TSTP Solution File: SEV431^1 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV431^1 : TPTP v6.1.0. Released v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n118.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 13:34:11 EDT 2014
% Result : Timeout 300.10s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV431^1 : TPTP v6.1.0. Released v5.2.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n118.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 09:13:11 CDT 2014
% % CPUTime : 300.10
% 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 0x192b908>, <kernel.Type object at 0x192b200>) of role type named a
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x192b5a8>, <kernel.Type object at 0x16bd560>) of role type named b
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x192b488>, <kernel.DependentProduct object at 0x16bd200>) of role type named f
% Using role type
% Declaring f:(a->b)
% FOF formula (forall (X:a) (Y:a), ((((eq b) (f X)) (f Y))->(((eq a) X) Y))) of role axiom named finj
% A new axiom: (forall (X:a) (Y:a), ((((eq b) (f X)) (f Y))->(((eq a) X) Y)))
% FOF formula ((ex (b->a)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) of role conjecture named invexists
% Conjecture to prove = ((ex (b->a)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% We need to prove ['((ex (b->a)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))']
% Parameter a:Type.
% Parameter b:Type.
% Parameter f:(a->b).
% Axiom finj:(forall (X:a) (Y:a), ((((eq b) (f X)) (f Y))->(((eq a) X) Y))).
% Trying to prove ((ex (b->a)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref000:=(eq_ref00 P):((P (x (f X)))->(P (x (f X))))
% Found (eq_ref00 P) as proof of ((P (x (f X)))->(P X))
% Found ((eq_ref0 (x (f X))) P) as proof of ((P (x (f X)))->(P X))
% Found (((eq_ref a) (x (f X))) P) as proof of ((P (x (f X)))->(P X))
% Found (((eq_ref a) (x (f X))) P) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> (((eq_ref a) (x (f X))) P)) as proof of ((P (x (f X)))->(P X))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_sym00 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found (((eq_sym0 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))):(((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found (eq_ref0 (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b0)
% Found ((eq_ref ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b0)
% Found ((eq_ref ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b0)
% Found ((eq_ref ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b0)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj000 ((eq_ref b) (f (x (f X))))) as proof of ((P (x (f X)))->(P X))
% Found (finj000 ((eq_ref b) (f (x (f X))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((finj00 x0) P)) ((eq_ref b) (f (x (f X))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> (((finj0 X) x0) P)) ((eq_ref b) (f (x (f X))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((((finj (x (f X))) X) x0) P)) ((eq_ref b) (f (x (f X))))) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((((finj (x (f X))) X) x0) P)) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (forall (X:a), (((eq a) (x (f X))) X)))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (forall (X:a), (((eq a) (x (f X))) X)))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (forall (X:a), (((eq a) (x (f X))) X)))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (forall (X:a), (((eq a) (x (f X))) X)))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f0 x)) (forall (X:a), (((eq a) (x (f X))) X))))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (forall (X:a), (((eq a) (x (f X))) X)))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (forall (X:a), (((eq a) (x (f X))) X)))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (forall (X:a), (((eq a) (x (f X))) X)))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (forall (X:a), (((eq a) (x (f X))) X)))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f0 x)) (forall (X:a), (((eq a) (x (f X))) X))))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_trans0000 ((eq_ref a) (x (f X)))) ((eq_ref a) b0)) as proof of (((eq a) (x (f X))) X)
% Found (((eq_trans000 X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b0:a)=> ((eq_trans00 b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b0:a)=> (((eq_trans0 (x (f X))) b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b0:a)=> ((((eq_trans a) (x (f X))) b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref000:=(eq_ref00 P):((P (x (f X)))->(P (x (f X))))
% Found (eq_ref00 P) as proof of ((P (x (f X)))->(P X))
% Found ((eq_ref0 (x (f X))) P) as proof of ((P (x (f X)))->(P X))
% Found (((eq_ref a) (x (f X))) P) as proof of ((P (x (f X)))->(P X))
