofType) Xv) (s Xy)))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c2)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) (s (s Xy)))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c3)) (((eq fofType) Xu) (s Xx)))) (((eq fofType) Xv) (s Xy))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c4)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) Xy)))))
% FOF formula (forall (Xx:fofType) (Xy:fofType) (Xu:fofType) (Xv:fofType), (((and ((((cCKB6_H Xx) Xy) Xu) Xv)) (((eq fofType) ((g Xu) Xv)) c3))->((and (((eq fofType) Xu) (s (s (s Xx))))) (((eq fofType) Xv) (s Xy))))) of role conjecture named cCKB_L43000
% Conjecture to prove = (forall (Xx:fofType) (Xy:fofType) (Xu:fofType) (Xv:fofType), (((and ((((cCKB6_H Xx) Xy) Xu) Xv)) (((eq fofType) ((g Xu) Xv)) c3))->((and (((eq fofType) Xu) (s (s (s Xx))))) (((eq fofType) Xv) (s Xy))))):Prop
% We need to prove ['(forall (Xx:fofType) (Xy:fofType) (Xu:fofType) (Xv:fofType), (((and ((((cCKB6_H Xx) Xy) Xu) Xv)) (((eq fofType) ((g Xu) Xv)) c3))->((and (((eq fofType) Xu) (s (s (s Xx))))) (((eq fofType) Xv) (s Xy)))))']
% Parameter fofType:Type.
% Parameter c1:fofType.
% Parameter c2:fofType.
% Parameter c3:fofType.
% Parameter c4:fofType.
% Parameter g:(fofType->(fofType->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)).
% Definition cCKB6_H:=(fun (Xx:fofType) (Xy:fofType) (Xu:fofType) (Xv:fofType)=> ((and ((cCKB6_BLACK Xx) Xy)) ((or ((or ((or ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c1)) (((eq fofType) Xu) (s (s (s Xx)))))) (((eq fofType) Xv) (s Xy)))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c2)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) (s (s Xy)))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c3)) (((eq fofType) Xu) (s Xx)))) (((eq fofType) Xv) (s Xy))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c4)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) Xy))))):(fofType->(fofType->(fofType->(fofType->Prop)))).
% Trying to prove (forall (Xx:fofType) (Xy:fofType) (Xu:fofType) (Xv:fofType), (((and ((((cCKB6_H Xx) Xy) Xu) Xv)) (((eq fofType) ((g Xu) Xv)) c3))->((and (((eq fofType) Xu) (s (s (s Xx))))) (((eq fofType) Xv) (s Xy)))))
% Found eq_ref000:=(eq_ref00 P):((P Xu)->(P Xu))
% Found (eq_ref00 P) as proof of (P0 Xu)
% Found ((eq_ref0 Xu) P) as proof of (P0 Xu)
% Found (((eq_ref fofType) Xu) P) as proof of (P0 Xu)
% Found (((eq_ref fofType) Xu) P) as proof of (P0 Xu)
% Found eq_ref000:=(eq_ref00 P):((P Xv)->(P Xv))
% Found (eq_ref00 P) as proof of (P0 Xv)
% Found ((eq_ref0 Xv) P) as proof of (P0 Xv)
% Found (((eq_ref fofType) Xv) P) as proof of (P0 Xv)
% Found (((eq_ref fofType) Xv) P) as proof of (P0 Xv)
% Found eq_ref00:=(eq_ref0 Xv):(((eq fofType) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found eq_ref00:=(eq_ref0 Xu):(((eq fofType) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found x10:=(x1 (fun (x2:fofType)=> (P Xu))):((P Xu)->(P Xu))
% Found (x1 (fun (x2:fofType)=> (P Xu))) as proof of (P0 Xu)
% Found (x1 (fun (x2:fofType)=> (P Xu))) as proof of (P0 Xu)
% Found x10:=(x1 (fun (x2:fofType)=> (P Xv))):((P Xv)->(P Xv))
% Found (x1 (fun (x2:fofType)=> (P Xv))) as proof of (P0 Xv)
% Found (x1 (fun (x2:fofType)=> (P Xv))) as proof of (P0 Xv)
% Found eq_ref00:=(eq_ref0 (((eq fofType) Xv) (s Xy))):(((eq Prop) (((eq fofType) Xv) (s Xy))) (((eq fofType) Xv) (s Xy)))
% Found (eq_ref0 (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found ((eq_ref Prop) (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found ((eq_ref Prop) (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found ((eq_ref Prop) (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq fofType) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found eq_ref00:=(eq_ref0 Xv):(((eq fofType) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found eq_ref00:=(eq_ref0 (((eq fofType) Xv) (s Xy))):(((eq Prop) (((eq fofType) Xv) (s Xy))) (((eq fofType) Xv) (s Xy)))
