pipes-3.1.0: Compositional pipelines

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Control.Proxy.Tutorial

Contents

Description

This module provides a brief introductory tutorial in the "Introduction" section followed by a lengthy discussion of the library's design and idioms.

Synopsis

Introduction

The pipes library replaces lazy IO with a safe, elegant, and theoretically principled alternative. Use this library if you:

  • want to write high-performance streaming programs
  • believe that lazy IO was a bad idea
  • enjoy composing modular and reusable components
  • love theory and elegant code

This library unifies many kinds of streaming abstractions, all of which are special cases of "proxies" (The pipes name is a legacy of one such abstraction).

Let's begin with the simplest Proxy: a Producer. The following Producer lazily streams lines from a Handle

 import Control.Monad
 import Control.Proxy
 import System.IO
 
 --                 Produces Strings ---+----------+
 --                                     |          |
 --                                     v          v
 lines' :: (Proxy p) => Handle -> () -> Producer p String IO ()
 lines' h () = runIdentityP loop where
     loop = do
         eof <- lift $ hIsEOF h
         if eof
         then return ()
         else do
             str <- lift $ hGetLine h
             respond str  -- Produce the string
             loop

 -- Ignore the 'runIdentityP' and '()' for now

But why limit ourselves to streaming lines from some file? Why not lazily generate values from an industrious user?

 --               Uses 'IO' as the base monad --+
 --                                             |
 --                                             v
 promptInt :: (Proxy p) => () -> Producer p Int IO r
 promptInt () = runIdentityP $ forever $ do
     lift $ putStrLn "Enter an Integer:"
     n <- lift readLn  -- 'lift' invokes an action in the base monad
     respond n

Now we need to hook our Producers up to a Consumer. The following Consumer endlessly requests a stream of Showable values and prints them:

 --                   Consumes 'a's ---+----------+    +-- Never terminates, so
 --                                    |          |    |   the return value is
 --                                    v          v    v   polymorphic
 printer :: (Proxy p, Show a) => () -> Consumer p a IO r
 printer () = runIdentityP $ forever $ do
     a <- request ()  -- Consume a value
     lift $ putStrLn "Received a value:"
     lift $ print a

You can compose a Producer and a Consumer using (>->), which produces a runnable Session:

 --                Self-contained session ---+         +--+-- These must match
 --                                          |         |  |   each component
 --                                          v         v  v
 lines' h  >-> printer :: (Proxy p) => () -> Session p IO ()

 promptInt >-> printer :: (Proxy p) => () -> Session p IO r

(>->) connects each request in printer with a respond in lines' or promptInt.

Finally, you use runProxy to run the Session and convert it back to the base monad. First we'll try our lines' Producer, which will stream lines from the following file:

 $ cat test.txt
 Line 1
 Line 2
 Line 3

The following program never brings more than a single line into memory (not that it matters for such a small file):

>>> withFile "test.txt" ReadMode $ \h -> runProxy $ lines' h >-> printer
Received a value:
"Line 1"
Received a value:
"Line 2"
Received a value:
"Line 3"

Similarly, we can lazily stream user input, requesting values from the user only when we need them:

>>> runProxy $ promptInt >-> printer :: IO r
Enter an Integer:
1<Enter>
Received a value:
1
Enter an Integer:
5<Enter>
Received a value:
5
...

The last example proceeds endlessly until we hit Ctrl-C to interrupt it.

We would like to limit the number of iterations, so lets define an intermediate Proxy that behaves like a verbose take. I will call it a Pipe (this library's namesake) since values flow through it:

                           'a's flow in ---+ +--- 'a's flow out
                                           | |
                                           v v
 take' :: (Proxy p) => Int -> () -> Pipe p a a IO ()
 take' n () = runIdentityP $ do
     replicateM_ n $ do
         a <- request ()
         respond a
     lift $ putStrLn "You shall not pass!"

This Pipe forwards the first n values it receives undisturbed, then it outputs a cute message. You can compose it between the Producer and Consumer using (>->):

>>> runProxy $ promptInt >-> take' 2 >-> printer :: IO ()
Enter an Integer:
9<Enter>
Received a value:
9
Enter an Integer:
2<Enter>
Received a value:
2
You shall not pass!

When take' 2 terminates, it brings down every Proxy composed with it.

Notice how promptInt behaves lazily and only responds with as many values as we request. We requested exactly two values, so it only prompts the user twice.

We can already spot several improvements upon traditional lazy IO:

  • You can define your own lazy components that have nothing to do with files
  • pipes never uses unsafePerformIO and never violates referential transparency.
  • You don't need strictness hacks to ensure the proper ordering of effects
  • You can interleave effects in downstream stages, too

However, this library can offer even more than that!

Bidirectionality

So far we've only defined proxies that send information downstream in the direction of the (>->) arrow. However, we don't need to limit ourselves to unidirectional communication and we can enhance these proxies with the ability to send information upstream with each request that determines how upstream stages respond.

For example, Clients generalize Consumers because they can supply an argument other than () with each request. The following Client sends three requests upstream, each of which provides an Int argument and expects a Bool result:

                      Sends out 'Int's ---+   +-- Receives back 'Bool's
                                          |   |
                                          v   v
 threeReqs :: (Proxy p) => () -> Client p Int Bool IO ()
 threeReqs () = runIdentityP $ forM_ [1, 3, 1] $ \argument -> do
     lift $ putStrLn $ "Client Sends:   " ++ show (argument :: Int)
     result <- request argument
     lift $ putStrLn $ "Client Receives:" ++ show (result :: Bool)
     lift $ putStrLn "*"

Notice how Clients use "request argument" instead of "request ()". This sends "argument" upstream to parametrize the request.

Servers similarly generalize Producers because they receive arguments other than (). The following Server receives Int requests and responds with Bool values:

                       Receives 'Int's ---+   +--- Replies with 'Bool's
                                          |   |
                                          v   v
 comparer :: (Proxy p) => Int -> Server p Int Bool IO r
 comparer = runIdentityK loop where
     loop argument = do
         lift $ putStrLn $ "Server Receives:" ++ show (argument :: Int)
         let result = argument > 2
         lift $ putStrLn $ "Server Sends:   " ++ show (result :: Bool)
         nextArgument <- respond result
         loop nextArgument

Notice how Servers receive their first argument as a parameter and bind each subsequent argument using respond. This library provides a combinator which abstracts away this common pattern:

 foreverK :: (Monad m) => (a -> m a) -> a -> m b
 foreverK f = loop where
     loop argument = do
          nextArgument <- f argument
          loop nextArgument

 -- or: foreverK f = f >=> foreverK f
 --                = f >=> f >=> f >=> f >=> ...

