# Idris: Totality, Dependent Types, Proofs

Idris is a programming language inspired by Haskell, with a set of innovative features to facilitate type safety and program correctness. Type Driven Development with Idris is a great introductory book which I highly recommend. In this post, I will try to cover the features I was most impressed with, providing some simple code samples. I will not cover syntax since most should be familiar from Haskell. If you are not familiar with Haskell syntax, here is a nice syntax cheat sheet. If you are not interested in either Haskell or Idris syntax, start with the last section of this post, Thoughts about the Future.

## Totality Checking

A total function in Idris is a function which is defined for all possible inputs and is guaranteed to produce a result in a finite amount of time [1]. The compiler obviously employs a heuristic, since the halting problem is undecidable, but is usually close enough to the truth to guarantee correctness from this point of view. It achieves this not by evaluating the function, but by ensuring that every recursive branch converges to a halting branch.

Natural numbers are defined in Idris using Peano axioms, so it is easy to prove things about them. Here is a minimal definition of natural numbers [2]:

```
data Nat' = Z | S Nat'
```

This defines Nat' as a data type which can be constructed either as Z (zero) or S Nat' (successor of another natural number). With this definition, the compiler can easily determine the following function is total:

```
f : Nat' -> ()
f Z = ()
f (S k) = g k
```

This function return a unit given Z, otherwise it recursively takes the predecessor of the argument. This converges to the Z case. The following function, on the other hand, is correctly identified as potentially non-terminating:

```
g : Nat' -> ()
g n = g (S n)
```

These are trivial examples, but in general, having a compile-time check for termination is a very powerful tool.

## Dependent Types

Dependent types are types computed from other types. To put it another way, Idris has first-order types, meaning functions can take types as arguments and return types as their output. Functions that compute types are evaluated at compile time. This is similar to C++ metaprogramming, but without employing a different syntax.

Before an example, we first need to define addition on naturals as follows:

```
(+) : Nat' -> Nat' -> Nat'
(+) Z r = r
(+) (S l) r = S (l + r)
```

Now we can declare a vector type consisting of a size (Nat') and a type argument:

```
data Vect' : Nat' -> Type -> Type where
Nil : Vect' Z a
(::) : (x : a) -> (xs : Vect' k a) -> Vect' (S k) a
```

Here a is a type argument. Vect' has two constructors: Nil, creating a Vect' of size Z containing elements of type a (0 elements) and (::), which concatenates an object of type a with a vector of size k of a and produces a vector of size S k containing a.

Now to see dependent types in action, we can define append', a function that appends a vector to another vector:

```
append' : Vect' n a -> Vect' m a -> Vect' (n + m) a
append' Nil ys = ys
append' (x :: xs) ys = x :: append' xs ys
```

The interesting part is the function signature - given a vector of size n and a vector of size m, the resulting vector will have size n + m. This information is captured in the declaration and the compiler knows to apply the (+) defined above and type-check that this is indeed true for a given pair of arguments.

## Proofs

We can also attempt to define a reverse' function, which recursively appends the head of the vector to the reversed tail:

```
reverse' : Vect' n a -> Vect' n a
reverse' Nil = Nil
reverse' (x :: xs) = append' (reverse' xs) [x]
```

This doesn’t compile though. We get the following error message:

```
When checking right hand side of reverse' with expected type
Vect' (S k) a
Type mismatch between
Vect' (k + S Z) a (Type of append' (reverse' xs) [x])
and
Vect' (S k) a (Expected type)
Specifically:
Type mismatch between
k + S Z
and
S k
```

We are claiming the function returns a vector of the same length as the input vector, but we haven’t proven enough theorems about our definition of natural numbers to convince the type checker. In this particular case, the problem is that the compiler expects an S k but finds an k + S Z. We need to prove that these are indeed equal (successor of k is the same as k + successor of Z). Here is the proof:

```
addOneProof : (n : Nat') -> S n = n + S Z
addOneProof Z = Refl
addOneProof (S k) = cong (addOneProof k)
```

Proofs are functions. There are a few things worth noting here: first, the return type of this function is an equality (our theorem). Given a natural n, the function proves that the equality holds. Refl is the built-in reflexivity constructor, which constructs x = x. For the Z case, we can use Refl to say that S Z = Z + S Z which is true by the definition of (+). For the (S k) case, we use cong. cong is a built in function that states that equality holds after function application. It’s signature is cong : (a = b) -> f a = f b, which basically means if a is equal to b, then f a is equal to f b. In our case, we are saying that if addOneProof k holds, then so does addOneProof (S k), which allows us to converge on the Z case.

