Corollary insert_at_wf' : forall v p t t' (Hinsert : insert_at v t p = Some t')
      (Hwf : wf_btree t) (Hp : leaf_path t (removelast p)),
      wf_btree t' \/ wf_btree_over t'.

(* Specifications: *)
Lemma wf_weak : forall t, wf_btree_aux t -> wf_btree t.
Lemma find_index_le : forall tgt keys i (Hfind : find_index tgt keys = NotFound i), (i <= length keys)%nat.
Lemma nth_error_succeeds : forall {A} n (l : list A) (Hlt : (n < length l)%nat), exists x, nth_error l n = Some x.
Lemma find_leaf_path : forall v t p (Hfind : find_leaf v t = Some p) (Hwf : wf_btree t), leaf_path t (removelast p).
Lemma insert_wf : forall v t t' (Hinsert : b_insert v t = Some t') (Hwf : wf_btree t), wf_btree t'.
Lemma insert_In2 : forall {A} i (x : A) l, In x (insert i x l).
Lemma insert_at_In : forall v t p t' (Hinsert : insert_at v t p = Some t'), In v (contents t').
Lemma insert_at_sorted : forall v t p (Hp : find_leaf v t = Some p) (Hsorted : sorted (contents t)), exists t', insert_at v t p = Some t' /\ sorted (contents t').
Lemma insert_sorted : forall v t t' (Hinsert : b_insert v t = Some t') (Hsorted : sorted (contents t)), sorted (contents t') /\ In v (contents t').