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_latest_singleton : forall v1 v2, log_latest (singleton v1 v2) v1 v2.