Posts tagged: idris

First impressions of Coq and “Certified Programming with Dependent Types”

A simplistic view of strongly typed functional programming ecosystem is that the two most influential languages are Haskell and ML. Haskell is my language of choice so when it came to learning dependently-typed programming I stayed on the Haskell side of the spectrum and went with Agda and Idris. I chose the two over the ML-inspired Coq, most often advertised as a proof assistant rather that a programming language, planning to learn it “when I find some free time”. I wouldn’t probably find that time if it wasn’t for a friend of mine who recently picked up a book “Certified Programming with Dependent Types” by Adam Chlipala of MIT. Having a companion to discuss the ideas in the book was a perfect opportunity to pick it up – the truth is I found out about the book well over a year ago and since then it stayed on my always too long must-read-this-some-day list. So far I have read around 1/3rd of the book and would like to share some of my first impressions, both about the book, which I will refer to as CPDT, and Coq.

(Note: In what follows I will compare Coq to Agda and Idris but you have to be aware that despite similarity in features of these languages they don’t aim to be the same. Coq is a proof assistant with an extra feature of code extraction that allows you to turn your proofs into code – if you ever heard about “programs as proofs” this is it. Idris is a programming language with extra features that allow you to prove your code correct. I’m not really sure how to classify Agda. It is definitely on the programming-language-end of the spectrum – it allows you to prove your code correct but does not provide any extra built-in proof support. At the same time turning Agda code into working programs is non-trivial.)

Let me start off by saying that I don’t have any real-life Coq project on the horizon, so my learning is not motivated by need to solve any practical problem. My main driving force for learning Coq is purely interest in programming languages and seeing how Coq compares to Agda and Idris. A common thing with dependently-typed languages is that the types can get too complicated for the programmer to comprehend and thus a language requires an interactive mode to provide programmer with compiler feedback about the types. This is true for Agda, Idris and Coq. Agda offers a great support for holes: programmer can insert question marks into the program code and once it is re-compiled in the editor (read: Emacs) question marks become holes ie. places where Agda compiler provides user with feedback about expected types, bindings available in the hole context as well as some nice inference features allowing to automatically fill in contents of holes. So in Agda one proves by constructing terms of appropriate types. Coq is different, as it relies on a mechanism called “tactics”. Once the user writes down a type (theorem) he is presented with a set of goals to prove. Applying a tactic transforms the current goal into a different goal (or several goals). Conducting consecutive steps of the proof (ie. applying several tactics) should lead to some trivial goal that follows from definition and ends the proof. To work with Coq I decided to use Proof General, an Emacs extension for working with proofs (many other proof assistants are supported besides Coq)1. It launches Coq process in the background and essentially integrates writing code with proving. With Proof General I can easily step through my proofs to see how the goals are transformed by usage of tactics. Idris falls somewhere between Agda and Coq. As stated earlier it is mostly a programming language but it also provides tactic-based proving. So for example when I write a definition that requires explicit proof to typecheck, idris-mode launches interactive REPL in which I can conduct a proof in a fashion similar to Proof General and once I’m finished the proof is inserted into the source code. the result looks something like this:

par : (n : Nat) -> Parity n
par Z = even {n=Z}
par (S Z) = odd {n=Z}
par (S (S k)) with (par k)
  par (S (S (j + j)))     | even ?= even {n = S j}
  par (S (S (S (j + j)))) | odd  ?= odd {n = S j}
 
---------- Proofs ----------
 
Basics.par_lemma_2 = proof
  intros
  rewrite sym (plusSuccRightSucc j j)
  trivial
 
 
Basics.par_lemma_1 = proof
  intros
  rewrite sym (plusSuccRightSucc j j)
  trivial

The last time I checked Idris once the proof was completed and added to the source code it was not possible to step through it back and forth to see how goals are transformed. (Things might have changed since I last checked.)

