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 update_reg_charact: forall n rd vd r v, wf_numbering n -> In r (PMap.get v (update_reg n rd vd)) -> PTree.get r (PTree.set rd vd n.(num_reg)) = Some v.