TSTP Solution File: SEU935^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU935^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n090.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 13:33:24 EDT 2014
% Result : Timeout 300.02s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU935^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n090.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 11:43:36 CDT 2014
% % CPUTime : 300.02
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x114bdd0>, <kernel.Type object at 0x114bef0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x10309e0>, <kernel.Type object at 0x114b710>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x114b908>, <kernel.Type object at 0x114b9e0>) of role type named c_type
% Using role type
% Declaring c:Type
% FOF formula (forall (F:(a->b)) (G:(b->c)), (((and (forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (F X)) Y))))) (forall (Y:c), ((ex b) (fun (X:b)=> (((eq c) (G X)) Y)))))->(forall (Y:c), ((ex a) (fun (X:a)=> (((eq c) (G (F X))) Y)))))) of role conjecture named cFN_THM_2_pme
% Conjecture to prove = (forall (F:(a->b)) (G:(b->c)), (((and (forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (F X)) Y))))) (forall (Y:c), ((ex b) (fun (X:b)=> (((eq c) (G X)) Y)))))->(forall (Y:c), ((ex a) (fun (X:a)=> (((eq c) (G (F X))) Y)))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% Parameter c_DUMMY:c.
% We need to prove ['(forall (F:(a->b)) (G:(b->c)), (((and (forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (F X)) Y))))) (forall (Y:c), ((ex b) (fun (X:b)=> (((eq c) (G X)) Y)))))->(forall (Y:c), ((ex a) (fun (X:a)=> (((eq c) (G (F X))) Y))))))']
% Parameter a:Type.
% Parameter b:Type.
% Parameter c:Type.
% Trying to prove (forall (F:(a->b)) (G:(b->c)), (((and (forall (Y:b), ((ex a) (fun (X:a)=> (((eq b) (F X)) Y))))) (forall (Y:c), ((ex b) (fun (X:b)=> (((eq c) (G X)) Y)))))->(forall (Y:c), ((ex a) (fun (X:a)=> (((eq c) (G (F X))) Y))))))
% Found eq_ref00:=(eq_ref0 (fun (X:a)=> (((eq c) (G (F X))) Y))):(((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found (eq_ref0 (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found ((eq_ref (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found ((eq_ref (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found ((eq_ref (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% 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 c) (G (F x0))) Y))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq c) (G (F x0))) Y))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq c) (G (F x0))) Y))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq c) (G (F x0))) Y))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq c) (G (F 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 c) (G (F x0))) Y))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq c) (G (F x0))) Y))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq c) (G (F x0))) Y))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq c) (G (F x0))) Y))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq c) (G (F x))) Y)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:a)=> (((eq c) (G (F X))) Y))):(((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) (fun (x:a)=> (((eq c) (G (F x))) Y)))
% Found (eta_expansion_dep00 (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x0)))->(P (G (F x0))))
% Found (eq_ref00 P) as proof of (P0 (G (F x0)))
% Found ((eq_ref0 (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% 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 c) (G (F x2))) Y))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq c) (G (F x2))) Y))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq c) (G (F x2))) Y))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq c) (G (F x2))) Y))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq c) (G (F 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 c) (G (F x2))) Y))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq c) (G (F x2))) Y))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (((eq c) (G (F x2))) Y))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (((eq c) (G (F x2))) Y))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq c) (G (F x))) Y)))
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x2)))->(P (G (F x2))))
% Found (eq_ref00 P) as proof of (P0 (G (F x2)))
% Found ((eq_ref0 (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found eq_ref000:=(eq_ref00 P):((P (G (F x0)))->(P (G (F x0))))
% Found (eq_ref00 P) as proof of (P0 (G (F x0)))
% Found ((eq_ref0 (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found eq_ref00:=(eq_ref0 (G (F x2))):(((eq c) (G (F x2))) (G (F x2)))
% Found (eq_ref0 (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: b0:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x0)))->(P (G (F x0))))
% Found (eq_ref00 P) as proof of (P0 (G (F x0)))
% Found ((eq_ref0 (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% 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) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: b0:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found x1:(P (G (F x0)))
% Instantiate: b0:=(G (F x0)):c
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x2)))->(P (G (F x2))))
% Found (eq_ref00 P) as proof of (P0 (G (F x2)))
% Found ((eq_ref0 (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% 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) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (X:a)=> (((eq c) (G (F X))) Y)))
% Found eq_ref000:=(eq_ref00 P):((P (G (F x0)))->(P (G (F x0))))
% Found (eq_ref00 P) as proof of (P0 (G (F x0)))
