# GADTs for dummies

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− | + | [[Category:Tutorials]] | |

− | + | [[Category:Language extensions]] | |

− | + | ||

− | Now I have an idea how it works, | + | For a long time, I didn't understand what GADTs were or how they could be used. It was sort of a conspiracy of silence — people who understood GADTs thought |

− | someone else. | + | that everything was obvious and didn't need further explanation, but I still |

+ | couldn't understand them. | ||

+ | |||

+ | Now that I have an idea of how it works, I think that it was really obvious. :) So, I want to share my understanding of GADTs. Maybe the way I realized how GADTs work could help | ||

+ | someone else. See also [[Generalised algebraic datatype]] | ||

== Type functions == | == Type functions == | ||

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but we can use it on type values. Type constructors declared with | but we can use it on type values. Type constructors declared with | ||

"data" statements and type functions declared with "type" statements | "data" statements and type functions declared with "type" statements | ||

− | used together to build arbitrarily complex types. In such | + | can be used together to build arbitrarily complex types. In such |

"computations" type constructors serves as basic "values" and type | "computations" type constructors serves as basic "values" and type | ||

functions as a way to process them. | functions as a way to process them. | ||

Line 43: | Line 46: | ||

== Hypothetical Haskell extension - Full-featured type functions == | == Hypothetical Haskell extension - Full-featured type functions == | ||

− | Let's build hypothetical Haskell extension | + | Let's build a hypothetical Haskell extension that mimics, for type |

− | functions the well-known ways to define ordinary functions, including | + | functions, the well-known ways to define ordinary functions, including |

pattern matching: | pattern matching: | ||

Line 51: | Line 54: | ||

</haskell> | </haskell> | ||

− | multiple statements (this is meaningful only in presence of pattern matching): | + | multiple statements (this is meaningful only in the presence of pattern matching): |

<haskell> | <haskell> | ||

Line 67: | Line 70: | ||

</haskell> | </haskell> | ||

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

<haskell> | <haskell> | ||

Line 78: | Line 81: | ||

</haskell> | </haskell> | ||

− | Let's | + | Let's not forget about statement guards: |

<haskell> | <haskell> | ||

Line 84: | Line 87: | ||

</haskell> | </haskell> | ||

− | Here we define type function F only for simple datatypes by using | + | Here we define type function F only for simple datatypes by using a |

guard type function "IsSimple": | guard type function "IsSimple": | ||

Line 110: | Line 113: | ||

</haskell> | </haskell> | ||

− | Here, we | + | Here, we defined a list of simple types. The implied result of all |

− | written statements for "IsSimple" is True | + | written statements for "IsSimple" is True, and False for |

− | + | everything else. Essentially, "IsSimple" is a TYPE PREDICATE! | |

I really love it! :) How about constructing a predicate that traverses a | I really love it! :) How about constructing a predicate that traverses a | ||

Line 126: | Line 129: | ||

or a type function that substitutes one type with another inside | or a type function that substitutes one type with another inside | ||

− | + | arbitrarily-deep types: | |

<haskell> | <haskell> | ||

Line 152: | Line 155: | ||

Pay attention to the last statement of the "Collection" definition, where | Pay attention to the last statement of the "Collection" definition, where | ||

− | we | + | we used the type variable "b" that was not mentioned on the left side, |

− | nor defined in any other way. | + | nor defined in any other way. Since it's perfectly possible for the |

− | "Collection" function | + | "Collection" function to have multiple values, using some free variable on |

− | + | the right side that can be replaced with any type is not a problem | |

− | at all | + | at all. "Map Bool Int", "Map [Int] Int" and "Map Int Int" all are |

possible values of "Collection Int" along with "[Int]" and "Set Int". | possible values of "Collection Int" along with "[Int]" and "Set Int". | ||

− | + | At first glance, it seems that multiple-value functions are meaningless - they | |

− | meaningless - they can't be used to define datatypes, because we need | + | can't be used to define datatypes, because we need concrete types here. But |

− | concrete types here. But | + | if we take another look, they can be useful to define type constraints and |

− | useful to define type constraints and type families. | + | type families. |

− | We can also represent multiple-value function as predicate: | + | We can also represent a multiple-value function as a predicate: |

<haskell> | <haskell> | ||

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</haskell> | </haskell> | ||

− | If you | + | If you're familiar with Prolog, you should know that a predicate, in contrast to |

− | function, is multi- | + | a function, is a multi-directional thing - it can be used to deduce any |

− | parameter from other ones. For example, in this hypothetical definition: | + | parameter from the other ones. For example, in this hypothetical definition: |

<haskell> | <haskell> | ||

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</haskell> | </haskell> | ||

− | we define 'head' function for any Collection containing Ints. | + | we define a 'head' function for any Collection containing Ints. |

