TSTP Solution File: SET616^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SET616^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n117.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 13:30:51 EDT 2014
% Result : Timeout 300.09s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SET616^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n117.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 10:22:11 CDT 2014
% % CPUTime : 300.09
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1243368>, <kernel.Type object at 0x1243560>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (X:(a->Prop)) (Y:(a->Prop)), ((((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> ((and (Y Xx)) ((X Xx)->False))))->(((eq (a->Prop)) X) Y))) of role conjecture named cBOOL_PROP_90_pme
% Conjecture to prove = (forall (X:(a->Prop)) (Y:(a->Prop)), ((((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> ((and (Y Xx)) ((X Xx)->False))))->(((eq (a->Prop)) X) Y))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:(a->Prop)) (Y:(a->Prop)), ((((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> ((and (Y Xx)) ((X Xx)->False))))->(((eq (a->Prop)) X) Y)))']
% Parameter a:Type.
% Trying to prove (forall (X:(a->Prop)) (Y:(a->Prop)), ((((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> ((and (Y Xx)) ((X Xx)->False))))->(((eq (a->Prop)) X) Y)))
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% 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 x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% 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 x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 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_expansion_dep000:=(eta_expansion_dep00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion_dep00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) 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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y 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 eta_expansion_dep000:=(eta_expansion_dep00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion_dep00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) 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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 x0:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq (a->Prop)) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) 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)) (Y x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Y x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Y x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (Y x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f 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)) (Y x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Y x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Y x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (Y x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (Y x)))
% Found x0:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq (a->Prop)) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% 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:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% 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)) (Y x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Y x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Y x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (Y x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f 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)) (Y x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Y x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (Y x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (Y x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (Y x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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 (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% 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:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% 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 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Y):(((eq (a->Prop)) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found x0:(P Y)
% Instantiate: b:=Y:(a->Prop)
% Found x0 as proof of (P0 b)
% 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 eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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 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 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 (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found ((eq_ref (a->Prop)) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Y):(((eq (a->Prop)) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eq_ref (a->Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found x0:(P Y)
% Instantiate: b:=Y:(a->Prop)
% Found x0 as proof of (P0 b)
% 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 x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:(P Y)
% Instantiate: f:=Y:(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 Y)
% Instantiate: f:=Y:(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 x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) 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 eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b0)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% 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 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) 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 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) 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 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) 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 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P1 Y))):((P1 Y)->(P1 Y))
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P1 Y))):((P1 Y)->(P1 Y))
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P1 Y))):((P1 Y)->(P1 Y))
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P1 Y))):((P1 Y)->(P1 Y))
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:(P Y)
% Instantiate: f:=Y:(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 Y)
% Instantiate: f:=Y:(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 x1:=(x (fun (x1:(a->Prop))=> (P1 (X x0)))):((P1 (X x0))->(P1 (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (X x0)))):((P1 (X x0))->(P1 (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (X x0)))):((P1 (X x0))->(P1 (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (X x0)))):((P1 (X x0))->(P1 (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y 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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) 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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) 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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) 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)) 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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% 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 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) 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 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) 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 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) 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 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P1 Y))):((P1 Y)->(P1 Y))
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P1 Y))):((P1 Y)->(P1 Y))
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P1 Y))):((P1 Y)->(P1 Y))
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P1 Y))):((P1 Y)->(P1 Y))
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found (x (fun (x0:(a->Prop))=> (P1 Y))) as proof of (P2 Y)
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (X x0)))):((P1 (X x0))->(P1 (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (X x0)))):((P1 (X x0))->(P1 (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (X x0)))):((P1 (X x0))->(P1 (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (X x0)))):((P1 (X x0))->(P1 (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (X x0)))) as proof of (P2 (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) 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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y 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 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b0)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b0)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b0)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b0)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b0)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) X)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) X)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) X)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) X)
