Agda is a dependently typed programming language. I’ve read about dependent types on and off for the past few years in magazine articles as well as discussed them a little with my collaborator Dan Ghica and others, but never really had the time to properly experiment until this Easter holiday. This book – by Aaron Stump – is interesting because it’s basic, practical, and jumps right into coding. I thought it would be good for someone like me who wants to play with the practice of dependent types and only delve into theoretical concerns as necessary or motivated to do so.

The book comes with its own Agda library, the Iowa Agda Library (IAL), a basic library of Booleans, lists, etc., simply written for the beginner to understand. It has a lot of exercises and code snippets based on that library which I found useful and relatively easy to follow.

The book assumes no functional programming expertise, and since I’ve done a (very) little bit of functional programming in Haskell in the distant past, this made the introductory material easy to follow and a useful revision aid.

Introductory Agda

The early parts of the book – Chapters 1 to 4 – are based on various data structures: Booleans, natural numbers, lists. Here the functional programming adds little to what I’ve seen before. The neat thing is the ability to use Agda as a theorem prover. This is what is described in the book as “external verification” – you write code and then you prove – separately – theorems about the code.

Firstly, let’s take a look at some example Agda code, taken from Chapter 4:


take : ∀{ℓ}{A : Set ℓ} → (n : ℕ) → 𝕃 A → 𝕃 A
take 0       _         = []
take (suc n) []        = []
take (suc n) (x :: xs) = x :: take n xs

There’s a few interesting things here already. Firstly, unicode! It feels surprisingly liberating to write unicode characters in code. Secondly the type declaration declares (via ∀), the function to be polymorphic – it operates on lists (𝕃 from IAL) of any type A (of any level ℓ within a hierarchy of type levels).

The real excitement comes when we start proving theorems. For example, given the function nthTail from IAL which returns all elements after the first n elements in a list (and the empty list if the list has fewer than n elements), we can write


ex46 : ∀{ℓ}{A : Set ℓ} (n : ℕ)(l : 𝕃 A) → (take n l) ++ (nthTail n l) ≡ l
ex46 0        _                           = refl
ex46 (suc n) []                           = refl
ex46 (suc n)  (x :: xs) rewrite ex46 n xs = refl

Here ex46 is just the name of my proof (it was Exercise 4.6 in the book). The type declaration for the function ex46 can be read as “I’ll produce a proof that concatenating (take n l) with (nthTail n l) just produces l for all natural numbers n and for all lists of elements of any type A. Here , taken from IAL, is propositional equality, and the only property of it we’re using is reflexivity; in fact the only inhabitant of this type is refl. The first two lines of the proof work because the definitions of take and nthTail are such that when the first argument is zero or the second argument is empty, the left and right hand side normalize to the same result, namely .l ≡ .l in the first case (.l being the name assigned to the second unnamed argument _) and [] ≡ [] in the second case. The third case explicitly uses induction: rewrite is a directive that applies the induction hypothesis ex46 n xs to the proof of ex46 (suc n) (x :: xs). Very pleasing!

Dependent Types Revealed

Chapter 5 is where we really start to see the power of dependent types. Up until this point, we’ve written functional programs and then proved properties about those programs. We have definitely used dependent types in the process, but they were somehow “invisible” to us as a user, and – at least for the exercises in the book – we could ignore the dependent-type based implementation without really affecting the usability of the approach. Chapter 5 shows how the proofs can be part of the programs themselves, using the type system to express the properties we wish to enforce. The simplest and clearest example given is a declaration of a vector datatype:


data 𝕍 {ℓ} (A : Set ℓ) : ℕ → Set ℓ where
[] : 𝕍 A 0
_::_ : {n : ℕ} (x : A) (xs : 𝕍 A n) → 𝕍 A (suc n)

Note here that a vector, unlike a list, has a specific length built into its type. As a result, you get various properties for free, for example Agda’s type system verifies that the length of the concatenation of two vectors is the sum of the individual lengths from a straight-forward definition of a concatenation operation; the code below type checks, and that’s all that necessary.


_++𝕍_ ∀{ℓ}{A : Set ℓ}{n m : ℕ} → 𝕍 A n → 𝕍 A m → 𝕍 A (n + m)
[] ++𝕍 ys = ys
(x :: xs) ++𝕍 ys = x :: (xs ++𝕍 ys)

A more complex example of Braun trees is described in detail in the book. These are binary trees where the data at a node has a value less than or equal to that of its children while the size of the node left and right subtrees are either equal or the left subtree is exactly one node larger than the right subtree. The author chooses to encode this latter property within the declaration of the Braun tree type, and then shows how to write code for tree insertion, deletion, etc.

I enjoyed this chapter, though it had me puzzling for quite some time over sigma types, a subtle and powerful idea that I think deserves a bit more discussion than the space given in the book; I ended up reading around this topic a bit before I felt comfortable. An example given considers a function that turns lists into vectors. Immediately we have a problem – what is the return type of this function? It should be (𝕍 A n) but n is not fixed! The solution presented is to return a type written in Agda as Σ ℕ (λ n → 𝕍 A n); an inhabitant of this type should be interpreted as “a pair consisting of a natural number and a function which, when applied to that natural number, returns the vector type we’re actually interested in.” This takes a little getting used to!

Type-Level Computation

Chapter 6, introduces the idea of proof by reflection, amongst other things. This looks super-powerful, and I’m motivated to dig further into this topic when I have the time. The author illustrates the concept by creating a list simplifier, which simplifies expressions on lists. By reflecting the operations on lists as corresponding constructors for list expressions, and then proving the soundness of the simplifier, he is able to use this soundness proof to, inter alia, prove properties about the operations operating on lists. The example is really interesting, though unlike in previous chapters of the book, it’s not really discussed in detail in Chapter 6, and you have to look carefully at the code accompanying the book to see how it all fits together.

