(***********************************************************************)
(** ** Translating nominal terms to LN terms *)
(***********************************************************************)

(** We decode named terms to LN terms through the use of the
    [close_exp_wrt_exp] function. This function replaces all occurrences of
    a given atom by a new bound variable.

    This function is defined in the [Stlc.Lemmas] module and can be
    automatically generated by LNgen.  *)

Fixpoint nom_to_exp (ne : n_exp) : exp :=
  match ne with
  | n_var x => var_f x
  | n_app e1 e2 => app (nom_to_exp e1) (nom_to_exp e2)
  | n_abs x e1 => abs (close_exp_wrt_exp x (nom_to_exp e1))
end.(* Note that the stack could have some heap definitions in it,
   so we apply the substitutions in the heap there too. *)

Fixpoint apply_stack h (s : list frame) (e :exp) : exp :=
  match s with
  | nil => e
  | n_app2 e' :: s' => apply_stack h s' (app e (apply_heap h (nom_to_exp e')))
  end.(** The full decode function puts all of these parts together. *)

Definition decode (c:configuration) : exp  :=
  match c with
  | (h,e,s) => apply_stack h s (apply_heap h (nom_to_exp e))
  end.Example decode1 : decode conf1 = (app (abs (var_b 0)) (var_f Z)).

Lemma apply_heap_open : forall h e e0, lc_exp e0 -> apply_heap h (open e e0) = open (apply_heap h e) (apply_heap h e0).