TSTP Solution File: SYN058^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SYN058^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n101.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 13:37:48 EDT 2014
% Result : Theorem 58.20s
% Output : Proof 58.20s
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SYN058^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n101.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 08:05:21 CDT 2014
% % CPUTime: 58.20
% 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 0x2250710>, <kernel.DependentProduct object at 0x21d5560>) of role type named cG
% Using role type
% Declaring cG:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x2250ab8>, <kernel.DependentProduct object at 0x1ce3c20>) of role type named cF
% Using role type
% Declaring cF:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x2250f80>, <kernel.DependentProduct object at 0x1ce32d8>) of role type named cP
% Using role type
% Declaring cP:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x21d51b8>, <kernel.DependentProduct object at 0x2252ea8>) of role type named cS
% Using role type
% Declaring cS:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x22503b0>, <kernel.DependentProduct object at 0x22522d8>) of role type named cQ
% Using role type
% Declaring cQ:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x1ce3c20>, <kernel.DependentProduct object at 0x1ce6050>) of role type named cR
% Using role type
% Declaring cR:(fofType->Prop)
% FOF formula (((and ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx)))))->(forall (Xx:fofType), (((and (cP Xx)) (cF Xx))->(cG Xx)))) of role conjecture named cPELL28
% Conjecture to prove = (((and ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx)))))->(forall (Xx:fofType), (((and (cP Xx)) (cF Xx))->(cG Xx)))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(((and ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx)))))->(forall (Xx:fofType), (((and (cP Xx)) (cF Xx))->(cG Xx))))']
% Parameter fofType:Type.
% Parameter cG:(fofType->Prop).
% Parameter cF:(fofType->Prop).
% Parameter cP:(fofType->Prop).
% Parameter cS:(fofType->Prop).
% Parameter cQ:(fofType->Prop).
% Parameter cR:(fofType->Prop).
% Trying to prove (((and ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx)))))->(forall (Xx:fofType), (((and (cP Xx)) (cF Xx))->(cG Xx))))
% Found x2:(cF Xx)
% Found x2 as proof of (cF Xx)
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x2:(cF Xx)
% Found x2 as proof of (cF Xx)
% Found x4:(cF Xx)
% Found (fun (x4:(cF Xx))=> x4) as proof of (cF Xx)
% Found (fun (x3:(cP Xx)) (x4:(cF Xx))=> x4) as proof of ((cF Xx)->(cF Xx))
% Found (fun (x3:(cP Xx)) (x4:(cF Xx))=> x4) as proof of ((cP Xx)->((cF Xx)->(cF Xx)))
% Found (and_rect10 (fun (x3:(cP Xx)) (x4:(cF Xx))=> x4)) as proof of (cF Xx)
% Found ((and_rect1 (cF Xx)) (fun (x3:(cP Xx)) (x4:(cF Xx))=> x4)) as proof of (cF Xx)
% Found (((fun (P:Type) (x3:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x3) x0)) (cF Xx)) (fun (x3:(cP Xx)) (x4:(cF Xx))=> x4)) as proof of (cF Xx)
% Found (((fun (P:Type) (x3:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x3) x0)) (cF Xx)) (fun (x3:(cP Xx)) (x4:(cF Xx))=> x4)) as proof of (cF Xx)
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found (fun (x6:(cF Xx))=> x6) as proof of (cF Xx)
% Found (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6) as proof of ((cF Xx)->(cF Xx))
% Found (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6) as proof of ((cP Xx)->((cF Xx)->(cF Xx)))
% Found (and_rect20 (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)) as proof of (cF Xx)
% Found ((and_rect2 (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)) as proof of (cF Xx)
% Found (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)) as proof of (cF Xx)
% Found (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)) as proof of (cF Xx)
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x3:(cP Xx))=> (x20 Xx)) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x3:(cP Xx))=> (x20 Xx)) as proof of ((cP Xx)->((cF Xx)->(cG Xx)))
% Found (and_rect10 (fun (x3:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found ((and_rect1 (cG Xx)) (fun (x3:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found (((fun (P:Type) (x3:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x3) x0)) (cG Xx)) (fun (x3:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found (((fun (P:Type) (x3:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x3) x0)) (cG Xx)) (fun (x3:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found x2:(cF Xx)
% Found x2 as proof of (cF Xx)
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cP Xx)->((cF Xx)->(cG Xx)))
% Found (and_rect20 (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found ((and_rect2 (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cP Xx)->((cF Xx)->(cG Xx)))
% Found (and_rect20 (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found ((and_rect2 (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found (fun (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx)))) as proof of (cG Xx)
% Found (fun (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx)))) as proof of (((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx))
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cP Xx)->((cF Xx)->(cG Xx)))
