One case where a reader monad can be useful is when there is some notion of the "current configuration" of the application that is passed through many recursive calls. An example of such a program is tree, which recursively prints the files in the current directory and its subdirectories, indicating their tree structure using characters. The version of tree in this chapter, called doug after the mighty Douglas Fir tree that adorns the west coast of North America, provides the option of Unicode box-drawing characters or their ASCII equivalents when indicating directory structure.

For example, the following commands create a directory structure and some empty files in a directory called doug-demo:

$cd doug-demo$ mkdir -p a/b/c
$mkdir -p a/d$ mkdir -p a/e/f
$touch a/b/hello$ touch a/d/another-file
$touch a/e/still-another-file-again  Running doug results in the following: $ doug
├── doug-demo/
│   ├── a/
│   │   ├── e/
│   │   │   ├── f/
│   │   │   ├── still-another-file-again
│   │   ├── d/
│   │   │   ├── another-file
│   │   ├── b/
│   │   │   ├── hello
│   │   │   ├── c/


## Implementation

Internally, doug passes a configuration value downwards as it recursively traverses the directory structure. This configuration contains two fields: useASCII determines whether to use Unicode box-drawing characters or ASCII vertical line and dash characters to indicate structure, and currentPrefix contains a string to prepend to each line of output. As the current directory deepens, the prefix string accumulates indicators of being in a directory. The configuration is a structure:

structure Config where
useASCII : Bool := false
currentPrefix : String := ""


This structure has default definitions for both fields. The default Config uses Unicode display with no prefix.

Users who invoke doug will need to be able to provide command-line arguments. The usage information is as follows:

def usage : String :=
"Usage: doug [--ascii]
Options:
\t--ascii\tUse ASCII characters to display the directory structure"


Accordingly, a configuration can be constructed by examining a list of command-line arguments:

def configFromArgs : List String → Option Config
| [] => some {} -- both fields default
| ["--ascii"] => some {useASCII := true}
| _ => none


The main function is a wrapper around an inner worker, called dirTree, that shows the contents of a directory using a configuration. Before calling dirTree, main is responsible for processing command-line arguments. It must also return the appropriate exit code to the operating system:

def main (args : List String) : IO UInt32 := do
match configFromArgs args with
| some config =>
dirTree config (← IO.currentDir)
pure 0
| none =>
IO.eprintln s!"Didn't understand argument(s) {" ".separate args}\n"
IO.eprintln usage
pure 1


Not all paths should be shown in the directory tree. In particular, files named . or .. should be skipped, as they are actually features used for navigation rather than files per se. Of those files that should be shown, there are two kinds: ordinary files and directories:

inductive Entry where
| file : String → Entry
| dir : String → Entry


To determine whether a file should be shown, along with which kind of entry it is, doug uses toEntry:

def toEntry (path : System.FilePath) : IO (Option Entry) := do
match path.components.getLast? with
| none => pure (some (.dir ""))
| some "." | some ".." => pure none
| some name =>
pure (some (if (← path.isDir) then .dir name else .file name))


System.FilePath.components converts a path into a list of path components, splitting the name at directory separators. If there is no last component, then the path is the root directory. If the last component is a special navigation file (. or ..), then the file should be excluded. Otherwise, directories and files are wrapped in the corresponding constructors.

Lean's logic has no way to know that directory trees are finite. Indeed, some systems allow the construction of circular directory structures. Thus, dirTree is declared partial:

partial def dirTree (cfg : Config) (path : System.FilePath) : IO Unit := do
match ← toEntry path with
| none => pure ()
| some (.file name) => showFileName cfg name
| some (.dir name) =>
showDirName cfg name
let newConfig := cfg.inDirectory
doList contents.toList fun d =>
dirTree newConfig d.path


The call to toEntry is a nested action—the parentheses are optional in positions where the arrow couldn't have any other meaning, such as match. When the filename doesn't correspond to an entry in the tree (e.g. because it is ..), dirTree does nothing. When the filename points to an ordinary file, dirTree calls a helper to show it with the current configuration. When the filename points to a directory, it is shown with a helper, and then its contents are recursively shown in a new configuration in which the prefix has been extended to account for being in a new directory.

Showing the names of files and directories is achieved with showFileName and showDirName:

def showFileName (cfg : Config) (file : String) : IO Unit := do
IO.println (cfg.fileName file)

def showDirName (cfg : Config) (dir : String) : IO Unit := do
IO.println (cfg.dirName dir)


Both of these helpers delegate to functions on Config that take the ASCII vs Unicode setting into account:

def Config.preFile (cfg : Config) :=
if cfg.useASCII then "|--" else "├──"

def Config.preDir (cfg : Config) :=
if cfg.useASCII then "|  " else "│  "

def Config.fileName (cfg : Config) (file : String) : String :=
s!"{cfg.currentPrefix}{cfg.preFile} {file}"

def Config.dirName (cfg : Config) (dir : String) : String :=
s!"{cfg.currentPrefix}{cfg.preFile} {dir}/"


