## Weight-biased leftist heaps verified in Haskell using dependent types

In January I announced my implementation of weight-biased leftist heaps verified with dependent types in Agda. This was part of my work on a paper submitted to CAV’14 conference. The paper got rejected and I decided not to resubmit it anywhere else. At this year’s ICFP listening to Stephanie Weirich’s keynote speech motivated me to finally port that implementation to Haskell, something that I had planned for a couple of months now. You can take a look at the result on github. Here I want to share some of my experiences and insights.

My overall impression is that porting from Agda to Haskell turned out to be fairly straightforward. It was definitely not a complete rewrite. More like syntax adjustments here and there. There were of course some surprises and bumps along the way but nothing too problematic. More precise details are given in the code comments.

When it comes to programming with dependent types Agda, being a fully-fledged dependently-typed language, beats Haskell in many aspects:

• Agda has the same language for terms and types. Haskell separates these languages, which means that if I want to have addition for natural numbers then I need to have two separate definitions for terms and types. Moreover, to tie types and terms together I need singleton types. And once I have singleton types then I need to write third definition of addition that works on singletons. All of this is troublesome to write and use. (This tedious process can be automated by  using singletons package.)
• interactive agda-mode for Emacs makes writing code much simpler in Agda. Here I was porting code that was already written so having an interactive Emacs mode for Haskell was not at all important. But if I were to write all that dependently-typed code from scratch in Haskell this would be painful. We definitely need better tools for dependently-typed programming in Haskell.
• Agda admits Unicode identifiers. This allows me to have type constructors like ? or variables like p?b. In Haskell I have GEq and pgeb, respectively. I find that less readable. (This is very subjective.)
• Agda has implicit arguments that can be deduced from types. Haskell does not, which makes some function calls more difficult. Surprisingly that was not as huge problem as I initially thought it will be.
• Agda is total, while Haskell is not. Since there are bottoms in Haskell it is not sound as a logic. In other words we can prove false eg. by using undefined.

The list is noticeably shorter:

• Haskell has much better term-level syntax. In many places this resulted in significantly shorter code than in Agda.
• Haskell is not total. As stated earlier this has its drawbacks but it also has a good side: we don’t need to struggle with convincing the termination checker that our code does actually terminate. This was painful in Agda since it required using sized types.
• Haskell’s gcastWith function is much better than Agda’s subst. Both these functions allow type-safe casts given the proof that the cast is safe. The difference is that Agda’s subst requires more explicit arguments (as I noted earlier the opposite is usually the case) and restricts the cast to the last type parameter (Haskell allows cast for any type parameter).

# Summary

While the list of wins is longer for Agda than it is for Haskell I’m actually very happy with Haskell’s performance in this task. The verification in Haskell is as powerful as it is in Agda. No compromises required.

It’s worth remarking that my implementation works with GHC 7.6, so you don’t need the latest fancy type-level features like closed type families. The really essential part are the promoted data types.

### 7 Responses to “Weight-biased leftist heaps verified in Haskell using dependent types”

1. Mikhail Glushenkov says:

Haskell also supports Unicode identifiers with {-# LANGUAGE UnicodeSyntax #-}. See e.g. base-unicode-symbols on Hackage.

2. Jan Stolarek says:

Yes, that’s certainly true. But with support from agda-mode Unicode in Agda becomes a natural thing. To the best of my knowledge we don’t have such support for writing Unicode in Haskell. To me in practice this means that Unicode in Haskell is not practically usable.

3. William Dhalgren says:

> But with support from agda-mode Unicode in Agda becomes a natural thing.

could you please expand on the kind of support you’re talking about?

I played with Unicode in my haskell code with a compose key defined – one I needed anyhow for diacritics in my native language if I wished to keep a us layout otherwise – and found it quite a natural thing as well; compositions were easy enough to type, default mnemonics sensible, and also easily customizable.

Another option I read some people preferred was to write ascii only but setup replacements in their display only. Guess most people don’t have much reason for dealing with unicode otherwise so this way you don’t force them to. Haven’t tried that variant.

4. Jan Stolarek says:

> could you please expand on the kind of support you?re talking about?

I meant support that lets you type Greek letters by writing \G and then a letter, write mathematical symbols like \forall and so on. I suppose this could be set up *somehow* in Emacs independently of haskell-mode but agda-mode just gives that for free.

5. Mikhail Glushenkov says:

> I meant support that lets you type Greek letters by writing \G and then a letter

This can be enabled with M-x set-input-method TeX. I suspect that agda-mode’s support for Unicode amounts to just turning on the TeX input method by default.

6. Jan Stolarek says:

I just tried that and:

a) to get ? in Agda I have to type \Gl, in haskell-mode I have to type \lambda, so these are not exactly the same input methods.

b) Agda accepts identifiers like “t?p”, Haskell does not.

So I sustain my claim that Agda’s support for Unicode beats Haskell.

7. Mikhail Glushenkov says:

Re: your point a) – it looks like Agda’s input method can be also used in other Emacs modes:

http://stackoverflow.com/questions/25340687/use-agdas-input-method-in-other-emacs-mode

?t?p? is not supported for the same reason ?t!!!!p? is not. You can use stuff like ?let ł???? x = …?, however.

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