TSTP Solution File: SEV167^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV167^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n089.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:49 EDT 2014
% Result : Timeout 300.00s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV167^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n089.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:18:36 CDT 2014
% % CPUTime : 300.00
% 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 0x110f320>, <kernel.Type object at 0x110e098>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xr:(a->(a->(a->(a->Prop))))), ((forall (Xp:((a->(a->a))->a)) (Xq:((a->(a->a))->a)), (((((Xr (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xp (fun (Xx:a) (Xy:a)=> Xy))) (Xq (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xy)))->(((eq ((a->(a->a))->a)) Xp) Xq)))->(forall (Xx1:a) (Xy1:a) (Xx2:a) (Xy2:a), (((((Xr Xx1) Xy1) Xx2) Xy2)->((and (((eq a) Xx1) Xx2)) (((eq a) Xy1) Xy2)))))) of role conjecture named cTHM189_pme
% Conjecture to prove = (forall (Xr:(a->(a->(a->(a->Prop))))), ((forall (Xp:((a->(a->a))->a)) (Xq:((a->(a->a))->a)), (((((Xr (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xp (fun (Xx:a) (Xy:a)=> Xy))) (Xq (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xy)))->(((eq ((a->(a->a))->a)) Xp) Xq)))->(forall (Xx1:a) (Xy1:a) (Xx2:a) (Xy2:a), (((((Xr Xx1) Xy1) Xx2) Xy2)->((and (((eq a) Xx1) Xx2)) (((eq a) Xy1) Xy2)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xr:(a->(a->(a->(a->Prop))))), ((forall (Xp:((a->(a->a))->a)) (Xq:((a->(a->a))->a)), (((((Xr (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xp (fun (Xx:a) (Xy:a)=> Xy))) (Xq (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xy)))->(((eq ((a->(a->a))->a)) Xp) Xq)))->(forall (Xx1:a) (Xy1:a) (Xx2:a) (Xy2:a), (((((Xr Xx1) Xy1) Xx2) Xy2)->((and (((eq a) Xx1) Xx2)) (((eq a) Xy1) Xy2))))))']
% Parameter a:Type.
% Trying to prove (forall (Xr:(a->(a->(a->(a->Prop))))), ((forall (Xp:((a->(a->a))->a)) (Xq:((a->(a->a))->a)), (((((Xr (Xp (fun (Xx:a) (Xy:a)=> Xx))) (Xp (fun (Xx:a) (Xy:a)=> Xy))) (Xq (fun (Xx:a) (Xy:a)=> Xx))) (Xq (fun (Xx:a) (Xy:a)=> Xy)))->(((eq ((a->(a->a))->a)) Xp) Xq)))->(forall (Xx1:a) (Xy1:a) (Xx2:a) (Xy2:a), (((((Xr Xx1) Xy1) Xx2) Xy2)->((and (((eq a) Xx1) Xx2)) (((eq a) Xy1) Xy2))))))
% Found eq_ref000:=(eq_ref00 P):((P Xx1)->(P Xx1))
% Found (eq_ref00 P) as proof of (P0 Xx1)
% Found ((eq_ref0 Xx1) P) as proof of (P0 Xx1)
% Found (((eq_ref a) Xx1) P) as proof of (P0 Xx1)
% Found (((eq_ref a) Xx1) P) as proof of (P0 Xx1)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref a) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref a) Xy1) P) as proof of (P0 Xy1)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found eq_ref00:=(eq_ref0 (((eq a) Xy1) Xy2)):(((eq Prop) (((eq a) Xy1) Xy2)) (((eq a) Xy1) Xy2))
% Found (eq_ref0 (((eq a) Xy1) Xy2)) as proof of (((eq Prop) (((eq a) Xy1) Xy2)) b)
% Found ((eq_ref Prop) (((eq a) Xy1) Xy2)) as proof of (((eq Prop) (((eq a) Xy1) Xy2)) b)
% Found ((eq_ref Prop) (((eq a) Xy1) Xy2)) as proof of (((eq Prop) (((eq a) Xy1) Xy2)) b)
% Found ((eq_ref Prop) (((eq a) Xy1) Xy2)) as proof of (((eq Prop) (((eq a) Xy1) Xy2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found x1:(P Xx1)
% Instantiate: b:=Xx1:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found x1:(P Xy1)
% Instantiate: b:=Xy1:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found eq_ref000:=(eq_ref00 P):((P Xx1)->(P Xx1))
% Found (eq_ref00 P) as proof of (P0 Xx1)
% Found ((eq_ref0 Xx1) P) as proof of (P0 Xx1)
% Found (((eq_ref a) Xx1) P) as proof of (P0 Xx1)
% Found (((eq_ref a) Xx1) P) as proof of (P0 Xx1)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref a) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref a) Xy1) P) as proof of (P0 Xy1)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found x1:(P Xy2)
% Instantiate: b:=Xy2:a
% Found x1 as proof of (P0 b)
% Found x1:(P Xx2)
