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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(* $Id$ *)
(** * Finite map library *)
(** This file proposes interfaces for finite maps *)
Require Export Bool DecidableType OrderedType.
Set Implicit Arguments.
Unset Strict Implicit.
(** When compared with Ocaml Map, this signature has been split in
several parts :
- The first parts [WSfun] and [WS] propose signatures for weak
maps, which are maps with no ordering on the key type nor the
data type. [WSfun] and [WS] are almost identical, apart from the
fact that [WSfun] is expressed in a functorial way whereas [WS]
is self-contained. For obtaining an instance of such signatures,
a decidable equality on keys in enough (see for example
[FMapWeakList]). These signatures contain the usual operators
(add, find, ...). The only function that asks for more is
[equal], whose first argument should be a comparison on data.
- Then comes [Sfun] and [S], that extend [WSfun] and [WS] to the
case where the key type is ordered. The main novelty is that
[elements] is required to produce sorted lists.
- Finally, [Sord] extends [S] with a complete comparison function. For
that, the data type should have a decidable total ordering as well.
If unsure, what you're looking for is probably [S]: apart from [Sord],
all other signatures are subsets of [S].
Some additional differences with Ocaml:
- no [iter] function, useless since Coq is purely functional
- [option] types are used instead of [Not_found] exceptions
- more functions are provided: [elements] and [cardinal] and [map2]
*)
(** ** Weak signature for maps
No requirements for an ordering on keys nor elements, only decidability
of equality on keys. First, a functorial signature: *)
Module Type WSfun (E : EqualityType).
(** The module E of base objects is meant to be a [DecidableType]
(and used to be so). But requiring only an [EqualityType] here
allows subtyping between weak and ordered maps. *)
Definition key := E.t.
Parameter t : Set -> Set.
(** the abstract type of maps *)
Section Types.
Variable elt:Set.
Parameter empty : t elt.
(** The empty map. *)
Parameter is_empty : t elt -> bool.
(** Test whether a map is empty or not. *)
Parameter add : key -> elt -> t elt -> t elt.
(** [add x y m] returns a map containing the same bindings as [m],
plus a binding of [x] to [y]. If [x] was already bound in [m],
its previous binding disappears. *)
Parameter find : key -> t elt -> option elt.
(** [find x m] returns the current binding of [x] in [m],
or [None] if no such binding exists. *)
Parameter remove : key -> t elt -> t elt.
(** [remove x m] returns a map containing the same bindings as [m],
except for [x] which is unbound in the returned map. *)
Parameter mem : key -> t elt -> bool.
(** [mem x m] returns [true] if [m] contains a binding for [x],
and [false] otherwise. *)
Variable elt' : Set.
Variable elt'': Set.
Parameter map : (elt -> elt') -> t elt -> t elt'.
(** [map f m] returns a map with same domain as [m], where the associated
value a of all bindings of [m] has been replaced by the result of the
application of [f] to [a]. Since Coq is purely functional, the order
in which the bindings are passed to [f] is irrelevant. *)
Parameter mapi : (key -> elt -> elt') -> t elt -> t elt'.
(** Same as [map], but the function receives as arguments both the
key and the associated value for each binding of the map. *)
Parameter map2 :
(option elt -> option elt' -> option elt'') -> t elt -> t elt' -> t elt''.
(** [map2 f m m'] creates a new map whose bindings belong to the ones
of either [m] or [m']. The presence and value for a key [k] is
determined by [f e e'] where [e] and [e'] are the (optional) bindings
of [k] in [m] and [m']. *)
Parameter elements : t elt -> list (key*elt).
(** [elements m] returns an assoc list corresponding to the bindings
of [m], in any order. *)
Parameter cardinal : t elt -> nat.
(** [cardinal m] returns the number of bindings in [m]. *)
Parameter fold : forall A: Type, (key -> elt -> A -> A) -> t elt -> A -> A.
(** [fold f m a] computes [(f kN dN ... (f k1 d1 a)...)],
where [k1] ... [kN] are the keys of all bindings in [m]
(in any order), and [d1] ... [dN] are the associated data. *)
Parameter equal : (elt -> elt -> bool) -> t elt -> t elt -> bool.
(** [equal cmp m1 m2] tests whether the maps [m1] and [m2] are equal,
that is, contain equal keys and associate them with equal data.
[cmp] is the equality predicate used to compare the data associated
with the keys. *)
Section Spec.
Variable m m' m'' : t elt.
Variable x y z : key.
Variable e e' : elt.
Parameter MapsTo : key -> elt -> t elt -> Prop.
Definition In (k:key)(m: t elt) : Prop := exists e:elt, MapsTo k e m.
Definition Empty m := forall (a : key)(e:elt) , ~ MapsTo a e m.
Definition eq_key (p p':key*elt) := E.eq (fst p) (fst p').
Definition eq_key_elt (p p':key*elt) :=
E.eq (fst p) (fst p') /\ (snd p) = (snd p').
(** Specification of [MapsTo] *)
Parameter MapsTo_1 : E.eq x y -> MapsTo x e m -> MapsTo y e m.
(** Specification of [mem] *)
Parameter mem_1 : In x m -> mem x m = true.
Parameter mem_2 : mem x m = true -> In x m.
(** Specification of [empty] *)
Parameter empty_1 : Empty empty.
(** Specification of [is_empty] *)
Parameter is_empty_1 : Empty m -> is_empty m = true.
Parameter is_empty_2 : is_empty m = true -> Empty m.
(** Specification of [add] *)
Parameter add_1 : E.eq x y -> MapsTo y e (add x e m).
