TSTP Solution File: ALG283^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : ALG283^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n115.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:18:22 EDT 2014
% Result : Timeout 300.06s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : ALG283^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n115.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 09:07:16 CDT 2014
% % CPUTime : 300.06
% 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 0x1cdd710>, <kernel.Type object at 0x1cdd6c8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x18acd40>, <kernel.DependentProduct object at 0x1cdd290>) of role type named cP
% Using role type
% Declaring cP:(a->(a->a))
% FOF formula (<kernel.Constant object at 0x18acd40>, <kernel.Constant object at 0x1cdd290>) of role type named cE
% Using role type
% Declaring cE:a
% FOF formula (<kernel.Constant object at 0x1cdd710>, <kernel.DependentProduct object at 0x1cddf38>) of role type named cJ
% Using role type
% Declaring cJ:(a->a)
% FOF formula (((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))->(forall (X:a) (Y:a), ((ex a) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))))) of role conjecture named cTHM20_pme
% Conjecture to prove = (((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))->(forall (X:a) (Y:a), ((ex a) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))))):Prop
% We need to prove ['(((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))->(forall (X:a) (Y:a), ((ex a) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))))']
% Parameter a:Type.
% Parameter cP:(a->(a->a)).
% Parameter cE:a.
% Parameter cJ:(a->a).
% Trying to prove (((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))->(forall (X:a) (Y:a), ((ex a) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))):(((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) (fun (x:a)=> (((eq a) ((cP X) x)) Y)))
% Found (eta_expansion00 (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found ((eta_expansion0 Prop) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found (((eta_expansion a) Prop) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found (((eta_expansion a) Prop) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found (((eta_expansion a) Prop) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) ((cP X) x0)) Y))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) ((cP X) x0)) Y))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) ((cP X) x0)) Y))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq a) ((cP X) x0)) Y))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) ((cP X) x)) Y)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) ((cP X) x0)) Y))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) ((cP X) x0)) Y))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq a) ((cP X) x0)) Y))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq a) ((cP X) x0)) Y))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) ((cP X) x)) Y)))
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x0))->(P ((cP X) x0)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x0))
% Found ((eq_ref0 ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))):(((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) (fun (x:a)=> (((eq a) ((cP X) x)) Y)))
% Found (eta_expansion_dep00 (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) ((cP X) x2)) Y))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) ((cP X) x2)) Y))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) ((cP X) x2)) Y))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq a) ((cP X) x2)) Y))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) ((cP X) x)) Y)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) ((cP X) x2)) Y))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) ((cP X) x2)) Y))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq a) ((cP X) x2)) Y))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq a) ((cP X) x2)) Y))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) ((cP X) x)) Y)))
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x2))->(P ((cP X) x2)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x2))
% Found ((eq_ref0 ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found (((eq_ref a) ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found (((eq_ref a) ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found eq_ref00:=(eq_ref0 (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))):(((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found (eq_ref0 (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found ((eq_ref (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found ((eq_ref (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found ((eq_ref (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) as proof of (((eq (a->Prop)) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x0))->(P ((cP X) x0)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x0))
% Found ((eq_ref0 ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (((eq a) ((cP X) x4)) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq a) ((cP X) x4)) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq a) ((cP X) x4)) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (((eq a) ((cP X) x4)) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) ((cP X) x)) Y)))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (((eq a) ((cP X) x4)) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq a) ((cP X) x4)) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq a) ((cP X) x4)) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (((eq a) ((cP X) x4)) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq a) ((cP X) x)) Y)))
% Found eq_ref00:=(eq_ref0 ((cP X) x2)):(((eq a) ((cP X) x2)) ((cP X) x2))
% Found (eq_ref0 ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x4))->(P ((cP X) x4)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x4))
% Found ((eq_ref0 ((cP X) x4)) P) as proof of (P0 ((cP X) x4))
% Found (((eq_ref a) ((cP X) x4)) P) as proof of (P0 ((cP X) x4))
% Found (((eq_ref a) ((cP X) x4)) P) as proof of (P0 ((cP X) x4))
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x2))->(P ((cP X) x2)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x2))
% Found ((eq_ref0 ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found (((eq_ref a) ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found (((eq_ref a) ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x0))->(P ((cP X) x0)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x0))
% Found ((eq_ref0 ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found eq_ref00:=(eq_ref0 ((cP X) x4)):(((eq a) ((cP X) x4)) ((cP X) x4))
% Found (eq_ref0 ((cP X) x4)) as proof of (((eq a) ((cP X) x4)) b)