% Instantiate: b:=Xx2:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref000:=(eq_ref00 P):((P Xx1)->(P Xx1))
% Found (eq_ref00 P) as proof of (P0 Xx1)
% Found ((eq_ref0 Xx1) P) as proof of (P0 Xx1)
% Found (((eq_ref a) Xx1) P) as proof of (P0 Xx1)
% Found (((eq_ref a) Xx1) P) as proof of (P0 Xx1)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref a) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref a) Xy1) P) as proof of (P0 Xy1)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref000:=(eq_ref00 P):((P Xx2)->(P Xx2))
% Found (eq_ref00 P) as proof of (P0 Xx2)
% Found ((eq_ref0 Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref000:=(eq_ref00 P):((P Xy2)->(P Xy2))
% Found (eq_ref00 P) as proof of (P0 Xy2)
% Found ((eq_ref0 Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) Xy1) Xy2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Xy1) Xy2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Xy1) Xy2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 P):((P Xx2)->(P Xx2))
% Found (eq_ref00 P) as proof of (P0 Xx2)
% Found ((eq_ref0 Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found eq_ref000:=(eq_ref00 P):((P Xy2)->(P Xy2))
% Found (eq_ref00 P) as proof of (P0 Xy2)
% Found ((eq_ref0 Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found x1:(P Xx1)
% Instantiate: b:=Xx1:a
% Found x1 as proof of (P0 b)
% Found x1:(P Xy1)
% Instantiate: b:=Xy1:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found eq_ref000:=(eq_ref00 P):((P Xx1)->(P Xx1))
% Found (eq_ref00 P) as proof of (P0 Xx1)
% Found ((eq_ref0 Xx1) P) as proof of (P0 Xx1)
% Found (((eq_ref a) Xx1) P) as proof of (P0 Xx1)
% Found (((eq_ref a) Xx1) P) as proof of (P0 Xx1)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found eq_ref000:=(eq_ref00 P):((P Xy1)->(P Xy1))
% Found (eq_ref00 P) as proof of (P0 Xy1)
% Found ((eq_ref0 Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref a) Xy1) P) as proof of (P0 Xy1)
% Found (((eq_ref a) Xy1) P) as proof of (P0 Xy1)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found x1:(P Xx1)
% Instantiate: a0:=Xx1:a
% Found x1 as proof of (P0 a0)
% Found x1:(P Xy1)
% Instantiate: a0:=Xy1:a
% Found x1 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found x1:(P Xx2)
% Instantiate: b:=Xx2:a
% Found x1 as proof of (P0 b)
% Found x1:(P Xy2)
% Instantiate: b:=Xy2:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found x1:(P1 Xx2)
% Instantiate: b:=Xx2:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xy2)
% Instantiate: b:=Xy2:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xx2)
% Instantiate: b:=Xx2:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xy2)
% Instantiate: b:=Xy2:a
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy2)->(P1 Xy2))
% Found (eq_ref00 P1) as proof of (P2 Xy2)
% Found ((eq_ref0 Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found x1:(P Xy2)
% Instantiate: b:=Xy2:a
% Found x1 as proof of (P0 b)
% Found x1:(P Xx2)
% Instantiate: b:=Xx2:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found x1:(P b)
% Found x1 as proof of (P0 Xx1)
% Found x1:(P b)
% Found x1 as proof of (P0 Xy1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref000:=(eq_ref00 P):((P Xy2)->(P Xy2))
% Found (eq_ref00 P) as proof of (P0 Xy2)
% Found ((eq_ref0 Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref000:=(eq_ref00 P):((P Xx2)->(P Xx2))
% Found (eq_ref00 P) as proof of (P0 Xx2)
% Found ((eq_ref0 Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref000:=(eq_ref00 P):((P Xy2)->(P Xy2))
% Found (eq_ref00 P) as proof of (P0 Xy2)
% Found ((eq_ref0 Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref000:=(eq_ref00 P):((P Xx2)->(P Xx2))
% Found (eq_ref00 P) as proof of (P0 Xx2)
% Found ((eq_ref0 Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found x1:(P Xy2)
% Instantiate: a0:=Xy2:a
% Found x1 as proof of (P0 a0)
% Found x1:(P Xx2)
% Instantiate: a0:=Xx2:a
% Found x1 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xx2)->(P0 Xx2))
% Found (eq_ref00 P0) as proof of (P1 Xx2)
% Found ((eq_ref0 Xx2) P0) as proof of (P1 Xx2)
