(** Once we have normalized, we will apply this instance to simplify the problem. *)
  
  Definition proper_flip_proper `(mor : Proper A R m) : Proper (flip R) m := mor.

(* Specifications: *)
Lemma reflexive_proper `{Reflexive A R} (x : A) : Proper R x.
Lemma proper_eq {A} (x : A) : Proper (@eq A) x.
Lemma proper_eq (x : A) : Proper (@eq A) x.