From: Jim Pryor <profjim@jimpryor.net>
Date: Sat, 11 Dec 2010 00:00:14 +0000 (-0500)
Subject: push monads library
X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=341631c0ca1850cc0e2bdaa459fc1bdc771a3175

push monads library

Signed-off-by: Jim Pryor <profjim@jimpryor.net>
---

diff --git a/code/monads.ml b/code/monads.ml
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+(*
+ * monads.ml
+ *
+ * Relies on features introduced in OCaml 3.12
+ *
+ * This library uses parameterized modules, see tree_monadize.ml for
+ * more examples and explanation.
+ *
+ * Some comparisons with the Haskell monadic libraries, which we mostly follow:
+ * In Haskell, the Reader 'a monadic type would be defined something like this:
+ *     newtype Reader a = Reader { runReader :: env -> a }
+ * (For simplicity, I'm suppressing the fact that Reader is also parameterized
+ * on the type of env.)
+ * This creates a type wrapper around `env -> a`, so that Haskell will
+ * distinguish between values that have been specifically designated as
+ * being of type `Reader a`, and common-garden values of type `env -> a`.
+ * To lift an aribtrary expression E of type `env -> a` into an `Reader a`,
+ * you do this:
+ *     Reader { runReader = E }
+ * or use any of the following equivalent shorthands:
+ *     Reader (E)
+ *     Reader $ E
+ * To drop an expression R of type `Reader a` back into an `env -> a`, you do
+ * one of these:
+ *     runReader (R)
+ *     runReader $ R
+ * The `newtype` in the type declaration ensures that Haskell does this all
+ * efficiently: though it regards E and R as type-distinct, their underlying
+ * machine implementation is identical and doesn't need to be transformed when
+ * lifting/dropping from one type to the other.
+ *
+ * Now, you _could_ also declare monads as record types in OCaml, too, _but_
+ * doing so would introduce an extra level of machine representation, and
+ * lifting/dropping from the one type to the other wouldn't be free like it is
+ * in Haskell.
+ *
+ * This library encapsulates the monadic types in another way: by
+ * making their implementations private. The interpreter won't let
+ * let you freely interchange the `'a Reader_monad.m`s defined below
+ * with `Reader_monad.env -> 'a`. The code in this library can see that
+ * those are equivalent, but code outside the library can't. Instead, you'll 
+ * have to use operations like `run` to convert the abstract monadic types
+ * to types whose internals you have free access to.
+ *
+ *)
+
+
+(* Some library functions used below. *)
+module Util = struct
+  let fold_right = List.fold_right
+  let map = List.map
+  let append = List.append
+  let reverse = List.rev
+  let concat = List.concat
+  let concat_map f lst = List.concat (List.map f lst)
+  (* let zip = List.combine *)
+  let unzip = List.split
+  let zip_with = List.map2
+  let replicate len fill =
+    let rec loop n accu =
+      if n == 0 then accu else loop (pred n) (fill :: accu)
+    in loop len []
+end
+
+
+
+(*
+ * This module contains factories that extend a base set of
+ * monadic definitions with a larger family of standard derived values.
+ *)
+
+module Monad = struct
+
+  (* type of base definitions *)
+  module type BASE = sig
+    (*
+     * The only constraints we impose here on how the monadic type
+     * is implemented is that it have a single type parameter 'a.
+     *)
+    type 'a m
+    val unit : 'a -> 'a m
+    val bind : 'a m -> ('a -> 'b m) -> 'b m
+    type 'a result
+    val run : 'a m -> 'a result
+    (* run_exn tries to provide a more ground-level result, but may fail *)
+    type 'a result_exn
+    val run_exn : 'a m -> 'a result_exn
+  end
+  module type S = sig
+    include BASE
+    val (>>=) : 'a m -> ('a -> 'b m) -> 'b m
+    val (>>) : 'a m -> 'b m -> 'b m
+    val join : ('a m) m -> 'a m
+    val apply : ('a -> 'b) m -> 'a m -> 'b m
+    val lift : ('a -> 'b) -> 'a m -> 'b m
+    val lift2 :  ('a -> 'b -> 'c) -> 'a m -> 'b m -> 'c m
+    val (>=>) : ('a -> 'b m) -> ('b -> 'c m) -> 'a -> 'c m
+    val do_when :  bool -> unit m -> unit m
+    val do_unless :  bool -> unit m -> unit m
+    val forever : 'a m -> 'b m
+    val sequence : 'a m list -> 'a list m
+    val sequence_ : 'a m list -> unit m
+  end
+
+  (* Standard, single-type-parameter monads. *)
+  module Make(B : BASE) : S with type 'a m = 'a B.m and type 'a result = 'a B.result and type 'a result_exn = 'a B.result_exn = struct
+    include B
+    let (>>=) = bind
+    let (>>) u v = u >>= fun _ -> v
+    let lift f u = u >>= fun a -> unit (f a)
+    (* lift is called listM, fmap, and <$> in Haskell *)
+    let join uu = uu >>= fun u -> u
+    (* u >>= f === join (lift f u) *)
+    let apply u v = u >>= fun f -> v >>= fun a -> unit (f a)
+    (* [f] <*> [x1,x2] = [f x1,f x2] *)
+    (* let apply u v = u >>= fun f -> lift f v *)
+    (* let apply = lift2 id *)
+    let lift2 f u v = u >>= fun a -> v >>= fun a' -> unit (f a a')
+    (* let lift f u === apply (unit f) u *)
+    (* let lift2 f u v = apply (lift f u) v *)
+
+    let (>=>) f g = fun a -> f a >>= g
+    let do_when test u = if test then u else unit ()
+    let do_unless test u = if test then unit () else u
+    let rec forever u = u >> forever u
+    let sequence ms =
+      let op u v = u >>= fun x -> v >>= fun xs -> unit (x :: xs) in
+        Util.fold_right op ms (unit [])
+    let sequence_ ms =
+      Util.fold_right (>>) ms (unit ())
+
+    (* Haskell defines these other operations combining lists and monads.
