Section sincl.
(* Strict inclusion between lists is a well founded relation *)
Variable (X : Type).
Implicit Type (l m : list X).
(* sincl l m if incl l m and there is a witness in m \ l *)
Definition sincl l m := incl l m /\ exists x, ~ In x l /\ In x m.
(* Any n-chain m ~~> l contains a duplication-free subset of l of size n which does not intersect m  *)
(* Hence, by the PHP, if there is a n-chain to l then n must be less than length l *)

Corollary sincl_chain_bounded l m n : chain sincl n m l -> n <= length l.