Fact dfbfall_spec : forall ν, df_pred dfbfall ν <-> forall i, i < ν 0 -> df_pred f (dv_change ν i).
End df_fall_lt.

Corollary dio_rel_fall_lt_0 (K : nat -> (nat -> nat) -> Prop) :
𝔻R (fun ν => K (ν 0) (fun n => ν (S n)))
-> 𝔻R (fun ν => forall x, x < ν 0 -> K x (fun n => ν (S n))).