We can use this to simplify the comparer Server:

 comparer = runIdentityK $ foreverK $ \argument -> do
     lift $ putStrLn $ "Server Receives:" ++ show argument
     let result = argument > 2
     lift $ putStrLn $ "Server Sends:   " ++ show result
     respond result

... which looks just like the way you might write a server's main loop in another programming language.

You can compose a Server and Client using (>->), and this also returns a runnable Session:

 comparer >-> threeReqs :: (Proxy p) => () -> Session p IO ()

Running this executes the client-server session:

>>> runProxy $ comparer >-> threeReqs :: IO ()
Client Sends:    1
Server Receives: 1
Server Sends:    False
Client Receives: False
*
Client Sends:    3
Server Receives: 3
Server Sends:    True
Client Receives: True
*
Client Sends:    1
Server Receives: 1
Server Sends:    False
Client Receives: False
*

Proxys generalize Pipes because they allow information to flow upstream. The following Proxy caches requests to reduce the load on the Server if the request matches a previous one:

 import qualified Data.Map as M

 -- 'p' is the Proxy, as the (Proxy p) constraint indicates

 cache :: (Proxy p, Ord key) => key -> p key val key val IO r
 cache = runIdentityK (loop M.empty) where
     loop _map key = case M.lookup key _map of
         Nothing -> do
             val  <- request key
             key2 <- respond val
             loop (M.insert key val _map) key2
         Just val -> do
             lift $ putStrLn "Used cache!"
             key2 <- respond val
             loop _map key2

You can compose the cache Proxy between the Server and Client using (>->):

>>> runProxy $ comparer >-> cache >-> threeReqs
Client Sends:    1
Server Receives: 1
Server Sends:    False
Client Receives: False
*
Client Sends:    3
Server Receives: 3
Server Sends:    True
Client Receives: True
*
Client Sends:    1
Used cache!
Client Receives: False
*

This bidirectional flow of information separates pipes from other streaming libraries which are unable to model Clients, Servers, or Proxys. Using pipes you can define interfaces to RPC interfaces, REST architectures, message buses, chat clients, web servers, network protocols ... you name it!

Type Synonyms

You might wonder why (>->) accepts Producers, Consumers, Pipes, Clients, Servers, and Proxys. It turns out that these type-check because they are all type synonyms that expand to the following central type:

 (Proxy p) => p a' a b' b m r

Like the name suggests, a Proxy exposes two interfaces: an upstream interface and a downstream interface. Each interface can both send and receive values:

 Upstream | Downstream
     +---------+
     |         |
 a' <==       <== b'
     |  Proxy  |
 a  ==>       ==> b
     |         |
     +---------+

Proxies are monad transformers that enrich the base monad with the ability to send or receive values upstream or downstream:

   | Sends    | Receives | Receives   | Sends      | Base  | Return
   | Upstream | Upstream | Downstream | Downstream | Monad | Value
 p   a'         a          b'           b            m       r

We can selectively close certain inputs or outputs to generate specialized proxies.

For example, a Producer is a Proxy that can only output values to its downstream interface:

 Upstream | Downstream
     +----------+
     |          |
 C  <==        <== ()
     | Producer |
 () ==>        ==> b
     |          |
     +----------+

 type Producer p b m r = p C () () b m r

 -- The 'C' type is uninhabited, so it 'C'loses an output end

A Consumer is a Proxy that can only receive values on its upstream interface:

 Upstream | Downstream
     +----------+
     |          |
 () <==        <== ()
     | Consumer |
 a  ==>        ==> C
     |          |
     +----------+

 type Consumer p a m r = p () a () C m r

A Pipe is a Proxy that can only receive values on its upstream interface and send values on its downstream interface:

 Upstream | Downstream
     +--------+
     |        |
 () <==      <== ()
     |  Pipe  |
 a  ==>      ==> b
     |        |
     +--------+

 type Pipe p a b m r = p () a () b m r

When we compose proxies, the type system ensures sure that their input and output types match:

       promptInt    >->    take' 2    >->    printer

     +-----------+       +---------+       +---------+
     |           |       |         |       |         |
 C  <==         <== ()  <==       <== ()  <==       <== ()
     |           |       |         |       |         |
     | promptInt |       | take' 2 |       | printer |
     |           |       |         |       |         |
 () ==>         ==> Int ==>       ==> Int ==>       ==> C
     |           |       |         |       |         |
     +-----------+       +---------+       +---------+

Composition fuses these into a new Proxy that has both ends closed, which is a Session:

     +-----------------------------------+
     |                                   |
 C  <==                                 <== ()
     |                                   |
     | promptInt >-> take' 2 >-> printer |
     |                                   |
 () ==>                                 ==> C
     |                                   |
     +-----------------------------------+

 type Session p m r = p C () () C m r

A Client is a Proxy that only uses its upstream interface:

 Upstream | Downstream
     +----------+
     |          |
 a' <==        <== ()
     |  Client  |
 a  ==>        ==> C
     |          |
     +----------+

 type Client p a' a m r = p a' a () C m r

A Server is a Proxy that only uses its downstream interface:

 Upstream | Downstream
     +----------+
     |          |
 C  <==        <== b'
     |  Server  |
 () ==>        ==> b
     |          |
     +----------+

 type Server p b' b m r = p C () b' b m r

The compiler ensures that the types match when we compose Servers, Proxys, and Clients.

        comparer   >->     cache   >->      threeReqs

     +----------+        +-------+        +-----------+
     |          |        |       |        |           |
 C  <==        <== Int  <==     <== Int  <==         <== ()
     |          |        |       |        |           |
     | comparer |        | cache |        | threeReqs |
     |          |        |       |        |           |
 () ==>        ==> Bool ==>     ==> Bool ==>         ==> C
     |          |        |       |        |           |
     +----------+        +-------+        +-----------+

This similarly fuses into a Session:

     +----------------------------------+
     |                                  |
 C  <==                                <== ()
     |                                  |
     | comparer >-> cache >-> threeReqs |
     |                                  |
 () ==>                                ==> C
     |                                  |
     +----------------------------------+

pipes encourages substantial code reuse by implementing all abstractions as type synonyms on top of a single type class: Proxy. This makes your life easier because:

  • You only use one composition operator: (>->)
  • You can mix multiple abstractions together as long as the types match

Request and Respond

There are only two ways to interact with other proxies: request and respond. Let's examine their type signatures to understand how they work:

 request :: (Monad m, Proxy p) => a' -> p a' a b' b m a
                                  ^                   ^
                                  |                   |
                       Argument --+          Result --+

request sends an argument of type a' upstream, and binds a result of type a. Whenever you request, you block until upstream responds with a value.

 respond :: (Monad m, Proxy p) => b -> p a' a b' b m b'
                                  ^                  ^
                                  |                  |
                         Result --+  Next Argument --+

respond replies with a result of type b, and then binds the next argument of type b'. Whenever you respond, you block until downstream requests a new value.