% Found (and_rect20 (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found ((and_rect2 (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found (fun (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx)))) as proof of (cG Xx)
% Found (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx)))) as proof of (((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx))
% Found (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx)))) as proof of ((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx)))
% Found x2:(cF Xx)
% Found x2 as proof of (cF Xx)
% Found x2:(cF Xx)
% Found x2 as proof of (cF Xx)
% Found x2:(cF Xx)
% Found x2 as proof of (cF Xx)
% Found x2:(cF Xx)
% Found x2 as proof of (cF Xx)
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x3:(cP Xx))=> (x20 Xx)) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x3:(cP Xx))=> (x20 Xx)) as proof of ((cP Xx)->((cF Xx)->(cG Xx)))
% Found x4000:=(x400 x2):(cG Xx)
% Found (x400 x2) as proof of (cG Xx)
% Found ((x40 Xx) x2) as proof of (cG Xx)
% Found (fun (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x40 Xx) x2)) as proof of (cG Xx)
% Found (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x40 Xx) x2)) as proof of (((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx))
% Found (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x40 Xx) x2)) as proof of ((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx)))
% Found (and_rect20 (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x40 Xx) x2))) as proof of (cG Xx)
% Found ((and_rect2 (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x40 Xx) x2))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x5) x3)) (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x40 Xx) x2))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x5) x3)) (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x40 Xx) x2))) as proof of (cG Xx)
% Found x2000:=(x200 x4):(cG Xx)
% Found (x200 x4) as proof of (cG Xx)
% Found ((x20 Xx) x4) as proof of (cG Xx)
% Found (fun (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)) as proof of (cG Xx)
% Found (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)) as proof of (((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx))
% Found (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)) as proof of ((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx)))
% Found (and_rect20 (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4))) as proof of (cG Xx)
% Found ((and_rect2 (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x5) x1)) (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x5) x1)) (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4))) as proof of (cG Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x2000:=(x200 x4):(cG Xx)
% Found (x200 x4) as proof of (cG Xx)
% Found ((x20 Xx) x4) as proof of (cG Xx)
% Found (fun (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)) as proof of (cG Xx)
% Found (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)) as proof of (((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx))
% Found (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)) as proof of ((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx)))
% Found (and_rect20 (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4))) as proof of (cG Xx)
% Found ((and_rect2 (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x5) x1)) (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4))) as proof of (cG Xx)
% Found (fun (x4:(cF Xx))=> (((fun (P:Type) (x5:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x5) x1)) (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)))) as proof of (cG Xx)
% Found (fun (x4:(cF Xx))=> (((fun (P:Type) (x5:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x5) x1)) (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)))) as proof of ((cF Xx)->(cG Xx))
% Found x2000:=(x200 x4):(cG Xx)
% Found (x200 x4) as proof of (cG Xx)
% Found ((x20 Xx) x4) as proof of (cG Xx)
% Found (fun (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)) as proof of (cG Xx)
% Found (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)) as proof of (((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx))
% Found (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)) as proof of ((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx)))
% Found (and_rect20 (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4))) as proof of (cG Xx)
% Found ((and_rect2 (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x5) x1)) (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4))) as proof of (cG Xx)
% Found (fun (x4:(cF Xx))=> (((fun (P:Type) (x5:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x5) x1)) (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)))) as proof of (cG Xx)
% Found (fun (x3:(cP Xx)) (x4:(cF Xx))=> (((fun (P:Type) (x5:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x5) x1)) (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)))) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x3:(cP Xx)) (x4:(cF Xx))=> (((fun (P:Type) (x5:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x5) x1)) (cG Xx)) (fun (x5:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x6:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> ((x20 Xx) x4)))) as proof of ((cP Xx)->((cF Xx)->(cG Xx)))