And in this, again, hypothetical definition: | And in this, again, hypothetical definition: | ||

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== Back to real Haskell - type classes == | == Back to real Haskell - type classes == | ||

− | + | After reading about all of these glorious examples, you may be wondering | |

− | + | "Why doesn't Haskell support full-featured type functions?" Hold your breath... | |

− | Haskell already contains them and | + | Haskell already contains them, and GHC has implemented all of the |

− | mentioned | + | capabilities mentioned above for more than 10 years! They were just named... |

− | TYPE CLASSES! Let's translate all our examples to their language: | + | TYPE CLASSES! Let's translate all of our examples to their language: |

<haskell> | <haskell> | ||

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</haskell> | </haskell> | ||

− | Haskell'98 standard supports type classes with only one parameter | + | The Haskell'98 standard supports type classes with only one parameter. |

− | limits us to defining | + | That limits us to only defining type predicates like this one. But GHC and |

− | Hugs | + | Hugs support multi-parameter type classes that allow us to define |

arbitrarily-complex type functions | arbitrarily-complex type functions | ||

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</haskell> | </haskell> | ||

− | All the "hypothetical" Haskell extensions we investigated earlier | + | All of the "hypothetical" Haskell extensions we investigated earlier are |

actually implemented at the type class level! | actually implemented at the type class level! | ||

Line 251: | Line 254: | ||

− | Let's define type class which contains any collection which uses Int | + | Let's define a type class which contains any collection which uses Int in |

its elements or indexes: | its elements or indexes: | ||

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− | + | Another example is a class that replaces all occurrences of 'a' with | |

− | 'b' in type 't' and return result as 'res': | + | 'b' in type 't' and return the result as 'res': |

<haskell> | <haskell> | ||

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</haskell> | </haskell> | ||

− | You can compare it to the hypothetical definition we | + | You can compare it to the hypothetical definition we gave earlier. |

− | It's important to note that type class instances, as | + | It's important to note that type class instances, as opposed to |

− | function statements, are not checked in order. Instead, most | + | function statements, are not checked in order. Instead, the most |

− | _specific_ instance automatically selected. So, in Replace case, the | + | _specific_ instance is automatically selected. So, in the Replace case, the |

− | last instance | + | last instance, which is the most general instance, will be selected only if all the others |

− | + | fail to match, which is what we want. | |

In many other cases this automatic selection is not powerful enough | In many other cases this automatic selection is not powerful enough | ||

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language developers. The two most well-known language extensions | language developers. The two most well-known language extensions | ||

proposed to solve such problems are instance priorities, which allow | proposed to solve such problems are instance priorities, which allow | ||

− | to explicitly specify instance selection order, and '/=' constraints, | + | us to explicitly specify instance selection order, and '/=' constraints, |

which can be used to explicitly prohibit unwanted matches: | which can be used to explicitly prohibit unwanted matches: | ||

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</haskell> | </haskell> | ||

− | You can check that these instances | + | You can check that these instances no longer overlap. |

− | + | In practice, type-level arithmetic by itself is not very useful. It becomes a | |

− | becomes | + | fantastic tool when combined with another feature that type classes provide - |

− | type classes provide - member functions. For example: | + | member functions. For example: |

<haskell> | <haskell> | ||

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− | I'll be | + | I'll also be glad to see the possibility of using type classes in data |

− | declarations like this: | + | declarations, like this: |

<haskell> | <haskell> | ||

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</haskell> | </haskell> | ||

− | but | + | but as far as I know, this is not yet implemented. |

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== Back to GADTs == | == Back to GADTs == | ||

− | If you are | + | If you are wondering how all of these interesting type manipulations relate to |

− | to GADTs, | + | GADTs, here is the answer. As you know, Haskell contains highly |

− | contains highly developed ways to express data-to-data functions. | + | developed ways to express data-to-data functions. We also know that |

− | + | Haskell contains rich facilities to write type-to-type functions in the form of | |

− | type-to-type functions in form of "type" statements and type classes. | + | "type" statements and type classes. But how do "data" statements fit into this |

− | But how "data" statements | + | infrastructure? |

− | My answer: they just | + | My answer: they just define a type-to-data constructor translation. Moreover, |

− | Moreover, this translation may give multiple results. Say, the | + | this translation may give multiple results. Say, the following definition: |

− | following definition: | + | |

<haskell> | <haskell> | ||

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</haskell> | </haskell> | ||

− | defines type-to-data constructors function "Maybe" that has parameter | + | defines type-to-data constructors function "Maybe" that has a parameter |

"a" and for each "a" has two possible results - "Just a" and | "a" and for each "a" has two possible results - "Just a" and | ||

"Nothing". We can rewrite it in the same hypothetical syntax that was | "Nothing". We can rewrite it in the same hypothetical syntax that was | ||

