(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \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.

(* Here we use axiom total_order_T, which is not constructive *)
Lemma in_holed_interval_dec (a b h : R) (u : nat -> R) (n : nat)
: {in_holed_interval a b h u n} + {~in_holed_interval a b h u n}.