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cP Xx)->((cF Xx)->(cG Xx)))
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cP Xx)->((cF Xx)->(cG Xx)))
% Found x6:(cF Xx)
% Found (fun (x6:(cF Xx))=> x6) as proof of (cF Xx)
% Found (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6) as proof of ((cF Xx)->(cF Xx))
% Found (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6) as proof of ((cP Xx)->((cF Xx)->(cF Xx)))
% Found (and_rect20 (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)) as proof of (cF Xx)
% Found ((and_rect2 (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)) as proof of (cF Xx)
% Found (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)) as proof of (cF Xx)
% Found (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)) as proof of (cF Xx)
% Found x5000:=(x500 x1):(cQ Xx0)
% Found (x500 x1) as proof of (cQ Xx0)
% Found ((x50 Xx) x1) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x2:(cF Xx)
% Found x2 as proof of (cF Xx)
% Found x2:(cF Xx)
% Found x2 as proof of (cF Xx)
% Found x3000:=(x300 x5):(cQ Xx0)
% Found (x300 x5) as proof of (cQ Xx0)
% Found ((x30 Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x5000:=(x500 x3):(cQ Xx0)
% Found (x500 x3) as proof of (cQ Xx0)
% Found ((x50 Xx) x3) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x3000:=(x300 x5):(cQ Xx0)
% Found (x300 x5) as proof of (cQ Xx0)
% Found ((x30 Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x2000:=(x200 x6):(cG Xx)
% Found (x200 x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found x2000:=(x200 x6):(cG Xx)
% Found (x200 x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found x2000:=(x200 x6):(cG Xx)
% Found (x200 x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found x2000:=(x200 x6):(cG Xx)
% Found (x200 x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found x200:=(x20 Xx):((cF Xx)->(cG Xx))
% Found (x20 Xx) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cF Xx)->(cG Xx))
% Found (fun (x5:(cP Xx))=> (x20 Xx)) as proof of ((cP Xx)->((cF Xx)->(cG Xx)))
% Found (and_rect20 (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found ((and_rect2 (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cG Xx)) (fun (x5:(cP Xx))=> (x20 Xx))) as proof of (cG Xx)
% Found x2000:=(x200 x6):(cG Xx)
% Found (x200 x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found x2000:=(x200 x6):(cG Xx)
% Found (x200 x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found ((x20 Xx) x6) as proof of (cG Xx)
% Found x2:(cF Xx)
% Found x2 as proof of (cF Xx)
% Found x2:(cF Xx)
% Found x2 as proof of (cF Xx)
% Found x5000:=(x500 x1):(cQ Xx0)
% Found (x500 x1) as proof of (cQ Xx0)
% Found ((x50 Xx) x1) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x5000:=(x500 x1):(cQ Xx0)
% Found (x500 x1) as proof of (cQ Xx0)
% Found ((x50 Xx) x1) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x3000:=(x300 x5):(cQ Xx0)
% Found (x300 x5) as proof of (cQ Xx0)
% Found ((x30 Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x4000:=(x400 x3):(cQ Xx0)
% Found (x400 x3) as proof of (cQ Xx0)
% Found ((x40 Xx) x3) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x4 Xx00) x6) Xx0)) Xx) x3) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x4 Xx00) x6) Xx0)) Xx) x3) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x4 Xx00) x6) Xx0)) Xx) x3)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x4 Xx00) x6) Xx0)) Xx) x3)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x4 Xx00) x6) Xx0)) Xx) x3)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x4 Xx00) x6) Xx0)) Xx) x3))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x4 Xx00) x6) Xx0)) Xx) x3))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x5000:=(x500 x1):(cQ Xx0)
% Found (x500 x1) as proof of (cQ Xx0)
% Found ((x50 Xx) x1) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x1))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x4:(cF Xx)
% Found x4 as proof of (cF Xx)
% Found x3000:=(x300 x5):(cQ Xx0)
% Found (x300 x5) as proof of (cQ Xx0)
% Found ((x30 Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x6:(cP Xx00))=> (((x3 Xx00) x6) Xx0)) Xx) x5))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x5000:=(x500 x3):(cQ Xx0)
% Found (x500 x3) as proof of (cQ Xx0)
% Found ((x50 Xx) x3) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x5 Xx00) x7) Xx0)) Xx) x3))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x3000:=(x300 x5):(cQ Xx0)
% Found (x300 x5) as proof of (cQ Xx0)
% Found ((x30 Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x3000:=(x300 x5):(cQ Xx0)
% Found (x300 x5) as proof of (cQ Xx0)
% Found ((x30 Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5) as proof of (cQ Xx0)
% Found (or_introl00 (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((or_introl0 (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5))) as proof of ((cF Xx)->((or (cQ Xx0)) (cR Xx0)))
% Found (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5))) as proof of ((cP Xx)->((cF Xx)->((or (cQ Xx0)) (cR Xx0))))
% Found (and_rect20 (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found ((and_rect2 ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5))))) as proof of ((or (cQ Xx0)) (cR Xx0))
% Found (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5))))) as proof of (forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x6:(cF Xx)
% Found x6 as proof of (cF Xx)
