In the previous installment we discussed inductive types and patern matching.

Today we’re going to talk about Dependent Types and Type Classes.

Latest revision: Sep 4, 2022

1. Dependent functions

1.1 Idea

A function $f: A \to B$ between two sets $A, B$ is commonly defined in mathematics as a collection of pairs $f := \{(a, b)\} \subset A \times B$ with the property that elements $a$ of $A$ appear exactly once.

A dependently typed function is a generalization of this concept:

Each element $f(a)$ is allowed to be a member of a different set that depends on $a$.

This construction gives rise to a set-level function $B$ that assigns a set $B(a)$ to each element $a \in A$.

Observe that if this function $B$ is constant (i.e. it maps all elements $a$ to a single set) then we are back to the regular definition of non-dependent functions.

Example: Vector fields

At every point $p$ of a circle $S^1$, consider a vector $V(p)$ on the tangent space $T_{p}S^{1}$ at this point:

Since each $V(p)$ lives in a different tangent space $T_{p}S^{1}$ (that depends on $p$), it makes sense to regard $V$ as a dependent function.

Example: operations on tuples

A very common example of dependent types in Scala 3 is obtaining components of tuples at specific positions:

val t: (Int, String, Boolean) = (0, "", false)

assert( t.apply(0) == 0 )
assert( t.apply(1) == "" )
assert( t.apply(2) == false )

We can see how apply produces a value of a type that depends on the input index:

In fact most generic tuple operations use dependent types.

1.2 Dependent functions in Lean

In Lean the type of a regular function $f$ from $A$ to $B$ is written as

f: A -> B

here the codomain B is the same for every element a of the domain A.

In a dependent function we need a way to indicate exactly how the codomain depends on each element of the domain.

The signature of a dependent function $f$ from $A$ to $B(a)$ is written in Lean like this:

f: (a: A) -> B a

Example: variable number of arguments

Consider a function adder that takes:

1. A number $n$ representing extra arguments
2. An initial value $acc$
3. $n$ arguments

and returns the sum of the initial value and extra arguments.

adder 2 acc 4 3 = acc + 4 + 3

For example

example: adder 0 5        = (5 : Nat) := rfl
example: adder 1 10 11    = (21: Nat) := rfl
example: adder 2 18 17 12 = (47: Nat) := rfl

i.e. the first argument is the number of extra arguments after the 2nd argument.

(This example is described in Edwin Brady’s Type-Driven Development with Idris)

We will proceed in two steps:

1. Define a type level function that calculates the return type: a function with $n$ parameters:

def Adder: Nat -> Type
  | 0     => Nat 
  | k + 1 => Nat -> Adder k


example: Adder 0 =  Nat                := rfl
example: Adder 1 = (Nat -> Nat)        := rfl
example: Adder 2 = (Nat -> Nat -> Nat) := rfl

2. Define a regular function, whose signature is computed by Adder based on the 1st argument:

def adder: (n: Nat) -> (acc: Nat) -> (Adder n)
  | 0,     acc => acc
  | k + 1, acc => fun n => adder k (n + acc)

-- some examples of types:

-- adder 0 : Nat -> Nat
-- adder 1 : Nat -> Nat -> Nat
-- adder 2 : Nat -> Nat -> Nat -> Nat

example: (adder 0) 5        = (5 : Nat) := rfl
example: (adder 1) 10 11    = (21: Nat) := rfl
example: (adder 2) 18 17 12 = (47: Nat) := rfl

So adder n produces a function whose type depends on n:

1.3 Dependent Functions in Scala 3

Match types

Scala 3 supports a similar form of dependent typing using match types.