We now have a proof that S n = n + S Z. With this, we can prove that the type Vect (k + (S Z)) a can be rewritten as Vect (S k) a:

```
reverseProof : Vect' (k + (S Z)) a -> Vect' (S k) a
reverseProof {k} result = rewrite addOneProof k in result
```

There is some Idris-specific syntax here: {k} brings k from the function declaration in scope, so we can refer to it in the function body even if it is not passed in as an argument. The rewrite ... in expression applies the equality in the proof above to the input, in this case effectively rewriting Vect (k + (S Z)) a to Vect (S k) a. Note these proofs are evaluated at compile time and simply provide information to the type checker. With this proof, we can implement reverse like this:

```
reverse' : Vect' n a -> Vect' n a
reverse' Nil = Nil
reverse' (x :: xs) = reverseProof (append' (reverse' xs) [x])
```

This is similar to the previous implementation, we just apply reverseProof to the result of append'. This definition compiles.

## Thoughts About the Future

Software development is generally driven by economics, where we more often than
not trade correctness for speed to market. But once the software is up and
running, correctness becomes an issue. As code increases in complexity, the
number of issues tends to increase, and the velocity with which changes can be
made without introducing regression drops dramatically. We have various
techniques that aim to maintain stability, like automated testing, but these are
not perfect: a test can prove that for a given input we get an expected output,
but cannot prove that for *any* input we would get the expected output.

On the other hand, we have solutions that do eliminate entire classes of issues.
An example is typing. Python, Ruby, and JavaScript, all dynamically typed, are
extremely expressive and make it very easy to whip up a proof of concept. But
there is an entire class of type errors which now turns into runtime issues. We
are notoriously bad at predicting what our code does, so the more help we get
from machines to ensure correctness, the better. In a strongly typed language,
even though it takes longer to convince the compiler that the code is type-safe,
this whole class of errors is eliminated. Language evolution over the years
tends to converge towards stronger typing: dynamic languages are augmented with
type checkers (Python has type hints, JavaScript has TypeScript etc.) and
statically typed languages are becoming less verbose as type inference evolves.
There will always be a need for a quick prototype, but code we want to deem
*reliable* should be typed. This includes a wide range of business-critical
applications where errors are very costly.

I see Idris as the next step beyond this. Totality checking allows the compiler
to guarantee termination, eliminating hangs from the code. First-order types
allows us to push more information to the type-checker, allowing for stricter
type-checking. Proofs, expressed as functions with regular syntax, allow the
compiler to provide formal verification of programs - here, as opposed to unit
tests, we are actually proving that we get the expected output for *any* input.
These are all tools for writing better, more correct code. As other functional
concepts got adopted over the years into more mainstream languages (for example
first-order functions, anonymous functions, algebraic types etc.), I expect (and
hopre) these features to eventually be adopted too.

There is still a lot of room for improvement: writing proofs is tedious, compiler errors are not always very clear, and, coming back to the speed to market tradeoff, I doubt we will ever get to entire large applications formally proven correct (barring some form of proof inference to speed things up by a couple of orders of magnitude). That being said, I would love to have these facilities as optional features in other languages and at least have the ability to prove that the core functionality of a component does what it is supposed to do, and get a compile break whenever a regression is introduced.

Programming languages are continuously evolving and the future looks exciting!

[1] | Idris also supports functions that produce an infinite stream of values which can be used with lazy evaluation. The full definition of totality includes functions which don’t terminate but produce Inf. This allows for non-terminating functions, but ensures non-termination is intentional. |

[2] | I am using ' to avoid naming conflicts with the built-in types and functions. Idris already provides Nat, Vect, append and reverse. |