% Found (((eq_ref a) (x (f X))) P) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> (((eq_ref a) (x (f X))) P)) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> (((eq_ref a) (x (f X))) P)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym0000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found (eq_sym0000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> ((eq_sym000 x0) P)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> (((eq_sym00 (x (f X))) x0) P)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> ((((eq_sym0 X) (x (f X))) x0) P)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> (((((eq_sym a) X) (x (f X))) x0) P)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> ((fun (x0:(((eq a) X) (x (f X))))=> (((((eq_sym a) X) (x (f X))) x0) P)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))))) as proof of ((P (x (f X)))->(P X))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_sym00 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((eq_sym0 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj000 ((eq_ref b) (f (x (f X))))) as proof of ((P (x (f X)))->(P X))
% Found (finj000 ((eq_ref b) (f (x (f X))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((finj00 x0) P)) ((eq_ref b) (f (x (f X))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> (((finj0 X) x0) P)) ((eq_ref b) (f (x (f X))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((((finj (x (f X))) X) x0) P)) ((eq_ref b) (f (x (f X))))) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((((finj (x (f X))) X) x0) P)) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((((finj (x (f X))) X) x0) P)) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((b->a)->Prop)) a0) (fun (x:(b->a))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:(b->a))=> Prop)) a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) a0) as proof of (((eq ((b->a)->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((b->a)->Prop)) b0) (fun (x:(b->a))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found (((eta_expansion (b->a)) Prop) b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found (((eta_expansion (b->a)) Prop) b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found (((eta_expansion (b->a)) Prop) b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((b->a)->Prop)) b0) (fun (x:(b->a))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found ((eta_expansion_dep0 (fun (x1:(b->a))=> Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found x0:(P (x (f X)))
% Instantiate: X0:=(x (f X)):a
% Found x0 as proof of (P0 X0)
% Found eq_ref00:=(eq_ref0 (f X0)):(((eq b) (f X0)) (f X0))
% Found (eq_ref0 (f X0)) as proof of (((eq b) (f X0)) (f X))
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) (f X))
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) (f X))
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) (f X))
% Found x0:(P (x (f X)))
% Instantiate: b0:=(x (f X)):a
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_trans0000 ((eq_ref a) (x (f X)))) ((eq_ref a) b0)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((eq_trans000 X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((fun (b0:a)=> ((eq_trans00 b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((fun (b0:a)=> (((eq_trans0 (x (f X))) b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((fun (b0:a)=> ((((eq_trans a) (x (f X))) b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((fun (b0:a)=> ((((eq_trans a) (x (f X))) b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym0000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found (eq_sym0000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> ((eq_sym000 x0) P)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> (((eq_sym00 (x (f X))) x0) P)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> ((((eq_sym0 X) (x (f X))) x0) P)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> (((((eq_sym a) X) (x (f X))) x0) P)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> ((fun (x0:(((eq a) X) (x (f X))))=> (((((eq_sym a) X) (x (f X))) x0) P)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))))) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> ((fun (x0:(((eq a) X) (x (f X))))=> (((((eq_sym a) X) (x (f X))) x0) P)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_trans00000 ((eq_ref a) (x (f X)))) ((eq_ref a) b0)) as proof of ((P (x (f X)))->(P X))
% Found ((eq_trans00000 ((eq_ref a) (x (f X)))) ((eq_ref a) b0)) as proof of ((P (x (f X)))->(P X))
% Found (((fun (x0:(((eq a) (x (f X))) b0)) (x00:(((eq a) b0) X))=> (((eq_trans0000 x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) b0)) as proof of ((P (x (f X)))->(P X))
% Found (((fun (x0:(((eq a) (x (f X))) X)) (x00:(((eq a) X) X))=> ((((eq_trans000 X) x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of ((P (x (f X)))->(P X))
% Found (((fun (x0:(((eq a) (x (f X))) X)) (x00:(((eq a) X) X))=> (((((fun (b0:a)=> ((eq_trans00 b0) X)) X) x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of ((P (x (f X)))->(P X))