% Found ((eq_ref0 (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x0)))->(P (G (F x0))))
% Found (eq_ref00 P) as proof of (P0 (G (F x0)))
% Found ((eq_ref0 (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x3:(P (G (F x2)))
% Instantiate: b0:=(G (F x2)):c
% Found x3 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x1:(P Y)
% Instantiate: b0:=Y:c
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 (G (F x2))):(((eq c) (G (F x2))) (G (F x2)))
% Found (eq_ref0 (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: b0:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found x00 as proof of (P b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:a)=> (((eq c) (G (F X))) Y))):(((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) (fun (x:a)=> (((eq c) (G (F x))) Y)))
% Found (eta_expansion_dep00 (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x3:(P (G (F x0)))
% Instantiate: b0:=(G (F x0)):c
% Found x3 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found x1:(P (G (F x0)))
% Instantiate: b0:=(G (F x0)):c
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found x00 as proof of (P f)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found x00 as proof of (P f)
% 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 c) (G (F x4))) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq c) (G (F 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 c) (G (F x4))) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq c) (G (F x))) Y)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:a)=> (((eq c) (G (F X))) Y))):(((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) (fun (x:a)=> (((eq c) (G (F x))) Y)))
% Found (eta_expansion_dep00 (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) b0)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) as proof of (((eq (a->Prop)) (fun (X:a)=> (((eq c) (G (F X))) Y))) 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 c) b0) P) as proof of (P0 b0)
% Found (((eq_ref c) b0) P) as proof of (P0 b0)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x2)))->(P (G (F x2))))
% Found (eq_ref00 P) as proof of (P0 (G (F x2)))
% Found ((eq_ref0 (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found eq_ref00:=(eq_ref0 (G (F x2))):(((eq c) (G (F x2))) (G (F x2)))
% Found (eq_ref0 (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 (G (F x2))):(((eq c) (G (F x2))) (G (F x2)))
% Found (eq_ref0 (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% 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 c) Y) P) as proof of (P0 Y)
% Found (((eq_ref c) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq c) (G (F 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 c) (G (F x4))) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) (((eq c) (G (F x4))) Y))
% Found (fun (x4:a)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:a), (((eq Prop) (f x)) (((eq c) (G (F x))) Y)))
% Found eq_ref000:=(eq_ref00 P):((P (G (F x0)))->(P (G (F x0))))
% Found (eq_ref00 P) as proof of (P0 (G (F x0)))
% Found ((eq_ref0 (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found x3:(P Y)
% Instantiate: b0:=Y:c
% Found x3 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (G (F x2))):(((eq c) (G (F x2))) (G (F x2)))
% Found (eq_ref0 (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x3:(P Y)
% Instantiate: b0:=Y:c
% Found x3 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq c) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% Found x3:(P Y)
% Instantiate: b0:=Y:c
% Found x3 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found x30:=(x3 (fun (x5:b)=> (P (G (F x4))))):((P (G (F x4)))->(P (G (F x4))))
% Found (x3 (fun (x5:b)=> (P (G (F x4))))) as proof of (P0 (G (F x4)))
% Found (x3 (fun (x5:b)=> (P (G (F x4))))) as proof of (P0 (G (F x4)))
% Found x30:=(x3 (fun (x5:c)=> (P (G (F x4))))):((P (G (F x4)))->(P (G (F x4))))
% Found (x3 (fun (x5:c)=> (P (G (F x4))))) as proof of (P0 (G (F x4)))
% Found (x3 (fun (x5:c)=> (P (G (F x4))))) as proof of (P0 (G (F x4)))
% Found x1:(P Y)
% Instantiate: b0:=Y:c
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found x3:(((eq c) (G x2)) Y0)
% Instantiate: Y0:=Y:c;b0:=(G x2):c
% Found x3 as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 (G (F x4))):(((eq c) (G (F x4))) (G (F x4)))
% Found (eq_ref0 (G (F x4))) as proof of (((eq c) (G (F x4))) b0)
% Found ((eq_ref c) (G (F x4))) as proof of (((eq c) (G (F x4))) b0)
% Found ((eq_ref c) (G (F x4))) as proof of (((eq c) (G (F x4))) b0)
% Found ((eq_ref c) (G (F x4))) as proof of (((eq c) (G (F x4))) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq b) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F 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 c) Y) P) as proof of (P0 Y)
% Found (((eq_ref c) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% 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 c) b0) P) as proof of (P0 b0)
% Found (((eq_ref c) b0) P) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x40:=(x4 (fun (x5:b)=> (P (G (F x2))))):((P (G (F x2)))->(P (G (F x2))))
% Found (x4 (fun (x5:b)=> (P (G (F x2))))) as proof of (P0 (G (F x2)))
% Found (x4 (fun (x5:b)=> (P (G (F x2))))) as proof of (P0 (G (F x2)))
% Found x40:=(x4 (fun (x5:c)=> (P (G (F x2))))):((P (G (F x2)))->(P (G (F x2))))
% Found (x4 (fun (x5:c)=> (P (G (F x2))))) as proof of (P0 (G (F x2)))
% Found (x4 (fun (x5:c)=> (P (G (F x2))))) as proof of (P0 (G (F x2)))
% Found x4:(((eq c) (G x3)) Y0)
% Instantiate: Y0:=Y:c;b0:=(G x3):c
% Found x4 as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 (G (F x2))):(((eq c) (G (F x2))) (G (F x2)))