+     * We don't, but notice that M.mapM == ListT(M).distribute
+     * There's also a parallel TreeT(M).distribute *)
+    (*
+    let mapM f alist = sequence (Util.map f alist)
+    let mapM_ f alist = sequence_ (Util.map f alist)
+    let rec filterM f lst = match lst with
+      | [] -> unit []
+      | x::xs -> f x >>= fun flag -> filterM f xs >>= fun ys -> unit (if flag then x :: ys else ys)
+    let forM alist f = mapM f alist
+    let forM_ alist f = mapM_ f alist
+    let map_and_unzipM f xs = sequence (Util.map f xs) >>= fun x -> unit (Util.unzip x)
+    let zip_withM f xs ys = sequence (Util.zip_with f xs ys)
+    let zip_withM_ f xs ys = sequence_ (Util.zip_with f xs ys)
+    let rec foldM f z lst = match lst with
+      | [] -> unit z
+      | x::xs -> f z x >>= fun z' -> foldM f z' xs
+    let foldM_ f z xs = foldM f z xs >> unit ()
+    let replicateM n x = sequence (Util.replicate n x)
+    let replicateM_ n x = sequence_ (Util.replicate n x)
+    *)
+  end
+
+  (* Single-type-parameter monads that also define `plus` and `zero`
+   * operations. These obey the following laws:
+   *     zero >>= f   ===  zero
+   *     plus zero u  ===  u
+   *     plus u zero  ===  u
+   * Additionally, these monads will obey one of the following laws:
+   *     (Catch)   plus (unit a) v  ===  unit a
+   *     (Distrib) plus u v >>= f   ===  plus (u >>= f) (v >>= f)
+   *)
+  module type PLUSBASE = sig
+    include BASE
+    val zero : unit -> 'a m
+    val plus : 'a m -> 'a m -> 'a m
+  end
+  module type PLUS = sig
+    type 'a m
+    val zero : unit -> 'a m
+    val plus : 'a m -> 'a m -> 'a m
+    val guard : bool -> unit m
+    val sum : 'a m list -> 'a m
+  end
+  (* MakeCatch and MakeDistrib have the same implementation; we just declare
+   * them twice to document which laws the client code is promising to honor. *)
+  module MakeCatch(B : PLUSBASE) : PLUS with type 'a m = 'a B.m = struct
+    type 'a m = 'a B.m
+    let zero = B.zero
+    let plus = B.plus
+    let guard test = if test then B.unit () else zero ()
+    let sum ms = Util.fold_right plus ms (zero ())
+  end
+  module MakeDistrib = MakeCatch
+
+  (* We have to define BASE, S, and Make again for double-type-parameter monads. *)
+  module type BASE2 = sig
+    type ('x,'a) m
+    val unit : 'a -> ('x,'a) m
+    val bind : ('x,'a) m -> ('a -> ('x,'b) m) -> ('x,'b) m
+    type ('x,'a) result
+    val run : ('x,'a) m -> ('x,'a) result
+    type ('x,'a) result_exn
+    val run_exn : ('x,'a) m -> ('x,'a) result
+  end
+  module type S2 = sig
+    include BASE2
+    val (>>=) : ('x,'a) m -> ('a -> ('x,'b) m) -> ('x,'b) m
+    val (>>) : ('x,'a) m -> ('x,'b) m -> ('x,'b) m
+    val join : ('x,('x,'a) m) m -> ('x,'a) m
+    val apply : ('x,'a -> 'b) m -> ('x,'a) m -> ('x,'b) m
+    val lift : ('a -> 'b) -> ('x,'a) m -> ('x,'b) m
+    val lift2 :  ('a -> 'b -> 'c) -> ('x,'a) m -> ('x,'b) m -> ('x,'c) m
+    val (>=>) : ('a -> ('x,'b) m) -> ('b -> ('x,'c) m) -> 'a -> ('x,'c) m
+    val do_when :  bool -> ('x,unit) m -> ('x,unit) m
+    val do_unless :  bool -> ('x,unit) m -> ('x,unit) m
+    val forever : ('x,'a) m -> ('x,'b) m
+    val sequence : ('x,'a) m list -> ('x,'a list) m
+    val sequence_ : ('x,'a) m list -> ('x,unit) m
+  end
+  module Make2(B : BASE2) : S2 with type ('x,'a) m = ('x,'a) B.m and type ('x,'a) result = ('x,'a) B.result and type ('x,'a) result_exn = ('x,'a) B.result_exn = struct
+    include B
+    let (>>=) = bind
+    let (>>) u v = u >>= fun _ -> v
+    let lift f u = u >>= fun a -> unit (f a)
+    let join uu = uu >>= fun u -> u
+    let apply u v = u >>= fun f -> v >>= fun a -> unit (f a)
+    let lift2 f u v = u >>= fun a -> v >>= fun a' -> unit (f a a')
+    let (>=>) f g = fun a -> f a >>= g
+    let do_when test u = if test then u else unit ()
+    let do_unless test u = if test then unit () else u
+    let rec forever u = u >> forever u
+    let sequence ms =
+      let op u v = u >>= fun x -> v >>= fun xs -> unit (x :: xs) in
+        Util.fold_right op ms (unit [])
+    let sequence_ ms =
+      Util.fold_right (>>) ms (unit ())
+  end
+
+  (* Signatures for MonadT *)
+  module type W = sig
+    include S
+  end
+  module type WP = sig
+    include W
+    val zero : unit -> 'a m
+    val plus : 'a m -> 'a m -> 'a m
+  end
+  module type TRANS = sig
+    type 'a m
+    val bind : 'a m -> ('a -> 'b m) -> 'b m
+    module Wrapped : W
+    type 'a result
+    val run : 'a m -> 'a result
+    type 'a result_exn
+    val run_exn : 'a m -> 'a result_exn
+    val elevate : 'a Wrapped.m -> 'a m
+    (* lift/elevate laws:
+     *     elevate (W.unit a) == unit a
+     *     elevate (W.bind w f) == elevate w >>= fun a -> elevate (f a)
+     *)
+  end
+  module MakeT(T : TRANS) = struct
+    include Make(struct
+        include T
+        let unit a = elevate (Wrapped.unit a)
+    end)
+    let elevate = T.elevate
+  end
+
+end
+
+
+
+
+
+module Identity_monad : sig
+  (* expose only the implementation of type `'a result` *)
+  type 'a result = 'a
+  type 'a result_exn = 'a
+  include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+end = struct
+  module Base = struct
+    type 'a m = 'a
+    let unit a = a
+    let bind a f = f a
+    type 'a result = 'a
+    let run a = a
+    type 'a result_exn = 'a
+    let run_exn a = a
+  end
+  include Monad.Make(Base)
+end
+
+
+module Maybe_monad : sig
+  (* expose only the implementation of type `'a result` *)
+  type 'a result = 'a option
+  type 'a result_exn = 'a
+  include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+  include Monad.PLUS with type 'a m := 'a m
+  (* MaybeT transformer *)
+  module T : functor (Wrapped : Monad.W) -> sig
+    type 'a result = 'a option Wrapped.result
+    type 'a result_exn = 'a Wrapped.result_exn
+    include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+    include Monad.PLUS with type 'a m := 'a m