% Found (((fun (x0:(((eq a) (x (f X))) X)) (x00:(((eq a) X) X))=> (((((fun (b0:a)=> (((eq_trans0 (x (f X))) b0) X)) X) x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of ((P (x (f X)))->(P X))
% Found (((fun (x0:(((eq a) (x (f X))) X)) (x00:(((eq a) X) X))=> (((((fun (b0:a)=> ((((eq_trans a) (x (f X))) b0) X)) X) x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> (((fun (x0:(((eq a) (x (f X))) X)) (x00:(((eq a) X) X))=> (((((fun (b0:a)=> ((((eq_trans a) (x (f X))) b0) X)) X) x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) X))) as proof of ((P (x (f X)))->(P X))
% Found x0:(P (x (f X)))
% Instantiate: X0:=(x (f X)):a
% Found x0 as proof of (P0 X0)
% Found eq_ref00:=(eq_ref0 (f X0)):(((eq b) (f X0)) (f X0))
% Found (eq_ref0 (f X0)) as proof of (((eq b) (f X0)) (f X))
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) (f X))
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) (f X))
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) (f X))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found x0:(P (x (f X)))
% Instantiate: b0:=(x (f X)):a
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_trans00000 ((eq_ref a) (x (f X)))) ((eq_ref a) b0)) as proof of ((P (x (f X)))->(P X))
% Found ((eq_trans00000 ((eq_ref a) (x (f X)))) ((eq_ref a) b0)) as proof of ((P (x (f X)))->(P X))
% Found (((fun (x0:(((eq a) (x (f X))) b0)) (x00:(((eq a) b0) X))=> (((eq_trans0000 x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) b0)) as proof of ((P (x (f X)))->(P X))
% Found (((fun (x0:(((eq a) (x (f X))) X)) (x00:(((eq a) X) X))=> ((((eq_trans000 X) x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of ((P (x (f X)))->(P X))
% Found (((fun (x0:(((eq a) (x (f X))) X)) (x00:(((eq a) X) X))=> (((((fun (b0:a)=> ((eq_trans00 b0) X)) X) x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of ((P (x (f X)))->(P X))
% Found (((fun (x0:(((eq a) (x (f X))) X)) (x00:(((eq a) X) X))=> (((((fun (b0:a)=> (((eq_trans0 (x (f X))) b0) X)) X) x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of ((P (x (f X)))->(P X))
% Found (((fun (x0:(((eq a) (x (f X))) X)) (x00:(((eq a) X) X))=> (((((fun (b0:a)=> ((((eq_trans a) (x (f X))) b0) X)) X) x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> (((fun (x0:(((eq a) (x (f X))) X)) (x00:(((eq a) X) X))=> (((((fun (b0:a)=> ((((eq_trans a) (x (f X))) b0) X)) X) x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) X))) as proof of ((P (x (f X)))->(P X))
% Found (fun (P:(a->Prop))=> (((fun (x0:(((eq a) (x (f X))) X)) (x00:(((eq a) X) X))=> (((((fun (b0:a)=> ((((eq_trans a) (x (f X))) b0) X)) X) x0) x00) P)) ((eq_ref a) (x (f X)))) ((eq_ref a) X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym100 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_sym10 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found (((eq_sym1 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_sym00 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found (((eq_sym0 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_sym00 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found (((eq_sym0 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq ((b->a)->Prop)) f0) (fun (x:(b->a))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found ((eta_expansion_dep0 (fun (x1:(b->a))=> Prop)) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((b->a)->Prop)) f0) (fun (x:(b->a))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((b->a)->Prop)) f0) (fun (x:(b->a))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((b->a)->Prop)) f0) (fun (x:(b->a))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((b->a)->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((b->a)->Prop)) b0) b00)
% Found ((eq_ref ((b->a)->Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) b00)
% Found ((eq_ref ((b->a)->Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) b00)
% Found ((eq_ref ((b->a)->Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) b00)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f0 x)) (b0 x)))
% Found x0:(P (x (f X)))
% Instantiate: a0:=(x (f X)):a
% Found x0 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))):(((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found (eq_ref0 (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b00)
% Found ((eq_ref ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b00)
% Found ((eq_ref ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b00)
% Found ((eq_ref ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b00)
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X0))->(P1 (f X0)))
% Found (eq_ref00 P1) as proof of (P2 (f X0))
% Found ((eq_ref0 (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X0))->(P1 (f X0)))
% Found (eq_ref00 P1) as proof of (P2 (f X0))
% Found ((eq_ref0 (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X0))->(P1 (f X0)))
% Found (eq_ref00 P1) as proof of (P2 (f X0))
% Found ((eq_ref0 (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found eq_ref00:=(eq_ref0 (f X0)):(((eq b) (f X0)) (f X0))
