Proof with eauto with val_renamings.
intros.
destruct v...
destruct Hvalid as [b' Hf].
specialize (Hid _ _ Hf);
subst...
Qed.

Lemma ren_cmp_bool:
forall f v v' v0 cmp,
val_obs f v v' ->
Val.cmp_bool cmp v v0 = Val.cmp_bool cmp v' v0.