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</haskell> | </haskell> | ||

− | But how | + | But how flexible are "data" definitions? As you should remember, "type" |

− | "type" definitions | + | definitions were very limited in their features, while type classes, |

− | classes, | + | on the other hand, were more developed than ordinary Haskell functions |

− | + | facilities. What about features of "data" definitions examined as sort of functions? | |

− | examined as sort of functions? | + | |

− | On the one | + | On the one hand, they supports multiple statements and multiple results and |

− | results and can be recursive, like the "List" definition above. On the | + | can be recursive, like the "List" definition above. On the other, that's all - |

− | other | + | no pattern matching or even type constants on the left side and no guards. |

− | the left side and no guards. | + | |

− | Lack of pattern matching means that left side can contain only free | + | Lack of pattern matching means that the left side can contain only free type |

− | type variables | + | variables. That in turn means that the left sides of all "data" statements for a |

− | statements for | + | type will be essentially the same. Therefore, repeated left sides in |

− | repeated left sides in multi-statement "data" definitions are omitted | + | multi-statement "data" definitions are omitted and instead of |

− | and instead of | + | |

<haskell> | <haskell> | ||

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− | And here finally | + | And here we finally come to GADTs! It's just a way to define data types using |

− | using pattern matching and constants on the left side of "data" | + | pattern matching and constants on the left side of "data" statements! |

− | statements! | + | Let's say we want to do this: |

<haskell> | <haskell> | ||

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</haskell> | </haskell> | ||

− | Amazed? After all, GADTs | + | |

− | extension to data type definition facilities. | + | We cannot do this using a standard data definition. So, now we must use a GADT definition: |

+ | |||

+ | <haskell> | ||

+ | data T a where | ||

+ | D1 :: Int -> T String | ||

+ | D2 :: T Bool | ||

+ | D3 :: (a,a) -> T [a] | ||

+ | </haskell> | ||

+ | |||

+ | Amazed? After all, GADTs seem to be a really simple and obvious extension to | ||

+ | data type definition facilities. | ||

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− | The idea is to allow a data constructor's return type to | + | The idea here is to allow a data constructor's return type to be specified |

− | be specified directly: | + | directly: |

<haskell> | <haskell> | ||

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</haskell> | </haskell> | ||

− | In a function that performs pattern matching on Term, the pattern | + | In a function that performs pattern matching on Term, the pattern match gives |

− | match gives type as well as value information. For example, | + | type as well as value information. For example, consider this function: |

− | consider this function: | + | |

<haskell> | <haskell> | ||

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</haskell> | </haskell> | ||

− | If the argument matches Lit, it must have been built with a | + | If the argument matches Lit, it must have been built with a Lit constructor, |

− | Lit constructor, so type 'a' must be Int, and hence we can return 'i' | + | so type 'a' must be Int, and hence we can return 'i' (an Int) in the right |

− | (an Int) in the right hand side. The same | + | hand side. The same thing applies to the Pair constructor. |

− | constructor. | + | |

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== Further reading == | == Further reading == | ||

− | The best paper on type level arithmetic using type classes | + | The best paper on type level arithmetic using type classes I've seen |

is "Faking it: simulating dependent types in Haskell" | is "Faking it: simulating dependent types in Haskell" | ||

− | ( http://www.cs.nott.ac.uk/~ctm/faking.ps.gz ). Most | + | ( http://www.cs.nott.ac.uk/~ctm/faking.ps.gz ). Most of |

− | article | + | this article comes from his work. |

− | + | A great demonstration of type-level arithmetic is in the TypeNats package, | |

which "defines type-level natural numbers and arithmetic operations on | which "defines type-level natural numbers and arithmetic operations on | ||

them including addition, subtraction, multiplication, division and GCD" | them including addition, subtraction, multiplication, division and GCD" | ||

( darcs get --partial --tag '0.1' http://www.eecs.tufts.edu/~rdocki01/typenats/ ) | ( darcs get --partial --tag '0.1' http://www.eecs.tufts.edu/~rdocki01/typenats/ ) | ||

− | I should also mention here Oleg Kiselyov page on type-level | + | I should also mention here Oleg Kiselyov's page on type-level |

programming in Haskell: http://okmij.org/ftp/Haskell/types.html | programming in Haskell: http://okmij.org/ftp/Haskell/types.html | ||

− | There are plenty of GADT-related papers, but best for beginners | + | There are plenty of GADT-related papers, but the best one for beginners |

− | + | is "Fun with phantom types" | |

(http://www.informatik.uni-bonn.de/~ralf/publications/With.pdf). | (http://www.informatik.uni-bonn.de/~ralf/publications/With.pdf). | ||

Phantom types is another name of GADT. You should also know that this | 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 | + | paper uses old GADT syntax. This paper is a must-read because it |

− | contains numerous examples of practical GADT usage - theme completely | + | contains numerous examples of practical GADT usage - a theme completely |

omitted from my article. | omitted from my article. | ||

− | Other GADT-related papers | + | Other GADT-related papers: |

"Dynamic Optimization for Functional Reactive Programming using | "Dynamic Optimization for Functional Reactive Programming using | ||

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

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## Revision as of 01:42, 20 July 2013

For a long time, I didn't understand what GADTs were or how they could be used. It was sort of a conspiracy of silence — people who understood GADTs thought
that everything was obvious and didn't need further explanation, but I still
couldn't understand them.

