lean4-htt/doc/typeclass.md

Type classes

Typeclasses were introduced as a principled way of enabling ad-hoc polymorphism in functional programming languages. We first observe that it would be easy to implement an ad-hoc polymorphic function (such as addition) if the function simply took the type-specific implementation of addition as an argument and then called that implementation on the remaining arguments. For example, suppose we declare a structure in Lean to hold implementations of addition

# namespace Ex
structure Add (a : Type) where
  add : a -> a -> a

#check @Add.add
-- Add.add : {a : Type} → Add a → a → a → a
# end Ex

In the above Lean code, the field add has type Add.add : {α : Type} → Add α → α → α → α where the curly braces around the type a mean that it is an implicit argument. We could implement double by

# namespace Ex
# structure Add (a : Type) where
#  add : a -> a -> a
def double (s : Add a) (x : a) : a :=
  s.add x x

#eval double { add := Nat.add } 10
-- 20

#eval double { add := Nat.mul } 10
-- 100

#eval double { add := Int.add } 10
-- 20

# end Ex

Note that you can double a natural number n by double { add := Nat.add } n. Of course, it would be highly cumbersome for users to manually pass the implementations around in this way. Indeed, it would defeat most of the potential benefits of ad-hoc polymorphism.

The main idea behind typeclasses is to make arguments such as Add a implicit, and to use a database of user-defined instances to synthesize the desired instances automatically through a process known as typeclass resolution. In Lean, by changing structure to class in the example above, the type of Add.add becomes

# namespace Ex
class Add (a : Type) where
  add : a -> a -> a

#check @Add.add
-- Add.add : {a : Type} → [self : Add a] → a → a → a
# end Ex

where the square brackets indicate that the argument of type Add a is instance implicit, i.e. that it should be synthesized using typeclass resolution. This version of add is the Lean analogue of the Haskell term add :: Add a => a -> a -> a. Similarly, we can register an instance by

# namespace Ex
# class Add (a : Type) where
#  add : a -> a -> a
instance : Add Nat where
  add := Nat.add

# end Ex

Then for n : Nat and m : Nat, the term Add.add n m triggers typeclass resolution with the goal of Add Nat, and typeclass resolution will synthesize the instance above. In general, instances may depend on other instances in complicated ways. For example, you can declare an (anonymous) instance stating that if a has addition, then Array a has addition:

instance [Add a] : Add (Array a) where
  add x y := Array.zipWith x y (. + .)

#eval Add.add #[1, 2] #[3, 4]
-- #[4, 6]

#eval #[1, 2] + #[3, 4]
-- #[4, 6]

Note that x + y is notation for Add.add x y in Lean.

The example above demonstrates how type classes are used to overload notation. Now, we explore another application. We often need an arbitrary element of a given type. Recall that types may not have any elements in Lean. It often happens that we would like a definition to return an arbitraryt element in a "corner case." For example, we may like the expression head xs to be of type a when xs is of type List a. Similarly, many theorems hold under the additional assumption that a type is not empty. For example, if a is a type, exists x : a, x = x is true only if a is not empty. The standard library defines a type class Inhabited to enable type class inference to infer a "default" or "arbitrary" element of an inhabited type. Let us start with the first step of the program above, declaring an appropriate class:

# namespace Ex
class Inhabited (a : Type _) where
  default : a

#check @Inhabited.default
-- Inhabited.default : {a : Type _} → [self : Inhabited a] → a
# end Ex

Note Inhabited.default doesn't have any explicit argument.

An element of the class Inhabited a is simply an expression of the form Inhabited.mk x, for some element x : a. The projection Inhabited.default will allow us to "extract" such an element of a from an element of Inhabited a. Now we populate the class with some instances:

# namespace Ex
# class Inhabited (a : Type _) where
#  default : a
instance : Inhabited Bool where
  default := true

instance : Inhabited Nat where
  default := 0

instance : Inhabited Unit where
  default := ()

instance : Inhabited Prop where
  default := True

#eval (Inhabited.default : Nat)
-- 0

#eval (Inhabited.default : Bool)
-- true
# end Ex

You can use the command export to create the alias default for Inhabited.default

# namespace Ex
# class Inhabited (a : Type _) where
#  default : a
# instance : Inhabited Bool where
#  default := true
# instance : Inhabited Nat where
#  default := 0
# instance : Inhabited Unit where
#  default := ()
# instance : Inhabited Prop where
#  default := True
export Inhabited (default)

#eval (default : Nat)
-- 0

#eval (default : Bool)
-- true
# end Ex

Sometimes we want to think of the default element of a type as being an arbitrary element, whose specific value should not play a role in our proofs. For that purpose, we can write arbitrary instead of default. We define arbitrary as an opaque constant. Opaque constants are never unfolded by the type checker.

# namespace Ex
# export Inhabited (default)
theorem defNatEq0 : (default : Nat) = 0 :=
  rfl

constant arbitrary [Inhabited a] : a :=
  Inhabited.default

-- theorem arbitraryNatEq0 : (arbitrary : Nat) = 0 :=
--   rfl
/-
error: type mismatch
  rfl
has type
  arbitrary = arbitrary
but is expected to have type
  arbitrary = 0
-/
# end Ex

The theorem defNatEq0 type checks because the type checker can unfold (default : Nat) and reduce it to 0. This is not the case in the theorem arbitraryNatEq0 because arbitrary is an opaque constant.

Chaining Instances

If that were the extent of type class inference, it would not be all that impressive; it would be simply a mechanism of storing a list of instances for the elaborator to find in a lookup table. What makes type class inference powerful is that one can chain instances. That is, an instance declaration can in turn depend on an implicit instance of a type class. This causes class inference to chain through instances recursively, backtracking when necessary, in a Prolog-like search.

For example, the following definition shows that if two types a and b are inhabited, then so is their product:

instance [Inhabited a] [Inhabited b] : Inhabited (a × b) where
  default := (arbitrary, arbitrary)

With this added to the earlier instance declarations, type class instance can infer, for example, a default element of Nat × Bool:

# namespace Ex
# class Inhabited (a : Type _) where
#  default : a
# instance : Inhabited Bool where
#  default := true
# instance : Inhabited Nat where
#  default := 0
# constant arbitrary [Inhabited a] : a :=
#  Inhabited.default
instance [Inhabited a] [Inhabited b] : Inhabited (a × b) where
  default := (arbitrary, arbitrary)

#eval (arbitrary : Nat × Bool)
-- (0, true)
# end Ex

Similarly, we can inhabit tyhe function with suitable constant functions:

instance [Inhabited b] : Inhabited (a -> b) where
  default := fun _ => arbitrary

As an exercise, try defining default instances for other types, such as List and Sum types.