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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Definition enumeration (A : Type) (u : nat -> A) (v : A -> nat) : Prop :=
(forall x : A, u (v x) = x) /\ (forall n : nat, v (u n) = n).
Definition in_holed_interval (a b hole : R) (u : nat -> R) (n : nat) : Prop :=
Rlt a (u n) /\ Rlt (u n) b /\ u n <> hole.
Definition point_in_holed_interval (a b h : R) : R :=
if Req_EM_T h (Rdiv (Rplus a b) (INR 2)) then (Rdiv (Rplus a h) (INR 2))
else (Rdiv (Rplus a b) (INR 2)).

Lemma middle_in_interval : forall a b : R, Rlt a b -> (a < (a + b) / INR 2 < b)%R.