---
title: Compiling a Functional Language Using C++, Part 10 - Polymorphism
date: 2020-02-29T20:09:37-08:00
tags: ["C and C++", "Functional Languages", "Compilers"]
draft: true
---

[In part 8]({{< relref "08_compiler_llvm.md" >}}), we wrote some pretty interesting programs in our little language.
We successfully expressed arithmetic and recursion. But there's one thing
that we cannot express in our language without further changes: an `if` statement.

Suppose we didn't want to add a special `if/else` expression into our language.
Thanks to lazy evaluation, we can express it using a function:

```
defn if c t e = {
    case c of {
        True -> { t }
        False -> { e }
    }
}
```

But an issue still remains: so far, our compiler remains __monomorphic__. That
is, a particular function can only have one possible type for each one of its
arguments. With our current setup, something like this
{{< sidenote "right" "if-note" "would not work:" >}}
In a polymorphically typed language, the inner <code>if</code> would just evaluate to
<code>False</code>, and the whole expression to 3.
{{< /sidenote >}}

```
if (if True False True) 11 3
```

This is because, for this to work, both of the following would need to hold (borrowing
some of our notation from the [typechecking]({{< relref "03_compiler_typechecking.md" >}}) post):

$$
\\text{if} : \\text{Int} \\rightarrow \\text{Int}
$$
$$
\\text{if} : \\text{Bool} \\rightarrow \\text{Bool}
$$

But using our rules so far, such a thing is impossible, since there is no way for
\\(\text{Int}\\) to be unified with \\(\text{Bool}\\). We need a more powerful
set of rules to describe our program's types. One such set of rules is
the [Hindley-Milner type system](https://en.wikipedia.org/wiki/Hindley%E2%80%93Milner_type_system),
which we have previously alluded to. In fact, the rules we came up
with were already very close to Hindley-Milner, with the exception of two:
__generalization__ and __instantiation__. It's been quite a while since the last time we worked on typechecking, so I'm going
to present a table with these new rules, as well as all of the ones that we previously used. I will also give a quick
summary of each of these rules.

Rule|Name and Description
-----|-------
$$\\frac{x:\\sigma \\in \\Gamma}{\\Gamma \\vdash x:\\sigma}$$| __Var__: If the variable \\(x\\) is known to have some polymorphic type \\(\\sigma\\) then an expression consisting only of that variable is of that type.
$$\\frac{\\Gamma \\vdash e\_1 : \\tau\_1 \\rightarrow \\tau\_2 \\quad \\Gamma \\vdash e\_2 : \\tau\_1}{\\Gamma \\vdash e\_1 \\; e\_2 : \\tau\_2}$$| __App__: If an expression \\(e\_1\\), which is a function from monomorphic type \\(\\tau\_1\\) to another monomorphic type \\(\\tau\_2\\), is applied to an argument \\(e\_2\\) of type \\(\\tau\_1\\), then the result is of type \\(\\tau\_2\\).
$$\\frac{\\Gamma, x:\\tau \\vdash e : \\tau'}{\\Gamma \\vdash \\lambda x.e : \\tau \\rightarrow \\tau'}$$| __Abs__: If the body \\(e\\) of a lambda abstraction \\(\\lambda x.e\\) is of type \\(\\tau'\\) when \\(x\\) is of type \\(\\tau\\) then the whole lambda abstraction is of type \\(\\tau \\rightarrow \\tau'\\).
$$\\frac{\\Gamma \\vdash e : \\tau \\quad \\text{matcht}(\\tau, p\_i) = b\_i \\quad \\Gamma,b\_i \\vdash e\_i : \\tau\_c}{\\Gamma \\vdash \\text{case} \\; e \\; \\text{of} \\; \\\{ (p\_1,e\_1) \\ldots (p\_n, e\_n) \\\} : \\tau\_c }$$ | __Case__: This rule is not part of Hindley-Milner, and is specific to our language. If the expression being case-analyzed is of type \\(\\tau\\) and each branch \\((p\_i, e\_i)\\) is of the same type \\(\\tau\_c\\) when the pattern \\(p\_i\\) works with type \\(\\tau\\) producing extra bindings \\(b\_i\\), the whole case expression is of type \\(\\tau\_c\\).
$$\\frac{\\Gamma \\vdash e : \\sigma \\quad \\sigma' \\sqsubseteq \\sigma}{\\Gamma \\vdash e : \\sigma'}$$| __Inst (New)__: If type \\(\\sigma'\\) is an instantiation of type \\(\\sigma\\) then an expression of type \\(\\sigma\\) is also an expression of type \\(\\sigma'\\).
$$\\frac{\\Gamma \\vdash e : \\sigma \\quad \\alpha \\not \\in \\text{free}(\\Gamma)}{\\Gamma \\vdash e : \\forall a . \\sigma}$$| __Gen (New)__: If an expression has a type with free variables, this rule allows us generalize it to allow all possible types to be used for these free variables.