We follow the same steps as in Lean:

1. Define a type level function that calculates the return type:

import compiletime.ops.int.S

type Adder[N <: Int] = N match
  case 0    =>  Int
  case S[k] => (Int => Adder[k & Singleton])


summon[ Adder[0] =:=  Int               ]
summon[ Adder[1] =:= (Int => Int)       ]
summon[ Adder[2] =:= (Int => Int => Int)]

2. Define a function, whose signature is computed by Adder based on the 1st argument:

inline def adder[N <: Int & Singleton](n: N)(acc: Int): Adder[N] =
  inline n match
    case _: 0    => acc
    case _: S[k] => n => adder(constValue[k & Singleton])(n + acc)


assert( adder(0)(5)          == 5  )
assert( adder(1)(10)(11)     == 21 )
assert( adder(2)(18)(17)(12) == 47 )

Here we can see how the fact that types are not first class in Scala introduces some complexities:

• Singleton is needed so that adder(0) is inferred as adder[0](0) and not adder[Int](0).
• After pattern matching on the type _: S[k] we obtain the type k of the predecessor n - 1.
• In order to continue the process we need to obtain the value associated with the singleton type k. For this we use constValue.
• But constValue only works for values known at compile time, so we have to use inline match and also inline the whole function.

2. Type Classes

Type Classes can be used as a mechanism to represent dependent types.

2.1 Type Classes in Lean

Lean has support for type classes by using the keywords class / instance.

Let’s implement the adder function using implicit type class resolution this time:

class Bundle (n: Nat) where
  Adder: Type
  adder: Nat -> Adder

This declaration defines a few things:

Note how both Bundle.Adder and Bundle.adder acquired two extra parameters:

• The class argument (n: Nat)
• An implicit instance [self : Bundle n] (indicated by the square brackets)

In order to use the function Bundle.adder the number n needs to be known, either by inference or explicitly providing it like so:

#eval Bundle.adder (n := 0) 5
-- 5
#eval Bundle.adder (n := 1) 10 11
-- 21

Alternatively we can create a helper function where n is required explicitly

def adder (n acc: Nat) [c: Bundle n]: Bundle.Adder n :=
    @Bundle.adder n c acc
    
#eval adder 0 5
-- 5
#eval adder 1 10 11
-- 21

We need two Bundle instances again, corresponding to the base case and the inductive step:

instance : Bundle 0 where
  Adder := Nat
  adder := id

instance (k: Nat) [predecessor: Bundle k] : Bundle (k + 1) where
  Adder := Nat -> predecessor.Adder
  adder := fun acc n => Bundle.adder (n + acc)

Notes:

• id is the identity function
• adder k (n + acc) is using the implicit instance in scope (predecessor), so we don’t need to pass it manually.

Sanity check:

example: (adder 0) 5        = (5 : Nat) := rfl
example: (adder 1) 10 11    = (21: Nat) := rfl
example: (adder 2) 18 17 12 = (47: Nat) := rfl

2.2 Type Classes in Scala

The same type class in Scala:

trait Bundle[N <: Int]:
  type Adder
  def adder: Int => Adder

Due to a idiosyncrasy of Scala we need to create a helper type function BundleWithAdder to manually propagate type information (this is the Aux pattern common in shapeless).

type BundleWithAdder[N <: Int, A] =
  Bundle[N] { type Adder = A }

// base case: Bundle[0]

given BundleWithAdder[0, Int] =
  new Bundle[0]:
    type Adder = Int
    def adder = identity

// inductive step: Bundle[K] implies Bundle[S[K]]

given [K <: Int](using predecessor: Bundle[K]): 
  BundleWithAdder[S[K], Int => predecessor.Adder] =
  new Bundle[S[K]]:
    type Adder = Int => predecessor.Adder
    def adder = acc => n => predecessor.adder(n + acc)

Sanity check:

val bundle0 = summon[Bundle[0]]
val bundle1 = summon[Bundle[1]]
val bundle2 = summon[Bundle[2]]

summon[ bundle0.Adder =:=  Int                ]
summon[ bundle1.Adder =:= (Int => Int)        ]
summon[ bundle2.Adder =:= (Int => Int => Int) ]

Taking the singleton type n.type of the term n will kickstart the process:

def adder(n: Int)(acc: Int)(using bundle: Bundle[n.type]) =
  bundle.adder(acc)

assert( adder(0)(5)          == 5  )
assert( adder(1)(10)(11)     == 21 )
assert( adder(2)(18)(17)(12) == 47 )

3. Propositions depending on values.

Propositions can be used as goals unto themselves (example 3.1) or as a way to create specifications (example 3.2).