% Found (eq_ref0 (f X0)) as proof of (((eq b) (f X0)) b0)
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) b0)
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) b0)
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (f X))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X))
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b1)
% Found ((eq_ref a) X) as proof of (((eq a) X) b1)
% Found ((eq_ref a) X) as proof of (((eq a) X) b1)
% Found ((eq_ref a) X) as proof of (((eq a) X) b1)
% Found eq_ref000:=(eq_ref00 P1):((P1 b0)->(P1 b0))
% Found (eq_ref00 P1) as proof of (P2 b0)
% Found ((eq_ref0 b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X))->(P1 (f X)))
% Found (eq_ref00 P1) as proof of (P2 (f X))
% Found ((eq_ref0 (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq b) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref000:=(eq_ref00 P1):((P1 X0)->(P1 X0))
% Found (eq_ref00 P1) as proof of (P2 X0)
% Found ((eq_ref0 X0) P1) as proof of (P2 X0)
% Found (((eq_ref a) X0) P1) as proof of (P2 X0)
% Found (((eq_ref a) X0) P1) as proof of (P2 X0)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X))->(P1 (f X)))
% Found (eq_ref00 P1) as proof of (P2 (f X))
% Found ((eq_ref0 (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X))->(P1 (f X)))
% Found (eq_ref00 P1) as proof of (P2 (f X))
% Found ((eq_ref0 (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found eq_ref000:=(eq_ref00 P1):((P1 X0)->(P1 X0))
% Found (eq_ref00 P1) as proof of (P2 X0)
% Found ((eq_ref0 X0) P1) as proof of (P2 X0)
% Found (((eq_ref a) X0) P1) as proof of (P2 X0)
% Found (((eq_ref a) X0) P1) as proof of (P2 X0)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f b0))->(P1 (f b0)))
% Found (eq_ref00 P1) as proof of (P2 (f b0))
% Found ((eq_ref0 (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found eq_ref00:=(eq_ref0 X0):(((eq a) X0) X0)
% Found (eq_ref0 X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found eq_ref000:=(eq_ref00 P1):((P1 b0)->(P1 b0))
% Found (eq_ref00 P1) as proof of (P2 b0)
% Found ((eq_ref0 b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found eq_ref00:=(eq_ref0 X0):(((eq a) X0) X0)
% Found (eq_ref0 X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f b0))->(P1 (f b0)))
% Found (eq_ref00 P1) as proof of (P2 (f b0))
% Found ((eq_ref0 (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found eq_ref00:=(eq_ref0 (f b0)):(((eq b) (f b0)) (f b0))
% Found (eq_ref0 (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq b) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref00:=(eq_ref0 b00):(((eq b) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found eq_ref00:=(eq_ref0 (f b0)):(((eq b) (f b0)) (f b0))
% Found (eq_ref0 (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f b0))->(P1 (f b0)))
% Found (eq_ref00 P1) as proof of (P2 (f b0))
% Found ((eq_ref0 (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f b0))->(P1 (f b0)))
% Found (eq_ref00 P1) as proof of (P2 (f b0))
% Found ((eq_ref0 (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f b0))->(P1 (f b0)))
% Found (eq_ref00 P1) as proof of (P2 (f b0))
% Found ((eq_ref0 (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found eq_ref00:=(eq_ref0 (f b0)):(((eq b) (f b0)) (f b0))
% Found (eq_ref0 (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq b) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f b0))->(P1 (f b0)))
% Found (eq_ref00 P1) as proof of (P2 (f b0))
% Found ((eq_ref0 (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found (((eq_ref b) (f b0)) P1) as proof of (P2 (f b0))
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym100 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_sym10 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((eq_sym1 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym100 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_sym10 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found (((eq_sym1 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_sym00 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((eq_sym0 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_sym00 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((eq_sym0 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X))->(P1 (f X)))
% Found (eq_ref00 P1) as proof of (P2 (f X))