% Found (eq_ref0 (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq c) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% 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 c) Y) P) as proof of (P0 Y)
% Found (((eq_ref c) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 (G (F x4))):(((eq c) (G (F x4))) (G (F x4)))
% Found (eq_ref0 (G (F x4))) as proof of (((eq c) (G (F x4))) b0)
% Found ((eq_ref c) (G (F x4))) as proof of (((eq c) (G (F x4))) b0)
% Found ((eq_ref c) (G (F x4))) as proof of (((eq c) (G (F x4))) b0)
% Found ((eq_ref c) (G (F x4))) as proof of (((eq c) (G (F x4))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% 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 c) b0) P) as proof of (P0 b0)
% Found (((eq_ref c) b0) P) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found x40:=(x4 (fun (x5:b)=> (P (G (F x0))))):((P (G (F x0)))->(P (G (F x0))))
% Found (x4 (fun (x5:b)=> (P (G (F x0))))) as proof of (P0 (G (F x0)))
% Found (x4 (fun (x5:b)=> (P (G (F x0))))) as proof of (P0 (G (F x0)))
% Found x40:=(x4 (fun (x5:c)=> (P (G (F x0))))):((P (G (F x0)))->(P (G (F x0))))
% Found (x4 (fun (x5:c)=> (P (G (F x0))))) as proof of (P0 (G (F x0)))
% Found (x4 (fun (x5:c)=> (P (G (F x0))))) as proof of (P0 (G (F x0)))
% Found x4:(((eq c) (G x3)) Y0)
% Instantiate: Y0:=Y:c;b0:=(G x3):c
% Found x4 as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq c) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% 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 c) Y) P) as proof of (P0 Y)
% Found (((eq_ref c) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 (G (F x2))):(((eq c) (G (F x2))) (G (F x2)))
% Found (eq_ref0 (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) 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 c) b0) P) as proof of (P0 b0)
% Found (((eq_ref c) b0) P) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (G (F x2))):(((eq c) (G (F x2))) (G (F x2)))
% Found (eq_ref0 (G (F x2))) as proof of (((eq c) (G (F x2))) b00)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b00)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b00)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq c) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% 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 c) Y) P) as proof of (P0 Y)
% Found (((eq_ref c) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 b00):(((eq c) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq c) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% Found ((eq_ref c) b00) as proof of (((eq c) b00) Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq b) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq b) a0) (F x2))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x2))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x2))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x2))
% 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 c) Y) P) as proof of (P0 Y)
% Found (((eq_ref c) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 b00):(((eq c) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found x3:(((eq c) (G x2)) Y0)
% Instantiate: Y0:=(G (F x4)):c;b0:=(G x2):c
% Found x3 as proof of (((eq c) b0) (G (F x4)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq b) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x0))
% 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 c) b0) P) as proof of (P0 b0)
% Found (((eq_ref c) b0) P) as proof of (P0 b0)
% Found x4:(((eq c) (G x3)) Y0)
% Instantiate: Y0:=(G (F x2)):c;b0:=(G x3):c
% Found x4 as proof of (((eq c) b0) (G (F x2)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq c) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq c) b1) Y)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) Y)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) Y)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) b1)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) b1)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) b1)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq b) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x0))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq c) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G x3))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G x3))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G x3))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G x3))
% Found eq_ref00:=(eq_ref0 (G (F x2))):(((eq c) (G (F x2))) (G (F x2)))
% Found (eq_ref0 (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found ((eq_ref c) (G (F x2))) as proof of (((eq c) (G (F x2))) b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x4)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x4)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x4)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x4)))
% Found eq_ref00:=(eq_ref0 a0):(((eq b) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F x0))
% Found ((eq_ref b) a0) as proof of (((eq b) a0) (F 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 c) Y) P) as proof of (P0 Y)
% Found (((eq_ref c) Y) P) as proof of (P0 Y)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x2)))->(P (G (F x2))))
% Found (eq_ref00 P) as proof of (P0 (G (F x2)))
% Found ((eq_ref0 (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found eq_ref00:=(eq_ref0 b00):(((eq c) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found x4:(((eq c) (G x3)) Y0)
% Instantiate: Y0:=(G (F x2)):c;b0:=(G x3):c
% Found x4 as proof of (((eq c) b0) (G (F x2)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x4:(((eq c) (G x3)) Y0)
% Instantiate: Y0:=(G (F x0)):c;b0:=(G x3):c