% Found (((eq_ref a) Xx2) P0) as proof of (P1 Xx2)
% Found (((eq_ref a) Xx2) P0) as proof of (P1 Xx2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xy2)->(P0 Xy2))
% Found (eq_ref00 P0) as proof of (P1 Xy2)
% Found ((eq_ref0 Xy2) P0) as proof of (P1 Xy2)
% Found (((eq_ref a) Xy2) P0) as proof of (P1 Xy2)
% Found (((eq_ref a) Xy2) P0) as proof of (P1 Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b1)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b1)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b1)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b1)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b1)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b1)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of a0
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy2)->(P1 Xy2))
% Found (eq_ref00 P1) as proof of (P2 Xy2)
% Found ((eq_ref0 Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy2)->(P1 Xy2))
% Found (eq_ref00 P1) as proof of (P2 Xy2)
% Found ((eq_ref0 Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref000:=(eq_ref00 P):((P Xy2)->(P Xy2))
% Found (eq_ref00 P) as proof of (P0 Xy2)
% Found ((eq_ref0 Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found eq_ref000:=(eq_ref00 P):((P Xx2)->(P Xx2))
% Found (eq_ref00 P) as proof of (P0 Xx2)
% Found ((eq_ref0 Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b0)->(P b0))
% Found (eq_ref00 P) as proof of (P0 b0)
% Found ((eq_ref0 b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found eq_ref000:=(eq_ref00 P):((P b0)->(P b0))
% Found (eq_ref00 P) as proof of (P0 b0)
% Found ((eq_ref0 b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (P Xx1)
% Found ((eq_ref a) Xx1) as proof of (P Xx1)
% Found ((eq_ref a) Xx1) as proof of (P Xx1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (P Xy1)
% Found ((eq_ref a) Xy1) as proof of (P Xy1)
% Found ((eq_ref a) Xy1) as proof of (P Xy1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found x1:(P Xx2)
% Instantiate: b0:=Xx2:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xy2)
% Instantiate: b0:=Xy2:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xy2)
% Instantiate: b0:=Xy2:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xx2)
% Instantiate: b0:=Xx2:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found x1:(P Xx2)
% Found x1 as proof of (P0 Xx2)
% Found x1:(P Xy2)
% Found x1 as proof of (P0 Xy2)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xx1)->(P0 Xx1))
% Found (eq_ref00 P0) as proof of (P1 Xx1)
% Found ((eq_ref0 Xx1) P0) as proof of (P1 Xx1)
% Found (((eq_ref a) Xx1) P0) as proof of (P1 Xx1)
% Found (((eq_ref a) Xx1) P0) as proof of (P1 Xx1)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy2)->(P1 Xy2))
% Found (eq_ref00 P1) as proof of (P2 Xy2)
% Found ((eq_ref0 Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy2)->(P1 Xy2))
% Found (eq_ref00 P1) as proof of (P2 Xy2)
% Found ((eq_ref0 Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xx1)->(P0 Xx1))
% Found (eq_ref00 P0) as proof of (P1 Xx1)
% Found ((eq_ref0 Xx1) P0) as proof of (P1 Xx1)
% Found (((eq_ref a) Xx1) P0) as proof of (P1 Xx1)
% Found (((eq_ref a) Xx1) P0) as proof of (P1 Xx1)
% Found eq_ref000:=(eq_ref00 P):((P Xy2)->(P Xy2))
% Found (eq_ref00 P) as proof of (P0 Xy2)
% Found ((eq_ref0 Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found eq_ref000:=(eq_ref00 P):((P Xx2)->(P Xx2))
% Found (eq_ref00 P) as proof of (P0 Xx2)
% Found ((eq_ref0 Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b1)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b1)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b1)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xy1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b1)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b1)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b1)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref000:=(eq_ref00 P):((P Xy2)->(P Xy2))
% Found (eq_ref00 P) as proof of (P0 Xy2)