% 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) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found x1:(P (Y x0))
% Instantiate: b:=(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 (Y x0))
% Instantiate: b:=(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 x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% 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 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 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b0)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b0)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b0)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b0)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b0)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x1:(P (Y x0))
% Instantiate: b:=(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 (Y x0))
% Instantiate: b:=(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 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 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found ((eq_ref Prop) (b x0)) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion0 Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion a) Prop) Y) as proof of (((eq (a->Prop)) Y) b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% 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) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) X)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) X)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) X)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) X)
% Found x1:(P (Y x0))
% Instantiate: b:=(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 (Y x0))
% Instantiate: b:=(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:=(x (fun (x1:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Y)
% 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 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 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 (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 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:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% 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 x1:(P (Y x0))
% Instantiate: b:=(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 (Y x0))
% Instantiate: b:=(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:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (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) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (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) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y x0))
% Found x0:=(x (fun (x0:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x0:=(x (fun (x0:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 x1:=(x (fun (x1:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) 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)) 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 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 (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 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 (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) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (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) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P1 (Y x0)))):((P1 (Y x0))->(P1 (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P1 (Y x0)))) as proof of (P2 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y 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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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) Y)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Y)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Y)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Y)
% 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) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% 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_expansion_dep000:=(eta_expansion_dep00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion_dep00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) 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_expansion_dep000:=(eta_expansion_dep00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion_dep00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P 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 (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P 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:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X 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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y 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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found x0:=(x (fun (x0:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) Y)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Y)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Y)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Y)
% 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) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% 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_expansion_dep000:=(eta_expansion_dep00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion_dep00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) 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_expansion_dep000:=(eta_expansion_dep00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion_dep00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% 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 x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found x0:=(x (fun (x0:(a->Prop))=> (P Y))):((P Y)->(P Y))
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% Found (x (fun (x0:(a->Prop))=> (P Y))) as proof of (P0 Y)
% 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 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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P 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 (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P 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:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (X x0)))):((P (X x0))->(P (X x0)))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found (x (fun (x1:(a->Prop))=> (P (X x0)))) as proof of (P0 (X x0))
% Found x1:(P (X x0))
% Instantiate: b:=X:(a->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion_dep00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found x1:(P (X x0))
% Instantiate: b:=X:(a->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion_dep00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) 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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x1:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x1:(a->Prop))=> (P b))) 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 (P b)
% Found ((eq_ref Prop) (X x0)) as proof of (P b)
% Found ((eq_ref Prop) (X x0)) as proof of (P b)
% Found ((eq_ref Prop) (X x0)) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion_dep00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (P b)
% Found ((eq_ref Prop) (X x0)) as proof of (P b)
% Found ((eq_ref Prop) (X x0)) as proof of (P b)
% Found ((eq_ref Prop) (X x0)) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Y):(((eq (a->Prop)) Y) (fun (x:a)=> (Y x)))
% Found (eta_expansion_dep00 Y) as proof of (((eq (a->Prop)) Y) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Y) as proof of (((eq (a->Prop)) Y) b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (b x0)))):((P (b x0))->(P (b x0)))
% Found (x (fun (x1:(a->Prop))=> (P (b x0)))) as proof of (P0 (b x0))
% Found (x (fun (x1:(a->Prop))=> (P (b x0)))) as proof of (P0 (b x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (b x0)))):((P (b x0))->(P (b x0)))
% Found (x (fun (x1:(a->Prop))=> (P (b x0)))) as proof of (P0 (b x0))
% Found (x (fun (x1:(a->Prop))=> (P (b x0)))) as proof of (P0 (b x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x0:=(x (fun (x0:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P (Y x0)))):((P (Y x0))->(P (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found (x (fun (x1:(a->Prop))=> (P (Y x0)))) as proof of (P0 (Y x0))
% Found x0:=(x (fun (x0:(a->Prop))=> (P b))):((P b)->(P b))
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% Found (x (fun (x0:(a->Prop))=> (P b))) as proof of (P0 b)
% 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 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) (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 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) (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 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) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x0)) as proof of (((eq Prop) (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 (Y x0)):(((eq Prop) (Y x0)) (Y x0))
% Found (eq_ref0 (Y x0)) as proof of (((eq Prop) (Y x0)) b0)
% Found ((eq_ref Prop) (Y x
% EOF
%------------------------------------------------------------------------------