%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV133^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n187.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 13:33:47 EDT 2014

% Result   : Timeout 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV133^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n187.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 08:11:56 CDT 2014
% % CPUTime  : 300.02 
% 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 0x27cbe60>, <kernel.Type object at 0x276bab8>) of role type named a_type
% Using role type
% Declaring atype:Type
% FOF formula (<kernel.Constant object at 0x27cb830>, <kernel.Constant object at 0x276b6c8>) of role type named a
% Using role type
% Declaring a:atype
% FOF formula (<kernel.Constant object at 0x27cbe60>, <kernel.Constant object at 0x276bf38>) of role type named b
% Using role type
% Declaring b:atype
% FOF formula (<kernel.Constant object at 0x27cb830>, <kernel.DependentProduct object at 0x276bb00>) of role type named cSTAR
% Using role type
% Declaring cSTAR:((atype->(atype->Prop))->(atype->(atype->Prop)))
% FOF formula (forall (Xr:(atype->(atype->Prop))), (((and ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b))->((ex atype) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))))) of role conjecture named cTC_INTERP_THIRD_pme
% Conjecture to prove = (forall (Xr:(atype->(atype->Prop))), (((and ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b))->((ex atype) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))))):Prop
% We need to prove ['(forall (Xr:(atype->(atype->Prop))), (((and ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b))->((ex atype) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))))']
% Parameter atype:Type.
% Parameter a:atype.
% Parameter b:atype.
% Parameter cSTAR:((atype->(atype->Prop))->(atype->(atype->Prop))).
% Trying to prove (forall (Xr:(atype->(atype->Prop))), (((and ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b))->((ex atype) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))))
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) b)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) b)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) b)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found (fun (x2:(((cSTAR Xr) a) b))=> x2) as proof of (((cSTAR Xr) x0) b)
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2) as proof of ((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) b))
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2) as proof of (((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) b)))
% Found (and_rect00 (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) b)
% Found ((and_rect0 (((cSTAR Xr) x0) b)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) b)
% Found (((fun (P:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P) x1) x)) (((cSTAR Xr) x0) b)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) b)
% Found (((fun (P:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P) x1) x)) (((cSTAR Xr) x0) b)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x6:=a:atype
% Found x1 as proof of (((cSTAR Xr) x6) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found (eta_expansion_dep00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found ((eta_expansion_dep0 (fun (x1:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x1:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x1:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x1:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr a) x0)) (((cSTAR Xr) x0) b)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr a) x0)) (((cSTAR Xr) x0) b)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr a) x0)) (((cSTAR Xr) x0) b)))
% Found (fun (x0:atype)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((Xr a) x0)) (((cSTAR Xr) x0) b)))
% Found (fun (x0:atype)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr a) x0)) (((cSTAR Xr) x0) b)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr a) x0)) (((cSTAR Xr) x0) b)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr a) x0)) (((cSTAR Xr) x0) b)))
% Found (fun (x0:atype)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((Xr a) x0)) (((cSTAR Xr) x0) b)))
% Found (fun (x0:atype)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found (eta_expansion_dep00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found ((eta_expansion_dep0 (fun (x3:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x3:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x3:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x3:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr a) x2)) (((cSTAR Xr) x2) b)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr a) x2)) (((cSTAR Xr) x2) b)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr a) x2)) (((cSTAR Xr) x2) b)))
% Found (fun (x2:atype)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((Xr a) x2)) (((cSTAR Xr) x2) b)))
% Found (fun (x2:atype)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr a) x2)) (((cSTAR Xr) x2) b)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr a) x2)) (((cSTAR Xr) x2) b)))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr a) x2)) (((cSTAR Xr) x2) b)))
% Found (fun (x2:atype)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((Xr a) x2)) (((cSTAR Xr) x2) b)))
% Found (fun (x2:atype)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found (eta_expansion00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion atype) Prop) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((Xr a) x4)) (((cSTAR Xr) x4) b)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((Xr a) x4)) (((cSTAR Xr) x4) b)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((Xr a) x4)) (((cSTAR Xr) x4) b)))
% Found (fun (x4:atype)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((Xr a) x4)) (((cSTAR Xr) x4) b)))
% Found (fun (x4:atype)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((Xr a) x4)) (((cSTAR Xr) x4) b)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((Xr a) x4)) (((cSTAR Xr) x4) b)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((Xr a) x4)) (((cSTAR Xr) x4) b)))