So far I’ve been doing rather basic stuff with Coq so I haven’t seen much that wouldn’t be also possible in Agda or Idris. The biggest difference is that Coq feels a much more grown up language than any of the mentioned two. One totally new thing I learned so far is co-induction, but I’m only starting with it and won’t go into details, rather leaving it for a separate post. (Agda also supports co-induction.)

As for the CPDT book I have to say it is a challenging read: it’s very dense and focuses on more advanced Coq techniques without going into details of basics. As such it is a great demonstration of what can be done in Coq but not a good explanation of how it can be done. Depending on what you are expecting this book might or might not be what you want. As stated earlier I don’t plan on applying Coq in any project but rather want to see a demo of Coq features and possibly pick up some interesting theoretical concepts. As such CPDT works quite well for me although I am not entirely happy with not being able to fully understand some of the demonstrated techniques. As such CPDT is definitely not a self-contained read, which I believe was a conscious decision on the author’s side. Discussing with people on #coq IRC channel and reading various posts on the internet leads me to a conclusion that CPDT is a great book for people that have been using Coq for some time and want to take their skills to a new level. The main theme of the book is proof automation, that is replacing tactic-based sequential proofs with automated decision procedures adjusted to the problem at hand that can construct proofs automatically. Indeed tactic-based proofs are difficult to understand and maintain. Consider this proof of a simple property that n + 0 = n:

Theorem n_plus_O : forall (n : nat), plus n O = n.
  intros.
  induction n.
  reflexivity.
  simpl.
  rewrite IHn.
  reflexivity.
Qed.

To understand that proof one has to step through it to see how goals are transformed or have enough knowledge of Coq to know that without the need of stepping through. Throughout the book Adam Chlipala demonstrates the power of proof automation by using his tactic called crush, which feels like a magic wand since it usually ends the proof immediately (sometimes it requires some minimal guidance before it ends the proof immediately). I admit this is a bit frustrating as I don’t feel I learn anything by seeing crush applied to magically finish a proof. Like I said, a good demo of what can be done but without an explanation. The worst thing is that crush does not seem to be explained anywhere in the book so readers wanting to understand it are left on their own (well, almost on their own).

What about those of you who want to learn Coq starting from the basics? It seems like Software Foundations is the introductory book about Coq and – given that the main author is Benjamin Pierce – it looks like you can’t go wrong with this book. I am not yet sure whether I’ll dive into SF but most likely not as this would mean giving up on CPDT and for me it’s more important to get a general coverage of more advanced topics rather than in-depth treatment of basics.

  1. Other choices of interactive mode are available for Coq, for example CoqIDE shipped by default with Coq installation []

Idris – first impressions

During last few weeks I got a pretty good grip of basics of dependent types and Agda. Programming in Agda is fun but nevertheless I decided to experiment with other dependently-typed programming languages. Back in March I attempted to learn Idris from one of Edwin Brady’s presentations, but having no knowledge of dependent types I had to give up after about 30 minutes of first video. Now that I know basics of Agda I decided to give Idris another try. This time it was much simpler. Reading official Idris tutorial and doing some experiments took me about 5 hours. Below are some of my first impressions (I’m underlining that phrase to make it clear that some of these opinions may change in the future).

