Section Add.
Variable A : Type.
(* [Add a l l'] means that [l'] is exactly [l], with [a] added once somewhere *)
Inductive Add (a:A) : list A -> list A -> Prop :=
| Add_head l : Add a l (a::l)
| Add_cons x l l' : Add a l l' -> Add a (x::l) (x::l').

Lemma Add_length a l l' : Add a l l' -> length l' = S (length l).