# Finite sets library : conversion to old Finite_sets

Require Import Ensembles Finite_sets.
Require Import FSetInterface FSetProperties OrderedTypeEx DecidableTypeEx.

# Going from FSets with usual Leibniz equality

to the good old Ensembles and Finite_sets theory.

Module WS_to_Finite_set (U:UsualDecidableType)(M: WSfun U).
Module MP:= WProperties_fun U M.
Import M MP FM Ensembles Finite_sets.

Definition mkEns : M.t -> Ensemble M.elt :=
fun s x => M.In x s.

Notation " !! " := mkEns.

Lemma In_In : forall s x, M.In x s <-> In _ (!!s) x.

Lemma Subset_Included : forall s s', s[<=]s' <-> Included _ (!!s) (!!s').

Notation " a === b " := (Same_set M.elt a b) (at level 70, no associativity).

Lemma Equal_Same_set : forall s s', s[=]s' <-> !!s === !!s'.

Lemma empty_Empty_Set : !!M.empty === Empty_set _.

Lemma Empty_Empty_set : forall s, Empty s -> !!s === Empty_set _.

Lemma singleton_Singleton : forall x, !!(M.singleton x) === Singleton _ x .

Lemma union_Union : forall s s', !!(union s s') === Union _ (!!s) (!!s').

Lemma inter_Intersection : forall s s', !!(inter s s') === Intersection _ (!!s) (!!s').

Lemma remove_Subtract : forall x s, !!(remove x s) === Subtract _ (!!s) x.

Lemma mkEns_Finite : forall s, Finite _ (!!s).

Lemma mkEns_cardinal : forall s, cardinal _ (!!s) (M.cardinal s).

we can even build a function from Finite Ensemble to FSet ... at least in Prop.

Lemma Ens_to_FSet : forall e : Ensemble M.elt, Finite _ e ->
exists s:M.t, !!s === e.

End WS_to_Finite_set.

Module S_to_Finite_set (U:UsualOrderedType)(M: Sfun U) :=
WS_to_Finite_set U M.