% Found (eq_ref0 (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found ((eq_ref Prop) (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found ((eq_ref Prop) (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found ((eq_ref Prop) (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found x10:=(x1 (fun (x4:fofType)=> (P Xv))):((P Xv)->(P Xv))
% Found (x1 (fun (x4:fofType)=> (P Xv))) as proof of (P0 Xv)
% Found (x1 (fun (x4:fofType)=> (P Xv))) as proof of (P0 Xv)
% Found x10:=(x1 (fun (x4:fofType)=> (P Xu))):((P Xu)->(P Xu))
% Found (x1 (fun (x4:fofType)=> (P Xu))) as proof of (P0 Xu)
% Found (x1 (fun (x4:fofType)=> (P Xu))) as proof of (P0 Xu)
% Found x10:=(x1 (fun (x2:fofType)=> (P Xv))):((P Xv)->(P Xv))
% Found (x1 (fun (x2:fofType)=> (P Xv))) as proof of (P0 Xv)
% Found (x1 (fun (x2:fofType)=> (P Xv))) as proof of (P0 Xv)
% Found x10:=(x1 (fun (x2:fofType)=> (P Xu))):((P Xu)->(P Xu))
% Found (x1 (fun (x2:fofType)=> (P Xu))) as proof of (P0 Xu)
% Found (x1 (fun (x2:fofType)=> (P Xu))) as proof of (P0 Xu)
% Found eq_ref00:=(eq_ref0 Xv):(((eq fofType) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found eq_ref00:=(eq_ref0 Xu):(((eq fofType) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found eq_ref00:=(eq_ref0 (s Xy)):(((eq fofType) (s Xy)) (s Xy))
% Found (eq_ref0 (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xv)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xv)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xv)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xv)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found eq_ref00:=(eq_ref0 (s (s (s Xx)))):(((eq fofType) (s (s (s Xx)))) (s (s (s Xx))))
% Found (eq_ref0 (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq fofType) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found eq_ref00:=(eq_ref0 Xv):(((eq fofType) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found eq_ref00:=(eq_ref0 (((eq fofType) Xv) (s Xy))):(((eq Prop) (((eq fofType) Xv) (s Xy))) (((eq fofType) Xv) (s Xy)))
% Found (eq_ref0 (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found ((eq_ref Prop) (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found ((eq_ref Prop) (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found ((eq_ref Prop) (((eq fofType) Xv) (s Xy))) as proof of (((eq Prop) (((eq fofType) Xv) (s Xy))) b)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xj:fofType) (Xk:fofType), (((and (((eq fofType) Xu) (s (s (s Xj))))) (((eq fofType) Xv) (s Xk)))->((and ((and (((eq fofType) Xu) (s (s (s (s (s Xj))))))) (((eq fofType) Xv) (s Xk)))) ((and (((eq fofType) Xu) (s (s (s (s Xj)))))) (((eq fofType) Xv) (s (s Xk)))))))):(((eq Prop) (forall (Xj:fofType) (Xk:fofType), (((and (((eq fofType) Xu) (s (s (s Xj))))) (((eq fofType) Xv) (s Xk)))->((and ((and (((eq fofType) Xu) (s (s (s (s (s Xj))))))) (((eq fofType) Xv) (s Xk)))) ((and (((eq fofType) Xu) (s (s (s (s Xj)))))) (((eq fofType) Xv) (s (s Xk)))))))) (forall (Xj:fofType) (Xk:fofType), (((and (((eq fofType) Xu) (s (s (s Xj))))) (((eq fofType) Xv) (s Xk)))->((and ((and (((eq fofType) Xu) (s (s (s (s (s Xj))))))) (((eq fofType) Xv) (s Xk)))) ((and (((eq fofType) Xu) (s (s (s (s Xj)))))) (((eq fofType) Xv) (s (s Xk))))))))
% Found (eq_ref0 (forall (Xj:fofType) (Xk:fofType), (((and (((eq fofType) Xu) (s (s (s Xj))))) (((eq fofType) Xv) (s Xk)))->((and ((and (((eq fofType) Xu) (s (s (s (s (s Xj))))))) (((eq fofType) Xv) (s Xk)))) ((and (((eq fofType) Xu) (s (s (s (s Xj)))))) (((eq fofType) Xv) (s (s Xk)))))))) as proof of (((eq Prop) (forall (Xj:fofType) (Xk:fofType), (((and (((eq fofType) Xu) (s (s (s Xj))))) (((eq fofType) Xv) (s Xk)))->((and ((and (((eq fofType) Xu) (s (s (s (s (s Xj))))))) (((eq fofType) Xv) (s Xk)))) ((and (((eq fofType) Xu) (s (s (s (s Xj)))))) (((eq fofType) Xv) (s (s Xk)))))))) b)
