Inductive Tree (A: Set): Set :=
  | Leaf: A -> Tree A
  | Node: list (Tree A) -> Tree A.Inductive Tree (A: Set): Set :=
  | Leaf: A -> Tree A
  | Node: list (Tree A) -> Tree A.Inductive Tree (A: Set): Set :=
  | Leaf: A -> Tree A
  | Node: list (Tree A) -> Tree A.Definition Tree_ind
  : forall (A: Set) (P : Tree A -> Prop),
    (forall n : A, P (Leaf n)) ->
    (forall l : list (Tree A), (forall t, In t l -> P t) -> P (Node l)) ->
  forall t, P t.Definition liftM (A B: Set) (f: A -> B) (M: Monad) (x: M A): M B :=
    xv <- x ; ret (f xv).Definition liftM2 (A B C: Set) (f: A -> B -> C) (M: Monad) (x: M A) (y: M B): M C :=
    xv <- x ; yv <- y ; ret (f xv yv).Fixpoint foldlM {A B: Set} {M: Monad} (f: A -> B -> M A) (x: A) (l: list B) {struct l}: M A :=
    match l with
    | nil => ret x
    | h :: t => fax <- f x h ; foldlM f fax t
    end.(* Haskell's foldM *)

  Fixpoint foldrM {A B: Set} {M: Monad} (f: B -> A -> M A) (x: A) (l: list B) {struct l}: M A :=
    match l with
    | nil => ret x
    | h :: t => t' <- foldrM f x t; f h t'
    end.

Lemma foldlM_cons (A B: Set) (M: Monad) (f: A -> B -> M A) (x: A) (h: B) (t: list B): foldlM f x (h :: t) = fax <- f x h ; foldlM f fax t.