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(*
* The Why certification tool
* Copyright (C) 2002 Jean-Christophe FILLIATRE
*
* This software is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public
* License version 2, as published by the Free Software Foundation.
*
* This software is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
*
* See the GNU General Public License version 2 for more details
* (enclosed in the file GPL).
*)
(* $Id: WhyArraysFMap.v,v 1.2 2008/04/15 15:02:37 regisgia Exp $ *)
(**************************************)
(* Functional arrays, for use in Why. *)
(**************************************)
(*
* The type (array N T) is the type of arrays ranging from 0 to N-1
* which elements are of type T.
*
* Arrays are created with new, accessed with access and modified with update.
*
* Operations of accessing and storing are not guarded, but axioms are.
* So these arrays can be viewed as arrays where accessing and storing
* out of the bounds has no effect.
*)
Require Export WhyInt.
Set Implicit Arguments.
Unset Strict Implicit.
Require Import Coq.FSets.FMapPositive.
Import FMapPositive.PositiveMap.
Require Import Coq.FSets.FMapFacts.
Module F := Coq.FSets.FMapFacts.Facts (FMapPositive.PositiveMap).
Import F.
(* The type of arrays
Arrays are implemented using functional maps of the standard library. *)
Record raw_array (T:Set) : Type := mk_raw_array {
default: T;
carrier: FMapPositive.PositiveMap.t T
}.
Definition array (T:Set) := prod Z (raw_array T).
(* Array length *)
Definition array_length (T:Set) (t:array T) : Z := let (n, _) := t in n.
Definition array_length_ := array_length.
(* Functions to create, access and modify arrays *)
Definition raw_new (T:Set) (x:T) : raw_array T := mk_raw_array x (empty T).
Definition new (T:Set) (n:Z) (a:T) : array T := (n, raw_new a).
Definition set_carrier T x c := mk_raw_array (T:=T) (default x) c.
Hint Unfold set_carrier.
Definition raw_access (T:Set) (x:raw_array T) (n:Z) : T :=
match n with
| Z0 =>
match PositiveMap.find xH (carrier x) with
| Some v => v
| None => default x
end
| Zpos n =>
match PositiveMap.find (Psucc n) (carrier x) with
| Some v => v
| None => default x
end
| _ => default x
end.
Definition access (T:Set) (t:array T) (i:Z) : T :=
let (_, r) := t in raw_access r i.
Hint Unfold access.
Definition raw_update : forall T:Set, raw_array T -> Z -> T -> raw_array T :=
fun T x n v =>
match n with
| Z0 => set_carrier (T:=T) x (add xH v (carrier x))
| Zpos k => set_carrier (T:=T) x (add (Psucc k) v (carrier x))
| _ => x
end.
Definition update (T:Set) (t:array T) (i:Z) (v:T) : array T :=
(array_length t, let (_, r) := t in raw_update r i v).
Hint Unfold update.
Definition elements (T:Set) (x:array T) := elements (carrier (snd x)).
Require Import List.
Definition from_list (T:Set) (default: T) (l: list T) :=
let n := List.length l in
let a := new (Z_of_nat n) default in
snd (List.fold_left (fun (accu : (Z * array T)) => fun x =>
let (i, a) := accu in
((i + 1)%Z, update a i x)) l (0%Z, a)).
(* Update does not change length *)
Lemma array_length_update :
forall (T:Set) (t:array T) (i:Z) (v:T),
array_length (update t i v) = array_length t.
Proof. trivial. Qed.
(* Properties *)
Lemma new_def : forall (T:Set) (n:Z) (v0:T) (i:Z),
(0 <= i < n)%Z ->
access (new n v0) i = v0.
Proof.
intros T n v0 i H;
destruct i; [|simpl; rewrite (gempty T)|]; auto.
Qed.
Lemma update_def_0 :
forall (T:Set) (t:array T) (v:T) (i:Z),
default (snd (update t i v)) = default (snd t).
Proof.
intros T u v i. case_eq u; destruct i; auto.
Qed.
Opaque find add.
Lemma update_def_1 :
forall (T:Set) (t:array T) (v:T) (i:Z),
(0 <= i < array_length t)%Z -> access (update t i v) i = v.
Proof.
intros T a v i H.
destruct i; simpl; case_eq a; intros z r Ha;
simpl;
[ rewrite (find_1 (A:=T) (x:=xH) (e:=v) (m := (add xH v (carrier r))))
| rewrite
(find_1 (A:=T) (x:=Psucc p) (e := v) (m := (add (Psucc p) v (carrier r))))
| assert False; intuition
]; auto; red; apply add_1; unfold E.eq; auto.
Qed.
Lemma aux_find: forall T (t:t T) (i j: positive) (v: T),
i <> j -> PositiveMap.find i t = PositiveMap.find i (add j v t).
Proof.
intros T a i j v Hdiff.
generalize (find_mapsto_iff a i).
generalize (find_mapsto_iff (add j v a) i).
intros H1 H2.
destruct (PositiveMap.find i a) as [ u | ]; [
generalize (add_neq_mapsto_iff a (x:=j) (y:=i) v u)
| destruct (PositiveMap.find i (add j v a)) as [ u | ];
[ generalize (add_neq_mapsto_iff a (x:=j) (y:=i) v u) | ]
]; intuition; firstorder.
Qed.
Lemma update_def_2 :
forall (T:Set) (t:array T) (v:T) (i j:Z),
(0 <= i < array_length t)%Z ->
(0 <= j < array_length t)%Z ->
i <> j -> access (update t i v) j = access t j.
Proof.
intros T a v i j Hi Hj diff;
generalize Psucc_not_one; intros;
destruct i; destruct j; destruct Hi; destruct Hj;
case_eq a; intros z r eq_a; simpl; (congruence; auto with zarith) || idtac;
[ rewrite <- (aux_find (carrier r) v (i := Psucc p) (j := xH))
| rewrite -> (aux_find (carrier r) v (i := xH) (j := Psucc p))
| rewrite -> (aux_find (carrier r) v (i := Psucc p0) (j := Psucc p)) ];
trivial; (congruence || idtac); assert (p <> p0); [ congruence |
intro He; generalize (Psucc_inj _ _ He); congruence
].
Qed.
Hint Resolve new_def update_def_1 update_def_2 : datatypes v62.
(* A tactic to simplify access in arrays *)
Ltac WhyArrays :=
repeat rewrite update_def_1; repeat rewrite array_length_update.
Ltac AccessStore i j H :=
elim (Z_eq_dec i j);
[ intro H; rewrite H; rewrite update_def_1; WhyArrays
| intro H; rewrite update_def_2; [ idtac | idtac | idtac | exact H ] ].
Ltac AccessSame := rewrite update_def_1; WhyArrays; try omega.
Ltac AccessOther := rewrite update_def_2; WhyArrays; try omega.
Ltac ArraySubst t := subst t; simpl; WhyArrays; try omega.
(* Syntax and pretty-print for arrays *)
(* <Warning> : Grammar is replaced by Notation *)
(***
Syntax constr level 0 :
array_access
[ << (access ($VAR $t) $c) >> ] -> [ $t $c:L ].
***)
|