  • Standard library in Idris feels friendlier than in Agda. It is bundled with the compiler and doesn’t require additional installation (unlike Agda’s). Prelude is by default imported into every module so programmer can use Nat, Bool, lists and so on out of the box. There are also some similarities with Haskell prelude. All in all, standard library in Idris is much less daunting than in Agda.
  • Idris is really a programming language, i.e. one can write programs that actually run. Agda feels more like a proof assistant. According to one of the tutorials I’ve read you can run programs written in Agda, but it is not as straightforward as in Idris. I personally I haven’t run a single Agda program – I’m perfectly happy that they typecheck.
  • Compared to Agda Idris has limited Unicode support. I’ve never felt the need to use Unicode in my source code until I started programming in Agda – after just a few weeks it feels like an essential thing. I think Idris allows Unicode only in identifiers, but doesn’t allow it in operators, which means I have to use awkward operators like <!= instead of ≤. I recall seeing some discussions about Unicode at #idris channel, so I wouldn’t be surprised if that changed soon.
  • One of the biggest differences between Agda and Idris is approach to proofs. In Agda a proof is part of function’s code. Programmer is assisted by agda-mode (in Emacs) which guides code writing according to types (a common feature in dependently typed languages). Over the past few weeks I’ve come to appreciate convenience offered by agda-mode: automatic generation of case analysis, refinement of holes, autocompletion of code based on types to name a few. Idris-mode for Emacs doesn’t support interactive development. One has to use interactive proof mode provided in Idris REPL – this means switching between terminal windows, which might be a bit inconvenient. Proofs in Idris can be separated from code they are proving. This allows to write code that is much clearer. In proof mode one can use tactics, which are methods used to convert proof terms in order to reach a certain goal. Generated proof can then be added to source file. It is hard for me to decide which method I prefer. The final result is more readable in Idris, but using tactics is not always straightforward. I also like interactive development offered by Agda. Tough choice.
  • Both languages are poorly documented. That said, Idris has much less documentation (mostly papers and presentations by Edwin Brady). I expect this to change, as the Idris community seems to be growing (slowly, but still).
  • One thing I didn’t like in Idris are visibility qualifiers used to define how functions and datatypes are exported from the module. There are three available: public (export name and implementation), private (don’t export anything) and abstract (export type signature, but don’t export implementation). This is slightly different than in Haskell – I think that difference comes from properties of dependent types. What I didn’t like are rules and syntax used to define export visibility. Visibility for a function or datatype can be defined by annotating it with one of three keywords: public, private, abstract. If all definitions in a module are not annotated then everything is public. But if there is at least one annotation everything without annotation is private. Unless you changed the default visibility, in which case everything without annotation can be abstract! In other words if you see a definition without annotation it means that: a) it can be public, but you have to check if all other definitions are without annotations; b) private, if at least one other definition is annotated – again, you have to check whole file; c) but it can be abstract as well – you need to check the file to see if the default export level was set. The only way to be sure – except for nuking the entire site from orbit – is annotating every function with an export modifier, but that feels very verbose. I prefer Haskell’s syntax for defining what is exported and what is not and I think it could be easily extended to support three possible levels of export visibility.
  • Unlike Agda, Idris has case expressions. They have some limitations however. I’m not sure whether these limitations come from properties of dependently typed languages or are they just simplifications in Idris implementation that could theoretically be avoided.
  • Idris has lots of other cool features. Idiom brackets are a syntactic sugar for applicative style: you can write [| f a b c |] instead of pure f <*> a <*> b <$*gt; c. Idris has syntax extensions designed to support development of EDSLs. Moreover tuples are available out of the box, there’s do-notation for monadic expressions, there are list comprehensions and Foreign Function Interface.
  • One feature that I’m a bit sceptical about are “implicit conversions” that allow to define implicit casts between arguments and write expressions like "Number " ++ x, where x is an Int. I can imagine this could be a misfeature.
  • Idris has “using” notation that allows to introduce definitions that are visible throughout a block of code. Most common use seems to be in definition of data types. Agda does it better IMO by introducing type parameters into scope of data constructors.
  • Idris seems to be developed more actively. The repos are stored on github so anyone can easily contribute. This is not the case with Agda, which has Darcs repos and the whole process feels closed (in a sense “not opened to community”). On the other hand mailing list for Idris is set up on Google lists, which is a blocker for me.

All in all programming in Idris is also fun although it is slightly different kind of fun than in Agda. I must say that I miss two features from Agda: interactive development in Emacs and Unicode support. Given how actively Idris is developed I imagine it could soon become more popular than Agda. Perhaps these “missing” features will also be added one day?

As an exercise I rewrote code from “Why dependent types matter” paper from Agda (see my previous post) to Idris. Code is available in on github.

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