+    val elevate : 'a Wrapped.m -> 'a m
+  end
+end = struct
+  module Base = struct
+   type 'a m = 'a option
+   let unit a = Some a
+   let bind u f = match u with Some a -> f a | None -> None
+   type 'a result = 'a option
+   let run u = u
+   type 'a result_exn = 'a
+   let run_exn u = match u with
+     | Some a -> a
+     | None -> failwith "no value"
+   let zero () = None
+   let plus u v = match u with None -> v | _ -> u
+  end
+  include Monad.Make(Base)
+  include (Monad.MakeCatch(Base) : Monad.PLUS with type 'a m := 'a m)
+  module T(Wrapped : Monad.W) = struct
+    module Trans = struct
+      include Monad.MakeT(struct
+        module Wrapped = Wrapped
+        type 'a m = 'a option Wrapped.m
+        let elevate w = Wrapped.bind w (fun a -> Wrapped.unit (Some a))
+        let bind u f = Wrapped.bind u (fun t -> match t with
+          | Some a -> f a
+          | None -> Wrapped.unit None)
+        type 'a result = 'a option Wrapped.result
+        let run u = Wrapped.run u
+        type 'a result_exn = 'a Wrapped.result_exn
+        let run_exn u =
+          let w = Wrapped.bind u (fun t -> match t with
+            | Some a -> Wrapped.unit a
+            | None -> failwith "no value")
+          in Wrapped.run_exn w
+      end)
+      let zero () = Wrapped.unit None
+      let plus u v = Wrapped.bind u (fun t -> match t with | None -> v | _ -> u)
+    end
+    include Trans
+    include (Monad.MakeCatch(Trans) : Monad.PLUS with type 'a m := 'a m)
+  end
+end
+
+
+module List_monad : sig
+  (* declare additional operation, while still hiding implementation of type m *)
+  type 'a result = 'a list
+  type 'a result_exn = 'a
+  include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+  include Monad.PLUS with type 'a m := 'a m
+  val permute : 'a m -> 'a m m
+  val select : 'a m -> ('a * 'a m) m
+  (* ListT transformer *)
+  module T : functor (Wrapped : Monad.W) -> sig
+    type 'a result = 'a list Wrapped.result
+    type 'a result_exn = 'a Wrapped.result_exn
+    include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+    include Monad.PLUS with type 'a m := 'a m
+    val elevate : 'a Wrapped.m -> 'a m
+    (* note that second argument is an 'a list, not the more abstract 'a m *)
+    (* type is ('a -> 'b W) -> 'a list -> 'b list W == 'b listT(W) *)
+    val distribute : ('a -> 'b Wrapped.m) -> 'a list -> 'b m
+(* TODO
+    val permute : 'a m -> 'a m m
+    val select : 'a m -> ('a * 'a m) m
+*)
+  end
+end = struct
+  module Base = struct
+   type 'a m = 'a list
+   let unit a = [a]
+   let bind u f = Util.concat_map f u
+   type 'a result = 'a list
+   let run u = u
+   type 'a result_exn = 'a
+   let run_exn u = match u with
+     | [] -> failwith "no values"
+     | [a] -> a
+     | many -> failwith "multiple values"
+   let zero () = []
+   let plus = Util.append
+  end
+  include Monad.Make(Base)
+  include (Monad.MakeDistrib(Base) : Monad.PLUS with type 'a m := 'a m)
+  (* let either u v = plus u v *)
+  (* insert 3 [1;2] ~~> [[3;1;2]; [1;3;2]; [1;2;3]] *)
+  let rec insert a u =
+    plus (unit (a :: u)) (match u with
+        | [] -> zero ()
+        | x :: xs -> (insert a xs) >>= fun v -> unit (x :: v)
+    )
+  (* permute [1;2;3] ~~> [1;2;3]; [2;1;3]; [2;3;1]; [1;3;2]; [3;1;2]; [3;2;1] *)
+  let rec permute u = match u with
+      | [] -> unit []
+      | x :: xs -> (permute xs) >>= (fun v -> insert x v)
+  (* select [1;2;3] ~~> [(1,[2;3]); (2,[1;3]), (3;[1;2])] *)
+  let rec select u = match u with
+    | [] -> zero ()
+    | x::xs -> plus (unit (x, xs)) (select xs >>= fun (x', xs') -> unit (x', x :: xs'))
+  let base_plus = plus
+  module T(Wrapped : Monad.W) = struct
+    module Trans = struct
+      let zero () = Wrapped.unit []
+      let plus u v =
+        Wrapped.bind u (fun us ->
+        Wrapped.bind v (fun vs ->
+        Wrapped.unit (base_plus us vs)))
+      (* Wrapped.sequence ms  ===  
+           let plus1 u v =
+             Wrapped.bind u (fun x ->
+             Wrapped.bind v (fun xs ->
+             Wrapped.unit (x :: xs)))
+           in Util.fold_right plus1 ms (Wrapped.unit []) *)
+      (* distribute  ===  Wrapped.mapM; copies alist to its image under f *)
+      let distribute f alist = Wrapped.sequence (Util.map f alist)
+      include Monad.MakeT(struct
+        module Wrapped = Wrapped
+        type 'a m = 'a list Wrapped.m
+        let elevate w = Wrapped.bind w (fun a -> Wrapped.unit [a])
+        let bind u f =
+          Wrapped.bind u (fun ts ->
+          Wrapped.bind (distribute f ts) (fun tts ->
+          Wrapped.unit (Util.concat tts)))
+        type 'a result = 'a list Wrapped.result
+        let run u = Wrapped.run u
+        type 'a result_exn = 'a Wrapped.result_exn
+        let run_exn u =
+          let w = Wrapped.bind u (fun ts -> match ts with
+            | [] -> failwith "no values"
+            | [a] -> Wrapped.unit a
+            | many -> failwith "multiple values"
+          ) in Wrapped.run_exn w
+      end)
+    end
+    include Trans
+    include (Monad.MakeDistrib(Trans) : Monad.PLUS with type 'a m := 'a m)
+(*
+    let permute : 'a m -> 'a m m
+    let select : 'a m -> ('a * 'a m) m
+*)
+  end
+end
+
+
+(* must be parameterized on (struct type err = ... end) *)
+module Error_monad(Err : sig
+  type err
+  exception Exc of err
+  (*
+  val zero : unit -> err
+  val plus : err -> err -> err
+  *)
+end) : sig
+  (* declare additional operations, while still hiding implementation of type m *)
+  type err = Err.err
+  type 'a error = Error of err | Success of 'a
+  type 'a result = 'a
+  type 'a result_exn = 'a
+  include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+  (* include Monad.PLUS with type 'a m := 'a m *)
+  val throw : err -> 'a m
+  val catch : 'a m -> (err -> 'a m) -> 'a m
+  (* ErrorT transformer *)
+  module T : functor (Wrapped : Monad.W) -> sig
+    type 'a result = 'a Wrapped.result
