Definition equiv (se1 se2: t) : Prop :=
     (forall id, find_symbol se2 id = find_symbol se1 id)
  /\ (forall id, public_symbol se2 id = public_symbol se1 id)
  /\ (forall b, block_is_volatile se2 b = block_is_volatile se1 b).(**r The type of information attached to variables *)

(** The type of global environments. *)

Record t: Type := mkgenv {
  genv_public: list ident;              (**r which symbol names are public *)
  genv_symb: PTree.t block;             (**r mapping symbol -> block *)
  genv_defs: PTree.t (globdef F V);     (**r mapping block -> definition *)
  genv_next: block;                     (**r next symbol pointer *)
  genv_symb_range: forall id b, PTree.get id genv_symb = Some b -> Plt b genv_next;
  genv_defs_range: forall b g, PTree.get b genv_defs = Some g -> Plt b genv_next;
  genv_vars_inj: forall id1 id2 b,
    PTree.get id1 genv_symb = Some b -> PTree.get id2 genv_symb = Some b -> id1 = id2
}.(**r The type of information attached to variables *)

(** The type of global environments. *)

Record t: Type := mkgenv {
  genv_public: list ident;              (**r which symbol names are public *)
  genv_symb: PTree.t block;             (**r mapping symbol -> block *)
  genv_defs: PTree.t (globdef F V);     (**r mapping block -> definition *)
  genv_next: block;                     (**r next symbol pointer *)
  genv_symb_range: forall id b, PTree.get id genv_symb = Some b -> Plt b genv_next;
  genv_defs_range: forall b g, PTree.get b genv_defs = Some g -> Plt b genv_next;
  genv_vars_inj: forall id1 id2 b,
    PTree.get id1 genv_symb = Some b -> PTree.get id2 genv_symb = Some b -> id1 = id2
}.(**r The type of information attached to variables *)

(** The type of global environments. *)

Record t: Type := mkgenv {
  genv_public: list ident;              (**r which symbol names are public *)
  genv_symb: PTree.t block;             (**r mapping symbol -> block *)
  genv_defs: PTree.t (globdef F V);     (**r mapping block -> definition *)
  genv_next: block;                     (**r next symbol pointer *)
  genv_symb_range: forall id b, PTree.get id genv_symb = Some b -> Plt b genv_next;
  genv_defs_range: forall b g, PTree.get b genv_defs = Some g -> Plt b genv_next;
  genv_vars_inj: forall id1 id2 b,
    PTree.get id1 genv_symb = Some b -> PTree.get id2 genv_symb = Some b -> id1 = id2
}.(**r The type of information attached to variables *)

(** The type of global environments. *)

Record t: Type := mkgenv {
  genv_public: list ident;              (**r which symbol names are public *)
  genv_symb: PTree.t block;             (**r mapping symbol -> block *)
  genv_defs: PTree.t (globdef F V);     (**r mapping block -> definition *)
  genv_next: block;                     (**r next symbol pointer *)
  genv_symb_range: forall id b, PTree.get id genv_symb = Some b -> Plt b genv_next;
  genv_defs_range: forall b g, PTree.get b genv_defs = Some g -> Plt b genv_next;
  genv_vars_inj: forall id1 id2 b,
    PTree.get id1 genv_symb = Some b -> PTree.get id2 genv_symb = Some b -> id1 = id2
}.(**r The type of information attached to variables *)

(** The type of global environments. *)

Record t: Type := mkgenv {
  genv_public: list ident;              (**r which symbol names are public *)
  genv_symb: PTree.t block;             (**r mapping symbol -> block *)
  genv_defs: PTree.t (globdef F V);     (**r mapping block -> definition *)
  genv_next: block;                     (**r next symbol pointer *)
  genv_symb_range: forall id b, PTree.get id genv_symb = Some b -> Plt b genv_next;
  genv_defs_range: forall b g, PTree.get b genv_defs = Some g -> Plt b genv_next;
  genv_vars_inj: forall id1 id2 b,
    PTree.get id1 genv_symb = Some b -> PTree.get id2 genv_symb = Some b -> id1 = id2
}.(**r The type of information attached to variables *)

(** The type of global environments. *)

Record t: Type := mkgenv {
  genv_public: list ident;              (**r which symbol names are public *)
  genv_symb: PTree.t block;             (**r mapping symbol -> block *)
  genv_defs: PTree.t (globdef F V);     (**r mapping block -> definition *)
  genv_next: block;                     (**r next symbol pointer *)
  genv_symb_range: forall id b, PTree.get id genv_symb = Some b -> Plt b genv_next;
  genv_defs_range: forall b g, PTree.get b genv_defs = Some g -> Plt b genv_next;
  genv_vars_inj: forall id1 id2 b,
    PTree.get id1 genv_symb = Some b -> PTree.get id2 genv_symb = Some b -> id1 = id2
}.(**r The type of information attached to variables *)

(** The type of global environments. *)

