TSTP Solution File: SEU876^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU876^5 : 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 13:33:19 EDT 2014
% Result : Theorem 6.51s
% Output : Proof 6.51s
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU876^5 : 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 11:39:26 CDT 2014
% % CPUTime : 6.51
% 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 0x1f186c8>, <kernel.Type object at 0x1f18fc8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x22f3518>, <kernel.DependentProduct object at 0x1f18830>) of role type named cE
% Using role type
% Declaring cE:(a->Prop)
% FOF formula (<kernel.Constant object at 0x1f18b90>, <kernel.DependentProduct object at 0x1f18560>) of role type named cA
% Using role type
% Declaring cA:((a->Prop)->Prop)
% FOF formula ((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))))))) of role conjecture named cDOMLEMMA4_pme
% Conjecture to prove = ((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))))))']
% Parameter a:Type.
% Parameter cE:(a->Prop).
% Parameter cA:((a->Prop)->Prop).
% Trying to prove ((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))))))
% Found x1:(cA Xx)
% Found (fun (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1) as proof of (cA Xx)
% Found (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1) as proof of ((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->(cA Xx))
% Found (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1) as proof of ((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->(cA Xx)))
% Found (and_rect00 (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1)) as proof of (cA Xx)
% Found ((and_rect0 (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1)) as proof of (cA Xx)
% Found (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1)) as proof of (cA Xx)
% Found (fun (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1))) as proof of (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))
% Found eq_ref000:=(eq_ref00 (and (cA Xy))):(((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))))
% Found (eq_ref00 (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))
% Found ((eq_ref0 (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))
% Found (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))
% Found (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))
% Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))
% Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))
% Found x:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))
% Instantiate: x1:=cE:(a->Prop)
% Found x as proof of (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))
% Found eq_ref000:=(eq_ref00 x1):((x1 Xx0)->(x1 Xx0))
% Found (eq_ref00 x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))
% Found x:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))
% Found x as proof of (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))
% Found x00:((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))
% Instantiate: x3:=cE:(a->Prop)
% Found (fun (x00:((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))))=> x00) as proof of ((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))
% Found (fun (Xy:(a->Prop)) (x00:((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))))=> x00) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))
% Found (fun (Xy:(a->Prop)) (x00:((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))))=> x00) as proof of (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))
% Found eq_ref000:=(eq_ref00 (and (cA Xy))):(((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))))
% Found (eq_ref00 (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))
% Found ((eq_ref0 (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))
% Found (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))
% Found (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))
% Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))
% Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))
% Found conj200:=(conj20 (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))):((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))
% Found (conj20 (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) as proof of ((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))))
% Found ((fun (B:Prop)=> ((conj2 B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) as proof of ((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))))
% Found ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) as proof of ((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))))
% Found (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) as proof of ((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))))
% Found (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) as proof of ((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))))
% Found (and_rect00 (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))) as proof of ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))
% Found ((and_rect0 ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))) as proof of ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))
% Found (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))) as proof of ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))
% Found (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))) as proof of ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))
% Found ((conj10 (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))))) as proof of ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))
% Found (((conj1 (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))))) as proof of ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))
% Found ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))))) as proof of ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))
% Found ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))))) as proof of ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))
% Found (ex_intro000 ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy)))))) as proof of ((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))
% Found ((ex_intro00 cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy)))))) as proof of ((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))
% Found (((ex_intro0 (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy)))))) as proof of ((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))
% Found ((((ex_intro (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy)))))) as proof of ((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))
% Found (fun (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy))))))) as proof of ((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))
% Found (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy))))))) as proof of (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))))
% Found (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy))))))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))))
% Found ((conj00 (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1)))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy)))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))))))
% Found (((conj0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1)))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy)))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))))))
% Found ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1)))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy)))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))))))
% Found (fun (x:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1)))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy))))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))))))
% Found (fun (x:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1)))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy))))))))) as proof of ((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))))))
% Got proof (fun (x:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1)))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy)))))))))
% Time elapsed = 6.151064s
% node=643 cost=820.000000 depth=24
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (x:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X cE))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))=> x1)))) (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))))) cE) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0)))) P) x2) x0)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))) (fun (x2:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((cE Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X cE)))) B) x)) (forall (Xx0:a), ((cE Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((cE Xx0)->(Xy Xx0)))) (and (cA Xy)))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------