The Monad Type Class

Rather than having to import an operator like ok or andThen for each type that is a monad, the Lean standard library contains a type class that allow them to be overloaded, so that the same operators can be used for any monad. Monads have two operations, which are the equivalent of ok and andThen:

class Monad (m : Type → Type) where
  pure : α → m α
  bind : m α → (α → m β) → m β

This definition is slightly simplified. The actual definition in the Lean library is somewhat more involved, and will be presented later.

The Monad instances for Option and Except can be created by adapting the definitions of their respective andThen operations:

instance : Monad Option where
  pure x := some x
  bind opt next :=
    match opt with
    | none => none
    | some x => next x

instance : Monad (Except ε) where
  pure x := Except.ok x
  bind attempt next :=
    match attempt with
    | Except.error e => Except.error e
    | Except.ok x => next x

As an example, firstThirdFifthSeventh was defined separately for Option α and Except String α return types. Now, it can be defined polymorphically for any monad. It does, however, require a lookup function as an argument, because different monads might fail to find a result in different ways. The infix version of bind is >>=, which plays the same role as ~~> in the examples.

def firstThirdFifthSeventh [Monad m] (lookup : List α → Nat → m α) (xs : List α) : m (α × α × α × α) :=
  lookup xs 0 >>= fun first =>
  lookup xs 2 >>= fun third =>
  lookup xs 4 >>= fun fifth =>
  lookup xs 6 >>= fun seventh =>
  pure (first, third, fifth, seventh)

Given example lists of slow mammals and fast birds, this implementation of firstThirdFifthSeventh can be used with Option:

def slowMammals : List String :=
  ["Three-toed sloth", "Slow loris"]

def fastBirds : List String := [
  "Peregrine falcon",
  "Saker falcon",
  "Golden eagle",
  "Gray-headed albatross",
  "Spur-winged goose",
  "Anna's hummingbird"

#eval firstThirdFifthSeventh (fun xs i => xs[i]?) slowMammals
#eval firstThirdFifthSeventh (fun xs i => xs[i]?) fastBirds
some ("Peregrine falcon", "Golden eagle", "Spur-winged goose", "Anna's hummingbird")

After renaming Except's lookup function get to something more specific, the very same implementation of firstThirdFifthSeventh can be used with Except as well:

def getOrExcept (xs : List α) (i : Nat) : Except String α :=
  match xs[i]? with
  | none => Except.error s!"Index {i} not found (maximum is {xs.length - 1})"
  | some x => Except.ok x

#eval firstThirdFifthSeventh getOrExcept slowMammals
Except.error "Index 2 not found (maximum is 1)"
#eval firstThirdFifthSeventh getOrExcept fastBirds
Except.ok ("Peregrine falcon", "Golden eagle", "Spur-winged goose", "Anna's hummingbird")

The fact that m must have a Monad instance means that the >>= and pure operations are available.

General Monad Operations

Because many different types are monads, functions that are polymorphic over any monad are very powerful. For example, the function mapM is a version of map that uses a Monad to sequence and combine the results of applying a function:

def mapM [Monad m] (f : α → m β) : List α → m (List β)
  | [] => pure []
  | x :: xs =>
    f x >>= fun hd =>
    mapM f xs >>= fun tl =>
    pure (hd :: tl)

The return type of the function argument f determines which Monad instance will be used. In other words, mapM can be used for functions that produce logs, for functions that can fail, or for functions that use mutable state. Because f's type determines the available effects, they can be tightly controlled by API designers.

