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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
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(** * 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]  *)
Definition Cmp (elt:Type)(cmp:elt->elt->bool) e1 e2 := cmp e1 e2 = true.
(** ** 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 : DecidableType).
Definition key := E.t.
#[global]
Hint Transparent key : core.
Parameter t : Type -> Type.
(** the abstract type of maps *)
Section Types.
Variable elt:Type.
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' elt'' : Type.
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.
(** Equality of maps *)
(** Caveat: there are at least three distinct equality predicates on maps. - The simplest (and maybe most natural) way is to consider keys up to their equivalence [E.eq], but elements up to Leibniz equality, in the spirit of [eq_key_elt] above. This leads to predicate [Equal]. - Unfortunately, this [Equal] predicate can't be used to describe the [equal] function, since this function (for compatibility with ocaml) expects a boolean comparison [cmp] that may identify more elements than Leibniz. So logical specification of [equal] is done via another predicate [Equivb] - This predicate [Equivb] is quite ad-hoc with its boolean [cmp], it can be generalized in a [Equiv] expecting a more general (possibly non-decidable) equality predicate on elements *)
Definition Equal m m' := forall y, find y m = find y m'.
Definition Equiv (eq_elt:elt->elt->Prop) m m' :=
(forall k, In k m <-> In k m') /\
(forall k e e', MapsTo k e m -> MapsTo k e' m' -> eq_elt e e').
Definition Equivb (cmp: elt->elt->bool) := Equiv (Cmp cmp).
(** Specification of [equal] *)
Variable cmp : elt -> elt -> bool.
Parameter equal_1 : Equivb cmp m m' -> equal cmp m m' = true.
Parameter equal_2 : equal cmp m m' = true -> Equivb cmp m m'.
End Spec.
End Types.
(** Specification of [map] *)
Parameter map_1 : forall (elt elt':Type)(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':Type)(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':Type)(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':Type)(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'':Type)(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'':Type)(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'.
#[global]
Hint Immediate MapsTo_1 mem_2 is_empty_2
map_2 mapi_2 add_3 remove_3 find_2
: map.
#[global]
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 : DecidableType.
Include WSfun E.
End WS.
(** ** Maps on ordered keys, functorial signature *)
Module Type Sfun (E : OrderedType).
Include WSfun E.
Section elt.
Variable elt:Type.
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 Sfun E.
End S.
(** ** Maps with ordering both on keys and datas *)
Module Type Sord.
Declare Module Data : OrderedType.
Declare Module MapS : S.
Definition t := MapS.t Data.t.
Parameter eq : t -> t -> Prop.
Parameter lt : t -> t -> Prop.

Axiom eq_sym : forall m1 m2 : t, eq m1 m2 -> eq m2 m1.