Hint Resolve dio_rel_fall_lt_bound.

Theorem dio_rel_fall_le a (K : nat -> (nat -> nat) -> Prop) :
𝔻P a
-> 𝔻R (fun ν => K (ν 0) (fun n => ν (S n)))
-> 𝔻R (fun ν => forall x, x <= a ν -> K x ν).