Library Coq.FSets.FMapWeakInterface
This file proposes an interface for finite maps over keys with decidable
equality, but no decidable order.
Set Implicit Arguments.
Unset Strict Implicit.
Require Import FSetInterface.
Require Import FSetWeakInterface.
Module Type S.
Declare Module E : DecidableType.
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 raises Not_found if no such binding exists.
NB: in Coq, the exception mechanism becomes a option type.
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.
Coq comment:
iter is useless in a purely functional world
val iter : (key -> 'a -> unit) -> 'a t -> unit
iter f m applies f to all bindings in map m. f receives the key as
first argument, and the associated value as second argument.
The bindings are passed to f in increasing order with respect to the
ordering over the type of the keys. Only current bindings are
presented to f: bindings hidden by more recent bindings are not
passed to f.
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value a of all bindings of m has been replaced by the result of the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. The bindings are passed to f in
increasing order with respect to the ordering over the type of the
keys.
Parameter mapi : (key -> elt -> elt') -> t elt -> t elt'.
Same as
S.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''.
Not present in Ocaml.
map 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).
Not present in Ocaml.
elements m returns an assoc list corresponding to the bindings of m.
Elements of this list are sorted with respect to their first components.
Useful to specify fold ...
Parameter fold : forall A: Set, (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 increasing 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.
Parameter elements_3 : NoDupA eq_key (elements m).
Specification of
fold
Parameter fold_1 :
forall (A : Set) (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.
Hint Resolve mem_1 is_empty_1 is_empty_2 add_1 add_2 add_3 remove_1
remove_2 remove_3 find_1 find_2 fold_1 map_1 map_2 mapi_1 mapi_2.
End S.