# Types and Functions

## Primitive Types

Idris defines several primitive types: `Int`

, `Integer`

and
`Double`

for numeric operations, `Char`

and `String`

for text
manipulation, and `Ptr`

which represents foreign pointers. There are
also several data types declared in the library, including `Bool`

,
with values `True`

and `False`

. We can declare some constants with
these types. Enter the following into a file `Prims.idr`

and load it
into the Idris interactive environment by typing `idris2 Prims.idr`

:

```
module Prims
x : Int
x = 94
foo : String
foo = "Sausage machine"
bar : Char
bar = 'Z'
quux : Bool
quux = False
```

An Idris file consists of an optional module declaration (here
`module Prims`

) followed by an optional list of imports and a
collection of declarations and definitions. In this example no imports
have been specified. However Idris programs can consist of several
modules and the definitions in each module each have their own
namespace. This is discussed further in Section
Modules and Namespaces. When writing Idris programs both the order in which
definitions are given and indentation are significant. Functions and
data types must be defined before use, incidentally each definition must
have a type declaration, for example see `x : Int`

, ```
foo :
String
```

, from the above listing. New declarations must begin at the
same level of indentation as the preceding declaration.
Alternatively, a semicolon `;`

can be used to terminate declarations.

A library module `prelude`

is automatically imported by every
Idris program, including facilities for IO, arithmetic, data
structures and various common functions. The prelude defines several
arithmetic and comparison operators, which we can use at the prompt.
Evaluating things at the prompt gives an answer, for example:

```
Prims> 13+9*9
94 : Integer
Prims> x == 9*9+13
True
```

All of the usual arithmetic and comparison operators are defined for
the primitive types. They are overloaded using interfaces, as we
will discuss in Section Interfaces and can be extended to
work on user defined types. Boolean expressions can be tested with the
`if...then...else`

construct, for example:

```
*prims> if x == 8 * 8 + 30 then "Yes!" else "No!"
"Yes!"
```

## Data Types

Data types are declared in a similar way and with similar syntax to Haskell. Natural numbers and lists, for example, can be declared as follows:

```
data Nat = Z | S Nat -- Natural numbers
-- (zero and successor)
data List a = Nil | (::) a (List a) -- Polymorphic lists
```

Data type names cannot begin with a lower case letter (we will see later
why not!). The above declarations are taken from the standard library. Unary
natural numbers can be either zero (`Z`

), or the successor of another natural
number (`S k`

). Lists can either be empty (`Nil`

) or a value added to the
front of another list (`x :: xs`

). In the declaration for `List`

, we used
an infix operator `::`

. New operators such as this can be added using a
fixity declaration, as follows:

```
infixr 10 ::
```

Functions, data constructors and type constructors may all be given
infix operators as names. They may be used in prefix form if enclosed
in brackets, e.g. `(::)`

. Infix operators can use any of the
symbols:

```
:+-*\/=.?|&><!@$%^~#
```

Some operators built from these symbols can’t be user defined. These are

`%`

, `\`

, `:`

, `=`

, `|`

, `|||`

, `<-`

, `->`

, `=>`

, `?`

,
`!`

, `&`

, `**`

, `..`

## Functions

Functions are implemented by pattern matching, again using a similar
syntax to Haskell. The main difference is that Idris requires type
declarations for all functions, using a single colon `:`

(rather
than Haskell’s double colon `::`

). Some natural number arithmetic
functions can be defined as follows, again taken from the standard
library:

```
-- Unary addition
plus : Nat -> Nat -> Nat
plus Z y = y
plus (S k) y = S (plus k y)
-- Unary multiplication
mult : Nat -> Nat -> Nat
mult Z y = Z
mult (S k) y = plus y (mult k y)
```

The standard arithmetic operators `+`

and `*`

are also overloaded
for use by `Nat`

, and are implemented using the above functions.
Unlike Haskell, there is no restriction on whether function
names must begin with a capital letter or not. Function names
(`plus`

and `mult`

above), data constructors (`Z`

, `S`

,
`Nil`

and `::`

) and type constructors (`Nat`

and `List`

) are
all part of the same namespace. By convention, however,
data types and constructor names typically begin with a capital letter.
We can test these functions at the Idris prompt:

```
Main> plus (S (S Z)) (S (S Z))
4
Main> mult (S (S (S Z))) (plus (S (S Z)) (S (S Z)))
12
```

Like arithmetic operations, integer literals are also overloaded using interfaces, meaning that we can also test the functions as follows:

```
Idris> plus 2 2
4
Idris> mult 3 (plus 2 2)
12
```

You may wonder, by the way, why we have unary natural numbers when our
computers have perfectly good integer arithmetic built in. The reason
is primarily that unary numbers have a very convenient structure which
is easy to reason about, and easy to relate to other data structures
as we will see later. Nevertheless, we do not want this convenience to
be at the expense of efficiency. Fortunately, Idris knows about
the relationship between `Nat`

(and similarly structured types) and
numbers. This means it can optimise the representation, and functions
such as `plus`

and `mult`

.

`where`

clauses

Functions can also be defined *locally* using `where`

clauses. For
example, to define a function which reverses a list, we can use an
auxiliary function which accumulates the new, reversed list, and which
does not need to be visible globally:

```
reverse : List a -> List a
reverse xs = revAcc [] xs where
revAcc : List a -> List a -> List a
revAcc acc [] = acc
revAcc acc (x :: xs) = revAcc (x :: acc) xs
```

Indentation is significant — functions in the `where`

block must be
indented further than the outer function.

Note

Scope

Any names which are visible in the outer scope are also visible in
the `where`

clause (unless they have been redefined, such as `xs`

here). A name which appears in the type will be in scope in the
`where`

clause.

As well as functions, `where`

blocks can include local data
declarations, such as the following where `MyLT`

is not accessible
outside the definition of `foo`

:

```
foo : Int -> Int
foo x = case isLT of
Yes => x*2
No => x*4
where
data MyLT = Yes | No
isLT : MyLT
isLT = if x < 20 then Yes else No
```

Functions defined in a `where`

clause need a type
declaration just like any top level function. Here is another example
of how this works in practice:

```
even : Nat -> Bool
even Z = True
even (S k) = odd k where
odd : Nat -> Bool
odd Z = False
odd (S k) = even k
test : List Nat
test = [c (S 1), c Z, d (S Z)]
where c : Nat -> Nat
c x = 42 + x
d : Nat -> Nat
d y = c (y + 1 + z y)
where z : Nat -> Nat
z w = y + w
```

### Totality and Covering

By default, functions in Idris must be `covering`

. That is, there must be
patterns which cover all possible values of the inputs types. For example,
the following definition will give an error:

```
fromMaybe : Maybe a -> a
fromMaybe (Just x) = x
```

This gives an error because `fromMaybe Nothing`

is not defined. Idris
reports:

```
frommaybe.idr:1:1--2:1:fromMaybe is not covering. Missing cases:
fromMaybe Nothing
```

You can override this with a `partial`

annotation:

```
partial fromMaybe : Maybe a -> a
fromMaybe (Just x) = x
```

However, this is not advisable, and in general you should only do this during
the initial development of a function, or during debugging. If you try to
evaluate `fromMaybe Nothing`

at run time you will get a run time error.

### Holes

Idris programs can contain *holes* which stand for incomplete parts of
programs. For example, we could leave a hole for the greeting in our
“Hello world” program:

```
main : IO ()
main = putStrLn ?greeting
```

The syntax `?greeting`

introduces a hole, which stands for a part of
a program which is not yet written. This is a valid Idris program, and you
can check the type of `greeting`

:

```
Main> :t greeting
-------------------------------------
greeting : String
```

Checking the type of a hole also shows the types of any variables in scope.
For example, given an incomplete definition of `even`

:

```
even : Nat -> Bool
even Z = True
even (S k) = ?even_rhs
```

We can check the type of `even_rhs`

and see the expected return type,
and the type of the variable `k`

:

```
Main> :t even_rhs
k : Nat
-------------------------------------
even_rhs : Bool
```

Holes are useful because they help us write functions *incrementally*.
Rather than writing an entire function in one go, we can leave some parts
unwritten and use Idris to tell us what is necessary to complete the
definition.

## Dependent Types

### First Class Types

In Idris, types are first class, meaning that they can be computed and manipulated (and passed to functions) just like any other language construct. For example, we could write a function which computes a type:

```
isSingleton : Bool -> Type
isSingleton True = Nat
isSingleton False = List Nat
```

This function calculates the appropriate type from a `Bool`

which flags
whether the type should be a singleton or not. We can use this function
to calculate a type anywhere that a type can be used. For example, it
can be used to calculate a return type:

```
mkSingle : (x : Bool) -> isSingleton x
mkSingle True = 0
mkSingle False = []
```

Or it can be used to have varying input types. The following function
calculates either the sum of a list of `Nat`

, or returns the given
`Nat`

, depending on whether the singleton flag is true:

```
sum : (single : Bool) -> isSingleton single -> Nat
sum True x = x
sum False [] = 0
sum False (x :: xs) = x + sum False xs
```

### Vectors

A standard example of a dependent data type is the type of “lists with
length”, conventionally called vectors in the dependent type
literature. They are available as part of the Idris library, by
importing `Data.Vect`

, or we can declare them as follows:

```
data Vect : Nat -> Type -> Type where
Nil : Vect Z a
(::) : a -> Vect k a -> Vect (S k) a
```

Note that we have used the same constructor names as for `List`

.
Ad-hoc name overloading such as this is accepted by Idris,
provided that the names are declared in different namespaces (in
practice, normally in different modules). Ambiguous constructor names
can normally be resolved from context.

This declares a family of types, and so the form of the declaration is
rather different from the simple type declarations above. We
explicitly state the type of the type constructor `Vect`

— it takes
a `Nat`

and a type as an argument, where `Type`

stands for the
type of types. We say that `Vect`

is *indexed* over `Nat`

and
*parameterised* by `Type`

. Each constructor targets a different part
of the family of types. `Nil`

can only be used to construct vectors
with zero length, and `::`

to construct vectors with non-zero
length. In the type of `::`

, we state explicitly that an element of
type `a`

and a tail of type `Vect k a`

(i.e., a vector of length
`k`

) combine to make a vector of length `S k`

.

We can define functions on dependent types such as `Vect`

in the same
way as on simple types such as `List`

and `Nat`

above, by pattern
matching. The type of a function over `Vect`

will describe what
happens to the lengths of the vectors involved. For example, `++`

,
defined as follows, appends two `Vect`

:

```
(++) : Vect n a -> Vect m a -> Vect (n + m) a
(++) Nil ys = ys
(++) (x :: xs) ys = x :: xs ++ ys
```

The type of `(++)`

states that the resulting vector’s length will be
the sum of the input lengths. If we get the definition wrong in such a
way that this does not hold, Idris will not accept the definition.
For example:

```
(++) : Vect n a -> Vect m a -> Vect (n + m) a
(++) Nil ys = ys
(++) (x :: xs) ys = x :: xs ++ xs -- BROKEN
```

When run through the Idris type checker, this results in the following:

```
$ idris2 Vect.idr --check
1/1: Building Vect (Vect.idr)
Vect.idr:7:26--8:1:While processing right hand side of Main.++ at Vect.idr:7:1--8:1:
When unifying plus k k and plus k m
Mismatch between:
k
and
m
```

This error message suggests that there is a length mismatch between
two vectors — we needed a vector of length `k + m`

, but provided a
vector of length `k + k`

.

### The Finite Sets

Finite sets, as the name suggests, are sets with a finite number of
elements. They are available as part of the Idris library, by
importing `Data.Fin`

, or can be declared as follows:

```
data Fin : Nat -> Type where
FZ : Fin (S k)
FS : Fin k -> Fin (S k)
```

From the signature, we can see that this is a type constructor that takes a `Nat`

, and produces a type.
So this is not a set in the sense of a collection that is a container of objects,
rather it is the canonical set of unnamed elements, as in “the set of 5 elements,” for example.
Effectively, it is a type that captures integers that fall into the range of zero to `(n - 1)`

where
`n`

is the argument used to instantiate the `Fin`

type.
For example, `Fin 5`

can be thought of as the type of integers between 0 and 4.

Let us look at the constructors in greater detail.

`FZ`

is the zeroth element of a finite set with `S k`

elements;
`FS n`

is the `n+1`

th element of a finite set with `S k`

elements. `Fin`

is indexed by a `Nat`

, which represents the number
of elements in the set. Since we can’t construct an element of an
empty set, neither constructor targets `Fin Z`

.

As mentioned above, a useful application of the `Fin`

family is to
represent bounded natural numbers. Since the first `n`

natural
numbers form a finite set of `n`

elements, we can treat `Fin n`

as
the set of integers greater than or equal to zero and less than `n`

.

For example, the following function which looks up an element in a
`Vect`

, by a bounded index given as a `Fin n`

, is defined in the
prelude:

```
index : Fin n -> Vect n a -> a
index FZ (x :: xs) = x
index (FS k) (x :: xs) = index k xs
```

This function looks up a value at a given location in a vector. The
location is bounded by the length of the vector (`n`

in each case),
so there is no need for a run-time bounds check. The type checker
guarantees that the location is no larger than the length of the
vector, and of course no less than zero.

Note also that there is no case for `Nil`

here. This is because it
is impossible. Since there is no element of `Fin Z`

, and the
location is a `Fin n`

, then `n`

can not be `Z`

. As a result,
attempting to look up an element in an empty vector would give a
compile time type error, since it would force `n`

to be `Z`

.

### Implicit Arguments

Let us take a closer look at the type of `index`

:

```
index : Fin n -> Vect n a -> a
```

It takes two arguments, an element of the finite set of `n`

elements,
and a vector with `n`

elements of type `a`

. But there are also two
names, `n`

and `a`

, which are not declared explicitly. These are
*implicit* arguments to `index`

. We could also write the type of
`index`

as:

```
index : forall a, n . Fin n -> Vect n a -> a
```

Implicit arguments, given with the `forall`

declaration,
are not given in applications of `index`

; their values can be
inferred from the types of the `Fin n`

and `Vect n a`

arguments. Any name beginning with a lower case letter which appears
as a parameter or index in a
type declaration, which is not applied to any arguments, will
*always* be automatically
bound as an implicit argument; this is why data type names cannot begin with
a lower case letter. Implicit arguments can still be given
explicitly in applications, using `{a=value}`

and `{n=value}`

, for
example:

```
index {a=Int} {n=2} FZ (2 :: 3 :: Nil)
```

In fact, any argument, implicit or explicit, may be given a name. We
could have declared the type of `index`

as:

```
index : (i : Fin n) -> (xs : Vect n a) -> a
```

It is a matter of taste whether you want to do this — sometimes it can help document a function by making the purpose of an argument more clear.

The names of implicit arguments are in scope in the body of the function, although they cannot be used at run time. There is much more to say about implicit arguments - we will discuss the question of what is available at run time, among other things, in Section Multiplicities

#### Note: Declaration Order and `mutual`

blocks

In general, functions and data types must be defined before use, since
dependent types allow functions to appear as part of types, and type
checking can rely on how particular functions are defined (though this
is only true of total functions; see Section Totality Checking).
However, this restriction can be relaxed by using a `mutual`

block,
which allows data types and functions to be defined simultaneously:

```
mutual
even : Nat -> Bool
even Z = True
even (S k) = odd k
odd : Nat -> Bool
odd Z = False
odd (S k) = even k
```

In a `mutual`

block, first all of the type declarations are added,
then the function bodies. As a result, none of the function types can
depend on the reduction behaviour of any of the functions in the
block.

Forward declarations can allow you to have more fine-grained control over the order in which mutually defined concepts are declared. This can be useful if you need to mention a datatype’s constructor in the type of a mutually defined function, or need to rely on the behaviour of a mutually defined function for something to typecheck.

```
data V : Type
T : V -> Type
data V : Type where
N : V
Pi : (a : V) -> (b : T a -> V) -> V
T N = Nat
T (Pi a b) = (x : T a) -> T (b x)
```

```
data Even : Nat -> Type
data Odd : Nat -> Type
data Even : Nat -> Type where
ZIsEven : Even Z
SOddIsEven : Odd n -> Even (S k)
data Odd : Nat -> Type where
SEvenIsOdd : Even n -> Odd (S k)
```

```
even : Nat -> Bool
odd : Nat -> Bool
-- or just ``even, odd : Nat -> Bool``
even Z = True
even (S k) = odd k
odd Z = False
odd (S k) = even k
```

Placing signature declarations forward can suggest Idris to detect their corresponding mutual definitions.

## I/O

Computer programs are of little use if they do not interact with the
user or the system in some way. The difficulty in a pure language such
as Idris — that is, a language where expressions do not have
side-effects — is that I/O is inherently side-effecting. So, Idris provides
a parameterised type `IO`

which *describes* the interactions that the
run-time system will perform when executing a function:

```
data IO a -- description of an IO operation returning a value of type a
```

We’ll leave the definition of `IO`

abstract, but effectively it
describes what the I/O operations to be executed are, rather than how
to execute them. The resulting operations are executed externally, by
the run-time system. We’ve already seen one I/O program:

```
main : IO ()
main = putStrLn "Hello world"
```

The type of `putStrLn`

explains that it takes a string, and returns
an I/O action which produces an element of the unit type `()`

. There is a
variant `putStr`

which decribes the output of a string without a newline:

```
putStrLn : String -> IO ()
putStr : String -> IO ()
```

We can also read strings from user input:

```
getLine : IO String
```

A number of other I/O operations are available. For example, by adding
`import System.File`

to your program, you get access to functions for
reading and writing files, including:

```
data File -- abstract
data Mode = Read | Write | ReadWrite
openFile : (f : String) -> (m : Mode) -> IO (Either FileError File)
closeFile : File -> IO ()
fGetLine : (h : File) -> IO (Either FileError String)
fPutStr : (h : File) -> (str : String) -> IO (Either FileError ())
fEOF : File -> IO Bool
```

Note that several of these return `Either`

, since they may fail.

## “`do`

” notation

I/O programs will typically need to sequence actions, feeding the
output of one computation into the input of the next. `IO`

is an
abstract type, however, so we can’t access the result of a computation
directly. Instead, we sequence operations with `do`

notation:

```
greet : IO ()
greet = do putStr "What is your name? "
name <- getLine
putStrLn ("Hello " ++ name)
```

The syntax `x <- iovalue`

executes the I/O operation `iovalue`

, of
type `IO a`

, and puts the result, of type `a`

into the variable
`x`

. In this case, `getLine`

returns an `IO String`

, so `name`

has type `String`

. Indentation is significant — each statement in
the do block must begin in the same column. The `pure`

operation
allows us to inject a value directly into an IO operation:

```
pure : a -> IO a
```

As we will see later, `do`

notation is more general than this, and
can be overloaded.

You can try executing `greet`

at the Idris 2 REPL by running the command
`:exec greet`

:

## Laziness

Normally, arguments to functions are evaluated before the function
itself (that is, Idris uses *eager* evaluation). However, this is
not always the best approach. Consider the following function:

```
ifThenElse : Bool -> a -> a -> a
ifThenElse True t e = t
ifThenElse False t e = e
```

This function uses one of the `t`

or `e`

arguments, but not both.
We would prefer if *only* the argument which was used was evaluated. To achieve
this, Idris provides a `Lazy`

primitive, which allows evaluation to be
suspended. It is a primitive, but conceptually we can think of it as follows:

```
data Lazy : Type -> Type where
Delay : (val : a) -> Lazy a
Force : Lazy a -> a
```

A value of type `Lazy a`

is unevaluated until it is forced by
`Force`

. The Idris type checker knows about the `Lazy`

type,
and inserts conversions where necessary between `Lazy a`

and `a`

,
and vice versa. We can therefore write `ifThenElse`

as follows,
without any explicit use of `Force`

or `Delay`

:

```
ifThenElse : Bool -> Lazy a -> Lazy a -> a
ifThenElse True t e = t
ifThenElse False t e = e
```

## Infinite data Types

Infinite data types (codata) allow us to define infinite data structures by marking recursive arguments as potentially infinite. One example of an infinite type is Stream, which is defined as follows.

```
data Stream : Type -> Type where
(::) : (e : a) -> Inf (Stream a) -> Stream a
```

The following is an example of how the codata type `Stream`

can be used to
form an infinite data structure. In this case we are creating an infinite stream
of ones.

```
ones : Stream Nat
ones = 1 :: ones
```

## Useful Data Types

Idris includes a number of useful data types and library functions
(see the `libs/`

directory in the distribution, and the
documentation). This
section describes a few of these, and how to import them.

`List`

and `Vect`

We have already seen the `List`

and `Vect`

data types:

```
data List a = Nil | (::) a (List a)
data Vect : Nat -> Type -> Type where
Nil : Vect Z a
(::) : a -> Vect k a -> Vect (S k) a
```

You can get access to `Vect`

with `import Data.Vect`

.
Note that the constructor names are the same for each — constructor
names (in fact, names in general) can be overloaded, provided that
they are declared in different namespaces (see Section
Modules and Namespaces), and will typically be resolved according to
their type. As syntactic sugar, any implementation of the names
`Nil`

and `::`

can be written in list form. For example:

`[]`

means`Nil`

`[1,2,3]`

means`1 :: 2 :: 3 :: Nil`

Similarly, any implementation of the names `Lin`

and `:<`

can be
written in **snoc**-list form:

`[<]`

mean`Lin`

`[< 1, 2, 3]`

means`Lin :< 1 :< 2 :< 3`

.

and the prelude includes a pre-defined datatype for snoc-lists:

```
data SnocList a = Lin | (:<) (SnocList a) a
```

The library also defines a number of functions for manipulating these
types. `map`

is overloaded both for `List`

and `Vect`

(we’ll see more
details of precisely how later when we cover interfaces in
Section Interfaces) and applies a function to every element of the
list or vector.

```
map : (a -> b) -> List a -> List b
map f [] = []
map f (x :: xs) = f x :: map f xs
map : (a -> b) -> Vect n a -> Vect n b
map f [] = []
map f (x :: xs) = f x :: map f xs
```

For example, given the following vector of integers, and a function to double an integer:

```
intVec : Vect 5 Int
intVec = [1, 2, 3, 4, 5]
double : Int -> Int
double x = x * 2
```

the function `map`

can be used as follows to double every element in
the vector:

```
*UsefulTypes> show (map double intVec)
"[2, 4, 6, 8, 10]" : String
```

For more details of the functions available on `List`

and
`Vect`

, look in the library files:

`libs/base/Data/List.idr`

`libs/base/Data/Vect.idr`

Functions include filtering, appending, reversing, and so on.

#### Aside: Anonymous functions and operator sections

There are neater ways to write the above expression. One way would be to use an anonymous function:

```
*UsefulTypes> show (map (\x => x * 2) intVec)
"[2, 4, 6, 8, 10]" : String
```

The notation `\x => val`

constructs an anonymous function which takes
one argument, `x`

and returns the expression `val`

. Anonymous
functions may take several arguments, separated by commas,
e.g. `\x, y, z => val`

. Arguments may also be given explicit types,
e.g. `\x : Int => x * 2`

, and can pattern match,
e.g. `\(x, y) => x + y`

. We could also use an operator section:

```
*UsefulTypes> show (map (* 2) intVec)
"[2, 4, 6, 8, 10]" : String
```

`(*2)`

is shorthand for a function which multiplies a number
by 2. It expands to `\x => x * 2`

. Similarly, `(2*)`

would expand
to `\x => 2 * x`

.

### Maybe

`Maybe`

, defined in the Prelude, describes an optional value. Either there is
a value of the given type, or there isn’t:

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

`Maybe`

is one way of giving a type to an operation that may
fail. For example, looking something up in a `List`

(rather than a
vector) may result in an out of bounds error:

```
list_lookup : Nat -> List a -> Maybe a
list_lookup _ Nil = Nothing
list_lookup Z (x :: xs) = Just x
list_lookup (S k) (x :: xs) = list_lookup k xs
```

The `maybe`

function is used to process values of type `Maybe`

,
either by applying a function to the value, if there is one, or by
providing a default value:

```
maybe : Lazy b -> Lazy (a -> b) -> Maybe a -> b
```

Note that the types of the first two arguments are wrapped in
`Lazy`

. Since only one of the two arguments will actually be used,
we mark them as `Lazy`

in case they are large expressions where it
would be wasteful to compute and then discard them.

### Tuples

Values can be paired with the following built-in data type:

```
data Pair a b = MkPair a b
```

As syntactic sugar, we can write `(a, b)`

which, according to
context, means either `Pair a b`

or `MkPair a b`

. Tuples can
contain an arbitrary number of values, represented as nested pairs:

```
fred : (String, Int)
fred = ("Fred", 42)
jim : (String, Int, String)
jim = ("Jim", 25, "Cambridge")
```

```
*UsefulTypes> fst jim
"Jim" : String
*UsefulTypes> snd jim
(25, "Cambridge") : (Int, String)
*UsefulTypes> jim == ("Jim", (25, "Cambridge"))
True : Bool
```

### Dependent Pairs

Dependent pairs allow the type of the second element of a pair to depend on the value of the first element:

```
data DPair : (a : Type) -> (p : a -> Type) -> Type where
MkDPair : {p : a -> Type} -> (x : a) -> p x -> DPair a p
```

Again, there is syntactic sugar for this. `(x : a ** p)`

is the type
of a pair of A and P, where the name `x`

can occur inside `p`

.
`( x ** p )`

constructs a value of this type. For example, we can
pair a number with a `Vect`

of a particular length:

```
vec : (n : Nat ** Vect n Int)
vec = (2 ** [3, 4])
```

If you like, you can write it out the long way; the two are equivalent:

```
vec : DPair Nat (\n => Vect n Int)
vec = MkDPair 2 [3, 4]
```

The type checker could infer the value of the first element
from the length of the vector. We can write an underscore `_`

in
place of values which we expect the type checker to fill in, so the
above definition could also be written as:

```
vec : (n : Nat ** Vect n Int)
vec = (_ ** [3, 4])
```

We might also prefer to omit the type of the first element of the pair, since, again, it can be inferred:

```
vec : (n ** Vect n Int)
vec = (_ ** [3, 4])
```

One use for dependent pairs is to return values of dependent types
where the index is not necessarily known in advance. For example, if
we filter elements out of a `Vect`

according to some predicate, we
will not know in advance what the length of the resulting vector will
be:

```
filter : (a -> Bool) -> Vect n a -> (p ** Vect p a)
```

If the `Vect`

is empty, the result is:

```
filter p Nil = (_ ** [])
```

In the `::`

case, we need to inspect the result of a recursive call
to `filter`

to extract the length and the vector from the result. To
do this, we use a `case`

expression, which allows pattern matching on
intermediate values:

```
filter : (a -> Bool) -> Vect n a -> (n' ** Vect n' a)
filter p Nil = (_ ** [])
filter p (x :: xs)
= case filter p xs of
(_ ** xs') => if p x then (_ ** x :: xs')
else (_ ** xs')
```

Dependent pairs are sometimes referred to as “Sigma types”.

### Records

*Records* are data types which collect several values (the record’s *fields*)
together. Idris provides syntax for defining records and automatically
generating field access and update functions. Unlike the syntax used for data
structures, records in Idris follow a different syntax to that seen with
Haskell. For example, we can represent a person’s name and age in a record:

```
record Person where
constructor MkPerson
firstName, middleName, lastName : String
age : Int
fred : Person
fred = MkPerson "Fred" "Joe" "Bloggs" 30
```

The constructor name is provided using the `constructor`

keyword, and the
*fields* are then given which are in an indented block following the where
keyword (here, `firstName`

, `middleName`

, `lastName`

, and `age`

). You
can declare multiple fields on a single line, provided that they have the same
type. The field names can be used to access the field values:

```
*Record> fred.firstName
"Fred" : String
*Record> fred.age
30 : Int
*Record> :t (.firstName)
Main.Person.(.firstName) : Person -> String
```

We can use prefix field projections, like in Haskell:

```
*Record> firstName fred
"Fred" : String
*Record> age fred
30 : Int
*Record> :t firstName
firstName : Person -> String
```

Prefix field projections can be disabled per record definition
using pragma `%prefix_record_projections off`

, which makes
all subsequently defined records generate only dotted projections.
This pragma has effect until the end of the module
or until the closest occurrence of `%prefix_record_projections on`

.

We can also use the field names to update a record (or, more precisely, produce a copy of the record with the given fields updated):

```
*Record> { firstName := "Jim" } fred
MkPerson "Jim" "Joe" "Bloggs" 30 : Person
*Record> { firstName := "Jim", age $= (+ 1) } fred
MkPerson "Jim" "Joe" "Bloggs" 31 : Person
```

The syntax `{ field := val, ... }`

generates a function which
updates the given fields in a record. `:=`

assigns a new value to a field,
and `$=`

applies a function to update its value.

Each record is defined in its own namespace, which means that field names can be reused in multiple records.

Records, and fields within records, can have dependent types. Updates are allowed to change the type of a field, provided that the result is well-typed.

```
record Class where
constructor ClassInfo
students : Vect n Person
className : String
```

It is safe to update the `students`

field to a vector of a different
length because it will not affect the type of the record:

```
addStudent : Person -> Class -> Class
addStudent p c = { students := p :: students c } c
```

```
*Record> addStudent fred (ClassInfo [] "CS")
ClassInfo [MkPerson "Fred" "Joe" "Bloggs" 30] "CS" : Class
```

We could also use `$=`

to define `addStudent`

more concisely:

```
addStudent' : Person -> Class -> Class
addStudent' p c = { students $= (p ::) } c
```

#### Nested record projection

Nested record fields can be accessed using the dot notation:

```
x.a.b.c
map (.a.b.c) xs
```

For the dot notation, there must be no spaces after the dots but there may be
spaces before the dots. The composite projection must be parenthesised,
otherwise `map .a.b.c xs`

would be understood as `map.a.b.c xs`

.

Nested record fields can be accessed using the prefix notation, too:

```
(c . b . a) x
map (c . b . a) xs
```

Dots with spaces around them stand for function composition operators.

#### Nested record update

Idris also provides a convenient syntax for accessing and updating
nested records. For example, if a field is accessible with the
expression `x.a.b.c`

, it can be updated using the following
syntax:

```
{ a.b.c := val } x
```

This returns a new record, with the field accessed by the path
`a.b.c`

set to `val`

. The syntax is first class, i.e. ```
{
a.b.c := val }
```

itself has a function type.

The `$=`

notation is also valid for nested record updates.

### Dependent Records

Records can also be dependent on values. Records have *parameters*, which
cannot be updated like the other fields. The parameters appear as arguments
to the resulting type, and are written following the record type
name. For example, a pair type could be defined as follows:

```
record Prod a b where
constructor Times
fst : a
snd : b
```

Using the `Class`

record from earlier, the size of the class can be
restricted using a `Vect`

and the size included in the type by parameterising
the record with the size. For example:

```
record SizedClass (size : Nat) where
constructor SizedClassInfo
students : Vect size Person
className : String
```

In the case of `addStudent`

earlier, we can still add a student to a
`SizedClass`

since the size is implicit, and will be updated when a student
is added:

```
addStudent : Person -> SizedClass n -> SizedClass (S n)
addStudent p c = { students := p :: students c } c
```

In fact, the dependent pair type we have just seen is, in practice, defined
as a record, with fields `fst`

and `snd`

which allow projecting values
out of the pair:

```
record DPair a (p : a -> Type) where
constructor MkDPair
fst : a
snd : p fst
```

It is possible to use record update syntax to update dependent fields, provided that all related fields are updated at once. For example:

```
cons : t -> (x : Nat ** Vect x t) -> (x : Nat ** Vect x t)
cons val xs
= { fst := S (fst xs),
snd := (val :: snd xs) } xs
```