+    type 'a result_exn = 'a Wrapped.result_exn
+    include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+    val elevate : 'a Wrapped.m -> 'a m
+    val throw : err -> 'a m
+    val catch : 'a m -> (err -> 'a m) -> 'a m
+  end
+end = struct
+  type err = Err.err
+  type 'a error = Error of err | Success of 'a
+  module Base = struct
+    type 'a m = 'a error
+    let unit a = Success a
+    let bind u f = match u with
+      | Success a -> f a
+      | Error e -> Error e (* input and output may be of different 'a types *)
+    type 'a result = 'a
+    (* TODO: should run refrain from failing? *)
+    let run u = match u with
+      | Success a -> a
+      | Error e -> raise (Err.Exc e)
+    type 'a result_exn = 'a
+    let run_exn = run
+    (*
+    let zero () = Error Err.zero
+    let plus u v = match (u, v) with
+      | Success _, _ -> u
+      (* to satisfy (Catch) laws, plus u zero = u, even if u = Error _
+       * otherwise, plus (Error _) v = v *)
+      | Error _, _ when v = zero -> u
+      (* combine errors *)
+      | Error e1, Error e2 when u <> zero -> Error (Err.plus e1 e2)
+      | Error _, _ -> v
+    *)
+  end
+  include Monad.Make(Base)
+  (* include (Monad.MakeCatch(Base) : Monad.PLUS with type 'a m := 'a m) *)
+  let throw e = Error e
+  let catch u handler = match u with
+    | Success _ -> u
+    | Error e -> handler e
+  module T(Wrapped : Monad.W) = struct
+    module Trans = struct
+      module Wrapped = Wrapped
+      type 'a m = 'a Base.m Wrapped.m
+      let elevate w = Wrapped.bind w (fun a -> Wrapped.unit (Success a))
+      let bind u f = Wrapped.bind u (fun t -> match t with
+        | Success a -> f a
+        | Error e -> Wrapped.unit (Error e))
+      type 'a result = 'a Wrapped.result
+      (* TODO: should run refrain from failing? *)
+      let run u =
+        let w = Wrapped.bind u (fun t -> match t with
+          | Success a -> Wrapped.unit a
+          (* | _ -> Wrapped.fail () *)
+          | Error e -> raise (Err.Exc e))
+        in Wrapped.run w
+      type 'a result_exn = 'a Wrapped.result_exn
+      let run_exn u =
+        let w = Wrapped.bind u (fun t -> match t with
+          | Success a -> Wrapped.unit a
+          (* | _ -> Wrapped.fail () *)
+          | Error e -> raise (Err.Exc e))
+        in Wrapped.run_exn w
+    end
+    include Monad.MakeT(Trans)
+    let throw e = Wrapped.unit (Error e)
+    let catch u handler = Wrapped.bind u (fun t -> match t with
+      | Success _ -> Wrapped.unit t
+      | Error e -> handler e)
+  end
+end
+
+(* pre-define common instance of Error_monad *)
+module Failure = Error_monad(struct
+  type err = string
+  exception Exc = Failure
+  (*
+  let zero = ""
+  let plus s1 s2 = s1 ^ "\n" ^ s2
+  *)
+end)
+
+(* must be parameterized on (struct type env = ... end) *)
+module Reader_monad(Env : sig type env end) : sig
+  (* declare additional operations, while still hiding implementation of type m *)
+  type env = Env.env
+  type 'a result = env -> 'a
+  type 'a result_exn = env -> 'a
+  include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+  val ask : env m
+  val asks : (env -> 'a) -> 'a m
+  val local : (env -> env) -> 'a m -> 'a m
+  (* ReaderT transformer *)
+  module T : functor (Wrapped : Monad.W) -> sig
+    type 'a result = env -> 'a Wrapped.result
+    type 'a result_exn = env -> 'a Wrapped.result_exn
+    include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+    val elevate : 'a Wrapped.m -> 'a m
+    val ask : env m
+    val asks : (env -> 'a) -> 'a m
+    val local : (env -> env) -> 'a m -> 'a m
+  end
+  (* ReaderT transformer when wrapped monad has plus, zero *)
+  module TP : functor (Wrapped : Monad.WP) -> sig
+    include module type of T(Wrapped)
+    include Monad.PLUS with type 'a m := 'a m
+  end
+end = struct
+  type env = Env.env
+  module Base = struct
+    type 'a m = env -> 'a
+    let unit a = fun e -> a
+    let bind u f = fun e -> let a = u e in let u' = f a in u' e
+    type 'a result = env -> 'a
+    let run u = fun e -> u e
+    type 'a result_exn = env -> 'a
+    let run_exn = run
+  end
+  include Monad.Make(Base)
+  let ask = fun e -> e
+  let asks selector = ask >>= (fun e -> unit (selector e)) (* may fail *)
+  let local modifier u = fun e -> u (modifier e)
+  module T(Wrapped : Monad.W) = struct
+    module Trans = struct
+      module Wrapped = Wrapped
+      type 'a m = env -> 'a Wrapped.m
+      let elevate w = fun e -> w
+      let bind u f = fun e -> Wrapped.bind (u e) (fun v -> f v e)
+      type 'a result = env -> 'a Wrapped.result
+      let run u = fun e -> Wrapped.run (u e)
+      type 'a result_exn = env -> 'a Wrapped.result_exn
+      let run_exn u = fun e -> Wrapped.run_exn (u e)
+    end
+    include Monad.MakeT(Trans)
+    let ask = fun e -> Wrapped.unit e
+    let asks selector = ask >>= (fun e -> unit (selector e)) (* may fail *)
+    let local modifier u = fun e -> u (modifier e)
+  end
+  module TP(Wrapped : Monad.WP) = struct
+    module TransP = struct
+      include T(Wrapped)
+      let plus u v = fun s -> Wrapped.plus (u s) (v s)
+      let zero () = elevate (Wrapped.zero ())
+      let asks selector = ask >>= (fun e ->
+        try unit (selector e)
+        with Not_found -> fun e -> Wrapped.zero ())
+    end
+    include TransP
+    include (Monad.MakeDistrib(TransP) : Monad.PLUS with type 'a m := 'a m)
+  end
+end
+
+
+(* must be parameterized on (struct type store = ... end) *)
+module State_monad(Store : sig type store end) : sig
+  (* declare additional operations, while still hiding implementation of type m *)
+  type store = Store.store
+  type 'a result = store -> 'a * store
+  type 'a result_exn = store -> 'a
+  include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+  val get : store m
+  val gets : (store -> 'a) -> 'a m
+  val put : store -> unit m
+  val puts : (store -> store) -> unit m