The other primary example of Chapter 6 is a statically typed version of printf, the archetypal type-evasive function of my youth

int printf( const char *restrict format, ... );

Here, the author presents an example credited to Augustsson, showing that dependent types can come to the rescue essentially by providing a type for C’s “…” which depends on the format string.

The integer example from this chapter is the closest to my own interests, showing how integers can be defined using naturals in an interesting way: an integer is a natural together with a sign which is unit (trivial) if the natural is zero but has Boolean type otherwise, i.e. the type of the sign is dependent on the value of the natural, leading to the following type declaration and definition of ℤ [taken from here].


ℤ-pos-t : ℕ → Set
ℤ-pos-t 0 = ⊤
ℤ-pos-t (suc _) = 𝔹

data ℤ : Set where
mkℤ : (n : ℕ) → ℤ-pos-t n → ℤ

I had followed through the discussion in the book easily, but came totally unstuck when trying to implement a seemingly obvious exercise I set myself: prove that integer multiplication commutes using a pre-existing proof of the commutativity of multiplication over the naturals.

I had defined integer multiplication as:


_*ℤ_ : ℤ → ℤ → ℤ
_*ℤ_ (mkℤ 0 _)        _                = mkℤ 0 triv
_*ℤ_ (mkℤ (suc n) _)  (mkℤ 0 _)        = mkℤ 0 triv
_*ℤ_ (mkℤ (suc n) sn) (mkℤ (suc m) sm) = mkℤ ((suc n) * (suc m)) (sn xor sm)

where triv is the single inhabitant of the type.  I had therefore naively assumed that one could simply write the proof as:

*ℤcommutes (mkℤ 0 _)        (mkℤ 0 _) = refl
*ℤcommutes (mkℤ 0 _) (mkℤ (suc bN) _) = refl
*ℤcommutes (mkℤ (suc aN) _) (mkℤ 0 _) = refl
*ℤcommutes (mkℤ (suc a) sa) (mkℤ (suc b) sb) rewrite xor-comm sa sb | (*comm (suc a) (suc b)) = refl

However, this fails – for the final pattern – for a very interesting reason that was, however, totally incomprehensible to me at first. Agda reports:

𝔹 !=< ℤ-pos-t w of type Set
when checking that the type
(b a w : ℕ) →
w ≡ suc (a + b * suc a) →
(sa sb : 𝔹) →
mkℤ w (sb xor sa) ≡ mkℤ (suc (a + b * suc a)) (sb xor sa)
of the generated with function is well-formed

“Huh? ‘Generated with function?'”

Thus began a small odyssey of Easter holiday evenings spent learning how rewrites are implemented as withs, and how withs in turn are implemented using generated auxiliary functions. I found Norrell and Chapman’s tutorial, together with the good quality documentation provided me with the necessary additional background required to teach myself what was going on here and to produce a final, working proof:


*ℤcommutes : ∀ (a b : ℤ) → a *ℤ b ≡ b *ℤ a
*ℤcommutes (mkℤ 0 _)        (mkℤ 0 _)        = refl
*ℤcommutes (mkℤ 0 _)        (mkℤ (suc m) sm) = refl
*ℤcommutes (mkℤ (suc n) _)  (mkℤ 0 _)        = refl
*ℤcommutes (mkℤ (suc n) sn) (mkℤ (suc m) sm)
with m + n * suc m | *comm (suc n) (suc m)
... | .(n + m * suc n) | refl rewrite xor-comm sn sm = refl

I have mixed feelings about this experience. On the one hand, I’ve learnt a lot about Agda specifically (and dependent types in general) by being forced to go through this process to understand what’s going on under the hood. On the other hand, this seems like a lot of hoops to jump through for this simple proof, and the book under review didn’t have all the answers I was looking for to solve the exercise. I don’t see the latter point as a big deal, but if you’re going to do your explorations on holiday, make sure you have a wifi connection available so you can look beyond the book for help!

Selected Topics

The remainder of the book is eclectic but interesting.

Chapters 7 and 8 combine to produce a larger example piece of code – a Huffman coder / decoder. It’s good to see the ideas put into practice in a larger case study, of course, though these chapters were less interesting to me personally.

Chapter 9 really recaptured my attention, though! The author uses Agda’s types to reason about termination. He explicitly sets up a general datatype representing what it means for a relation DAG to not have any infinite chain of successor nodes starting from a given named node, something he calls a terminating relation. He then proves in Agda that > over ℕ is terminating in this sense. The chapter then goes on to apply this to producing a termination proof for a recursively defined division function. I’m aware of this kind of reasoning, and have even played a bit with tools making use of these ideas like Byron Cook et al.‘s Terminator. So it’s interesting to see this in the context of the type system of Agda where in fact it becomes rather important since all functions must terminate in Agda – otherwise none of the proofs discussed above make sense!

The final chapter of the book, Chapter 10, formalises some ideas in a particular logic within Agda, using Agda to prove soundness and completeness of the logic.

Conclusion

The book ends rather abruptly with Chapter 10 on intuitionistic logic and Kripke semantics. There is no rounding off, summarising lessons learnt or directions for future research or indeed practical programming advice.

Nevertheless, I learnt a lot from reading this book – and a lot more from doing its exercises! It’s given me an initial understanding of one of the most recently talked-about developments in programming. I’m left keen to try to apply this to some real-world problems in my research! Well recommended holiday reading.

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