Remark in_remove:
forall (A: Type) (eq: forall (x y: A), {x=y}+{x<>y}) x y l,
In y (List.remove eq x l) <-> x <> y /\ In y l.

Lemma forget_reg_charact:
forall n rd r v,
wf_numbering n ->
In r (PMap.get v (forget_reg n rd)) -> r <> rd /\ In r (PMap.get v n.(num_val)).