Wait, if respond always binds the next argument, where does the first argument come from? Well, it turns out that every Proxy receives this initial argument as an ordinary parameter, as if they all began blocked on a respond statement.

We can see this if we take all the previous proxies we defined and fully expand every type synonym. The initial argument of each Proxy matches the type parameter corresponding to the return value of respond:

                                          These
                                    +--  Columns  ---+
                                    |     Match      |
                                    v                v
 promptInt :: (Proxy p)          => ()  -> p C   ()  ()  Int  IO r
 printer   :: (Proxy p, Show a)  => ()  -> p ()  a   ()  C    IO r
 take'     :: (Proxy p)   => Int -> ()  -> p ()  a   ()  a    IO ()
 comparer  :: (Proxy p)          => Int -> p C   ()  Int Bool IO r
 cache     :: (Proxy p, Ord key) => key -> p key val key val  IO r

You can also study the type of composition, which follows this same pattern. Composition requires two Proxys blocked on a respond, and produces a new Proxy similarly blocked on a respond:

 (>->) :: (Monad m, Proxy p)
  => (b' -> p a' a b' b m r)
  -> (c' -> p b' b c' c m r)
  -> (c' -> p a' a c' c m r)
      ^            ^
      |   These    |
      +---Match----+

This is why Producers, Consumers, and Clients all take () as their initial argument, because their corresponding respond commands all have a return value of ().

This library also provides (>~>), which is the dual of the (>->) composition operator. (>~>) composes two Proxys blocked on a request and returns a new Proxy blocked on a request:

 (>~>)
  :: (Monad m, Proxy p)
  => (a -> p a' a b' b m r)
  -> (b -> p b' b c' c m r)
  -> (a -> p a' a c' c m r)

Conceptually, (>->) composes pull-based systems and (>~>) composes push-based systems.

In fact, if you went back through the previous code and systematically replaced every:

... then everything would still work and produce identical behavior, except the compiler would now infer the symmetric types with all interfaces reversed. We can therefore conclude the obvious: pull-based systems are symmetric to push-based systems.

Since these two composition operators are perfectly symmetric, I arbitrarily standardize on using (>->) and I provide all standard library proxies blocked on respond so that they work with (>->). This gives behavior more familiar to Haskell programmers that work with lazy pull-based functions. I only include the (>~>) composition operator for theoretical completeness.

Composition

When we compose (p1 >-> p2), composition ensures that p1's downstream interface matches p2's upstream interface. This follows from the type of (>->):

 (>->) :: (Monad m, Proxy p)
  => (b' -> p a' a b' b m r)
  -> (c' -> p b' b c' c m r)
  -> (c' -> p a' a c' c m r)

Diagramatically, this looks like:

         p1     >->      p2

     +--------+      +--------+
     |        |      |        |
 a' <==      <== b' <==      <== c'
     |   p1   |      |   p2   |
 a  ==>      ==> b  ==>      ==> c
     |        |      |        |
     +--------+      +--------+

p1's downstream (b', b) interface matches p2's upstream (b', b) interface, so composition connects them on this shared interface. This fuses away the (b', b) interface, leaving behind p1's upstream (a', a) interface and p2's downstream (c', c) interface:

     +-----------------+
     |                 |
 a' <==               <== c'
     |   p1  >->  p2   |
 a  ==>               ==> c
     |                 |
     +-----------------+

Proxy composition has the very nice property that it is associative, meaning that it behaves the exact same way no matter how you group composition:

 (p1 >-> p2) >-> p3 = p1 >-> (p2 >-> p3)

... so you can safely elide the parentheses:

 p1 >-> p2 >-> p3

Also, we can define a 'T'ransparent Proxy that auto-forwards values both ways:

 idT :: (Monad m, Proxy p) => a' -> p a' a a' a m r
 idT = runIdentityK loop where
     loop a' = do
         a   <- request a'
         a'2 <- respond a
         loop a'2

 -- or: idT = runIdentityK $ foreverK $ request >=> respond
 --         = runIdentityK $ request >=> respond >=> request >=> respond ...

Diagramatically, this looks like:

     +-----+
     |     |
 a' <======== a'   <- All values pass
     | idT |          straight through
 a  ========> a    <- immediately
     |     |
     +-----+

Transparency means that:

 idT >-> p = p

 p >-> idT = p

In other words, idT is an identity of composition.

This means that proxies form a true Category where (>->) is composition and idT is the identity. The associativity law and the two identity laws are just the Category laws. The objects of the category are the Proxy interfaces.

These Category laws guarantee the following important properties:

  • You can reason about each proxy's behavior independently of other proxies
  • You don't encounter weird behavior at the interface between two components
  • You don't encounter corner cases at the Server or Client ends of a Session

The Proxy Class

All the proxy code we wrote was generic over the Proxy type class, which defines the three central operations of this library's API:

  • (>->): Proxy composition
  • request: Request input from upstream
  • respond: Respond with output to downstream

pipes defines everything in terms of these three operations, which is why all the library's utilities are polymorphic over the Proxy type class.

Let's look at some example instances of the Proxy type class:

 instance Proxy ProxyFast     -- Fastest implementation
 instance Proxy ProxyCorrect  -- Strict monad transformer laws

These two types provide the two alternative base implementations:

  • ProxyFast: This runs significantly faster on pure code segments and employs several rewrite rules to optimize your code into the equivalent hand-tuned code.
  • ProxyCorrect: This uses a monad transformer implementation that is correct by construction, but runs about 8x slower on pure code segments. However, for IO-bound code, the performance gap is small.

These two implementations differ only in the runProxy function that they export, which is how the compiler selects which Proxy implementation to use.

Control.Proxy automatically selects the fast implementation for you, but you can always choose the correct implementation instead by replacing Control.Proxy with the following two imports:

 import Control.Proxy.Core         -- Everything except the base implementation
 import Control.Proxy.Core.Correct -- The alternative base implementation

These are not the only instances of the Proxy type class! This library also provides several "proxy transformers", which are like monad transformers except that they also correctly lift the Proxy type class:

 instance (Proxy p) => Proxy (IdentityP p)
 instance (Proxy p) => Proxy (EitherP e p)
 instance (Proxy p) => Proxy (MaybeP    p)
 instance (Proxy p) => Proxy (ReaderP i p)
 instance (Proxy p) => Proxy (StateP  s p)
 instance (Proxy p) => Proxy (WriterP w p)

All of the Proxy code we wrote so far also works seamlessly with all of these proxy transformers. The Proxy class abstracts over the implementation details and extensions so that you can reuse the same library code for any feature set.