% Found ((eq_ref0 Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref000:=(eq_ref00 P):((P Xx2)->(P Xx2))
% Found (eq_ref00 P) as proof of (P0 Xx2)
% Found ((eq_ref0 Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b0)
% Found ((eq_ref a) b) as proof of (P b0)
% Found ((eq_ref a) b) as proof of (P b0)
% Found ((eq_ref a) b) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b0)
% Found ((eq_ref a) b) as proof of (P b0)
% Found ((eq_ref a) b) as proof of (P b0)
% Found ((eq_ref a) b) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found eq_ref000:=(eq_ref00 P):((P Xx2)->(P Xx2))
% Found (eq_ref00 P) as proof of (P0 Xx2)
% Found ((eq_ref0 Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found eq_ref000:=(eq_ref00 P):((P Xy2)->(P Xy2))
% Found (eq_ref00 P) as proof of (P0 Xy2)
% Found ((eq_ref0 Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (P Xx2)
% Found ((eq_ref a) Xx2) as proof of (P Xx2)
% Found ((eq_ref a) Xx2) as proof of (P Xx2)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (P Xy2)
% Found ((eq_ref a) Xy2) as proof of (P Xy2)
% Found ((eq_ref a) Xy2) as proof of (P Xy2)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b1)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b1)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b1)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b1)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b1)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b1)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b00)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b00)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b00)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b00)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b00)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b00)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found x1:(P Xx1)
% Instantiate: a0:=Xx1:a
% Found x1 as proof of (P0 a0)
% Found x1:(P Xy1)
% Instantiate: a0:=Xy1:a
% Found x1 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy2)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx2)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xx1)->(P0 Xx1))
% Found (eq_ref00 P0) as proof of (P1 Xx1)
% Found ((eq_ref0 Xx1) P0) as proof of (P1 Xx1)
% Found (((eq_ref a) Xx1) P0) as proof of (P1 Xx1)
% Found (((eq_ref a) Xx1) P0) as proof of (P1 Xx1)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xy1)->(P0 Xy1))
% Found (eq_ref00 P0) as proof of (P1 Xy1)
% Found ((eq_ref0 Xy1) P0) as proof of (P1 Xy1)
% Found (((eq_ref a) Xy1) P0) as proof of (P1 Xy1)
% Found (((eq_ref a) Xy1) P0) as proof of (P1 Xy1)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref000:=(eq_ref00 P):((P b0)->(P b0))
% Found (eq_ref00 P) as proof of (P0 b0)
% Found ((eq_ref0 b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found eq_ref000:=(eq_ref00 P):((P b0)->(P b0))
% Found (eq_ref00 P) as proof of (P0 b0)
% Found ((eq_ref0 b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b1)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b1)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b1)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xy1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy1)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b1)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b1)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b1)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found x1:(P1 Xy2)
% Instantiate: b:=Xy2:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xx2)
% Instantiate: b:=Xx2:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xx2)
% Instantiate: b:=Xx2:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xy2)
% Instantiate: b:=Xy2:a
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 (((eq a) Xy1) Xy2)):(((eq Prop) (((eq a) Xy1) Xy2)) (((eq a) Xy1) Xy2))
% Found (eq_ref0 (((eq a) Xy1) Xy2)) as proof of (((eq Prop) (((eq a) Xy1) Xy2)) b0)
% Found ((eq_ref Prop) (((eq a) Xy1) Xy2)) as proof of (((eq Prop) (((eq a) Xy1) Xy2)) b0)
