+Do you see the pattern? Where before `unit` was implemented by a function that returned an `'a * store` value, now we instead use `M.unit` to return an `('a * store) M` value. Where before `bind` supplied an `'a state` value `(u s)` as an argument to a function, now we instead `M.bind` it to that function.
+
+Once again, what do you think you'd get if you wrapped StateT monadic packaging around an Identity monad?
+
+
+We spell out all the common monads, their common dedicated operations (such as `lookup`- and `shift`-like operations for the Reader monad), and monad transformer cousins of all of these, in an OCaml [[monad library]]. Read the linked page for details about how to use the library, and some design choices we made. Our [[State Monad Tutorial]] gives some more examples of using the library.
+
+When a T monadic layer encloses an inner M monad, the T's interface is the most exposed one. To use operations defined in the inner M monad, you'll have to "elevate" them into the outer T packaging. Haskell calls this operation `lift`, but we call it `elevate` because the term "lift" is already now too overloaded. In our usage, `lift` (and `lift2`) are functions that bring non-monadic operations into a monad; `elevate` brings monadic operations from a wrapped monad out into the wrapping.
+
+Here's an example. Suppose `S` is an instance of a State monad:
+
+ # #use "path/to/monads.ml";;
+ # module S = State_monad(struct type store = int end);;
+
+and `MS` is a MaybeT wrapped around `S`:
+
+ # module MS = Maybe_monad.T(S);;
+
+Then if you want to use an `S`-specific monad like `puts succ` inside `MS`, you'll have to use `MS`'s `elevate` function, like this:
+
+ # MS.(...elevate (S.puts succ) ...)
+
+Each monad transformer's `elevate` function will be defined differently. They have to obey the following laws:
+
+* `Outer.elevate (Inner.unit a) <~~> Outer.unit a`
+* `Outer.elevate (Inner.bind u f) <~~> Outer.bind (Outer.elevate u) (fun a -> Outer.elevate (f a))`
+
+We said that when T encloses M, you can rely on T's interface to be most exposed. That is intuitive. What you cannot also assume is that the implementing type has a Tish structure surrounding an Mish structure. Often it will be reverse: a ListT(Maybe) is implemented by a `'a list option`, not by an `'a option list`. Until you've tried to write the code to a monadic transformer library yourself, this will probably remain counter-intuitive. But you don't need to concern yourself with it in practise. Think of what you have as a ListT(Maybe); don't worry about whether the underlying implementation is as an `'a list option` or an `'a option list` or as something more complicated.
+
+Notice from the code for StateT, above, that an `'a stateT(M)` is not an `('a M) state`; neither is it an `('a state) M`. The pattern by which we transform the types from a Blah monad to a BlahT monad transformer is:
+
+ 't0 ---> 't0 M
+ 't1 -> 't0 ---> 't1 -> 't0 M
+ ('t1 -> 't0) -> 't0 ---> ('t1 -> 't0 M) -> 't0 M
+
+Ken Shan's paper [Monads for natural language semantics](http://arxiv.org/abs/cs/0205026v1) (2001) discusses how to systematically move from some base monads to the corresponding monad transformers. But as he notes, his algorithm isn't the only one possible, and it only applies to monads whose type has a certain form. (Reader and State have that form; List for example doesn't.)
+
+As best we know, figuring out how a monad transformer should be defined is still something of an art, not something that can be done mechanically. However, you can think that all of the art goes into deciding what StateT and so on should be; having figured that out, plain State would follow as the simple case where StateT is parameterized on the Identity monad.
+
+Apart from whose interface is outermost, the behavior of a StateT(Maybe) and a MaybeT(State) will partly coincide. But in certain crucial respects they will diverge, and you need to think carefully about which behavior you want and what the appropriate layering is for your needs. Consider these examples:
+
+ # module MS = Maybe_monad.T(S);;
+ # module SM = S.T(Maybe_monad);;
+ # MS.(run (elevate (S.puts succ) >> zero () >> elevate S.get >>= fun cur -> unit (cur+10) )) 0;;
+ - : int option * S.store = (None, 1)
+ # MS.(run (elevate (S.puts succ) >> zero () >> elevate (S.put 5) )) 0;;
+ - : unit option * S.store = (None, 1)
+
+Although we have a wrapped `None`, notice that the store (as it was at the point of failure) is still retrievable.
