%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV105^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n099.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 13:33:44 EDT 2014

% Result   : Timeout 300.09s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV105^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n099.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:05:41 CDT 2014
% % CPUTime  : 300.09 
% 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 0x1246680>, <kernel.Type object at 0x1246cf8>) of role type named a_type
% Using role type
% Declaring atype:Type
% FOF formula (<kernel.Constant object at 0x1518518>, <kernel.Constant object at 0x1246c20>) of role type named a
% Using role type
% Declaring a:atype
% FOF formula (<kernel.Constant object at 0x1246b90>, <kernel.Constant object at 0x1246c20>) of role type named b
% Using role type
% Declaring b:atype
% FOF formula (forall (Xr:(atype->(atype->Prop))) (T:(atype->(atype->Prop))), (((and ((and ((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))) (forall (S:(atype->(atype->Prop))), (((and ((and (forall (Xx:atype), ((S Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((S Xx) Xy))))->(forall (Xx:atype) (Xy:atype), (((T Xx) Xy)->((S Xx) Xy))))))->(((and (not (((eq atype) a) b))) ((T a) b))->((ex atype) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b))))))) of role conjecture named cTC_INTERP_OTHER_pme
% Conjecture to prove = (forall (Xr:(atype->(atype->Prop))) (T:(atype->(atype->Prop))), (((and ((and ((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))) (forall (S:(atype->(atype->Prop))), (((and ((and (forall (Xx:atype), ((S Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((S Xx) Xy))))->(forall (Xx:atype) (Xy:atype), (((T Xx) Xy)->((S Xx) Xy))))))->(((and (not (((eq atype) a) b))) ((T a) b))->((ex atype) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b))))))):Prop
% We need to prove ['(forall (Xr:(atype->(atype->Prop))) (T:(atype->(atype->Prop))), (((and ((and ((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))) (forall (S:(atype->(atype->Prop))), (((and ((and (forall (Xx:atype), ((S Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((S Xx) Xy))))->(forall (Xx:atype) (Xy:atype), (((T Xx) Xy)->((S Xx) Xy))))))->(((and (not (((eq atype) a) b))) ((T a) b))->((ex atype) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))))))']
% Parameter atype:Type.
% Parameter a:atype.
% Parameter b:atype.
% Trying to prove (forall (Xr:(atype->(atype->Prop))) (T:(atype->(atype->Prop))), (((and ((and ((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))) (forall (S:(atype->(atype->Prop))), (((and ((and (forall (Xx:atype), ((S Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((S Xx) Xy))))->(forall (Xx:atype) (Xy:atype), (((T Xx) Xy)->((S Xx) Xy))))))->(((and (not (((eq atype) a) b))) ((T a) b))->((ex atype) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))))))
% Found x2:((T a) b)
% Instantiate: x3:=a:atype
% Found x2 as proof of ((T x3) b)
% Found x3:((T a) b)
% Instantiate: x1:=a:atype
% Found x3 as proof of ((T x1) b)
% Found x2:((T a) b)
% Instantiate: x5:=a:atype
% Found x2 as proof of ((T x5) b)
% Found x4:((T a) b)
% Instantiate: x5:=a:atype
% Found x4 as proof of ((T x5) b)
% Found eq_ref000:=(eq_ref00 (T a)):(((T a) b)->((T a) b))
% Found (eq_ref00 (T a)) as proof of (((T a) b)->((T x1) b))
% Found ((eq_ref0 b) (T a)) as proof of (((T a) b)->((T x1) b))
% Found (((eq_ref atype) b) (T a)) as proof of (((T a) b)->((T x1) b))
% Found (((eq_ref atype) b) (T a)) as proof of (((T a) b)->((T x1) b))
% Found (fun (x2:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of (((T a) b)->((T x1) b))
% Found (fun (x2:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of ((not (((eq atype) a) b))->(((T a) b)->((T x1) b)))
% Found (and_rect00 (fun (x2:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T x1) b)
% Found ((and_rect0 ((T x1) b)) (fun (x2:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T x1) b)
% Found (((fun (P:Type) (x2:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x2) x0)) ((T x1) b)) (fun (x2:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T x1) b)
% Found (((fun (P:Type) (x2:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x2) x0)) ((T x1) b)) (fun (x2:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T x1) b)
% Found x2:((T a) b)
% Instantiate: x3:=a:atype
% Found x2 as proof of ((T x3) b)
% Found x5:((T a) b)
% Instantiate: x3:=a:atype
% Found x5 as proof of ((T x3) b)
% Found x3:((T a) b)
% Instantiate: x1:=a:atype
% Found x3 as proof of ((T x1) b)
% Found x5:((T a) b)
% Instantiate: x1:=a:atype
% Found x5 as proof of ((T x1) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) ((T x) b))))
% Found (eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found x2:((T a) b)
% Instantiate: x7:=a:atype
% Found x2 as proof of ((T x7) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((Xr a) x1)) ((T x1) b)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((Xr a) x1)) ((T x1) b)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((Xr a) x1)) ((T x1) b)))
% Found (fun (x1:atype)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((Xr a) x1)) ((T x1) b)))
% Found (fun (x1:atype)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((Xr a) x1)) ((T x1) b)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((Xr a) x1)) ((T x1) b)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((Xr a) x1)) ((T x1) b)))
% Found (fun (x1:atype)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((Xr a) x1)) ((T x1) b)))
% Found (fun (x1:atype)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found x50:=(x5 x7):((T x7) x7)
