(** The main operation over environments is update: it assigns
  a value [v] to a register [r] and preserves the values of other
  registers. *)

Definition update (r: reg) (v: val) (e: env) : env :=
  fun r' => if reg_eq r' r then v else e r'.

Lemma update_o: forall r v e r', r' <> r -> update r v e r' = e r'.