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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Certification of Imperative Programs / Jean-Christophe Fillitre *)
(* $Id: Arrays.v,v 1.9.2.1 2004/07/16 19:29:59 herbelin Exp $ *)
(**********************************************)
(* Functional arrays, for use in Correctness. *)
(**********************************************)
(* This is an axiomatization of arrays.
*
* 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 store.
*
* 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 ProgInt.
Set Implicit Arguments.
(* The type of arrays *)
Parameter array : Z -> Set -> Set.
(* Functions to create, access and modify arrays *)
Parameter new : forall (n:Z) (T:Set), T -> array n T.
Parameter access : forall (n:Z) (T:Set), array n T -> Z -> T.
Parameter store : forall (n:Z) (T:Set), array n T -> Z -> T -> array n T.
(* Axioms *)
Axiom
new_def :
forall (n:Z) (T:Set) (v0:T) (i:Z),
(0 <= i < n)%Z -> access (new n v0) i = v0.
Axiom
store_def_1 :
forall (n:Z) (T:Set) (t:array n T) (v:T) (i:Z),
(0 <= i < n)%Z -> access (store t i v) i = v.
Axiom
store_def_2 :
forall (n:Z) (T:Set) (t:array n T) (v:T) (i j:Z),
(0 <= i < n)%Z ->
(0 <= j < n)%Z -> i <> j -> access (store t i v) j = access t j.
Hint Resolve new_def store_def_1 store_def_2: datatypes v62.
(* A tactic to simplify access in arrays *)
Ltac array_access i j H :=
elim (Z_eq_dec i j);
[ intro H; rewrite H; rewrite store_def_1
| intro H; rewrite store_def_2; [ idtac | idtac | idtac | exact H ] ].
(* Symbolic notation for access *)
Notation "# t [ c ]" := (access t c) (at level 0, t at level 0).
|