% Found (x5 x7) as proof of ((T x7) b)
% Found (x5 x7) as proof of ((T x7) b)
% Found (x5 x7) as proof of ((T x7) b)
% Found x2:((T a) b)
% Instantiate: x3:=a:atype
% Found x2 as proof of ((T x3) b)
% Found x2:((T a) b)
% Instantiate: x5:=a:atype
% Found x2 as proof of ((T x5) b)
% Found x5:((T a) b)
% Instantiate: x3:=a:atype
% Found (fun (x5:((T a) b))=> x5) as proof of ((T x3) b)
% Found (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5) as proof of (((T a) b)->((T x3) b))
% Found (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5) as proof of ((not (((eq atype) a) b))->(((T a) b)->((T x3) b)))
% Found (and_rect10 (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T x3) b)
% Found ((and_rect1 ((T x3) b)) (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T x3) b)
% Found (((fun (P:Type) (x4:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x4) x0)) ((T x3) b)) (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T x3) b)
% Found (((fun (P:Type) (x4:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x4) x0)) ((T x3) b)) (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T x3) b)
% Found x6:((T a) b)
% Instantiate: x7:=a:atype
% Found x6 as proof of ((T x7) b)
% Found x4:((T a) b)
% Instantiate: x7:=a:atype
% Found x4 as proof of ((T x7) b)
% Found x60:=(x6 x1):((T x1) x1)
% Found (x6 x1) as proof of ((T x1) b)
% Found (x6 x1) as proof of ((T x1) b)
% Found (x6 x1) as proof of ((T x1) b)
% Found x60:=(x6 x3):((T x3) x3)
% Found (x6 x3) as proof of ((T x3) b)
% Found (x6 x3) as proof of ((T x3) b)
% Found (x6 x3) as proof of ((T x3) b)
% Found x60:=(x6 x5):((T x5) x5)
% Found (x6 x5) as proof of ((T x5) b)
% Found (x6 x5) as proof of ((T x5) b)
% Found (x6 x5) as proof of ((T x5) b)
% Found eq_ref00:=(eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b))))
% Found (eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b))))
% Found (eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found x3:((T a) b)
% Instantiate: x1:=a:atype
% Found x3 as proof of ((T x1) b)
% Found eq_ref000:=(eq_ref00 (T a)):(((T a) b)->((T a) b))
% Found (eq_ref00 (T a)) as proof of (((T a) b)->((T x1) b))
% Found ((eq_ref0 b) (T a)) as proof of (((T a) b)->((T x1) b))
% Found (((eq_ref atype) b) (T a)) as proof of (((T a) b)->((T x1) b))
% Found (((eq_ref atype) b) (T a)) as proof of (((T a) b)->((T x1) b))
% Found (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of (((T a) b)->((T x1) b))
% Found (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of ((not (((eq atype) a) b))->(((T a) b)->((T x1) b)))
% Found (and_rect10 (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T x1) b)
% Found ((and_rect1 ((T x1) b)) (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T x1) b)
% Found (((fun (P:Type) (x4:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x4) x0)) ((T x1) b)) (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T x1) b)
% Found (((fun (P:Type) (x4:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x4) x0)) ((T x1) b)) (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T x1) b)
% Found x4:((T a) b)
% Instantiate: x5:=a:atype
% Found x4 as proof of ((T x5) b)
% Found x7:((T a) b)
% Instantiate: x5:=a:atype
% Found x7 as proof of ((T x5) b)
% Found x2:((T a) b)
% Instantiate: x5:=b:atype
% Found x2 as proof of ((T a) x5)
% Found x4:((T a) b)
% Instantiate: x5:=b:atype
% Found x4 as proof of ((T a) x5)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found (fun (x3:atype)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found (fun (x3:atype)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found (fun (x3:atype)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found (fun (x3:atype)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found x5:((T a) b)
% Instantiate: x1:=a:atype
% Found x5 as proof of ((T x1) b)
% Found x7:((T a) b)
% Instantiate: x1:=a:atype
% Found x7 as proof of ((T x1) b)
% Found x5:((T a) b)
% Instantiate: x3:=a:atype
% Found x5 as proof of ((T x3) b)
% Found x7:((T a) b)
% Instantiate: x3:=a:atype
% Found x7 as proof of ((T x3) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found (fun (x3:atype)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found (fun (x3:atype)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found (fun (x3:atype)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((Xr a) x3)) ((T x3) b)))
% Found (fun (x3:atype)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found x2:((T a) b)
% Instantiate: x3:=b:atype
% Found x2 as proof of ((T a) x3)
% Found x5:((T a) b)
% Instantiate: x3:=b:atype
% Found x5 as proof of ((T a) x3)
% Found x2:((T a) b)
% Instantiate: x3:=b:atype
% Found x2 as proof of ((T a) x3)
% Found x5:((T a) b)
% Instantiate: x3:=b:atype
% Found x5 as proof of ((T a) x3)
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found x2:((T a) b)
% Instantiate: x9:=a:atype
% Found x2 as proof of ((T x9) b)
% Found eq_ref00:=(eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b))))
% Found (eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found x60:=(x6 x1):((T x1) x1)
% Found (x6 x1) as proof of ((T x1) b)
% Found (x6 x1) as proof of ((T x1) b)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x1)) as proof of ((T x1) b)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x1)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T x1) b))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x1)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T x1) b)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x1))) as proof of ((T x1) b)
