(** * Neededness for register environments *)
Module NVal <: SEMILATTICE.
Definition t := nval.
Definition eq (x y: t) := (x = y).
Definition eq_refl: forall x, eq x x := (@eq_refl t).
Definition eq_sym: forall x y, eq x y -> eq y x := (@eq_sym t).
Definition eq_trans: forall x y z, eq x y -> eq y z -> eq x z := (@eq_trans t).
Definition beq (x y: t) : bool := proj_sumbool (eq_nval x y).

Lemma eagree_update_dead:
forall e1 e2 ne v1 r,
nreg ne r = Nothing ->
eagree e1 e2 ne -> eagree (e1#r <- v1) e2 ne.