This post is part of a series. Click here to go to the beginning of the series.

Last week, we added inexhaustive matches and record extension to cubiml. Today we will tackle let polymorphism.

Let polymorphism

Up until now, everything in the type system has been monomorphic, meaning that a single type is inferred for each expression. However, in some cases, this is unduly restrictive. Consider the following code example

let id = fun x -> x;

(id 1) + 3;
(id "x") ^ "y";

Currently, this results in a type error because id is forced to be monomorphic, giving it the inferred type (int & string) -> (int | string). Essentially, monomorphism means that the compiler considers the arguments from any call to a function to potentially flow to the result of every call to that function. Therefore, if we call the function with ints and later with strings, they get mixed together in the function type, resulting in an error.

In order to make the above code typecheck, we need to add context sensitivity to the typechecker, making the compiler keep different calls to the same function separate and understand that only the arguments to a specific call may flow to the result of that specific call.

The Hindley-Milner system handles this via a feature known as let polymorphism. Basically, when inferring the type of a let bound variable (i.e. let v = e1 in e2), instead of inferring a single, monomorphic type for v, we instead infer a type scheme.

Type schemes

A type scheme is basically a template that can be used to stamp out new types on demand. For let-bound variables, we store the type scheme in the typing environment instead of a single type. Whenever the variable is referenced, the scheme is used to stamp out a new type which is specific to that particular usage of the variable.

In the code example above, without let polymorphism, id would have the inferred type T -> T. In this case, T is a single type variable, which means every use of id must share the same value of T leading to the errors above. With let polymorphism, we instead infer the type scheme forall T => (T -> T). This says that whenever id is referenced, we create a new type variable and then return the type T -> T where T is the newly created variable. This means that different uses of id will have different type variables T that don’t interact at all, meaning that one could be substituted with int and another with string without causing any spurious type errors.

Lazy let polymorphism

There are a number of ways to implement let polymorphism, but for now, we’ll just go with the simplest and laziest possible implementation to make things easier to understand, even if it is slower than more optimized implementations.

Recall that the problem in the previous code example is that we want to be able to reuse the (let-bound) id function with different types every time it is referenced. Without polymorphism, we can still work around that by simply duplicating the code of the function whenever it is referenced. For example, if the above code were rewritten as follows, it would type check even without let polymorphism:

let id = fun x -> x
    in  (id 1) + 3;

let id = fun x -> x
    in  (id "x") ^ "y";   

Basically, the goal of let polymorphism is to be able to write a function and have the typechecker treat it as if it were duplicated whenever it is referenced for type checking purposes, but still have only one actual copy of the function at runtime. The simplest way to achieve this is to just rerun the typechecker on the definition code whenever the type scheme is instantiated (with some caveats we’ll cover later).


Previously, we stored plain types in the frontend’s typing environment (the map of variable names to previously inferred types). So to start, we need to modify it to store type schemes instead.

enum Scheme {
    Poly(Rc<dyn Fn(&mut TypeCheckerCore) -> Result<Value>>),

struct Bindings {
    m: HashMap<String, Scheme>,
    changes: Vec<(String, Option<Scheme>)>,

Previously, bindings stored a String -> Value map. We define a new enum Scheme and change it to a String -> Scheme map. Note that the terminology here is slightly different than above. Our Scheme enum can store either a plain, monomorphic type (for function arguments and recursive variables) or a polymorphic type scheme (for let bound variables).

Poly(Rc<dyn Fn(&mut TypeCheckerCore) -> Result<Value>>),

In the later case, our enum stores a closure object, which can be called to stamp out new types on demand. It takes in a reference to the typechecker core, uses it to create any new type variables as applicable, and returns a new monomorphic type.

Next up, we have to update the code for handling Variable nodes in the AST:

Variable((name, span)) => {
    if let Some(scheme) = bindings.get(name.as_str()) {
        match scheme {
            Scheme::Mono(v) => Ok(*v),
            Scheme::Poly(cb) => cb(engine),
    } else {
        Err(SyntaxError::new1(format!("SyntaxError: Undefined variable {}", name), *span))

Previously, we just looked up the type in the bindings map. Now we look up the Scheme enum in the bindings map. If it contains a type, we use that, otherwise we call the contained closure to generate a fresh new type.

Inferring type schemes

With that out of the way, it’s time to actually create some type schemes. Recall that previously, our Let handling code was duplicated between handling let expressions and top level let definitions. Since we’ll be making more complicated changes to it, I’ve gone ahead and refactored it to the check_let function to avoid duplication.

Let((name, var_expr), rest_expr) => {
    let var_scheme = check_let(engine, bindings, var_expr)?;
    bindings.in_child_scope(|bindings| {
        bindings.insert_scheme(name.clone(), var_scheme);
        check_expr(engine, bindings, rest_expr)

Now comes the interesting part— check_let itself.

fn check_let(engine: &mut TypeCheckerCore, bindings: &mut Bindings, expr: &ast::Expr) -> Result<Scheme> {
    let saved_bindings = RefCell::new(Bindings {
        m: bindings.m.clone(),
        changes: Vec::new(),
    let saved_expr = expr.clone();

    let f: Rc<dyn Fn(&mut TypeCheckerCore) -> Result<Value>> =
        Rc::new(move |engine| check_expr(engine, &mut saved_bindings.borrow_mut(), &saved_expr));


This might look a bit scary, but all it’s doing is creating a closure that typechecks the given Expr and returns the resulting type.

let saved_bindings = RefCell::new(Bindings {
    m: bindings.m.clone(),
    changes: Vec::new(),
let saved_expr = expr.clone();

First, we need to copy the typing environment and the ast subtree so we can store them in the closure. This isn’t strictly necessary. It’s possible to eliminate the AST copies and most of the binding copies by making a few changes elsewhere in the code, but for now we’re going with the really lazy implementation approach to minimize the code changes required. (We do however have to mark the AST node type as cloneable. Failing to do so results in a very confusing compiler error in Rust, so watch out for that.)

let f: Rc<dyn Fn(&mut TypeCheckerCore) -> Result<Value>> =
    Rc::new(move |engine| check_expr(engine, &mut saved_bindings.borrow_mut(), &saved_expr));

After that, we just create the closure object. Sadly, Rust requires an explicit type annotation here.


Finally, we invoke the closure and then return a Scheme containing the closure. The first point is a bit subtle. Recall that our current implementation basically just typechecks the body of a let definition whenever the variable is referenced. However, cubiml has strict execution semantics, which means we need to typecheck the definition at least once, even if the variable is never referenced.

For example, consider the following code:

let x = false + 34 in
    "hello, world!";

Without the extra f(engine)? call above, the erroneous false + 34 addition would pass the typechecker instead of returning an error, because the variable x is never referenced, and hence its scheme callback is never invoked.

Note that if you do variable resolution in a separate pass beforehand (which is a good idea anyway), you can eliminate the need for this check by marking (or even optimizing away) unused variables during variable resolution.

The value restriction

Unfortunately, we’re not quite done. The above code mostly works, but it is unsound when combined with mutation. Consider the following example:

let p = ref (fun x -> x);
(* 42 is a callable function, right? *)
p := 42;
!p "arg"

This example falsely passes the typechecker and crashes at runtime.

An error occurred during evaluation in the repl: TypeError: $.v0.$p is not a function

Next week, we’ll see how to fix this issue using the value restriction.

Next post: Subtype Inference by Example Part 11: The Value Restriction and Polymorphic Recursion

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