Similarly, Config.inDirectory extends the prefix with a directory marker:

def Config.inDirectory (cfg : Config) : Config :=
{cfg with currentPrefix := cfg.preDir ++ " " ++ cfg.currentPrefix}


Iterating an IO action over a list of directory contents is achieved using doList. Because doList carries out all the actions in a list and does not base control-flow decisions on the values returned by any of the actions, the full power of Monad is not necessary, and it will work for any Applicative:

def doList [Applicative f] : List α → (α → f Unit) → f Unit
| [], _ => pure ()
| x :: xs, action =>
action x *>
doList xs action


While this implementation of doug works, manually passing the configuration around is verbose and error-prone. The type system will not catch it if the wrong configuration is passed downwards, for instance. A reader effect ensures that the same configuration is passed to all recursive calls, unless it is manually overridden, and it helps make the code less verbose.

To create a version of IO that is also a reader of Config, first define the type and its Monad instance, following the recipe from the evaluator example:

def ConfigIO (α : Type) : Type :=
Config → IO α

pure x := fun _ => pure x
bind result next := fun cfg => do
let v ← result cfg
next v cfg


The difference between this Monad instance and the one for Reader is that this one uses do-notation in the IO monad as the body of the function that bind returns, rather than applying next directly to the value returned from result. Any IO effects performed by result must occur before next is invoked, which is ensured by the IO monad's bind operator. ConfigIO is not universe polymorphic because the underlying IO type is also not universe polymorphic.

Running a ConfigIO action involves transforming it into an IO action by providing it with a configuration:

def ConfigIO.run (action : ConfigIO α) (cfg : Config) : IO α :=
action cfg


This function is not really necessary, as a caller could simply provide the configuration directly. However, naming the operation can make it easier to see which parts of the code are intended to run in which monad.

The next step is to define a means of accessing the current configuration as part of ConfigIO:

def currentConfig : ConfigIO Config :=
fun cfg => pure cfg


This is just like read from the evaluator example, except it uses IO's pure to return its value rather than doing so directly. Because entering a directory modifies the current configuration for the scope of a recursive call, it will be necessary to have a way to override a configuration:

def locally (change : Config → Config) (action : ConfigIO α) : ConfigIO α :=
fun cfg => action (change cfg)


Much of the code used in doug has no need for configurations, and doug calls ordinary Lean IO actions from the standard library that certainly don't need a Config. Ordinary IO actions can be run using runIO, which ignores the configuration argument:

def runIO (action : IO α) : ConfigIO α :=
fun _ => action


With these components, showFileName and showDirName can be updated to take their configuration arguments implicitly through the ConfigIO monad. They use nested actions to retrieve the configuration, and runIO to actually execute the call to IO.println:

def showFileName (file : String) : ConfigIO Unit := do
runIO (IO.println ((← currentConfig).fileName file))

def showDirName (dir : String) : ConfigIO Unit := do
runIO (IO.println ((← currentConfig).dirName dir))


In the new version of dirTree, the calls to toEntry and System.FilePath.readDir are wrapped in runIO. Additionally, instead of building a new configuration and then requiring the programmer to keep track of which one to pass to recursive calls, it uses locally to naturally delimit the modified configuration to only a small region of the program, in which it is the only valid configuration:

partial def dirTree (path : System.FilePath) : ConfigIO Unit := do
match ← runIO (toEntry path) with
| none => pure ()
| some (.file name) => showFileName name
| some (.dir name) =>
showDirName name
locally (·.inDirectory)
(doList contents.toList fun d =>
dirTree d.path)


The new version of main uses ConfigIO.run to invoke dirTree with the initial configuration:

def main (args : List String) : IO UInt32 := do
match configFromArgs args with
| some config =>
(dirTree (← IO.currentDir)).run config
pure 0
| none =>
IO.eprintln s!"Didn't understand argument(s) {" ".separate args}\n"
IO.eprintln usage
pure 1


This custom monad has a number of advantages over passing configurations manually:

1. It is easier to ensure that configurations are passed down unchanged, except when changes are desired
2. The concern of passing the configuration onwards is more clearly separated from the concern of printing directory contents
3. As the program grows, there will be more and more intermediate layers that do nothing with configurations except propagate them, and these layers don't need to be rewritten as the configuration logic changes

However, there are also some clear downsides:

1. As the program evolves and the monad requires more features, each of the basic operators such as locally and currentConfig will need to be updated
2. Wrapping ordinary IO actions in runIO is noisy and distracts from the flow of the program
3. Writing monads instances by hand is repetitive, and the technique for adding a reader effect to another monad is a design pattern that requires documentation and communication overhead

1. A definition of the transformer itself, which is typically a function from types to types
2. A Monad instance that assumes the inner type is already a monad
3. An operator to "lift" an action from the inner monad to the transformed monad, akin to runIO

Adding a reader effect to IO was accomplished in ConfigIO by wrapping IO α in a function type. The Lean standard library contains a function that can do this to any polymorphic type, called ReaderT:

def ReaderT (ρ : Type u) (m : Type u → Type v) (α : Type u) : Type (max u v) :=
ρ → m α


Its arguments are as follows:

• ρ is the environment that is accessible to the reader
• m is the monad that is being transformed, such as IO
• α is the type of values being returned by the monadic computation Both α and ρ are in the same universe because the operator that retrieves the environment in the monad will have type m ρ.