As described in this chapter's introduction, State σ α represents programs that make use of a mutable variable of type σ and return a value of type α. These programs are actually functions from a starting state to a pair of a value and a final state. The Monad class requires that its parameter expect a single type argument—that is, it should be a Type → Type. This means that the instance for State should mention the state type σ, which becomes a parameter to the instance:

instance : Monad (State σ) where
  pure x := fun s => (s, x)
  bind first next :=
    fun s =>
      let (s', x) := first s
      next x s'

This means that the type of the state cannot change between calls to get and set that are sequenced using bind, which is a reasonable rule for stateful computations. The operator increment increases a saved state by a given amount, returning the old value:

def increment (howMuch : Int) : State Int Int :=
  get >>= fun i =>
  set (i + howMuch) >>= fun () =>
  pure i

Using mapM with increment results in a program that computes the sum of the entries in a list. More specifically, the mutable variable contains the sum so far, while the resulting list contains a running sum. In other words, mapM increment has type List Int → State Int (List Int), and expanding the definition of State yields List Int → Int → (Int × List Int). It takes an initial sum as an argument, which should be 0:

#eval mapM increment [1, 2, 3, 4, 5] 0
(15, [0, 1, 3, 6, 10])

A logging effect can be represented using WithLog. Just like State, its Monad instance is polymorphic with respect to the type of the logged data:

instance : Monad (WithLog logged) where
  pure x := {log := [], val := x}
  bind result next :=
    let {log := thisOut, val := thisRes} := result
    let {log := nextOut, val := nextRes} := next thisRes
    {log := thisOut ++ nextOut, val := nextRes}

saveIfEven is a function that logs even numbers but returns its argument unchanged:

def saveIfEven (i : Int) : WithLog Int Int :=
  (if isEven i then
    save i
   else pure ()) >>= fun () =>
  pure i

Using this function with mapM results in a log containing even numbers paired with an unchanged input list:

#eval mapM saveIfEven [1, 2, 3, 4, 5]
{ log := [2, 4], val := [1, 2, 3, 4, 5] }

The Identity Monad

Monads encode programs with effects, such as failure, exceptions, or logging, into explicit representations as data and functions. Sometimes, however, an API will be written to use a monad for flexibility, but the API's client may not require any encoded effects. The identity monad is a monad that has no effects, and allows pure code to be used with monadic APIs:

def Id (t : Type) : Type := t

instance : Monad Id where
  pure x := x
  bind x f := f x

The type of pure should be α → Id α, but Id α reduces to just α. Similarly, the type of bind should be α → (α → Id β) → Id β. Because this reduces to α → (α → β) → β, the second argument can be applied to the first to find the result.

With the identity monad, mapM becomes equivalent to map. To call it this way, however, Lean requires a hint that the intended monad is Id:

#eval mapM (m := Id) (· + 1) [1, 2, 3, 4, 5]
[2, 3, 4, 5, 6]

Omitting the hint results in an error:

#eval mapM (· + 1) [1, 2, 3, 4, 5]
failed to synthesize instance
  HAdd Nat Nat (?m.9063 ?m.9065)

In this error, the application of one metavariable to another indicates that Lean doesn't run the type-level computation backwards. The return type of the function is expected to be the monad applied to some other type. Similarly, using mapM with a function whose type doesn't provide any specific hints about which monad is to be used results in an "instance problem stuck" message:

#eval mapM (fun x => x) [1, 2, 3, 4, 5]
typeclass instance problem is stuck, it is often due to metavariables
  Monad ?m.9063

The Monad Contract

Just as every pair of instances of BEq and Hashable should ensure that any two equal values have the same hash, there is a contract that each instance of Monad should obey. First, pure should be a left identity of bind. That is, bind (pure v) f should be the same as f v. Secondly, pure should be a right identity of bind, so bind v pure is the same as v. Finally, bind should be associative, so bind (bind v f) g is the same as bind v (fun x => bind (f x) g).

This contract specifies the expected properties of programs with effects more generally. Because pure has no effects, sequencing its effects with bind shouldn't change the result. The associative property of bind basically says that the sequencing bookkeeping itself doesn't matter, so long as the order in which things are happening is preserved.


Mapping on a Tree

Define a function BinTree.mapM. By analogy to mapM for lists, this function should apply a monadic function to each data entry in a tree, as a preorder traversal. The type signature should be:

def BinTree.mapM [Monad m] (f : α → m β) : BinTree α → m (BinTree β)

The Option Monad Contract

First, write a convincing argument that the Monad instance for Option satisfies the monad contract. Then, consider the following instance:

instance : Monad Option where
  pure x := some x
  bind opt next := none

Both methods have the correct type. Why does this instance violate the monad contract?