(* Max

  Given an array `a` of natural numbers with length `n`, 
  return the maximum element of the array.
  
  You should stengthen the loop invariant.
*)

module Max

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

  let max (a: array int) (n: int) : (max: int)
    requires { n = length a }
    requires { forall i. 0 <= i < n -> a[i] >= 0 }
    ensures  { forall i. 0 <= i < n -> a[i] <= max }
    ensures { exists i. 0 <= i < n -> a[i] = max }
  = let ref max = 0 in
    for i = 0 to n - 1 do
      (* IMPORTANT: DON'T MODIFY THE ABOVE LINES *)
      invariant { true (* TODO: Replace `true` with your solution *) }
      (* IMPORTANT: DON'T MODIFY THE BELOW LINES *)
      if max < a[i] then max <- a[i];
    done;
    max

end