% Found (fun (x4:atype)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((Xr a) x4)) (((cSTAR Xr) x4) b)))
% Found (fun (x4:atype)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found eq_ref00:=(eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (eq_ref0 (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found ((eq_ref (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and ((Xr a) x6)) (((cSTAR Xr) x6) b)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((Xr a) x6)) (((cSTAR Xr) x6) b)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((Xr a) x6)) (((cSTAR Xr) x6) b)))
% Found (fun (x6:atype)=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((and ((Xr a) x6)) (((cSTAR Xr) x6) b)))
% Found (fun (x6:atype)=> ((eq_ref Prop) (f x6))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and ((Xr a) x6)) (((cSTAR Xr) x6) b)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((Xr a) x6)) (((cSTAR Xr) x6) b)))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and ((Xr a) x6)) (((cSTAR Xr) x6) b)))
% Found (fun (x6:atype)=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((and ((Xr a) x6)) (((cSTAR Xr) x6) b)))
% Found (fun (x6:atype)=> ((eq_ref Prop) (f x6))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x8:=a:atype
% Found x1 as proof of (((cSTAR Xr) x8) b)
% Found x8:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x9:(Xx Xy))=> x8) as proof of (Xx Xz)
% Found (fun (x8:((Xr Xy) Xz)) (x9:(Xx Xy))=> x8) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x8:((Xr Xy) Xz)) (x9:(Xx Xy))=> x8) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x8:((Xr Xy) Xz)) (x9:(Xx Xy))=> x8)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x8:((Xr Xy) Xz)) (x9:(Xx Xy))=> x8)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x8:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x8) x7)) (Xx Xz)) (fun (x8:((Xr Xy) Xz)) (x9:(Xx Xy))=> x8)) as proof of (Xx Xz)
% Found (fun (x7:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x8:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x8) x7)) (Xx Xz)) (fun (x8:((Xr Xy) Xz)) (x9:(Xx Xy))=> x8))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x7:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x8:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x8) x7)) (Xx Xz)) (fun (x8:((Xr Xy) Xz)) (x9:(Xx Xy))=> x8))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x6:=a:atype
% Found x1 as proof of (((cSTAR Xr) x6) b)
% Found x9:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x10:(Xx Xy))=> x9) as proof of (Xx Xz)
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (fun (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x9:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x10:(Xx Xy))=> x9) as proof of (Xx Xz)
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (fun (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found ex_intro000:=(ex_intro00 b):((((cSTAR Xr) a) b)->((ex atype) ((cSTAR Xr) a)))
% Found (ex_intro00 b) as proof of ((((cSTAR Xr) a) b)->((ex atype) b0))
% Found ((ex_intro0 ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) b0))
% Found (((ex_intro atype) ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) b0))
% Found (((ex_intro atype) ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) b0))
% Found (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b)) as proof of ((((cSTAR Xr) a) b)->((ex atype) b0))
% Found (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b)) as proof of (((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->((ex atype) b0)))
% Found (and_rect00 (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) b0)
% Found ((and_rect0 ((ex atype) b0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) b0)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) b0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) b0)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) b0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) b0)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) b0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of (P b0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) b)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) b)
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found ex_intro000:=(ex_intro00 b):((((cSTAR Xr) a) b)->((ex atype) ((cSTAR Xr) a)))
% Found (ex_intro00 b) as proof of ((((cSTAR Xr) a) b)->((ex atype) f))
% Found ((ex_intro0 ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) f))
% Found (((ex_intro atype) ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) f))
% Found (((ex_intro atype) ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) f))
% Found (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b)) as proof of ((((cSTAR Xr) a) b)->((ex atype) f))
% Found (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b)) as proof of (((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->((ex atype) f)))
% Found (and_rect00 (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) f)
% Found ((and_rect0 ((ex atype) f)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) f)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) f)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) f)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) f)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) f)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) f)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of (P f)
% Found ex_intro000:=(ex_intro00 b):((((cSTAR Xr) a) b)->((ex atype) ((cSTAR Xr) a)))
% Found (ex_intro00 b) as proof of ((((cSTAR Xr) a) b)->((ex atype) f))
% Found ((ex_intro0 ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) f))
% Found (((ex_intro atype) ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) f))
% Found (((ex_intro atype) ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) f))
% Found (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b)) as proof of ((((cSTAR Xr) a) b)->((ex atype) f))