% Found x9:(cS x5)
% Instantiate: x7:=x5:fofType
% Found (fun (x9:(cS x5))=> x9) as proof of (cS x7)
% Found (fun (x8:(cQ x5)) (x9:(cS x5))=> x9) as proof of ((cS x5)->(cS x7))
% Found (fun (x8:(cQ x5)) (x9:(cS x5))=> x9) as proof of ((cQ x5)->((cS x5)->(cS x7)))
% Found (and_rect20 (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)) as proof of (cS x7)
% Found ((and_rect2 (cS x7)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)) as proof of (cS x7)
% Found (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x7)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)) as proof of (cS x7)
% Found (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x7)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)) as proof of (cS x7)
% Found (ex_intro000 (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x7)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9))) as proof of ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))
% Found ((ex_intro00 x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9))) as proof of ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))
% Found (((ex_intro0 (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9))) as proof of ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))
% Found ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9))) as proof of ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))
% Found (fun (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))) as proof of ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))
% Found (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))) as proof of (((and (cQ x5)) (cS x5))->((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))
% Found (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))) as proof of (forall (x:fofType), (((and (cQ x)) (cS x))->((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))))
% Found (ex_ind00 (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9))))) as proof of ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))
% Found ((ex_ind0 ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9))))) as proof of ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))
% Found (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) x40)) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9))))) as proof of ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))
% Found (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) x40)) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9))))) as proof of ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))
% Found ((x20 (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) x40)) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))) as proof of (cG Xx)
% Found (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) x40)) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))) as proof of (cG Xx)
% Found (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) x40)) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))) as proof of (cG Xx)
% Found (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))) as proof of (cG Xx)
% Found (fun (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)))) as proof of (cG Xx)
% Found (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)))) as proof of (((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx))
% Found (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)))) as proof of ((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->(cG Xx)))
% Found (and_rect10 (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))))) as proof of (cG Xx)
% Found ((and_rect1 (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))))) as proof of (cG Xx)
% Found (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))))) as proof of (cG Xx)
% Found (fun (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)))))) as proof of (cG Xx)
% Found (fun (x1:((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))) (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)))))) as proof of ((((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0))))->(cG Xx))
% Found (fun (x1:((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))) (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)))))) as proof of (((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))->((((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0))))->(cG Xx)))
% Found (and_rect00 (fun (x1:((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))) (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))))))) as proof of (cG Xx)
% Found ((and_rect0 (cG Xx)) (fun (x1:((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))) (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x:fofType), (((and (cQ x)) (cS x))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))))))) as proof of (cG Xx)
% Found (((fun (P:Type) (x1:(((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))->((((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))->P)))=> (((((and_rect ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))) P) x1) x)) (cG Xx)) (fun (x1:((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))) (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x0:fofType), (((and (cQ x0)) (cS x0))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))))))) as proof of (cG Xx)
