For a long time, I didn't understandd what GADTs are and how they can be used. It was a sort of conspiracy of silence - people who understand GADTs think it is all obvious, and don't need any explanation, but I still couldn't understand.

Now I have an idea how it works, and think that it was really obvious :) and I want to share my understanding - may be my way to realize GADTs can help someone else. So:

## Type functions

A "data" declaration is a way to declare both type constructor and data constructors. For example,

```data Either a b = Left a | Right b
```

declares type constructor "Either" and two data constructors "Left" and "Right". Ordinary Haskell functions works with data constructors:

```isLeft (Left a) = True
isLeft (Right b) = False
```

but there is also an analogous way to work with type constructors!

```type X a = Either a a
```

declares a TYPE FUNCTION named "X". Its parameter "a" must be some type and it returns some type as its result. We can't use "X" on data values, but we can use it on type values. Type constructors declared with "data" statements and type functions declared with "type" statements used together to build arbitrarily complex types. In such "computations" type constructors serves as basic "values" and type functions as a way to process them.

Indeed, type functions in Haskell are very limited compared to ordinary functions - they don't support pattern matching, nor multiple statements, nor recursion.

## Hypothetical Haskell extension - Full-featured type functions

Let's build hypothetical Haskell extension, that mimics for type functions the well-known ways to define ordinary functions, including pattern matching:

```type F [a] = Set a
```

multiple statements (this is meaningful only in presence of pattern matching):

```type F Bool   = Char
F String = Int
```

and recursion (which again needs pattern matching and multiple statements):

```type F [a]       = F a
F (Map a b) = F b
F (Set a)   = F a
F a         = a
```

As you may already have guessed, this last definition calculates a simple base type of arbitrarily-nested collections, e.g:

```F [[[Set Int]]] =
F [[Set Int]] =
F [Set Int] =
F (Set Int) =
F Int =
Int
```

Let's don't forget about statement guards:

```type F a | IsSimple a == TrueType  = a
```

Here we define type function F only for simple datatypes by using in guard type function "IsSimple":

```type IsSimple Bool = TrueType
IsSimple Int = TrueType
....
IsSimple Double = TrueType
IsSimple a = FalseType

data TrueType = T
data FalseType = F
```

These definitions seem a bit odd, and while we are in imaginary land, let's consider a way to write this shorter:

```type F a | IsSimple a  = a

type IsSimple Bool
IsSimple Int
....
IsSimple Double
```

Here, we just defined list of simple types, the implied result of all written statements for "IsSimple" is True value, and False value for anything else. Essentially, "IsSimple" is no less than TYPE PREDICATE!

I really love it! :) How about constructing a predicate that traverses a complex type trying to decide whether it contains "Int" anywhere?

```type HasInt Int
HasInt [a] = HasInt a
HasInt (Set a) = HasInt a
HasInt (Map a b) | HasInt a
HasInt (Map a b) | HasInt b
```

or a type function that substitutes one type with another inside arbitrary-deep types:

```type Replace t a b | t==a    =  b
Replace [t] a b         =  [Replace t a b]
Replace (Set t) a b     =  Set (Replace t a b)
Replace (Map t1 t2) a b =  Map (Replace t1 a b) (Replace t2 a b)
Replace t a b           =  t
```

## One more hypothetical extension - multi-value type functions

Let's add more fun! We will introduce one more hypothetical Haskell extension - type functions that may have MULTIPLE VALUES. Say,

```type Collection a = [a]
Collection a = Set a
Collection a = Map b a
```

So, "Collection Int" has "[Int]", "Set Int" and "Map String Int" as its values, i.e. different collection types with elements of type "Int".

Pay attention to the last statement of the "Collection" definition, where we've used type variable "b" that was not mentioned on the left side nor defined in any other way. It's perfectly possible - anyway "Collection" function has multiple values, so using on the right side some free variable that can be replaced with any type is not a problem at all - the "Map Bool Int", "Map [Int] Int" and "Map Int Int" all are possible values of "Collection Int" along with "[Int]" and "Set Int".

On the first look, it seems that multiple-value functions are meaningless - they can't be used to define datatypes, because we need concrete types here. But on the second look :) we can find them useful to define type constraints and type families.

We can also represent multiple-value function as predicate:

```type Collection a [a]
Collection a (Set a)
Collection a (Map b a)
```

If you remember Prolog, you should guess that predicate, in contrast to function, is multi-purpose thing - it can be used to deduce any parameter from other ones. For example, in this hypothetical definition:

```head | Collection Int a  ::  a -> Int
```

we define 'head' function for any Collection containing Ints.