% Found ((eq_ref Prop) (((eq a) Xy1) Xy2)) as proof of (((eq Prop) (((eq a) Xy1) Xy2)) b0)
% Found ((eq_ref Prop) (((eq a) Xy1) Xy2)) as proof of (((eq Prop) (((eq a) Xy1) Xy2)) b0)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xx2)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xx2)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xx2)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xy2)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xy2)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xy2)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy2)->(P1 Xy2))
% Found (eq_ref00 P1) as proof of (P2 Xy2)
% Found ((eq_ref0 Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found x1:(P Xy2)
% Instantiate: b:=Xy2:a
% Found x1 as proof of (P0 b)
% Found x1:(P Xx2)
% Instantiate: b:=Xx2:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found x1:(P b)
% Found x1 as proof of (P0 Xy1)
% Found x1:(P b)
% Found x1 as proof of (P0 Xx1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref000:=(eq_ref00 P):((P Xy2)->(P Xy2))
% Found (eq_ref00 P) as proof of (P0 Xy2)
% Found ((eq_ref0 Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref000:=(eq_ref00 P):((P Xx2)->(P Xx2))
% Found (eq_ref00 P) as proof of (P0 Xx2)
% Found ((eq_ref0 Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found x1:(P Xy2)
% Instantiate: a0:=Xy2:a
% Found x1 as proof of (P0 a0)
% Found x1:(P Xy2)
% Instantiate: a0:=Xy2:a
% Found x1 as proof of (P0 a0)
% Found x1:(P Xx2)
% Instantiate: a0:=Xx2:a
% Found x1 as proof of (P0 a0)
% Found x1:(P Xx2)
% Instantiate: a0:=Xx2:a
% Found x1 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xy2)->(P0 Xy2))
% Found (eq_ref00 P0) as proof of (P1 Xy2)
% Found ((eq_ref0 Xy2) P0) as proof of (P1 Xy2)
% Found (((eq_ref a) Xy2) P0) as proof of (P1 Xy2)
% Found (((eq_ref a) Xy2) P0) as proof of (P1 Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b1)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b1)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b1)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b1)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b1)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b1)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xx2)->(P0 Xx2))
% Found (eq_ref00 P0) as proof of (P1 Xx2)
% Found ((eq_ref0 Xx2) P0) as proof of (P1 Xx2)
% Found (((eq_ref a) Xx2) P0) as proof of (P1 Xx2)
% Found (((eq_ref a) Xx2) P0) as proof of (P1 Xx2)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b1)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b1)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b1)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b1)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b1)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b1)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b1)
% Found x1:(P b)
% Found x1 as proof of (P0 Xy1)
% Found x1:(P b)
% Found x1 as proof of (P0 Xx1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found x1:(P1 Xy2)
% Instantiate: b:=Xy2:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xx2)
% Instantiate: b:=Xx2:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xy2)
% Instantiate: b:=Xy2:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xx2)
% Instantiate: b:=Xx2:a
% Found x1 as proof of (P2 b)
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of b
% Found eta_expansion as proof of a0
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy2)->(P1 Xy2))
% Found (eq_ref00 P1) as proof of (P2 Xy2)
% Found ((eq_ref0 Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy2)->(P1 Xy2))
% Found (eq_ref00 P1) as proof of (P2 Xy2)
% Found ((eq_ref0 Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy2)->(P1 Xy2))
% Found (eq_ref00 P1) as proof of (P2 Xy2)
% Found ((eq_ref0 Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy2)->(P1 Xy2))
% Found (eq_ref00 P1) as proof of (P2 Xy2)
% Found ((eq_ref0 Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xx2)->(P1 Xx2))
% Found (eq_ref00 P1) as proof of (P2 Xx2)