% Found (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b)) as proof of (((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->((ex atype) f)))
% Found (and_rect00 (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) f)
% Found ((and_rect0 ((ex atype) f)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) f)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) f)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) f)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) f)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) f)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) f)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of (P f)
% Found eq_ref00:=(eq_ref0 a0):(((eq (atype->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found ((eq_ref (atype->Prop)) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found ((eq_ref (atype->Prop)) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found ((eq_ref (atype->Prop)) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (atype->Prop)) b0) (fun (x:atype)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found ((eta_expansion_dep0 (fun (x1:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion_dep atype) (fun (x1:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion_dep atype) (fun (x1:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion_dep atype) (fun (x1:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found eq_ref00:=(eq_ref0 (((cSTAR Xr) x0) b)):(((eq Prop) (((cSTAR Xr) x0) b)) (((cSTAR Xr) x0) b))
% Found (eq_ref0 (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (atype->Prop)) a0) (fun (x:atype)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found (((eta_expansion atype) Prop) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found (((eta_expansion atype) Prop) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found (((eta_expansion atype) Prop) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (atype->Prop)) b0) (fun (x:atype)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found ((eta_expansion_dep0 (fun (x3:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion_dep atype) (fun (x3:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion_dep atype) (fun (x3:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion_dep atype) (fun (x3:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (atype->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found ((eq_ref (atype->Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found ((eq_ref (atype->Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found ((eq_ref (atype->Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b0:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b0
% Found eq_ref00:=(eq_ref0 (((cSTAR Xr) x2) b)):(((eq Prop) (((cSTAR Xr) x2) b)) (((cSTAR Xr) x2) b))
% Found (eq_ref0 (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 (((cSTAR Xr) x0) b)):(((eq Prop) (((cSTAR Xr) x0) b)) (((cSTAR Xr) x0) b))
% Found (eq_ref0 (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (atype->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found ((eq_ref (atype->Prop)) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found ((eq_ref (atype->Prop)) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found ((eq_ref (atype->Prop)) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (atype->Prop)) b0) (fun (x:atype)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found ((eta_expansion_dep0 (fun (x5:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion_dep atype) (fun (x5:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion_dep atype) (fun (x5:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion_dep atype) (fun (x5:atype)=> Prop)) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of b0
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 (((cSTAR Xr) x4) b)):(((eq Prop) (((cSTAR Xr) x4) b)) (((cSTAR Xr) x4) b))
% Found (eq_ref0 (((cSTAR Xr) x4) b)) as proof of (((eq Prop) (((cSTAR Xr) x4) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x4) b)) as proof of (((eq Prop) (((cSTAR Xr) x4) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x4) b)) as proof of (((eq Prop) (((cSTAR Xr) x4) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x4) b)) as proof of (((eq Prop) (((cSTAR Xr) x4) b)) b0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of b0
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found x61:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 (((cSTAR Xr) x0) b)):(((eq Prop) (((cSTAR Xr) x0) b)) (((cSTAR Xr) x0) b))
% Found (eq_ref0 (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found eq_ref00:=(eq_ref0 (((cSTAR Xr) x2) b)):(((eq Prop) (((cSTAR Xr) x2) b)) (((cSTAR Xr) x2) b))
% Found (eq_ref0 (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of b0
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found x71:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x71 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x71:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x71 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found eq_ref00:=(eq_ref0 a0):(((eq (atype->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found ((eq_ref (atype->Prop)) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found ((eq_ref (atype->Prop)) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found ((eq_ref (atype->Prop)) a0) as proof of (((eq (atype->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (atype->Prop)) b0) (fun (x:atype)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion atype) Prop) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion atype) Prop) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found (((eta_expansion atype) Prop) b0) as proof of (((eq (atype->Prop)) b0) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b))))
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found x61:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype;a0:=b:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype;a0:=b:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found ex_intro0:=(ex_intro atype):(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P)))