% Found ((eq_ref0 (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_sym00 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found (((eq_sym0 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_sym00 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found (((eq_sym0 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X0)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found eq_ref00:=(eq_ref0 b00):(((eq b) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq b) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq b) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((b->a)->Prop)) f0) (fun (x:(b->a))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq ((b->a)->Prop)) f0) (fun (x:(b->a))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion (b->a)) Prop) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 f0):(((eq ((b->a)->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found ((eq_ref ((b->a)->Prop)) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found ((eq_ref ((b->a)->Prop)) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found ((eq_ref ((b->a)->Prop)) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq ((b->a)->Prop)) f0) (fun (x:(b->a))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found ((eta_expansion_dep0 (fun (x1:(b->a))=> Prop)) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found (((eta_expansion_dep (b->a)) (fun (x1:(b->a))=> Prop)) f0) as proof of (((eq ((b->a)->Prop)) f0) b0)
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((b->a)->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((b->a)->Prop)) b0) b00)
% Found ((eq_ref ((b->a)->Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) b00)
% Found ((eq_ref ((b->a)->Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) b00)
% Found ((eq_ref ((b->a)->Prop)) b0) as proof of (((eq ((b->a)->Prop)) b0) b00)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f0 x)) (b0 x)))
% Found x0:(P (x (f X)))
% Instantiate: a0:=(x (f X)):a
% Found x0 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj000 ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (finj000 ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((finj00 x0) P0)) ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> (((finj0 X) x0) P0)) ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((((finj (x (f X))) X) x0) P0)) ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((((finj (x (f X))) X) x0) P0)) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (b0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj000 ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (finj000 ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((finj00 x0) P0)) ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> (((finj0 X) x0) P0)) ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((((finj (x (f X))) X) x0) P0)) ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((((finj (x (f X))) X) x0) P0)) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj000 ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (finj000 ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((finj00 x0) P0)) ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> (((finj0 X) x0) P0)) ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((((finj (x (f X))) X) x0) P0)) ((eq_ref b) (f (x (f X))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> ((fun (x0:(((eq b) (f (x (f X)))) (f X)))=> ((((finj (x (f X))) X) x0) P0)) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found eq_ref00:=(eq_ref0 X0):(((eq a) X0) X0)
% Found (eq_ref0 X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(b->a))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(b->a)), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found eq_ref00:=(eq_ref0 X0):(((eq a) X0) X0)
% Found (eq_ref0 X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found ((eq_ref a) X0) as proof of (((eq a) X0) b0)
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref000:=(eq_ref00 P1):((P1 b0)->(P1 b0))
% Found (eq_ref00 P1) as proof of (P2 b0)
% Found ((eq_ref0 b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X0))->(P1 (f X0)))
% Found (eq_ref00 P1) as proof of (P2 (f X0))
% Found ((eq_ref0 (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found eq_ref000:=(eq_ref00 P1):((P1 b0)->(P1 b0))
% Found (eq_ref00 P1) as proof of (P2 b0)
% Found ((eq_ref0 b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))):(((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X))))
% Found (eq_ref0 (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b00)
% Found ((eq_ref ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b00)
% Found ((eq_ref ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b00)
% Found ((eq_ref ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) as proof of (((eq ((b->a)->Prop)) (fun (G:(b->a))=> (forall (X:a), (((eq a) (G (f X))) X)))) b00)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X0))->(P1 (f X0)))
% Found (eq_ref00 P1) as proof of (P2 (f X0))
% Found ((eq_ref0 (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found eq_ref00:=(eq_ref0 b00):(((eq b) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f X))
% Found eq_ref00:=(eq_ref0 (f b0)):(((eq b) (f b0)) (f b0))
% Found (eq_ref0 (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found ((eq_ref b) (f b0)) as proof of (((eq b) (f b0)) b00)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X0))->(P1 (f X0)))
% Found (eq_ref00 P1) as proof of (P2 (f X0))
% Found ((eq_ref0 (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X0))->(P1 (f X0)))
