(** Size of integers, in bits. *)

Definition Zsize (x: Z) : Z :=
  match x with
  | Zpos p => Zpos (Pos.size p)
  | _ => 0
  end.Definition size (x: int) : Z := Zsize (unsigned x).Remark Zsize_pos: forall x, 0 <= Zsize x.Remark Zsize_pos': forall x, 0 < x -> 0 < Zsize x.

(* Specifications: *)
Lemma Zsize_shiftin: forall b x, 0 < x -> Zsize (Zshiftin b x) = Z.succ (Zsize x).
Lemma Ztestbit_size_1: forall x, 0 < x -> Z.testbit x (Z.pred (Zsize x)) = true.
Lemma Ztestbit_size_2: forall x, 0 <= x -> forall i, i >= Zsize x -> Z.testbit x i = false.
Lemma Zsize_interval_1: forall x, 0 <= x -> 0 <= x < two_p (Zsize x).
Lemma Zsize_interval_2: forall x n, 0 <= n -> 0 <= x < two_p n -> n >= Zsize x.
Lemma Zsize_monotone: forall x y, 0 <= x <= y -> Zsize x <= Zsize y.
Theorem size_zero: size zero = 0.
Theorem bits_size_1: forall x, x = zero \/ testbit x (Z.pred (size x)) = true.
Theorem bits_size_2: forall x i, size x <= i -> testbit x i = false.
Theorem size_range: forall x, 0 <= size x <= zwordsize.
Theorem bits_size_3: forall x n, 0 <= n -> (forall i, n <= i < zwordsize -> testbit x i = false) -> size x <= n.
Theorem bits_size_4: forall x n, 0 <= n -> testbit x (Z.pred n) = true -> (forall i, n <= i < zwordsize -> testbit x i = false) -> size x = n.
Theorem size_interval_1: forall x, 0 <= unsigned x < two_p (size x).
Theorem size_interval_2: forall x n, 0 <= n -> 0 <= unsigned x < two_p n -> n >= size x.
Theorem size_and: forall a b, size (and a b) <= Z.min (size a) (size b).
Lemma Zsize_shiftin: forall b x, 0 < x -> Zsize (Zshiftin b x) = Z.succ (Zsize x).
Lemma Ztestbit_size_1: forall x, 0 < x -> Z.testbit x (Z.pred (Zsize x)) = true.
Lemma Ztestbit_size_2: forall x, 0 <= x -> forall i, i >= Zsize x -> Z.testbit x i = false.
Lemma Zsize_interval_1: forall x, 0 <= x -> 0 <= x < two_p (Zsize x).
Lemma Zsize_interval_2: forall x n, 0 <= n -> 0 <= x < two_p n -> n >= Zsize x.
Lemma Zsize_monotone: forall x y, 0 <= x <= y -> Zsize x <= Zsize y.
Theorem size_zero: size zero = 0.
Theorem bits_size_1: forall x, x = zero \/ testbit x (Z.pred (size x)) = true.
Theorem bits_size_2: forall x i, size x <= i -> testbit x i = false.
Theorem size_range: forall x, 0 <= size x <= zwordsize.
Theorem bits_size_3: forall x n, 0 <= n -> (forall i, n <= i < zwordsize -> testbit x i = false) -> size x <= n.
Theorem bits_size_4: forall x n, 0 <= n -> testbit x (Z.pred n) = true -> (forall i, n <= i < zwordsize -> testbit x i = false) -> size x = n.
Theorem size_interval_1: forall x, 0 <= unsigned x < two_p (size x).
Theorem size_interval_2: forall x n, 0 <= n -> 0 <= unsigned x < two_p n -> n >= size x.
Theorem size_and: forall a b, size (and a b) <= Z.min (size a) (size b).