### Type Checking and Cyclic Proof

Standard functional programming languages like Haskell and SML use polymorphic type systems. These type systems however, are not sound. What this means is that the type-systems themselves can not be used to prove properties of the software in the sense of "total correctness". To see that this is true, we can get a "proof" (read program) of inhabitation of any arbitrary type A, by simply using the program:

data Bot {- Look Ma! No constructors! -}
bot :: Bot
bot = bot

Clearly when we say that bot is an inhabitant of Bot, we dont mean that it actually produces a value of type Bot, since there aren't any as Bot has no constructors! We can easily use this type of proof to prove something like A ∧ ¬ A which leads to a pretty degenerate logic. However, the type system is still *useful* in the sense that if the program ever *does* terminate, it's sure to do so with the appropriate type. This means we can get the full class of Turing complete programs, a very useful benefit.

In Constructive Type Theory, we need to keep stronger guarantees. For programming languages such as Coq, we use syntactic restrictions to ensure that programs terminate (or coterminate). This however has the annoying feature that a lot of programs which are obviously (co)terminating will be rejected simply because of syntax.

Cyclic proofs give a method of describing inductive or coinductive proofs without requiring that we demonstrate the fact that our term (co)terminates up front. We can defer the proof until later. The huge advantage to this is that we can use ideas from supercompilation, as a form of proof normalisation and then show that the syntactic termination criteria are satisfied for the resulting transformed proof, rather than for the original.

As an example, take the following program (with the usual list, cons notation from Haskell):
```codata CoNat = Z | S CoNat
data List = [] | (:) CoNat List

plus :: CoNat -> CoNat
plus Z y = y
plus (S x) y = S (plus x y)

sum :: List -> CoNat
sum [] = Z
sum (x:xs) = plus x (sum xs)
```

Now, we can associate with this the following (infered) pre-proof type tree:

Now, we can use the usual super-compilation manipulations on this type tree, including beta-reduction, etc. to arrive at this new tree:

This is actually a proof, rather than a pre-proof as can be verified syntactically. It satisfies the guardedness condition of coinduction, and the structural recursion condition for induction.

From the proof above, we can derive the following program, which is syntactically sound.

```sum [] = Z
sum (Z:xs) = sum xs
sum (S:xs) = S(f x xs)

f Z xs = sum xs
f (S x) xs = S(f x xs)
```

This process is basically an extended form of cut-elimination where we can extend the applicability of cut-elimination since we don't directly use induction rules, but instead we use cycles in the proof. We can then work with transformations over a larger class of things which are similar to normalisation.

There are a lot of advantages to this approach. In the first program our function 'sum' did not meet the guardedness condition, which means it would not be admissible in Coq, despite being perfectly correct (as it is in fact productive). Using pre-proofs we can defer proof, which gives us better compositionality. We can even use higher order functions which are not in general correct, on particular functions to derive programs which are totally correct.

In addition, we can decide only to show total correctness for regions of a program, rather than the entire program. We could decide that only certain regions require total correctness, and freely mix total correctness with partial correctness.

There is still a ton of work to be done in this area. It would be nice to know what proof transformation rules coupled with which algorithms can solve various classes of problems. Kamendantskaya has a very interesting class of productive functions which, I believe, could be found using a particular proof transformation algorithm. I'd like to have this algorithm and a proof that it works. In addition, I'd like to have more examples where this can be used to enhance compositionality (I'm thinking of filter functions in particular, where this might come in handy).

Sorry if this blog post is a bit "whirl-wind". I intend to lay out the entire theory in a slower and better motivated way later.

### 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 som…

### Formalisation of Tables in a Dependent Language

I've had an idea kicking about in my head for a while of making query plans explicit in SQL in such a way that one can be assured that the query plan corresponds to the SQL statement desired. The idea is something like a Curry-Howard in a relational setting. One could infer the plan from the SQL, the SQL from the plan, or do a sort of "type-checking" to make sure that the plan corresponds to the SQL.

The devil is always in the details however. When I started looking at the primitives that I would need, it turns out that the low level table joining operations are actually not that far from primitive SQL statement themselves. I decided to go ahead and formalise some of what would be necessary in Agda in order get a better feel for the types of objects I would need and the laws which would be required to demonstrate that a plan corresponded with a statement.

Dependent types are very powerful and give you plenty of rope to hang yourself. It's always something of…

### Plotkin, the LGG and the MGU

Legend has it that a million years ago Plotkin was talking to his professor Popplestone, who said that unification (finding the most general unifier or the MGU) might have an interesting dual, and that Plotkin should find it. It turns out that the dual *is* interesting and it is known as the Least General Generalisation (LGG). Plotkin apparently described both the LGG for terms, and for clauses. I say apparently because I can't find his paper on-line.

The LGG for clauses is more complicated so we'll get back to it after we look at the LGG of terms. We can see how the MGU is related to the LGG by looking at a couple of examples and the above image. We use the prolog convention that function symbols start with lower case, and variables start with uppercase. The image above is organised as a DAG (Directed Acyclic Graph). DAGs are a very important structure in mathematics since DAGs are lattices.

Essentially what we have done is drawn an (incomplete) Hasse diagram f…