% Instantiate: b0:=(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P))):Prop
% Found ex_intro0 as proof of b0
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x8:=a:atype
% Found x1 as proof of (((cSTAR Xr) x8) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x8:=a:atype
% Found x1 as proof of (((cSTAR Xr) x8) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found ex_intro0:=(ex_intro atype):(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P)))
% Instantiate: b0:=(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P))):Prop
% Found ex_intro0 as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found ex_intro0:=(ex_intro atype):(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P)))
% Instantiate: b0:=(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P))):Prop
% Found ex_intro0 as proof of b0
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x6:=a:atype
% Found x1 as proof of (((cSTAR Xr) x6) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x6:=a:atype
% Found x1 as proof of (((cSTAR Xr) x6) b)
% Found eq_ref00:=(eq_ref0 x4):(((eq atype) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of b0
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x6:=a:atype
% Found x1 as proof of (((cSTAR Xr) x6) b)
% Found eq_ref00:=(eq_ref0 (((cSTAR Xr) x6) b)):(((eq Prop) (((cSTAR Xr) x6) b)) (((cSTAR Xr) x6) b))
% Found (eq_ref0 (((cSTAR Xr) x6) b)) as proof of (((eq Prop) (((cSTAR Xr) x6) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x6) b)) as proof of (((eq Prop) (((cSTAR Xr) x6) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x6) b)) as proof of (((eq Prop) (((cSTAR Xr) x6) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x6) b)) as proof of (((eq Prop) (((cSTAR Xr) x6) b)) b0)
% Found eq_ref00:=(eq_ref0 x4):(((eq atype) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found eq_ref00:=(eq_ref0 x4):(((eq atype) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) b)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) b)
% Found ex_intro0:=(ex_intro atype):(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P)))
% Instantiate: b0:=(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P))):Prop
% Found ex_intro0 as proof of b0
% Found ex_intro0:=(ex_intro atype):(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P)))
% Instantiate: b0:=(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P))):Prop
% Found ex_intro0 as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) b)
% Found ex_intro0:=(ex_intro atype):(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P)))
% Instantiate: b0:=(forall (P:(atype->Prop)) (x:atype), ((P x)->((ex atype) P))):Prop
% Found ex_intro0 as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found eq_ref00:=(eq_ref0 (((cSTAR Xr) x0) b)):(((eq Prop) (((cSTAR Xr) x0) b)) (((cSTAR Xr) x0) b))
% Found (eq_ref0 (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x0) b)) as proof of (((eq Prop) (((cSTAR Xr) x0) b)) b0)
% Found eq_ref00:=(eq_ref0 (((cSTAR Xr) x2) b)):(((eq Prop) (((cSTAR Xr) x2) b)) (((cSTAR Xr) x2) b))
% Found (eq_ref0 (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x2) b)) as proof of (((eq Prop) (((cSTAR Xr) x2) b)) b0)
% Found eq_ref00:=(eq_ref0 (((cSTAR Xr) x4) b)):(((eq Prop) (((cSTAR Xr) x4) b)) (((cSTAR Xr) x4) b))
% Found (eq_ref0 (((cSTAR Xr) x4) b)) as proof of (((eq Prop) (((cSTAR Xr) x4) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x4) b)) as proof of (((eq Prop) (((cSTAR Xr) x4) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x4) b)) as proof of (((eq Prop) (((cSTAR Xr) x4) b)) b0)
% Found ((eq_ref Prop) (((cSTAR Xr) x4) b)) as proof of (((eq Prop) (((cSTAR Xr) x4) b)) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found (fun (x2:(((cSTAR Xr) a) b))=> x2) as proof of b0
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2) as proof of ((((cSTAR Xr) a) b)->b0)
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2) as proof of (((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->b0))
% Found (and_rect00 (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of b0
% Found ((and_rect0 b0) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of b0
% Found (((fun (P0:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x1) x)) b0) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of b0
% Found (((fun (P0:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x1) x)) b0) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of b0
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x8:=a:atype
% Found x1 as proof of (((cSTAR Xr) x8) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x8:=a:atype
% Found x1 as proof of (((cSTAR Xr) x8) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x8:=a:atype
% Found x1 as proof of (((cSTAR Xr) x8) b)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) a0)
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x9:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x10:(Xx Xy))=> x9) as proof of (Xx Xz)
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (fun (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype;a0:=b:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype;a0:=b:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x6:=a:atype
% Found x1 as proof of b0
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x9:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x10:(Xx Xy))=> x9) as proof of (Xx Xz)
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (fun (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x9:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x10:(Xx Xy))=> x9) as proof of (Xx Xz)
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (fun (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x9:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x10:(Xx Xy))=> x9) as proof of (Xx Xz)