+  (* StateT transformer *)
+  module T : functor (Wrapped : Monad.W) -> sig
+    type 'a result = store -> ('a * store) Wrapped.result
+    type 'a result_exn = store -> 'a Wrapped.result_exn
+    include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+    val elevate : 'a Wrapped.m -> 'a m
+    val get : store m
+    val gets : (store -> 'a) -> 'a m
+    val put : store -> unit m
+    val puts : (store -> store) -> unit m
+  end
+  (* StateT transformer when wrapped monad has plus, zero *)
+  module TP : functor (Wrapped : Monad.WP) -> sig
+    include module type of T(Wrapped)
+    include Monad.PLUS with type 'a m := 'a m
+  end
+end = struct
+  type store = Store.store
+  module Base = struct
+    type 'a m = store -> 'a * store
+    let unit a = fun s -> (a, s)
+    let bind u f = fun s -> let (a, s') = u s in let u' = f a in u' s'
+    type 'a result = store -> 'a * store
+    let run u = fun s -> (u s)
+    type 'a result_exn = store -> 'a
+    let run_exn u = fun s -> fst (u s)
+  end
+  include Monad.Make(Base)
+  let get = fun s -> (s, s)
+  let gets viewer = fun s -> (viewer s, s) (* may fail *)
+  let put s = fun _ -> ((), s)
+  let puts modifier = fun s -> ((), modifier s)
+  module T(Wrapped : Monad.W) = struct
+    module Trans = struct
+      module Wrapped = Wrapped
+      type 'a m = store -> ('a * store) Wrapped.m
+      let elevate w = fun s ->
+        Wrapped.bind w (fun a -> Wrapped.unit (a, s))
+      let bind u f = fun s ->
+        Wrapped.bind (u s) (fun (a, s') -> f a s')
+      type 'a result = store -> ('a * store) Wrapped.result
+      let run u = fun s -> Wrapped.run (u s)
+      type 'a result_exn = store -> 'a Wrapped.result_exn
+      let run_exn u = fun s ->
+        let w = Wrapped.bind (u s) (fun (a,s) -> Wrapped.unit a)
+        in Wrapped.run_exn w
+    end
+    include Monad.MakeT(Trans)
+    let get = fun s -> Wrapped.unit (s, s)
+    let gets viewer = fun s -> Wrapped.unit (viewer s, s) (* may fail *)
+    let put s = fun _ -> Wrapped.unit ((), s)
+    let puts modifier = fun s -> Wrapped.unit ((), modifier s)
+  end
+  module TP(Wrapped : Monad.WP) = struct
+    module TransP = struct
+      include T(Wrapped)
+      let plus u v = fun s -> Wrapped.plus (u s) (v s)
+      let zero () = elevate (Wrapped.zero ())
+    end
+    let gets viewer = fun s ->
+      try Wrapped.unit (viewer s, s)
+      with Not_found -> Wrapped.zero ()
+    include TransP
+    include (Monad.MakeDistrib(TransP) : Monad.PLUS with type 'a m := 'a m)
+  end
+end
+
+(* State monad with different interface (structured store) *)
+module Ref_monad(V : sig
+  type value
+end) : sig
+  type ref
+  type value = V.value
+  type 'a result = 'a
+  type 'a result_exn = 'a
+  include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+  val newref : value -> ref m
+  val deref : ref -> value m
+  val change : ref -> value -> unit m
+  (* RefT transformer *)
+  module T : functor (Wrapped : Monad.W) -> sig
+    type 'a result = 'a Wrapped.result
+    type 'a result_exn = 'a Wrapped.result_exn
+    include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+    val elevate : 'a Wrapped.m -> 'a m
+    val newref : value -> ref m
+    val deref : ref -> value m
+    val change : ref -> value -> unit m
+  end
+  (* RefT transformer when wrapped monad has plus, zero *)
+  module TP : functor (Wrapped : Monad.WP) -> sig
+    include module type of T(Wrapped)
+    include Monad.PLUS with type 'a m := 'a m
+  end
+end = struct
+  type ref = int
+  type value = V.value
+  module D = Map.Make(struct type t = ref let compare = compare end)
+  type dict = { next: ref; tree : value D.t }
+  let empty = { next = 0; tree = D.empty }
+  let alloc (value : value) (d : dict) =
+    (d.next, { next = succ d.next; tree = D.add d.next value d.tree })
+  let read (key : ref) (d : dict) =
+    D.find key d.tree
+  let write (key : ref) (value : value) (d : dict) =
+    { next = d.next; tree = D.add key value d.tree }
+  module Base = struct
+    type 'a m = dict -> 'a * dict
+    let unit a = fun s -> (a, s)
+    let bind u f = fun s -> let (a, s') = u s in let u' = f a in u' s'
+    type 'a result = 'a
+    let run u = fst (u empty)
+    type 'a result_exn = 'a
+    let run_exn = run
+  end
+  include Monad.Make(Base)
+  let newref value = fun s -> alloc value s
+  let deref key = fun s -> (read key s, s) (* shouldn't fail because key will have an abstract type, and we never garbage collect *)
+  let change key value = fun s -> ((), write key value s) (* shouldn't allocate because key will have an abstract type *)
+  module T(Wrapped : Monad.W) = struct
+    module Trans = struct
+      module Wrapped = Wrapped
+      type 'a m = dict -> ('a * dict) Wrapped.m
+      let elevate w = fun s ->
+        Wrapped.bind w (fun a -> Wrapped.unit (a, s))
+      let bind u f = fun s ->
+        Wrapped.bind (u s) (fun (a, s') -> f a s')
+      type 'a result = 'a Wrapped.result
+      let run u =
+        let w = Wrapped.bind (u empty) (fun (a,s) -> Wrapped.unit a)
+        in Wrapped.run w
+      type 'a result_exn = 'a Wrapped.result_exn
+      let run_exn u =
+        let w = Wrapped.bind (u empty) (fun (a,s) -> Wrapped.unit a)
+        in Wrapped.run_exn w
+    end
+    include Monad.MakeT(Trans)
+    let newref value = fun s -> Wrapped.unit (alloc value s)
+    let deref key = fun s -> Wrapped.unit (read key s, s)
+    let change key value = fun s -> Wrapped.unit ((), write key value s)
+  end
+  module TP(Wrapped : Monad.WP) = struct
+    module TransP = struct
+      include T(Wrapped)
+      let plus u v = fun s -> Wrapped.plus (u s) (v s)
+      let zero () = elevate (Wrapped.zero ())
+    end
+    include TransP
+    include (Monad.MakeDistrib(TransP) : Monad.PLUS with type 'a m := 'a m)
+  end
+end
+
+
+(* must be parameterized on (struct type log = ... end) *)
