Module ISF := FSetFacts.Facts(IS).Module ISP := FSetProperties.Properties(IS).Definition ref_fundef (fd: fundef) (id: ident) : Prop :=
  match fd with Internal f => ref_function f id | External ef => False end.Definition ref_init (il: list init_data) (id: ident) : Prop :=
  exists ofs, In (Init_addrof id ofs) il.Definition ref_def (gd: globdef fundef unit) (id: ident) : Prop :=
  match gd with
  | Gfun fd => ref_fundef fd id
  | Gvar gv => ref_init gv.(gvar_init) id
  end.Definition match_prog (p tp: program) : Prop :=
  exists u: IS.t, valid_used_set p u /\ match_prog_1 u p tp.(** * Properties of the static analysis *)

(** Monotonic evolution of the workset. *)

Inductive workset_incl (w1 w2: workset) : Prop :=
  workset_incl_intro:
    forall (SEEN: IS.Subset w1.(w_seen) w2.(w_seen))
           (TODO: List.incl w1.(w_todo) w2.(w_todo))
           (TRACK: forall id, IS.In id w2.(w_seen) ->
                      IS.In id w1.(w_seen) \/ List.In id w2.(w_todo)),
    workset_incl w1 w2.(** * Properties of the static analysis *)

(** Monotonic evolution of the workset. *)

Inductive workset_incl (w1 w2: workset) : Prop :=
  workset_incl_intro:
    forall (SEEN: IS.Subset w1.(w_seen) w2.(w_seen))
           (TODO: List.incl w1.(w_todo) w2.(w_todo))
           (TRACK: forall id, IS.In id w2.(w_seen) ->
                      IS.In id w1.(w_seen) \/ List.In id w2.(w_todo)),
    workset_incl w1 w2.

Theorem used_globals_sound: forall u, used_globals p pm = Some u -> used_set_closed u.