Proof with eauto with val_renamings.
intros.
destruct v1, v2; simpl in *; try discriminate.
match goal with
| [H: match ?Expr with _ => _ end = _ |- _] =>
destruct Expr
end; try discriminate.
inversion H...
Qed.

Lemma val_obs_mods_id:
forall f v1 v2 v,
Val.mods v1 v2 = Some v ->
val_obs f v v.