+module Writer_monad(Log : sig
+  type log
+  val zero : log
+  val plus : log -> log -> log
+end) : sig
+  (* declare additional operations, while still hiding implementation of type m *)
+  type log = Log.log
+  type 'a result = 'a * log
+  type 'a result_exn = 'a * log
+  include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+  val tell : log -> unit m
+  val listen : 'a m -> ('a * log) m
+  val listens : (log -> 'b) -> 'a m -> ('a * 'b) m
+  (* val pass : ('a * (log -> log)) m -> 'a m *)
+  val censor : (log -> log) -> 'a m -> 'a m
+end = struct
+  type log = Log.log
+  module Base = struct
+    type 'a m = 'a * log
+    let unit a = (a, Log.zero)
+    let bind (a, w) f = let (a', w') = f a in (a', Log.plus w w')
+    type 'a result = 'a * log
+    let run u = u
+    type 'a result_exn = 'a * log
+    let run_exn = run
+  end
+  include Monad.Make(Base)
+  let tell entries = ((), entries) (* add entries to log *)
+  let listen (a, w) = ((a, w), w)
+  let listens selector u = listen u >>= fun (a, w) -> unit (a, selector w) (* filter listen through selector *)
+  let pass ((a, f), w) = (a, f w) (* usually use censor helper *)
+  let censor f u = pass (u >>= fun a -> unit (a, f))
+end
+
+(* pre-define simple Writer *)
+module Writer1 = Writer_monad(struct
+  type log = string
+  let zero = ""
+  let plus s1 s2 = s1 ^ "\n" ^ s2
+end)
+
+(* slightly more efficient Writer *)
+module Writer2 = struct
+  include Writer_monad(struct
+    type log = string list
+    let zero = []
+    let plus w w' = Util.append w' w
+  end)
+  let tell_string s = tell [s]
+  let tell entries = tell (Util.reverse entries)
+  let run u = let (a, w) = run u in (a, Util.reverse w)
+  let run_exn = run
+end
+
+
+module IO_monad : sig
+  (* declare additional operation, while still hiding implementation of type m *)
+  type 'a result = 'a
+  type 'a result_exn = 'a
+  include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+  val printf : ('a, unit, string, unit m) format4 -> 'a
+  val print_string : string -> unit m
+  val print_int : int -> unit m
+  val print_hex : int -> unit m
+  val print_bool : bool -> unit m
+end = struct
+  module Base = struct
+    type 'a m = { run : unit -> unit; value : 'a }
+    let unit a = { run = (fun () -> ()); value = a }
+    let bind (a : 'a m) (f: 'a -> 'b m) : 'b m =
+     let fres = f a.value in
+       { run = (fun () -> a.run (); fres.run ()); value = fres.value }
+    type 'a result = 'a
+    let run a = let () = a.run () in a.value
+    type 'a result_exn = 'a
+    let run_exn = run
+  end
+  include Monad.Make(Base)
+  let printf fmt =
+    Printf.ksprintf (fun s -> { Base.run = (fun () -> Pervasives.print_string s); value = () }) fmt
+  let print_string s = { Base.run = (fun () -> Printf.printf "%s\n" s); value = () }
+  let print_int i = { Base.run = (fun () -> Printf.printf "%d\n" i); value = () }
+  let print_hex i = { Base.run = (fun () -> Printf.printf "0x%x\n" i); value = () }
+  let print_bool b = { Base.run = (fun () -> Printf.printf "%B\n" b); value = () }
+end
+
+module Continuation_monad : sig
+  (* expose only the implementation of type `('r,'a) result` *)
+  type 'a m
+  type 'a result = 'a m
+  type 'a result_exn = 'a m
+  include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn and type 'a m := 'a m
+  (* misses that the answer types of all the cont's must be the same *)
+  val callcc : (('a -> 'b m) -> 'a m) -> 'a m
+  val reset : 'a m -> 'a m
+  (* misses that the answer types of second and third continuations must be b *)
+  val shift : (('a -> 'b m) -> 'b m) -> 'a m
+  (* overwrite the run declaration in S, because I can't declare 'a result =
+   * this polymorphic type (complains that 'r is unbound *)
+  val runk : 'a m -> ('a -> 'r) -> 'r
+  val run0 : 'a m -> 'a
+end = struct
+  let id = fun i -> i
+  module Base = struct
+    (* 'r is result type of whole computation *)
+    type 'a m = { cont : 'r. ('a -> 'r) -> 'r }
+    let unit a =
+      let cont : 'r. ('a -> 'r) -> 'r =
+        fun k -> k a
+      in { cont }
+    let bind u f =
+      let cont : 'r. ('a -> 'r) -> 'r =
+        fun k -> u.cont (fun a -> (f a).cont k)
+      in { cont }
+    type 'a result = 'a m
+    let run (u : 'a m) : 'a result = u
+    type 'a result_exn = 'a m
+    let run_exn (u : 'a m) : 'a result_exn = u
+    let callcc f =
+      let cont : 'r. ('a -> 'r) -> 'r = fun k ->
+          (* Can't figure out how to make the type polymorphic enough
+           * to satisfy the OCaml type-checker (it's ('a -> 'r) -> 'r
+           * instead of 'r. ('a -> 'r) -> 'r); so we have to fudge
+           * with Obj.magic... which tells OCaml's type checker to
+           * relax, the supplied value has whatever type the context
+           * needs it to have. *)
+          let usek a = { cont = Obj.magic (fun _ -> k a) }
+          in (f usek).cont k
+      in { cont }
+    let reset u = unit (u.cont id)
+    let shift (f : ('a -> 'b m) -> 'b m) : 'a m =
+      let cont =
+        fun k -> (f (fun a -> unit (k a))).cont id
+      in { cont = Obj.magic cont }
+    let runk u k = u.cont k
+    let run0 u = u.cont id
+  end
+  include Monad.Make(Base)
+  let callcc = Base.callcc
+  let reset = Base.reset
+  let shift = Base.shift
+  let runk = Base.runk
+  let run0 = Base.run0
+end
+
+(*
+(* This two-type parameter version works without Obj.magic *)
+
+module Continuation_monad2 : sig
+  (* expose only the implementation of type `('r,'a) result` *)
+  type ('r,'a) result = ('a -> 'r) -> 'r
+  type ('r,'a) result_exn = ('a -> 'r) -> 'r
+  include Monad.S2 with type ('r,'a) result := ('r,'a) result and type ('r,'a) result_exn := ('r,'a) result_exn
+  val callcc : (('a -> ('r,'b) m) -> ('r,'a) m) -> ('r,'a) m
+  val reset : ('a,'a) m -> ('r,'a) m
+  val shift : (('a -> ('q,'r) m) -> ('r,'r) m) -> ('r,'a) m
+
+end = struct
+  let id = fun i -> i
+  module Base = struct
