(* extract a value from a distribution arbitrarily, in this case taking the first elt of the underlying list *)
Definition peek {X} (d : dist X) : X :=
match d with
| {| dist_pmf := dist; dist_law := law |} =>
match dist as l return ((sum_dist l == 1)%Q -> X) with
| [] => fun law =>
match zero_one_neq law with end
| (x,_) :: _ => fun _ => x
end law
end.
(* WARNING: distribution should contain non-zero weighted support *)
Definition dist_lift {X} (P : X -> Prop) (d : dist X) : Prop.

(* Specifications: *)
Lemma dist_lift_ret : forall (X : Type) (x : X) (P : X -> Prop),
P x -> dist_lift P (mreturn x).
Lemma dist_lift_bind (X Y : Type) (P : X -> Prop)
(Q : Y -> Prop) (mx : dist X) (f : X -> dist Y) :
dist_lift P mx ->
(forall x : X, P x -> dist_lift Q (f x)) ->
dist_lift Q (x <- mx;; f x).
Lemma dist_lift_weaken {X} (P Q : X -> Prop) :
(forall x, P x -> Q x) -> forall d,
dist_lift P d -> dist_lift Q d.
Lemma dist_split {X} (P Q : X -> Prop) d :
dist_lift P d ->
dist_lift Q d ->
dist_lift (fun x => P x /\ Q x) d.
Lemma dist_true {X} (d : dist X) : dist_lift (fun _ => True) d.
Lemma dist_comm {X Y} (P : X -> Y -> Prop) (m1 : dist X) (m2 : dist Y) :
dist_lift (fun y => dist_lift (fun x => P x y) m1) m2 ->
dist_lift (fun x => dist_lift (P x) m2) m1.