With ReaderT, ConfigIO becomes:

abbrev ConfigIO (α : Type) : Type := ReaderT Config IO α


It is an abbrev because ReaderT has many useful features defined in the standard library that a non-reducible definition would hide. Rather than taking responsibility for making these work directly for ConfigIO, it's easier to simply have ConfigIO behave identically to ReaderT Config IO.

The manually-written currentConfig obtained the environment out of the reader. This effect can be defined in a generic form for all uses of ReaderT, under the name read:

def read [Monad m] : ReaderT ρ m ρ :=
fun env => pure env


However, not every monad that provides a reader effect is built with ReaderT. The type class MonadReader allows any monad to provide a read operator:

class MonadReader (ρ : outParam (Type u)) (m : Type u → Type v) : Type (max (u + 1) v) where

read := fun env => pure env



The type ρ is an output parameter because any given monad typically only provides a single type of environment through a reader, so automatically selecting it when the monad is known makes programs more convenient to write.

The Monad instance for ReaderT is essentially the same as the Monad instance for ConfigIO, except IO has been replaced by some arbitrary monad argument m:

instance [Monad m] : Monad (ReaderT ρ m) where
pure x := fun _ => pure x
bind result next := fun env => do
let v ← result env
next v env


The next step is to eliminate uses of runIO. When Lean encounters a mismatch in monad types, it automatically attempts to use a type class called MonadLift to transform the actual monad into the expected monad. This process is similar to the use of coercions. MonadLift is defined as follows:

class MonadLift (m : Type u → Type v) (n : Type u → Type w) where
monadLift : {α : Type u} → m α → n α


The method monadLift translates from the monad m to the monad n. The process is called "lifting" because it takes an action in the embedded monad and makes it into an action in the surrounding monad. In this case, it will be used to "lift" from IO to ReaderT Config IO, though the instance works for any inner monad m:

instance : MonadLift m (ReaderT ρ m) where
monadLift action := fun _ => action


The implementation of monadLift is very similar to that of runIO. Indeed, it is enough to define showFileName and showDirName without using runIO:

def showFileName (file : String) : ConfigIO Unit := do

def showDirName (dir : String) : ConfigIO Unit := do


One final operation from the original ConfigIO remains to be translated to a use of ReaderT: locally. The definition can be translated directly to ReaderT, but the Lean standard library provides a more general version. The standard version is called withReader, and it is part of a type class called MonadWithReader:

class MonadWithReader (ρ : outParam (Type u)) (m : Type u → Type v) where
withReader {α : Type u} : (ρ → ρ) → m α → m α


Just as in MonadReader, the environment ρ is an outParam. The withReader operation is exported, so that it doesn't need to be written with the type class name before it:

export MonadWithReader (withReader)


The instance for ReaderT is essentially the same as the definition of locally:

instance : MonadWithReader ρ (ReaderT ρ m) where
fun cfg => action (change cfg)


With these definitions in place, the new version of dirTree can be written:

partial def dirTree (path : System.FilePath) : ConfigIO Unit := do
match ← toEntry path with
| none => pure ()
| some (.file name) => showFileName name
| some (.dir name) =>
showDirName name
(doList contents.toList fun d =>
dirTree d.path)


Aside from replacing locally with withReader, it is the same as before.

Replacing the custom ConfigIO type with ReaderT did not save a large number of lines of code in this section. However, rewriting the code using components from the standard library does have long-term benefits. First, readers who know about ReaderT don't need to take time to understand the Monad instance for ConfigIO, working backwards to the meaning of monad itself. Instead, they can be confident in their initial understanding. Next, adding further effects to the monad (such as a state effect to count the files in each directory and display a count at the end) requires far fewer changes to the code, because the monad transformers and MonadLift instances provided in the library work well together. Finally, using a set of type classes included in the standard library, polymorphic code can be written in such a way that it can work with a variety of monads without having to care about details like the order in which the monad transformers were applied. Just as some functions work in any monad, others can work in any monad that provides a certain type of state, or a certain type of exceptions, without having to specifically describe the way in which a particular concrete monad provides the state or exceptions.

## Exercises

### Controlling the Display of Dotfiles

Files whose names begin with a dot character ('.') typically represent files that should usually be hidden, such as source-control metadata and configuration files. Modify doug with an option to show or hide filenames that begin with a dot. This option should be controlled with a -a command-line option.

### Starting Directory as Argument

Modify doug so that it takes a starting directory as an additional command-line argument.