Fact not_divides_eq p x : ~ divides p x <-> exists r, r = rem x p /\ r <> 0.

(* Specifications: *)
Lemma dio_rel_not_divides x p : 𝔻P x -> 𝔻P p -> 𝔻R (fun ν => ~ divides (x ν) (p ν)).
Lemma dio_rel_compose : 𝔻R (fun ν => R (f ν) ν).