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (fun (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))):(((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) (fun (x:atype)=> ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found (eta_expansion_dep00 (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found ((eta_expansion_dep0 (fun (x9:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x9:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x9:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found (((eta_expansion_dep atype) (fun (x9:atype)=> Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) as proof of (((eq (atype->Prop)) (fun (Xc:atype)=> ((and ((Xr a) Xc)) (((cSTAR Xr) Xc) b)))) b0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of b0
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of b0
% Found eq_ref00:=(eq_ref0 x6):(((eq atype) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq atype) x6) b0)
% Found ((eq_ref atype) x6) as proof of (((eq atype) x6) b0)
% Found ((eq_ref atype) x6) as proof of (((eq atype) x6) b0)
% Found ((eq_ref atype) x6) as proof of (((eq atype) x6) b0)
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq atype) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found ((eq_ref atype) a0) as proof of (((eq atype) a0) b)
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found eq_ref00:=(eq_ref0 x6):(((eq atype) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq atype) x6) b0)
% Found ((eq_ref atype) x6) as proof of (((eq atype) x6) b0)
% Found ((eq_ref atype) x6) as proof of (((eq atype) x6) b0)
% Found ((eq_ref atype) x6) as proof of (((eq atype) x6) b0)
% Found eq_ref00:=(eq_ref0 x6):(((eq atype) x6) x6)
% Found (eq_ref0 x6) as proof of (((eq atype) x6) b0)
% Found ((eq_ref atype) x6) as proof of (((eq atype) x6) b0)
% Found ((eq_ref atype) x6) as proof of (((eq atype) x6) b0)
% Found ((eq_ref atype) x6) as proof of (((eq atype) x6) b0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype;a0:=b:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype;a0:=b:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found (fun (x2:(((cSTAR Xr) a) b))=> x2) as proof of (((cSTAR Xr) x0) a0)
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2) as proof of ((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0))
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2) as proof of (((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0)))
% Found (and_rect00 (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found ((and_rect0 (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found (((fun (P0:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x1) x)) (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found (((fun (P0:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x1) x)) (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found (fun (x2:(((cSTAR Xr) a) b))=> x2) as proof of (((cSTAR Xr) x0) a0)
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2) as proof of ((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0))
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2) as proof of (((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0)))
% Found (and_rect00 (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found ((and_rect0 (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found (((fun (P0:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x1) x)) (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found (((fun (P0:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x1) x)) (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found (fun (x2:(((cSTAR Xr) a) b))=> x2) as proof of (((cSTAR Xr) x0) a0)
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2) as proof of ((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0))
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2) as proof of (((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0)))
% Found (and_rect00 (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found ((and_rect0 (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found (((fun (P0:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x1) x)) (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found (((fun (P0:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x1) x)) (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (x2:(((cSTAR Xr) a) b))=> x2)) as proof of (((cSTAR Xr) x0) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found eq_ref00:=(eq_ref0 x4):(((eq atype) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found eq_ref000:=(eq_ref00 ((cSTAR Xr) a)):((((cSTAR Xr) a) b)->(((cSTAR Xr) a) b))
% Found (eq_ref00 ((cSTAR Xr) a)) as proof of ((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0))
% Found ((eq_ref0 b) ((cSTAR Xr) a)) as proof of ((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0))
% Found (((eq_ref atype) b) ((cSTAR Xr) a)) as proof of ((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0))
% Found (((eq_ref atype) b) ((cSTAR Xr) a)) as proof of ((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0))
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((eq_ref atype) b) ((cSTAR Xr) a))) as proof of ((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0))
% Found (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((eq_ref atype) b) ((cSTAR Xr) a))) as proof of (((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->(((cSTAR Xr) x0) a0)))
% Found (and_rect00 (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((eq_ref atype) b) ((cSTAR Xr) a)))) as proof of (((cSTAR Xr) x0) a0)
% Found ((and_rect0 (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((eq_ref atype) b) ((cSTAR Xr) a)))) as proof of (((cSTAR Xr) x0) a0)
% Found (((fun (P0:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x1) x)) (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((eq_ref atype) b) ((cSTAR Xr) a)))) as proof of (((cSTAR Xr) x0) a0)
% Found (((fun (P0:Type) (x1:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x1) x)) (((cSTAR Xr) x0) a0)) (fun (x1:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((eq_ref atype) b) ((cSTAR Xr) a)))) as proof of (((cSTAR Xr) x0) a0)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of b0
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of b0
