Notation nthe n A := (nth_error A n).

(* Specifications: *)
Lemma nthe_length A n :
length A > n -> { x | nthe n A = Some x }.
Lemma pos_nthe x A n :
pos x A = Some n -> nthe n A = Some x.
Lemma nthe_app_l x n A B :
nthe n A = Some x -> nthe n (A ++ B) = Some x.
Lemma nthe_length A n :
length A > n -> { x | nthe n A = Some x }.
Lemma pos_nthe x A n :
pos x A = Some n -> nthe n A = Some x.
Lemma nthe_app_l x n A B :
nthe n A = Some x -> nthe n (A ++ B) = Some x.