Notation "h1 =~ h2" := (loweq h1 h2) (at level 80).

(* Specifications: *)
Lemma low_refl h : h =~ h.
Lemma low_sym h1 h2 : (h1 =~ h2) = (h2 =~ h1).
Lemma low_trans h2 h1 h3 : h1 =~ h2 -> h2 =~ h3 -> h1 =~ h3.
Lemma loweqP h1 h2 : reflect (ldom h1 =i ldom h2) (h1 =~ h2).
Lemma loweqE h1 h2 : h1 =~ h2 -> lfresh h1 = lfresh h2.