% Found ((eq_ref0 Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found (((eq_ref a) Xx2) P1) as proof of (P2 Xx2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy2)->(P1 Xy2))
% Found (eq_ref00 P1) as proof of (P2 Xy2)
% Found ((eq_ref0 Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found (((eq_ref a) Xy2) P1) as proof of (P2 Xy2)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b)
% Found eq_ref000:=(eq_ref00 P):((P Xy2)->(P Xy2))
% Found (eq_ref00 P) as proof of (P0 Xy2)
% Found ((eq_ref0 Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found (((eq_ref a) Xy2) P) as proof of (P0 Xy2)
% Found eq_ref000:=(eq_ref00 P):((P Xx2)->(P Xx2))
% Found (eq_ref00 P) as proof of (P0 Xx2)
% Found ((eq_ref0 Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found (((eq_ref a) Xx2) P) as proof of (P0 Xx2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref000:=(eq_ref00 P):((P b0)->(P b0))
% Found (eq_ref00 P) as proof of (P0 b0)
% Found ((eq_ref0 b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found eq_ref000:=(eq_ref00 P):((P b0)->(P b0))
% Found (eq_ref00 P) as proof of (P0 b0)
% Found ((eq_ref0 b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found eq_ref000:=(eq_ref00 P):((P b0)->(P b0))
% Found (eq_ref00 P) as proof of (P0 b0)
% Found ((eq_ref0 b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found eq_ref000:=(eq_ref00 P):((P b0)->(P b0))
% Found (eq_ref00 P) as proof of (P0 b0)
% Found ((eq_ref0 b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found (((eq_ref a) b0) P) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (P Xx1)
% Found ((eq_ref a) Xx1) as proof of (P Xx1)
% Found ((eq_ref a) Xx1) as proof of (P Xx1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (P Xy1)
% Found ((eq_ref a) Xy1) as proof of (P Xy1)
% Found ((eq_ref a) Xy1) as proof of (P Xy1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (P Xx1)
% Found ((eq_ref a) Xx1) as proof of (P Xx1)
% Found ((eq_ref a) Xx1) as proof of (P Xx1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (P Xy1)
% Found ((eq_ref a) Xy1) as proof of (P Xy1)
% Found ((eq_ref a) Xy1) as proof of (P Xy1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found x1:(P Xy2)
% Instantiate: b0:=Xy2:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xx2)
% Instantiate: b0:=Xx2:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xy2)
% Instantiate: b0:=Xy2:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xx2)
% Instantiate: b0:=Xx2:a
% Found x1 as proof of (P0 b0)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xy2)
% Instantiate: b0:=Xy2:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xx2)
% Instantiate: b0:=Xx2:a
% Found x1 as proof of (P0 b0)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xy2)->(P0 Xy2))
% Found (eq_ref00 P0) as proof of (P1 Xy2)
% Found ((eq_ref0 Xy2) P0) as proof of (P1 Xy2)
% Found (((eq_ref a) Xy2) P0) as proof of (P1 Xy2)
% Found (((eq_ref a) Xy2) P0) as proof of (P1 Xy2)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xx2)->(P0 Xx2))
% Found (eq_ref00 P0) as proof of (P1 Xx2)
% Found ((eq_ref0 Xx2) P0) as proof of (P1 Xx2)
% Found (((eq_ref a) Xx2) P0) as proof of (P1 Xx2)
% Found (((eq_ref a) Xx2) P0) as proof of (P1 Xx2)
% Found eq_ref00:=(eq_ref0 Xx1):(((eq a) Xx1) Xx1)
% Found (eq_ref0 Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found ((eq_ref a) Xx1) as proof of (((eq a) Xx1) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx1)
% Found eq_ref00:=(eq_ref0 Xx2):(((eq a) Xx2) Xx2)
% Found (eq_ref0 Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found ((eq_ref a) Xx2) as proof of (((eq a) Xx2) b0)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 Xy2):(((eq a) Xy2) Xy2)
% Found (eq_ref0 Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found ((eq_ref a) Xy2) as proof of (((eq a) Xy2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx1)
% Found ((eq_ref a) b0) as proof of (((eq a) b
% EOF
%------------------------------------------------------------------------------