Inductive SearchTree' : key -> tree -> key -> Prop :=
| ST_E : forall lo hi, lo <= hi -> SearchTree' lo E hi
| ST_T: forall lo l k v r hi,
    SearchTree' lo l k ->
    SearchTree' (S k) r hi ->
    SearchTree' lo (T l k v r) hi.Inductive SearchTree' : key -> tree -> key -> Prop :=
| ST_E : forall lo hi, lo <= hi -> SearchTree' lo E hi
| ST_T: forall lo l k v r hi,
    SearchTree' lo l k ->
    SearchTree' (S k) r hi ->
    SearchTree' lo (T l k v r) hi.Inductive SearchTree' : key -> tree -> key -> Prop :=
| ST_E : forall lo hi, lo <= hi -> SearchTree' lo E hi
| ST_T: forall lo l k v r hi,
    SearchTree' lo l k ->
    SearchTree' (S k) r hi ->
    SearchTree' lo (T l k v r) hi.Inductive SearchTree: tree -> Prop :=
| ST_intro: forall t hi, SearchTree' 0 t hi -> SearchTree t.Inductive SearchTree: tree -> Prop :=
| ST_intro: forall t hi, SearchTree' 0 t hi -> SearchTree t.

Lemma list2map_not_in_default: forall (al: list (key*V)) (i: key), ~(exists v, In (i,v) al) -> list2map al i = default.