% Found ((eq_ref a) ((cP X) x4)) as proof of (((eq a) ((cP X) x4)) b)
% Found ((eq_ref a) ((cP X) x4)) as proof of (((eq a) ((cP X) x4)) b)
% Found ((eq_ref a) ((cP X) x4)) as proof of (((eq a) ((cP X) x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 ((cP X) x2)):(((eq a) ((cP X) x2)) ((cP X) x2))
% Found (eq_ref0 ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x4))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x4))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x4))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x4))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x0))->(P ((cP X) x0)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x0))
% Found ((eq_ref0 ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found x1:(P ((cP X) x0))
% Instantiate: b:=((cP X) x0):a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x2))->(P ((cP X) x2)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x2))
% Found ((eq_ref0 ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found (((eq_ref a) ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found (((eq_ref a) ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x0))->(P ((cP X) x0)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x0))
% Found ((eq_ref0 ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x0))->(P ((cP X) x0)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x0))
% Found ((eq_ref0 ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% 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 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found x1:(P Y)
% Instantiate: b:=Y:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found x3:(P ((cP X) x2))
% Instantiate: b:=((cP X) x2):a
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 ((cP X) x2)):(((eq a) ((cP X) x2)) ((cP X) x2))
% Found (eq_ref0 ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found x3:(P ((cP X) x0))
% Instantiate: b:=((cP X) x0):a
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found x1:(P ((cP X) x0))
% Instantiate: b:=((cP X) x0):a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Z:a)=> (((eq a) ((cP X) Z)) Y)))
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x4))->(P ((cP X) x4)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x4))
% Found ((eq_ref0 ((cP X) x4)) P) as proof of (P0 ((cP X) x4))
% Found (((eq_ref a) ((cP X) x4)) P) as proof of (P0 ((cP X) x4))
% Found (((eq_ref a) ((cP X) x4)) P) as proof of (P0 ((cP X) x4))
% 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 ((cP X) x2))->(P ((cP X) x2)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x2))
% Found ((eq_ref0 ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found (((eq_ref a) ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found (((eq_ref a) ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found eq_ref00:=(eq_ref0 ((cP X) x2)):(((eq a) ((cP X) x2)) ((cP X) x2))
% Found (eq_ref0 ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 ((cP X) x2)):(((eq a) ((cP X) x2)) ((cP X) x2))
% Found (eq_ref0 ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x2))->(P ((cP X) x2)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x2))
% Found ((eq_ref0 ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found (((eq_ref a) ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% Found (((eq_ref a) ((cP X) x2)) P) as proof of (P0 ((cP X) x2))
% 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 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x0))->(P ((cP X) x0)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x0))
% Found ((eq_ref0 ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found eq_ref000:=(eq_ref00 P):((P Y)->(P Y))
% Found (eq_ref00 P) as proof of (P0 Y)
% Found ((eq_ref0 Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found x3:(P Y)
% Instantiate: b:=Y:a
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) x2)):(((eq a) ((cP X) x2)) ((cP X) x2))
% Found (eq_ref0 ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found eq_ref000:=(eq_ref00 P):((P ((cP X) x0))->(P ((cP X) x0)))
% Found (eq_ref00 P) as proof of (P0 ((cP X) x0))
% Found ((eq_ref0 ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found (((eq_ref a) ((cP X) x0)) P) as proof of (P0 ((cP X) x0))
% Found eq_ref00:=(eq_ref0 ((cP X) x4)):(((eq a) ((cP X) x4)) ((cP X) x4))
% Found (eq_ref0 ((cP X) x4)) as proof of (((eq a) ((cP X) x4)) b)
% Found ((eq_ref a) ((cP X) x4)) as proof of (((eq a) ((cP X) x4)) b)
% Found ((eq_ref a) ((cP X) x4)) as proof of (((eq a) ((cP X) x4)) b)
% Found ((eq_ref a) ((cP X) x4)) as proof of (((eq a) ((cP X) x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found x5:(P ((cP X) x4))
% Instantiate: b:=((cP X) x4):a
% Found x5 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% 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 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x2))
% Found x3:(P Y)
% Instantiate: b:=Y:a
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found eq_ref00:=(eq_ref0 ((cP X) x2)):(((eq a) ((cP X) x2)) ((cP X) x2))
% Found (eq_ref0 ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found ((eq_ref a) ((cP X) x2)) as proof of (((eq a) ((cP X) x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found x5:(P ((cP X) x2))
% Instantiate: b:=((cP X) x2):a
% Found x5 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found x5:(P ((cP X) x0))
% Instantiate: b:=((cP X) x0):a
% Found x5 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found x1:(P ((cP X) x0))
% Instantiate: b:=((cP X) x0):a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found x3:(P ((cP X) x2))
% Instantiate: b:=((cP X) x2):a
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found x3:(P ((cP X) x0))
% Instantiate: b:=((cP X) x0):a
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) 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 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) x0))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b0)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b0)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b0)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Y)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Y)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Y)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Y)
% Found x3:(P Y)
% Instantiate: b:=Y:a
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found x1:(P Y)
% Instantiate: b:=Y:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) x0)):(((eq a) ((cP X) x0)) ((cP X) x0))
% Found (eq_ref0 ((cP X) x0)) as proof of (((eq a) ((cP X) x0)) b)
% Found ((eq_ref a) ((cP X) x0)) as proof of (((eq a) ((cP
% EOF
%------------------------------------------------------------------------------