%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV050^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n106.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 13:33:40 EDT 2014

% Result   : Theorem 0.85s
% Output   : Proof 0.85s
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV050^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n106.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 07:44:31 CDT 2014
% % CPUTime  : 0.85 
% 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 0x12e4b90>, <kernel.Type object at 0x12e4368>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xr:(a->(a->Prop))), ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))) of role conjecture named cTHM599_pme
% Conjecture to prove = (forall (Xr:(a->(a->Prop))), ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xr:(a->(a->Prop))), ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))))']
% Parameter a:Type.
% Trying to prove (forall (Xr:(a->(a->Prop))), ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))))
% Found x00:((x Xa) Xb)
% Instantiate: x:=Xt:(a->(a->Prop))
% Found x00 as proof of ((Xt Xa) Xb)
% Found (fun (x00:((x Xa) Xb))=> x00) as proof of ((Xt Xa) Xb)
% Found (fun (Xb:a) (x00:((x Xa) Xb))=> x00) as proof of (((x Xa) Xb)->((Xt Xa) Xb))
% Found (fun (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> x00) as proof of (forall (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))
% Found (fun (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> x00) as proof of (forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))
% Found x0000:=(x000 x0):((Xt Xa) Xb)
% Found (x000 x0) as proof of ((Xt Xa) Xb)
% Found ((x00 Xt) x0) as proof of ((Xt Xa) Xb)
% Found ((x00 Xt) x0) as proof of ((Xt Xa) Xb)
% Found (fun (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of ((Xt Xa) Xb)
% Found (fun (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (((x Xa) Xb)->((Xt Xa) Xb))
% Found (fun (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (forall (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))
% Found (fun (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))
% Found (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb))))
% Found (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))
% Found x0:((Xr Xa) Xb)
% Instantiate: x:=Xr:(a->(a->Prop))
% Found (fun (x0:((Xr Xa) Xb))=> x0) as proof of ((x Xa) Xb)
% Found (fun (Xb:a) (x0:((Xr Xa) Xb))=> x0) as proof of (((Xr Xa) Xb)->((x Xa) Xb))
% Found (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb))=> x0) as proof of (forall (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))
% Found (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb))=> x0) as proof of (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))
% Found x20:=(x2 Xx):((Xt Xx) Xx)
% Found (x2 Xx) as proof of ((Xt Xx) Xx)
% Found (fun (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)) as proof of ((Xt Xx) Xx)
% Found (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)) as proof of ((forall (Xx0:a), ((Xt Xx0) Xx0))->((Xt Xx) Xx))
% Found (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)) as proof of ((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->((Xt Xx) Xx)))
% Found (and_rect00 (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))) as proof of ((Xt Xx) Xx)
% Found ((and_rect0 ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))) as proof of ((Xt Xx) Xx)
% Found (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))) as proof of ((Xt Xx) Xx)
% Found (fun (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))) as proof of ((Xt Xx) Xx)
% Found (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))) as proof of (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0)))->((Xt Xx) Xx))
% Found (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))) as proof of ((x Xx) Xx)
% Found (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))) as proof of (forall (Xx:a), ((x Xx) Xx))
% Found x1000:=(x100 x0):((Xt Xa) Xb)
% Found (x100 x0) as proof of ((Xt Xa) Xb)
% Found ((x10 Xb) x0) as proof of ((Xt Xa) Xb)
% Found (((x1 Xa) Xb) x0) as proof of ((Xt Xa) Xb)
% Found (fun (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)) as proof of ((Xt Xa) Xb)
% Found (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)) as proof of ((forall (Xx:a), ((Xt Xx) Xx))->((Xt Xa) Xb))
% Found (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)) as proof of ((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->((Xt Xa) Xb)))
% Found (and_rect00 (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))) as proof of ((Xt Xa) Xb)
% Found ((and_rect0 ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))) as proof of ((Xt Xa) Xb)
% Found (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))) as proof of ((Xt Xa) Xb)
% Found (fun (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of ((Xt Xa) Xb)
% Found (fun (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of (((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt Xa) Xb))
% Found (fun (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of ((x Xa) Xb)
% Found (fun (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of (((Xr Xa) Xb)->((x Xa) Xb))
% Found (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of (forall (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))
% Found (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0)))) as proof of (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))
% Found ((conj10 (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))
% Found (((conj1 (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))
% Found ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))
% Found ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx))))) as proof of ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))
% Found ((conj00 ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb))))))
% Found (((conj0 (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb))))))
% Found ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb))))))
% Found ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0))) as proof of ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb))))))
% Found (ex_intro000 ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((x Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:a), ((x Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:((x Xa) Xb))=> ((x00 Xt) x0)))) as proof of ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))
% Found ((ex_intro00 (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0)))) as proof of ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))
% Found (((ex_intro0 (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0)))) as proof of ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))
% Found ((((ex_intro (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0)))) as proof of ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))
% Found (fun (Xr:(a->(a->Prop)))=> ((((ex_intro (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0))))) as proof of ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))
% Found (fun (Xr:(a->(a->Prop)))=> ((((ex_intro (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0))))) as proof of (forall (Xr:(a->(a->Prop))), ((ex (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))))
% Got proof (fun (Xr:(a->(a->Prop)))=> ((((ex_intro (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0)))))
% Time elapsed = 0.533292s
% node=67 cost=876.000000 depth=28
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (Xr:(a->(a->Prop)))=> ((((ex_intro (a->(a->Prop))) (fun (Xs:(a->(a->Prop)))=> ((and ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:a), ((Xs Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))) (fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1))))) ((((conj ((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx)))) (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->(forall (Xa:a) (Xb:a), ((((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)->((Xt Xa) Xb)))))) ((((conj (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb)))) (forall (Xx:a), (((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xx) Xx))) (fun (Xa:a) (Xb:a) (x0:((Xr Xa) Xb)) (Xt:(a->(a->Prop))) (x00:((and (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))))=> (((fun (P:Type) (x1:((forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:a), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:a), ((Xt Xx) Xx))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:(forall (Xa0:a) (Xb0:a), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x2:(forall (Xx:a), ((Xt Xx) Xx)))=> (((x1 Xa) Xb) x0))))) (fun (Xx:a) (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))))=> (((fun (P:Type) (x1:((forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:a), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:a), ((Xt Xx0) Xx0))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:(forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x2:(forall (Xx0:a), ((Xt Xx0) Xx0)))=> (x2 Xx)))))) (fun (Xt:(a->(a->Prop))) (x0:((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))) (Xa:a) (Xb:a) (x00:(((fun (a0:a) (a1:a)=> (forall (Xt:(a->(a->Prop))), (((and (forall (Xa:a) (Xb:a), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:a), ((Xt Xx) Xx)))->((Xt a0) a1)))) Xa) Xb))=> ((x00 Xt) x0)))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------