% Found ((and_rect2 ((T x1) b)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x1))) as proof of ((T x1) b)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T x1) b)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x1))) as proof of ((T x1) b)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T x1) b)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x1))) as proof of ((T x1) b)
% Found x60:=(x6 x3):((T x3) x3)
% Found (x6 x3) as proof of ((T x3) b)
% Found (x6 x3) as proof of ((T x3) b)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x3)) as proof of ((T x3) b)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x3)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T x3) b))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x3)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T x3) b)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x3))) as proof of ((T x3) b)
% Found ((and_rect2 ((T x3) b)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x3))) as proof of ((T x3) b)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T x3) b)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x3))) as proof of ((T x3) b)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T x3) b)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x3))) as proof of ((T x3) b)
% Found x60:=(x6 x5):((T x5) x5)
% Found (x6 x5) as proof of ((T x5) b)
% Found (x6 x5) as proof of ((T x5) b)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x5)) as proof of ((T x5) b)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x5)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T x5) b))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x5)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T x5) b)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x5))) as proof of ((T x5) b)
% Found ((and_rect2 ((T x5) b)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x5))) as proof of ((T x5) b)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x3)) ((T x5) b)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x5))) as proof of ((T x5) b)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x3)) ((T x5) b)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 x5))) as proof of ((T x5) b)
% Found x2:((T a) b)
% Instantiate: x3:=a:atype
% Found x2 as proof of ((T x3) b)
% Found x2:((T a) b)
% Instantiate: x5:=a:atype
% Found x2 as proof of ((T x5) b)
% Found x2:((T a) b)
% Instantiate: x7:=a:atype
% Found x2 as proof of ((T x7) b)
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found eta_expansion000:=(eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) ((T x) b))))
% Found (eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) ((T x) b))))
% Found (eta_expansion_dep00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eta_expansion_dep0 (fun (x6:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x6:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x6:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x6:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found x4:((T a) b)
% Instantiate: x9:=a:atype
% Found x4 as proof of ((T x9) b)
% Found x3:((T a) b)
% Instantiate: x1:=a:atype
% Found x3 as proof of ((T x1) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found x4:((T a) b)
% Instantiate: x5:=a:atype
% Found x4 as proof of ((T x5) b)
% Found x4:((T a) b)
% Instantiate: x7:=a:atype
% Found x4 as proof of ((T x7) b)
% Found x2:((T a) b)
% Instantiate: x7:=b:atype
% Found x2 as proof of ((T a) x7)
% Found x8:((T a) b)
% Instantiate: x9:=a:atype
% Found x8 as proof of ((T x9) b)
% Found x6:((T a) b)
% Instantiate: x9:=a:atype
% Found x6 as proof of ((T x9) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found x50:=(x5 a):((T a) a)
% Found (x5 a) as proof of ((T a) x7)
% Found (x5 a) as proof of ((T a) x7)
% Found (x5 a) as proof of ((T a) x7)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((Xr a) x5)) ((T x5) b)))
% Found (fun (x5:atype)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found x5:((T a) b)
% Instantiate: x1:=a:atype
% Found x5 as proof of ((T x1) b)
% Found x5:((T a) b)
% Instantiate: x3:=a:atype
% Found x5 as proof of ((T x3) b)
% Found x2:((T a) b)
% Instantiate: x3:=b:atype
% Found x2 as proof of ((T a) x3)
% Found x2:((T a) b)
% Instantiate: x5:=b:atype
% Found x2 as proof of ((T a) x5)
% Found x6:((T a) b)
% Instantiate: x7:=a:atype
% Found x6 as proof of ((T x7) b)
% Found x9:((T a) b)
% Instantiate: x7:=a:atype
% Found x9 as proof of ((T x7) b)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x3)
% Found (x6 a) as proof of ((T a) x3)
% Found x4:((T a) b)
% Instantiate: x7:=b:atype
% Found x4 as proof of ((T a) x7)
% Found x6:((T a) b)
% Instantiate: x7:=b:atype
% Found x6 as proof of ((T a) x7)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x3)
% Found (x6 a) as proof of ((T a) x3)
% Found (x6 a) as proof of ((T a) x3)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x5)
% Found (x6 a) as proof of ((T a) x5)
% Found (x6 a) as proof of ((T a) x5)
% Found x5:((T a) b)
% Instantiate: x3:=b:atype
% Found (fun (x5:((T a) b))=> x5) as proof of ((T a) x3)
% Found (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5) as proof of (((T a) b)->((T a) x3))
% Found (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5) as proof of ((not (((eq atype) a) b))->(((T a) b)->((T a) x3)))
% Found (and_rect10 (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T a) x3)
% Found ((and_rect1 ((T a) x3)) (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T a) x3)
% Found (((fun (P:Type) (x4:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x4) x0)) ((T a) x3)) (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T a) x3)