And in this, again, hypothetical definition:

```data Safe c  | Collection c a =  Safe c a
```

we deduced element type 'a' from collection type 'c' passed as the parameter to the type constructor.

## Back to real Haskell - type classes

Reading all those glorious examples you may be wondering - why Haskell don't yet supports full-featured type functions? Hold your breath... Haskell already contains them and at least GHC implements all the mentioned abilities more than 10 years ago! They just was named... TYPE CLASSES! Let's translate all our examples to their language:

```class IsSimple a
instance IsSimple Bool
instance IsSimple Int
....
instance IsSimple Double
```

Haskell'98 standard supports type classes with only one parameter that limits us to defining only type predicates like this one. But GHC and Hugs supports multi-parameter type classes that allows us to define arbitrarily-complex type functions

```class Collection a c
instance Collection a [a]
instance Collection a (Set a)
instance Collection a (Map b a)
```

All the "hypothetical" Haskell extensions we investigated earlier - actually implemented at the type class level!

Pattern matching:

```instance Collection a [a]
```

Multiple statements:

```instance Collection a [a]
instance Collection a (Set a)
```

Recursion:

```instance (Collection a c) => Collection a [c]
```

Pattern guards:

```instance (IsSimple a) => Collection a (UArray a)
```

Let's define type class which contains any collection which uses Int as its elements or indexes:

```class HasInt a
instance HasInt Int
instance (HasInt a) => HasInt [a]
instance (HasInt a) => HasInt (Map a b)
instance (HasInt b) => HasInt (Map a b)
```

Anther example is a class that replaces all occurrences of 'a' with 'b' in type 't' and return result as 'res':

```class Replace t a b res
instance Replace t a a t
instance Replace [t] a b [Replace t a b]
instance (Replace t a b res)
=> Replace (Set t) a b (Set res)
instance (Replace t1 a b res1, Replace t2 a b res2)
=> Replace (Map t1 t2) a b (Map res1 res2)
instance Replace t a b t
```

You can compare it to the hypothetical definition we given earlier. It's important to note that type class instances, as opposite to function statements, are not checked in order. Instead, most _specific_ instance automatically selected. So, in Replace case, the last instance that is most general will be selected only if all other are failed to match and that is that we want.

In many other cases this automatic selection is not powerful enough and we are forced to use some artificial tricks or complain to the language developers. The two most well-known language extensions proposed to solve such problems are instance priorities, which allow to explicitly specify instance selection order, and '/=' constraints, which can be used to explicitly prohibit unwanted matches:

```instance Replace t a a t
instance (a/=b)   => Replace [t] a b [Replace t a b]
instance (a/=b, t/=[_]) => Replace t a b t
```

You can check that these instances are no more overlaps.

At practice, type-level arithmetics by itself is not very useful. It becomes really strong weapon when combined with another feature that type classes provide - member functions. For example:

```class Collection a c where
foldr1 :: (a -> a -> a) -> c -> a

class Num a where
(+) :: a -> a -> a

sum :: (Num a, Collection a c) =>  c -> a
sum = foldr1 (+)
```

I'll be also glad to see possibility to use type classes in data declarations like this:

```data Safe c = (Collection c a) => Safe c a
```

but afaik this is also not yet implemented

UNIFICATION ...

If you are wonder how relates all these interesting type manipulations to GADTs, now is the time to give you answer. As you know, Haskell contains highly developed ways to express data-to-data functions. Now we also know that Haskell contains rich facilities to write type-to-type functions in form of "type" statements and type classes. But how "data" statements fits in this infrastructure?

My answer: they just defines type-to-data constructors translation. Moreover, this translation may give multiple results. Say, the following definition:

```data Maybe a = Just a | Nothing
```

defines type-to-data constructors function "Maybe" that has parameter "a" and for each "a" has two possible results - "Just a" and "Nothing". We can rewrite it in the same hypothetical syntax that was used above for multi-value type functions:

```data Maybe a = Just a
Maybe a = Nothing
```

```data List a = Cons a (List a)
List a = Nil
```

and this:

```data Either a b = Left a
Either a b = Right b
```

But how are flexible "data" definitions? As you should remember, "type" definitions was very limited in their features, while type classes, vice versa, much more developed than ordinary Haskell functions facilities. What about features of "data" definitions examined as sort of functions?

On the one side, they supports multiple statements and multiple results and can be recursive, like the "List" definition above. On the other side, that's all - no pattern matching or even type constants on the left side and no guards.