% Found (eq_ref00 P1) as proof of (P2 (f X0))
% Found ((eq_ref0 (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X0))->(P1 (f X0)))
% Found (eq_ref00 P1) as proof of (P2 (f X0))
% Found ((eq_ref0 (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X0))->(P1 (f X0)))
% Found (eq_ref00 P1) as proof of (P2 (f X0))
% Found ((eq_ref0 (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found (((eq_ref b) (f X0)) P1) as proof of (P2 (f X0))
% Found eq_ref00:=(eq_ref0 (f X0)):(((eq b) (f X0)) (f X0))
% Found (eq_ref0 (f X0)) as proof of (((eq b) (f X0)) b0)
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) b0)
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) b0)
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (f X))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X))
% Found eq_ref00:=(eq_ref0 (f X0)):(((eq b) (f X0)) (f X0))
% Found (eq_ref0 (f X0)) as proof of (((eq b) (f X0)) b0)
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) b0)
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) b0)
% Found ((eq_ref b) (f X0)) as proof of (((eq b) (f X0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (f X))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) b1)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b1)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b1)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_trans0000 ((eq_ref a) (x (f X)))) ((eq_ref a) b1)) as proof of (((eq a) (x (f X))) X)
% Found (((eq_trans000 X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b1:a)=> ((eq_trans00 b1) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b1:a)=> (((eq_trans0 (x (f X))) b1) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b1:a)=> ((((eq_trans a) (x (f X))) b1) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b1)
% Found ((eq_ref a) X) as proof of (((eq a) X) b1)
% Found ((eq_ref a) X) as proof of (((eq a) X) b1)
% Found ((eq_ref a) X) as proof of (((eq a) X) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b1)
% Found ((eq_ref a) X) as proof of (((eq a) X) b1)
% Found ((eq_ref a) X) as proof of (((eq a) X) b1)
% Found ((eq_ref a) X) as proof of (((eq a) X) b1)
% Found eq_ref000:=(eq_ref00 P1):((P1 b0)->(P1 b0))
% Found (eq_ref00 P1) as proof of (P2 b0)
% Found ((eq_ref0 b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found eq_ref000:=(eq_ref00 P1):((P1 b0)->(P1 b0))
% Found (eq_ref00 P1) as proof of (P2 b0)
% Found ((eq_ref0 b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found (((eq_ref a) b0) P1) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_trans0000 ((eq_ref a) (x (f X)))) ((eq_ref a) b0)) as proof of (((eq a) (x (f X))) X)
% Found (((eq_trans000 X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b0:a)=> ((eq_trans00 b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b0:a)=> (((eq_trans0 (x (f X))) b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b0:a)=> ((((eq_trans a) (x (f X))) b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found ((eq_ref a) (x (f X))) as proof of (((eq a) (x (f X))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_trans0000 ((eq_ref a) (x (f X)))) ((eq_ref a) b0)) as proof of (((eq a) (x (f X))) X)
% Found (((eq_trans000 X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b0:a)=> ((eq_trans00 b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b0:a)=> (((eq_trans0 (x (f X))) b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found ((((fun (b0:a)=> ((((eq_trans a) (x (f X))) b0) X)) X) ((eq_ref a) (x (f X)))) ((eq_ref a) X)) as proof of (((eq a) (x (f X))) X)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X))->(P1 (f X)))
% Found (eq_ref00 P1) as proof of (P2 (f X))
% Found ((eq_ref0 (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X))->(P1 (f X)))
% Found (eq_ref00 P1) as proof of (P2 (f X))
% Found ((eq_ref0 (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X))->(P1 (f X)))
% Found (eq_ref00 P1) as proof of (P2 (f X))
% Found ((eq_ref0 (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((eq_sym00 (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found (((eq_sym0 X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found ((((eq_sym a) X) (x (f X))) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of (((eq a) (x (f X))) X)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (x (f X))):(((eq a) (x (f X))) (x (f X)))
% Found (eq_ref0 (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((eq_ref a) (x (f X))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref000:=(eq_ref00 P1):((P1 X)->(P1 X))
% Found (eq_ref00 P1) as proof of (P2 X)
% Found ((eq_ref0 X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found (((eq_ref a) X) P1) as proof of (P2 X)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref000:=(eq_ref00 P0):((P0 (x (f X)))->(P0 (x (f X))))