% Found (((fun (P:Type) (x4:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x4) x0)) ((T a) x3)) (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x2:((T a) b)
% Instantiate: x3:=b:atype
% Found x2 as proof of ((T a) x3)
% Found x2:((T a) b)
% Instantiate: x3:=b:atype
% Found x2 as proof of ((T a) x3)
% Found x2:((T a) b)
% Instantiate: x5:=b:atype
% Found x2 as proof of ((T a) x5)
% Found x7:((T a) b)
% Instantiate: x1:=a:atype
% Found x7 as proof of ((T x1) b)
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found x9:((T a) b)
% Instantiate: x1:=a:atype
% Found x9 as proof of ((T x1) b)
% Found eq_ref00:=(eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b))))
% Found (eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found x7:((T a) b)
% Instantiate: x3:=a:atype
% Found x7 as proof of ((T x3) b)
% Found x9:((T a) b)
% Instantiate: x3:=a:atype
% Found x9 as proof of ((T x3) b)
% Found x9:((T a) b)
% Instantiate: x5:=a:atype
% Found x9 as proof of ((T x5) b)
% Found x7:((T a) b)
% Instantiate: x5:=a:atype
% Found x7 as proof of ((T x5) b)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x4:((T a) b)
% Instantiate: x5:=b:atype
% Found x4 as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x3)
% Found (x6 a) as proof of ((T a) x3)
% Found (x6 a) as proof of ((T a) x3)
% Found x7:((T a) b)
% Instantiate: x5:=b:atype
% Found x7 as proof of ((T a) x5)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x3)
% Found (x6 a) as proof of ((T a) x3)
% Found (x6 a) as proof of ((T a) x3)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x5)
% Found (x6 a) as proof of ((T a) x5)
% Found (x6 a) as proof of ((T a) x5)
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found (fun (x5:((T a) b))=> x5) as proof of ((T a) x1)
% Found (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5) as proof of (((T a) b)->((T a) x1))
% Found (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5) as proof of ((not (((eq atype) a) b))->(((T a) b)->((T a) x1)))
% Found (and_rect10 (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T a) x1)
% Found ((and_rect1 ((T a) x1)) (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T a) x1)
% Found (((fun (P:Type) (x4:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x4) x0)) ((T a) x1)) (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T a) x1)
% Found (((fun (P:Type) (x4:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x4) x0)) ((T a) x1)) (fun (x4:(not (((eq atype) a) b))) (x5:((T a) b))=> x5)) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref00:=(eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b))))
% Found (eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found x4:((T a) b)
% Instantiate: x5:=b:atype
% Found x4 as proof of ((T a) x5)
% Found x7:((T a) b)
% Instantiate: x5:=b:atype
% Found x7 as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found x5:((T a) b)
% Instantiate: x3:=b:atype
% Found x5 as proof of ((T a) x3)
% Found x7:((T a) b)
% Instantiate: x1:=b:atype
% Found x7 as proof of ((T a) x1)
% Found x7:((T a) b)
% Instantiate: x3:=b:atype
% Found x7 as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (T a)):(((T a) b)->((T a) b))
% Found (eq_ref00 (T a)) as proof of (((T a) b)->((T a) x1))
% Found ((eq_ref0 b) (T a)) as proof of (((T a) b)->((T a) x1))
% Found (((eq_ref atype) b) (T a)) as proof of (((T a) b)->((T a) x1))
% Found (((eq_ref atype) b) (T a)) as proof of (((T a) b)->((T a) x1))
% Found (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of (((T a) b)->((T a) x1))
% Found (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of ((not (((eq atype) a) b))->(((T a) b)->((T a) x1)))
% Found (and_rect10 (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x1)
% Found ((and_rect1 ((T a) x1)) (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x4:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x4) x0)) ((T a) x1)) (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x4:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x4) x0)) ((T a) x1)) (fun (x4:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found x7:((T a) b)
% Found x7 as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x7:((T a) b)
% Instantiate: x1:=b:atype
% Found x7 as proof of ((T a) x1)
% Found x5:((T a) b)
% Instantiate: x3:=b:atype
% Found x5 as proof of ((T a) x3)
% Found x7:((T a) b)
% Instantiate: x3:=b:atype
% Found x7 as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref00:=(eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b))))
% Found (eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) ((T x) b))))
% Found (eta_expansion_dep00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eta_expansion_dep0 (fun (x8:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x8:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x8:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x8:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x7:((T a) b)
% Found x7 as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x7:((T a) b)
% Found x7 as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x5:((Xr Xx) Xy)
% Instantiate: x3:=Xy:atype
% Found (fun (x5:((Xr Xx) Xy))=> x5) as proof of ((Xr Xx) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x3))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x3))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x3))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x3))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found x7:((T a) b)
% Instantiate: x1:=b:atype
% Found x7 as proof of ((T a) x1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))):(((eq Prop) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy)))))))