% Found x4 as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq c) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G x3))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G x3))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G x3))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G x3))
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x0)))->(P (G (F x0))))
% Found (eq_ref00 P) as proof of (P0 (G (F x0)))
% Found ((eq_ref0 (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq c) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq c) b00) (G (F x2)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x2)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x2)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x2)))
% 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 c) Y) P) as proof of (P0 Y)
% Found (((eq_ref c) Y) P) as proof of (P0 Y)
% Found x4:(((eq c) (G x3)) Y0)
% Instantiate: Y0:=(G (F x0)):c;b0:=(G x3):c
% Found x4 as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: b0:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq c) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x2)))->(P (G (F x2))))
% Found (eq_ref00 P) as proof of (P0 (G (F x2)))
% Found ((eq_ref0 (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found eq_ref000:=(eq_ref00 P):((P (G (F x2)))->(P (G (F x2))))
% Found (eq_ref00 P) as proof of (P0 (G (F x2)))
% Found ((eq_ref0 (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found (((eq_ref c) (G (F x2))) P) as proof of (P0 (G (F x2)))
% Found x4:(((eq c) (G x3)) Y0)
% Instantiate: Y0:=(G (F x0)):c;b0:=(G x3):c
% Found x4 as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: b0:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq c) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: b0:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x0)))->(P (G (F x0))))
% Found (eq_ref00 P) as proof of (P0 (G (F x0)))
% Found ((eq_ref0 (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found eq_ref00:=(eq_ref0 b00):(((eq c) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found ((eq_ref c) b00) as proof of (((eq c) b00) (G (F x0)))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b00)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x0)))->(P (G (F x0))))
% Found (eq_ref00 P) as proof of (P0 (G (F x0)))
% Found ((eq_ref0 (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found x1:(P (G (F x0)))
% Instantiate: b0:=(G (F x0)):c
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found x00:=(x0 Y0):((ex a) (fun (X:a)=> (((eq b) (F X)) Y0)))
% Instantiate: f:=(fun (X:a)=> (((eq b) (F X)) Y0)):(a->Prop)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x0)))
% Found eq_ref00:=(eq_ref0 b1):(((eq c) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq c) b1) Y)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) Y)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) Y)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) b1)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) b1)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) b1)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq c) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) b0)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b1)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b1)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b1)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b1)
% 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 c) Y) P) as proof of (P0 Y)
% Found (((eq_ref c) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref000:=(eq_ref00 P):((P (G (F x0)))->(P (G (F x0))))
% Found (eq_ref00 P) as proof of (P0 (G (F x0)))
% Found ((eq_ref0 (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found (((eq_ref c) (G (F x0))) P) as proof of (P0 (G (F x0)))
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found ((eq_ref c) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq c) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq c) b1) Y)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) Y)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) Y)
% Found ((eq_ref c) b1) as proof of (((eq c) b1) Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) b1)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) b1)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) b1)
% Found ((eq_ref c) b0) as proof of (((eq c) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G x3))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G x3))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G x3))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G x3))
% Found eq_ref00:=(eq_ref0 Y):(((eq c) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found ((eq_ref c) Y) as proof of (((eq c) Y) b0)
% Found eq_ref00:=(eq_ref0 (G x3)):(((eq c) (G x3)) (G x3))
% Found (eq_ref0 (G x3)) as proof of (((eq c) (G x3)) b0)
% Found ((eq_ref c) (G x3)) as proof of (((eq c) (G x3)) b0)
% Found ((eq_ref c) (G x3)) as proof of (((eq c) (G x3)) b0)
% Found ((eq_ref c) (G x3)) as proof of (((eq c) (G x3)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq c) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found ((eq_ref c) b0) as proof of (((eq c) b0) (G (F x2)))
% Found x1:(P Y)
% Instantiate: b0:=Y:c
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (G (F x0))):(((eq c) (G (F x0))) (G (F x0)))
% Found (eq_ref0 (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c) (G (F x0))) b0)
% Found ((eq_ref c) (G (F x0))) as proof of (((eq c)
% EOF
%------------------------------------------------------------------------------