Local Notation power := (mscal mult 1).Local Notation "∑" := (msum plus 0).Fixpoint df_mconj k f :=
    match k with 
      | 0   => df_true
      | S k => df_conj (f 0) (df_mconj k (fun i => f (S i)))
    end.Fact df_mconj_spec k f ν : df_pred (df_mconj k f) ν <-> forall i, i < k -> df_pred (f i) ν.

(* Specifications: *)
Lemma dio_rel_mconj k (P : nat -> (nat -> nat) -> Prop) : (forall i, i < k -> dio_rel (P i)) -> dio_rel (fun ν => forall i, i < k -> P i ν).