Section Logs.
(* A log is a map from nat, and a finite log has a latest element. *)
Context {B : Type}.
Local Close Scope Z.
Definition log_latest s (v1 : nat) (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.