Section MEM_BETWEEN.
(** [mem_between above below s] is [true] iff there exists an element of [s] that belongs to the interval described by the predicates [above] and [below]. Using the monotonicity of [above] and [below], the implementation of [mem_between] avoids traversing the subtrees of [s] that lie entirely outside the interval of interest. *)
Variable above_low_bound: elt -> bool.
Variable below_high_bound: elt -> bool.
Fixpoint raw_mem_between (m: Raw.tree) : bool :=
match m with
| Raw.Leaf => false
| Raw.Node _ l x r =>
if above_low_bound x
then if below_high_bound x
then true
else raw_mem_between l
else raw_mem_between r
end.
Definition mem_between (m: t) : bool := raw_mem_between m.(MSet.this).
Hypothesis above_monotone:
forall x1 x2, X.eq x1 x2 \/ X.lt x1 x2 -> above_low_bound x1 = true -> above_low_bound x2 = true.
Hypothesis below_monotone:
forall x1 x2, X.eq x1 x2 \/ X.lt x2 x1 -> below_high_bound x1 = true -> below_high_bound x2 = true.

Theorem mem_between_1:
forall s,
mem_between s = true ->
exists x, In x s /\ above_low_bound x = true /\ below_high_bound x = true.