This polymorphism comes at a price: you must embed your Proxy code in at least one proxy transformer if you want clean type class constraints. If you don't use extensions then you embed your code in the identity proxy transformer: IdentityP. This is why all the examples use runIdentityP or runIdentityK to embed their code in IdentityP. Control.Proxy.Class provides a longer discussion on this subject.

Without this IdentityP embedding, the compiler infers uglier constraints, which are also significantly less polymorphic. We can show this by removing the runIdentityP call from promptInt and see what type the compiler infers:

 promptInt () = forever $ do
     lift $ putStrLn "Enter an Integer:"
     n <- lift readLn
     respond n
>>> :t promptInt -- I've substantially cleaned up the inferred type
promptInt
  :: (Monad (Producer p Int IO), MonadTrans (Producer p Int), Proxy p) =>
     () -> Producer p Int IO r

All Proxy instances are already monads and monad transformers, but the compiler cannot infer that without the IdentityP embedding. When we embed promptInt in IdentityP, the compiler collapses the Monad and MonadTrans constraints into the Proxy constraint.

Fortunately, you do not pay any performance price for this IdentityP embedding or the type class polymorphism. Your polymorphic code will still run very rapidly, as fast as if you had specialized it to a concrete Proxy instance without the IdentityP embedding. I've taken great care to ensure that all optimizations and rewrite rules always see through these abstractions without any assistance on your part.

Interleaving Effects

When you compose two proxies, you interleave their effects in the base monad. The following two proxies demonstrate this interleaving of effects:

 downstream :: (Proxy p) => () -> Consumer p () IO ()
 downstream () = runIdentityP $ do
     lift $ print 1
     request ()  -- Switch to upstream
     lift $ print 3
     request ()  -- Switch to upstream

 upstream :: (Proxy p) => () -> Producer p () IO ()
 upstream () = runIdentityP $ do
     lift $ print 2
     respond () -- Switch to downstraem
     lift $ print 4

Control.Proxy.Class enumerates the Proxy laws, which equationally define how all Proxy instances must behave. These laws require that (upstream >-> downstream) must reduce to the following:

 upstream >-> downstream  -- This is true no matter what feature
 =                        -- set or 'Proxy' instance you select
 \() -> lift $ do
     print 1
     print 2
     print 3
     print 4

Conceptually, runProxy just applies this to () and removes the lift:

 runProxy $ upstream >-> downstream
 =
 do print 1
    print 2
    print 3
    print 4

Let's test this:

>>> runProxy $ upstream >-> downstream
1
2
3
4

The Proxy laws let you reason about how proxies interleave effects without knowing any specifics about the underlying implementation. Intuitively, the Proxy laws say that:

Several of the utilities in Control.Proxy.Prelude.Base use these equational laws to rigorously prove things about their behavior. For example, consider the mapD proxy, which applies a function f to all values flowing downstream:

 mapD :: (Monad m, Proxy p) => (a -> b) -> x -> p x a x b m r
 mapD f = runIdentityK loop where
     loop x = do
         a  <- request x
         x2 <- respond (f a)
         loop x2

 -- or: mapD f = runIdentityK $ foreverK $ request >=> respond . f

We can use the Proxy laws to prove that:

 mapD f >-> mapD g = mapD (g . f)

 mapD id = idT

... which is what we expect. We can fuse two consecutive mapDs into one by composing their functions, and mapping id does nothing at all, just like the identity proxy: idT.

In fact, these are just the functor laws in disguise, where mapD defines a functor between the category of Haskell function composition and the category of Proxy composition. Control.Proxy.Prelude.Base is full of utilities like this that are simultaneously practical and theoretically elegant.

Mixing Base Monads

Composition can't interleave two proxies if their base monads do not match. For instance, I might try to modify promptInt to use EitherT String to report the error instead of using exceptions:

 import Control.Monad.Trans.Either -- from the "either" package
 import Safe (readMay)

 promptInt2 :: (Proxy p) => () -> Producer p Int (EitherT String IO) r
 promptInt2 () = runIdentityP $ forever $ do
     str <- lift $ lift $ do
         putStrLn "Enter an Integer:"
         getLine
     case readMay str of
         Nothing -> lift $ left "Could not read Integer"
         Just n  -> respond n

However, if I try to compose it with printer, I receive a type error:

>>> runEitherT $ runProxy $ promptInt2 >-> printer
<interactive>:2:40:
    Couldn't match expected type `EitherT String IO'
                with actual type `IO'
    ...

The type error says that promptInt2 uses (EitherT String IO) for its base monad, but printer uses IO for its base monad, so composition can't interleave their effects.

You can easily fix this using the hoist function from the MFunctor type class in Control.MFunctor, which transforms the base monad of any monad transformer, including the Proxy monad transformer. Control.MFunctor really belongs in the transformers package, however it currently resides here because it requires the Rank2Types extension.

You will commonly use hoist to lift one proxy's base monad to match another proxy's base monad, like so:

>>> runEitherT $ runProxy $ promptInt2 >-> (hoist lift . printer)
Enter an Integer:
Hello<Enter>
Left "Could not read Integer"

This library provides three syntactic conveniences for making this easier to write.

First, (.) has higher precedence than (>->), so you can drop the parentheses:

>>> runEitherT $ runProxy $ promptInt2 >-> hoist lift . printer
...

Second, lift is such a common argument to hoist that Control.MFunctor provides the raise function:

 raise = hoist lift
>>> runEitherT $ runProxy $ promptInt2 >-> raise . printer
...

Third, Control.Proxy.Prelude.Kleisli provides the hoistK and raiseK functions in case you think composition looks ugly:

 hoistK f = (hoist f .)

 raiseK = (raise .)
>>> runEitherT $ runProxy $ promptInt2 >-> raiseK printer
...

Note that Control.MFunctor also provides MFunctor instances for all the monad transformers in the transformers package. This means that you can fix any incompatibility between two monad transformer stacks just using various combinations of hoist and lift.

To see how, consider the following contrived pathological example where I want to mix two very different monad transformer stacks:

 m1 :: StateT s (ReaderT i IO) r
 m2 :: MaybeT   (WriterT w IO) r

I can interleave their transformers through judicious use of hoist and lift

 mBoth :: StateT s (MaybeT (ReaderT i (WriterT w IO))) r
 mBoth = do
     hoist (lift . hoist lift) m1
     lift (hoist lift m2)

Utilities

The Control.Proxy.Prelude heirarchy provides several utility functions for common tasks. We can redefine the previous example functions just by composing these utilities.

For example, readLnS reads values from user input, so we can read Ints just by specializing its type:

 readLnS :: (Proxy p, Read a) => () -> Producer p a IO r

 readIntS :: (Proxy p) => () -> Producer p Int IO r
 readIntS = readLnS

The S suffix indicates that it belongs in the 'S'erver position.

