# Finite map library

This file proposes interfaces for finite maps

Require Export Bool DecidableType OrderedType.

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.

• 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.

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.

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 simpliest (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'.

Hint Immediate MapsTo_1 mem_2 is_empty_2
: map.
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 Type WSfun E.
End WS.

## Maps on ordered keys, functorial signature

Module Type Sfun (E : OrderedType).
Include Type WSfun E.
Section elt.
Variable elt:Type.
Definition lt_key (p p':key*elt) := E.lt (fst p) (fst p').
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', Equivb cmp m m' -> eq m m'.
Parameter eq_2 : forall m m', eq m m' -> Equivb 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.