Corollary find_dom : forall A x v dom (m : t A dom), find x m = Some v -> Dom.In x dom.

Theorem add_Some : forall A x y v u dom (m : t A dom), find y (add x v m) = Some u <-> X.eq y x ∧ u = v ∨ ¬X.eq y x ∧ find y m = Some u.