TSTP Solution File: SEV414^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV414^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n100.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 13:34:10 EDT 2014
% Result : Theorem 1.06s
% Output : Proof 1.06s
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV414^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n100.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 09:10:36 CDT 2014
% % CPUTime : 1.06
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x923cb0>, <kernel.DependentProduct object at 0x923c68>) of role type named cS
% Using role type
% Declaring cS:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0xb21290>, <kernel.Single object at 0x923b00>) of role type named c0
% Using role type
% Declaring c0:fofType
% FOF formula ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) of role conjecture named cTHM594_pme
% Conjecture to prove = ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))):Prop
% We need to prove ['((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx))))))))']
% Parameter fofType:Type.
% Parameter cS:(fofType->fofType).
% Parameter c0:fofType.
% Trying to prove ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx))))))))
% Found x00:(x Xx)
% Instantiate: x:=Xp:(fofType->Prop)
% Found x00 as proof of (Xp Xx)
% Found (fun (x00:(x Xx))=> x00) as proof of (Xp Xx)
% Found (fun (Xx:fofType) (x00:(x Xx))=> x00) as proof of ((x Xx)->(Xp Xx))
% Found (fun (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> x00) as proof of (forall (Xx:fofType), ((x Xx)->(Xp Xx)))
% Found x0000:=(x000 x0):(Xp Xx)
% Found (x000 x0) as proof of (Xp Xx)
% Found ((x00 Xp) x0) as proof of (Xp Xx)
% Found ((x00 Xp) x0) as proof of (Xp Xx)
% Found (fun (x00:(x Xx))=> ((x00 Xp) x0)) as proof of (Xp Xx)
% Found (fun (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0)) as proof of ((x Xx)->(Xp Xx))
% Found (fun (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0)) as proof of (forall (Xx:fofType), ((x Xx)->(Xp Xx)))
% Found (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0)) as proof of (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx))))
% Found (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0)) as proof of (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))
% Found x1:(Xp c0)
% Found (fun (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1) as proof of (Xp c0)
% Found (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1) as proof of ((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->(Xp c0))
% Found (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1) as proof of ((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->(Xp c0)))
% Found (and_rect00 (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)) as proof of (Xp c0)
% Found ((and_rect0 (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)) as proof of (Xp c0)
% Found (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)) as proof of (Xp c0)
% Found (fun (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1))) as proof of (Xp c0)
% Found (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1))) as proof of (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp c0))
% Found (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1))) as proof of (x c0)
% Found x010:=(x01 x00):(Xp Xw)
% Found (x01 x00) as proof of (Xp Xw)
% Found ((x0 Xp) x00) as proof of (Xp Xw)
% Found ((x0 Xp) x00) as proof of (Xp Xw)
% Found (x20 ((x0 Xp) x00)) as proof of (Xp (cS Xw))
% Found ((x2 Xw) ((x0 Xp) x00)) as proof of (Xp (cS Xw))
% Found (fun (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))) as proof of (Xp (cS Xw))
% Found (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))) as proof of ((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->(Xp (cS Xw)))
% Found (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))) as proof of ((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->(Xp (cS Xw))))
% Found (and_rect00 (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))) as proof of (Xp (cS Xw))
% Found ((and_rect0 (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))) as proof of (Xp (cS Xw))
% Found (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))) as proof of (Xp (cS Xw))
% Found (fun (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))) as proof of (Xp (cS Xw))
% Found (fun (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))) as proof of (((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))->(Xp (cS Xw)))
% Found (fun (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))) as proof of (x (cS Xw))
% Found (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))) as proof of ((x Xw)->(x (cS Xw)))
% Found (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))) as proof of (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))
% Found ((conj10 (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))))) as proof of ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))
% Found (((conj1 (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))))) as proof of ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))
% Found ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))))) as proof of ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))
% Found ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00)))))) as proof of ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))
% Found ((conj00 ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0))) as proof of ((and ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx))))))
% Found (((conj0 (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0))) as proof of ((and ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx))))))
% Found ((((conj ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0))) as proof of ((and ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx))))))
% Found ((((conj ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0))) as proof of ((and ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx))))))
% Found (ex_intro000 ((((conj ((and (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((x Xx)->(Xp Xx)))))) ((((conj (x c0)) (forall (Xw:fofType), ((x Xw)->(x (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:(x Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:(x Xx))=> ((x00 Xp) x0)))) as proof of ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx))))))))
% Found ((ex_intro00 (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0)))) as proof of ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx))))))))
% Found (((ex_intro0 (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0)))) as proof of ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx))))))))
% Found ((((ex_intro (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0)))) as proof of ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx))))))))
% Found ((((ex_intro (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0)))) as proof of ((ex (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx))))))))
% Got proof ((((ex_intro (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0))))
% Time elapsed = 0.738446s
% node=108 cost=884.000000 depth=28
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% ((((ex_intro (fofType->Prop)) (fun (Xv:(fofType->Prop))=> ((and ((and (Xv c0)) (forall (Xw:fofType), ((Xv Xw)->(Xv (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), ((Xv Xx)->(Xp Xx)))))))) (fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0))))) ((((conj ((and ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw)))))) (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(forall (Xx:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx)->(Xp Xx)))))) ((((conj ((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) c0)) (forall (Xw:fofType), (((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)->((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) (cS Xw))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw))))) P) x1) x0)) (Xp c0)) (fun (x1:(Xp c0)) (x2:(forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))=> x1)))) (fun (Xw:fofType) (x0:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xw)) (Xp:(fofType->Prop)) (x00:((and (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))))=> (((fun (P:Type) (x1:((Xp c0)->((forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))->P)))=> (((((and_rect (Xp c0)) (forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0))))) P) x1) x00)) (Xp (cS Xw))) (fun (x1:(Xp c0)) (x2:(forall (Xw0:fofType), ((Xp Xw0)->(Xp (cS Xw0)))))=> ((x2 Xw) ((x0 Xp) x00))))))) (fun (Xp:(fofType->Prop)) (x0:((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))) (Xx:fofType) (x00:((fun (a0:fofType)=> (forall (Xp:(fofType->Prop)), (((and (Xp c0)) (forall (Xw:fofType), ((Xp Xw)->(Xp (cS Xw)))))->(Xp a0)))) Xx))=> ((x00 Xp) x0))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------