% Found ((eq_ref Prop) (forall (Xj:fofType) (Xk:fofType), (((and (((eq fofType) Xu) (s (s (s Xj))))) (((eq fofType) Xv) (s Xk)))->((and ((and (((eq fofType) Xu) (s (s (s (s (s Xj))))))) (((eq fofType) Xv) (s Xk)))) ((and (((eq fofType) Xu) (s (s (s (s Xj)))))) (((eq fofType) Xv) (s (s Xk)))))))) as proof of (((eq Prop) (forall (Xj:fofType) (Xk:fofType), (((and (((eq fofType) Xu) (s (s (s Xj))))) (((eq fofType) Xv) (s Xk)))->((and ((and (((eq fofType) Xu) (s (s (s (s (s Xj))))))) (((eq fofType) Xv) (s Xk)))) ((and (((eq fofType) Xu) (s (s (s (s Xj)))))) (((eq fofType) Xv) (s (s Xk)))))))) b)
% Found ((eq_ref Prop) (forall (Xj:fofType) (Xk:fofType), (((and (((eq fofType) Xu) (s (s (s Xj))))) (((eq fofType) Xv) (s Xk)))->((and ((and (((eq fofType) Xu) (s (s (s (s (s Xj))))))) (((eq fofType) Xv) (s Xk)))) ((and (((eq fofType) Xu) (s (s (s (s Xj)))))) (((eq fofType) Xv) (s (s Xk)))))))) as proof of (((eq Prop) (forall (Xj:fofType) (Xk:fofType), (((and (((eq fofType) Xu) (s (s (s Xj))))) (((eq fofType) Xv) (s Xk)))->((and ((and (((eq fofType) Xu) (s (s (s (s (s Xj))))))) (((eq fofType) Xv) (s Xk)))) ((and (((eq fofType) Xu) (s (s (s (s Xj)))))) (((eq fofType) Xv) (s (s Xk)))))))) b)
% Found ((eq_ref Prop) (forall (Xj:fofType) (Xk:fofType), (((and (((eq fofType) Xu) (s (s (s Xj))))) (((eq fofType) Xv) (s Xk)))->((and ((and (((eq fofType) Xu) (s (s (s (s (s Xj))))))) (((eq fofType) Xv) (s Xk)))) ((and (((eq fofType) Xu) (s (s (s (s Xj)))))) (((eq fofType) Xv) (s (s Xk)))))))) as proof of (((eq Prop) (forall (Xj:fofType) (Xk:fofType), (((and (((eq fofType) Xu) (s (s (s Xj))))) (((eq fofType) Xv) (s Xk)))->((and ((and (((eq fofType) Xu) (s (s (s (s (s Xj))))))) (((eq fofType) Xv) (s Xk)))) ((and (((eq fofType) Xu) (s (s (s (s Xj)))))) (((eq fofType) Xv) (s (s Xk)))))))) b)
% Found x10:=(x1 (fun (x4:fofType)=> (P Xu))):((P Xu)->(P Xu))
% Found (x1 (fun (x4:fofType)=> (P Xu))) as proof of (P0 Xu)
% Found (x1 (fun (x4:fofType)=> (P Xu))) as proof of (P0 Xu)
% Found x10:=(x1 (fun (x4:fofType)=> (P Xv))):((P Xv)->(P Xv))
% Found (x1 (fun (x4:fofType)=> (P Xv))) as proof of (P0 Xv)
% Found (x1 (fun (x4:fofType)=> (P Xv))) as proof of (P0 Xv)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xv)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xv)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xv)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xv)
% Found eq_ref00:=(eq_ref0 (s Xy)):(((eq fofType) (s Xy)) (s Xy))
% Found (eq_ref0 (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found eq_ref00:=(eq_ref0 (s (s (s Xx)))):(((eq fofType) (s (s (s Xx)))) (s (s (s Xx))))
% Found (eq_ref0 (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found x0:(P Xv)
% Instantiate: b:=Xv:fofType
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (s Xy)):(((eq fofType) (s Xy)) (s Xy))
% Found (eq_ref0 (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found x0:(P Xu)
% Instantiate: b:=Xu:fofType
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (s (s (s Xx)))):(((eq fofType) (s (s (s Xx)))) (s (s (s Xx))))
% Found (eq_ref0 (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq fofType) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found eq_ref00:=(eq_ref0 Xv):(((eq fofType) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b
% Found x3:((or ((or ((or ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c1)) (((eq fofType) Xu) (s (s (s Xx)))))) (((eq fofType) Xv) (s Xy)))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c2)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) (s (s Xy)))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c3)) (((eq fofType) Xu) (s Xx)))) (((eq fofType) Xv) (s Xy))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c4)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) Xy)))