% Found (eq_ref0 (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) as proof of (((eq Prop) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) as proof of (((eq Prop) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) as proof of (((eq Prop) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) as proof of (((eq Prop) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) b0)
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found x5:((T a) b)
% Instantiate: x3:=b:atype
% Found x5 as proof of ((T a) x3)
% Found x7:((T a) b)
% Instantiate: x1:=b:atype
% Found x7 as proof of ((T a) x1)
% Found x7:((T a) b)
% Instantiate: x3:=b:atype
% Found x7 as proof of ((T a) x3)
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x2:((T a) b)
% Instantiate: x9:=b:atype
% Found x2 as proof of ((T a) x9)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x1))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x1))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x1))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x1))
% Found x5:((Xr Xx) Xy)
% Instantiate: x1:=Xy:atype
% Found (fun (x5:((Xr Xx) Xy))=> x5) as proof of ((Xr Xx) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x1)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found ((and_rect2 ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x1)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found ((and_rect2 ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x3)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x3)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x3))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x3)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found ((and_rect2 ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x70:=(x7 a):((T a) a)
% Found (x7 a) as proof of ((T a) x9)
% Found (x7 a) as proof of ((T a) x9)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr a) x7)) ((T x7) b)))
% Found (fun (x7:atype)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x5:((Xr Xx) Xy)
% Instantiate: x1:=Xy:atype
% Found (fun (x5:((Xr Xx) Xy))=> x5) as proof of ((Xr Xx) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x1))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x1))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x1))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x1))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x2:((T a) b)
% Instantiate: x3:=b:atype
% Found x2 as proof of ((T a) x3)
% Found x2:((T a) b)
% Instantiate: x5:=b:atype
% Found x2 as proof of ((T a) x5)
% Found x2:((T a) b)
% Instantiate: x7:=b:atype
% Found x2 as proof of ((T a) x7)
% Found eq_ref000:=(eq_ref00 (T a)):(((T a) b)->((T a) b))
% Found (eq_ref00 (T a)) as proof of (((T a) b)->((T a) x5))
% Found ((eq_ref0 b) (T a)) as proof of (((T a) b)->((T a) x5))
% Found (((eq_ref atype) b) (T a)) as proof of (((T a) b)->((T a) x5))
% Found (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of (((T a) b)->((T a) x5))
% Found (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of ((not (((eq atype) a) b))->(((T a) b)->((T a) x5)))
% Found (and_rect20 (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x5)
% Found ((and_rect2 ((T a) x5)) (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x5)
% Found (((fun (P:Type) (x6:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x6) x0)) ((T a) x5)) (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x5)
% Found (((fun (P:Type) (x6:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x6) x0)) ((T a) x5)) (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x1)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found ((and_rect2 ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x4:((T a) b)
% Instantiate: x9:=b:atype
% Found x4 as proof of ((T a) x9)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x1)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found ((and_rect2 ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x1)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found ((and_rect2 ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x3)
% Found (x6 a) as proof of ((T a) x3)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x3)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x3))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x3)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found ((and_rect2 ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x3)
% Found (x6 a) as proof of ((T a) x3)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x3)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x3))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x3)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found ((and_rect2 ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x5)
% Found (x6 a) as proof of ((T a) x5)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x5)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x5))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x5)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x5)
% Found ((and_rect2 ((T a) x5)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x5)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x3)) ((T a) x5)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x5)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x3)) ((T a) x5)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x3)
% Found (x8 a) as proof of ((T a) x3)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x5)
% Found (x8 a) as proof of ((T a) x5)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x7)
% Found (x8 a) as proof of ((T a) x7)
% Found x70:=(x7 a):((T a) a)
% Found (x7 a) as proof of ((T a) x9)
% Found (x7 a) as proof of ((T a) x9)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x2:((T a) b)
