Definition nat_of := fun (x : atom) => x.(** We make [atom], [fresh], [fresh_not_in] and [atom_fresh_for_list] available
    without qualification. *)

Notation atom := Atom.atom.Notation fresh := Atom.fresh.Notation fresh_not_in := Atom.fresh_not_in.Notation atom_fresh_for_list := Atom.atom_fresh_for_list.(** It is trivial to declare an instance of [EqDec] for [atom]. *)

Instance EqDec_atom : @EqDec atom eq eq_equivalence.(* ********************************************************************** *)
(** * Finite sets of atoms *)

(** We use our implementation of atoms to obtain an implementation of
    finite sets of atoms.  We give the resulting type an intuitive
    name, as well as import names of set operations for use within
    this library.  In order to avoid polluting Coq's namespace, we do
    not use [Module Export]. *)

Module Import AtomSetImpl : FSetExtra.WSfun Atom :=
  FSetExtra.Make Atom.Notation atoms :=
  AtomSetImpl.t.(** The [AtomSetDecide] module provides the [fsetdec] tactic for
    solving facts about finite sets of atoms. *)

Module Export AtomSetDecide := Coq.FSets.FSetDecide.WDecide_fun Atom AtomSetImpl.(** The [AtomSetNotin] module provides the [destruct_notin] and
    [solve_notin] for reasoning about non-membership in finite sets of
    atoms, as well as a variety of lemmas about non-membership. *)

Module Export AtomSetNotin := FSetWeakNotin.Notin_fun Atom AtomSetImpl.(** Given the [fsetdec] tactic, we typically do not need to refer to
    specific lemmas about finite sets.  However, instantiating
    functors from the FSets library makes a number of setoid rewrites
    available.  These rewrites are crucial to developments since they
    allow us to replace a set with an extensionally equal set (see the
    [Equal] relation on finite sets) in propositions about finite
    sets. *)

Module AtomSetFacts := FSetFacts.WFacts_fun Atom AtomSetImpl.Module AtomSetProperties := FSetProperties.WProperties_fun Atom AtomSetImpl.

Lemma atom_fresh : forall L : atoms, { x : atom | ~ In x L }.