Program Definition empty_genv (pub: list ident): t :=
  @mkgenv pub (PTree.empty _) (PTree.empty _) 1%positive _ _ _.Definition globalenv (p: program F V) :=
  add_globals (empty_genv p.(prog_public)) p.(prog_defs).

(* Specifications: *)
Lemma add_globals_preserves: forall gl ge, (forall ge id g, P ge -> In (id, g) gl -> P (add_global ge (id, g))) -> P ge -> P (add_globals ge gl).
Lemma add_globals_ensures: forall id g gl ge, (forall ge id g, P ge -> In (id, g) gl -> P (add_global ge (id, g))) -> (forall ge, P (add_global ge (id, g))) -> In (id, g) gl -> P (add_globals ge gl).
Lemma add_globals_unique_preserves: forall id gl ge, (forall ge id1 g, P ge -> In (id1, g) gl -> id1 <> id -> P (add_global ge (id1, g))) -> ~In id (map fst gl) -> P ge -> P (add_globals ge gl).
Lemma add_globals_unique_ensures: forall gl1 id g gl2 ge, (forall ge id1 g1, P ge -> In (id1, g1) gl2 -> id1 <> id -> P (add_global ge (id1, g1))) -> (forall ge, P (add_global ge (id, g))) -> ~In id (map fst gl2) -> P (add_globals ge (gl1 ++ (id, g) :: gl2)).
Lemma add_globals_preserves: forall gl ge, (forall ge id g, P ge -> In (id, g) gl -> P (add_global ge (id, g))) -> P ge -> P (add_globals ge gl).
Lemma add_globals_ensures: forall id g gl ge, (forall ge id g, P ge -> In (id, g) gl -> P (add_global ge (id, g))) -> (forall ge, P (add_global ge (id, g))) -> In (id, g) gl -> P (add_globals ge gl).
Lemma add_globals_unique_preserves: forall id gl ge, (forall ge id1 g, P ge -> In (id1, g) gl -> id1 <> id -> P (add_global ge (id1, g))) -> ~In id (map fst gl) -> P ge -> P (add_globals ge gl).
Lemma add_globals_unique_ensures: forall gl1 id g gl2 ge, (forall ge id1 g1, P ge -> In (id1, g1) gl2 -> id1 <> id -> P (add_global ge (id1, g1))) -> (forall ge, P (add_global ge (id, g))) -> ~In id (map fst gl2) -> P (add_globals ge (gl1 ++ (id, g) :: gl2)).