% Instantiate: x3:=b:atype
% Found x2 as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x2:((T a) b)
% Instantiate: x3:=b:atype
% Found x2 as proof of ((T a) x3)
% Found x2:((T a) b)
% Instantiate: x5:=b:atype
% Found x2 as proof of ((T a) x5)
% Found x2:((T a) b)
% Instantiate: x3:=b:atype
% Found x2 as proof of ((T a) x3)
% Found x2:((T a) b)
% Instantiate: x5:=b:atype
% Found x2 as proof of ((T a) x5)
% Found x2:((T a) b)
% Instantiate: x7:=b:atype
% Found x2 as proof of ((T a) x7)
% Found x5:((Xr Xx) Xy)
% Instantiate: x3:=Xy:atype
% Found x5 as proof of ((Xr Xx) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (T a)):(((T a) b)->((T a) b))
% Found (eq_ref00 (T a)) as proof of (((T a) b)->((T a) x1))
% Found ((eq_ref0 b) (T a)) as proof of (((T a) b)->((T a) x1))
% Found (((eq_ref atype) b) (T a)) as proof of (((T a) b)->((T a) x1))
% Found (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of (((T a) b)->((T a) x1))
% Found (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of ((not (((eq atype) a) b))->(((T a) b)->((T a) x1)))
% Found (and_rect20 (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x1)
% Found ((and_rect2 ((T a) x1)) (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x6) x0)) ((T a) x1)) (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x6) x0)) ((T a) x1)) (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (T a)):(((T a) b)->((T a) b))
% Found (eq_ref00 (T a)) as proof of (((T a) b)->((T a) x3))
% Found ((eq_ref0 b) (T a)) as proof of (((T a) b)->((T a) x3))
% Found (((eq_ref atype) b) (T a)) as proof of (((T a) b)->((T a) x3))
% Found (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of (((T a) b)->((T a) x3))
% Found (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a))) as proof of ((not (((eq atype) a) b))->(((T a) b)->((T a) x3)))
% Found (and_rect20 (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x3)
% Found ((and_rect2 ((T a) x3)) (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x3)
% Found (((fun (P:Type) (x6:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x6) x0)) ((T a) x3)) (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x3)
% Found (((fun (P:Type) (x6:((not (((eq atype) a) b))->(((T a) b)->P)))=> (((((and_rect (not (((eq atype) a) b))) ((T a) b)) P) x6) x0)) ((T a) x3)) (fun (x6:(not (((eq atype) a) b)))=> (((eq_ref atype) b) (T a)))) as proof of ((T a) x3)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x3)
% Found (x8 a) as proof of ((T a) x3)
% Found x20:=(x2 (fun (x7:atype) (x60:atype)=> ((ex atype) (fun (Xc:atype)=> ((and ((Xr x7) Xc)) ((T Xc) x60)))))):(((and ((and (forall (Xx:atype), ((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xx)))))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))) ((ex atype) (fun (Xc:atype)=> ((and ((Xr Xy) Xc)) ((T Xc) Xz)))))->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xz)))))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy)))))))->(forall (Xx:atype) (Xy:atype), (((T Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy)))))))
% Instantiate: b0:=(((and ((and (forall (Xx:atype), ((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xx)))))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))) ((ex atype) (fun (Xc:atype)=> ((and ((Xr Xy) Xc)) ((T Xc) Xz)))))->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xz)))))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy)))))))->(forall (Xx:atype) (Xy:atype), (((T Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))):Prop
% Found x20 as proof of b0
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x3)
% Found (x8 a) as proof of ((T a) x3)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x3)
% Found (x8 a) as proof of ((T a) x3)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x5)
% Found (x8 a) as proof of ((T a) x5)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x5)
% Found (x8 a) as proof of ((T a) x5)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x7)
% Found (x8 a) as proof of ((T a) x7)
% Found x4:((T a) b)
% Instantiate: x5:=b:atype
% Found x4 as proof of ((T a) x5)
% Found x4:((T a) b)
% Instantiate: x7:=b:atype
% Found x4 as proof of ((T a) x7)
% Found eta_expansion000:=(eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) ((T x) b))))
% Found (eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x1)
% Found (x8 a) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x8:((T a) b)
% Instantiate: x9:=b:atype
% Found x8 as proof of ((T a) x9)
% Found x6:((T a) b)
% Instantiate: x9:=b:atype
% Found x6 as proof of ((T a) x9)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x5)
% Found (x8 a) as proof of ((T a) x5)
% Found x70:=(x7 a):((T a) a)
% Found (x7 a) as proof of ((T a) x1)
% Found (x7 a) as proof of ((T a) x1)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x7)
% Found (x8 a) as proof of ((T a) x7)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found x5:((Xr Xx) Xy)
% Instantiate: x1:=Xy:atype
% Found x5 as proof of ((Xr Xx) x1)
% Found x50:=(x5 a):((T a) a)
% Found (x5 a) as proof of ((T a) x9)
% Found (x5 a) as proof of ((T a) x9)
% Found x70:=(x7 a):((T a) a)
% Found (x7 a) as proof of ((T a) x9)
% Found (x7 a) as proof of ((T a) x9)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x4:((T a) b)
% Instantiate: x5:=b:atype
% Found x4 as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x4:((T a) b)
% Instantiate: x5:=b:atype
% Found x4 as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x4:((T a) b)
% Instantiate: x7:=b:atype
% Found x4 as proof of ((T a) x7)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x1)