(takeB_ n) allows at most n values to pass through it in 'B'oth directions:

 takeB_ :: (Monad m, Proxy p) => Int -> a' -> p a' a a' a m ()

takeB_ has a more general type than take' because it allows any type of value to flow upstream.

printD prints all values flowing 'D'ownstream:

 printD :: (Proxy p, Show a) => x -> p x a x a IO r

printD has a more general type than our original printer because it forwards all values further downstream after printing them. This means that you could use it as an intermediate stage as well. However, printD still type-checks as the most downstream stage, too, since runProxy just discards any unused outbound values.

These utilities do not clash with the Prelude namespace or common libraries because they all end with a capital letter suffix that indicates their directionality:

  • 'D' suffix: interacts with values flowing 'D'ownstream
  • 'U' suffix: interacts with values flowing 'U'pstream
  • 'B' suffix: interacts with values flowing 'B'oth ways (or: 'B'idirectional)
  • 'S' suffix: belongs furthest upstream in the 'S'erver position
  • 'C' suffix: belongs furthest downstream in the 'C'lient position

We can assemble these functions into a silent version of our previous Session:

>>> runProxy $ readIntS >-> takeB_ 2 >-> printD
4<Enter>
4
39<Enter>
39

Fortunately, we don't have to give up our previous useful diagnostics. We can use execU, which executes an action each time values flow upstream through it, and execD, which executes an action each time values flow downstream through it:

 promptInt :: (Proxy p) => () -> Producer p Int IO r
 promptInt = readLnS >-> execU (putStrLn "Enter an Integer:")

 printer :: (Proxy p, Show a) => x -> p x a x a IO r
 printer = execD (putStrLn "Received a value:") >-> printD

Similarly, we can build our old take' on top of takeB_:

 take' :: (Proxy p) => Int -> a' -> p a' a a' a m ()
 take' n a' = runIdentityP $ do  -- Remember, we need 'runIdentityP' if
     takeB_ n a'                 -- we use 'do' notation or 'lift'
     lift $ putStrLn "You shall not pass!"
>>> runProxy $ promptInt >-> take' 2 >-> printer
<Exact same behavior>

Or perhaps I want to skip user input for testing and mock promptInt by replacing it with a predefined set of values:

>>> runProxy $ fromListS [4, 37, 1] >-> take'2 >-> printer
Received a value:
4
Received a value:
37

What about our original lines' function? That's just hGetLineS:

 hGetLineS :: (Proxy p) => Handle -> () -> Producer p String IO ()

You could hand-write loops that accomplish these same tasks, but proxies let you:

  • Rapidly swap in and out components for testing, debugging, and fast prototyping
  • Factor out common patterns into modular components
  • Mix and match simple stages to build sophisticated programs

This compositional programming style emphasizes building a library of reusable components and connecting them like Unix pipes to assemble the desired streaming program.

Mix Monads and Composition

Composition isn't the only way to assemble proxies. You can also sequence predefined proxies using do notation to generate more elaborate behaviors.

Most commonly, you will sequence sources to combine their outputs, very similar to how the Unix cat utility behaves:

 threeSources () = do
     source1 ()
     source2 ()
     source3 ()

 -- or: threeSources = source1 >=> source2 >=> source3

As a concrete example, we could create a Producer where our first source presets the first few values and then we let the user take over to generate the remaining values:

 source1 :: (Proxy p) => () -> Producer p Int IO r
 source1 () = runIdentityP $ do
     fromListS [4, 4] ()  -- Source 1
     readLnS ()           -- Source 2

 -- or: source1 = runIdentityK (fromListS [4, 4] >=> readLnS)
>>> runProxy $ source1 >-> printD
4
4
70<Enter>
70
34<Enter>
34
...

What if we only want the user to provide three values? We can selectively throttle it with takeB_:

 source2 :: (Proxy p) => () -> Producer p Int IO ()
 source2 () = runIdentityP $ do
     fromListS [4, 4] ()
     (readLnS >-> takeB_ 3) () -- You can compose inside a do block!

 -- or: source2 = runIdentityK (fromListS [4, 4] >=> (readLnS >-> takeB_ 3))

Notice that composition works inside of a do block! This is a very handy trick!

>>> runProxy $ source2 >-> printD
4
4
56<Enter>
56
41<Enter>
41
80<Enter>
80

You can also concatenate sinks, too:

 sink1 :: (Proxy p) => () -> Consumer p Int IO ()
 sink1 () = do
     (takeB_ 3         >-> printD) () -- Sink 1
     (takeWhileD (< 4) >-> printD) () -- Sink 2

 -- or: sink1 = (takeB_ 3 >-> printD) >=> (takeWhileD (< 4) >-> printD)
>>> runProxy $ source2 >-> sink1
4          -- The first sink
4          -- handles these
68<Enter>  --
68
1<Enter>   -- The second sink
1          -- handles these
5<Enter>   --

... but the above example is gratuitous because you can simply concatenate the intermediate stages:

 sink2 :: (Proxy p) => () -> Consumer p Int IO ()
 sink2 () = intermediate >-> printD where
     intermediate () = do
         takeB_ 3 ()       -- Intermediate stage 1
         takeWhileD (< 4)  -- Intermediate stage 2

 -- or: sink2 = (takeB_ 3 >=> takeWhileD (< 4)) >-> printD
>>> runProxy $ source2 >-> sink2
<Exact same behavior>

These examples demonstrate the two principal ways to combine proxies:

  • "Vertical" composition, using (>=>) from the Kleisli category
  • "Horizontal" composition: using (>->) from the Proxy category

You assemble most proxies simply by composing them in one or both of these two categories.

Folds

You can fold a stream of values in two ways, both of which use the base monad:

  • Use WriterT in the base monad and tell the values to fold
  • Use StateT in the base monad and put strict values

WriterT is more elegant in principle but leaks space for a large number of values to fold. StateT does not leak space if you keep the accumulator strict, but is less elegant and doesn't guarantee write-only behavior. To remedy this, I am currently working on a stricter WriterT implementation that does not leak space to add to the transformers package.

