End FOR_ALL_BETWEEN.
Section PARTITION_BETWEEN.
Variable above_low_bound: elt -> bool.
Variable below_high_bound: elt -> bool.
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)).