% Found (x8 a) as proof of ((T a) x1)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x1)
% Found (x8 a) as proof of ((T a) x1)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x1)
% Found (x8 a) as proof of ((T a) x1)
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x5:((T a) b)
% Instantiate: x3:=b:atype
% Found x5 as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x50:=(x5 a):((T a) a)
% Found (x5 a) as proof of ((T a) x7)
% Found (x5 a) as proof of ((T a) x7)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x5)
% Found (x8 a) as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x5)
% Found (x8 a) as proof of ((T a) x5)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x7)
% Found (x8 a) as proof of ((T a) x7)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x6:((T a) b)
% Instantiate: x7:=b:atype
% Found x6 as proof of ((T a) x7)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x9:((T a) b)
% Instantiate: x7:=b:atype
% Found x9 as proof of ((T a) x7)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x3:((T a) b)
% Instantiate: x1:=b:atype
% Found x3 as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) ((T x) b))))
% Found (eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x1)
% Found (x8 a) as proof of ((T a) x1)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x3)
% Found (x8 a) as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x1)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found ((and_rect2 ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (fun (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)))) as proof of ((T a) x1)
% Found (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)))) as proof of ((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->((T a) x1))
% Found (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)))) as proof of (((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))->((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->((T a) x1)))
% Found (and_rect10 (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x1)
% Found ((and_rect1 ((T a) x1)) (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x4:(((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))->((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))) P) x4) x2)) ((T a) x1)) (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x4:(((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))->((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))) P) x4) x2)) ((T a) x1)) (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x1)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x1)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found ((and_rect2 ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x1)
% Found (fun (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)))) as proof of ((T a) x1)
% Found (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)))) as proof of ((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->((T a) x1))
% Found (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)))) as proof of (((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))->((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->((T a) x1)))
% Found (and_rect10 (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x1)
% Found ((and_rect1 ((T a) x1)) (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x4:(((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))->((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))) P) x4) x2)) ((T a) x1)) (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x1)
% Found (((fun (P:Type) (x4:(((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))->((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))) P) x4) x2)) ((T a) x1)) (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x1)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x3)
% Found (fun (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((T a) x3)
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x3))
% Found (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)) as proof of ((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->((T a) x3)))
% Found (and_rect20 (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found ((and_rect2 ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))) as proof of ((T a) x3)
% Found (fun (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)))) as proof of ((T a) x3)
% Found (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)))) as proof of ((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->((T a) x3))
% Found (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a)))) as proof of (((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))->((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->((T a) x3)))
% Found (and_rect10 (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x3)
% Found ((and_rect1 ((T a) x3)) (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x3)
% Found (((fun (P:Type) (x4:(((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))->((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))) P) x4) x1)) ((T a) x3)) (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x3)
% Found (((fun (P:Type) (x4:(((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))->((forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))->P)))=> (((((and_rect ((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy)))) P) x4) x1)) ((T a) x3)) (fun (x4:((and (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (x5:(forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((T Xx) Xy))))=> (((fun (P:Type) (x6:((forall (Xx:atype), ((T Xx) Xx))->((forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))->P)))=> (((((and_rect (forall (Xx:atype), ((T Xx) Xx))) (forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz)))) P) x6) x4)) ((T a) x3)) (fun (x6:(forall (Xx:atype), ((T Xx) Xx))) (x7:(forall (Xx:atype) (Xy:atype) (Xz:atype), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))=> (x6 a))))) as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x4:((Xr Xx) Xy)
