(* *********************************************************************)
(*                                                                     *)
(*              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 INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)
(** The abstract domains for value analysis *)
Inductive block_class : Type :=
| BCinvalid
| BCglob (id: ident)
| BCstack
| BCother.
Definition block_class_eq: forall (x y: block_class), {x=y} + {x<>y}.
Hint Extern 2 (_ = _) => congruence : va.
Hint Extern 2 (_ <> _) => congruence : va.
Hint Extern 2 (_ < _) => xomega : va.
Hint Extern 2 (_ <= _) => xomega : va.
Hint Extern 2 (_ > _) => xomega : va.
Hint Extern 2 (_ >= _) => xomega : va.
Section MATCH.
Variable bc: block_classification.
(** * Abstracting the result of conditions (type [option bool]) *)
Inductive abool :=
| Bnone               (**r always [None] (undefined) *)
| Just (b: bool)      (**r always [Some b] (defined and known to be [b]) *)
| Maybe (b: bool)     (**r either [None] or [Some b] (known to be [b] if defined) *)
| Btop.               (**r unknown, all results are possible *)
Inductive cmatch: option bool -> abool -> Prop :=
| cmatch_none: cmatch None Bnone
| cmatch_just: forall b, cmatch (Some b) (Just b)
| cmatch_maybe_none: forall b, cmatch None (Maybe b)
| cmatch_maybe_some: forall b, cmatch (Some b) (Maybe b)
| cmatch_top: forall ob, cmatch ob Btop.
Hint Constructors cmatch : va.
Definition club (x y: abool) : abool :=
match x, y with
| Bnone, Bnone => Bnone
| Bnone, (Just b | Maybe b) => Maybe b
| (Just b | Maybe b), Bnone => Maybe b
| Just b1, Just b2 => if eqb b1 b2 then x else Btop
| Maybe b1, Maybe b2 => if eqb b1 b2 then x else Btop
| Maybe b1, Just b2 => if eqb b1 b2 then x else Btop
| Just b1, Maybe b2 => if eqb b1 b2 then y else Btop
| _, _ => Btop
end.

(* Specifications: *)
Lemma cmatch_lub_l:
forall ob x y, cmatch ob x -> cmatch ob (club x y).
Lemma cmatch_lub_r:
forall ob x y, cmatch ob y -> cmatch ob (club x y).