Skip to main content

We don't need no stinking modules!

In my last post I described how unbelievably cool modules are in Coq. Well it turns out that there is some disagreement in the community about whether one should even use them!

Why? Because the type system in Coq is so powerful we hardly even need them. In fact with a bit of syntactic sugar for inductively defined types and the generation of constructors we magically get something like first class modules (first-class meaning modules that are actually values that can be passed around to functions etc...). Enter the Record.

Record MonadBind : Type :=
  { M : forall (A : Type), Type;
    bind : forall (A B : Type), M A -> (A -> M B) -> M B; 
    ret : forall (A : Type), A -> M A; 
    left_unit : forall (A B : Type) (f : A -> M B) (a : A), 
      bind A B (ret A a) f = f a;
    right_unit : forall (A B : Type) (m : M A), 
      bind A A m (ret A) = m;
    bind_assoc : forall (A B C : Type) (m : M A) (f : A -> M B) (g : B -> M C) (x : B), 
      bind B C (bind A B m f) g = bind A C m (fun x => bind B C (f x) g)
  }.

This implements exactly the same strict controls on the instantiation of a Monad that were given with our MONAD signature in the previous post. Each type can rely on previously defined types in the record giving us the full dependent type system needed to specify a proper monad. With a little bit of work we can rewrite our module to be an instantiation of a record.

Look here for the example code

How cool is that!

Comments

Popular posts from this blog

Managing state in Prolog monadically, using DCGs.

Prolog is a beautiful language which makes a lot of irritating rudimentary rule application and search easy. I have found it is particularly nice when trying to deal with compilers which involve rule based transformation from a source language L to a target language L'. However, the management of these rules generally requires keeping track of a context, and this context has to be explicitly threaded through the entire application, which involves a lot of irritating and error prone sequence variables. This often leads to your code looking something a bit like this: compile(seq(a,b),(ResultA,ResultB),S0,S2) :- compile(a,ResultA,S0,S1), compile(b,ResultB,S1,S2). While not the worst thing, I've found it irritating and ugly, and I've made a lot of mistakes with incorrectly sequenced variables. It's much easier to see sequence made explicitly textually in the code. While they were not designed for this task, but rather for parsing, DCGs turn out to be a conveni

Decidable Equality in Agda

So I've been playing with typing various things in System-F which previously I had left with auxiliary well-formedness conditions. This includes substitutions and contexts, both of which are interesting to have well typed versions of. Since I've been learning Agda, it seemed sensible to carry out this work in that language, as there is nothing like a problem to help you learn a language. In the course of proving properties, I ran into the age old problem of showing that equivalence is decidable between two objects. In this particular case, I need to be able to show the decidability of equality over types in System F in order to have formation rules for variable contexts. We'd like a context Γ to have (x:A) only if (x:B) does not occur in Γ when (A ≠ B). For us to have statements about whether two types are equal or not, we're going to need to be able to decide if that's true using a terminating procedure. And so we arrive at our story. In Coq, equality is

Teagrey

I was ironing my shirt today, which I almost never do. Because of this I don't have an ironing board so I tried to make a make-shift ironing board on my floor using a towel and some books. I grabbed the heaviest books on the nearest shelf, which happened to be Organic Chemistry, Stalingrad and an annotated study bible containing the old and new testament. As I pulled out the bible, a flower fell out which had been there for over 17 years now. I know that because it was put there by my first wife, Daniel, who killed herself in April about 17 years ago. It fell from Thessalonians to which it had been opened partially. Highlighted was the passage: "Ye are all sons of the light and sons of the day." I guess the passage gave her solace. Daniel was a complicated woman. She had serious mental health issues which plagued her for her entire life. None of them were her fault. She was dealt an absolutely awful hand in life, some truly nasty cards. She had some considerable c