Type Inference by Example, Part 5

Joakim Ahnfelt-Rønne
3 min readApr 13, 2020


First demo—type inference for lambda calculus.

Continuing where we left off in part 4, let’s finish implementing a first version of the type inference — and see a small demo.

The first thing we’ll need is a language to infer types for. Let’s start with the lambda calculus:

sealed abstract class Expression
case class ELambda(x : String, e : Expression) extends Expression
case class EApply(e1 : Expression, e2 : Expression) extends Expression
case class EVariable(x : String) extends Expression

We’ll also need a representation of constraints. For now, we only have equality constraints:

sealed abstract class Constraint
case class CEquality(t1 : Type, t2 : Type) extends Constraint

As we go though the syntax tree and generate constraints, we’ll need somewhere to store them:

val typeConstraints = ArrayBuffer[Constraint]()

For generating fresh type variables, recall that we chose a representation where each type variable is initially bound to itself in the substitution:

def freshTypeVariable() : TVariable = {
val result = TVariable(substitution.length)
substitution += result

Inferring types of expressions

Now we need to go through the expression and generate the constraints that are to be solved, and find out what the type of the expression is:

def inferType(
expression : Expression,
environment : Map[String, Type]
) : Type = expression match {

This function takes in an expression and an environment and returns a type. The environment is used to keep track of the type of variables during inference.

For lambda functions, we generate a fresh type variable t1 for the variable and add it to the environment. Then we infer the type t2 of the lambda body in this environment. The final type is the function type t1 => t2:

case ELambda(x, e) =>
val t1 = freshTypeVariable()
val environment2 = environment.updated(x, t1)
val t2 = inferType(e, environment2)
TConstructor("Function1", List(t1, t2))

When we encounter a variable, we look it up in the environment and return that type:

case EVariable(x) =>

For application (as in, calling a function with an argument) we first infer the type of the function and the argument, and generate a fresh type variable for the return type. Then we constrain the function type to be, well, a function type from the type of the argument to the return type:

case EApply(e1, e2) =>
val t1 = inferType(e1, environment)
val t2 = inferType(e2, environment)
val t3 = freshTypeVariable()
typeConstraints +=
CEquality(t1, TConstructor("Function1", List(t2, t3)))

That’s it for the inferType function:


Finishing up

The type we get back from inferType is likely a type variable at this point. In order to find out the concrete type, we’ll need to solve the constraints and then apply the substitution to the type.

Since we only have equality constraints right now, we can simply use unification we developed in part 4 to solve them:

def solveConstraints() : Unit = {
for(CEquality(t1, t2) <- typeConstraints) unify(t1, t2)

Applying the substitution is done by following the chain of subsitutions and substituting the generics recursively:

def substitute(t : Type) : Type = t match {
case TVariable(i) if substitution(i) != TVariable(i) =>
case TConstructor(name, generics) =>
TConstructor(name, generics.map(t => substitute(t)))
case _ => t

And we’re done.


Now that we have the first version of our type inference, it’s time to see it in action. View and run the code so far here.

Stay tuned for part 6, where we’ll extend the type inference to a bigger language and insert the missing types into the syntax tree.



Joakim Ahnfelt-Rønne

MSc Computer Science, working with functional programming in the industry — github.com/ahnfelt