---
title: Meaningfully Typechecking a Language in Idris
date: 2020-02-27T21:58:55-08:00
draft: true
tags: ["Haskell", "Idris"]
---

This term, I'm a TA for Oregon State University's Programming Languages course.
The students in the course are tasked with using Haskell to implement a programming
language of their own design. One of the things they can do to gain points for the
project is implement type checking, rejecting
{{< sidenote "right" "ill-typed-note" "ill-typed programs or expressions" >}}
Whether or not the below example is ill-typed actually depends on your language.
Many languages (even those with a static type system, like C++ or Crystal)
have a notion of "truthy" and "falsy" values. These values can be used 
in the condition of an if-expression, and will be equivalent to "true" or "false",
respectively. However, for simplicity, I will avoid including
truthy and falsy values into the languages in this post. For the same reason, I will avoid
reasoning about
<a href="https://developer.mozilla.org/en-US/docs/Glossary/Type_coercion">type coercions</a>,
which make expressions like <code>"Hello"+3</code> valid.
{{< /sidenote >}} such as:

```Haskell
if "Hello" then 0 else 1
```

For instance, a student may have a function `typecheck`, with the following
signature (in Haskell):

```Haskell
typecheck :: Expr -> Either TypeError ExprType
```

The function will return an error if something goes wrong, or, if everything
goes well, the type of the given expression. So far, so good.

A student asked, however:

> Now that I ran type checking on my program, surely I don't need to include errors
in my {{< sidenote "right" "valuation-function-note" "valuation function!" >}}
I'm using "valuation function" here in the context of 
<a href="https://en.wikibooks.org/wiki/Haskell/Denotational_semantics">denotational semantics</a>.
In short, a
<a href="http://www.inf.ed.ac.uk/teaching/courses/inf2a/readings/semantics-note.pdf">valuation function</a>
takes an expression and assigns to it some
representation of its meaning. For a language of arithmetic expression, the
"meaning" of an expression is just a number (the result of simplifying the expression).
For a language of booleans, <code>and</code>, and <code>or</code>, the "meaning" is a boolean
for the same reason. Since an expression in the language can be ill-formed (like
<code>list(5)</code> in Python), the "meaning" (<em>semantic domain</em>) of a
complicated language tends to include the possibility of errors.
{{< /sidenote >}} I should be able to make my function be of type `Expr -> Val`, and not
`Expr -> Maybe Val`!

Unfortunately, this is not quite true. It is true that if the student's type checking
function is correct, then there will be no way for a type error to occur during
the evaluation of an expression "validated" by said function. The issue is, though,
that __the type system does not know about the expression's type-correctness__. Haskell
doesn't know that an expression has been type checked; worse, since the function's type
indicates that it accepts `Expr`, it must handle invalid expressions to avoid being [partial](https://wiki.haskell.org/Partial_functions). In short, even if we __know__ that the
expressions we give to a function are type safe, we have no way of enforcing this.

A potential solution offered in class was to separate the expressions into several
data types, `BoolExpr`, `ArithExpr`, and finally, a more general `Expr'` that can
be constructed from the first two. Operations such as `and` and `or`
will then only be applicable to boolean expressions:

```Haskell
data BoolExpr = BoolLit Bool | And BoolExpr BoolExpr | Or BoolExpr BoolExpr
```

It will be a type error to represent an expression such as `true or 5`. Then,
`Expr'` may have a constructor such as `IfElse` that only accepts a boolean
expression as the first argument:

```Haskell
data Expr' = IfElse BoolExpr Expr' Expr' | ...
```

All seems well. Now, it's impossible to have a non-boolean condition, and thus,
this error has been eliminated from the evaluator. Maybe we can even have
our type checking function translate an unsafe, potentially incorrect `Expr` into
a more safe `Expr'`:

```Haskell
typecheck :: Expr -> Either TypeError (Expr', ExprType)
```

However, we typically also want the branches of an if expression to both have the same
type - `if x then 3 else False` may work sometimes, but not always, depending of the
value of `x`. How do we encode this? Do we have two constructors, `IfElseBool` and
`IfElseInt`, with one `BoolExpr` and the other in `ArithExpr`?  What if we add strings?
We'll be copying functionality back and forth, and our code will suffer. Wouldn't it be
nice if we could somehow tag our expressions with the type they produce? Instead of
`BoolExpr` and `ArithExpr`, we would be able to have `Expr BoolType` and `Expr IntType`,
which would share the `IfElse` constructor...

It's not easy to do this in canonical Haskell, but it can be done in Idris!

### Enter Dependent Types
Idris is a language with support for [dependent types](https://en.wikipedia.org/wiki/Dependent_type). Wikipedia gives the following definition for "dependent type":

> In computer science and logic, a dependent type is a type whose definition depends on a value.

This is exactly what we want. In Idris, we can define the possible set of types in our
language:

{{< codelines "Idris" "typesafe-interpreter/TypesafeIntr.idr" 1 4>}}

Then, we can define a `SafeExpr` type family, which is indexed by `ExprType`.
Here's the
{{< sidenote "right" "gadt-note" "code," >}}
I should probably note that the definition of <code>SafeExpr</code> is that of
a
<a href="https://en.wikipedia.org/wiki/Generalized_algebraic_data_type">Generalized Algebraic Data Type</a>,
or GADT for short. This is what allows each of our constructors to produce
values of a different type: <code>IntLiteral</code> builds <code>SafeExpr IntType</code>,
while <code>BoolLiteral</code> builds <code>SafeExpr BoolType</code>.
{{</ sidenote >}} which we will discuss below:

{{< codelines "Idris" "typesafe-interpreter/TypesafeIntr.idr" 23 27 >}}

The first line of the above snippet says, "`SafeExpr` is a type constructor
that requires a value of type `ExprType`". For example, we can have
`SafeExpr IntType`, or `SafeExpr BoolType`. Next, we have to define constructors
for `SafeExpr`. One such constructor is `IntLiteral`, which takes a value of
type `Int` (which represents the value of the integer literal), and builds
a value of `SafeExpr IntType`, that is, an expression that __we know evaluates
to an integer__.

The same is the case for `BoolLiteral` and `StringLiteral`, only they build
values of type `SafeExpr BoolType` and `SafeExpr StringType`, respectively.

The more complicated case is that of `BinOperation`. Put simply, it takes
a binary function of type `a->b->c` (kind of), two `SafeExpr`s producing `a` and `b`,
and combines the values of those expressions using the function to generate
a value of type `c`. Since the whole expression returns `c`, `BinOperation`
builds a value of type `SafeExpr c`.

That's almost it. Except, what's up with `repr`? We need it because `SafeExpr`
is parameterized by a __value__ of type `ExprType`. Thus, `a`, `b`, and `c` are
all values in the definition of `BinOperation`. However, in a function
`input->output`, both `input` and `output` have to be __types__, not values.
Thus, we define a function `repr` which converts values such as `IntType` into
the actual type that `eval` would yield when running our expression:

{{< codelines "Idris" "typesafe-interpreter/TypesafeIntr.idr" 6 9 >}}

The power of dependent types allows us to run `repr` inside the type
of `BinOp` to compute the type of the function it must accept.