3.1 Properties of natural numbers

Consider the statement:

For all natural numbers, $n + 0 = n$

The expression $n + 0 = n$ is a proposition that depends on $n$, and so the statement can be modeled as a dependent function $f$ from natural numbers to a family of propositions:

example (n : Nat) : n + 0 = n :=
	rfl

3.2 Algebraic structures

Here’s how one could define a monoid in Lean

class Monoid (M) where
  mul : M -> M -> M
  one : M
  mul_assoc : ∀ (a b c : M), mul (mul a b) c = mul a (mul b c)
  one_mul   : ∀ (a : M), mul one a = a
  mul_one   : ∀ (a : M), mul a one = a

In this kind of situations we rely heavily on dependent types to encode all the laws associated with the structure at hand. In this example each law is a dependent function from M to propositions.

Here’s a monoid instance for addition of natural numbers:

instance : Monoid Nat where
  mul := (· + ·)
  one := 0
  mul_assoc := Nat.add_assoc
  one_mul   := Nat.zero_add
  mul_one   := Nat.add_zero

Luckily in this case the basic properties have already been proved in the std library, so we can just delegate to them.

• The expression (· + ·) is the short form of the anonymous function fun a b => a + b, similar to Scala’s _ + _.

4. Dependent pairs

There is another important example of dependent types called Dependent pairs or $\Sigma$-types (or even dependent products).

Consider the regular cartesian product of two sets $A$, $B$:

This represents all possible combinations of elements from both sets:

The Dependent pair type $\Sigma (x: A), B(x)$ denotes the type of pairs $(a, b)$ where $a \in A$ and $b \in B(a)$

The diagram depicts a situation where $x$ and $y$ are elements of $B(a)$ and $w \in B(b)$. In this case the inhabitants of the dependent product $\Sigma (x: A), B(x)$ are the pairs:

4.1 Dependent pairs in Lean

In Lean dependent pairs are defined as the inductive type:

universes u v

structure Sigma {A: Type u} (B: A -> Type v) where
  Fst: A
  snd: B Fst

Dependent pairs can be constructed like so:

Sigma.mk a b

For example, consider a type function like this:

def IntOrString: Bool -> Type
  | true => Int
  | false => String

With it we can create a list of values that are either Int or String, and nothing else:

def elems: List (Sigma IntOrString) := [
    Sigma.mk true (1: Int), 
    Sigma.mk true (2: Int), 
    Sigma.mk false "a"
  ]

Another use case for dependent pairs is to define mathematical structures. For example we could define a Magma type and two instances like so

def Magma := Sigma (fun M => M -> M -> M)

def additiveMagma: Magma := Sigma.mk Nat (fun a b => a + b)
def multiplMagma : Magma := Sigma.mk Nat (fun a b => a * b)

The type $\Sigma (x: A), B(x)$ can also be interpreted as the proposition “exists $a$ in $A$ such that $B(a)$“:

which can be proved by exhibiting at least one element of the set $B(a)$ for some $a: A$.

(Although Lean has a type Exists just for this scenario)

4.2 Dependent products in Scala

Dependent products can be encoded directly with path-dependent types.

trait Fn0:
  type B

class Sigma[A <: Fn0](fst: A, snd: fst.B)

In general, every function that uses path-dependent types in its parameters list is an example of Sigma types in action:

def f[A <: Fn0](a: A, b: a.B) = ???

def g[A <: Fn0](s: Sigma[A]) = ???

Next time we’ll talk about proofs and tactics. Stayed tunned!

Resources

• Type-Driven Development with Idris

• Theorem Proving in Lean:

  • Dependent Types
  • Type Classes

• Dependent Function Types: Sections of a Bundle

• Scala 3 manual:

  • Match Types

  • Inline