Notation "s #1" := (fst s) (at level 9, format "s '#1'") : pair_scope.Notation "s #2" := (snd s) (at level 9, format "s '#2'") : pair_scope.(** Auxiliary function for initialization of global variables. *)

Function store_zeros (m: mem) (b: block) (p: Z) (n: Z) {wf (Zwf 0) n}: option mem :=
  if zle n 0 then Some m else
    match Mem.store Mint8unsigned m b p Vzero with
    | Some m' => store_zeros m' b (p + 1) (n - 1)
    | None => None
    end.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Record t: Type := mksenv {
  (** Operations *)
  find_symbol: ident -> option block;
  public_symbol: ident -> bool;
  invert_symbol: block -> option ident;
  block_is_volatile: block -> bool;
  nextblock: block;
  (** Properties *)
  find_symbol_injective:
    forall id1 id2 b, find_symbol id1 = Some b -> find_symbol id2 = Some b -> id1 = id2;
  invert_find_symbol:
    forall id b, invert_symbol b = Some id -> find_symbol id = Some b;
  find_invert_symbol:
    forall id b, find_symbol id = Some b -> invert_symbol b = Some id;
  public_symbol_exists:
    forall id, public_symbol id = true -> exists b, find_symbol id = Some b;
  find_symbol_below:
    forall id b, find_symbol id = Some b -> Plt b nextblock;
  block_is_volatile_below:
    forall b, block_is_volatile b = true -> Plt b nextblock
}.Definition symbol_address (ge: t) (id: ident) (ofs: ptrofs) : val :=
  match find_symbol ge id with
  | Some b => Vptr b ofs
  | None => Vundef
  end.

Theorem shift_symbol_address: forall ge id ofs delta, symbol_address ge id (Ptrofs.add ofs delta) = Val.offset_ptr (symbol_address ge id ofs) delta.