(* Two programs to compute the factorial

   Questions:

   1. In module FactRecursive:

      a. Implement the program that satisfies specification.

   2. In module FactLoop:

      a. Strengthen the invariant to prove correctness of the given implementation.

      b. Select a correct variant to prove the termination.
     
*)

module FactRecursive

  use int.Int
  use int.Fact

  let rec fact_rec (n: int) : int
    requires { n >= 0 }
    ensures  { result = fact n }
    variant { n }
  = (* IMPORTANT: DON'T MODIFY THE ABOVE LINES *) 
    0 (*TODO*)
    
end

module FactLoop

  use int.Int
  use int.Fact

  let fact_loop (n: int) : int
    requires { 0 < n }
    ensures  { result = fact n }
  = let ref m = 0 in
    let ref r = 1 in
    while m < n do
      invariant { true (* TODO: Replace `true` with your solution  *) }
      variant { n (* TODO: Replace `n` with your solution  *) }
      m <- m + 1;
      r <- r * m
    done;
    r

end