Definition PdivBound bound p base := Pdivless bound p base p.

Theorem PdivBound_correct : forall bound p base, 1 < nat_of_P base -> nat_of_P p = oZ (fst (fst (PdivBound bound p base))) * exp (nat_of_P base) (snd (PdivBound bound p base)) + oZ (snd (fst (PdivBound bound p base))) /\ (oZ (fst (fst (PdivBound bound p base))) < nat_of_P bound /\ oZ (snd (fst (PdivBound bound p base))) < exp (nat_of_P base) (snd (PdivBound bound p base))) /\ (forall bound', nat_of_P bound = nat_of_P base * bound' -> nat_of_P bound <= nat_of_P p -> nat_of_P bound <= nat_of_P base * oZ (fst (fst (PdivBound bound p base)))).