End ListMaps.
Section Logs.
(* A log is a map from Z, and a finite log has a latest element. *)
Context {B : Type}.
Definition log_latest s (v1 : Z) (v2 : B) := s v1 = Some v2 /\ forall v', v1 < v' -> s v' = None.

Lemma log_incl_latest : forall k1 k2 v1 v2 log1 log2 (Hincl : map_incl log1 log2)
(Hv1 : log1 k1 = Some v1) (Hlatest : log_latest log2 k2 v2), k1 <= k2.