Record t: Type := mkgenv {
  genv_public: list ident;              (**r which symbol names are public *)
  genv_symb: PTree.t block;             (**r mapping symbol -> block *)
  genv_defs: PTree.t (globdef F V);     (**r mapping block -> definition *)
  genv_next: block;                     (**r next symbol pointer *)
  genv_symb_range: forall id b, PTree.get id genv_symb = Some b -> Plt b genv_next;
  genv_defs_range: forall b g, PTree.get b genv_defs = Some g -> Plt b genv_next;
  genv_vars_inj: forall id1 id2 b,
    PTree.get id1 genv_symb = Some b -> PTree.get id2 genv_symb = Some b -> id1 = id2
}.(**r The type of information attached to variables *)

(** The type of global environments. *)

Record t: Type := mkgenv {
  genv_public: list ident;              (**r which symbol names are public *)
  genv_symb: PTree.t block;             (**r mapping symbol -> block *)
  genv_defs: PTree.t (globdef F V);     (**r mapping block -> definition *)
  genv_next: block;                     (**r next symbol pointer *)
  genv_symb_range: forall id b, PTree.get id genv_symb = Some b -> Plt b genv_next;
  genv_defs_range: forall b g, PTree.get b genv_defs = Some g -> Plt b genv_next;
  genv_vars_inj: forall id1 id2 b,
    PTree.get id1 genv_symb = Some b -> PTree.get id2 genv_symb = Some b -> id1 = id2
}.(**r The type of information attached to variables *)

(** The type of global environments. *)

Record t: Type := mkgenv {
  genv_public: list ident;              (**r which symbol names are public *)
  genv_symb: PTree.t block;             (**r mapping symbol -> block *)
  genv_defs: PTree.t (globdef F V);     (**r mapping block -> definition *)
  genv_next: block;                     (**r next symbol pointer *)
  genv_symb_range: forall id b, PTree.get id genv_symb = Some b -> Plt b genv_next;
  genv_defs_range: forall b g, PTree.get b genv_defs = Some g -> Plt b genv_next;
  genv_vars_inj: forall id1 id2 b,
    PTree.get id1 genv_symb = Some b -> PTree.get id2 genv_symb = Some b -> id1 = id2
}.(** ** Lookup functions *)

(** [find_symbol ge id] returns the block associated with the given name, if any *)

Definition find_symbol (ge: t) (id: ident) : option block :=
  PTree.get id ge.(genv_symb).(** [symbol_address ge id ofs] returns a pointer into the block associated
  with [id], at byte offset [ofs].  [Vundef] is returned if no block is associated
  to [id]. *)

Definition symbol_address (ge: t) (id: ident) (ofs: ptrofs) : val :=
  match find_symbol ge id with
  | Some b => Vptr b ofs
  | None => Vundef
  end.(** [public_symbol ge id] says whether the name [id] is public and defined. *)

Definition public_symbol (ge: t) (id: ident) : bool :=
  match find_symbol ge id with
  | None => false
  | Some _ => In_dec ident_eq id ge.(genv_public)
  end.(** [find_def ge b] returns the global definition associated with the given address. *)

Definition find_def (ge: t) (b: block) : option (globdef F V) :=
  PTree.get b ge.(genv_defs).(** [find_funct_ptr ge b] returns the function description associated with
    the given address. *)

Definition find_funct_ptr (ge: t) (b: block) : option F :=
  match find_def ge b with Some (Gfun f) => Some f | _ => None end.(** [find_funct] is similar to [find_funct_ptr], but the function address
    is given as a value, which must be a pointer with offset 0. *)

Definition find_funct (ge: t) (v: val) : option F :=
  match v with
  | Vptr b ofs => if Ptrofs.eq_dec ofs Ptrofs.zero then find_funct_ptr ge b else None
  | _ => None
  end.(** [invert_symbol ge b] returns the name associated with the given block, if any *)

Definition invert_symbol (ge: t) (b: block) : option ident :=
  PTree.fold
    (fun res id b' => if eq_block b b' then Some id else res)
    ge.(genv_symb) None.(** [find_var_info ge b] returns the information attached to the variable
   at address [b]. *)

Definition find_var_info (ge: t) (b: block) : option (globvar V) :=
  match find_def ge b with Some (Gvar v) => Some v | _ => None end.(** [block_is_volatile ge b] returns [true] if [b] points to a global variable
  of volatile type, [false] otherwise. *)

Definition block_is_volatile (ge: t) (b: block) : bool :=
  match find_var_info ge b with
  | None => false
  | Some gv => gv.(gvar_volatile)
  end.(** ** Constructing the global environment *)

Program Definition add_global (ge: t) (idg: ident * globdef F V) : t :=
  @mkgenv
    ge.(genv_public)
    (PTree.set idg#1 ge.(genv_next) ge.(genv_symb))
    (PTree.set ge.(genv_next) idg#2 ge.(genv_defs))
    (Pos.succ ge.(genv_next))
    _ _ _.Definition add_globals (ge: t) (gl: list (ident * globdef F V)) : t :=
  List.fold_left add_global gl ge.

Lemma add_globals_app: forall gl2 gl1 ge, add_globals ge (gl1 ++ gl2) = add_globals (add_globals ge gl1) gl2.