Fixpoint raw_partition_between (m: Raw.tree) : Raw.tree * Raw.tree :=
  match m with
  | Raw.Leaf => (Raw.Leaf, Raw.Leaf)
  | Raw.Node _ l x r =>
      if above_low_bound x then
        if below_high_bound x then
          (let (l1, l2) := raw_partition_between l in
           let (r1, r2) := raw_partition_between r in
           (Raw.join l1 x r1, Raw.concat l2 r2))
        else
          (let (l1, l2) := raw_partition_between l in
           (l1, Raw.join l2 x r))
      else
        (let (r1, r2) := raw_partition_between r in
         (r1, Raw.join l x r2))
  end.Remark In_raw_partition_between_1:
  forall x m,
  Raw.In x (fst (raw_partition_between m)) -> Raw.In x m.Remark In_raw_partition_between_2:
  forall x m,
  Raw.In x (snd (raw_partition_between m)) -> Raw.In x m.

Lemma raw_partition_between_ok: forall m, Raw.bst m -> Raw.bst (fst (raw_partition_between m)) /\ Raw.bst (snd (raw_partition_between m)).