Parameter add_2 : ~ E.eq x y -> MapsTo y e m -> MapsTo y e (add x e' m).
Parameter add_3 : ~ E.eq x y -> MapsTo y e (add x e' m) -> MapsTo y e m.
(** Specification of [remove] *)
Parameter remove_1 : E.eq x y -> ~ In y (remove x m).
Parameter remove_2 : ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).
Parameter remove_3 : MapsTo y e (remove x m) -> MapsTo y e m.
(** Specification of [find] *)
Parameter find_1 : MapsTo x e m -> find x m = Some e.
Parameter find_2 : find x m = Some e -> MapsTo x e m.
(** Specification of [elements] *)
Parameter elements_1 :
MapsTo x e m -> InA eq_key_elt (x,e) (elements m).
Parameter elements_2 :
InA eq_key_elt (x,e) (elements m) -> MapsTo x e m.
(** When compared with ordered maps, here comes the only
property that is really weaker: *)
Parameter elements_3w : NoDupA eq_key (elements m).
(** Specification of [cardinal] *)
Parameter cardinal_1 : cardinal m = length (elements m).
(** Specification of [fold] *)
Parameter fold_1 :
forall (A : Type) (i : A) (f : key -> elt -> A -> A),
fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (elements m) i.
Definition Equal cmp m m' :=
(forall k, In k m <-> In k m') /\
(forall k e e', MapsTo k e m -> MapsTo k e' m' -> cmp e e' = true).
Variable cmp : elt -> elt -> bool.
(** Specification of [equal] *)
Parameter equal_1 : Equal cmp m m' -> equal cmp m m' = true.
Parameter equal_2 : equal cmp m m' = true -> Equal cmp m m'.
End Spec.
End Types.
(** Specification of [map] *)
Parameter map_1 : forall (elt elt':Set)(m: t elt)(x:key)(e:elt)(f:elt->elt'),
MapsTo x e m -> MapsTo x (f e) (map f m).
Parameter map_2 : forall (elt elt':Set)(m: t elt)(x:key)(f:elt->elt'),
In x (map f m) -> In x m.
(** Specification of [mapi] *)
Parameter mapi_1 : forall (elt elt':Set)(m: t elt)(x:key)(e:elt)
(f:key->elt->elt'), MapsTo x e m ->
exists y, E.eq y x /\ MapsTo x (f y e) (mapi f m).
Parameter mapi_2 : forall (elt elt':Set)(m: t elt)(x:key)
(f:key->elt->elt'), In x (mapi f m) -> In x m.
(** Specification of [map2] *)
Parameter map2_1 : forall (elt elt' elt'':Set)(m: t elt)(m': t elt')
(x:key)(f:option elt->option elt'->option elt''),
In x m \/ In x m' ->
find x (map2 f m m') = f (find x m) (find x m').
Parameter map2_2 : forall (elt elt' elt'':Set)(m: t elt)(m': t elt')
(x:key)(f:option elt->option elt'->option elt''),
In x (map2 f m m') -> In x m \/ In x m'.
Hint Immediate MapsTo_1 mem_2 is_empty_2
map_2 mapi_2 add_3 remove_3 find_2
: map.
Hint Resolve mem_1 is_empty_1 is_empty_2 add_1 add_2 remove_1
remove_2 find_1 fold_1 map_1 mapi_1 mapi_2
: map.
End WSfun.
(** ** Static signature for Weak Maps
Similar to [WSfun] but expressed in a self-contained way. *)
Module Type WS.
Declare Module E : EqualityType.
Include Type WSfun E.
End WS.
(** ** Maps on ordered keys, functorial signature *)
Module Type Sfun (E : OrderedType).
Include Type WSfun E.
Section elt.
Variable elt:Set.
Definition lt_key (p p':key*elt) := E.lt (fst p) (fst p').
(* Additional specification of [elements] *)
Parameter elements_3 : forall m, sort lt_key (elements m).
(** Remark: since [fold] is specified via [elements], this stronger
specification of [elements] has an indirect impact on [fold],
which can now be proved to receive elements in increasing order. *)
End elt.
End Sfun.
(** ** Maps on ordered keys, self-contained signature *)
Module Type S.
Declare Module E : OrderedType.
Include Type Sfun E.
End S.
(** ** Maps with ordering both on keys and datas *)
Module Type Sord.
Declare Module Data : OrderedType.
Declare Module MapS : S.
Import MapS.
Definition t := MapS.t Data.t.
Parameter eq : t -> t -> Prop.
Parameter lt : t -> t -> Prop.
Axiom eq_refl : forall m : t, eq m m.
Axiom eq_sym : forall m1 m2 : t, eq m1 m2 -> eq m2 m1.
Axiom eq_trans : forall m1 m2 m3 : t, eq m1 m2 -> eq m2 m3 -> eq m1 m3.
Axiom lt_trans : forall m1 m2 m3 : t, lt m1 m2 -> lt m2 m3 -> lt m1 m3.
Axiom lt_not_eq : forall m1 m2 : t, lt m1 m2 -> ~ eq m1 m2.
Definition cmp e e' := match Data.compare e e' with EQ _ => true | _ => false end.
Parameter eq_1 : forall m m', Equal cmp m m' -> eq m m'.
Parameter eq_2 : forall m m', eq m m' -> Equal cmp m m'.
Parameter compare : forall m1 m2, Compare lt eq m1 m2.
(** Total ordering between maps. [Data.compare] is a total ordering
used to compare data associated with equal keys in the two maps. *)
End Sord.
|
