
(** {1 Minimum excludant, aka mex}

   Author: Jean-Christophe Filliâtre (CNRS)

   Given a finite set of integers, find the smallest nonnegative integer
   that does not belong to this set.

   In the following, the set is given as an array A.
   If N is the size of this array, it is clear that we have 0 <= mex <= N
   for we cannot have the N+1 first natural numbers in the N cells of
   array A (pigeon hole principle).
*)

(**
   A simple algorithm is thus to mark values that belong to [0..N[ in
   some external Boolean array of length N, ignoring any value that is
   negative or greater or equal than N. Then a second loop scans the
   marks until we find some unused value. If don't find any, then it means
   that A contains exactly the integers 0,...,N-1 and the answer is N.

   The very last step in this reasoning requires to invoke the pigeon hole
   principle (imported from the standard library).

   Time O(N) and space O(N).
*)
module MexArray

  use int.Int
  use map.Map
  use array.Array
  use ref.Refint
  use pigeon.Pigeonhole

  predicate mem (x: int) (a: array int) =
    exists i. 0 <= i < length a && a[i] = x

  let mex (a: array int) : int
    ensures { 0 <= result <= length a }
    ensures { not (mem result a) }
    ensures { forall x. 0 <= x < result -> mem x a }
  = let n = length a in
    let used = make n false in
    let ghost idx = ref (fun i -> i) in (* the position of each marked value *)
    for i = 0 to n - 1 do
      invariant { forall x. 0 <= x < n -> used[x] ->
                    mem x a && 0 <= !idx x < n && a[!idx x] = x }
      invariant { forall j. 0 <= j < i -> 0 <= a[j] < n ->
                    used[a[j]] && 0 <= !idx a[j] < n && a[!idx a[j]] = a[j] }
      let x = a[i] in
      if 0 <= x && x < n then begin used[x] <- true; idx := set !idx x i end
    done;
    let r = ref 0 in
    let ghost posn = ref (-1) in
    while !r < n && used[!r] do
      invariant { 0 <= !r <= n }
      invariant { forall j. 0 <= j < !r -> used[j] && 0 <= !idx j < n }
      invariant { if !posn >= 0 then 0 <= !posn < n && a[!posn] = n
                                else forall j. 0 <= j < !r -> a[j] <> n }
      variant   { n - !r }
      if a[!r] = n then posn := !r;
      incr r
    done;
    (* we cannot have !r=n (all values marked) and !posn>=0 at the same time *)
    if !r = n && !posn >= 0 then pigeonhole (n+1) n (set !idx n !posn);
    !r

end

(**
  In this second implementation, we assume we are free to mutate array A.

  The idea is then to scan the array from left to right, while
  swapping elements to put any value in 0..N-1 at its place in the
  array.  When we are done, a second loop looks for the mex, advancing
  as long as a[i]=i holds.

  Since we perform only swaps, it is obvious that the mex of the final
  array is equal to the mex of the original array.

  Time O(N) and space O(1). The argument for a linear time complexity is as
  follows: whenever we do not advance, we swap the element to its place,
  which is further and did not contain that element; so we can do this only
  N times. Note that the variant below is bounded by 2N and thus shows
  the linear complexity.

  Surprinsingly, proving that N is the answer whenever array A contains a
  permutation of 0..N-1 is now easy (no need for a pigeon hole
  principle or any kind of proof by induction).
 *)
module MexArrayInPlace

  use int.Int
  use int.NumOf
  use array.Array
  use array.ArraySwap
  use ref.Refint

  predicate mem (x: int) (a: array int) =
    exists i. 0 <= i < length a && a[i] = x

  function placed (a: array int) : int -> bool =
    fun i -> a[i] = i

  let mex (a: array int) : int
    ensures { 0 <= result <= length a }
    ensures { not (mem result (old a)) }
    ensures { forall x. 0 <= x < result -> mem x (old a) }
  = let n = length a in
    let i = ref 0 in
    while !i < n do
      invariant { 0 <= !i <= n }
      invariant { forall x. mem x a <-> mem x (old a) }
      invariant { forall j. 0 <= j < !i -> 0 <= a[j] < n -> a[a[j]] = a[j] }
      variant   { n - !i + n - numof (placed a) 0 n }
      let x = a[!i] in
      if x < 0 || x >= n || a[x] = x then
        incr i
      else if x < !i then begin
        swap a !i x; incr i
      end else
        swap a !i x
    done;
    assert { forall j. 0 <= j < n -> let x = (old a)[j] in
             0 <= x < n -> a[x] = x };
    for i = 0 to n - 1 do
      invariant { forall j. 0 <= j < i -> a[j] = j }
      if a[i] <> i then return i
    done;
    n

end