Control.Proxy.Prelude.Base provides several common folds using WriterT as the base monad, such as:

  • lengthD: Count how many values flow downstream
 lengthD :: (Monad m, Proxy p) => x -> p x a x a (WriterT (Sum Int) m) r
  • toListD: Fold the values flowing downstream into a list.
 toListD :: (Monad m, Proxy p) => x -> p x a x a (WriterT [a] m) r
  • anyD: Determine whether any values satisfy the predicate
 anyD :: (Monad m, Proxy p) => (a -> Bool) -> x -> p x a x a (WriterT Any m) r

These WriterT versions demonstrate how the elegant approach should work in principle and they should be okay for folding a medium number of values until I release the fixed WriterT. If space leaks cause problems, you can temporarily rewrite the WriterT folds using the following two strict StateT folds:

  • foldlD': Strictly fold values flowing downstream
 foldlD'
  :: (Monad m, Proxy p) => (b -> a -> b) -> x -> p x a x a (StateT b m) r
  • foldlU': Strictly fold values flowing upstream
 foldU'
  :: (Monad m, Proxy p) => (b -> a' -> b) -> a' -> p a' x a' x (StateT b m) r

Now, let's try these folds out and see if we can build a list from user input:

>>> runWriterT $ runProxy $ raiseK promptInt >-> takeB_ 3 >-> toListD
Enter an Integer:
1<Enter>
Enter an Integer:
66<Enter>
Enter an Integer:
5<Enter>
((), [1, 66, 5])

Notice that promptInt uses IO as its base monad, but toListD uses (WriterT [Int] m) as its base monad, so I use raiseK to get the base monads to match.

You can insert these folds anywhere in the middle of a pipeline and they still work:

>>> runWriterT $ runProxy $ fromListS [5, 7, 4] >-> lengthD >-> raiseK printD
5
7
4
((), Sum 3)

You can also run multiple folds at the same time just by adding more WriterT layers to your base monad:

>>> runWriterT $ runWriterT $ runProxy $ fromListS [9, 10] >-> anyD even >-> raiseK sumD
(((), Any {getAny = True},Sum {getSum = 19})

I designed certain special folds to terminate the Session early if they can compute their result prematurely, in order to draw as little input as possible. These folds end with an underscore, such as headD_, which terminates the stream once it receives an input:

 headD_ :: (Monad m, Proxy p) => x -> p x a x a (WriterT (First a) m) ()
>>> runWriterT $ runProxy $ fromListS [3, 4, 9] >-> raiseK printD >-> headD_
3
((), First {getFirst = Just 3})

Compare this to headD without underscore, which folds the entire input:

>>> runWriterT $ runProxy $ fromListS [3, 4, 9] >-> raiseK printD >-> headD
3
4
9
((), First {getFirst = Just 3})

Use the versions that don't prematurely terminate if you are running multiple folds or if you want to continue to use the rest of the input when the fold is done. Use the versions that do prematurely terminate if collecting that single fold is the entire purpose of the session.

Resource Management

This core library provides utilities for lazily streaming from resources, but does not provide utilities for lazily managing resource allocation and deallocation. To frame the problem, let's assume that we try to be clever and write a streaming utility that lazily opens a file only in response to a request, such as the following Producer:

 readFileS :: FilePath -> () -> Producer p String IO
 readFileS file () = runIdentityP $ do
     h <- lift $ openFile file ReadMode
     lift $ putStrLn "Opening file"
     hGetLineS h ()
     lift $ putStrLn "Closing file"
     lift $ hClose h

This works well if we fully demand the file:

>>> runProxy $ readFileS "test.txt" >-> printD
Opening file
"Line 1"
"Line 2"
"Line 3"
Closing file

This also works well if we never demand the file at all, in which case we never open it:

>>> runProxy $ readFileS "test.txt" >-> return
-- Outputs nothing

But it gives exactly the wrong behavior if we partially demand the file:

>>> runProxy $ readFileS "test.txt" >-> takeB_ 1 >-> printD
Opening file
"Line 1"

Notice that this does not close the file, because once takeB_ 1 terminates it terminates the entire Session and readFileS does not get a chance to finalize the file.

The pipes-safe library solves this problem by providing resource management abstractions built on top of pipes and offers several other nice features:

  • It is completely exception safe, even against asynchronous exceptions
  • It is backwards compatible with "unmanaged" ordinary proxies

Backwards compatibility means that you don't need to buy in to the pipes-safe way of doing things. For example, another common approach is to just open all resources before running the session and close them all afterward. For example, if I wanted to emulate the Unix cp command, streaming one line at a time, I might write:

 import System.IO

 cp :: FilePath -> FilePath -> IO ()
 cp inFile outFile =
     withFile file1 ReadMode  $ \hIn  ->
     withFile file2 WriteMode $ \hOut ->
     runProxy $ hGetLineS hIn >-> hPutLineS hOut

Some people prefer that approach because it:

  • is straightforward, and
  • can reuse functions from Control.Exception

The disadvantage is that this does not lazily allocate resources, nor does this promptly deallocate them. Also, there is no way to recover from exceptions and resume the Session. On the other hand, pipes-safe lets you do all of these.

Fortunately, you can choose whichever approach you prefer and rest assured that the two approaches safely interoperate. Control.Proxy.Safe.Tutorial from the pipes-safe package provides a separate tutorial on how to:

  • extend pipes with resource management,
  • handle exceptions natively within proxies, and
  • interoperate with unmanaged code.

Extensions

This library provides several extensions that add features on top of the base Proxy API. These extensions behave like monad transformers, except that they also lift the Proxy class through the extension so that the extended proxy can still request, respond, and compose with other proxies:

 instance (Proxy p) => Proxy (IdentityP p)  -- Equivalent to IdentityT
 instance (Proxy p) => Proxy (EitherP e p)  -- Equivalent to EitherT
 instance (Proxy p) => Proxy (MaybeP    p)  -- Equivalent to MaybeT
 instance (Proxy p) => Proxy (StateP  s p)  -- Equivalent to StateT
 instance (Proxy p) => Proxy (WriterP w p)  -- Equivalent to WriterT

Each of these proxy transformers provides the same API as the equivalent monad transformer (sometimes even more). The following sections show some common problems that these proxy transformers solve.

Error handling

Our previous promptInt example suffered from one major flaw:

 promptInt2 :: (Proxy p) => () -> Producer p Int (EitherT String IO) r
 promptInt2 () = runIdentityP $ forever $ do
     str <- lift $ lift $ do
         putStrLn "Enter an Integer:"
         getLine
     case readMay str of
         Nothing -> lift $ left "Could not read Integer"
         Just n  -> respond n

There is no way to recover from the error and resume streaming data. You can only handle the Left value after using runProxy, but by then it is too late.

We can solve this by switching the order of the two monad transformers, but using EitherP this time instead of EitherT:

 import qualified Control.Proxy.Trans.Either as E

 --               Proxy transformers play
 --               nice with type synonyms --+
 --                                         |
 --                                         v
 promptInt3 :: (Proxy p) => () -> Producer (E.EitherP String p) Int IO r
 -- i.e.       (Proxy p) => () -> EitherP String p C () () Int IO r

 promptInt3 () = forever $ do
     str <- lift $ do
         putStrLn "Enter an Integer:"
         getLine
     case readMay str of
         Nothing -> E.throw "Could not read Integer"
         Just n' -> respond n

This example does not need runIdentityP (nor would that type-check) because the EitherP proxy transformer gives the compiler enough information to generalize the constraints.