% Found (eq_ref00 P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((eq_ref0 (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (((eq_ref a) (x (f X))) P0) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> (((eq_ref a) (x (f X))) P0)) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym1000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (eq_sym1000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> ((eq_sym100 x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> (((eq_sym10 (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> ((((eq_sym1 X) (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> (((((eq_sym a) X) (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> ((fun (x0:(((eq a) X) (x (f X))))=> (((((eq_sym a) X) (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq b) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq b) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq b) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found ((eq_ref b) b00) as proof of (((eq b) b00) (f b0))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym0000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (eq_sym0000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> ((eq_sym000 x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> (((eq_sym00 (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> ((((eq_sym0 X) (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> (((((eq_sym a) X) (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> ((fun (x0:(((eq a) X) (x (f X))))=> (((((eq_sym a) X) (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj__eq_sym00 ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found ((finj__eq_sym0 X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (((eq a) X) (x (f X)))
% Found (eq_sym0000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (eq_sym0000 (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> ((eq_sym000 x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> (((eq_sym00 (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> ((((eq_sym0 X) (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found ((fun (x0:(((eq a) X) (x (f X))))=> (((((eq_sym a) X) (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X)))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found (fun (P0:(a->Prop))=> ((fun (x0:(((eq a) X) (x (f X))))=> (((((eq_sym a) X) (x (f X))) x0) P0)) (((finj__eq_sym (x (f X))) X) ((eq_ref b) (f (x (f X))))))) as proof of ((P0 (x (f X)))->(P0 X))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) X)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref00:=(eq_ref0 (f (x (f X)))):(((eq b) (f (x (f X)))) (f (x (f X))))
% Found (eq_ref0 (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found ((eq_ref b) (f (x (f X)))) as proof of (((eq b) (f (x (f X)))) (f X))
% Found (finj00 ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found ((finj0 X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found (((finj (x (f X))) X) ((eq_ref b) (f (x (f X))))) as proof of (forall (P:(a->Prop)), ((P (x (f X)))->(P X)))
% Found eq_ref000:=(eq_ref00 P1):((P1 X0)->(P1 X0))
% Found (eq_ref00 P1) as proof of (P2 X0)
% Found ((eq_ref0 X0) P1) as proof of (P2 X0)
% Found (((eq_ref a) X0) P1) as proof of (P2 X0)
% Found (((eq_ref a) X0) P1) as proof of (P2 X0)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X))->(P1 (f X)))
% Found (eq_ref00 P1) as proof of (P2 (f X))
% Found ((eq_ref0 (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found eq_ref000:=(eq_ref00 P1):((P1 X0)->(P1 X0))
% Found (eq_ref00 P1) as proof of (P2 X0)
% Found ((eq_ref0 X0) P1) as proof of (P2 X0)
% Found (((eq_ref a) X0) P1) as proof of (P2 X0)
% Found (((eq_ref a) X0) P1) as proof of (P2 X0)
% Found eq_ref000:=(eq_ref00 P1):((P1 X0)->(P1 X0))
% Found (eq_ref00 P1) as proof of (P2 X0)
% Found ((eq_ref0 X0) P1) as proof of (P2 X0)
% Found (((eq_ref a) X0) P1) as proof of (P2 X0)
% Found (((eq_ref a) X0) P1) as proof of (P2 X0)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X))->(P1 (f X)))
% Found (eq_ref00 P1) as proof of (P2 (f X))
% Found ((eq_ref0 (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X))->(P1 (f X)))
% Found (eq_ref00 P1) as proof of (P2 (f X))
% Found ((eq_ref0 (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found eq_ref000:=(eq_ref00 P1):((P1 (f X))->(P1 (f X)))
% Found (eq_ref00 P1) as proof of (P2 (f X))
% Found ((eq_ref0 (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found (((eq_ref b) (f X)) P1) as proof of (P2 (f X))
% Found eq_ref00:=(eq_ref0 (f X)):(((eq b) (f X)) (f X))
% Found (eq_ref0 (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found ((eq_ref b) (f X)) as proof of (((eq b) (f X)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found ((eq_ref b) b0) as proof of (((eq b) b0) (f X0))
% Found eq_ref000:=(eq_ref00 P1):((P1 X0)->(P1 X0))
% Found (eq_ref00 P1) as proof of (P2 X0)
% Found ((eq_ref0 X0) P1) as proof of (P2 X0)
% Found (((eq_ref a) X0) P1) as proof o
% EOF
%------------------------------------------------------------------------------