Or even:

```
cons' : t -> (x : Nat ** Vect x t) -> (x : Nat ** Vect x t)
cons' val
= { fst $= S,
snd $= (val ::) }
```

## More Expressions

`let`

bindings

Intermediate values can be calculated using `let`

bindings:

```
mirror : List a -> List a
mirror xs = let xs' = reverse xs in
xs ++ xs'
```

We can do pattern matching in `let`

bindings too. For
example, we can extract fields from a record as follows, as well as by
pattern matching at the top level:

```
data Person = MkPerson String Int
showPerson : Person -> String
showPerson p = let MkPerson name age = p in
name ++ " is " ++ show age ++ " years old"
```

These let bindings can be annotated with a type:

```
mirror : List a -> List a
mirror xs = let xs' : List a = reverse xs in
xs ++ xs'
```

We can also use the symbol `:=`

instead of `=`

to, among other things,
avoid ambiguities with propositional equality:

```
Diag : a -> Type
Diag v = let ty : Type := v = v in ty
```

Local definitions can also be introduced using `let`

. Just like top level
ones and ones defined in a `where`

clause you need to:

declare the function and its type

define the function by pattern matching

```
foldMap : Monoid m => (a -> m) -> Vect n a -> m
foldMap f = let fo : m -> a -> m
fo ac el = ac <+> f el
in foldl fo neutral
```

The symbol `:=`

cannot be used in a local function definition. Which means
that it can be used to interleave let bindings and local definitions without
introducing ambiguities.

```
foldMap : Monoid m => (a -> m) -> Vect n a -> m
foldMap f = let fo : m -> a -> m
fo ac el = ac <+> f el
initial := neutral
-- ^ this indicates that `initial` is a separate binding,
-- not relevant to definition of `fo`
in foldl fo initial
```

### List comprehensions

Idris provides *comprehension* notation as a convenient shorthand
for building lists. The general form is:

```
[ expression | qualifiers ]
```

This generates the list of values produced by evaluating the
`expression`

, according to the conditions given by the comma
separated `qualifiers`

. For example, we can build a list of
Pythagorean triples as follows:

```
pythag : Int -> List (Int, Int, Int)
pythag n = [ (x, y, z) | z <- [1..n], y <- [1..z], x <- [1..y],
x*x + y*y == z*z ]
```

The `[a..b]`

notation is another shorthand which builds a list of
numbers between `a`

and `b`

. Alternatively `[a,b..c]`

builds a
list of numbers between `a`

and `c`

with the increment specified
by the difference between `a`

and `b`

. This works for type `Nat`

,
`Int`

and `Integer`

, using the `enumFromTo`

and `enumFromThenTo`

function from the prelude.

`case`

expressions

Another way of inspecting intermediate values is to use a `case`

expression.
The following function, for example, splits a string into two at a given
character:

```
splitAt : Char -> String -> (String, String)
splitAt c x = case break (== c) x of
(x, y) => (x, strTail y)
```

`break`

is a library function which breaks a string into a pair of
strings at the point where the given function returns true. We then
deconstruct the pair it returns, and remove the first character of the
second string.

A `case`

expression can match several cases, for example, to inspect
an intermediate value of type `Maybe a`

. Recall `list_lookup`

which looks up an index in a list, returning `Nothing`

if the index
is out of bounds. We can use this to write `lookup_default`

, which
looks up an index and returns a default value if the index is out of
bounds:

```
lookup_default : Nat -> List a -> a -> a
lookup_default i xs def = case list_lookup i xs of
Nothing => def
Just x => x
```

If the index is in bounds, we get the value at that index, otherwise we get a default value:

```
*UsefulTypes> lookup_default 2 [3,4,5,6] (-1)
5 : Integer
*UsefulTypes> lookup_default 4 [3,4,5,6] (-1)
-1 : Integer
```

## Totality

Idris distinguishes between *total* and *partial* functions.
A total function is a function that either:

Terminates for all possible inputs, or

Produces a non-empty, finite, prefix of a possibly infinite result

If a function is total, we can consider its type a precise description of what
that function will do. For example, if we have a function with a return
type of `String`

we know something different, depending on whether or not
it’s total:

If it’s total, it will return a value of type

`String`

in finite time;If it’s partial, then as long as it doesn’t crash or enter an infinite loop, it will return a

`String`

.

Idris makes this distinction so that it knows which functions are safe to evaluate while type checking (as we’ve seen with First Class Types). After all, if it tries to evaluate a function during type checking which doesn’t terminate, then type checking won’t terminate! Therefore, only total functions will be evaluated during type checking. Partial functions can still be used in types, but will not be evaluated further.