+    (* 'r is result type of whole computation *)
+    type ('r,'a) m = ('a -> 'r) -> 'r
+    let unit a = fun k -> k a
+    let bind u f = fun k -> u (fun a -> (f a) k)
+    type ('r,'a) result = ('a -> 'r) -> 'r
+    let run u = u
+    type ('r,'a) result_exn = ('a -> 'r) -> 'r
+    let run_exn = run
+  end
+  include Monad.Make2(Base)
+  let callcc f = fun k ->
+    let usek a = fun _ -> k a
+    in f usek k
+  (*
+  val callcc : (('a -> 'r) -> ('r,'a) m) -> ('r,'a) m
+  val throw : ('a -> 'r) -> 'a -> ('r,'b) m
+  let callcc f = fun k -> f k k
+  let throw k a = fun _ -> k a
+  *)
+  (* from http://www.haskell.org/haskellwiki/MonadCont_done_right *)
+  let reset u = unit (u id)
+  let shift u = fun k -> u (fun a -> unit (k a)) id
+end
+ *)
+
+
+(*
+ * Scheme:
+ * (define (example n)
+ *    (let ([u (let/cc k ; type int -> int pair
+ *               (let ([v (if (< n 0) (k 0) (list (+ n 100)))])
+ *                 (+ 1 (car v))))]) ; int
+ *      (cons u 0))) ; int pair
+ * ; (example 10) ~~> '(111 . 0)
+ * ; (example -10) ~~> '(0 . 0)
+ *
+ * OCaml monads:
+ * let example n : (int * int) =
+ *   Continuation_monad.(let u = callcc (fun k ->
+ *       (if n < 0 then k 0 else unit [n + 100])
+ *       (* all of the following is skipped by k 0; the end type int is k's input type *)
+ *       >>= fun [x] -> unit (x + 1)
+ *   )
+ *   (* k 0 starts again here, outside the callcc (...); the end type int * int is k's output type *)
+ *   >>= fun x -> unit (x, 0)
+ *   in run u)
+ *
+ *
+ * (* (+ 1000 (prompt (+ 100 (shift k (+ 10 1))))) ~~> 1011 *)
+ * let example1 () : int =
+ *   Continuation_monad.(let v = reset (
+ *       let u = shift (fun k -> unit (10 + 1))
+ *       in u >>= fun x -> unit (100 + x)
+ *     ) in let w = v >>= fun x -> unit (1000 + x)
+ *     in run w)
+ *
+ * (* (+ 1000 (prompt (+ 100 (shift k (k (+ 10 1)))))) ~~> 1111 *)
+ * let example2 () =
+ *   Continuation_monad.(let v = reset (
+ *       let u = shift (fun k -> k (10 :: [1]))
+ *       in u >>= fun x -> unit (100 :: x)
+ *     ) in let w = v >>= fun x -> unit (1000 :: x)
+ *     in run w)
+ *
+ * (* (+ 1000 (prompt (+ 100 (shift k (+ 10 (k 1)))))) ~~> 1111 but added differently *)
+ * let example3 () =
+ *   Continuation_monad.(let v = reset (
+ *       let u = shift (fun k -> k [1] >>= fun x -> unit (10 :: x))
+ *       in u >>= fun x -> unit (100 :: x)
+ *     ) in let w = v >>= fun x -> unit (1000 :: x)
+ *     in run w)
+ *
+ * (* (+ 100 ((prompt (+ 10 (shift k k))) 1)) ~~> 111 *)
+ * (* not sure if this example can be typed without a sum-type *)
+ *
+ * (* (+ 100 (prompt (+ 10 (shift k (k (k 1)))))) ~~> 121 *)
+ * let example5 () : int =
+ *   Continuation_monad.(let v = reset (
+ *       let u = shift (fun k -> k 1 >>= fun x -> k x)
+ *       in u >>= fun x -> unit (10 + x)
+ *     ) in let w = v >>= fun x -> unit (100 + x)
+ *     in run w)
+ *
+ *)
+
+
+module Leaf_monad : sig
+  (* We implement the type as `'a tree option` because it has a natural`plus`,
+   * and the rest of the library expects that `plus` and `zero` will come together. *)
+  type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
+  type 'a result = 'a tree option
+  type 'a result_exn = 'a tree
+  include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+  include Monad.PLUS with type 'a m := 'a m
+  (* LeafT transformer *)
+  module T : functor (Wrapped : Monad.W) -> sig
+    type 'a result = 'a tree option Wrapped.result
+    type 'a result_exn = 'a tree Wrapped.result_exn
+    include Monad.S with type 'a result := 'a result and type 'a result_exn := 'a result_exn
+    include Monad.PLUS with type 'a m := 'a m
+    val elevate : 'a Wrapped.m -> 'a m
+    (* note that second argument is an 'a tree?, not the more abstract 'a m *)
+    (* type is ('a -> 'b W) -> 'a tree? -> 'b tree? W == 'b treeT(W) *)
+    val distribute : ('a -> 'b Wrapped.m) -> 'a tree option -> 'b m
+  end
+end = struct
+  type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
+  (* uses supplied plus and zero to copy t to its image under f *)
+  let mapT (f : 'a -> 'b) (t : 'a tree option) (zero : unit -> 'b) (plus : 'b -> 'b -> 'b) : 'b = match t with
+      | None -> zero ()
+      | Some ts -> let rec loop ts = (match ts with
+                     | Leaf a -> f a
+                     | Node (l, r) ->
+                         (* recursive application of f may delete a branch *)
+                         plus (loop l) (loop r)
+                   ) in loop ts
+  module Base = struct
+    type 'a m = 'a tree option
+    let unit a = Some (Leaf a)
+    let zero () = None
+    let plus u v = match (u, v) with
+      | None, _ -> v
+      | _, None -> u
+      | Some us, Some vs -> Some (Node (us, vs))
+    let bind u f = mapT f u zero plus
+    type 'a result = 'a tree option
+    let run u = u
+    type 'a result_exn = 'a tree
+    let run_exn u = match u with
+      | None -> failwith "no values"
+      (*
+      | Some (Leaf a) -> a
+      | many -> failwith "multiple values"
+      *)
+      | Some us -> us
+  end
+  include Monad.Make(Base)
+  include (Monad.MakeDistrib(Base) : Monad.PLUS with type 'a m := 'a m)
+  let base_plus = plus
+  let base_lift = lift
+  module T(Wrapped : Monad.W) = struct
+    module Trans = struct
+      let zero () = Wrapped.unit None
+      let plus u v =
+        Wrapped.bind u (fun us ->
+        Wrapped.bind v (fun vs ->
+        Wrapped.unit (base_plus us vs)))
+      include Monad.MakeT(struct
+        module Wrapped = Wrapped
+        type 'a m = 'a Base.m Wrapped.m
+        let elevate w = Wrapped.bind w (fun a -> Wrapped.unit (Some (Leaf a)))
+        let bind u f = Wrapped.bind u (fun t -> mapT f t zero plus)
+        type 'a result = 'a tree option Wrapped.result
+        let run u = Wrapped.run u
+        type 'a result_exn = 'a tree Wrapped.result_exn
+        let run_exn u =
+            let w = Wrapped.bind u (fun t -> match t with
