TSTP Solution File: SET183^5 by cocATP---0.2.0
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%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SET183^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n115.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 13:30:20 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 : SET183^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n115.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 09:19:46 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 0x21bbab8>, <kernel.Type object at 0x21bb4d0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (X:(a->Prop)) (Y:(a->Prop)), ((forall (Xx:a), ((X Xx)->(Y Xx)))->(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) X))) of role conjecture named cBOOL_PROP_42_pme
% Conjecture to prove = (forall (X:(a->Prop)) (Y:(a->Prop)), ((forall (Xx:a), ((X Xx)->(Y Xx)))->(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) X))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:(a->Prop)) (Y:(a->Prop)), ((forall (Xx:a), ((X Xx)->(Y Xx)))->(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) X)))']
% Parameter a:Type.
% Trying to prove (forall (X:(a->Prop)) (Y:(a->Prop)), ((forall (Xx:a), ((X Xx)->(Y Xx)))->(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) X)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x0:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: b:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: b:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found x00:(P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P2 X)
% Found x00:(P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P2 X)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b0)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P ((and (X x0)) (Y x0)))
% Found (fun (x10:(P ((and (X x0)) (Y x0))))=> x10) as proof of (P ((and (X x0)) (Y x0)))
% Found (fun (x10:(P ((and (X x0)) (Y x0))))=> x10) as proof of (P0 ((and (X x0)) (Y x0)))
% Found x10:(P ((and (X x0)) (Y x0)))
% Found (fun (x10:(P ((and (X x0)) (Y x0))))=> x10) as proof of (P ((and (X x0)) (Y x0)))
% Found (fun (x10:(P ((and (X x0)) (Y x0))))=> x10) as proof of (P0 ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found x10:(P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P2 (X x0))
% Found x10:(P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P2 (X x0))
% Found x10:(P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P2 (X x0))
% Found x10:(P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P2 (X x0))
% Found x10:(P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P2 (X x0))
% Found x10:(P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P1 (X x0))
% Found (fun (x10:(P1 (X x0)))=> x10) as proof of (P2 (X x0))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found x1:(P0 b)
% Instantiate: b:=(X x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (X x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (X x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found x1:(P0 b)
% Instantiate: b:=(X x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (X x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (X x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (Y x0))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: b:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: b:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: b:=(fun (x2:a)=> ((and (X x2)) (Y x2))):(a->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: b:=(fun (x2:a)=> ((and (X x2)) (Y x2))):(a->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x0)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x10:(P ((and (X x0)) (Y x0)))
% Found (fun (x10:(P ((and (X x0)) (Y x0))))=> x10) as proof of (P ((and (X x0)) (Y x0)))
% Found (fun (x10:(P ((and (X x0)) (Y x0))))=> x10) as proof of (P0 ((and (X x0)) (Y x0)))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P ((and (X x0)) (Y x0)))
% Found (fun (x10:(P ((and (X x0)) (Y x0))))=> x10) as proof of (P ((and (X x0)) (Y x0)))
% Found (fun (x10:(P ((and (X x0)) (Y x0))))=> x10) as proof of (P0 ((and (X x0)) (Y x0)))
% Found x10:(P ((and (X x0)) (Y x0)))
% Found (fun (x10:(P ((and (X x0)) (Y x0))))=> x10) as proof of (P ((and (X x0)) (Y x0)))
% Found (fun (x10:(P ((and (X x0)) (Y x0))))=> x10) as proof of (P0 ((and (X x0)) (Y x0)))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: a0:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found x0:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: b:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: b:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: b:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: b:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x0:(P1 X)
% Instantiate: b:=X:(a->Prop)
% Found x0 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: b0:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x00:(P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P2 X)
% Found x00:(P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P2 X)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x0 as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x00:(P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P2 X)
% Found x00:(P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P1 X)
% Found (fun (x00:(P1 X))=> x00) as proof of (P2 X)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (b x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x0:(P X)
% Instantiate: a0:=X:(a->Prop)
% Found x0 as proof of (P0 a0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found x0:(P1 X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P2 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found x0:(P1 X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P2 f)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (X x1)) (Y x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (X x)) (Y x))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b0)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found x10:(P1 X)
% Found (fun (x10:(P1 X))=> x10) as proof of (P1 X)
% Found (fun (x10:(P1 X))=> x10) as proof of (P2 X)
% Found x1:(P1 ((and (X x0)) (Y x0)))
% Instantiate: b:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found x1:(P1 ((and (X x0)) (Y x0)))
% Instantiate: b:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b0)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b0)
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: a0:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P0 a0)
% Found x1:(P ((and (X x0)) (Y x0)))
% Instantiate: a0:=((and (X x0)) (Y x0)):Prop
% Found x1 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found x0:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x1))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x1))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x00:(P1 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P1 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P1 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (fun (x00:(P1 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))=> x00) as proof of (P2 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_re
% EOF
%------------------------------------------------------------------------------