% Instantiate: x5:=Xy:atype
% Found x4 as proof of ((Xr Xx) x5)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x7)
% Found (x8 a) as proof of ((T a) x7)
% Found x50:=(x5 a):((T a) a)
% Found (x5 a) as proof of ((T a) x7)
% Found (x5 a) as proof of ((T a) x7)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x1)
% Found (x8 a) as proof of ((T a) x1)
% Found x70:=(x7 a):((T a) a)
% Found (x7 a) as proof of ((T a) x1)
% Found (x7 a) as proof of ((T a) x1)
% Found x70:=(x7 a):((T a) a)
% Found (x7 a) as proof of ((T a) x3)
% Found (x7 a) as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found x5:((T a) b)
% Instantiate: x3:=b:atype
% Found x5 as proof of ((T a) x3)
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x5:((T a) b)
% Instantiate: x3:=b:atype
% Found x5 as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x9:((T a) b)
% Instantiate: x7:=b:atype
% Found x9 as proof of ((T a) x7)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x6:((T a) b)
% Instantiate: x7:=b:atype
% Found x6 as proof of ((T a) x7)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x3)
% Found (x6 a) as proof of ((T a) x3)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x1)
% Found (x8 a) as proof of ((T a) x1)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x5)
% Found (x6 a) as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x3)
% Found (x8 a) as proof of ((T a) x3)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x1)
% Found (x8 a) as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x3)
% Found (x8 a) as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x9:((T a) b)
% Instantiate: x1:=b:atype
% Found x9 as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x7:((T a) b)
% Instantiate: x1:=b:atype
% Found x7 as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x7:((T a) b)
% Instantiate: x3:=b:atype
% Found x7 as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x9:((T a) b)
% Instantiate: x3:=b:atype
% Found x9 as proof of ((T a) x3)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x9:((T a) b)
% Instantiate: x5:=b:atype
% Found x9 as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x7:((T a) b)
% Instantiate: x5:=b:atype
% Found x7 as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref00:=(eq_ref0 (f x9)):(((eq Prop) (f x9)) (f x9))
% Found (eq_ref0 (f x9)) as proof of (((eq Prop) (f x9)) ((and ((Xr a) x9)) ((T x9) b)))
% Found ((eq_ref Prop) (f x9)) as proof of (((eq Prop) (f x9)) ((and ((Xr a) x9)) ((T x9) b)))
% Found ((eq_ref Prop) (f x9)) as proof of (((eq Prop) (f x9)) ((and ((Xr a) x9)) ((T x9) b)))
% Found (fun (x9:atype)=> ((eq_ref Prop) (f x9))) as proof of (((eq Prop) (f x9)) ((and ((Xr a) x9)) ((T x9) b)))
% Found (fun (x9:atype)=> ((eq_ref Prop) (f x9))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref00:=(eq_ref0 (f x9)):(((eq Prop) (f x9)) (f x9))
% Found (eq_ref0 (f x9)) as proof of (((eq Prop) (f x9)) ((and ((Xr a) x9)) ((T x9) b)))
% Found ((eq_ref Prop) (f x9)) as proof of (((eq Prop) (f x9)) ((and ((Xr a) x9)) ((T x9) b)))
% Found ((eq_ref Prop) (f x9)) as proof of (((eq Prop) (f x9)) ((and ((Xr a) x9)) ((T x9) b)))
% Found (fun (x9:atype)=> ((eq_ref Prop) (f x9))) as proof of (((eq Prop) (f x9)) ((and ((Xr a) x9)) ((T x9) b)))
% Found (fun (x9:atype)=> ((eq_ref Prop) (f x9))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) ((T x) b))))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x7)
% Found (x8 a) as proof of ((T a) x7)
% Found x9:((T a) b)
% Found x9 as proof of ((T a) x7)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) ((T x) b))))
% Found (eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) ((T x) b))))
% Found (eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) ((T Xc) b)))) b0)
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found x5:((T a) b)
% Instantiate: x1:=b:atype
% Found x5 as proof of ((T a) x1)
% Found x5:((T a) b)
% Instantiate: x3:=b:atype
% Found x5 as proof of ((T a) x3)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x1)
% Found (x6 a) as proof of ((T a) x1)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x1)
% Found (x8 a) as proof of ((T a) x1)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x3)
% Found (x8 a) as proof of ((T a) x3)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x3)
% Found (x6 a) as proof of ((T a) x3)
% Found x80:=(x8 a):((T a) a)
% Found (x8 a) as proof of ((T a) x5)
% Found (x8 a) as proof of ((T a) x5)
% Found x60:=(x6 a):((T a) a)
% Found (x6 a) as proof of ((T a) x5)
% Found (x6 a) as proof of ((T a) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (((eq_ref atype) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found (fun (Xx:atype) (Xy:atype)=> (((eq_ref atype) Xy) (Xr Xx))) as proof of (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((Xr Xx) Xy)))
% Found eq_ref00:=(eq_ref0 (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))):(((eq Prop) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy)))))))
% Found (eq_ref0 (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) as proof of (((eq Prop) (forall (Xx:atype) (Xy:atype), (((Xr Xx) Xy)->((ex atype) (fun (Xc:atype)=> ((and ((Xr Xx) Xc)) ((T Xc) Xy))))))) b0)
% Found ((eq_ref Prop) (forall (X
% EOF
%------------------------------------------------------------------------------