(* Pascal

  Prove that the Pascal's triangle indeed computes combinations.
  HINT: https://en.wikipedia.org/wiki/Pascal%27s_triangle
*)

module Pascal

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

 (* The mathematical combination, defined recursively. *)
  let rec function comb (n k: int) : int
    requires { 0 <= k <= n }
    variant  { n }
    ensures  { result >= 1 }
  = if k = 0 || k = n then 1 else comb (n-1) k + comb (n-1) (k-1)

  (* Computes the Pascal's triangle and returns the `n`th row of it. *)
  (* Insert an appropriate invariant so that Why3 can verify this function. *)
  (* You SHOULD understand the Pascal's triangle first to find good invariants. *) 
  let chooses (n : int) : array int
    requires { n > 0 }
    ensures  { forall i: int.
      0 <= i < length result -> result[i] = comb n i }
  =
    let ref row = Array.make 1 1 in
    for r = 1 to n do
      invariant { length row = r }
      invariant { forall c: int. 0 <= c < r -> row[c] = comb (r-1) c } 
      let new_row = Array.make (r+1) 1 in
      for c = 1 to r-1 do
        (* IMPORTANT: DON'T MODIFY THE ABOVE LINES *)
        invariant { true (* TODO: Replace `true` with your solution. Your solution MUST be a single line, at line number 35. DON'T add another lines. *) }
        (* IMPORTANT: DON'T MODIFY THE BELOW LINES *)
        new_row[c] <- row[c-1] + row[c]
      done;
      row <- new_row
    done;
    row
    
end