(** * Additional notions over maps *)

  Definition Disjoint (m m' : t elt) :=
   forall k, ~(In k m /\ In k m').Definition Partition (m m1 m2 : t elt) :=
    Disjoint m1 m2 /\
    (forall k e, MapsTo k e m <-> MapsTo k e m1 \/ MapsTo k e m2).(** * Emulation of some functions lacking in the interface *)

  Definition filter (f : key -> elt -> bool)(m : t elt) :=
   fold (fun k e m => if f k e then add k e m else m) m (empty _).Definition for_all (f : key -> elt -> bool)(m : t elt) :=
   fold (fun k e b => if f k e then b else false) m true.Definition exists_ (f : key -> elt -> bool)(m : t elt) :=
   fold (fun k e b => if f k e then true else b) m false.Definition partition (f : key -> elt -> bool)(m : t elt) :=
   (filter f m, filter (fun k e => negb (f k e)) m).(** [update] adds to [m1] all the bindings of [m2]. It can be seen as
     an [union] operator which gives priority to its 2nd argument
     in case of binding conflit. *)

  Definition update (m1 m2 : t elt) := fold (@add _) m2 m1.(** [restrict] keeps from [m1] only the bindings whose key is in [m2].
      It can be seen as an [inter] operator, with priority to its 1st argument
      in case of binding conflit. *)

  Definition restrict (m1 m2 : t elt) := filter (fun k _ => mem k m2) m1.(** [diff] erases from [m1] all bindings whose key is in [m2]. *)

  Definition diff (m1 m2 : t elt) := filter (fun k _ => negb (mem k m2)) m1.

Lemma partition_iff_1 : forall m m1 k e, m1 = fst (partition f m) -> (MapsTo k e m1 <-> MapsTo k e m /\ f k e = true).