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of b0
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of b0
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of b0
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x4):(((eq atype) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of b0
% Found eq_ref00:=(eq_ref0 x4):(((eq atype) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x4):(((eq atype) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found eq_ref00:=(eq_ref0 (f x8)):(((eq Prop) (f x8)) (f x8))
% Found (eq_ref0 (f x8)) as proof of (((eq Prop) (f x8)) ((and ((Xr a) x8)) (((cSTAR Xr) x8) b)))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) ((and ((Xr a) x8)) (((cSTAR Xr) x8) b)))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) ((and ((Xr a) x8)) (((cSTAR Xr) x8) b)))
% Found (fun (x8:atype)=> ((eq_ref Prop) (f x8))) as proof of (((eq Prop) (f x8)) ((and ((Xr a) x8)) (((cSTAR Xr) x8) b)))
% Found (fun (x8:atype)=> ((eq_ref Prop) (f x8))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found eq_ref00:=(eq_ref0 (f x8)):(((eq Prop) (f x8)) (f x8))
% Found (eq_ref0 (f x8)) as proof of (((eq Prop) (f x8)) ((and ((Xr a) x8)) (((cSTAR Xr) x8) b)))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) ((and ((Xr a) x8)) (((cSTAR Xr) x8) b)))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) ((and ((Xr a) x8)) (((cSTAR Xr) x8) b)))
% Found (fun (x8:atype)=> ((eq_ref Prop) (f x8))) as proof of (((eq Prop) (f x8)) ((and ((Xr a) x8)) (((cSTAR Xr) x8) b)))
% Found (fun (x8:atype)=> ((eq_ref Prop) (f x8))) as proof of (forall (x:atype), (((eq Prop) (f x)) ((and ((Xr a) x)) (((cSTAR Xr) x) b))))
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found eq_ref00:=(eq_ref0 x0):(((eq atype) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found ((eq_ref atype) x0) as proof of (((eq atype) x0) b0)
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found eq_ref00:=(eq_ref0 x2):(((eq atype) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found ((eq_ref atype) x2) as proof of (((eq atype) x2) b0)
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found eq_ref00:=(eq_ref0 x4):(((eq atype) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found eq_ref00:=(eq_ref0 x4):(((eq atype) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found ((eq_ref atype) x4) as proof of (((eq atype) x4) b0)
% Found x9:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x10:(Xx Xy))=> x9) as proof of (Xx Xz)
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect50 (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found ((and_rect5 (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (fun (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x9:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x10:(Xx Xy))=> x9) as proof of (Xx Xz)
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect50 (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found ((and_rect5 (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9)) as proof of (Xx Xz)
% Found (fun (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x8:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x9:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x9) x8)) (Xx Xz)) (fun (x9:((Xr Xy) Xz)) (x10:(Xx Xy))=> x9))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x6:=a:atype
% Found x1 as proof of (((cSTAR Xr) x6) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x6:=a:atype
% Found x1 as proof of (((cSTAR Xr) x6) a0)
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x62:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x62 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x6:=a:atype
% Found x1 as proof of (((cSTAR Xr) x6) a0)
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x6:=a:atype
% Found x1 as proof of (((cSTAR Xr) x6) a0)
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found ex_intro000:=(ex_intro00 b):((((cSTAR Xr) a) b)->((ex atype) ((cSTAR Xr) a)))
% Found (ex_intro00 b) as proof of ((((cSTAR Xr) a) b)->((ex atype) a0))
% Found ((ex_intro0 ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) a0))
% Found (((ex_intro atype) ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) a0))
% Found (((ex_intro atype) ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) a0))
% Found (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b)) as proof of ((((cSTAR Xr) a) b)->((ex atype) a0))
% Found (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b)) as proof of (((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->((ex atype) a0)))
% Found (and_rect00 (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) a0)
% Found ((and_rect0 ((ex atype) a0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) a0)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) a0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) a0)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) a0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) a0)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) a0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of (P a0)
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect50 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect5 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect50 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect5 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype;a0:=b:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype;a0:=b:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: a0:=b:atype;x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) a0)
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx0:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx0 Xy))=> x10) as proof of (Xx0 Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx0 Xy))=> x10) as proof of ((Xx0 Xy)->(Xx0 Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx0 Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx0 Xy)->(Xx0 Xz)))
% Found (and_rect50 (fun (x10:((Xr Xy) Xz)) (x11:(Xx0 Xy))=> x10)) as proof of (Xx0 Xz)
