(* Binary search

   A classical example. Searches a sorted array for a given value v. *)

module BinarySearch

  use int.Int
  use int.ComputerDivision
  use ref.Ref
  use array.Array

  (* the code and its specification *)

  exception Break int (* raised to exit the loop *)
  exception Not_found (* raised to signal a search failure *)

  val mydiv(x:int):int
  ensures { 2*result = x }

  let binary_search [@bddinfer] [@infer](a : array int) (v : int)
    requires { forall i1 i2 : int. 0 <= i1 <= i2 < length a -> a[i1] <= a[i2] }
    ensures  { 0 <= result < length a /\ a[result] = v }
    raises   { Not_found -> forall i:int. 0 <= i < length a -> a[i] <> v }
  = try
      let l = ref 0 in
	  l := 0;
      let u = ref (length a - 1) in
      while !l <= !u do
        invariant { !u < length a }
        invariant {
          forall i : int. 0 <= i < length a -> a[i] = v -> !l <= i <= !u }
        variant { !u - !l }
        let m = !l + mydiv (!u - !l)  in
        assert { !l <= m <= !u };
        if a[m] < v then
          l := m + 1
        else if a[m] > v then
          u := m - 1
        else
          raise (Break m)
      done;
      raise Not_found
    with Break i ->
      i
    end

  let binary_search2 [@bddinfer] [@infer:oct](a : array int) (v : int)
    requires { forall i1 i2 : int. 0 <= i1 <= i2 < length a -> a[i1] <= a[i2] }
    ensures  { 0 <= result < length a /\ a[result] = v }
    raises   { Not_found -> forall i:int. 0 <= i < length a -> a[i] <> v }
  = try
      let l = ref 0 in
	  l := 0;
      let u = ref (length a - 1) in
      while !l <= !u do
        invariant { !u < length a }
        invariant {
          forall i : int. 0 <= i < length a -> a[i] = v -> !l <= i <= !u }
        variant { !u - !l }
        let m = !l + mydiv (!u - !l)  in
        assert { !l <= m <= !u };
        if a[m] < v then
          l := m + 1
        else if a[m] > v then
          u := m - 1
        else
          raise (Break m)
      done;
      raise Not_found
    with Break i ->
      i
    end

  let binary_search3 [@bddinfer] [@infer:box](a : array int) (v : int)
    requires { forall i1 i2 : int. 0 <= i1 <= i2 < length a -> a[i1] <= a[i2] }
    ensures  { 0 <= result < length a /\ a[result] = v }
    raises   { Not_found -> forall i:int. 0 <= i < length a -> a[i] <> v }
  = try
      let l = ref 0 in
	  l := 0;
      let u = ref (length a - 1) in
      while !l <= !u do
        invariant { !u < length a }
        invariant {
          forall i : int. 0 <= i < length a -> a[i] = v -> !l <= i <= !u }
        variant { !u - !l }
        let m = !l + mydiv (!u - !l)  in
        assert { !l <= m <= !u };
        if a[m] < v then
          l := m + 1
        else if a[m] > v then
          u := m - 1
        else
          raise (Break m)
      done;
      raise Not_found
    with Break i ->
      i
    end

end