Lack of pattern matching means that left side can contain only free type variables, that in turn means that left sides of all "data" statements for one type will be essentially the same. Therefore, repeated left sides in multi-statement "data" definitions are omitted and instead of

```data Either a b = Left a
Either a b = Right b
```

we write just

```data Either a b = Left a
| Right b
```

And here finally comes the GADTs! It's just a way to define data types using pattern matching and constants on the left side of "data" statements! How about this:

```data T String = D1 Int
T Bool   = D2
T [a]    = D3 (a,a)
```

Amazed? After all, GADTs seems really very simple and obvious extension to data type definition facilities.

The idea is to allow a data constructor's return type to be specified directly:

```data Term a where
Lit :: Int ->  Term Int
Pair :: Term a -> Term b -> Term (a,b)
...
```

In a function that performs pattern matching on Term, the pattern match gives type as well as value information. For example, consider this function:

```eval :: Term a -> a
eval (Lit i)     = i
eval (Pair a b)  = (eval a, eval b)
...
```

If the argument matches Lit, it must have been built with a Lit constructor, so type 'a' must be Int, and hence we can return 'i' (an Int) in the right hand side. The same objections applies to the Pair constructor.

The best paper on type level arithmetic using type classes i've seen is "Faking it: simulating dependent types in Haskell" ( http://www.cs.nott.ac.uk/~ctm/faking.ps.gz ). Most part of my article is just duplicates his work.

The great demonstration of type-level arithmetic is TypeNats package which "defines type-level natural numbers and arithmetic operations on them including addition, subtraction, multiplication, division and GCD" ( darcs get --partial --tag '0.1' http://www.eecs.tufts.edu/~rdocki01/typenats/ )

I should also mention here Oleg Kiselyov page on type-level programming in Haskell: http://okmij.org/ftp/Haskell/types.html

There are plenty of GADT-related papers, but best for beginners remains the "Fun with phantom types" (http://www.informatik.uni-bonn.de/~ralf/publications/With.pdf). Phantom types is another name of GADT. You should also know that this paper uses old GADT syntax. This paper is must-read because it contains numerous examples of practical GADT usage - theme completely omitted from my article.

"Dynamic Optimization for Functional Reactive Programming using Generalized Algebraic Data Types" http://www.cs.nott.ac.uk/~nhn/Publications/icfp2005.pdf

"Phantom types" (actually more scientific version of "Fun with phantom types") http://citeseer.ist.psu.edu/rd/0,606209,1,0.25,Download/http:qSqqSqwww.informatik.uni-bonn.deqSq~ralfqSqpublicationsqSqPhantom.ps.gz

"Phantom types and subtyping" http://arxiv.org/ps/cs.PL/0403034

"Existentially quantified type classes" by Stuckey, Sulzmann and Wazny (URL?)

## Random rubbish from previous versions of article

```data family Map k :: * -> *

data instance Map ()     v = MapUnit (Maybe v)
data instance Map (a, b) v = MapPair (Map a (Map b v))
```

let's consider well-known 'data' declarations:

```data T a = D a a Int
```

it can be seen as function 'T' from type 'a' to some data constructor.

'T Bool', for example, gives result 'D Bool Bool Int', while

'T [Int]' gives result 'D [Int] [Int] Int'.

'data' declaration can also have several "results", say

```data Either a b = Left a | Right b
```

and "result" of 'Either Int String' can be either "Left Int" or "Right String"

Well, to give compiler confidence that 'a' can be deduced in just one way from 'c', we can add some form of hint:

```type Collection :: a c | c->a
Collection a [a]
Collection a (Set a)
Collection a (Map b a)
```

The first line i added tell the compiler that Collection predicate has two parameters and the second parameter determines the first. Based on this restriction, compiler can detect and prohibit attempts to define different element types for the same collection:

```type Collection :: a c | c->a
Collection a (Map b a)
Collection b (Map b a)  -- error! prohibits functional dependency
```

Of course, Collection is just a function from 'c' to 'a', but if we will define it directly as a function:

```type Collection [a]        = a
Collection (Set a)    = a
Collection (Map b a)  = a
```

- it can't be used in 'head' definition above. Moreover, using functional dependencies we can define bi-directional functions:

```type TwoTimesBigger :: a b | a->b, b->a
TwoTimesBigger Int8 Int16
TwoTimesBigger Int16 Int32
TwoTimesBigger Int32 Int64
```

or predicates with 3, 4 or more parameters with any relations between them. It's a great power!