Remark select_reg_l_monotone:
  forall r q1 q2,
  OrderedEquation.eq q1 q2 \/ OrderedEquation.lt q1 q2 ->
  select_reg_l r q1 = true -> select_reg_l r q2 = true.Remark select_reg_h_monotone:
  forall r q1 q2,
  OrderedEquation.eq q1 q2 \/ OrderedEquation.lt q2 q1 ->
  select_reg_h r q1 = true -> select_reg_h r q2 = true.Remark select_reg_charact:
  forall r q, select_reg_l r q = true /\ select_reg_h r q = true <-> ereg q = r.

(* Specifications: *)
Lemma reg_unconstrained_sound: forall r e q, reg_unconstrained r e = true -> EqSet.In q e -> ereg q <> r.