% Found ((and_rect5 (Xx0 Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx0 Xy))=> x10)) as proof of (Xx0 Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx0 Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx0 Xy)) P) x10) x9)) (Xx0 Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx0 Xy))=> x10)) as proof of (Xx0 Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx0 Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx0 Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx0 Xy)) P) x10) x9)) (Xx0 Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx0 Xy))=> x10))) as proof of (Xx0 Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx0 Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx0 Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx0 Xy)) P) x10) x9)) (Xx0 Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx0 Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz))
% Found ex_intro000:=(ex_intro00 b):((((cSTAR Xr) a) b)->((ex atype) ((cSTAR Xr) a)))
% Found (ex_intro00 b) as proof of ((((cSTAR Xr) a) b)->((ex atype) a0))
% Found ((ex_intro0 ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) a0))
% Found (((ex_intro atype) ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) a0))
% Found (((ex_intro atype) ((cSTAR Xr) a)) b) as proof of ((((cSTAR Xr) a) b)->((ex atype) a0))
% Found (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b)) as proof of ((((cSTAR Xr) a) b)->((ex atype) a0))
% Found (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b)) as proof of (((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->((ex atype) a0)))
% Found (and_rect00 (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) a0)
% Found ((and_rect0 ((ex atype) a0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) a0)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) a0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) a0)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) a0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of ((ex atype) a0)
% Found (((fun (P0:Type) (x0:(((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))->((((cSTAR Xr) a) b)->P0)))=> (((((and_rect ((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b)))) (((cSTAR Xr) a) b)) P0) x0) x)) ((ex atype) a0)) (fun (x0:((and ((and (forall (Xx:(atype->Prop)), ((iff (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))))) (forall (Xa0:atype) (Xb0:atype), ((iff (((cSTAR Xr) Xa0) Xb0)) (forall (Xx:(atype->Prop)), ((forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xa0)->(Xx Xb0)))))))) (not (((eq atype) a) b))))=> (((ex_intro atype) ((cSTAR Xr) a)) b))) as proof of (P a0)
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x90:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x90 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x90:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x90 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x10:((Xr Xy) Xz)
% Instantiate: Xx:=(Xr Xy):(atype->Prop)
% Found (fun (x11:(Xx Xy))=> x10) as proof of (Xx Xz)
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of ((Xx Xy)->(Xx Xz))
% Found (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10) as proof of (((Xr Xy) Xz)->((Xx Xy)->(Xx Xz)))
% Found (and_rect40 (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found ((and_rect4 (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10)) as proof of (Xx Xz)
% Found (fun (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (Xx Xz)
% Found (fun (Xz:atype) (x9:((and ((Xr Xy) Xz)) (Xx Xy)))=> (((fun (P:Type) (x10:(((Xr Xy) Xz)->((Xx Xy)->P)))=> (((((and_rect ((Xr Xy) Xz)) (Xx Xy)) P) x10) x9)) (Xx Xz)) (fun (x10:((Xr Xy) Xz)) (x11:(Xx Xy))=> x10))) as proof of (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x2:(((cSTAR Xr) a) b)
% Instantiate: x0:=a:atype
% Found x2 as proof of (((cSTAR Xr) x0) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) a0)
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x2:=a:atype
% Found x1 as proof of (((cSTAR Xr) x2) a0)
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x90:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x90 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x90:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x90 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x1:(((cSTAR Xr) a) b)
% Instantiate: x4:=a:atype
% Found x1 as proof of (((cSTAR Xr) x4) a0)
% Found x90:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x90 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x90:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x90 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Instantiate: Xx:=Xx0:(atype->Prop)
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x80:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x80 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Instantiate: Xx0:=Xx:(atype->Prop)
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x71:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x71 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x61 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x70 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x72:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x72 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x72:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x72 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x60 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x62:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x62 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x71:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x71 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x71:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x71 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x71:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x71 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x62:(forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))
% Found x62 as proof of (forall (Xy:atype) (Xz:atype), (((and ((Xr Xy) Xz)) (Xx Xy
% EOF
%------------------------------------------------------------------------------