We've swapped the order of the transformers, so now we use runEitherK first to unwrap the EitherP followed by runProxy.

 runEitherK
  :: (q -> EitherP p a' a b' b m r) -> (q -> p a' a b' b m (Either e r))
>>> runProxy $ runEitherK $ promptInt3 >-> printer :: IO (Either String r)
Enter an Integer:
Hello<Enter>
Left "Could not read Integer"

Notice how we can directly compose printer with promptInt. This works because printer's base proxy type is completely polymorphic over the Proxy type class and doesn't use any features specific to any proxy transformers:

                  'p' type-checks as anything --+
                   that implements 'Proxy'      |
                                                v
 printer :: (Proxy p, Show a) => () -> Consumer p a IO r

This means that you can compose printer with anything that implements the Proxy type class, including EitherP, without any lifting.

EitherP lets us catch and handle errors locally without disturbing other proxies. For example, I can define a heartbeat function that just restarts a given proxy each time it raises an error:

 heartbeat
  :: (Proxy p)
  => E.EitherP String p a' a b' b IO r
  -> E.EitherP String p a' a b' b IO r
 heartbeat p = p `E.catch` \err -> do
     lift $ putStrLn err  -- Print the error
     heartbeat p          -- Restart 'p'

This uses the catch function from Control.Proxy.Trans.Either, which lets you catch and handle errors locally without disturbing other proxies.

>>> runProxy $ E.runEitherK $ (heartbeat . promptInt3) >-> takeB_ 2 >-> printer
Enter an Integer:
Hello<Enter>
Could not read Integer
Enter an Integer
8
Received a value:
8
Enter an Integer
0
Received a value:
0

It's very easy to prove that EitherP has only a local effect. In fact, we can run it entirely locally like so:

>>> runProxy $ (E.runEitherK $ heartbeat . promptInt3) >-> takeB_ 2 >-> printer

Proxy transformers do not use the base monad at all, so you can use them to isolate effects from other proxies, as the next section demonstrates.

Local state

The StateP proxy lets you embed local state into any Proxy computation. For example, we might want to gratuitously use state to generate successive numbers:

 import qualified Control.Proxy.Trans.State as S

 increment :: (Monad m, Proxy p) => () -> Producer (S.StateP Int p) Int m r
 increment () = forever $ do
     n <- S.get
     respond n
     S.put (n + 1)

We could then embed it locally into any Proxy, such as the following one:

 numbers :: (Monad m, Proxy p) => () -> Producer p Int m ()
 numbers () = runIdentityP $ do
     (takeB_ 5 <-< S.evalStateK 10 increment) ()
     S.evalStateK 1  (takeB_ 3 <-< increment) () -- This works, too!
>>> runProxy $ numbers >-> printD
10
11
12
13
14
1
2
3

We can also prove the effect is local even when you directly compose two StateP proxies before running them. Let's define a stateful consumer:

 increment2 :: (Proxy p) => () -> Consumer (S.StateP Int p) Int IO r
 increment2 () = forever $ do
     nOurs   <- S.get
     nTheirs <- request ()
     lift $ print (nTheirs, nOurs)
     S.put (nOurs + 2)

.. and hook it up directly to increment:

>>> runProxy $ S.evalStateK 0 $ increment >-> takeB_ 3 >-> increment2
(0, 0)
(1, 2)
(2, 4)

They each share the same initial state, but they isolate their own side effects completely from each other.

Branching, zips, and merges

So far we've only considered linear chains of proxies, but pipes allows you to branch these chains and generate more sophisticated topologies. The trick is to simply nest the Proxy monad transformer within itself.

For example, if I want to zip two inputs, I can just define the following triply nested proxy:

 zipD
  :: (Monad m, Proxy p1, Proxy p2, Proxy p3)
  => () -> Consumer p1 a (Consumer p2 b (Producer p3 (a, b) m)) r
 zipD () =
     runIdentityP . hoist (runIdentityP . hoist runIdentityP) $ forever $ do
     -- Yes, this 'runIdentityP' mess is necessary.  Sorry!

         a <- request ()               -- Request from the outer 'Consumer'
         b <- lift $ request ()        -- Request from the inner 'Consumer'
         lift $ lift $ respond (a, b)  -- Respond to the 'Producer'

zipD behaves analogously to a curried function. We partially apply it to each layer using composition and runProxyK or runProxy:

 -- 1st application
 p1 = runProxyK $ zipD <-< fromListS [1..3]

 -- 2nd application
 p2 = runProxyK $ p1 <-< fromListS [4..]

 -- 3rd application
 p3 = runProxy $ printD <-< p2
>>> p3
(1, 4)
(2, 5)
(3, 6)

You can use this trick to fork output, too:

 fork
  :: (Monad m, Proxy p1, Proxy p2, Proxy p3)
  => () -> Consumer p1 a (Producer p2 a (Producer p3 a m)) r
 fork () =
     runIdentityP . hoist (runIdentityP . hoist runIdentityP) $ forever $ do
         a <- request ()          -- Request output from the 'Consumer'
         lift $ respond a         -- Send output to the outer 'Producer'
         lift $ lift $ respond a  -- Send output to the inner 'Producer'

Again, we just keep partially applying it until it is fully applied:

 -- 1st application
 p1 = runProxyK $ fork <-< fromListS [1..3]

 -- 2nd application
 p2 = runProxyK $ raiseK printD <-< mapD (> 2) <-< p1

 -- 3rd application
 p3 = runProxy  $ printD <-< mapD show <-< p2
>>> p3
False
"1"
False
"2"
True
"3"

You can even merge or fork proxies that use entirely different feature sets:

 p1 = runProxyK $ S.evalStateK 0 $ fork <-< increment

 p2 = runProxyK $ raiseK printD <-< mapD (+ 10) <-< p1

 p3 = runProxy  $ E.runEitherK $ printD <-< (takeB_ 3 >=> E.throw) <-< p2
>>> p3
10
0
11
1
12
2
Left ()

We just forked a (StateP p1) proxy and read out the result in both a generic p2 proxy and an (EitherP p3) proxy. That's pretty crazy, but it gives you a sense of how versatile and robust proxies can be.

You can implement arbitrary branching topologies using this trick. However, I want to mention a few caveats:

  • The intermediate partially applied type signatures will be ugly as sin. I warned you.
  • You cannot implement cyclic topologies (and cyclic topologies do not make sense for proxies anyway)
  • You cannot use this trick to implement a polymorphic zip function of the following form:
 zip'  -- You can't define this
  :: (Monad m, Proxy p)
  => (() -> Producer p a      m r)
  -> (() -> Producer p b      m r)
  -> (() -> Producer p (a, b) m r)

Partial application requires selecting a Proxy instance, which is why you cannot define zip'. You can define a zip' specialized to a concrete Proxy instance, but I don't really recommend doing that since you should always strive to write polymorphic proxies to avoid locking your user into a particular feature set.

