Remark In_raw_partition_between_3:
  forall x m,
  Raw.In x (fst (raw_partition_between m)) -> above_low_bound x = true /\ below_high_bound x = true.Remark In_raw_partition_between_4:
  forall x m,
  Raw.bst m ->
  Raw.In x (snd (raw_partition_between m)) -> above_low_bound x = false \/ below_high_bound x = false.Remark In_raw_partition_between_5:
  forall x m,
  Raw.bst m ->
  Raw.In x m -> above_low_bound x = true -> below_high_bound x = true ->
  Raw.In x (fst (raw_partition_between m)).Remark In_raw_partition_between_6:
  forall x m,
  Raw.bst m ->
  Raw.In x m -> above_low_bound x = false \/ below_high_bound x = false ->
  Raw.In x (snd (raw_partition_between m)).Definition partition_between (s: t) : t * t :=
  let p := raw_partition_between s.(MSet.this) in
  (@MSet.Mkt (fst p) (proj1 (raw_partition_between_ok s.(MSet.this) s.(MSet.is_ok))),
   @MSet.Mkt (snd p) (proj2 (raw_partition_between_ok s.(MSet.this) s.(MSet.is_ok)))).

(* Specifications: *)
Theorem partition_between_iff_1: forall x s, In x (fst (partition_between s)) <-> In x s /\ above_low_bound x = true /\ below_high_bound x = true.
Theorem partition_between_iff_2: forall x s, In x (snd (partition_between s)) <-> In x s /\ (above_low_bound x = false \/ below_high_bound x = false).
Theorem partition_between_iff_1: forall x s, In x (fst (partition_between s)) <-> In x s /\ above_low_bound x = true /\ below_high_bound x = true.
Theorem partition_between_iff_2: forall x s, In x (snd (partition_between s)) <-> In x s /\ (above_low_bound x = false \/ below_high_bound x = false).