(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the GNU General Public License as published by  *)
(*  the Free Software Foundation, either version 2 of the License, or  *)
(*  (at your option) any later version.  This file is also distributed *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)
(** Finite sets of integer intervals *)
Module ISet.
(** "Raw", non-dependent implementation.  A set of intervals is a list of nonempty semi-open intervals [(lo, hi)], sorted in increasing order of the low bound, and with no overlap nor adjacency between elements. In particular, if the list contains [(lo1, hi1)] followed by [(lo2, hi2)], we have [lo1 < hi1 < lo2 < hi2]. *)
Module R.
Inductive t : Type := Nil | Cons (lo hi: Z) (tl: t).
Fixpoint In (x: Z) (s: t) :=
match s with
| Nil => False
| Cons l h s' => l <= x < h \/ In x s'
end.
Inductive ok: t -> Prop :=
| ok_nil: ok Nil
| ok_cons: forall l h s
(BOUNDS: l < h)
(BELOW: forall x, In x s -> h < x)
(OK: ok s),
ok (Cons l h s).
Fixpoint mem (x: Z) (s: t) : bool :=
match s with
| Nil => false
| Cons l h s' => if zlt x h then zle l x else mem x s'
end.

Lemma mem_In:
forall x s, ok s -> (mem x s = true <-> In x s).