With those caveats out of the way, this approach affords many indispensable features that other approaches do not allow:

  • It does not require extending the Proxy type class
  • It handles almost every branching scenario, including more complicated situations like concurrent interleavings
  • You can branch and merge mixtures of Servers, Clients, and Proxys
  • You can branch and merge heterogeneous feature sets
  • It is completely polymorphic over the Proxy class and uses no implementation-specific details

Proxy Transformers

There is one last scenario that you will eventually encounter: mixing proxies that have incompatible proxy transformer stacks. You solve this the exact same way you mix different monad transformer stacks, except that instead of using lift and hoist you use liftP and hoistP.

For example, we might want to mix promptInt3 and increment2:

 promptInt3 :: (Proxy p) => () -> Producer (E.EitherP String p) Int IO r

 increment2 :: (Proxy p) => () -> Consumer (S.StateP Int p) Int IO r

Unfortunately, they use two different feature sets so neither one is fully polymorphic over the Proxy class and we cannot directly compose them.

Fortunately, all proxy transformers implement the ProxyTrans class, analogous to the MonadTrans class for transformers:

 class ProxyTrans t where
     liftP
       :: (Monad m, Proxy p)
       => p a' a b' b m r -> t p a' a b' b m r

 -- mapP is slightly more elegant
 mapP
  :: (Monad m, Proxy p, ProxyTrans t)
  => (q -> p a' a b' b m r) -> (q -> t p a' a b' b m r)
 mapP = (liftP . )

It's very easy to use. Just use mapP to lift one proxy transformer to match another one. For example, we can mapP increment2 to match promptInt3:

 promptInt3 >-> mapP increment2
  :: (Proxy p) => () -> Session (EitherP String (StateP Int p)) IO r
>>> runProxy $ S.evalStateK 0 $ E.runEitherK $ promptInt3 >-> mapP increment2
Enter an Integer:
4<Enter>
(4, 0)
Enter an Integer:
5<Enter>
(5, 2)
Enter an Integer:
Hello<Enter>
Left "Could not read Integer"

... or we could instead mapP promptInt3 to match increment2 and switch the order of the two proxy transformers:

 mapP promptInt3 >-> increment2
  :: (Proxy p) => () -> Session (StateP Int (EitherP String p)) IO r
>>> runProxy $ E.runEitherK $ S.evalStateK 0 $ mapP promptInt3 >-> increment2
Enter an Integer:
4<Enter>
(4, 0)
Enter an Integer:
5<Enter>
(5, 2)
Enter an Integer:
Hello<Enter>
Left "Could not read Integer"

Like monad transformers, proxy transformers lift a base Monad instance to an extended Monad instance. liftP exactly mirrors the lift function from MonadTrans. liftP takes some base proxy, p, that implements Monad and "lift"s it to an extended proxy, (t p), that also implements Monad.

So for example, I could do something like:

 do liftP $ actionInBaseProxy
    actionInExtendedProxy

Monad transformers impose certain laws to ensure that this lifting is correct. These are known as the monad transformer laws;

 (lift .) (f >=> g) = (lift .) f >=> (lift .) g

 (lift .) return = return

If you convert these laws to do notation, they just say:

 do  x <- lift m  =  lift $ do x <- m
     lift (f x)                f x

 lift (return r) = return r

Proxy transformers require the exact same laws to ensure that they lift the base monad to the extended monad correctly. Just replace lift with liftP:

 (liftP .) (f >=> g) = (liftP .) f >=> (liftP .) g

 (liftP .) return = return

The only difference is mapP sweetens these laws a little bit:

 mapP = (liftP .)

 mapP (f >=> g) = mapP f >=> mapP g  -- These are functor laws!

 mapP return = return

However, proxy transformers do one extra thing above and beyond ordinary monad transformers. Proxy transformers lift the Proxy type class, meaning that if the base type implements Proxy, so does the extended type.

This means that we need a set of laws that guarantee that the proxy transformer lifts the Proxy instance correctly. I call these laws the "proxy morphism laws":

 mapP (f >-> g) = mapP f >-> mapP g  -- These are functor laws, too!

 mapP idT = idT

In other words, a proxy transformer defines a functor from the base composition to the extended composition! Neat!

But we're not even done, because proxies actually form three other categories, only one of which I have mentioned so far, and proxy transformers lift these three other categories, too:

 -- The push-based category

 mapP (f >~> g) = mapP f >~> mapP g

 mapP coidT = coidT
 -- The "request" category

 mapP (f \>\ g) = mapP f \>\ mapP g

 mapP request = request
 -- The "respond" category

 mapP (f />/ g) = mapP f />/ mapP g

 mapP respond = respond

I never even mentioned those last two categories because they are more exotic and you probably never need to use them. However, even if we never use those categories they still guarantee two really important laws that we should remember:

 mapP request = request

 mapP respond = respond

We can translate those back to liftP to get:

 liftP (request a') = request a'

 liftP (respond b)  = respond b

In other words, request and respond in the extended proxy must behave exactly the same as lifting request and respond from the base proxy.

All the proxy transformers in this library obey the proxy morphism laws, which ensure that liftP / mapP always do "the right thing".

Proxy transformers also implement hoistP from the PFunctor class in Control.PFunctor. This exactly parallels hoist for monad transformers.

Just like monad transformers, we can mix two completely exotic proxy transformer stacks using a combination of liftP and hoistP. Here's the proxy transformer equivalent of the previous example I gave:

 p1 :: (Proxy p) => a' -> StateP s (ReaderP i p) a' a a' a m r
 p2 :: (Proxy p) => a' -> MaybeP   (WriterP w p) a' a a' a m r

As before, I can interleave their proxy transformers through judicious use of hoistP and liftP

 pSequence
  :: (Proxy p) => StateP s (MaybeP (ReaderP i (WriterP w p))) a' a a' a r
 pSequence a' = do
     hoistP (liftP . hoistP liftP) (p1 a')
     liftP (hoistP liftP (p2 a'))

... but unlike ordinary monad transformers I could instead mix them by composition, too!

 pCompose
  :: (Proxy p) => StateP s (MaybeP (ReaderP i (WriterP w p))) a' a a' a r
 pCompose =
      hoistP (liftP . hoistP liftP) . p1
  >-> liftP . hoistP liftP . p2

Conclusion

The pipes library emphasizes the reuse of a small set of core abstractions grounded in theory to implement all functionality:

However, I don't expect everybody to immediately understand how so few primitives can implement such a wide variety of features. This tutorial gives a taste of how many interesting ways you can combine these few abstractions, but these examples barely scratch the surface, despite this tutorial's length. So if you don't know how to implement something using pipes, just ask me and I will be happy to help.