Now that I have an idea of how it works, I think that it was really obvious. :) So, I want to share my understanding of GADTs. Maybe the way I realized how GADTs work could help someone else. See also Generalised algebraic datatype

## Contents |

## 1 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 can be 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.

## 2 Hypothetical Haskell extension - Full-featured type functions

Let's build a 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 the 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 not forget about statement guards:

type F a | IsSimple a == TrueType = a

Here we define type function F only for simple datatypes by using a 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 defined a list of simple types. The implied result of all written statements for "IsSimple" is True, and False for everything else. Essentially, "IsSimple" is a 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 arbitrarily-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

## 3 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 used the type variable "b" that was not mentioned on the left side, nor defined in any other way. Since it's perfectly possible for the "Collection" function to have multiple values, using some free variable on the right side that can be replaced with any type is not a problem at all. "Map Bool Int", "Map [Int] Int" and "Map Int Int" all are possible values of "Collection Int" along with "[Int]" and "Set Int".

At first glance, it seems that multiple-value functions are meaningless - they can't be used to define datatypes, because we need concrete types here. But if we take another look, they can be useful to define type constraints and type families.

We can also represent a multiple-value function as a predicate:

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

If you're familiar with Prolog, you should know that a predicate, in contrast to a function, is a multi-directional thing - it can be used to deduce any parameter from the other ones. For example, in this hypothetical definition:

head | Collection Int a :: a -> Int

we define a '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.

## 4 Back to real Haskell - type classes

After reading about all of these glorious examples, you may be wondering "Why doesn't Haskell support full-featured type functions?" Hold your breath... Haskell already contains them, and GHC has implemented all of the capabilities mentioned above for more than 10 years! They were just named... TYPE CLASSES! Let's translate all of our examples to their language:

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

The Haskell'98 standard supports type classes with only one parameter. That limits us to only defining type predicates like this one. But GHC and Hugs support multi-parameter type classes that allow 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 of the "hypothetical" Haskell extensions we investigated earlier are 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 a type class which contains any collection which uses Int in 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)

Another example is a class that replaces all occurrences of 'a' with
'b' in type 't' and return the 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 gave earlier. It's important to note that type class instances, as opposed to function statements, are not checked in order. Instead, the most _specific_ instance is automatically selected. So, in the Replace case, the last instance, which is the most general instance, will be selected only if all the others fail to match, which is what 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 us 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 no longer overlap.

In practice, type-level arithmetic by itself is not very useful. It becomes a
fantastic tool 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 also be glad to see the possibility of using type classes in data
declarations, like this:

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

but as far as I know, this is not yet implemented.

UNIFICATION
...

## 5 Back to GADTs

If you are wondering how all of these interesting type manipulations relate to GADTs, here is the answer. As you know, Haskell contains highly developed ways to express data-to-data functions. We also know that Haskell contains rich facilities to write type-to-type functions in the form of "type" statements and type classes. But how do "data" statements fit into this infrastructure?

My answer: they just define a type-to-data constructor 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 a 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

Or how about this:

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 flexible are "data" definitions? As you should remember, "type" definitions were very limited in their features, while type classes, on the other hand, were more developed than ordinary Haskell functions facilities. What about features of "data" definitions examined as sort of functions?

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

Lack of pattern matching means that the left side can contain only free type variables. That in turn means that the left sides of all "data" statements for a 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 we finally come to GADTs! It's just a way to define data types using
pattern matching and constants on the left side of "data" statements!
Let's say we want to do this:

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

We cannot do this using a standard data definition. So, now we must use a GADT definition:

data T a where D1 :: Int -> T String D2 :: T Bool D3 :: (a,a) -> T [a]

Amazed? After all, GADTs seem to be a really simple and obvious extension to data type definition facilities.

The idea here 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 thing applies to the Pair constructor.

## 6 Further reading

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 of this article comes from his work.

A great demonstration of type-level arithmetic is in the 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's page on type-level programming in Haskell: http://okmij.org/ftp/Haskell/types.html

There are plenty of GADT-related papers, but the best one for beginners
is "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 a must-read because it
contains numerous examples of practical GADT usage - a theme completely
omitted from my article.

Other GADT-related papers:

"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?)