+              | None -> failwith "no values"
+              | Some ts -> Wrapped.unit ts)
+            in Wrapped.run_exn w
+      end)
+    end
+    include Trans
+    include (Monad.MakeDistrib(Trans) : Monad.PLUS with type 'a m := 'a m)
+    (* let distribute f t = mapT (fun a -> a) (base_lift (fun a -> elevate (f a)) t) zero plus *)
+    let distribute f t = mapT (fun a -> elevate (f a)) t zero plus
+  end
+end
+
+
+module L = List_monad;;
+module R = Reader_monad(struct type env = int -> int end);;
+module S = State_monad(struct type store = int end);;
+module T = Leaf_monad;;
+module LR = L.T(R);;
+module LS = L.T(S);;
+module TL = T.T(L);;
+module TR = T.T(R);;
+module TS = T.T(S);;
+
+let t1 = Some (T.Node (T.Node (T.Leaf 2, T.Leaf 3), T.Node (T.Leaf 5, T.Node (T.Leaf 7, T.Leaf 11))));;
+
+(*
+let ts = TS.distribute (fun i -> S.(puts succ >> unit i)) t1;;
+TS.run ts 0;;
+(*
+- : int T.tree option * S.store =
+(Some
+  (T.Node
+    (T.Node (T.Leaf 2, T.Leaf 3),
+     T.Node (T.Leaf 5, T.Node (T.Leaf 7, T.Leaf 11)))),
+ 5)
+*)
+
+let ts2 = TS.distribute (fun i -> S.(puts succ >> get >>= fun n -> unit (i,n))) t1;;
+TS.run_exn ts2 0;;
+(*
+- : (int * S.store) T.tree option * S.store =
+(Some
+  (T.Node
+    (T.Node (T.Leaf (2, 1), T.Leaf (3, 2)),
+     T.Node (T.Leaf (5, 3), T.Node (T.Leaf (7, 4), T.Leaf (11, 5))))),
+ 5)
+*)
+
+let tr = TR.distribute (fun i -> R.asks (fun e -> e i)) t1;;
+TR.run_exn tr (fun i -> i+i);;
+(*
+- : int T.tree option =
+Some
+ (T.Node
+   (T.Node (T.Leaf 4, T.Leaf 6),
+    T.Node (T.Leaf 10, T.Node (T.Leaf 14, T.Leaf 22))))
+*)
+
+let tl = TL.distribute (fun i -> L.(unit (i,i+1))) t1;;
+TL.run_exn tl;;
+(*
+- : (int * int) TL.result =
+[Some
+  (T.Node
+    (T.Node (T.Leaf (2, 3), T.Leaf (3, 4)),
+     T.Node (T.Leaf (5, 6), T.Node (T.Leaf (7, 8), T.Leaf (11, 12)))))]
+*)
+
+let l2 = [1;2;3;4;5];;
+let t2 = Some (T.Node (T.Leaf 1, (T.Node (T.Node (T.Node (T.Leaf 2, T.Leaf 3), T.Leaf 4), T.Leaf 5))));;
+
+LR.(run (distribute (fun i -> R.(asks (fun e -> e i))) l2 >>= fun j -> LR.(plus (unit j) (unit (succ j))))) (fun i -> i*10);;
+(* int list = [10; 11; 20; 21; 30; 31; 40; 41; 50; 51] *)
+
+TR.(run_exn (distribute (fun i -> R.(asks (fun e -> e i))) t2 >>= fun j -> TR.(plus (unit j) (unit (succ j))))) (fun i -> i*10);;
+(*
+int T.tree option =
+Some
+ (T.Node
+   (T.Node (T.Leaf 10, T.Leaf 11),
+    T.Node
+     (T.Node
+       (T.Node (T.Node (T.Leaf 20, T.Leaf 21), T.Node (T.Leaf 30, T.Leaf 31)),
+        T.Node (T.Leaf 40, T.Leaf 41)),
+      T.Node (T.Leaf 50, T.Leaf 51))))
+ *)
+
+LS.run (LS.distribute (fun i -> if i = -1 then S.get else if i < 0 then S.(puts succ >> unit 0) else S.unit i) [10;-1;-2;-1;20]) 0;;
+(*
+- : S.store list * S.store = ([10; 0; 0; 1; 20], 1)
+*)
+
+*)
+
+let id : 'z. 'z -> 'z = fun x -> x
+
+let example n : (int * int) =
+  Continuation_monad.(let u = callcc (fun k ->
+      (if n < 0 then k 0 else unit [n + 100])
+      (* all of the following is skipped by k 0; the end type int is k's input type *)
+      >>= fun [x] -> unit (x + 1)
+  )
+  (* k 0 starts again here, outside the callcc (...); the end type int * int is k's output type *)
+  >>= fun x -> unit (x, 0)
+  in run0 u)
+
+
+(* (+ 1000 (prompt (+ 100 (shift k (+ 10 1))))) ~~> 1011 *)
+let example1 () : int =
+  Continuation_monad.(let v = reset (
+      let u = shift (fun k -> unit (10 + 1))
+      in u >>= fun x -> unit (100 + x)
+    ) in let w = v >>= fun x -> unit (1000 + x)
+    in run0 w)
+
+(* (+ 1000 (prompt (+ 100 (shift k (k (+ 10 1)))))) ~~> 1111 *)
+let example2 () =
+  Continuation_monad.(let v = reset (
+      let u = shift (fun k -> k (10 :: [1]))
+      in u >>= fun x -> unit (100 :: x)
+    ) in let w = v >>= fun x -> unit (1000 :: x)
+    in run0 w)
+
+(* (+ 1000 (prompt (+ 100 (shift k (+ 10 (k 1)))))) ~~> 1111 but added differently *)
+let example3 () =
+  Continuation_monad.(let v = reset (
+      let u = shift (fun k -> k [1] >>= fun x -> unit (10 :: x))
+      in u >>= fun x -> unit (100 :: x)
+    ) in let w = v >>= fun x -> unit (1000 :: x)
+    in run0 w)
+
+(* (+ 100 ((prompt (+ 10 (shift k k))) 1)) ~~> 111 *)
+(* not sure if this example can be typed without a sum-type *)
+
+(* (+ 100 (prompt (+ 10 (shift k (k (k 1)))))) ~~> 121 *)
+let example5 () : int =
+  Continuation_monad.(let v = reset (
+      let u = shift (fun k -> k 1 >>= fun x -> k x)
+      in u >>= fun x -> unit (10 + x)
+    ) in let w = v >>= fun x -> unit (100 + x)
+    in run0 w)
+
+
+;;
+
+(1011, 1111, 1111, 121);;
+(example1(), example2(), example3(), example5());;
+((111,0), (0,0));;
+(example ~+10, example ~-10);;
+
+module C = Continuation_monad
+module TC = T.T(C)
+
+let testc df ic =
+    C.runk TC.(run_exn (distribute df t1)) ic;;
+
+
+(*
+(* do nothing *)
+let initial_continuation = fun t -> t in
+TreeCont.monadize t1 Continuation_monad.unit initial_continuation;;
+*)
+testc (C.unit) id;;
+
+(*
+(* count leaves, using continuation *)
+let initial_continuation = fun t -> 0 in
+TreeCont.monadize t1 (fun a k -> 1 + k a) initial_continuation;;
+*)
+
+testc C.(fun a -> shift (fun k -> k a >>= fun v -> unit (1 + v))) (fun t -> 0);;
+
+(*
+(* convert tree to list of leaves *)
+let initial_continuation = fun t -> [] in
+TreeCont.monadize t1 (fun a k -> a :: k a) initial_continuation;;
+*)
+
+testc C.(fun a -> shift (fun k -> k a >>= fun v -> unit (a::v))) (fun t -> ([] : int list));;
+
+(*
+(* square each leaf using continuation *)
+let initial_continuation = fun t -> t in
+TreeCont.monadize t1 (fun a k -> k (a*a)) initial_continuation;;
+*)
+
+testc C.(fun a -> shift (fun k -> k (a*a))) (fun t -> t);;
+
+
+(*
+(* replace leaves with list, using continuation *)
+let initial_continuation = fun t -> t in
+TreeCont.monadize t1 (fun a k -> k [a; a*a]) initial_continuation;;
+*)
+
+testc C.(fun a -> shift (fun k -> k (a,a+1))) (fun t -> t);;
+