Fact rem_equiv p x r : r = rem x p <-> (p = 0 /\ x = r)
\/ (p <> 0 /\ r < p /\ exists n, x = n*p + r).

Lemma dio_rel_congruence x y p : 𝔻P x -> 𝔻P y -> 𝔻P p
-> 𝔻R (fun ν => rem (x ν) (p ν) = rem (y ν) (p ν)).