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(*         *      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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Local Open Scope nat_scope.
Implicit Types k l p q r : nat.
Section Between.
Variables P Q : nat -> Prop.
(** The [between] type expresses the concept [forall i: nat, k <= i < l -> P i.]. *)
Inductive between k : nat -> Prop :=
| bet_emp : between k k
| bet_S : forall l, between k l -> P l -> between k (S l).
#[local]
Hint Constructors between: core.

(* Specifications: *)
Lemma bet_eq : forall k l, l = k -> between k l.
#[local]
Hint Resolve bet_eq: core.
Lemma between_le : forall k l, between k l -> k <= l.
#[local]
Hint Immediate between_le: core.
Lemma between_Sk_l : forall k l, between k l -> S k <= l -> between (S k) l.
#[local]
Hint Resolve between_Sk_l: core.
Lemma between_restr :
forall k l (m:nat), k <= l -> l <= m -> between k m -> between l m.