# What is Agda?

Agda is a dependently typed programming language. It is an extension of Martin-Löf’s type theory and is the latest in the tradition of languages developed in the programming logic group at Chalmers. Other languages in this tradition are Alf, Alfa, Agda 1, Cayenne. Some other loosely related languages are Coq, Epigram, Idris, and Lean.

Because of strong typing and dependent types, Agda can be used as a proof assistant, allowing one to prove mathematical theorems (in a constructive setting) and to run such proofs as algorithms.

## Dependent types

### Typing for programmers

Type theory is concerned
both with programming and logic. We see the type system as a way to
express syntactic correctness. A type correct program has a meaning.
Lisp
is a totally untyped programming language, and so are its derivatives
like
Scheme. In
such languages, if `f`

is a function, one can apply it to anything,
including itself. This makes it easy to write programs (almost all
programs are well formed), but it also makes it easy to write erroneous
programs. Programs will raise exceptions or loop forever. And it is
very difficult to analyze where the problems are.

Haskell or ML and its derivatives like Standard ML and Caml are typed languages, where functions come with a type expressing what type of arguments the program expects and what the result type is.

Between these two families of languages come languages, which may or may not have a typing discipline. Most imperative languages do not come with a rich type system. For example, C is typed, but very loosely (almost everything is an integer or a variant thereof). Moreover, the typing system does not allow the definition of trees or graphs without using pointers.

All these languages are examples of **partial languages**, i.e., the
result of computing the value of an expression `e`

of type `T`

is
one of the following:

the program terminates with a value in the type

`T`

the program

`e`

does not terminatethe program raises an exception which has been caused by an incomplete definition – for instance, a function is only defined for positive integers but is applied to a negative integer.

Agda and other languages based on type theory are **total languages**
in the sense that a program `e`

of type `T`

will always terminate
with a value in `T`

. No runtime error can occur, and no
nonterminating programs can be written (unless explicitly requested by
the programmer).

### Dependent types

Dependent types are
introduced by having families of types indexed by objects in another type.
For instance, we can define the type `Vec n`

of vectors of length `n`

.
This is a family of types indexed by objects in `Nat`

(a type
parameterized by natural numbers).

Having dependent types, we must generalize the type of functions and the type of pairs.

The **dependent function space** `(a : A) -> (B a)`

is the type of the
functions taking an argument `a`

in a type `A`

and returning a result in ```
B
a
```

. Here, `A`

is a type, and `B`

is a family of types indexed by
elements in `A`

.

For example, we could define the type of `n x m`

matrices as a type
indexed by two natural numbers. Call this type `Mat n m`

. The
function `identity`

, which takes a natural number `n`

as an argument
and produces the `n x n`

identity matrix, is then a function of type
`identity : (n : Nat) -> (Mat n n)`

.

**Remark**: We could, of course, just specify the `identity`

function
with the type `Nat -> Mat`

, where `Mat`

is the type of
matrices, but this is not as precise as the dependent version.

The advantage of using dependent types is that it makes it possible to
express properties of programs in the typing system. We saw above that
it is possible to express the type of square matrices of length `n`

.
It is also possible to define the type of operations on matrices so
that the lengths are correct. For instance, the type of matrix
multiplication is

```
∀ {i j k} → (Mat i j) -> (Mat j k) -> (Mat i k)
```

and the type system can check that a program for matrix multiplication really produces matrices of the correct size. It can also check that matrix multiplication is only applied to matrices, where the number of columns of the first argument is the same as the number of rows in the second argument.

### Dependent types and logic

Thanks to the Curry-Howard correspondence, one can express a logical specification using dependent types. For example, using only typing it is possible to define:

equality on natural numbers

properties of arithmetical operations

the type

`(n : Nat) -> (PrimRoot n)`

consisting of functions computing primitive roots in modular arithmetic.

Of course, a program of the above type will be more difficult to write
than the corresponding program of type `Nat -> Nat`

, which produces a
natural number which is a primitive root. However, the difficulty can
be compensated by the fact that the program is guaranteed to work: it
cannot produce something which is not a primitive root.

On a more mathematical level, we can express formulas and prove them
using an algorithm. For example, a function of type ```
(n : Nat) ->
(PrimRoot n)
```

is also a proof that every natural number has a
primitive root.