% Found (fun (x0:((and (cP Xx)) (cF Xx)))=> (((fun (P:Type) (x1:(((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))->((((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))->P)))=> (((((and_rect ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))) P) x1) x)) (cG Xx)) (fun (x1:((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))) (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x0:fofType), (((and (cQ x0)) (cS x0))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)))))))) as proof of (cG Xx)
% Found (fun (Xx:fofType) (x0:((and (cP Xx)) (cF Xx)))=> (((fun (P:Type) (x1:(((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))->((((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))->P)))=> (((((and_rect ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))) P) x1) x)) (cG Xx)) (fun (x1:((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))) (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x0:fofType), (((and (cQ x0)) (cS x0))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)))))))) as proof of (((and (cP Xx)) (cF Xx))->(cG Xx))
% Found (fun (x:((and ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx)))))) (Xx:fofType) (x0:((and (cP Xx)) (cF Xx)))=> (((fun (P:Type) (x1:(((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))->((((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))->P)))=> (((((and_rect ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))) P) x1) x)) (cG Xx)) (fun (x1:((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))) (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x0:fofType), (((and (cQ x0)) (cS x0))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)))))))) as proof of (forall (Xx:fofType), (((and (cP Xx)) (cF Xx))->(cG Xx)))
% Found (fun (x:((and ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx)))))) (Xx:fofType) (x0:((and (cP Xx)) (cF Xx)))=> (((fun (P:Type) (x1:(((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))->((((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))->P)))=> (((((and_rect ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))) P) x1) x)) (cG Xx)) (fun (x1:((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))) (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x0:fofType), (((and (cQ x0)) (cS x0))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6)))))))) as proof of (((and ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx)))))->(forall (Xx:fofType), (((and (cP Xx)) (cF Xx))->(cG Xx))))
% Got proof (fun (x:((and ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx)))))) (Xx:fofType) (x0:((and (cP Xx)) (cF Xx)))=> (((fun (P:Type) (x1:(((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))->((((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))->P)))=> (((((and_rect ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))) P) x1) x)) (cG Xx)) (fun (x1:((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))) (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x0:fofType), (((and (cQ x0)) (cS x0))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))))))))
% Time elapsed = 57.289949s
% node=9183 cost=2058.000000 depth=37
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (x:((and ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx)))))) (Xx:fofType) (x0:((and (cP Xx)) (cF Xx)))=> (((fun (P:Type) (x1:(((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))->((((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))->P)))=> (((((and_rect ((and (forall (Xx:fofType), ((cP Xx)->(forall (Xx0:fofType), (cQ Xx0))))) ((forall (Xx:fofType), ((or (cQ Xx)) (cR Xx)))->((ex fofType) (fun (Xx:fofType)=> ((and (cQ Xx)) (cS Xx))))))) (((ex fofType) (fun (Xx:fofType)=> (cS Xx)))->(forall (Xx:fofType), ((cF Xx)->(cG Xx))))) P) x1) x)) (cG Xx)) (fun (x1:((and (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))) (x2:(((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))->(forall (Xx0:fofType), ((cF Xx0)->(cG Xx0)))))=> (((fun (P:Type) (x3:((forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))->(((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))->P)))=> (((((and_rect (forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) ((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))))) P) x3) x1)) (cG Xx)) (fun (x3:(forall (Xx0:fofType), ((cP Xx0)->(forall (Xx00:fofType), (cQ Xx00))))) (x4:((forall (Xx0:fofType), ((or (cQ Xx0)) (cR Xx0)))->((ex fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0))))))=> (((fun (x5:((ex fofType) (fun (Xx0:fofType)=> (cS Xx0))))=> ((x2 x5) Xx)) (((fun (P:Prop) (x5:(forall (x0:fofType), (((and (cQ x0)) (cS x0))->P)))=> (((((ex_ind fofType) (fun (Xx0:fofType)=> ((and (cQ Xx0)) (cS Xx0)))) P) x5) (x4 (fun (Xx0:fofType)=> (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) ((or (cQ Xx0)) (cR Xx0))) (fun (x5:(cP Xx)) (x6:(cF Xx))=> (((or_introl (cQ Xx0)) (cR Xx0)) (((fun (Xx00:fofType) (x7:(cP Xx00))=> (((x3 Xx00) x7) Xx0)) Xx) x5)))))))) ((ex fofType) (fun (Xx0:fofType)=> (cS Xx0)))) (fun (x5:fofType) (x6:((and (cQ x5)) (cS x5)))=> ((((ex_intro fofType) (fun (Xx0:fofType)=> (cS Xx0))) x5) (((fun (P:Type) (x8:((cQ x5)->((cS x5)->P)))=> (((((and_rect (cQ x5)) (cS x5)) P) x8) x6)) (cS x5)) (fun (x8:(cQ x5)) (x9:(cS x5))=> x9)))))) (((fun (P:Type) (x5:((cP Xx)->((cF Xx)->P)))=> (((((and_rect (cP Xx)) (cF Xx)) P) x5) x0)) (cF Xx)) (fun (x5:(cP Xx)) (x6:(cF Xx))=> x6))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------