Definition mbind_dist_pmf {A B : Type} (xM : dist A) (f : A -> dist B) : list (B * Q) :=
  flat_map
   (fun (xq : A * Q) => 
     let (x , q) := xq in 
     (update_probs q (dist_pmf (f x))))
   (dist_pmf xM).Definition mbind_dist {A B : Type} (xM : dist A) (f : A -> dist B) : dist B.#[export] Instance Monad_dist : Monad dist := { mreturn {_} x := mreturn_dist x ; mbind {_ _} := mbind_dist }.Definition coin_flip : dist bool := Dist [ (true, 1 / 2) ; (false , 1 / 2) ] eq_refl.Fixpoint filter_dist {A} (l: list (A * Q))
  (f: A -> bool): list (A * Q) :=
  match l with
  | [] => []
  | h :: t => 
      match h with
        | pair k v => 
            if f k
            then h :: (filter_dist t f)
            else filter_dist t f
      end
  end.

Lemma zero_one_neq : ~ (0 == 1)%Q.