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 *)
(* Hence sincl is well-founded because n-chains to l have length bounded by length l *)

Theorem wf_sincl : well_founded sincl.