% Instantiate: b:=((or ((or ((or ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c1)) (((eq fofType) Xu) (s (s (s Xx)))))) (((eq fofType) Xv) (s Xy)))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c2)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) (s (s Xy)))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c3)) (((eq fofType) Xu) (s Xx)))) (((eq fofType) Xv) (s Xy))))) ((and ((and (((eq fofType) ((g (s (s Xx))) (s Xy))) c4)) (((eq fofType) Xu) (s (s Xx))))) (((eq fofType) Xv) Xy))):Prop
% Found x3 as proof of b
% Found eq_ref000:=(eq_ref00 P):((P Xu)->(P Xu))
% Found (eq_ref00 P) as proof of (P0 Xu)
% Found ((eq_ref0 Xu) P) as proof of (P0 Xu)
% Found (((eq_ref fofType) Xu) P) as proof of (P0 Xu)
% Found (((eq_ref fofType) Xu) P) as proof of (P0 Xu)
% Found x10:=(x1 (fun (x4:fofType)=> (P Xv))):((P Xv)->(P Xv))
% Found (x1 (fun (x4:fofType)=> (P Xv))) as proof of (P0 Xv)
% Found (x1 (fun (x4:fofType)=> (P Xv))) as proof of (P0 Xv)
% Found x2:(P Xv)
% Instantiate: b:=Xv:fofType
% Found x2 as proof of (P0 b)
% Found x2:(P Xu)
% Instantiate: b:=Xu:fofType
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (s Xy)):(((eq fofType) (s Xy)) (s Xy))
% Found (eq_ref0 (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found eq_ref00:=(eq_ref0 (s (s (s Xx)))):(((eq fofType) (s (s (s Xx)))) (s (s (s Xx))))
% Found (eq_ref0 (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found eq_ref00:=(eq_ref0 Xu):(((eq fofType) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found eq_ref00:=(eq_ref0 Xv):(((eq fofType) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found eq_ref00:=(eq_ref0 Xv):(((eq fofType) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found eq_ref00:=(eq_ref0 Xu):(((eq fofType) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found eq_ref000:=(eq_ref00 P):((P Xu)->(P Xu))
% Found (eq_ref00 P) as proof of (P0 Xu)
% Found ((eq_ref0 Xu) P) as proof of (P0 Xu)
% Found (((eq_ref fofType) Xu) P) as proof of (P0 Xu)
% Found (((eq_ref fofType) Xu) P) as proof of (P0 Xu)
% Found eq_ref00:=(eq_ref0 Xu):(((eq fofType) Xu) Xu)
% Found (eq_ref0 Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found ((eq_ref fofType) Xu) as proof of (((eq fofType) Xu) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s (s (s Xx))))
% Found eq_ref000:=(eq_ref00 P):((P Xv)->(P Xv))
% Found (eq_ref00 P) as proof of (P0 Xv)
% Found ((eq_ref0 Xv) P) as proof of (P0 Xv)
% Found (((eq_ref fofType) Xv) P) as proof of (P0 Xv)
% Found (((eq_ref fofType) Xv) P) as proof of (P0 Xv)
% Found eq_ref00:=(eq_ref0 Xv):(((eq fofType) Xv) Xv)
% Found (eq_ref0 Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found ((eq_ref fofType) Xv) as proof of (((eq fofType) Xv) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (s Xy))
% Found eq_ref00:=(eq_ref0 (s Xy)):(((eq fofType) (s Xy)) (s Xy))
% Found (eq_ref0 (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found ((eq_ref fofType) (s Xy)) as proof of (((eq fofType) (s Xy)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xv)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xv)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xv)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xv)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found eq_ref00:=(eq_ref0 (s (s (s Xx)))):(((eq fofType) (s (s (s Xx)))) (s (s (s Xx))))
% Found (eq_ref0 (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found ((eq_ref fofType) (s (s (s Xx)))) as proof of (((eq fofType) (s (s (s Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xu)
% Found eq_ref00:=(eq_ref0 (s (s (s Xx)))):(((eq fofType) (s (s (s Xx
% EOF
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