+
+ # SM.(run (puts succ >> elevate (Maybe_monad.zero ()) >> get >>= fun cur -> unit (cur+10) )) 0;;
+ - : ('a, int * S.store) Maybe_monad.result = None
+
+When Maybe is on the inside, on the other hand, a failure means the whole computation has failed, and even the store is no longer available.
+
+<!--
+ # ES.(run( elevate (S.puts succ) >> throw "bye" >> elevate S.get >>= fun i -> unit(i+10) )) 0;;
+ - : int Failure.error * S.store = (Failure.Error "bye", 1)
+ # SE.(run( puts succ >> elevate (Failure.throw "bye") >> get >>= fun i -> unit(i+10) )) 0;;
+ - : (int * S.store) Failure.result = Failure.Error "bye"
+ # ES.(run_exn( elevate (S.puts succ) >> throw "bye" >> elevate S.get >>= fun i -> unit(i+10) )) 0;;
+ Exception: Failure "bye".
+ # SE.(run_exn( puts succ >> elevate (Failure.throw "bye") >> get >>= fun i -> unit(i+10) )) 0;;
+ Exception: Failure "bye".
+-->
+
+Here's an example wrapping Maybe around List, and vice versa:
+
+ # module LM = List_monad.T(Maybe_monad);;
+ # module ML = Maybe_monad.T(List_monad);;
+ # ML.(run (plus (zero ()) (unit 20) >>= fun i -> unit (i+10)));;
+ - : ('_a, int) ML.result = [Some 30]
+
+When List is on the inside, the failed results just get dropped and the computation proceeds without them.
+
+ # LM.(run (plus (elevate (Maybe_monad.zero ())) (unit 20) >>= fun i -> unit (i+10)));;
+ - : ('_a, int) LM.result = None
+
+On the other hand, when Maybe is on the inside, failures abort the whole computation.
+
+<!--
+ # EL.(run( plus (throw "bye") (unit 20) >>= fun i -> unit(i+10)));;
+ - : int EL.result = [Failure.Error "bye"; Failure.Success 30]
+ # LE.(run( plus (elevate (Failure.throw "bye")) (unit 20) >>= fun i -> unit(i+10)));;
+ - : int LE.result = Failure.Error "bye"
+ # EL.(run_exn( plus (throw "bye") (unit 20) >>= fun i -> unit(i+10)));;
+ Exception: Failure "bye".
+ # LE.(run_exn( plus (elevate (Failure.throw "bye")) (unit 20) >>= fun i -> unit(i+10)));;
+ Exception: Failure "bye".
+-->
+
+This is fun. Notice the difference it makes whether the second `plus` is native to the outer `List_monad`, or whether it's the inner `List_monad`'s `plus` elevated into the outer wrapper:
+
+ # module LL = List_monad.T(List_monad);;
+
+ # LL.(run(plus (unit 1) (unit 2) >>= fun i -> plus (unit i) (unit(10*i)) ));;
+ - : ('_a, int) LL.result = \[[1; 10; 2; 20]]
+ # LL.(run(plus (unit 1) (unit 2) >>= fun i -> elevate L.(plus (unit i) (unit(10*i)) )));;
+ - : ('_a, int) LL.result = [[1; 2]; [1; 20]; [10; 2]; [10; 20]]
+
+
+
+Further Reading
+---------------
+
+* This is excellent, everyone should read: [Monad Transformers Step by Step](http://www.grabmueller.de/martin/www/pub/Transformers.pdf)
+
+* Read Part III of [All About Monads](http://web.archive.org/web/20071106232016/haskell.org/all_about_monads/html/introIII.html). This link is to an archived version, the main link to haskell.org seems to be broken. Some but not all of this site has been [absorbed into the Haskell wikibook](http://en.wikibooks.org/wiki/Haskell/Monad_transformers).
+
+
+Tree Monads
+===========
+
+Our [[monad library]] includes a `Tree_monad`, for binary, leaf-labeled trees. There are other kinds of trees you might want to monadize, but we took the name `Tree_monad` for this one. Like the Haskell [SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree) monad, our `Tree_monad` also incorporates an Optionish layer. (See the comments in our library code about `plus` and `zero` for discussion of why.)
+
+So how does our `Tree_monad` behave? Simplified, its implementation looks something like this: