# Documentation for the Idris Language¶

Note

The documentation for Idris has been published under the Creative
Commons CC0 License. As such to the extent possible under law, *The
Idris Community* has waived all copyright and related or neighboring
rights to Documentation for Idris.

More information concerning the CC0 can be found online at: http://creativecommons.org/publicdomain/zero/1.0/

## The Idris Tutorial¶

The is the Idris Tutorial. It will teach you about programming in the Idris Language.

Note

The documentation for Idris has been published under the Creative
Commons CC0 License. As such to the extent possible under law, *The
Idris Community* has waived all copyright and related or neighboring
rights to Documentation for Idris.

More information concerning the CC0 can be found online at: http://creativecommons.org/publicdomain/zero/1.0/

### Introduction¶

In conventional programming languages, there is a clear distinction
between *types* and *values*. For example, in Haskell, the following are types, representing
integers, characters, lists of characters, and lists of any value
respectively:

`Int`

,`Char`

,`[Char]`

,`[a]`

Correspondingly, the following values are examples of inhabitants of those types:

`42`

,`’a’`

,`"Hello world!"`

,`[2,3,4,5,6]`

In a language with *dependent types*, however, the distinction is less
clear. Dependent types allow types to “depend” on values — in other
words, types are a *first class* language construct and can be
manipulated like any other value. The standard example is the type of
lists of a given length [1], `Vect n a`

, where `a`

is the element
type and `n`

is the length of the list and can be an arbitrary term.

When types can contain values, and where those values describe
properties, for example the length of a list, the type of a function
can begin to describe its own properties. Take for example the
concatenation of two lists. This operation has the property that the
resulting list’s length is the sum of the lengths of the two input
lists. We can therefore give the following type to the `app`

function, which concatenates vectors:

```
app : Vect n a -> Vect m a -> Vect (n + m) a
```

This tutorial introduces Idris, a general purpose functional
programming language with dependent types. The goal of the Idris
project is to build a dependently typed language suitable for
verifiable *systems* programming. To this end, Idris is a compiled
language which aims to generate efficient executable code. It also has
a lightweight foreign function interface which allows easy interaction
with external `C`

libraries.

#### Intended Audience¶

This tutorial is intended as a brief introduction to the language, and is aimed at readers already familiar with a functional language such as Haskell or OCaml. In particular, a certain amount of familiarity with Haskell syntax is assumed, although most concepts will at least be explained briefly. The reader is also assumed to have some interest in using dependent types for writing and verifying systems software.

#### Example Code¶

This tutorial includes some example code, which has been tested with
against Idris. These files are available with the Idris distribution,
so that you can try them out easily. They can be found under
`samples`

. It is, however, strongly recommended that you type
them in yourself, rather than simply loading and reading them.

[1] | Typically, and perhaps confusingly, referred to in the dependently typed programming literature as “vectors” |

### Getting Started¶

#### Prerequisites¶

Before installing Idris, you will need to make sure you have all of the necessary libraries and tools. You will need:

- A fairly recent Haskell platform. Version
`2013.2.0.0`

should be sufficiently recent, though it is better to be completely up to date. - The
*GNU Multiple Precision Arithmetic Library*(GMP) is available from MacPorts/Homebrew and all major Linux distributions.

#### Downloading and Installing¶

The easiest way to install Idris, if you have all of the prerequisites, is to type:

```
cabal update; cabal install idris
```

This will install the latest version released on Hackage, along with any dependencies. If, however, you would like the most up to date development version you can find it, as well as build instructions, on GitHub at: https://github.com/idris-lang/Idris-dev.

If you haven’t previously installed anything using Cabal, then Idris
may not be on your path. Should the Idris executable not be found
please ensure that you have added `~/.cabal/bin`

to your `$PATH`

environment variable. Mac OS X users may find they need to add
`~/Library/Haskell/bin`

instead, and Windows users will typically
find that Cabal installs programs in `%HOME%\AppData\Roaming\cabal\bin`

.

To check that installation has succeeded, and to write your first
Idris program, create a file called `hello.idr`

containing the
following text:

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

If you are familiar with Haskell, it should be fairly clear what the
program is doing and how it works, but if not, we will explain the
details later. You can compile the program to an executable by
entering `idris hello.idr -o hello`

at the shell prompt. This will
create an executable called `hello`

, which you can run:

```
$ idris hello.idr -o hello
$ ./hello
Hello world
```

Please note that the dollar sign `$`

indicates the shell prompt!
Some useful options to the Idris command are:

`-o prog`

to compile to an executable called`prog`

.`--check`

type check the file and its dependencies without starting the interactive environment.`--package pkg`

add package as dependency, e.g.`--package contrib`

to make use of the contrib package.`--help`

display usage summary and command line options.

#### The Interactive Environment¶

Entering `idris`

at the shell prompt starts up the interactive
environment. You should see something like the following:

```
$ idris
____ __ _
/ _/___/ /____(_)____
/ // __ / ___/ / ___/ Version 0.9.17
_/ // /_/ / / / (__ ) http://www.idris-lang.org/
/___/\__,_/_/ /_/____/ Type :? for help
Idris>
```

This gives a `ghci`

style interface which allows evaluation of, as
well as type checking of, expressions; theorem proving, compilation;
editing; and various other operations. The command `:?`

gives a list
of supported commands. Below, we see an example run in
which `hello.idr`

is loaded, the type of `main`

is checked and
then the program is compiled to the executable `hello`

. Type
checking a file, if successful, creates a bytecode version of the file
(in this case `hello.ibc`

) to speed up loading in future. The
bytecode is regenerated if the source file changes.

```
$ idris hello.idr
____ __ _
/ _/___/ /____(_)____
/ // __ / ___/ / ___/ Version 0.9.17
_/ // /_/ / / / (__ ) http://www.idris-lang.org/
/___/\__,_/_/ /_/____/ Type :? for help
Type checking ./hello.idr
*hello> :t main
Main.main : IO ()
*hello> :c hello
*hello> :q
Bye bye
$ ./hello
Hello world
```

### Types and Functions¶

#### Primitive Types¶

Idris defines several primitive types: `Int`

, `Integer`

and
`Float`

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 ```
idris
Prims.idr
```

:

```
module Prims
x : Int
x = 42
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, and the type of the
answer. For example:

```
*prims> 6*6+6
42 : Integer
*prims> x == 6*6+6
True : Bool
```

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 == 6 * 6 + 6 then "The answer!" else "Not the answer"
"The answer!" : String
```

#### 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
```

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
`:`

, `=>`

, `->`

, `<-`

, `=`

, `?=`

, `|`

, `**`

,
`==>`

, `\`

, `%`

, `~`

, `?`

, and `!`

.

#### 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 types and 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. We can test these functions at the
Idris prompt:

```
Idris> plus (S (S Z)) (S (S Z))
4 : Nat
Idris> mult (S (S (S Z))) (plus (S (S Z)) (S (S Z)))
12 : Nat
```

Note

When displaying an element of `Nat`

such as `(S (S (S (S Z))))`

,
Idris displays it as `4`

.
The result of `plus (S (S Z)) (S (S Z))`

is actually `(S (S (S (S Z))))`

which is the natural number `4`

.
This can be checked at the Idris prompt:

```
Idris> (S (S (S (S Z))))
4 : Nat
```

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 : Nat
Idris> mult 3 (plus 2 2)
12 : Nat
```

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 only in the type will be in scope in the
`where`

clause if it is a *parameter* to one of the types, i.e. it
is fixed across the entire structure.

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
```

In general, functions defined in a `where`

clause need a type
declaration just like any top level function. However, the type
declaration for a function `f`

*can* be omitted if:

`f`

appears in the right hand side of the top level definition- The type of
`f`

can be completely determined from its first application

So, for example, the following definitions are legal:

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

#### Dependent Types¶

##### First Class Types¶

In Idris, types are a first class language construct, 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:

```
$ idris vbroken.idr --check
vbroken.idr:9:23:When elaborating right hand side of Vect.++:
When elaborating an application of constructor Vect.:::
Type mismatch between
Vect (k + k) a (Type of xs ++ xs)
and
Vect (plus k m) a (Expected type)
Specifically:
Type mismatch between
plus k k
and
plus k 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 : {a:Type} -> {n:Nat} -> Fin n -> Vect n a -> a
```

Implicit arguments, given in braces `{}`

in the type 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. 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.

##### “`using`

” notation¶

Sometimes it is useful to provide types of implicit arguments,
particularly where there is a dependency ordering, or where the
implicit arguments themselves have dependencies. For example, we may
wish to state the types of the implicit arguments in the following
definition, which defines a predicate on vectors (this is also defined
in `Data.Vect`

, under the name `Elem`

):

```
data IsElem : a -> Vect n a -> Type where
Here : {x:a} -> {xs:Vect n a} -> IsElem x (x :: xs)
There : {x,y:a} -> {xs:Vect n a} -> IsElem x xs -> IsElem x (y :: xs)
```

An instance of `IsElem x xs`

states that `x`

is an element of
`xs`

. We can construct such a predicate if the required element is
`Here`

, at the head of the vector, or `There`

, in the tail of the
vector. For example:

```
testVec : Vect 4 Int
testVec = 3 :: 4 :: 5 :: 6 :: Nil
inVect : IsElem 5 testVec
inVect = There (There Here)
```

If the same implicit arguments are being used a lot, it can make a
definition difficult to read. To avoid this problem, a `using`

block
gives the types and ordering of any implicit arguments which can
appear within the block:

```
using (x:a, y:a, xs:Vect n a)
data IsElem : a -> Vect n a -> Type where
Here : IsElem x (x :: xs)
There : IsElem x xs -> IsElem x (y :: xs)
```

###### 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 their
reduction behaviour to affect type 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.

#### 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. Therefore in
Idris, such interactions are encapsulated in the type `IO`

:

```
data IO a -- 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 IO program:

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

The type of `putStrLn`

explains that it takes a string, and returns
an element of the unit type `()`

via an I/O action. There is a
variant `putStr`

which outputs 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 defined in the prelude, for example 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 `return`

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

```
return : a -> IO a
```

As we will see later, `do`

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

#### 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
(in fact, this is used to implement the `if...then...else`

construct
as we will see later. We would prefer if *only* the argument which was
used was evaluated. To achieve this, Idris provides a `Lazy`

data type, which allows evaluation to be suspended:

```
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;
```

#### Useful Data Types¶

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

directory in the distribution). This chapter
describes a few of these. The functions described here are imported
automatically by every Idris program, as part of `Prelude.idr`

.

`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
```

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 type with the constructor names
`Nil`

and `::`

can be written in list form. For example:

`[]`

means`Nil`

`[1,2,3]`

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

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

is overloaded both for `List`

and `Vect`

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/prelude/Prelude/List.idr`

`libs/base/Data/List.idr`

`libs/base/Data/Vect.idr`

`libs/base/Data/VectType.idr`

Functions include filtering, appending, reversing, and so on. Also remember that Idris is still in development, so if you don’t see the function you need, please feel free to add it and submit a patch!

###### Aside: Anonymous functions and operator sections¶

There are actually 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`

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. Traditionally, these are referred to as “sigma types”:

```
data Sigma : (a : Type) -> (P : a -> Type) -> Type where
MkSigma : {P : a -> Type} -> (x : a) -> P x -> Sigma a P
```

Again, there is syntactic sugar for this. `(a : A ** P)`

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

can occur inside `P`

.
`( a ** 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 precisely equivalent.

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

The type checker could of course 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 easy:

```
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 `with`

notation, which allows pattern matching on
intermediate values:

```
filter p (x :: xs) with (filter p xs)
| ( _ ** xs' ) = if (p x) then ( _ ** x :: xs' ) else ( _ ** xs' )
```

We will see more on `with`

notation later.

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

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> record { firstName = "Jim" } fred
MkPerson "Jim" "Joe" "Bloggs" 30 : Person
*record> record { firstName = "Jim", age = 20 } fred
MkPerson "Jim" "Joe" "Bloggs" 20 : Person
```

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

generates a function which
updates the given fields in a record.

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 = record { students = p :: students c } c
```

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

###### 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 `c (b (a x))`

, it can be updated using the following
syntax:

```
record { 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. ```
record {
a->b->c = val }
```

itself has a function type. Symmetrically, the field
can also be accessed with the following syntax:

```
record { a->b->c } x
```

##### Dependent Records¶

Records can also be dependent on values. Records *parameters*, which
are not subject to field updates. 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 the original introduction to records. The
size of the class can be restricted using a `Vect`

and the size
promoted to the type level by parameterising the record with the size.
For example:

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

**Note** that it is no longer possible to use the `addStudent`

function from earlier, since that would change the size of the class. A
function to add a student must now specify in the type that the
size of the class has been increased by one. As the size is specified
using natural numbers, the new value can be incremented using the
`S`

constructor.

```
addStudent : Person -> SizedClass n -> SizedClass (S n)
addStudent p c = SizedClassInfo (p :: students c) (className c)
```

#### 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 simple 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"
```

##### 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 any numeric
type, using the `count`

function from the prelude.

`case`

expressions¶

Another way of inspecting intermediate values of *simple* types 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
```

**Restrictions:** The `case`

construct is intended for simple
analysis of intermediate expressions to avoid the need to write
auxiliary functions, and is also used internally to implement pattern
matching `let`

and lambda bindings. It will *only* work if:

- Each branch
*matches*a value of the same type, and*returns*a value of the same type. - The type of the result is “known”. i.e. the type of the expression
can be determined
*without*type checking the`case`

-expression itself.

### Interfaces¶

We often want to define functions which work across several different
data types. For example, we would like arithmetic operators to work on
`Int`

, `Integer`

and `Float`

at the very least. We would like
`==`

to work on the majority of data types. We would like to be able
to display different types in a uniform way.

To achieve this, we use *interfaces*, which are similar to type classes in
Haskell or traits in Rust. To define an interface, we provide a collection of
overloadable functions. A simple example is the `Show`

interface, which is defined in the prelude and provides an interface for
converting values to `String`

:

```
interface Show a where
show : a -> String
```

This generates a function of the following type (which we call a
*method* of the `Show`

interface):

```
show : Show a => a -> String
```

We can read this as: “under the constraint that `a`

has an implementation
of `Show`

, take an input `a`

and return a `String`

.” An implementation
of an interface is defined by giving definitions of the methods of the interface.
For example, the
`Show`

implementation for `Nat`

could be defined as:

```
Show Nat where
show Z = "Z"
show (S k) = "s" ++ show k
```

```
Idris> show (S (S (S Z)))
"sssZ" : String
```

Only one implementation of an interface can be given for a type — implementations may
not overlap. Implementation declarations can themselves have constraints.
To help with resolution, the arguments of an implementation must be
constructors (either data or type constructors), variables or
constants (i.e. you cannot give an implementation for a function). For
example, to define a `Show`

implementation for vectors, we need to know
that there is a `Show`

implementation for the element type, because we are
going to use it to convert each element to a `String`

:

```
Show a => Show (Vect n a) where
show xs = "[" ++ show' xs ++ "]" where
show' : Vect n a -> String
show' Nil = ""
show' (x :: Nil) = show x
show' (x :: xs) = show x ++ ", " ++ show' xs
```

#### Default Definitions¶

The library defines an `Eq`

interface which provides methods for
comparing values for equality or inequality, with implementations for all of
the built-in types:

```
interface Eq a where
(==) : a -> a -> Bool
(/=) : a -> a -> Bool
```

To declare an implementation for a type, we have to give definitions of all
of the methods. For example, for an implementation of `Eq`

for `Nat`

:

```
Eq Nat where
Z == Z = True
(S x) == (S y) = x == y
Z == (S y) = False
(S x) == Z = False
x /= y = not (x == y)
```

It is hard to imagine many cases where the `/=`

method will be
anything other than the negation of the result of applying the `==`

method. It is therefore convenient to give a default definition for
each method in the interface declaration, in terms of the other method:

```
interface Eq a where
(==) : a -> a -> Bool
(/=) : a -> a -> Bool
x /= y = not (x == y)
x == y = not (x /= y)
```

A minimal complete implementation of `Eq`

requires either
`==`

or `/=`

to be defined, but does not require both. If a method
definition is missing, and there is a default definition for it, then
the default is used instead.

#### Extending Interfaces¶

Interfaces can also be extended. A logical next step from an equality
relation `Eq`

is to define an ordering relation `Ord`

. We can
define an `Ord`

interface which inherits methods from `Eq`

as well as
defining some of its own:

```
data Ordering = LT | EQ | GT
```

```
interface Eq a => Ord a where
compare : a -> a -> Ordering
(<) : a -> a -> Bool
(>) : a -> a -> Bool
(<=) : a -> a -> Bool
(>=) : a -> a -> Bool
max : a -> a -> a
min : a -> a -> a
```

The `Ord`

interface allows us to compare two values and determine their
ordering. Only the `compare`

method is required; every other method
has a default definition. Using this we can write functions such as
`sort`

, a function which sorts a list into increasing order,
provided that the element type of the list is in the `Ord`

interface. We
give the constraints on the type variables left of the fat arrow
`=>`

, and the function type to the right of the fat arrow:

```
sort : Ord a => List a -> List a
```

Functions, interfaces and implementations can have multiple constraints. Multiple constraints are written in brackets in a comma separated list, for example:

```
sortAndShow : (Ord a, Show a) => List a -> String
sortAndShow xs = show (sort xs)
```

#### Functors and Applicatives¶

So far, we have seen single parameter interfaces, where the parameter
is of type `Type`

. In general, there can be any number of parameters
(even zero), and the parameters can have *any* type. If the type
of the parameter is not `Type`

, we need to give an explicit type
declaration. For example, the `Functor`

interface is defined in the
prelude:

```
interface Functor (f : Type -> Type) where
map : (m : a -> b) -> f a -> f b
```

A functor allows a function to be applied across a structure, for
example to apply a function to every element in a `List`

:

```
Functor List where
map f [] = []
map f (x::xs) = f x :: map f xs
```

```
Idris> map (*2) [1..10]
[2, 4, 6, 8, 10, 12, 14, 16, 18, 20] : List Integer
```

Having defined `Functor`

, we can define `Applicative`

which
abstracts the notion of function application:

```
infixl 2 <*>
interface Functor f => Applicative (f : Type -> Type) where
pure : a -> f a
(<*>) : f (a -> b) -> f a -> f b
```

#### Monads and `do`

-notation¶

The `Monad`

interface allows us to encapsulate binding and computation,
and is the basis of `do`

-notation introduced in Section
“do” notation. It extends `Applicative`

as defined above, and is
defined as follows:

```
interface Applicative m => Monad (m : Type -> Type) where
(>>=) : m a -> (a -> m b) -> m b
```

Inside a `do`

block, the following syntactic transformations are
applied:

`x <- v; e`

becomes`v >>= (\x => e)`

`v; e`

becomes`v >>= (\_ => e)`

`let x = v; e`

becomes`let x = v in e`

`IO`

has an implementation of `Monad`

, defined using primitive functions.
We can also define an implementation for `Maybe`

, as follows:

```
Monad Maybe where
Nothing >>= k = Nothing
(Just x) >>= k = k x
```

Using this we can, for example, define a function which adds two
`Maybe Int`

, using the monad to encapsulate the error handling:

```
m_add : Maybe Int -> Maybe Int -> Maybe Int
m_add x y = do x' <- x -- Extract value from x
y' <- y -- Extract value from y
return (x' + y') -- Add them
```

This function will extract the values from `x`

and `y`

, if they
are both available, or return `Nothing`

if one or both are not (“fail fast”). Managing the
`Nothing`

cases is achieved by the `>>=`

operator, hidden by the
`do`

notation.

```
*ifaces> m_add (Just 20) (Just 22)
Just 42 : Maybe Int
*ifaces> m_add (Just 20) Nothing
Nothing : Maybe Int
```

##### Pattern Matching Bind¶

Sometimes we want to pattern match immediately on the result of a function
in `do`

notation. For example, let’s say we have a function `readNumber`

which reads a number from the console, returning a value of the form
`Just x`

if the number is valid, or `Nothing`

otherwise:

```
readNumber : IO (Maybe Nat)
readNumber = do
input <- getLine
if all isDigit (unpack input)
then pure (Just (cast input))
else pure Nothing
```

If we then use it to write a function to read two numbers, returning
`Nothing`

if neither are valid, then we would like to pattern match
on the result of `readNumber`

:

```
readNumbers : IO (Maybe (Nat, Nat))
readNumbers =
do x <- readNumber
case x of
Nothing => pure Nothing
Just x_ok => do y <- readNumber
case y of
Nothing => pure Nothing
Just y_ok => pure (Just (x_ok, y_ok))
```

If there’s a lot of error handling, this could get deeply nested very quickly!
So instead, we can combine the bind and the pattern match in one line. For example,
we could try pattern matching on values of the form `Just x_ok`

:

```
readNumbers : IO (Maybe (Nat, Nat))
readNumbers =
do Just x_ok <- readNumber
Just y_ok <- readNumber
pure (Just (x_ok, y_ok))
```

There is still a problem, however, because we’ve now omitted the case for
`Nothing`

so `readNumbers`

is no longer total! We can add the `Nothing`

case back as follows:

```
readNumbers : IO (Maybe (Nat, Nat))
readNumbers =
do Just x_ok <- readNumber | Nothing => pure Nothing
Just y_ok <- readNumber | Nothing => pure Nothing
pure (Just (x_ok, y_ok))
```

The effect of this version of `readNumbers`

is identical to the first (in
fact, it is syntactic sugar for it and directly translated back into that form).
The first part of each statement (`Just x_ok <-`

and `Just y_ok <-`

) gives
the preferred binding - if this matches, execution will continue with the rest
of the `do`

block. The second part gives the alternative bindings, of which
there may be more than one.

`!`

-notation¶

In many cases, using `do`

-notation can make programs unnecessarily
verbose, particularly in cases such as `m_add`

above where the value
bound is used once, immediately. In these cases, we can use a
shorthand version, as follows:

```
m_add : Maybe Int -> Maybe Int -> Maybe Int
m_add x y = return (!x + !y)
```

The notation `!expr`

means that the expression `expr`

should be
evaluated and then implicitly bound. Conceptually, we can think of
`!`

as being a prefix function with the following type:

```
(!) : m a -> a
```

Note, however, that it is not really a function, merely syntax! In
practice, a subexpression `!expr`

will lift `expr`

as high as
possible within its current scope, bind it to a fresh name `x`

, and
replace `!expr`

with `x`

. Expressions are lifted depth first, left
to right. In practice, `!`

-notation allows us to program in a more
direct style, while still giving a notational clue as to which
expressions are monadic.

For example, the expression:

```
let y = 42 in f !(g !(print y) !x)
```

is lifted to:

```
let y = 42 in do y' <- print y
x' <- x
g' <- g y' x'
f g'
```

##### Monad comprehensions¶

The list comprehension notation we saw in Section
More Expressions is more general, and applies to anything which
has an implementation of both `Monad`

and `Alternative`

:

```
interface Applicative f => Alternative (f : Type -> Type) where
empty : f a
(<|>) : f a -> f a -> f a
```

In general, a comprehension takes the form ```
[ exp | qual1, qual2, …,
qualn ]
```

where `quali`

can be one of:

- A generator
`x <- e`

- A
*guard*, which is an expression of type`Bool`

- A let binding
`let x = e`

To translate a comprehension `[exp | qual1, qual2, …, qualn]`

, first
any qualifier `qual`

which is a *guard* is translated to ```
guard
qual
```

, using the following function:

```
guard : Alternative f => Bool -> f ()
```

Then the comprehension is converted to `do`

notation:

```
do { qual1; qual2; ...; qualn; return exp; }
```

Using monad comprehensions, an alternative definition for `m_add`

would be:

```
m_add : Maybe Int -> Maybe Int -> Maybe Int
m_add x y = [ x' + y' | x' <- x, y' <- y ]
```

#### Idiom brackets¶

While `do`

notation gives an alternative meaning to sequencing,
idioms give an alternative meaning to *application*. The notation and
larger example in this section is inspired by Conor McBride and Ross
Paterson’s paper “Applicative Programming with Effects” [1].

First, let us revisit `m_add`

above. All it is really doing is
applying an operator to two values extracted from `Maybe Int`

. We
could abstract out the application:

```
m_app : Maybe (a -> b) -> Maybe a -> Maybe b
m_app (Just f) (Just a) = Just (f a)
m_app _ _ = Nothing
```

Using this, we can write an alternative `m_add`

which uses this
alternative notion of function application, with explicit calls to
`m_app`

:

```
m_add' : Maybe Int -> Maybe Int -> Maybe Int
m_add' x y = m_app (m_app (Just (+)) x) y
```

Rather than having to insert `m_app`

everywhere there is an
application, we can use idiom brackets to do the job for us.
To do this, we can give `Maybe`

an implementation of `Applicative`

as follows, where `<*>`

is defined in the same way as `m_app`

above (this is defined in the Idris library):

```
Applicative Maybe where
pure = Just
(Just f) <*> (Just a) = Just (f a)
_ <*> _ = Nothing
```

Using `<*>`

we can use this implementation as follows, where a function
application `[| f a1 …an |]`

is translated into ```
pure f <*> a1 <*>
… <*> an
```

:

```
m_add' : Maybe Int -> Maybe Int -> Maybe Int
m_add' x y = [| x + y |]
```

##### An error-handling interpreter¶

Idiom notation is commonly useful when defining evaluators. McBride and Paterson describe such an evaluator [1], for a language similar to the following:

```
data Expr = Var String -- variables
| Val Int -- values
| Add Expr Expr -- addition
```

Evaluation will take place relative to a context mapping variables
(represented as `String`

s) to `Int`

values, and can possibly fail.
We define a data type `Eval`

to wrap an evaluator:

```
data Eval : Type -> Type where
MkEval : (List (String, Int) -> Maybe a) -> Eval a
```

Wrapping the evaluator in a data type means we will be able to provide implementations of interfaces for it later. We begin by defining a function to retrieve values from the context during evaluation:

```
fetch : String -> Eval Int
fetch x = MkEval (\e => fetchVal e) where
fetchVal : List (String, Int) -> Maybe Int
fetchVal [] = Nothing
fetchVal ((v, val) :: xs) = if (x == v)
then (Just val)
else (fetchVal xs)
```

When defining an evaluator for the language, we will be applying functions in
the context of an `Eval`

, so it is natural to give `Eval`

an implementation
of `Applicative`

. Before `Eval`

can have an implementation of
`Applicative`

it is necessary for `Eval`

to have an implementation of
`Functor`

:

```
Functor Eval where
map f (MkEval g) = MkEval (\e => map f (g e))
Applicative Eval where
pure x = MkEval (\e => Just x)
(<*>) (MkEval f) (MkEval g) = MkEval (\x => app (f x) (g x)) where
app : Maybe (a -> b) -> Maybe a -> Maybe b
app (Just fx) (Just gx) = Just (fx gx)
app _ _ = Nothing
```

Evaluating an expression can now make use of the idiomatic application to handle errors:

```
eval : Expr -> Eval Int
eval (Var x) = fetch x
eval (Val x) = [| x |]
eval (Add x y) = [| eval x + eval y |]
runEval : List (String, Int) -> Expr -> Maybe Int
runEval env e = case eval e of
MkEval envFn => envFn env
```

#### Named Implementations¶

It can be desirable to have multiple implementations of an interface for the
same type, for example to provide alternative methods for sorting or printing
values. To achieve this, implementations can be *named* as follows:

```
[myord] Ord Nat where
compare Z (S n) = GT
compare (S n) Z = LT
compare Z Z = EQ
compare (S x) (S y) = compare @{myord} x y
```

This declares an implementation as normal, but with an explicit name,
`myord`

. The syntax `compare @{myord}`

gives an explicit implementation to
`compare`

, otherwise it would use the default implementation for `Nat`

. We
can use this, for example, to sort a list of `Nat`

in reverse.
Given the following list:

```
testList : List Nat
testList = [3,4,1]
```

We can sort it using the default `Ord`

implementation, then the named
implementation `myord`

as follows, at the Idris prompt:

```
*named_impl> show (sort testList)
"[sO, sssO, ssssO]" : String
*named_impl> show (sort @{myord} testList)
"[ssssO, sssO, sO]" : String
```

#### Determining Parameters¶

When an interface has more than one parameter, it can help resolution if the parameters used to find an implementation are restricted. For example:

```
interface Monad m => MonadState s (m : Type -> Type) | m where
get : m s
put : s -> m ()
```

In this interface, only `m`

needs to be known to find an implementation of
this interface, and `s`

can then be determined from the implementation. This
is declared with the `| m`

after the interface declaration. We call `m`

a
*determining parameter* of the `MonadState`

interface, because it is the
parameter used to find an implementation.

[1] | (1, 2) Conor McBride and Ross Paterson. 2008. Applicative programming
with effects. J. Funct. Program. 18, 1 (January 2008),
1-13. DOI=10.1017/S0956796807006326
http://dx.doi.org/10.1017/S0956796807006326 |

### Modules and Namespaces¶

An Idris program consists of a collection of modules. Each module
includes an optional `module`

declaration giving the name of the
module, a list of `import`

statements giving the other modules which
are to be imported, and a collection of declarations and definitions of
types, interfaces and functions. For example, the listing below gives a
module which defines a binary tree type `BTree`

(in a file
`Btree.idr`

):

```
module Btree
data BTree a = Leaf
| Node (BTree a) a (BTree a)
insert : Ord a => a -> BTree a -> BTree a
insert x Leaf = Node Leaf x Leaf
insert x (Node l v r) = if (x < v) then (Node (insert x l) v r)
else (Node l v (insert x r))
toList : BTree a -> List a
toList Leaf = []
toList (Node l v r) = Btree.toList l ++ (v :: Btree.toList r)
toTree : Ord a => List a -> BTree a
toTree [] = Leaf
toTree (x :: xs) = insert x (toTree xs)
```

Then, this gives a main program (in a file
`bmain.idr`

) which uses the `Btree`

module to sort a list:

```
module Main
import Btree
main : IO ()
main = do let t = toTree [1,8,2,7,9,3]
print (Btree.toList t)
```

The same names can be defined in multiple modules. This is possible
because in practice names are *qualified* with the name of the module.
The names defined in the `Btree`

module are, in full:

`Btree.BTree`

`Btree.Leaf`

`Btree.Node`

`Btree.insert`

`Btree.toList`

`Btree.toTree`

If names are otherwise unambiguous, there is no need to give the fully qualified name. Names can be disambiguated either by giving an explicit qualification, or according to their type.

There is no formal link between the module name and its filename,
although it is generally advisable to use the same name for each. An
`import`

statement refers to a filename, using dots to separate
directories. For example, `import foo.bar`

would import the file
`foo/bar.idr`

, which would conventionally have the module declaration
`module foo.bar`

. The only requirement for module names is that the
main module, with the `main`

function, must be called
`Main`

—although its filename need not be `Main.idr`

.

#### Export Modifiers¶

By default, all names defined in a module are exported for use by other
modules. However, it is good practice only to export a minimal interface
and keep internal details abstract. Idris allows functions, types,
and interfaces to be marked as: `public`

, `abstract`

or `private`

:

`public`

means that both the name and definition are exported. For functions, this means that the implementation is exported (which means, for example, it can be used in a dependent type). For data types, this means that the type name and the constructors are exported. For interfaces, this means that the interface name and method names are exported.`abstract`

means that only the name is exported. For functions, this means that the implementation is not exported. For data types, this means that the type name is exported but not the constructors. For interfaces, this means that the interface name is exported but not the method names.`private`

means that neither the name nor the definition is exported.

Note

If any definition is given an export modifier, then all names with no modifier are assumed to be `private`

.

For our `Btree`

module, it makes sense for the tree data type and the
functions to be exported as `abstract`

, as we see below:

```
module Btree
abstract data BTree a = Leaf
| Node (BTree a) a (BTree a)
abstract
insert : Ord a => a -> BTree a -> BTree a
insert x Leaf = Node Leaf x Leaf
insert x (Node l v r) = if (x < v) then (Node (insert x l) v r)
else (Node l v (insert x r))
abstract
toList : BTree a -> List a
toList Leaf = []
toList (Node l v r) = Btree.toList l ++ (v :: Btree.toList r)
abstract
toTree : Ord a => List a -> BTree a
toTree [] = Leaf
toTree (x :: xs) = insert x (toTree xs)
```

Finally, the default export mode can be changed with the `%access`

directive, for example:

```
module Btree
%access abstract
data BTree a = Leaf
| Node (BTree a) a (BTree a)
insert : Ord a => a -> BTree a -> BTree a
insert x Leaf = Node Leaf x Leaf
insert x (Node l v r) = if (x < v) then (Node (insert x l) v r)
else (Node l v (insert x r))
toList : BTree a -> List a
toList Leaf = []
toList (Node l v r) = Btree.toList l ++ (v :: Btree.toList r)
toTree : Ord a => List a -> BTree a
toTree [] = Leaf
toTree (x :: xs) = insert x (toTree xs)
```

In this case, any function with no access modifier will be exported as
`abstract`

, rather than left `private`

.

Additionally, a module can re-export a module it has imported, by using
the `public`

modifier on an `import`

. For example:

```
module A
import B
import public C
public a : AType a = ...
```

The module `A`

will export the name `a`

, as well as any public or
abstract names in module `C`

, but will not re-export anything from
module `B`

.

#### Explicit Namespaces¶

Defining a module also defines a namespace implicitly. However,
namespaces can also be given *explicitly*. This is most useful if you
wish to overload names within the same module:

```
module Foo
namespace x
test : Int -> Int
test x = x * 2
namespace y
test : String -> String
test x = x ++ x
```

This (admittedly contrived) module defines two functions with fully
qualified names `foo.x.test`

and `foo.y.test`

, which can be
disambiguated by their types:

```
*foo> test 3
6 : Int
*foo> test "foo"
"foofoo" : String
```

#### Parameterised blocks¶

Groups of functions can be parameterised over a number of arguments
using a `parameters`

declaration, for example:

```
parameters (x : Nat, y : Nat)
addAll : Nat -> Nat
addAll z = x + y + z
```

The effect of a `parameters`

block is to add the declared parameters
to every function, type and data constructor within the
block. Specifically, adding the parameters to the front of the
argument list. Outside the block, the parameters must be given
explicitly. The `addAll`

function, when called from the REPL, will
thus have the following type signature.

```
*params> :t addAll
addAll : Nat -> Nat -> Nat -> Nat
```

and the following definition.

```
addAll : (x : Nat) -> (y : Nat) -> (z : Nat) -> Nat
addAll x y z = x + y + z
```

Parameters blocks can be nested, and can also include data declarations, in which case the parameters are added explicitly to all type and data constructors. They may also be dependent types with implicit arguments:

```
parameters (y : Nat, xs : Vect x a)
data Vects : Type -> Type where
MkVects : Vect y a -> Vects a
append : Vects a -> Vect (x + y) a
append (MkVects ys) = xs ++ ys
```

To use `Vects`

or `append`

outside the block, we must also give the
`xs`

and `y`

arguments. Here, we can use placeholders for the values
which can be inferred by the type checker:

```
*params> show (append _ _ (MkVects _ [1,2,3] [4,5,6]))
"[1, 2, 3, 4, 5, 6]" : String
```

### Packages¶

Idris includes a simple build system for building packages and executables from a named package description file. These files can be used with the Idris compiler to manage the development process .

#### Package Descriptions¶

A package description includes the following:

- A header, consisting of the keyword package followed by the package name.
- Fields describing package contents,
`<field> = <value>`

At least one field must be the modules field, where the value is a
comma separated list of modules. For example, given an idris package
`maths`

that has modules `Maths.idr`

, `Maths.NumOps.idr`

,
`Maths.BinOps.idr`

, and `Maths.HexOps.idr`

, the corresponding
package file would be:

```
package maths
modules = Maths
, Maths.NumOps
, Maths.BinOps
, Maths.HexOps
```

Other examples of package files can be found in the `libs`

directory
of the main Idris repository, and in third-party libraries.
More details including a complete listing of available fields can be found in Packages.

#### Using Package files¶

Given an Idris package file `maths.ipkg`

it can be used with the Idris compiler as follows:

`idris --build maths.ipkg`

will build all modules in the package`idris --install maths.ipkg`

will install the package, making it accessible by other Idris libraries and programs.`idris --clean maths.ipkg`

will delete all intermediate code and executable files generated when building.`idris --mkdoc maths.ipkg`

will build HTML documentation for your package in the folder`maths_doc`

in your project’s root directory.`idris --checkpkg maths.ipkg`

will type check all modules in the package only. This differs from build that type checks**and**generates code.`idris --testpkg maths.ipkg`

will compile and run any embedded tests you have specified in the`tests`

parameter. More information about testing is given in the next section.

Once the maths package has been installed, the command line option
`--package maths`

makes it accessible (abbreviated to `-p maths`

).
For example:

```
idris -p maths Main.idr
```

### Testing Idris Packages¶

The integrated build system includes a simple testing framework.
This framework collects functions listed in the `ipkg`

file under `tests`

.
All test functions must return `IO ()`

.

When you enter `idris --testpkg yourmodule.ipkg`

,
the build system creates a temporary file in a fresh environment on your machine
by listing the `tests`

functions under a single `main`

function.
It compiles this temporary file to an executable and then executes it.

The tests themselves are responsible for reporting their success or failure.
Test functions commonly use `putStrLn`

to report test results.
The test framework does not impose any standards for reporting and consequently
does not aggregate test results.

For example, lets take the following list of functions that are defined in a module called `NumOps`

for a sample package `maths`

.

```
module Maths.NumOps
double : Num a => a -> a
double a = a + a
triple : Num a => a -> a
triple a = a + double a
```

A simple test module, with a qualified name of `Test.NumOps`

can be declared as

```
module Test.NumOps
import Maths.NumOps
assertEq : Eq a => (given : a) -> (expected : a) -> IO ()
assertEq g e = if g == e
then putStrLn "Test Passed"
else putStrLn "Test Failed"
assertNotEq : Eq a => (given : a) -> (expected : a) -> IO ()
assertNotEq g e = if not (g == e)
then putStrLn "Test Passed"
else putStrLn "Test Failed"
testDouble : IO ()
testDouble = assertEq (double 2) 4
testTriple : IO ()
testTriple = assertNotEq (triple 2) 5
```

The functions `assertEq`

and `assertNotEq`

are used to run expected passing, and failing, equality tests.
The actual tests are `testDouble`

and `testTriple`

, and are declared in the `maths.ipkg`

file as follows:

```
package maths
modules = Maths.NumOps
, Test.NumOps
tests = Test.NumOps.testDouble
, Test.NumOps.testTriple
```

The testing framework can then be invoked using `idris --testpkg maths.ipkg`

:

```
> idris --testpkg maths.ipkg
Type checking ./Maths/NumOps.idr
Type checking ./Test/NumOps.idr
Type checking /var/folders/63/np5g0d5j54x1s0z12rf41wxm0000gp/T/idristests144128232716531729.idr
Test Passed
Test Passed
```

Note how both tests have reported success by printing `Test Passed`

as we arranged for with the `assertEq`

and `assertNoEq`

functions.

### Example: The Well-Typed Interpreter¶

In this section, we’ll use the features we’ve seen so far to write a
larger example, an interpreter for a simple functional programming
language, with variables, function application, binary operators and
an `if...then...else`

construct. We will use the dependent type
system to ensure that any programs which can be represented are
well-typed.

#### Representing Languages¶

First, let us define the types in the language. We have integers,
booleans, and functions, represented by `Ty`

:

```
data Ty = TyInt | TyBool | TyFun Ty Ty
```

We can write a function to translate these representations to a concrete Idris type — remember that types are first class, so can be calculated just like any other value:

```
interpTy : Ty -> Type
interpTy TyInt = Int
interpTy TyBool = Bool
interpTy (TyFun A T) = interpTy A -> interpTy T
```

We’re going to define a representation of our language in such a way
that only well-typed programs can be represented. We’ll index the
representations of expressions by their type, **and** the types of
local variables (the context). The context can be represented using
the `Vect`

data type, and as it will be used regularly it will be
represented as an implicit argument. To do so we define everything in
a `using`

block (keep in mind that everything after this point needs
to be indented so as to be inside the `using`

block):

```
using (G:Vect n Ty)
```

Expressions are indexed by the types of the local variables, and the type of the expression itself:

```
data Expr : Vect n Ty -> Ty -> Type
```

The full representation of expressions is:

```
data HasType : (i : Fin n) -> Vect n Ty -> Ty -> Type where
Stop : HasType FZ (t :: G) t
Pop : HasType k G t -> HasType (FS k) (u :: G) t
data Expr : Vect n Ty -> Ty -> Type where
Var : HasType i G t -> Expr G t
Val : (x : Int) -> Expr G TyInt
Lam : Expr (a :: G) t -> Expr G (TyFun a t)
App : Expr G (TyFun a t) -> Expr G a -> Expr G t
Op : (interpTy a -> interpTy b -> interpTy c) ->
Expr G a -> Expr G b -> Expr G c
If : Expr G TyBool ->
Lazy (Expr G a) ->
Lazy (Expr G a) -> Expr G a
```

The code above makes use of the `Vect`

and `Fin`

types from the
Idris standard library. We import them because they are not provided
in the prelude:

```
import Data.Vect
import Data.Fin
```

Since expressions are indexed by their type, we can read the typing rules of the language from the definitions of the constructors. Let us look at each constructor in turn.

We use a nameless representation for variables — they are *de Bruijn
indexed*. Variables are represented by a proof of their membership in
the context, `HasType i G T`

, which is a proof that variable `i`

in context `G`

has type `T`

. This is defined as follows:

```
data HasType : (i : Fin n) -> Vect n Ty -> Ty -> Type where
Stop : HasType FZ (t :: G) t
Pop : HasType k G t -> HasType (FS k) (u :: G) t
```

We can treat *Stop* as a proof that the most recently defined variable
is well-typed, and *Pop n* as a proof that, if the `n`

th most
recently defined variable is well-typed, so is the `n+1`

th. In
practice, this means we use `Stop`

to refer to the most recently
defined variable, `Pop Stop`

to refer to the next, and so on, via
the `Var`

constructor:

```
Var : HasType i G t -> Expr G t
```

So, in an expression `\x,\y. x y`

, the variable `x`

would have a
de Bruijn index of 1, represented as `Pop Stop`

, and `y 0`

,
represented as `Stop`

. We find these by counting the number of
lambdas between the definition and the use.

A value carries a concrete representation of an integer:

```
Val : (x : Int) -> Expr G TyInt
```

A lambda creates a function. In the scope of a function of type ```
a ->
t
```

, there is a new local variable of type `a`

, which is expressed
by the context index:

```
Lam : Expr (a :: G) t -> Expr G (TyFun a t)
```

Function application produces a value of type `t`

given a function
from `a`

to `t`

and a value of type `a`

:

```
App : Expr G (TyFun a t) -> Expr G a -> Expr G t
```

We allow arbitrary binary operators, where the type of the operator informs what the types of the arguments must be:

```
Op : (interpTy a -> interpTy b -> interpTy c) ->
Expr G a -> Expr G b -> Expr G c
```

Finally, if expressions make a choice given a boolean. Each branch must have the same type, and we will evaluate the branches lazily so that only the branch which is taken need be evaluated:

```
If : Expr G TyBool ->
Lazy (Expr G a) ->
Lazy (Expr G a) ->
Expr G a
```

#### Writing the Interpreter¶

When we evaluate an `Expr`

, we’ll need to know the values in scope,
as well as their types. `Env`

is an environment, indexed over the
types in scope. Since an environment is just another form of list,
albeit with a strongly specified connection to the vector of local
variable types, we use the usual `::`

and `Nil`

constructors so
that we can use the usual list syntax. Given a proof that a variable
is defined in the context, we can then produce a value from the
environment:

```
data Env : Vect n Ty -> Type where
Nil : Env Nil
(::) : interpTy a -> Env G -> Env (a :: G)
lookup : HasType i G t -> Env G -> interpTy t
lookup Stop (x :: xs) = x
lookup (Pop k) (x :: xs) = lookup k xs
```

Given this, an interpreter is a function which
translates an `Expr`

into a concrete Idris value with respect to a
specific environment:

```
interp : Env G -> Expr G t -> interpTy t
```

The complete interpreter is defined as follows, for reference. For each constructor, we translate it into the corresponding Idris value:

```
interp env (Var i) = lookup i env
interp env (Val x) = x
interp env (Lam sc) = \x => interp (x :: env) sc
interp env (App f s) = interp env f (interp env s)
interp env (Op op x y) = op (interp env x) (interp env y)
interp env (If x t e) = if interp env x then interp env t
else interp env e
```

Let us look at each case in turn. To translate a variable, we simply look it up in the environment:

```
interp env (Var i) = lookup i env
```

To translate a value, we just return the concrete representation of the value:

```
interp env (Val x) = x
```

Lambdas are more interesting. In this case, we construct a function which interprets the scope of the lambda with a new value in the environment. So, a function in the object language is translated to an Idris function:

```
interp env (Lam sc) = \x => interp (x :: env) sc
```

For an application, we interpret the function and its argument and apply
it directly. We know that interpreting `f`

must produce a function,
because of its type:

```
interp env (App f s) = interp env f (interp env s)
```

Operators and conditionals are, again, direct translations into the
equivalent Idris constructs. For operators, we apply the function to
its operands directly, and for `If`

, we apply the Idris
`if...then...else`

construct directly.

```
interp env (Op op x y) = op (interp env x) (interp env y)
interp env (If x t e) = if interp env x then interp env t
else interp env e
```

#### Testing¶

We can make some simple test functions. Firstly, adding two inputs
`\x. \y. y + x`

is written as follows:

```
add : Expr G (TyFun TyInt (TyFun TyInt TyInt))
add = Lam (Lam (Op (+) (Var Stop) (Var (Pop Stop))))
```

More interestingly, a factorial function `fact`

(e.g. `\x. if (x == 0) then 1 else (fact (x-1) * x)`

),
can be written as:

```
fact : Expr G (TyFun TyInt TyInt)
fact = Lam (If (Op (==) (Var Stop) (Val 0))
(Val 1)
(Op (*) (App fact (Op (-) (Var Stop) (Val 1)))
(Var Stop)))
```

#### Running¶

To finish, we write a `main`

program which interprets the factorial
function on user input:

```
main : IO ()
main = do putStr "Enter a number: "
x <- getLine
print (interp [] fact (cast x))
```

Here, `cast`

is an overloaded function which converts a value from
one type to another if possible. Here, it converts a string to an
integer, giving 0 if the input is invalid. An example run of this
program at the Idris interactive environment is:

```
$ idris interp.idr
____ __ _
/ _/___/ /____(_)____
/ // __ / ___/ / ___/ Version 0.9.17
_/ // /_/ / / / (__ ) http://www.idris-lang.org/
/___/\__,_/_/ /_/____/ Type :? for help
Type checking ./interp.idr
*interp> :exec
Enter a number: 6
720
*interp>
```

##### Aside: `cast`

¶

The prelude defines an interface `Cast`

which allows conversion
between types:

```
interface Cast from to where
cast : from -> to
```

It is a *multi-parameter* interface, defining the source type and
object type of the cast. It must be possible for the type checker to
infer *both* parameters at the point where the cast is applied. There
are casts defined between all of the primitive types, as far as they
make sense.

### Views and the “`with`

” rule¶

#### Dependent pattern matching¶

Since types can depend on values, the form of some arguments can be
determined by the value of others. For example, if we were to write
down the implicit length arguments to `(++)`

, we’d see that the form
of the length argument was determined by whether the vector was empty
or not:

```
(++) : Vect n a -> Vect m a -> Vect (n + m) a
(++) {n=Z} [] ys = ys
(++) {n=S k} (x :: xs) ys = x :: xs ++ ys
```

If `n`

was a successor in the `[]`

case, or zero in the `::`

case, the definition would not be well typed.

#### The `with`

rule — matching intermediate values¶

Very often, we need to match on the result of an intermediate
computation. Idris provides a construct for this, the `with`

rule, inspired by views in `Epigram`

[1], which takes account of
the fact that matching on a value in a dependently typed language can
affect what we know about the forms of other values. In its simplest
form, the `with`

rule adds another argument to the function being
defined, e.g. we have already seen a vector filter function, defined
as follows:

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

Here, the `with`

clause allows us to deconstruct the result of
`filter p xs`

. Effectively, it adds this value as an extra argument,
which we place after the vertical bar.

If the intermediate computation itself has a dependent type, then the
result can affect the forms of other arguments — we can learn the form
of one value by testing another. For example, a `Nat`

is either even
or odd. If it’s even it will be the sum of two equal `Nat`

.
Otherwise, it is the sum of two equal `Nat`

plus one:

```
data Parity : Nat -> Type where
Even : Parity (n + n)
Odd : Parity (S (n + n))
```

We say `Parity`

is a *view* of `Nat`

. It has a *covering function*
which tests whether it is even or odd and constructs the predicate
accordingly.

```
parity : (n:Nat) -> Parity n
```

We’ll come back to the definition of `parity`

shortly. We can use it
to write a function which converts a natural number to a list of
binary digits (least significant first) as follows, using the `with`

rule:

```
natToBin : Nat -> List Bool
natToBin Z = Nil
natToBin k with (parity k)
natToBin (j + j) | Even = False :: natToBin j
natToBin (S (j + j)) | Odd = True :: natToBin j
```

The value of the result of `parity k`

affects the form of `k`

,
because the result of `parity k`

depends on `k`

. So, as well as
the patterns for the result of the intermediate computation (`Even`

and `odd`

) right of the `|`

, we also write how the results
affect the other patterns left of the `|`

. Note that there is
a function in the patterns (`+`

) and repeated occurrences of
`j`

—this is allowed because another argument has determined the form
of these patterns.

We will return to this function in Section Provisional Definitions to
complete the definition of `parity`

.

[1] | Conor McBride and James McKinna. 2004. The view from the left. J. Funct. Program. 14, 1 (January 2004), 69-111. DOI=10.1017/S0956796803004829 http://dx.doi.org/10.1017/S0956796803004829ñ |

### Theorem Proving¶

#### Equality¶

Idris allows propositional equalities to be declared, allowing theorems about programs to be stated and proved. Equality is built in, but conceptually has the following definition:

```
data (=) : a -> b -> Type where
Refl : x = x
```

Equalities can be proposed between any values of any types, but the only way to construct a proof of equality is if values actually are equal. For example:

```
fiveIsFive : 5 = 5
fiveIsFive = Refl
twoPlusTwo : 2 + 2 = 4
twoPlusTwo = Refl
```

#### The Empty Type¶

There is an empty type, , which has no constructors. It is therefore impossible to construct an element of the empty type, at least without using a partially defined or general recursive function (see Section Totality Checking for more details). We can therefore use the empty type to prove that something is impossible, for example zero is never equal to a successor:

```
disjoint : (n : Nat) -> Z = S n -> Void
disjoint n p = replace {P = disjointTy} p ()
where
disjointTy : Nat -> Type
disjointTy Z = ()
disjointTy (S k) = Void
```

There is no need to worry too much about how this function works —
essentially, it applies the library function `replace`

, which uses an
equality proof to transform a predicate. Here we use it to transform a
value of a type which can exist, the empty tuple, to a value of a type
which can’t, by using a proof of something which can’t exist.

Once we have an element of the empty type, we can prove anything.
`void`

is defined in the library, to assist with proofs by
contradiction.

```
void : Void -> a
```

#### Simple Theorems¶

When type checking dependent types, the type itself gets *normalised*.
So imagine we want to prove the following theorem about the reduction
behaviour of `plus`

:

```
plusReduces : (n:Nat) -> plus Z n = n
```

We’ve written down the statement of the theorem as a type, in just the same way as we would write the type of a program. In fact there is no real distinction between proofs and programs. A proof, as far as we are concerned here, is merely a program with a precise enough type to guarantee a particular property of interest.

We won’t go into details here, but the Curry-Howard correspondence [1]
explains this relationship. The proof itself is trivial, because
`plus Z n`

normalises to `n`

by the definition of `plus`

:

```
plusReduces n = Refl
```

It is slightly harder if we try the arguments the other way, because
plus is defined by recursion on its first argument. The proof also works
by recursion on the first argument to `plus`

, namely `n`

.

```
plusReducesZ : (n:Nat) -> n = plus n Z
plusReducesZ Z = Refl
plusReducesZ (S k) = cong (plusReducesZ k)
```

`cong`

is a function defined in the library which states that equality
respects function application:

```
cong : {f : t -> u} -> a = b -> f a = f b
```

We can do the same for the reduction behaviour of plus on successors:

```
plusReducesS : (n:Nat) -> (m:Nat) -> S (plus n m) = plus n (S m)
plusReducesS Z m = Refl
plusReducesS (S k) m = cong (plusReducesS k m)
```

Even for trivial theorems like these, the proofs are a little tricky to construct in one go. When things get even slightly more complicated, it becomes too much to think about to construct proofs in this ‘batch mode’.

Idris provides interactive editing capabilities, which can help with building proofs. For more details on building proofs interactively in an editor, see Theorem Proving.

#### Totality Checking¶

If we really want to trust our proofs, it is important that they are
defined by *total* functions — that is, a function which is defined for
all possible inputs and is guaranteed to terminate. Otherwise we could
construct an element of the empty type, from which we could prove
anything:

```
-- making use of 'hd' being partially defined
empty1 : Void
empty1 = hd [] where
hd : List a -> a
hd (x :: xs) = x
-- not terminating
empty2 : Void
empty2 = empty2
```

Internally, Idris checks every definition for totality, and we can check at
the prompt with the `:total`

command. We see that neither of the above
definitions is total:

```
*theorems> :total empty1
possibly not total due to: empty1#hd
not total as there are missing cases
*theorems> :total empty2
possibly not total due to recursive path empty2
```

Note the use of the word “possibly” — a totality check can, of course, never be certain due to the undecidability of the halting problem. The check is, therefore, conservative. It is also possible (and indeed advisable, in the case of proofs) to mark functions as total so that it will be a compile time error for the totality check to fail:

```
total empty2 : Void
empty2 = empty2
```

```
Type checking ./theorems.idr
theorems.idr:25:empty2 is possibly not total due to recursive path empty2
```

Reassuringly, our proof in Section The Empty Type that the zero and successor constructors are disjoint is total:

```
*theorems> :total disjoint
Total
```

The totality check is, necessarily, conservative. To be recorded as
total, a function `f`

must:

- Cover all possible inputs
- Be
*well-founded*— i.e. by the time a sequence of (possibly mutually) recursive calls reaches`f`

again, it must be possible to show that one of its arguments has decreased. - Not use any data types which are not
*strictly positive* - Not call any non-total functions

##### Directives and Compiler Flags for Totality¶

By default, Idris allows all well-typed definitions, whether total or not. However, it is desirable for functions to be total as far as possible, as this provides a guarantee that they provide a result for all possible inputs, in finite time. It is possible to make total functions a requirement, either:

- By using the
`--total`

compiler flag. - By adding a
`%default total`

directive to a source file. All definitions after this will be required to be total, unless explicitly flagged as`partial`

.

All functions *after* a `%default total`

declaration are required to
be total. Correspondingly, after a `%default partial`

declaration, the
requirement is relaxed.

Finally, the compiler flag `--warnpartial`

causes to print a warning
for any undeclared partial function.

##### Totality checking issues¶

Please note that the totality checker is not perfect! Firstly, it is
necessarily conservative due to the undecidability of the halting
problem, so many programs which *are* total will not be detected as
such. Secondly, the current implementation has had limited effort put
into it so far, so there may still be cases where it believes a function
is total which is not. Do not rely on it for your proofs yet!

##### Hints for totality¶

In cases where you believe a program is total, but Idris does not agree, it is
possible to give hints to the checker to give more detail for a termination
argument. The checker works by ensuring that all chains of recursive calls
eventually lead to one of the arguments decreasing towards a base case, but
sometimes this is hard to spot. For example, the following definition cannot be
checked as `total`

because the checker cannot decide that `filter (<= x) xs`

will always be smaller than `(x :: xs)`

:

```
qsort : Ord a => List a -> List a
qsort [] = []
qsort (x :: xs)
= qsort (filter (< x) xs) ++
(x :: qsort (filter (>= x) xs))
```

The function `assert_smaller`

, defined in the Prelude, is intended to
address this problem:

```
assert_smaller : a -> a -> a
assert_smaller x y = y
```

It simply evaluates to its second argument, but also asserts to the
totality checker that `y`

is structurally smaller than `x`

. This can
be used to explain the reasoning for totality if the checker cannot work
it out itself. The above example can now be written as:

```
total
qsort : Ord a => List a -> List a
qsort [] = []
qsort (x :: xs)
= qsort (assert_smaller (x :: xs) (filter (< x) xs)) ++
(x :: qsort (assert_smaller (x :: xs) (filter (>= x) xs)))
```

The expression `assert_smaller (x :: xs) (filter (<= x) xs)`

asserts
that the result of the filter will always be smaller than the pattern
`(x :: xs)`

.

In more extreme cases, the function `assert_total`

marks a
subexpression as always being total:

```
assert_total : a -> a
assert_total x = x
```

In general, this function should be avoided, but it can be very useful when reasoning about primitives or externally defined functions (for example from a C library) where totality can be shown by an external argument.

[1] | Timothy G. Griffin. 1989. A formulae-as-type notion of control. In Proceedings of the 17th ACM SIGPLAN-SIGACT symposium on Principles of programming languages (POPL ‘90). ACM, New York, NY, USA, 47-58. DOI=10.1145/96709.96714 http://doi.acm.org/10.1145/96709.96714 |

### Provisional Definitions¶

Sometimes when programming with dependent types, the type required by
the type checker and the type of the program we have written will be
different (in that they do not have the same normal form), but
nevertheless provably equal. For example, recall the `parity`

function:

```
data Parity : Nat -> Type where
Even : Parity (n + n)
Odd : Parity (S (n + n))
```

We’d like to implement this as follows:

```
parity : (n:Nat) -> Parity n
parity Z = Even {n=Z}
parity (S Z) = Odd {n=Z}
parity (S (S k)) with (parity k)
parity (S (S (j + j))) | Even = Even {n=S j}
parity (S (S (S (j + j)))) | Odd = Odd {n=S j}
```

This simply states that zero is even, one is odd, and recursively, the
parity of `k+2`

is the same as the parity of `k`

. Explicitly marking
the value of `n`

is even and odd is necessary to help type inference.
Unfortunately, the type checker rejects this:

```
viewsbroken.idr:12:10:When elaborating right hand side of ViewsBroken.parity:
Type mismatch between
Parity (plus (S j) (S j))
and
Parity (S (S (plus j j)))
Specifically:
Type mismatch between
plus (S j) (S j)
and
S (S (plus j j))
```

The type checker is telling us that `(j+1)+(j+1)`

and `2+j+j`

do not
normalise to the same value. This is because `plus`

is defined by
recursion on its first argument, and in the second value, there is a
successor symbol on the second argument, so this will not help with
reduction. These values are obviously equal — how can we rewrite the
program to fix this problem?

#### Provisional definitions¶

*Provisional definitions* help with this problem by allowing us to defer
the proof details until a later point. There are two main reasons why
they are useful.

- When
*prototyping*, it is useful to be able to test programs before finishing all the details of proofs. - When
*reading*a program, it is often much clearer to defer the proof details so that they do not distract the reader from the underlying algorithm.

Provisional definitions are written in the same way as ordinary
definitions, except that they introduce the right hand side with a
`?=`

rather than `=`

. We define `parity`

as follows:

```
parity : (n:Nat) -> Parity n
parity Z = Even {n=Z}
parity (S Z) = Odd {n=Z}
parity (S (S k)) with (parity k)
parity (S (S (j + j))) | Even ?= Even {n=S j}
parity (S (S (S (j + j)))) | Odd ?= Odd {n=S j}
```

When written in this form, instead of reporting a type error, Idris will insert a hole standing for a theorem which will correct the type error. Idris tells us we have two proof obligations, with names generated from the module and function names:

```
*views> :m
Global holes:
[views.parity_lemma_2,views.parity_lemma_1]
```

The first of these has the following type:

```
*views> :p views.parity_lemma_1
---------------------------------- (views.parity_lemma_1) --------
{hole0} : (j : Nat) -> (Parity (plus (S j) (S j))) -> Parity (S (S (plus j j)))
-views.parity_lemma_1>
```

The two arguments are `j`

, the variable in scope from the pattern
match, and `value`

, which is the value we gave in the right hand side
of the provisional definition. Our goal is to rewrite the type so that
we can use this value. We can achieve this using the following theorem
from the prelude:

```
plusSuccRightSucc : (left : Nat) -> (right : Nat) ->
S (left + right) = left + (S right)
```

We need to use `compute`

again to unfold the definition of `plus`

:

```
-views.parity_lemma_1> compute
---------------------------------- (views.parity_lemma_1) --------
{hole0} : (j : Nat) -> (Parity (S (plus j (S j)))) -> Parity (S (S (plus j j)))
```

After applying `intros`

we have:

```
-views.parity_lemma_1> intros
j : Nat
value : Parity (S (plus j (S j)))
---------------------------------- (views.parity_lemma_1) --------
{hole2} : Parity (S (S (plus j j)))
```

Then we apply the `plusSuccRightSucc`

rewrite rule, symmetrically, to
`j`

and `j`

, giving:

```
-views.parity_lemma_1> rewrite sym (plusSuccRightSucc j j)
j : Nat
value : Parity (S (plus j (S j)))
---------------------------------- (views.parity_lemma_1) --------
{hole3} : Parity (S (plus j (S j)))
```

`sym`

is a function, defined in the library, which reverses the order
of the rewrite:

```
sym : l = r -> r = l
sym Refl = Refl
```

We can complete this proof using the `trivial`

tactic, which finds
`value`

in the premises. The proof of the second lemma proceeds in
exactly the same way.

We can now test the `natToBin`

function from Section The with rule — matching intermediate values
at the prompt. The number 42 is 101010 in binary. The binary digits are
reversed:

```
*views> show (natToBin 42)
"[False, True, False, True, False, True]" : String
```

#### Suspension of Disbelief¶

Idris requires that proofs be complete before compiling programs (although evaluation at the prompt is possible without proof details). Sometimes, especially when prototyping, it is easier not to have to do this. It might even be beneficial to test programs before attempting to prove things about them — if testing finds an error, you know you had better not waste your time proving something!

Therefore, Idris provides a built-in coercion function, which allows you to use a value of the incorrect types:

```
believe_me : a -> b
```

Obviously, this should be used with extreme caution. It is useful when
prototyping, and can also be appropriate when asserting properties of
external code (perhaps in an external C library). The “proof” of
`views.parity_lemma_1`

using this is:

```
views.parity_lemma_2 = proof {
intro;
intro;
exact believe_me value;
}
```

The `exact`

tactic allows us to provide an exact value for the proof.
In this case, we assert that the value we gave was correct.

#### Example: Binary numbers¶

Previously, we implemented conversion to binary numbers using the
`Parity`

view. Here, we show how to use the same view to implement a
verified conversion to binary. We begin by indexing binary numbers over
their `Nat`

equivalent. This is a common pattern, linking a
representation (in this case `Binary`

) with a meaning (in this case
`Nat`

):

```
data Binary : Nat -> Type where
BEnd : Binary Z
BO : Binary n -> Binary (n + n)
BI : Binary n -> Binary (S (n + n))
```

`BO`

and `BI`

take a binary number as an argument and effectively
shift it one bit left, adding either a zero or one as the new least
significant bit. The index, `n + n`

or `S (n + n)`

states the result
that this left shift then add will have to the meaning of the number.
This will result in a representation with the least significant bit at
the front.

Now a function which converts a Nat to binary will state, in the type, that the resulting binary number is a faithful representation of the original Nat:

```
natToBin : (n:Nat) -> Binary n
```

The `Parity`

view makes the definition fairly simple — halving the
number is effectively a right shift after all — although we need to use
a provisional definition in the Odd case:

```
natToBin : (n:Nat) -> Binary n
natToBin Z = BEnd
natToBin (S k) with (parity k)
natToBin (S (j + j)) | Even = BI (natToBin j)
natToBin (S (S (j + j))) | Odd ?= BO (natToBin (S j))
```

The problem with the Odd case is the same as in the definition of
`parity`

, and the proof proceeds in the same way:

```
natToBin_lemma_1 = proof {
intro;
intro;
rewrite sym (plusSuccRightSucc j j);
trivial;
}
```

To finish, we’ll implement a main program which reads an integer from the user and outputs it in binary.

```
main : IO ()
main = do putStr "Enter a number: "
x <- getLine
print (natToBin (fromInteger (cast x)))
```

For this to work, of course, we need a `Show`

implementation for
`Binary n`

:

```
Show (Binary n) where
show (BO x) = show x ++ "0"
show (BI x) = show x ++ "1"
show BEnd = ""
```

### Interactive Editing¶

By now, we have seen several examples of how Idris’ dependent type
system can give extra confidence in a function’s correctness by giving
a more precise description of its intended behaviour in its *type*. We
have also seen an example of how the type system can help with EDSL
development by allowing a programmer to describe the type system of an
object language. However, precise types give us more than verification
of programs — we can also exploit types to help write programs which
are *correct by construction*.

The Idris REPL provides several commands for inspecting and modifying parts of programs, based on their types, such as case splitting on a pattern variable, inspecting the type of a hole, and even a basic proof search mechanism. In this section, we explain how these features can be exploited by a text editor, and specifically how to do so in Vim. An interactive mode for Emacs is also available.

#### Editing at the REPL¶

The REPL provides a number of commands, which we will describe shortly, which generate new program fragments based on the currently loaded module. These take the general form

```
:command [line number] [name]
```

That is, each command acts on a specific source line, at a specific
name, and outputs a new program fragment. Each command has an
alternative form, which *updates* the source file in-place:

```
:command! [line number] [name]
```

When the REPL is loaded, it also starts a background process which
accepts and responds to REPL commands, using `idris --client`

. For
example, if we have a REPL running elsewhere, we can execute commands
such as:

```
$ idris --client ':t plus'
Prelude.Nat.plus : Nat -> Nat -> Nat
$ idris --client '2+2'
4 : Integer
```

A text editor can take advantage of this, along with the editing commands, in order to provide interactive editing support.

#### Editing Commands¶

##### :addclause¶

The `:addclause n f`

command (abbreviated `:ac n f`

) creates a
template definition for the function named `f`

declared on line
`n`

. For example, if the code beginning on line 94 contains:

```
vzipWith : (a -> b -> c) ->
Vect n a -> Vect n b -> Vect n c
```

then `:ac 94 vzipWith`

will give:

```
vzipWith f xs ys = ?vzipWith_rhs
```

The names are chosen according to hints which may be given by a programmer, and then made unique by the machine by adding a digit if necessary. Hints can be given as follows:

```
%name Vect xs, ys, zs, ws
```

This declares that any names generated for types in the `Vect`

family
should be chosen in the order `xs`

, `ys`

, `zs`

, `ws`

.

##### :casesplit¶

The `:casesplit n x`

command, abbreviated `:cs n x`

, splits the
pattern variable `x`

on line `n`

into the various pattern forms it
may take, removing any cases which are impossible due to unification
errors. For example, if the code beginning on line 94 is:

```
vzipWith : (a -> b -> c) ->
Vect n a -> Vect n b -> Vect n c
vzipWith f xs ys = ?vzipWith_rhs
```

then `:cs 96 xs`

will give:

```
vzipWith f [] ys = ?vzipWith_rhs_1
vzipWith f (x :: xs) ys = ?vzipWith_rhs_2
```

That is, the pattern variable `xs`

has been split into the two
possible cases `[]`

and `x :: xs`

. Again, the names are chosen
according to the same heuristic. If we update the file (using
`:cs!`

) then case split on `ys`

on the same line, we get:

```
vzipWith f [] [] = ?vzipWith_rhs_3
```

That is, the pattern variable `ys`

has been split into one case
`[]`

, Idris having noticed that the other possible case ```
y ::
ys
```

would lead to a unification error.

##### :addmissing¶

The `:addmissing n f`

command, abbreviated `:am n f`

, adds the
clauses which are required to make the function `f`

on line `n`

cover all inputs. For example, if the code beginning on line 94 is

```
vzipWith : (a -> b -> c) ->
Vect n a -> Vect n b -> Vect n c
vzipWith f [] [] = ?vzipWith_rhs_1
```

then `:am 96 vzipWith`

gives:

```
vzipWith f (x :: xs) (y :: ys) = ?vzipWith_rhs_2
```

That is, it notices that there are no cases for non-empty vectors, generates the required clauses, and eliminates the clauses which would lead to unification errors.

##### :proofsearch¶

The `:proofsearch n f`

command, abbreviated `:ps n f`

, attempts to
find a value for the hole `f`

on line `n`

by proof search,
trying values of local variables, recursive calls and constructors of
the required family. Optionally, it can take a list of *hints*, which
are functions it can try applying to solve the hole. For
example, if the code beginning on line 94 is:

```
vzipWith : (a -> b -> c) ->
Vect n a -> Vect n b -> Vect n c
vzipWith f [] [] = ?vzipWith_rhs_1
vzipWith f (x :: xs) (y :: ys) = ?vzipWith_rhs_2
```

then `:ps 96 vzipWith_rhs_1`

will give

```
[]
```

This works because it is searching for a `Vect`

of length 0, of
which the empty vector is the only possibility. Similarly, and perhaps
surprisingly, there is only one possibility if we try to solve ```
:ps
97 vzipWith_rhs_2
```

:

```
f x y :: (vzipWith f xs ys)
```

This works because `vzipWith`

has a precise enough type: The
resulting vector has to be non-empty (a `::`

); the first element
must have type `c`

and the only way to get this is to apply `f`

to
`x`

and `y`

; finally, the tail of the vector can only be built
recursively.

##### :makewith¶

The `:makewith n f`

command, abbreviated `:mw n f`

, adds a
`with`

to a pattern clause. For example, recall `parity`

. If line
10 is:

```
parity (S k) = ?parity_rhs
```

then `:mw 10 parity`

will give:

```
parity (S k) with (_)
parity (S k) | with_pat = ?parity_rhs
```

If we then fill in the placeholder `_`

with `parity k`

and case
split on `with_pat`

using `:cs 11 with_pat`

we get the following
patterns:

```
parity (S (plus n n)) | even = ?parity_rhs_1
parity (S (S (plus n n))) | odd = ?parity_rhs_2
```

Note that case splitting has normalised the patterns here (giving
`plus`

rather than `+`

). In any case, we see that using
interactive editing significantly simplifies the implementation of
dependent pattern matching by showing a programmer exactly what the
valid patterns are.

#### Interactive Editing in Vim¶

The editor mode for Vim provides syntax highlighting, indentation and interactive editing support using the commands described above. Interactive editing is achieved using the following editor commands, each of which update the buffer directly:

`\d`

adds a template definition for the name declared on thecurrent line (using

`:addclause`

).

`\c`

case splits the variable at the cursor (using`:casesplit`

).

`\m`

adds the missing cases for the name at the cursor (using`:addmissing`

).

`\w`

adds a`with`

clause (using`:makewith`

).`\o`

invokes a proof search to solve the hole under thecursor (using

`:proofsearch`

).

`\p`

invokes a proof search with additional hints to solve thehole under the cursor (using

`:proofsearch`

).

There are also commands to invoke the type checker and evaluator:

`\t`

displays the type of the (globally visible) name under thecursor. In the case of a hole, this displays the context and the expected type.

`\e`

prompts for an expression to evaluate.`\r`

reloads and type checks the buffer.

Corresponding commands are also available in the Emacs mode. Support
for other editors can be added in a relatively straightforward manner
by using `idris –client`

.

### Syntax Extensions¶

Idris supports the implementation of *Embedded Domain Specific
Languages* (EDSLs) in several ways [1]. One way, as we have already
seen, is through extending `do`

notation. Another important way is
to allow extension of the core syntax. In this section we describe two
ways of extending the syntax: `syntax`

rules and `dsl`

notation.

`syntax`

rules¶

We have seen `if...then...else`

expressions, but these are not built
in. Instead, we can define a function in the prelude as follows (we
have already seen this function in Section Laziness):

```
ifThenElse : (x:Bool) -> Lazy a -> Lazy a -> a;
ifThenElse True t e = t;
ifThenElse False t e = e;
```

and then extend the core syntax with a `syntax`

declaration:

```
syntax if [test] then [t] else [e] = ifThenElse test t e;
```

The left hand side of a `syntax`

declaration describes the syntax
rule, and the right hand side describes its expansion. The syntax rule
itself consists of:

**Keywords**— here,`if`

,`then`

and`else`

, which must be valid identifiers**Non-terminals**— included in square brackets,`[test]`

,`[t]`

and`[e]`

here, which stand for arbitrary expressions. To avoid parsing ambiguities, these expressions cannot use syntax extensions at the top level (though they can be used in parentheses).**Names**— included in braces, which stand for names which may be bound on the right hand side.**Symbols**— included in quotations marks, e.g.`:=`

. This can also be used to include reserved words in syntax rules, such as`let`

or`in`

.

The limitations on the form of a syntax rule are that it must include at least one symbol or keyword, and there must be no repeated variables standing for non-terminals. Any expression can be used, but if there are two non-terminals in a row in a rule, only simple expressions may be used (that is, variables, constants, or bracketed expressions). Rules can use previously defined rules, but may not be recursive. The following syntax extensions would therefore be valid:

```
syntax [var] ":=" [val] = Assign var val;
syntax [test] "?" [t] ":" [e] = if test then t else e;
syntax select [x] from [t] "where" [w] = SelectWhere x t w;
syntax select [x] from [t] = Select x t;
```

Syntax macros can be further restricted to apply only in patterns (i.e.,
only on the left hand side of a pattern match clause) or only in terms
(i.e. everywhere but the left hand side of a pattern match clause) by
being marked as `pattern`

or `term`

syntax rules. For example, we
might define an interval as follows, with a static check that the lower
bound is below the upper bound using `so`

:

```
data Interval : Type where
MkInterval : (lower : Float) -> (upper : Float) ->
so (lower < upper) -> Interval
```

We can define a syntax which, in patterns, always matches `oh`

for
the proof argument, and in terms requires a proof term to be provided:

```
pattern syntax "[" [x] "..." [y] "]" = MkInterval x y oh
term syntax "[" [x] "..." [y] "]" = MkInterval x y ?bounds_lemma
```

In terms, the syntax `[x...y]`

will generate a proof obligation
`bounds_lemma`

(possibly renamed).

Finally, syntax rules may be used to introduce alternative binding
forms. For example, a `for`

loop binds a variable on each iteration:

```
syntax for {x} in [xs] ":" [body] = forLoop xs (\x => body)
main : IO ()
main = do for x in [1..10]:
putStrLn ("Number " ++ show x)
putStrLn "Done!"
```

Note that we have used the `{x}`

form to state that `x`

represents
a bound variable, substituted on the right hand side. We have also put
`in`

in quotation marks since it is already a reserved word.

`dsl`

notation¶

The well-typed interpreter in Section Example: The Well-Typed Interpreter is a simple
example of a common programming pattern with dependent types. Namely:
describe an *object language* and its type system with dependent types
to guarantee that only well-typed programs can be represented, then
program using that representation. Using this approach we can, for
example, write programs for serialising binary data [2] or running
concurrent processes safely [3].

Unfortunately, the form of object language programs makes it rather
hard to program this way in practice. Recall the factorial program in
`Expr`

for example:

```
fact : Expr G (TyFun TyInt TyInt)
fact = Lam (If (Op (==) (Var Stop) (Val 0))
(Val 1) (Op (*) (App fact (Op (-) (Var Stop) (Val 1)))
(Var Stop)))
```

Since this is a particularly useful pattern, Idris provides syntax overloading [1] to make it easier to program in such object languages:

```
mkLam : TTName -> Expr (t::g) t' -> Expr g (TyFun t t')
mkLam _ body = Lam body
dsl expr
variable = Var
index_first = Stop
index_next = Pop
lambda = mkLam
```

A `dsl`

block describes how each syntactic construct is represented
in an object language. Here, in the `expr`

language, any variable is
translated to the `Var`

constructor, using `Pop`

and `Stop`

to
construct the de Bruijn index (i.e., to count how many bindings since
the variable itself was bound); and any lambda is translated to a
`Lam`

constructor. The `mkLam`

function simply ignores its first
argument, which is the name that the user chose for the variable. It
is also possible to overload `let`

and dependent function syntax
(`pi`

) in this way. We can now write `fact`

as follows:

```
fact : Expr G (TyFun TyInt TyInt)
fact = expr (\x => If (Op (==) x (Val 0))
(Val 1) (Op (*) (app fact (Op (-) x (Val 1))) x))
```

In this new version, `expr`

declares that the next expression will
be overloaded. We can take this further, using idiom brackets, by
declaring:

```
(<*>) : (f : Lazy (Expr G (TyFun a t))) -> Expr G a -> Expr G t
(<*>) f a = App f a
pure : Expr G a -> Expr G a
pure = id
```

Note that there is no need for these to be part of an implementation of
`Applicative`

, since idiom bracket notation translates directly to
the names `<*>`

and `pure`

, and ad-hoc type-directed overloading
is allowed. We can now say:

```
fact : Expr G (TyFun TyInt TyInt)
fact = expr (\x => If (Op (==) x (Val 0))
(Val 1) (Op (*) [| fact (Op (-) x (Val 1)) |] x))
```

With some more ad-hoc overloading and use of interfaces, and a new syntax rule, we can even go as far as:

```
syntax "IF" [x] "THEN" [t] "ELSE" [e] = If x t e
fact : Expr G (TyFun TyInt TyInt)
fact = expr (\x => IF x == 0 THEN 1 ELSE [| fact (x - 1) |] * x)
```

[1] | (1, 2) Edwin Brady and Kevin Hammond. 2012. Resource-Safe systems
programming with embedded domain specific languages. In
Proceedings of the 14th international conference on Practical
Aspects of Declarative Languages (PADL‘12), Claudio Russo and
Neng-Fa Zhou (Eds.). Springer-Verlag, Berlin, Heidelberg,
242-257. DOI=10.1007/978-3-642-27694-1_18
http://dx.doi.org/10.1007/978-3-642-27694-1_18 |

[2] | Edwin C. Brady. 2011. IDRIS —: systems programming meets full dependent types. In Proceedings of the 5th ACM workshop on Programming languages meets program verification (PLPV ‘11). ACM, New York, NY, USA, 43-54. DOI=10.1145/1929529.1929536 http://doi.acm.org/10.1145/1929529.1929536 |

[3] | Edwin Brady and Kevin Hammond. 2010. Correct-by-Construction Concurrency: Using Dependent Types to Verify Implementations of Effectful Resource Usage Protocols. Fundam. Inf. 102, 2 (April 2010), 145-176. http://dl.acm.org/citation.cfm?id=1883636 |

### Miscellany¶

In this section we discuss a variety of additional features:

- auto, implicit, and default arguments;
- literate programming;
- interfacing with external libraries through the foreign function
- interface;
- type providers;
- code generation; and
- the universe hierarchy.

#### Auto implicit arguments¶

We have already seen implicit arguments, which allows arguments to be omitted when they can be inferred by the type checker, e.g.

```
index : {a:Type} -> {n:Nat} -> Fin n -> Vect n a -> a
```

In other situations, it may be possible to infer arguments not by type
checking but by searching the context for an appropriate value, or
constructing a proof. For example, the following definition of `head`

which requires a proof that the list is non-empty:

```
isCons : List a -> Bool
isCons [] = False
isCons (x :: xs) = True
head : (xs : List a) -> (isCons xs = True) -> a
head (x :: xs) _ = x
```

If the list is statically known to be non-empty, either because its
value is known or because a proof already exists in the context, the
proof can be constructed automatically. Auto implicit arguments allow
this to happen silently. We define `head`

as follows:

```
head : (xs : List a) -> {auto p : isCons xs = True} -> a
head (x :: xs) = x
```

The `auto`

annotation on the implicit argument means that Idris
will attempt to fill in the implicit argument by searching for a value
of the appropriate type. It will try the following, in order:

- Local variables, i.e. names bound in pattern matches or
`let`

bindings, with exactly the right type. - The constructors of the required type. If they have arguments, it will search recursively up to a maximum depth of 100.
- Local variables with function types, searching recursively for the arguments.
- Any function with the appropriate return type which is marked with the
`%hint`

annotation.

In the case that a proof is not found, it can be provided explicitly as normal:

```
head xs {p = ?headProof}
```

More generally, we can fill in implicit arguments with a default value
by annotating them with `default`

. The definition above is equivalent
to:

```
head : (xs : List a) ->
{default proof { trivial; } p : isCons xs = True} -> a
head (x :: xs) = x
```

#### Implicit conversions¶

Idris supports the creation of *implicit conversions*, which allow
automatic conversion of values from one type to another when required to
make a term type correct. This is intended to increase convenience and
reduce verbosity. A contrived but simple example is the following:

```
implicit intString : Int -> String
intString = show
test : Int -> String
test x = "Number " ++ x
```

In general, we cannot append an `Int`

to a `String`

, but the
implicit conversion function `intString`

can convert `x`

to a
`String`

, so the definition of `test`

is type correct. An implicit
conversion is implemented just like any other function, but given the
`implicit`

modifier, and restricted to one explicit argument.

Only one implicit conversion will be applied at a time. That is, implicit conversions cannot be chained. Implicit conversions of simple types, as above, are however discouraged! More commonly, an implicit conversion would be used to reduce verbosity in an embedded domain specific language, or to hide details of a proof. Such examples are beyond the scope of this tutorial.

#### Literate programming¶

Like Haskell, Idris supports *literate* programming. If a file has
an extension of `.lidr`

then it is assumed to be a literate file. In
literate programs, everything is assumed to be a comment unless the line
begins with a greater than sign `>`

, for example:

```
> module literate
This is a comment. The main program is below
> main : IO ()
> main = putStrLn "Hello literate world!\n"
```

An additional restriction is that there must be a blank line between a
program line (beginning with `>`

) and a comment line (beginning with
any other character).

#### Foreign function calls¶

For practical programming, it is often necessary to be able to use
external libraries, particularly for interfacing with the operating
system, file system, networking, *et cetera*. Idris provides a
lightweight foreign function interface for achieving this, as part of
the prelude. For this, we assume a certain amount of knowledge of C and
the `gcc`

compiler. First, we define a datatype which describes the
external types we can handle:

```
data FTy = FInt | FFloat | FChar | FString | FPtr | FUnit
```

Each of these corresponds directly to a C type. Respectively: `int`

,
`double`

, `char`

, `char*`

, `void*`

and `void`

. There is also a
translation to a concrete Idris type, described by the following
function:

```
interpFTy : FTy -> Type
interpFTy FInt = Int
interpFTy FFloat = Float
interpFTy FChar = Char
interpFTy FString = String
interpFTy FPtr = Ptr
interpFTy FUnit = ()
```

A foreign function is described by a list of input types and a return type, which can then be converted to an Idris type:

```
ForeignTy : (xs:List FTy) -> (t:FTy) -> Type
```

A foreign function is assumed to be impure, so `ForeignTy`

builds an
`IO`

type, for example:

```
Idris> ForeignTy [FInt, FString] FString
Int -> String -> IO String : Type
Idris> ForeignTy [FInt, FString] FUnit
Int -> String -> IO () : Type
```

We build a call to a foreign function by giving the name of the
function, a list of argument types and the return type. The built in
construct `mkForeign`

converts this description to a function callable
by Idris:

```
data Foreign : Type -> Type where
FFun : String -> (xs:List FTy) -> (t:FTy) ->
Foreign (ForeignTy xs t)
mkForeign : Foreign x -> x
```

Note that the compiler expects `mkForeign`

to be fully applied to
build a complete foreign function call. For example, the `putStr`

function is implemented as follows, as a call to an external function
`putStr`

defined in the run-time system:

```
putStr : String -> IO ()
putStr x = mkForeign (FFun "putStr" [FString] FUnit) x
```

##### Include and linker directives¶

Foreign function calls are translated directly to calls to C functions, with appropriate conversion between the Idris representation of a value and the C representation. Often this will require extra libraries to be linked in, or extra header and object files. This is made possible through the following directives:

`%lib target x`

— include the`libx`

library. If the target is`C`

this is equivalent to passing the`-lx`

option to`gcc`

. If the target is Java the library will be interpreted as a`groupId:artifactId:packaging:version`

dependency coordinate for maven.`%include target x`

— use the header file or import`x`

for the given back end target.`%link target x.o`

— link with the object file`x.o`

when using the given back end target.`%dynamic x.so`

— dynamically link the interpreter with the shared object`x.so`

.

##### Testing foreign function calls¶

Normally, the Idris interpreter (used for typechecking and at the REPL)
will not perform IO actions. Additionally, as it neither generates C
code nor compiles to machine code, the `%lib`

, `%include`

and
`%link`

directives have no effect. IO actions and FFI calls can be
tested using the special REPL command `:x EXPR`

, and C libraries can
be dynamically loaded in the interpreter by using the `:dynamic`

command or the `%dynamic`

directive. For example:

```
Idris> :dynamic libm.so
Idris> :x unsafePerformIO ((mkForeign (FFun "sin" [FFloat] FFloat)) 1.6)
0.9995736030415051 : Float
```

#### Type Providers¶

Idris type providers, inspired by F#’s type providers, are a means of making our types be “about” something in the world outside of Idris. For example, given a type that represents a database schema and a query that is checked against it, a type provider could read the schema of a real database during type checking.

Idris type providers use the ordinary execution semantics of Idris to run an IO action and extract the result. This result is then saved as a constant in the compiled code. It can be a type, in which case it is used like any other type, or it can be a value, in which case it can be used as any other value, including as an index in types.

Type providers are still an experimental extension. To enable the
extension, use the `%language`

directive:

```
%language TypeProviders
```

A provider `p`

for some type `t`

is simply an expression of type
`IO (Provider t)`

. The `%provide`

directive causes the type checker
to execute the action and bind the result to a name. This is perhaps
best illustrated with a simple example. The type provider `fromFile`

reads a text file. If the file consists of the string `Int`

, then the
type `Int`

will be provided. Otherwise, it will provide the type
`Nat`

.

```
strToType : String -> Type
strToType "Int" = Int
strToType _ = Nat
fromFile : String -> IO (Provider Type)
fromFile fname = do str <- readFile fname
return (Provide (strToType (trim str)))
```

We then use the `%provide`

directive:

```
%provide (T1 : Type) with fromFile "theType"
foo : T1
foo = 2
```

If the file named `theType`

consists of the word `Int`

, then `foo`

will be an `Int`

. Otherwise, it will be a `Nat`

. When Idris
encounters the directive, it first checks that the provider expression
`fromFile theType`

has type `IO (Provider Type)`

. Next, it executes
the provider. If the result is `Provide t`

, then `T1`

is defined as
`t`

. Otherwise, the result is an error.

Our datatype `Provider t`

has the following definition:

```
data Provider a = Error String
| Provide a
```

We have already seen the `Provide`

constructor. The `Error`

constructor allows type providers to return useful error messages. The
example in this section was purposefully simple. More complex type
provider implementations, including a statically-checked SQLite binding,
are available in an external collection [1].

#### C Target¶

The default target of Idris is C. Compiling via :

```
$ idris hello.idr -o hello
```

is equivalent to :

```
$ idris --codegen C hello.idr -o hello
```

When the command above is used, a temporary C source is generated, which
is then compiled into an executable named `hello`

.

In order to view the generated C code, compile via :

```
$ idris hello.idr -S -o hello.c
```

To turn optimisations on, use the `%flag C`

pragma within the code, as
is shown below :

```
module Main
%flag C "-O3"
factorial : Int -> Int
factorial 0 = 1
factorial n = n * (factorial (n-1))
main : IO ()
main = do
putStrLn $ show $ factorial 3
```

#### JavaScript Target¶

Idris is capable of producing *JavaScript* code that can be run in a
browser as well as in the *NodeJS* environment or alike. One can use the
FFI to communicate with the *JavaScript* ecosystem.

##### Code Generation¶

Code generation is split into two separate targets. To generate code that is tailored for running in the browser issue the following command:

```
$ idris --codegen javascript hello.idr -o hello.js
```

The resulting file can be embedded into your HTML just like any other
*JavaScript* code.

Generating code for *NodeJS* is slightly different. Idris outputs a
*JavaScript* file that can be directly executed via `node`

.

```
$ idris --codegen node hello.idr -o hello
$ ./hello
Hello world
```

Take into consideration that the *JavaScript* code generator is using
`console.log`

to write text to `stdout`

, this means that it will
automatically add a newline to the end of each string. This behaviour
does not show up in the *NodeJS* code generator.

##### Using the FFI¶

To write a useful application we need to communicate with the outside
world. Maybe we want to manipulate the DOM or send an Ajax request. For
this task we can use the FFI. Since most *JavaScript* APIs demand
callbacks we need to extend the FFI so we can pass functions as
arguments.

The *JavaScript* FFI works a little bit differently than the regular
FFI. It uses positional arguments to directly insert our arguments into
a piece of *JavaScript* code.

One could use the primitive addition of *JavaScript* like so:

```
module Main
primPlus : Int -> Int -> IO Int
primPlus a b = mkForeign (FFun "%0 + %1" [FInt, FInt] FInt) a b
main : IO ()
main = do
a <- primPlus 1 1
b <- primPlus 1 2
print (a, b)
```

Notice that the `%n`

notation qualifies the position of the `n`

-th
argument given to our foreign function starting from 0. When you need a
percent sign rather than a position simply use `%%`

instead.

Passing functions to a foreign function is very similar. Let’s assume
that we want to call the following function from the *JavaScript* world:

```
function twice(f, x) {
return f(f(x));
}
```

We obviously need to pass a function `f`

here (we can infer it from
the way we use `f`

in `twice`

, it would be more obvious if
*JavaScript* had types).

The *JavaScript* FFI is able to understand functions as arguments when
you give it something of type `FFunction`

. The following example code
calls `twice`

in *JavaScript* and returns the result to our Idris
program:

```
module Main
twice : (Int -> Int) -> Int -> IO Int
twice f x = mkForeign (
FFun "twice(%0,%1)" [FFunction FInt FInt, FInt] FInt
) f x
main : IO ()
main = do
a <- twice (+1) 1
print a
```

The program outputs `3`

, just like we expected.

##### Including external *JavaScript* files¶

Whenever one is working with *JavaScript* one might want to include
external libraries or just some functions that she or he wants to call
via FFI which are stored in external files. The *JavaScript* and
*NodeJS* code generators understand the `%include`

directive. Keep in
mind that *JavaScript* and *NodeJS* are handled as different code
generators, therefore you will have to state which one you want to
target. This means that you can include different files for *JavaScript*
and *NodeJS* in the same Idris source file.

So whenever you want to add an external *JavaScript* file you can do
this like so:

For *NodeJS*:

```
%include Node "path/to/external.js"
```

And for use in the browser:

```
%include JavaScript "path/to/external.js"
```

The given files will be added to the top of the generated code.

##### Including *NodeJS* modules¶

The *NodeJS* code generator can also include modules with the `%lib`

directive.

```
%lib Node "fs"
```

This directive compiles into the following *JavaScript*

```
var fs = require("fs");
```

##### Shrinking down generated *JavaScript*¶

Idris can produce very big chunks of *JavaScript* code. However, the
generated code can be minified using the `closure-compiler`

from
Google. Any other minifier is also suitable but `closure-compiler`

offers advanced compilation that does some aggressive inlining and code
elimination. Idris can take full advantage of this compilation mode
and it’s highly recommended to use it when shipping a *JavaScript*
application written in Idris.

#### Cumulativity¶

Since values can appear in types and *vice versa*, it is natural that
types themselves have types. For example:

```
*universe> :t Nat
Nat : Type
*universe> :t Vect
Vect : Nat -> Type -> Type
```

But what about the type of `Type`

? If we ask Idris it reports

```
*universe> :t Type
Type : Type 1
```

If `Type`

were its own type, it would lead to an inconsistency due to
Girard’s paradox , so internally there is a
*hierarchy* of types (or *universes*):

```
Type : Type 1 : Type 2 : Type 3 : ...
```

Universes are *cumulative*, that is, if `x : Type n`

we can also have
that `x : Type m`

, as long as `n < m`

. The typechecker generates
such universe constraints and reports an error if any inconsistencies
are found. Ordinarily, a programmer does not need to worry about this,
but it does prevent (contrived) programs such as the following:

```
myid : (a : Type) -> a -> a
myid _ x = x
idid : (a : Type) -> a -> a
idid = myid _ myid
```

The application of `myid`

to itself leads to a cycle in the universe
hierarchy — `myid`

’s first argument is a `Type`

, which cannot be
at a lower level than required if it is applied to itself.

[1] | https://github.com/david-christiansen/idris-type-providers |

### Further Reading¶

Further information about Idris programming, and programming with dependent types in general, can be obtained from various sources:

The Idris web site (http://www.idris-lang.org/) and by asking questions on the mailing list.

The IRC channel

`#idris`

, on chat.freenode.net.- The wiki (https://github.com/idris-lang/Idris-dev/wiki/) has further
user provided information, in particular:

- Examining the prelude and exploring the
`samples`

in the distribution. The Idris source can be found online at: https://github.com/idris-lang/Idris-dev.

- Examining the prelude and exploring the
Existing projects on the

`Idris Hackers`

web space: http://idris-hackers.github.io.

[1] | Edwin Brady and Kevin Hammond. 2012. Resource-Safe systems programming with embedded domain specific languages. In Proceedings of the 14th international conference on Practical Aspects of Declarative Languages (PADL‘12), Claudio Russo and Neng-Fa Zhou (Eds.). Springer-Verlag, Berlin, Heidelberg, 242-257. DOI=10.1007/978-3-642-27694-1_18 http://dx.doi.org/10.1007/978-3-642-27694-1_18 |

[2] | Edwin C. Brady. 2011. IDRIS —: systems programming meets full dependent types. In Proceedings of the 5th ACM workshop on Programming languages meets program verification (PLPV ‘11). ACM, New York, NY, USA, 43-54. DOI=10.1145/1929529.1929536 http://doi.acm.org/10.1145/1929529.1929536 |

[3] | Edwin C. Brady and Kevin Hammond. 2010. Scrapping your inefficient engine: using partial evaluation to improve domain-specific language implementation. In Proceedings of the 15th ACM SIGPLAN international conference on Functional programming (ICFP ‘10). ACM, New York, NY, USA, 297-308. DOI=10.1145/1863543.1863587 http://doi.acm.org/10.1145/1863543.1863587 |

## Frequently Asked Questions¶

### What are the differences between Agda and Idris?¶

Like Idris, Agda is a functional language with dependent types, supporting dependent pattern matching. Both can be used for writing programs and proofs. However, Idris has been designed from the start to emphasise general purpose programming rather than theorem proving. As such, it supports interoperability with systems libraries and C programs, and language constructs for domain specific language implementation. It also includes higher level programming constructs such as interfaces (similar to type classes) and do notation.

Idris supports multiple back ends (C and JavaScript by default, with the ability to add more via plugins) and has a reference run time system, written in C, with a garbage collector and built-in message passing concurrency.

### Is Idris production ready?¶

Idris is primarily a research tool for exploring the possibilities of software development with dependent types, meaning that the primary goal is not (yet) to make a system which could be used in production. As such, there are a few rough corners, and lots of missing libraries. Nobody is working on Idris full time, and we don’t have the resources at the moment to polish the system on our own. Therefore, we don’t recommend building your business around it!

Having said that, contributions which help towards making Idris suitable for use in production would be very welcome - this includes (but is not limited to) extra library support, polishing the run-time system (and ensuring it is robust), providing and maintaining a JVM back end, etc.

### Why does Idris use eager evaluation rather than lazy?¶

Idris uses eager evaluation for more predictable performance, in particular
because one of the longer term goals is to be able to write efficient and
verified low level code such as device drivers and network infrastructure.
Furthermore, the Idris type system allows us to state precisely the type
of each value, and therefore the run-time form of each value. In a lazy
language, consider a value of type `Int`

:

```
thing : Int
```

What is the representation of `thing`

at run-time? Is it a bit pattern
representing an integer, or is it a pointer to some code which will compute
an integer? In Idris, we have decided that we would like to make this
distinction precise, in the type:

```
thing_val : Int
thing_comp : Lazy Int
```

Here, it is clear from the type that `thing_val`

is guaranteed to be a
concrete `Int`

, whereas `thing_comp`

is a computation which will produce an
`Int`

.

### How can I make lazy control structures?¶

You can make control structures using the special Lazy type. For
example, `if...then...else...`

in Idris expands to an application of
a function named `ifThenElse`

. The default implementation for
Booleans is defined as follows in the library:

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

The type `Lazy a`

for `t`

and `e`

indicates that those arguments will
only be evaluated if they are used, that is, they are evaluated lazily.

### Evaluation at the REPL doesn’t behave as I expect. What’s going on?¶

Being a fully dependently typed language, Idris has two phases where it evaluates things, compile-time and run-time. At compile-time it will only evaluate things which it knows to be total (i.e. terminating and covering all possible inputs) in order to keep type checking decidable. The compile-time evaluator is part of the Idris kernel, and is implemented in Haskell using a HOAS (higher order abstract syntax) style representation of values. Since everything is known to have a normal form here, the evaluation strategy doesn’t actually matter because either way it will get the same answer, and in practice it will do whatever the Haskell run-time system chooses to do.

The REPL, for convenience, uses the compile-time notion of evaluation. As well as being easier to implement (because we have the evaluator available) this can be very useful to show how terms evaluate in the type checker. So you can see the difference between:

```
Idris> \n, m => (S n) + m
\n => \m => S (plus n m) : Nat -> Nat -> Nat
Idris> \n, m => n + (S m)
\n => \m => plus n (S m) : Nat -> Nat -> Nat
```

### Why can’t I use a function with no arguments in a type?¶

If you use a name in a type which begins with a lower case letter, and which is not applied to any arguments, then Idris will treat it as an implicitly bound argument. For example:

```
append : Vect n ty -> Vect m ty -> Vect (n + m) ty
```

Here, `n`

, `m`

, and `ty`

are implicitly bound. This rule applies even
if there are functions defined elsewhere with any of these names. For example,
you may also have:

```
ty : Type
ty = String
```

Even in this case, `ty`

is still considered implicitly bound in the definition
of `append`

, rather than making the type of `append`

equivalent to...

```
append : Vect n String -> Vect m String -> Vect (n + m) String
```

...which is probably not what was intended! The reason for this rule is so
that it is clear just from looking at the type of `append`

, and no other
context, what the implicitly bound names are.

If you want to use an unapplied name in a type, you have two options. You
can either explicitly qualify it, for example, if `ty`

is defined in the
namespace `Main`

you can do the following:

```
append : Vect n Main.ty -> Vect m Main.ty -> Vect (n + m) Main.ty
```

Alternatively, you can use a name which does not begin with a lower case letter, which will never be implicitly bound:

```
Ty : Type
Ty = String
append : Vect n Ty -> Vect m Ty -> Vect (n + m) Ty
```

As a convention, if a name is intended to be used as a type synonym, it is best for it to begin with a capital letter to avoid this restriction.

### I have an obviously terminating program, but Idris says it possibly isn’t total. Why is that?¶

Idris can’t decide in general whether a program is terminating due to the undecidability of the Halting Problem. It is possible, however, to identify some programs which are definitely terminating. Idris does this using “size change termination” which looks for recursive paths from a function back to itself. On such a path, there must be at least one argument which converges to a base case.

- Mutually recursive functions are supported
- However, all functions on the path must be fully applied. In particular, higher order applications are not supported
- Idris identifies arguments which converge to a base case by looking for
recursive calls to syntactically smaller arguments of inputs. e.g.
`k`

is syntactically smaller than`S (S k)`

because`k`

is a subterm of`S (S k)`

, but`(k, k)`

is not syntactically smaller than`(S k, S k)`

.

If you have a function which you believe to be terminating, but Idris does
not, you can either restructure the program, or use the `assert_total`

function.

### When will Idris be self-hosting?¶

It’s not a priority, though not a bad idea in the long run. It would be a worthwhile effort in the short term to implement libraries to support self-hosting, such as a good parsing library.

### Does Idris have Universe Polymorphism? What is the type of `Type`

?¶

Rather than Universe polymorphism, Idris has a cumulative hierarchy of
universes; `Type : Type 1`

, `Type 1 : Type 2`

, etc.
Cumulativity means that if `x : Type n`

then also `x : Type m`

,
provided that `n <= m`

.

### Why does Idris use `Float`

and `Double`

instead of `Float32`

and `Float64`

?¶

Historically the C language and many other languages have used the
names `Float`

and `Double`

to represent floating point numbers of
size 32 and 64 respectively. Newer languages such as Rust and Julia
have begun to follow the naming scheme described in IEEE Standard for
Floating-Point Arithmetic (IEEE 754). This describes
single and double precision numbers as `Float32`

and `Float64`

;
the size is described in the type name.

Due to developer familiarity with the older naming convention, and
choice by the developers of Idris, Idris uses the C style convention.
That is, the names `Float`

and `Double`

are used to describe
single and double precision numbers.

### What is -ffreestanding?¶

The freestanding flag is used to build Idris binaries which have their libs and compiler in a relative path. This is useful for building binaries where the install directory is unknown at build time. When passing this flag, the IDRIS_LIB_DIR environment variable needs to be set to the path where the Idris libs reside relative to the idris executable. The IDRIS_TOOLCHAIN_DIR environment variable is optional, if that is set, Idris will use that path to find the C compiler.

Example:

```
IDRIS_LIB_DIR="./libs" IDRIS_TOOLCHAIN_DIR="./mingw/bin" CABALFLAGS="-fffi -ffreestanding -frelease" make
```

### What does the name ‘Idris’ mean?¶

British people of a certain age may be familiar with this singing dragon. If that doesn’t help, maybe you can invent a suitable acronym :-) .

### Where can I find more answers?¶

There is an Unofficial FAQ on the wiki on GitHub which answers more technical questions and may be updated more often.

## The Effects Tutorial¶

A tutorial on the Effects package in Idris.

Note

The documentation for Idris has been published under the Creative
Commons CC0 License. As such to the extent possible under law, *The
Idris Community* has waived all copyright and related or neighbouring
rights to Documentation for Idris.

More information concerning the CC0 can be found online at: http://creativecommons.org/publicdomain/zero/1.0/

### Introduction¶

Pure functional languages with dependent types such as Idris support reasoning about programs directly
in the type system, promising that we can *know* a program will run
correctly (i.e. according to the specification in its type) simply
because it compiles. Realistically, though, things are not so simple:
programs have to interact with the outside world, with user input,
input from a network, mutable state, and so on. In this tutorial I
will introduce the library, which is included with the distribution
and supports programming and reasoning with side-effecting programs,
supporting mutable state, interaction with the outside world,
exceptions, and verified resource management.

This tutorial assumes familiarity with pure programming in Idris, as described in Sections 1–6 of the main tutorial [1]. The examples presented are tested with Idris and can be found in the examples directory of the Idris repository.

Consider, for example, the following introductory function which illustrates the kind of properties which can be expressed in the type system:

```
vadd : Vect n Int -> Vect n Int -> Vect n Int
vadd [] [] = []
vadd (x :: xs) (y :: ys) = x + y :: vadd xs ys
```

This function adds corresponding elements in a pair of vectors. The type
guarantees that the vectors will contain only elements of type `Int`

,
and that the input lengths and the output length all correspond. A
natural question to ask here, which is typically neglected by
introductory tutorials, is “How do I turn this into a program?” That is,
given some lists entered by a user, how do we get into a position to be
able to apply the `vadd`

function? Before doing so, we will have to:

- Read user input, either from the keyboard, a file, or some other input device.
- Check that the user inputs are valid, i.e. contain only
`Int`

and are the same length, and report an error if not. - Write output

The complete program will include side-effects for I/O and error
handling, before we can get to the pure core functionality. In this
tutorial, we will see how Idris supports side-effects.
Furthermore, we will see how we can use the dependent type system to
*reason* about stateful and side-effecting programs. We will return to
this specific example later.

#### Hello world¶

To give an idea of how programs with effects look in , here is the
ubiquitous “Hello world” program, written using the `Effects`

library:

```
module Main
import Effects
import Effect.StdIO
hello : Eff () [STDIO]
hello = putStrLn “Hello world!”
main : IO ()
main = run hello
```

As usual, the entry point is `main`

. All `main`

has to do is invoke the
`hello`

function which supports the `STDIO`

effect for console I/O, and
returns the unit value. All programs using the `Effects`

library must
`import Effects`

. The details of the `Eff`

type will be presented in the
remainder of this tutorial.

To compile and run this program, Idris needs to be told to include
the `Effects`

package, using the `-p effects`

flag (this flag is
required for all examples in this tutorial):

```
idris hello.idr -o hello -p effects
./hello Hello world!
```

#### Outline¶

The tutorial is structured as follows: first, in Section
State, we will discuss state management, describing why it
is important and introducing the `effects`

library to show how it
can be used to manage state. This section also gives an overview of
the syntax of effectful programs. Section Simple Effects then
introduces a number of other effects a program may have: I/O;
Exceptions; Random Numbers; and Non-determinism, giving examples for
each, and an extended example combining several effects in one
complete program. Section Dependent Effects introduces *dependent*
effects, showing how states and resources can be managed in
types. Section Creating New Effects shows how new effects can be
implemented. Section Example: A “Mystery Word” Guessing Game gives an extended example
showing how to implement a “mystery word” guessing game, using effects
to describe the rules of the game and ensure they are implemented
accurately. References to further reading are given in Section
Further Reading.

[1] | You do not, however, need to know what a monad is! |

### State¶

Many programs, even pure programs, can benefit from locally mutable state. For example, consider a program which tags binary tree nodes with a counter, by an inorder traversal (i.e. counting depth first, left to right). This would perform something like the following:

We can describe binary trees with the following data type `BTree`

and `testTree`

to represent the example input above:

```
data BTree a = Leaf
| Node (BTree a) a (BTree a)
testTree : BTree String
testTree = Node (Node Leaf "Jim" Leaf)
"Fred"
(Node (Node Leaf "Alice" Leaf)
"Sheila"
(Node Leaf "Bob" Leaf))
```

Then our function to implement tagging, beginning to tag with a
specific value `i`

, has the following type:

```
treeTag : (i : Int) -> BTree a -> BTree (Int, a)
```

#### First attempt¶

Naïvely, we can implement `treeTag`

by implementing a helper
function which propagates a counter, returning the result of the count
for each subtree:

```
treeTagAux : (i : Int) -> BTree a -> (Int, BTree (Int, a))
treeTagAux i Leaf = (i, Leaf)
treeTagAux i (Node l x r)
= let (i', l') = treeTagAux i l in
let x' = (i', x) in
let (i'', r') = treeTagAux (i' + 1) r in
(i'', Node l' x' r')
treeTag : (i : Int) -> BTree a -> BTree (Int, a)
treeTag i x = snd (treeTagAux i x)
```

This gives the expected result when run at the REPL prompt:

```
*TreeTag> treeTag 1 testTree
Node (Node Leaf (1, "Jim") Leaf)
(2, "Fred")
(Node (Node Leaf (3, "Alice") Leaf)
(4, "Sheila")
(Node Leaf (5, "Bob") Leaf)) : BTree (Int, String)
```

This works as required, but there are several problems when we try to
scale this to larger programs. It is error prone, because we need to
ensure that state is propagated correctly to the recursive calls (i.e.
passing the appropriate `i`

or `i’`

). It is hard to read, because
the functional details are obscured by the state propagation. Perhaps
most importantly, there is a common programming pattern here which
should be abstracted but instead has been implemented by hand. There
is local mutable state (the counter) which we have had to make
explicit.

#### Introducing `Effects`

¶

Idris provides a library, `Effects`

[3], which captures this
pattern and many others involving effectful computation [1]. An
effectful program `f`

has a type of the following form:

```
f : (x1 : a1) -> (x2 : a2) -> ... -> Eff t effs
```

That is, the return type gives the effects that `f`

supports
(`effs`

, of type `List EFFECT`

) and the type the computation
returns `t`

. So, our `treeTagAux`

helper could be written with the
following type:

```
treeTagAux : BTree a -> Eff (BTree (Int, a)) [STATE Int]
```

That is, `treeTagAux`

has access to an integer state, because the
list of available effects includes `STATE Int`

. `STATE`

is
declared as follows in the module `Effect.State`

(that is, we must
`import Effect.State`

to be able to use it):

```
STATE : Type -> EFFECT
```

It is an effect parameterised by a type (by convention, we write
effects in all capitals). The `treeTagAux`

function is an effectful
program which builds a new tree tagged with `Ints`

, and is
implemented as follows:

```
treeTagAux Leaf = pure Leaf
treeTagAux (Node l x r)
= do l' <- treeTagAux l
i <- get
put (i + 1)
r' <- treeTagAux r
pure (Node l' (i, x) r')
```

There are several remarks to be made about this implementation.
Essentially, it hides the state, which can be accessed using `get`

and updated using `put`

, but it introduces several new features.
Specifically, it uses `do`

-notation, binding variables with `<-`

,
and a `pure`

function. There is much to be said about these
features, but for our purposes, it suffices to know the following:

`do`

blocks allow effectful operations to be sequenced.`x <- e`

binds the result of an effectful operation`e`

to avariable

`x`

. For example, in the above code,`treeTagAux l`

is an effectful operation returning`BTree (Int, a)`

, so`l’`

has type`BTree (Int, a)`

.

`pure e`

turns a pure value`e`

into the result of an effectfuloperation.

The `get`

and `put`

functions read and write a state `t`

,
assuming that the `STATE t`

effect is available. They have the
following types, polymorphic in the state `t`

they manage:

```
get : Eff t [STATE t]
put : t -> Eff () [STATE t]
```

A program in `Eff`

can call any other function in `Eff`

provided
that the calling function supports at least the effects required by
the called function. In this case, it is valid for `treeTagAux`

to
call both `get`

and `put`

because all three functions support the
`STATE Int`

effect.

Programs in `Eff`

are run in some underlying *computation context*,
using the `run`

or `runPure`

function. Using `runPure`

, which
runs an effectful program in the identity context, we can write the
`treeTag`

function as follows, using `put`

to initialise the
state:

```
treeTag : (i : Int) -> BTree a -> BTree (Int, a)
treeTag i x = runPure (do put i
treeTagAux x)
```

We could also run the program in an impure context such as `IO`

,
without changing the definition of `treeTagAux`

, by using `run`

instead of `runPure`

:

```
treeTagAux : BTree a -> Eff (BTree (Int, a)) [STATE Int]
...
treeTag : (i : Int) -> BTree a -> IO (BTree (Int, a))
treeTag i x = run (do put i
treeTagAux x)
```

Note that the definition of `treeTagAux`

is exactly as before. For
reference, this complete program (including a `main`

to run it) is
shown in Listing [introprog].

```
module Main
import Effects
import Effect.State
data BTree a = Leaf
| Node (BTree a) a (BTree a)
Show a => Show (BTree a) where
show Leaf = "[]"
show (Node l x r) = "[" ++ show l ++ " "
++ show x ++ " "
++ show r ++ "]"
testTree : BTree String
testTree = Node (Node Leaf "Jim" Leaf)
"Fred"
(Node (Node Leaf "Alice" Leaf)
"Sheila"
(Node Leaf "Bob" Leaf))
treeTagAux : BTree a -> Eff (BTree (Int, a)) [STATE Int]
treeTagAux Leaf = pure Leaf
treeTagAux (Node l x r) = do l' <- treeTagAux l
i <- get
put (i + 1)
r' <- treeTagAux r
pure (Node l' (i, x) r')
treeTag : (i : Int) -> BTree a -> BTree (Int, a)
treeTag i x = runPure (do put i; treeTagAux x)
main : IO ()
main = print (treeTag 1 testTree)
```

#### Effects and Resources¶

Each effect is associated with a *resource*, which is initialised
before an effectful program can be run. For example, in the case of
`STATE Int`

the corresponding resource is the integer state itself.
The types of `runPure`

and `run`

show this (slightly simplified
here for illustrative purposes):

```
runPure : {env : Env id xs} -> Eff a xs -> a
run : Applicative m => {env : Env m xs} -> Eff a xs -> m a
```

The `env`

argument is implicit, and initialised automatically where
possible using default values given by implementations of the following
interface:

```
interface Default a where
default : a
```

Implementations of `Default`

are defined for all primitive types, and many
library types such as `List`

, `Vect`

, `Maybe`

, pairs, etc.
However, where no default value exists for a resource type (for
example, you may want a `STATE`

type for which there is no
`Default`

implementation) the resource environment can be given explicitly
using one of the following functions:

```
runPureInit : Env id xs -> Eff a xs -> a
runInit : Applicative m => Env m xs -> Eff a xs -> m a
```

To be well-typed, the environment must contain resources corresponding
exactly to the effects in `xs`

. For example, we could also have
implemented `treeTag`

by initialising the state as follows:

```
treeTag : (i : Int) -> BTree a -> BTree (Int, a)
treeTag i x = runPureInit [i] (treeTagAux x)
```

#### Labelled Effects¶

What if we have more than one state, especially more than one state of
the same type? How would `get`

and `put`

know which state they
should be referring to? For example, how could we extend the tree
tagging example such that it additionally counts the number of leaves
in the tree? One possibility would be to change the state so that it
captured both of these values, e.g.:

```
treeTagAux : BTree a -> Eff (BTree (Int, a)) [STATE (Int, Int)]
```

Doing this, however, ties the two states together throughout (as well as not indicating which integer is which). It would be nice to be able to call effectful programs which guaranteed only to access one of the states, for example. In a larger application, this becomes particularly important.

The library therefore allows effects in general to be *labelled* so
that they can be referred to explicitly by a particular name. This
allows multiple effects of the same type to be included. We can count
leaves and update the tag separately, by labelling them as follows:

```
treeTagAux : BTree a -> Eff (BTree (Int, a))
['Tag ::: STATE Int,
'Leaves ::: STATE Int]
```

The `:::`

operator allows an arbitrary label to be given to an
effect. This label can be any type—it is simply used to identify an
effect uniquely. Here, we have used a symbol type. In general
`’name`

introduces a new symbol, the only purpose of which is to
disambiguate values [2].

When an effect is labelled, its operations are also labelled using the
`:-`

operator. In this way, we can say explicitly which state we
mean when using `get`

and `put`

. The tree tagging program which
also counts leaves can be written as follows:

```
treeTagAux Leaf = do
'Leaves :- update (+1)
pure Leaf
treeTagAux (Node l x r) = do
l' <- treeTagAux l
i <- 'Tag :- get
'Tag :- put (i + 1)
r' <- treeTagAux r
pure (Node l' (i, x) r')
```

The `update`

function here is a combination of `get`

and `put`

,
applying a function to the current state.

```
update : (x -> x) -> Eff () [STATE x]
```

Finally, our top level `treeTag`

function now returns a pair of the
number of leaves, and the new tree. Resources for labelled effects are
initialised using the `:=`

operator (reminiscent of assignment in an
imperative language):

```
treeTag : (i : Int) -> BTree a -> (Int, BTree (Int, a))
treeTag i x = runPureInit ['Tag := i, 'Leaves := 0]
(do x' <- treeTagAux x
leaves <- 'Leaves :- get
pure (leaves, x'))
```

To summarise, we have:

`:::`

to convert an effect to a labelled effect.`:-`

to convert an effectful operation to a labelled effectful operation.`:=`

to initialise a resource for a labelled effect.

Or, more formally with their types (slightly simplified to account only for the situation where available effects are not updated):

```
(:::) : lbl -> EFFECT -> EFFECT
(:-) : (l : lbl) -> Eff a [x] -> Eff a [l ::: x]
(:=) : (l : lbl) -> res -> LRes l res
```

Here, `LRes`

is simply the resource type associated with a labelled
effect. Note that labels are polymorphic in the label type `lbl`

.
Hence, a label can be anything—a string, an integer, a type, etc.

`!`

-notation¶

In many cases, using `do`

-notation can make programs unnecessarily
verbose, particularly in cases where the value bound is used once,
immediately. The following program returns the length of the
`String`

stored in the state, for example:

```
stateLength : Eff Nat [STATE String]
stateLength = do x <- get
pure (length x)
```

This seems unnecessarily verbose, and it would be nice to program in a
more direct style in these cases. provides `!`

-notation to help with
this. The above program can be written instead as:

```
stateLength : Eff Nat [STATE String]
stateLength = pure (length !get)
```

The notation `!expr`

means that the expression `expr`

should be
evaluated and then implicitly bound. Conceptually, we can think of
`!`

as being a prefix function with the following type:

```
(!) : Eff a xs -> a
```

Note, however, that it is not really a function, merely syntax! In
practice, a subexpression `!expr`

will lift `expr`

as high as
possible within its current scope, bind it to a fresh name `x`

, and
replace `!expr`

with `x`

. Expressions are lifted depth first, left
to right. In practice, `!`

-notation allows us to program in a more
direct style, while still giving a notational clue as to which
expressions are effectful.

For example, the expression:

```
let y = 42 in f !(g !(print y) !x)
```

is lifted to:

```
let y = 42 in do y' <- print y
x' <- x
g' <- g y' x'
f g'
```

#### The Type `Eff`

¶

Underneath, `Eff`

is an overloaded function which translates to an
underlying type `EffM`

:

```
EffM : (m : Type -> Type) -> (t : Type)
-> (List EFFECT)
-> (t -> List EFFECT) -> Type
```

This is more general than the types we have been writing so far. It is
parameterised over an underlying computation context `m`

, a
result type `t`

as we have already seen, as well as a `List EFFECT`

and a
function type `t -> List EFFECT`

.

These additional parameters are the list of *input* effects, and a
list of *output* effects, computed from the result of an effectful
operation. That is: running an effectful program can change the set
of effects available! This is a particularly powerful idea, and we
will see its consequences in more detail later. Some examples of
operations which can change the set of available effects are:

- Updating a state containing a dependent type (for example adding an element to a vector).
- Opening a file for reading is an effect, but whether the file really
*is*open afterwards depends on whether the file was successfully opened. - Closing a file means that reading from the file should no longer be possible.

While powerful, this can make uses of the `EffM`

type hard to read.
Therefore the library provides an overloaded function `Eff`

There are the following three versions:

```
SimpleEff.Eff : (t : Type) -> (input_effs : List EFFECT) -> Type
TransEff.Eff : (t : Type) -> (input_effs : List EFFECT) ->
(output_effs : List EFFECT) -> Type
DepEff.Eff : (t : Type) -> (input_effs : List EFFECT) ->
(output_effs_fn : x -> List EFFECT) -> Type
```

So far, we have used only the first version, `SimpleEff.Eff`

, which
is defined as follows:

```
Eff : (x : Type) -> (es : List EFFECT) -> Type
Eff x es = {m : Type -> Type} -> EffM m x es (\v => es)
```

i.e. the set of effects remains the same on output. This suffices for
the `STATE`

example we have seen so far, and for many useful
side-effecting programs. We could also have written `treeTagAux`

with the expanded type:

```
treeTagAux : BTree a ->
EffM m (BTree (Int, a)) [STATE Int] (\x => [STATE Int])
```

Later, we will see programs which update effects:

```
Eff a xs xs'
```

which is expanded to

```
EffM m a xs (\_ => xs')
```

i.e. the set of effects is updated to `xs’`

(think of a transition
in a state machine). There is, for example, a version of `put`

which
updates the type of the state:

```
putM : y -> Eff () [STATE x] [STATE y]
```

Also, we have:

```
Eff t xs (\res => xs')
```

which is expanded to

```
EffM m t xs (\res => xs')
```

i.e. the set of effects is updated according to the result of the
operation `res`

, of type `t`

.

Parameterising `EffM`

over an underlying computation context allows us
to write effectful programs which are specific to one context, and in some
cases to write programs which *extend* the list of effects available using
the `new`

function, though this is beyond the scope of this tutorial.

[1] | The earlier paper [3] describes the essential implementation details, although the library presented there is an earlier version which is less powerful than that presented in this tutorial. |

[2] | In practice, `’name` simply introduces a new empty type |

[3] | (1, 2) Edwin Brady. 2013. Programming and reasoning with algebraic
effects and dependent types. SIGPLAN Not. 48, 9 (September
2013), 133-144. DOI=10.1145/2544174.2500581
http://dl.acm.org/citation.cfm?doid=2544174.2500581 |

### Simple Effects¶

So far we have seen how to write programs with locally mutable state
using the `STATE`

effect. To recap, we have the definitions below
in a module `Effect.State`

```
module Effect.State
STATE : Type -> EFFECT
get : Eff x [STATE x]
put : x -> Eff () [STATE x]
putM : y -> Eff () [STATE x] [STATE y]
update : (x -> x) -> Eff () [STATE x]
Handler State m where { ... }
```

The last line, `Handler State m where { ... }`

, means that the `STATE`

effect is usable in any computation context `m`

. That is, a program
which uses this effect and returns something of type `a`

can be
evaluated to something of type `m a`

using `run`

, for any
`m`

. The lower case `State`

is a data type describing the
operations which make up the `STATE`

effect itself—we will go into
more detail about this in Section Creating New Effects.

In this section, we will introduce some other supported effects,
allowing console I/O, exceptions, random number generation and
non-deterministic programming. For each effect we introduce, we will
begin with a summary of the effect, its supported operations, and the
contexts in which it may be used, like that above for `STATE`

, and
go on to present some simple examples. At the end, we will see some
examples of programs which combine multiple effects.

All of the effects in the library, including those described in this section, are summarised in Appendix Effects Summary.

#### Console I/O¶

Console I/O is supported with the `STDIO`

effect, which allows reading and writing characters and strings to and
from standard input and standard output. Notice that there is a
constraint here on the computation context `m`

, because it only
makes sense to support console I/O operations in a context where we
can perform (or at the very least simulate) console I/O:

```
module Effect.StdIO
STDIO : EFFECT
putChar : Char -> Eff () [STDIO]
putStr : String -> Eff () [STDIO]
putStrLn : String -> Eff () [STDIO]
getStr : Eff String [STDIO]
getChar : Eff Char [STDIO]
Handler StdIO IO where { ... }
Handler StdIO (IOExcept a) where { ... }
```

##### Examples¶

A program which reads the user’s name, then says hello, can be written as follows:

```
hello : Eff () [STDIO]
hello = do putStr "Name? "
x <- getStr
putStrLn ("Hello " ++ trim x ++ "!")
```

We use `trim`

here to remove the trailing newline from the
input. The resource associated with `STDIO`

is simply the empty
tuple, which has a default value `()`

, so we can run this as
follows:

```
main : IO ()
main = run hello
```

In `hello`

we could also use `!`

-notation instead of ```
x <-
getStr
```

, since we only use the string that is read once:

```
hello : Eff () [STDIO]
hello = do putStr "Name? "
putStrLn ("Hello " ++ trim !getStr ++ "!")
```

More interestingly, we can combine multiple effects in one program. For example, we can loop, counting the number of people we’ve said hello to:

```
hello : Eff () [STATE Int, STDIO]
hello = do putStr "Name? "
putStrLn ("Hello " ++ trim !getStr ++ "!")
update (+1)
putStrLn ("I've said hello to: " ++ show !get ++ " people")
hello
```

The list of effects given in `hello`

means that the function can
call `get`

and `put`

on an integer state, and any functions which
read and write from the console. To run this, `main`

does not need
to be changed.

##### Aside: Resource Types¶

To find out the resource type of an effect, if necessary (for example
if we want to initialise a resource explicitly with `runInit`

rather
than using a default value with `run`

) we can run the
`resourceType`

function at the REPL:

```
*ConsoleIO> resourceType STDIO
() : Type
*ConsoleIO> resourceType (STATE Int)
Int : Type
```

#### Exceptions¶

The `EXCEPTION`

effect is declared in module `Effect.Exception`

. This allows programs
to exit immediately with an error, or errors to be handled more
generally:

```
module Effect.Exception
EXCEPTION : Type -> EFFECT
raise : a -> Eff b [EXCEPTION a]
Handler (Exception a) Maybe where { ... }
Handler (Exception a) List where { ... }
Handler (Exception a) (Either a) where { ... }
Handler (Exception a) (IOExcept a) where { ... }
Show a => Handler (Exception a) IO where { ... }
```

##### Example¶

Suppose we have a `String`

which is expected to represent an integer
in the range `0`

to `n`

. We can write a function `parseNumber`

which returns an `Int`

if parsing the string returns a number in the
appropriate range, or throws an exception otherwise. Exceptions are
parameterised by an error type:

```
data Err = NotANumber | OutOfRange
parseNumber : Int -> String -> Eff Int [EXCEPTION Err]
parseNumber num str
= if all isDigit (unpack str)
then let x = cast str in
if (x >=0 && x <= num)
then pure x
else raise OutOfRange
else raise NotANumber
```

Programs which support the `EXCEPTION`

effect can be run in any
context which has some way of throwing errors, for example, we can run
`parseNumber`

in the `Either Err`

context. It returns a value of
the form `Right x`

if successful:

```
*Exception> the (Either Err Int) $ run (parseNumber 42 "20")
Right 20 : Either Err Int
```

Or `Left e`

on failure, carrying the appropriate exception:

```
*Exception> the (Either Err Int) $ run (parseNumber 42 "50")
Left OutOfRange : Either Err Int
*Exception> the (Either Err Int) $ run (parseNumber 42 "twenty")
Left NotANumber : Either Err Int
```

In fact, we can do a little bit better with `parseNumber`

, and have
it return a *proof* that the integer is in the required range along
with the integer itself. One way to do this is define a type of
bounded integers, `Bounded`

:

```
Bounded : Int -> Type
Bounded x = (n : Int ** So (n >= 0 && n <= x))
```

Recall that `So`

is parameterised by a `Bool`

, and only ```
So
True
```

is inhabited. We can use `choose`

to construct such a value
from the result of a dynamic check:

```
data So : Bool -> Type = Oh : So True
choose : (b : Bool) -> Either (So b) (So (not b))
```

We then write `parseNumber`

using `choose`

rather than an
`if/then/else`

construct, passing the proof it returns on success as
the boundedness proof:

```
parseNumber : (x : Int) -> String -> Eff (Bounded x) [EXCEPTION Err]
parseNumber x str
= if all isDigit (unpack str)
then let num = cast str in
case choose (num >=0 && num <= x) of
Left p => pure (num ** p)
Right p => raise OutOfRange
else raise NotANumber
```

#### Random Numbers¶

Random number generation is also implemented by the library, in module
`Effect.Random`

:

```
module Effect.Random
RND : EFFECT
srand : Integer -> Eff () [RND]
rndInt : Integer -> Integer -> Eff Integer [RND]
rndFin : (k : Nat) -> Eff (Fin (S k)) [RND]
Handler Random m where { ... }
```

Random number generation is considered side-effecting because its
implementation generally relies on some external source of randomness.
The default implementation here relies on an integer *seed*, which can
be set with `srand`

. A specific seed will lead to a predictable,
repeatable sequence of random numbers. There are two functions which
produce a random number:

`rndInt`

, which returns a random integer between the given lowerand upper bounds.

`rndFin`

, which returns a random element of a finite set(essentially a number with an upper bound given in its type).

##### Example¶

We can use the `RND`

effect to implement a simple guessing game. The
`guess`

function, given a target number, will repeatedly ask the
user for a guess, and state whether the guess is too high, too low, or
correct:

```
guess : Int -> Eff () [STDIO]
```

For reference, the code for `guess`

is given below:

```
guess : Int -> Eff () [STDIO]
guess target
= do putStr "Guess: "
case run {m=Maybe} (parseNumber 100 (trim !getStr)) of
Nothing => do putStrLn "Invalid input"
guess target
Just (v ** _) =>
case compare v target of
LT => do putStrLn "Too low"
guess target
EQ => putStrLn "You win!"
GT => do putStrLn "Too high"
guess target
```

Note that we use `parseNumber`

as defined previously to read user input, but
we don’t need to list the `EXCEPTION`

effect because we use a nested `run`

to invoke `parseNumber`

, independently of the calling effectful program.

To invoke this, we pick a random number within the range 0–100,
having set up the random number generator with a seed, then run
`guess`

:

```
game : Eff () [RND, STDIO]
game = do srand 123456789
guess (fromInteger !(rndInt 0 100))
main : IO ()
main = run game
```

If no seed is given, it is set to the `default`

value. For a less
predictable game, some better source of randomness would be required,
for example taking an initial seed from the system time. To see how to
do this, see the `SYSTEM`

effect described in Effects Summary.

#### Non-determinism¶

The listing below gives the definition of the non-determinism effect, which allows a program to choose a value non-deterministically from a list of possibilities in such a way that the entire computation succeeds:

```
import Effects
import Effect.Select
SELECT : EFFECT
select : List a -> Eff a [SELECT]
Handler Selection Maybe where { ... }
Handler Selection List where { ... }
```

##### Example¶

The `SELECT`

effect can be used to solve constraint problems, such
as finding Pythagorean triples. The idea is to use `select`

to give
a set of candidate values, then throw an exception for any combination
of values which does not satisfy the constraint:

```
triple : Int -> Eff (Int, Int, Int) [SELECT, EXCEPTION String]
triple max = do z <- select [1..max]
y <- select [1..z]
x <- select [1..y]
if (x * x + y * y == z * z)
then pure (x, y, z)
else raise "No triple"
```

This program chooses a value for `z`

between `1`

and `max`

, then
values for `y`

and `x`

. In operation, after a `select`

, the
program executes the rest of the `do`

-block for every possible
assignment, effectively searching depth-first. If the list is empty
(or an exception is thrown) execution fails.

There are handlers defined for `Maybe`

and `List`

contexts, i.e.
contexts which can capture failure. Depending on the context `m`

,
`triple`

will either return the first triple it finds (if in
`Maybe`

context) or all triples in the range (if in `List`

context). We can try this as follows:

```
main : IO ()
main = do print $ the (Maybe _) $ run (triple 100)
print $ the (List _) $ run (triple 100)
```

`vadd`

revisited¶

We now return to the `vadd`

program from the introduction. Recall the
definition:

```
vadd : Vect n Int -> Vect n Int -> Vect n Int
vadd [] [] = []
vadd (x :: xs) (y :: ys) = x + y :: vadd xs ys
```

Using , we can set up a program so that it reads input from a user,
checks that the input is valid (i.e both vectors contain integers, and
are the same length) and if so, pass it on to `vadd`

. First, we
write a wrapper for `vadd`

which checks the lengths and throw an
exception if they are not equal. We can do this for input vectors of
length `n`

and `m`

by matching on the implicit arguments `n`

and
`m`

and using `decEq`

to produce a proof of their equality, if
they are equal:

```
vadd_check : Vect n Int -> Vect m Int ->
Eff (Vect m Int) [EXCEPTION String]
vadd_check {n} {m} xs ys with (decEq n m)
vadd_check {n} {m=n} xs ys | (Yes Refl) = pure (vadd xs ys)
vadd_check {n} {m} xs ys | (No contra) = raise "Length mismatch"
```

To read a vector from the console, we implement a function of the following type:

```
read_vec : Eff (p ** Vect p Int) [STDIO]
```

This returns a dependent pair of a length, and a vector of that
length, because we cannot know in advance how many integers the user
is going to input. We can use `-1`

to indicate the end of input:

```
read_vec : Eff (p ** Vect p Int) [STDIO]
read_vec = do putStr "Number (-1 when done): "
case run (parseNumber (trim !getStr)) of
Nothing => do putStrLn "Input error"
read_vec
Just v => if (v /= -1)
then do (_ ** xs) <- read_vec
pure (_ ** v :: xs)
else pure (_ ** [])
where
parseNumber : String -> Eff Int [EXCEPTION String]
parseNumber str
= if all (\x => isDigit x || x == '-') (unpack str)
then pure (cast str)
else raise "Not a number"
```

This uses a variation on `parseNumber`

which does not require a
number to be within range.

Finally, we write a program which reads two vectors and prints the result of pairwise addition of them, throwing an exception if the inputs are of differing lengths:

```
do_vadd : Eff () [STDIO, EXCEPTION String]
do_vadd = do putStrLn "Vector 1"
(_ ** xs) <- read_vec
putStrLn "Vector 2"
(_ ** ys) <- read_vec
putStrLn (show !(vadd_check xs ys))
```

By having explicit lengths in the type, we can be sure that `vadd`

is only being used where the lengths of inputs are guaranteed to be
equal. This does not stop us reading vectors from user input, but it
does require that the lengths are checked and any discrepancy is dealt
with gracefully.

#### Example: An Expression Calculator¶

To show how these effects can fit together, let us consider an evaluator for a simple expression language, with addition and integer values.

```
data Expr = Val Integer
| Add Expr Expr
```

An evaluator for this language always returns an `Integer`

, and
there are no situations in which it can fail!

```
eval : Expr -> Integer
eval (Val x) = x
eval (Add l r) = eval l + eval r
```

If we add variables, however, things get more interesting. The evaluator will need to be able to access the values stored in variables, and variables may be undefined.

```
data Expr = Val Integer
| Var String
| Add Expr Expr
```

To start, we will change the type of `eval`

so that it is effectful,
and supports an exception effect for throwing errors, and a state
containing a mapping from variable names (as `String`

) to their
values:

```
Env : Type
Env = List (String, Integer)
eval : Expr -> Eff Integer [EXCEPTION String, STATE Env]
eval (Val x) = return x
eval (Add l r) = return $ !(eval l) + !(eval r)
```

Note that we are using `!`

-notation to avoid having to bind
subexpressions in a `do`

block. Next, we add a case for evaluating
`Var`

:

```
eval (Var x) = case lookup x !get of
Nothing => raise $ "No such variable " ++ x
Just val => return val
```

This retrieves the state (with `get`

, supported by the `STATE Env`

effect) and raises an exception if the variable is not in the
environment (with `raise`

, supported by the `EXCEPTION String`

effect).

To run the evaluator on a particular expression in a particular
environment of names and their values, we can write a function which
sets the state then invokes `eval`

:

```
runEval : List (String, Integer) -> Expr -> Maybe Integer
runEval args expr = run (eval' expr)
where eval' : Expr -> Eff Integer [EXCEPTION String, STATE Env]
eval' e = do put args
eval e
```

We have picked `Maybe`

as a computation context here; it needs to be
a context which is available for every effect supported by
`eval`

. In particular, because we have exceptions, it needs to be a
context which supports exceptions. Alternatively, `Either String`

or
`IO`

would be fine, for example.

What if we want to extend the evaluator further, with random number
generation? To achieve this, we add a new constructor to `Expr`

,
which gives a random number up to a maximum value:

```
data Expr = Val Integer
| Var String
| Add Expr Expr
| Random Integer
```

Then, we need to deal with the new case, making sure that we extend
the list of events to include `RND`

. It doesn’t matter where `RND`

appears in the list, as long as it is present:

```
eval : Expr -> Eff Integer [EXCEPTION String, RND, STATE Env]
eval (Random upper) = rndInt 0 upper
```

For test purposes, we might also want to print the random number which has been generated:

```
eval (Random upper) = do val <- rndInt 0 upper
putStrLn (show val)
return val
```

If we try this without extending the effects list, we would see an error something like the following:

```
Expr.idr:28:6:When elaborating right hand side of eval:
Can't solve goal
SubList [STDIO]
[(EXCEPTION String), RND, (STATE (List (String, Integer)))]
```

In other words, the `STDIO`

effect is not available. We can correct
this simply by updating the type of `eval`

to include `STDIO`

.

```
eval : Expr -> Eff Integer [STDIO, EXCEPTION String, RND, STATE Env]
```

Note that using `STDIO`

will restrict the number of contexts in
which `eval`

can be `run`

to those which support `STDIO`

, such
as `IO`

. Once effect lists get longer, it can be a good idea instead
to encapsulate sets of effects in a type synonym. This is achieved as
follows, simply by defining a function which computes a type, since
types are first class in Idris:

```
EvalEff : Type -> Type
EvalEff t = Eff t [STDIO, EXCEPTION String, RND, STATE Env]
eval : Expr -> EvalEff Integer
```

### Dependent Effects¶

In the programs we have seen so far, the available effects have remained
constant. Sometimes, however, an operation can *change* the available
effects. The simplest example occurs when we have a state with a
dependent type—adding an element to a vector also changes its type, for
example, since its length is explicit in the type. In this section, we
will see how the library supports this. Firstly, we will see how states
with dependent types can be implemented. Secondly, we will see how the
effects can depend on the *result* of an effectful operation. Finally,
we will see how this can be used to implement a type-safe and
resource-safe protocol for file management.

#### Dependent States¶

Suppose we have a function which reads input from the console, converts
it to an integer, and adds it to a list which is stored in a `STATE`

.
It might look something like the following:

```
readInt : Eff () [STATE (List Int), STDIO]
readInt = do let x = trim !getStr
put (cast x :: !get)
```

But what if, instead of a list of integers, we would like to store a
`Vect`

, maintaining the length in the type?

```
readInt : Eff () [STATE (Vect n Int), STDIO]
readInt = do let x = trim !getStr
put (cast x :: !get)
```

This will not type check! Although the vector has length `n`

on entry
to `readInt`

, it has length `S n`

on exit. The library allows us to
express this as follows:

```
readInt : Eff ()[STATE (Vect n Int), STDIO]
[STATE (Vect (S n) Int), STDIO]
readInt = do let x = trim !getStr
putM (cast x :: !get)
```

The type `Eff a xs xs'`

means that the operation
begins with effects `xs`

available, and ends with effects `xs’`

available. We have used `putM`

to update the state, where the `M`

suffix indicates that the *type* is being updated as well as the value.
It has the following type:

```
putM : y -> Eff () [STATE x] [STATE y]
```

#### Result-dependent Effects¶

Often, whether a state is updated could depend on the success or
otherwise of an operation. In our `readInt`

example, we might wish to
update the vector only if the input is a valid integer (i.e. all
digits). As a first attempt, we could try the following, returning a
`Bool`

which indicates success:

```
readInt : Eff Bool [STATE (Vect n Int), STDIO]
[STATE (Vect (S n) Int), STDIO]
readInt = do let x = trim !getStr
case all isDigit (unpack x) of
False => pure False
True => do putM (cast x :: !get)
pure True
```

Unfortunately, this will not type check because the vector does not get
extended in both branches of the `case`

!

```
MutState.idr:18:19:When elaborating right hand side of Main.case
block in readInt:
Unifying n and S n would lead to infinite value
```

Clearly, the size of the resulting vector depends on whether or not the value read from the user was valid. We can express this in the type:

```
readInt : Eff Bool [STATE (Vect n Int), STDIO]
(\ok => if ok then [STATE (Vect (S n) Int), STDIO]
else [STATE (Vect n Int), STDIO])
readInt = do let x = trim !getStr
case all isDigit (unpack x) of
False => pureM False
True => do putM (cast x :: !get)
pureM True
```

Using `pureM`

rather than `pure`

allows the output effects to be
calculated from the value given. Its type is:

```
pureM : (val : a) -> EffM m a (xs val) xs
```

When using `readInt`

, we will have to check its return
value in order to know what the new set of effects is. For example, to
read a set number of values into a vector, we could write the following:

```
readN : (n : Nat) ->
Eff () [STATE (Vect m Int), STDIO]
[STATE (Vect (n + m) Int), STDIO]
readN Z = pure ()
readN {m} (S k) = case !readInt of
True => rewrite plusSuccRightSucc k m in readN k
False => readN (S k)
```

The `case`

analysis on the result of `readInt`

means that we know in
each branch whether reading the integer succeeded, and therefore how
many values still need to be read into the vector. What this means in
practice is that the type system has verified that a necessary dynamic
check (i.e. whether reading a value succeeded) has indeed been done.

Note

Only `case`

will work here. We cannot use `if/then/else`

because the `then`

and `else`

branches must have the same
type. The `case`

construct, however, abstracts over the value
being inspected in the type of each branch.

#### File Management¶

A practical use for dependent effects is in specifying resource usage protocols and verifying that they are executed correctly. For example, file management follows a resource usage protocol with the following (informally specified) requirements:

- It is necessary to open a file for reading before reading it
- Opening may fail, so the programmer should check whether opening was successful
- A file which is open for reading must not be written to, and vice versa
- When finished, an open file handle should be closed
- When a file is closed, its handle should no longer be used

These requirements can be expressed formally in , by creating a
`FILE_IO`

effect parameterised over a file handle state, which is
either empty, open for reading, or open for writing. The `FILE_IO`

effect’s definition is given below. Note that this
effect is mainly for illustrative purposes—typically we would also like
to support random access files and better reporting of error conditions.

```
module Effect.File
import Effects
import Control.IOExcept
FILE_IO : Type -> EFFECT
data OpenFile : Mode -> Type
open : (fname : String)
-> (m : Mode)
-> Eff Bool [FILE_IO ()]
(\res => [FILE_IO (case res of
True => OpenFile m
False => ())])
close : Eff () [FILE_IO (OpenFile m)] [FILE_IO ()]
readLine : Eff String [FILE_IO (OpenFile Read)]
writeLine : String -> Eff () [FILE_IO (OpenFile Write)]
eof : Eff Bool [FILE_IO (OpenFile Read)]
Handler FileIO IO where { ... }
```

In particular, consider the type of `open`

:

```
open : (fname : String)
-> (m : Mode)
-> Eff Bool [FILE_IO ()]
(\res => [FILE_IO (case res of
True => OpenFile m
False => ())])
```

This returns a `Bool`

which indicates whether opening the file was
successful. The resulting state depends on whether the operation was
successful; if so, we have a file handle open for the stated purpose,
and if not, we have no file handle. By `case`

analysis on the result,
we continue the protocol accordingly.

```
readFile : Eff (List String) [FILE_IO (OpenFile Read)]
readFile = readAcc [] where
readAcc : List String -> Eff (List String) [FILE_IO (OpenFile Read)]
readAcc acc = if (not !eof)
then readAcc (!readLine :: acc)
else pure (reverse acc)
```

Given a function `readFile`

, above, which reads from
an open file until reaching the end, we can write a program which opens
a file, reads it, then displays the contents and closes it, as follows,
correctly following the protocol:

```
dumpFile : String -> Eff () [FILE_IO (), STDIO]
dumpFile name = case !(open name Read) of
True => do putStrLn (show !readFile)
close
False => putStrLn ("Error!")
```

The type of `dumpFile`

, with `FILE_IO ()`

in its effect list,
indicates that any use of the file resource will follow the protocol
correctly (i.e. it both begins and ends with an empty resource). If we
fail to follow the protocol correctly (perhaps by forgetting to close
the file, failing to check that `open`

succeeded, or opening the file
for writing) then we will get a compile-time error. For example,
changing `open name Read`

to `open name Write`

yields a compile-time
error of the following form:

```
FileTest.idr:16:18:When elaborating right hand side of Main.case
block in testFile:
Can't solve goal
SubList [(FILE_IO (OpenFile Read))]
[(FILE_IO (OpenFile Write)), STDIO]
```

In other words: when reading a file, we need a file which is open for
reading, but the effect list contains a `FILE_IO`

effect carrying a
file open for writing.

#### Pattern-matching bind¶

It might seem that having to test each potentially failing operation
with a `case`

clause could lead to ugly code, with lots of
nested case blocks. Many languages support exceptions to improve this,
but unfortunately exceptions may not allow completely clean resource
management—for example, guaranteeing that any `open`

which did succeed
has a corresponding close.

Idris supports *pattern-matching* bindings, such as the following:

```
dumpFile : String -> Eff () [FILE_IO (), STDIO]
dumpFile name = do True <- open name Read
putStrLn (show !readFile)
close
```

This also has a problem: we are no longer dealing with the case where opening a file failed! The solution is to extend the pattern-matching binding syntax to give brief clauses for failing matches. Here, for example, we could write:

```
dumpFile : String -> Eff () [FILE_IO (), STDIO]
dumpFile name = do True <- open name Read | False => putStrLn "Error"
putStrLn (show !readFile)
close
```

This is exactly equivalent to the definition with the explicit `case`

.
In general, in a `do`

-block, the syntax:

```
do pat <- val | <alternatives>
p
```

is desugared to

```
do x <- val
case x of
pat => p
<alternatives>
```

There can be several `alternatives`

, separated by a vertical bar
`|`

. For example, there is a `SYSTEM`

effect which supports
reading command line arguments, among other things (see Appendix
Effects Summary). To read command line arguments, we can use
`getArgs`

:

```
getArgs : Eff (List String) [SYSTEM]
```

A main program can read command line arguments as follows, where in the
list which is returned, the first element `prog`

is the executable
name and the second is an expected argument:

```
emain : Eff () [SYSTEM, STDIO]
emain = do [prog, arg] <- getArgs
putStrLn $ "Argument is " ++ arg
{- ... rest of function ... -}
```

Unfortunately, this will not fail gracefully if no argument is given, or if too many arguments are given. We can use pattern matching bind alternatives to give a better (more informative) error:

```
emain : Eff () [SYSTEM, STDIO]
emain = do [prog, arg] <- getArgs | [] => putStrLn "Can't happen!"
| [prog] => putStrLn "No arguments!"
| _ => putStrLn "Too many arguments!"
putStrLn $ "Argument is " ++ arg
{- ... rest of function ... -}
```

If `getArgs`

does not return something of the form `[prog, arg]`

the
alternative which does match is executed instead, and that value
returned.

### Creating New Effects¶

We have now seen several side-effecting operations provided by the
`Effects`

library, and examples of their use in Section
Simple Effects. We have also seen how operations may *modify*
the available effects by changing state in Section
Dependent Effects. We have not, however, yet seen how these
operations are implemented. In this section, we describe how a
selection of the available effects are implemented, and show how new
effectful operations may be provided.

#### State¶

Effects are described by *algebraic data types*, where the
constructors describe the operations provided when the effect is
available. Stateful operations are described as follows:

```
data State : Effect where
Get : State a a (\x => a)
Put : b -> State () a (\x => b)
```

`Effect`

itself is a type synonym, giving the required type for an
effect signature:

```
Effect : Type
Effect = (result : Type) ->
(input_resource : Type) ->
(output_resource : result -> Type) -> Type
```

Each effect is associated with a *resource*. The second argument to
an effect signature is the resource type on *input* to an operation,
and the third is a function which computes the resource type on
*output*. Here, it means:

`Get`

takes no arguments. It has a resource of type`a`

, which is not updated, and running the`Get`

operation returns something of type`a`

.`Put`

takes a`b`

as an argument. It has a resource of type`a`

on input, which is updated to a resource of type`b`

. Running the`Put`

operation returns the element of the unit type.

The effects library provides an overloaded function `sig`

which can make effect signatures more concise, particularly when the
result has no effect on the resource type. For `State`

, we can
write:

```
data State : Effect where
Get : sig State a a
Put : b -> sig State () a b
```

There are four versions of `sig`

, depending on whether we
are interested in the resource type, and whether we are updating the
resource. Idris will infer the appropriate version from usage.

```
NoResourceEffect.sig : Effect -> Type -> Type
NoUpdateEffect.sig : Effect -> (ret : Type) ->
(resource : Type) -> Type
UpdateEffect.sig : Effect -> (ret : Type) ->
(resource_in : Type) ->
(resource_out : Type) -> Type
DepUpdateEffect.sig : Effect -> (ret : Type) ->
(resource_in : Type) ->
(resource_out : ret -> Type) -> Type
```

In order to convert `State`

(of type `Effect`

) into something
usable in an effects list, of type `EFFECT`

, we write the following:

```
STATE : Type -> EFFECT
STATE t = MkEff t State
```

`MkEff`

constructs an `EFFECT`

by taking the resource type (here,
the `t`

which parameterises `STATE`

) and the effect signature
(here, `State`

). For reference, `EFFECT`

is declared as follows:

```
data EFFECT : Type where
MkEff : Type -> Effect -> EFFECT
```

Recall that to run an effectful program in `Eff`

, we use one of the
`run`

family of functions to run the program in a particular
computation context `m`

. For each effect, therefore, we must explain
how it is executed in a particular computation context for `run`

to
work in that context. This is achieved with the following interface:

```
interface Handler (e : Effect) (m : Type -> Type) where
handle : resource -> (eff : e t resource resource') ->
((x : t) -> resource' x -> m a) -> m a
```

We have already seen some implementation declarations in the effect
summaries in Section Simple Effects. An implementation of ```
Handler e
m
```

means that the effect declared with signature `e`

can be run in
computation context `m`

. The `handle`

function takes:

- The
`resource`

on input (so, the current value of the state for`State`

) - The effectful operation (either
`Get`

or`Put x`

for`State`

) - A
*continuation*, which we conventionally call`k`

, and should be passed the result value of the operation, and an updated resource.

There are two reasons for taking a continuation here: firstly, this is neater because there are multiple return values (a new resource and the result of the operation); secondly, and more importantly, the continuation can be called zero or more times.

A `Handler`

for `State`

simply passes on the value of the state,
in the case of `Get`

, or passes on a new state, in the case of
`Put`

. It is defined the same way for all computation contexts:

```
Handler State m where
handle st Get k = k st st
handle st (Put n) k = k () n
```

This gives enough information for `Get`

and `Put`

to be used
directly in `Eff`

programs. It is tidy, however, to define top level
functions in `Eff`

, as follows:

```
get : Eff x [STATE x]
get = call Get
put : x -> Eff () [STATE x]
put val = call (Put val)
putM : y -> Eff () [STATE x] [STATE y]
putM val = call (Put val)
```

**An implementation detail (aside):** The `call`

function converts
an `Effect`

to a function in `Eff`

, given a proof that the effect
is available. This proof can be constructed automatically by , since
it is essentially an index into a statically known list of effects:

```
call : {e : Effect} ->
(eff : e t a b) -> {auto prf : EffElem e a xs} ->
Eff t xs (\v => updateResTy v xs prf eff)
```

This is the reason for the `Can’t solve goal`

error when an effect
is not available: the implicit proof `prf`

has not been solved
automatically because the required effect is not in the list of
effects `xs`

.

Such details are not important for using the library, or even writing new effects, however.

##### Summary¶

The following listing summarises what is required to define the
`STATE`

effect:

```
data State : Effect where
Get : sig State a a
Put : b -> sig State () a b
STATE : Type -> EFFECT
STATE t = MkEff t State
Handler State m where
handle st Get k = k st st
handle st (Put n) k = k () n
get : Eff x [STATE x]
get = call Get
put : x -> Eff () [STATE x]
put val = call (Put val)
putM : y -> Eff () [STATE x] [STATE y]
putM val = call (Put val)
```

#### Console I/O¶

Then listing below gives the definition of the `STDIO`

effect, including handlers for `IO`

and `IOExcept`

. We omit the
definition of the top level `Eff`

functions, as this merely invoke
the effects `PutStr`

, `GetStr`

, `PutCh`

and `GetCh`

directly.

Note that in this case, the resource is the unit type in every case,
since the handlers merely apply the `IO`

equivalents of the effects
directly.

```
data StdIO : Effect where
PutStr : String -> sig StdIO ()
GetStr : sig StdIO String
PutCh : Char -> sig StdIO ()
GetCh : sig StdIO Char
Handler StdIO IO where
handle () (PutStr s) k = do putStr s; k () ()
handle () GetStr k = do x <- getLine; k x ()
handle () (PutCh c) k = do putChar c; k () ()
handle () GetCh k = do x <- getChar; k x ()
Handler StdIO (IOExcept a) where
handle () (PutStr s) k = do ioe_lift $ putStr s; k () ()
handle () GetStr k = do x <- ioe_lift $ getLine; k x ()
handle () (PutCh c) k = do ioe_lift $ putChar c; k () ()
handle () GetCh k = do x <- ioe_lift $ getChar; k x ()
STDIO : EFFECT
STDIO = MkEff () StdIO
```

#### Exceptions¶

The listing below gives the definition of the `Exception`

effect, including two of its handlers for `Maybe`

and `List`

. The
only operation provided is `Raise`

. The key point to note in the
definitions of these handlers is that the continuation `k`

is not
used. Running `Raise`

therefore means that computation stops with an
error.

```
data Exception : Type -> Effect where
Raise : a -> sig (Exception a) b
Handler (Exception a) Maybe where
handle _ (Raise e) k = Nothing
Handler (Exception a) List where
handle _ (Raise e) k = []
EXCEPTION : Type -> EFFECT
EXCEPTION t = MkEff () (Exception t)
```

#### Non-determinism¶

The following listing gives the definition of the `Select`

effect for writing non-deterministic programs, including a handler for
`List`

context which returns all possible successful values, and a
handler for `Maybe`

context which returns the first successful
value.

```
data Selection : Effect where
Select : List a -> sig Selection a
Handler Selection Maybe where
handle _ (Select xs) k = tryAll xs where
tryAll [] = Nothing
tryAll (x :: xs) = case k x () of
Nothing => tryAll xs
Just v => Just v
Handler Selection List where
handle r (Select xs) k = concatMap (\x => k x r) xs
SELECT : EFFECT
SELECT = MkEff () Selection
```

Here, the continuation is called multiple times in each handler, for
each value in the list of possible values. In the `List`

handler, we
accumulate all successful results, and in the `Maybe`

handler we try
the first value in the list, and try later values only if that fails.

#### File Management¶

Result-dependent effects are no different from non-dependent effects
in the way they are implemented. The listing below
illustrates this for the `FILE_IO`

effect. The syntax for state
transitions `{ x ==> {res} x’ }`

, where the result state `x’`

is
computed from the result of the operation `res`

, follows that for
the equivalent `Eff`

programs.

```
data FileIO : Effect where
Open : (fname: String)
-> (m : Mode)
-> sig FileIO Bool () (\res => case res of
True => OpenFile m
False => ())
Close : sig FileIO () (OpenFile m)
ReadLine : sig FileIO String (OpenFile Read)
WriteLine : String -> sig FileIO () (OpenFile Write)
EOF : sig FileIO Bool (OpenFile Read)
Handler FileIO IO where
handle () (Open fname m) k = do h <- openFile fname m
if !(validFile h)
then k True (FH h)
else k False ()
handle (FH h) Close k = do closeFile h
k () ()
handle (FH h) ReadLine k = do str <- fread h
k str (FH h)
handle (FH h) (WriteLine str) k = do fwrite h str
k () (FH h)
handle (FH h) EOF k = do e <- feof h
k e (FH h)
FILE_IO : Type -> EFFECT
FILE_IO t = MkEff t FileIO
```

Note that in the handler for `Open`

, the types passed to the
continuation `k`

are different depending on whether the result is
`True`

(opening succeeded) or `False`

(opening failed). This uses
`validFile`

, defined in the `Prelude`

, to test whether a file
handler refers to an open file or not.

### Example: A “Mystery Word” Guessing Game¶

In this section, we will use the techniques and specific effects discussed in the tutorial so far to implement a larger example, a simple text-based word-guessing game. In the game, the computer chooses a word, which the player must guess letter by letter. After a limited number of wrong guesses, the player loses [1].

We will implement the game by following these steps:

- Define the game state, in enough detail to express the rules
- Define the rules of the game (i.e. what actions the player may take, and how these actions affect the game state)
- Implement the rules of the game (i.e. implement state updates for each action)
- Implement a user interface which allows a player to direct actions

Step 2 may be achieved by defining an effect which depends on the state
defined in step 1. Then step 3 involves implementing a `Handler`

for
this effect. Finally, step 4 involves implementing a program in `Eff`

using the newly defined effect (and any others required to implement the
interface).

#### Step 1: Game State¶

First, we categorise the game states as running games (where there are a number of guesses available, and a number of letters still to guess), or non-running games (i.e. games which have not been started, or games which have been won or lost).

```
data GState = Running Nat Nat | NotRunning
```

Notice that at this stage, we say nothing about what it means to make a guess, what the word to be guessed is, how to guess letters, or any other implementation detail. We are only interested in what is necessary to describe the game rules.

We will, however, parameterise a concrete game state `Mystery`

over
this data:

```
data Mystery : GState -> Type
```

#### Step 2: Game Rules¶

We describe the game rules as a dependent effect, where each action has
a *precondition* (i.e. what the game state must be before carrying out
the action) and a *postcondition* (i.e. how the action affects the game
state). Informally, these actions with the pre- and postconditions are:

- Guess
Guess a letter in the word.

- Precondition: The game must be running, and there must be both guesses still available, and letters still to be guessed.
- Postcondition: If the guessed letter is in the word and not yet guessed, reduce the number of letters, otherwise reduce the number of guesses.

- Won
Declare victory

- Precondition: The game must be running, and there must be no letters still to be guessed.
- Postcondition: The game is no longer running.

- Lost
Accept defeat

- Precondition: The game must be running, and there must be no guesses left.
- Postcondition: The game is no longer running.

- NewWord
Set a new word to be guessed

- Precondition: The game must not be running.
- Postcondition: The game is running, with 6 guesses available (the choice of 6 is somewhat arbitrary here) and the number of unique letters in the word still to be guessed.

- Get
- Get a string representation of the game state. This is for display purposes; there are no pre- or postconditions.

We can make these rules precise by declaring them more formally in an effect signature:

```
data MysteryRules : Effect where
Guess : (x : Char) ->
sig MysteryRules Bool
(Mystery (Running (S g) (S w)))
(\inword => if inword
then Mystery (Running (S g) w)
else Mystery (Running g (S w)))
Won : sig MysteryRules () (Mystery (Running g 0))
(Mystery NotRunning)
Lost : sig MysteryRules () (Mystery (Running 0 g))
(Mystery NotRunning)
NewWord : (w : String) ->
sig MysteryRules () (Mystery NotRunning) (Mystery (Running 6 (length (letters w))))
Get : sig MysteryRules String (Mystery h)
```

This description says nothing about how the rules are implemented. In
particular, it does not specify *how* to tell whether a guessed letter
was in a word, just that the result of `Guess`

depends on it.

Nevertheless, we can still create an `EFFECT`

from this, and use it in
an `Eff`

program. Implementing a `Handler`

for `MysteryRules`

will
then allow us to play the game.

```
MYSTERY : GState -> EFFECT
MYSTERY h = MkEff (Mystery h) MysteryRules
```

#### Step 3: Implement Rules¶

To *implement* the rules, we begin by giving a concrete definition of
game state:

```
data Mystery : GState -> Type where
Init : Mystery NotRunning
GameWon : (word : String) -> Mystery NotRunning
GameLost : (word : String) -> Mystery NotRunning
MkG : (word : String) ->
(guesses : Nat) ->
(got : List Char) ->
(missing : Vect m Char) ->
Mystery (Running guesses m)
```

If a game is `NotRunning`

, that is either because it has not yet
started (`Init`

) or because it is won or lost (`GameWon`

and
`GameLost`

, each of which carry the word so that showing the game
state will reveal the word to the player). Finally, `MkG`

captures a
running game’s state, including the target word, the letters
successfully guessed, and the missing letters. Using a `Vect`

for the
missing letters is convenient since its length is used in the type.

To initialise the state, we implement the following functions:
`letters`

, which returns a list of unique letters in a `String`

(ignoring spaces) and `initState`

which sets up an initial state
considered valid as a postcondition for `NewWord`

.

```
letters : String -> List Char
initState : (x : String) -> Mystery (Running 6 (length (letters x)))
```

When checking if a guess is in the vector of missing letters, it is
convenient to return a *proof* that the guess is in the vector, using
`isElem`

below, rather than merely a `Bool`

:

```
data IsElem : a -> Vect n a -> Type where
First : IsElem x (x :: xs)
Later : IsElem x xs -> IsElem x (y :: xs)
isElem : DecEq a => (x : a) -> (xs : Vect n a) -> Maybe (IsElem x xs)
```

The reason for returning a proof is that we can use it to remove an element from the correct position in a vector:

```
shrink : (xs : Vect (S n) a) -> IsElem x xs -> Vect n a
```

We leave the definitions of `letters`

, `init`

, `isElem`

and
`shrink`

as exercises. Having implemented these, the `Handler`

implementation for `MysteryRules`

is surprisingly straightforward:

```
Handler MysteryRules m where
handle (MkG w g got []) Won k = k () (GameWon w)
handle (MkG w Z got m) Lost k = k () (GameLost w)
handle st Get k = k (show st) st
handle st (NewWord w) k = k () (initState w)
handle (MkG w (S g) got m) (Guess x) k =
case isElem x m of
Nothing => k False (MkG w _ got m)
(Just p) => k True (MkG w _ (x :: got) (shrink m p))
```

Each case simply involves directly updating the game state in a way
which is consistent with the declared rules. In particular, in
`Guess`

, if the handler claims that the guessed letter is in the word
(by passing `True`

to `k`

), there is no way to update the state in
such a way that the number of missing letters or number of guesses does
not follow the rules.

#### Step 4: Implement Interface¶

Having described the rules, and implemented state transitions which
follow those rules as an effect handler, we can now write an interface
for the game which uses the `MYSTERY`

effect:

```
game : Eff () [MYSTERY (Running (S g) w), STDIO]
[MYSTERY NotRunning, STDIO]
```

The type indicates that the game must start in a running state, with some guesses available, and eventually reach a not-running state (i.e. won or lost). The only way to achieve this is by correctly following the stated rules.

Note that the type of `game`

makes no assumption that there are
letters to be guessed in the given word (i.e. it is `w`

rather than
`S w`

). This is because we will be choosing a word at random from a
vector of `String`

, and at no point have we made it explicit that
those `String`

are non-empty.

Finally, we need to initialise the game by picking a word at random from
a list of candidates, setting it as the target using `NewWord`

, then
running `game`

:

```
runGame : Eff () [MYSTERY NotRunning, RND, SYSTEM, STDIO]
runGame = do srand !time
let w = index !(rndFin _) words
call $ NewWord w
game
putStrLn !(call Get)
```

We use the system time (`time`

from the `SYSTEM`

effect; see
Appendix Effects Summary) to initialise the random number
generator, then pick a random `Fin`

to index into a list of
`words`

. For example, we could initialise a word list as follows:

```
words : ?wtype
words = with Vect ["idris","agda","haskell","miranda",
"java","javascript","fortran","basic",
"coffeescript","rust"]
wtype = proof search
```

Note

Rather than have to explicitly declare a type with the vector’s
length, it is convenient to give a hole `?wtype`

and let
Idris’s proof search mechanism find the type. This is a
limited form of type inference, but very useful in practice.

A possible complete implementation of `game`

is
presented below:

```
game : Eff () [MYSTERY (Running (S g) w), STDIO]
[MYSTERY NotRunning, STDIO]
game {w=Z} = Won
game {w=S _}
= do putStrLn !Get
putStr "Enter guess: "
let guess = trim !getStr
case choose (not (guess == "")) of
(Left p) => processGuess (strHead' guess p)
(Right p) => do putStrLn "Invalid input!"
game
where
processGuess : Char -> Eff () [MYSTERY (Running (S g) (S w)), STDIO]
[MYSTERY NotRunning, STDIO]
processGuess {g} {w} c
= case !(Main.Guess c) of
True => do putStrLn "Good guess!"
case w of
Z => Won
(S k) => game
False => do putStrLn "No, sorry"
case g of
Z => Lost
(S k) => game
```

#### Discussion¶

Writing the rules separately as an effect, then an implementation
which uses that effect, ensures that the implementation must follow
the rules. This has practical applications in more serious contexts;
`MysteryRules`

for example can be though of as describing a
*protocol* that a game player most follow, or alternative a
*precisely-typed API*.

In practice, we wouldn’t really expect to write rules first then
implement the game once the rules were complete. Indeed, I didn’t do
so when constructing this example! Rather, I wrote down a set of
likely rules making any assumptions *explicit* in the state
transitions for `MysteryRules`

. Then, when implementing `game`

at
first, any incorrect assumption was caught as a type error. The
following errors were caught during development:

- Not realising that allowing
`NewWord`

to be an arbitrary string would mean that`game`

would have to deal with a zero-length word as a starting state. - Forgetting to check whether a game was won before recursively calling
`processGuess`

, thus accidentally continuing a finished game. - Accidentally checking the number of missing letters, rather than the number of remaining guesses, when checking if a game was lost.

These are, of course, simple errors, but were caught by the type checker before any testing of the game.

[1] | Readers may recognise this game by the name “Hangman”. |

### Further Reading¶

This tutorial has given an introduction to writing and reasoning about
side-effecting programs in Idris, using the `Effects`

library.
More details about the *implementation* of the library, such as how
`run`

works, how handlers are invoked, etc, are given in a separate
paper [1].

Some libraries and programs which use `Effects`

can be found in the
following places:

- https://github.com/edwinb/SDL-idris — some bindings for the SDL media library, supporting graphics in particular.
- https://github.com/edwinb/idris-demos — various demonstration programs, including several examples from this tutorial, and a “Space Invaders” game.
- https://github.com/SimonJF/IdrisNet2 — networking and socket libraries.
- https://github.com/edwinb/Protocols — a high level communication protocol description language.

The inspiration for the `Effects`

library was Bauer and Pretnar’s
Eff language [2], which describes a language based on algebraic
effects and handlers. Other recent languages and libraries have also
been built on this ideas, for example [3] and [4]. The theoretical
foundations are also well-studied see [5], [6], [7], [8].

[1] | Edwin Brady. 2013. Programming and reasoning with algebraic effects and dependent types. SIGPLAN Not. 48, 9 (September 2013), 133-144. DOI=10.1145/2544174.2500581 https://dl.acm.org/citation.cfm?doid=2544174.2500581 |

[2] | Andrej Bauer, Matija Pretnar, Programming with algebraic effects and handlers, Journal of Logical and Algebraic Methods in Programming, Volume 84, Issue 1, January 2015, Pages 108-123, ISSN 2352-2208, http://math.andrej.com/wp-content/uploads/2012/03/eff.pdf |

[3] | Ben Lippmeier. 2009. Witnessing Purity, Constancy and Mutability. In Proceedings of the 7th Asian Symposium on Programming Languages and Systems (APLAS ‘09), Zhenjiang Hu (Ed.). Springer-Verlag, Berlin, Heidelberg, 95-110. DOI=10.1007/978-3-642-10672-9_9 http://link.springer.com/chapter/10.1007%2F978-3-642-10672-9_9 |

[4] | Ohad Kammar, Sam Lindley, and Nicolas Oury. 2013. Handlers in action. SIGPLAN Not. 48, 9 (September 2013), 145-158. DOI=10.1145/2544174.2500590 https://dl.acm.org/citation.cfm?doid=2544174.2500590 |

[5] | Martin Hyland, Gordon Plotkin, John Power, Combining effects: Sum and tensor, Theoretical Computer Science, Volume 357, Issues 1–3, 25 July 2006, Pages 70-99, ISSN 0304-3975, (https://www.sciencedirect.com/science/article/pii/S0304397506002659) |

[6] | Paul Blain Levy. 2004. Call-By-Push-Value: A Functional/Imperative Synthesis (Semantics Structures in Computation, V. 2). Kluwer Academic Publishers, Norwell, MA, USA. |

[7] | Plotkin, Gordon, and Matija Pretnar. “Handlers of algebraic effects.” Programming Languages and Systems. Springer Berlin Heidelberg, 2009. 80-94. |

[8] | Pretnar, Matija. “Logic and handling of algebraic effects.” (2010). |

### Effects Summary¶

This appendix gives interfaces for the core effects provided by the library.

#### EXCEPTION¶

```
module Effect.Exception
import Effects
import System
import Control.IOExcept
EXCEPTION : Type -> EFFECT
raise : a -> Eff b [EXCEPTION a]
Handler (Exception a) Maybe where { ... }
Handler (Exception a) List where { ... }
Handler (Exception a) (Either a) where { ... }
Handler (Exception a) (IOExcept a) where { ... }
Show a => Handler (Exception a) IO where { ... }
```

#### FILE_IO¶

```
module Effect.File
import Effects
import Control.IOExcept
FILE_IO : Type -> EFFECT
data OpenFile : Mode -> Type
open : (fname : String)
-> (m : Mode)
-> Eff Bool [FILE_IO ()]
(\res => [FILE_IO (case res of
True => OpenFile m
False => ())])
close : Eff () [FILE_IO (OpenFile m)] [FILE_IO ()]
readLine : Eff String [FILE_IO (OpenFile Read)]
writeLine : String -> Eff () [FILE_IO (OpenFile Write)]
eof : Eff Bool [FILE_IO (OpenFile Read)]
Handler FileIO IO where { ... }
```

#### RND¶

```
module Effect.Random
import Effects
import Data.Vect
import Data.Fin
RND : EFFECT
srand : Integer -> Eff m () [RND]
rndInt : Integer -> Integer -> Eff m Integer [RND]
rndFin : (k : Nat) -> Eff m (Fin (S k)) [RND]
Handler Random m where { ... }
```

#### SELECT¶

```
module Effect.Select
import Effects
SELECT : EFFECT
select : List a -> Eff m a [SELECT]
Handler Selection Maybe where { ... }
Handler Selection List where { ... }
```

#### STATE¶

```
module Effect.State
import Effects
STATE : Type -> EFFECT
get : Eff m x [STATE x]
put : x -> Eff m () [STATE x]
putM : y -> Eff m () [STATE x] [STATE y]
update : (x -> x) -> Eff m () [STATE x]
Handler State m where { ... }
```

#### STDIO¶

```
module Effect.StdIO
import Effects
import Control.IOExcept
STDIO : EFFECT
putChar : Handler StdIO m => Char -> Eff m () [STDIO]
putStr : Handler StdIO m => String -> Eff m () [STDIO]
putStrLn : Handler StdIO m => String -> Eff m () [STDIO]
getStr : Handler StdIO m => Eff m String [STDIO]
getChar : Handler StdIO m => Eff m Char [STDIO]
Handler StdIO IO where { ... }
Handler StdIO (IOExcept a) where { ... }
```

#### SYSTEM¶

```
module Effect.System
import Effects
import System
import Control.IOExcept
SYSTEM : EFFECT
getArgs : Handler System e => Eff e (List String) [SYSTEM]
time : Handler System e => Eff e Int [SYSTEM]
getEnv : Handler System e => String -> Eff e (Maybe String) [SYSTEM]
Handler System IO where { ... }
Handler System (IOExcept a) where { ... }
```

## Theorem Proving¶

A tutorial on theorem proving in Idris.

Note

The documentation for Idris has been published under the Creative
Commons CC0 License. As such to the extent possible under law, *The
Idris Community* has waived all copyright and related or neighboring
rights to Documentation for Idris.

### Running example: Addition of Natural Numbers¶

Throughout this tutorial, we will be working with the following function, defined in the Idris prelude, which defines addition on natural numbers:

```
plus : Nat -> Nat -> Nat
plus Z m = m
plus (S k) m = S (plus k m)
```

It is defined by the above equations, meaning that we have for free the
properties that adding `m`

to zero always results in `m`

, and that
adding `m`

to any non-zero number `S k`

always results in
`S (plus k m)`

. We can see this by evaluation at the Idris REPL (i.e.
the prompt, the read-eval-print loop):

```
Idris> \m => plus Z m
\m => m : Nat -> Nat
Idris> \k,m => plus (S k) m
\k => \m => S (plus k m) : Nat -> Nat -> Nat
```

Note that unlike many other language REPLs, the Idris REPL performs
evaluation on *open* terms, meaning that it can reduce terms which
appear inside lambda bindings, like those above. Therefore, we can
introduce unknowns `k`

and `m`

as lambda bindings and see how
`plus`

reduces.

The `plus`

function has a number of other useful properties, for
example:

- It is
*commutative*, that is for all`Nat`

inputs`n`

and`m`

, we know that`plus n m = plus m n`

. - It is
*associative*, that is for all`Nat`

inputs`n`

,`m`

and`p`

, we know that`plus n (plus m p) = plus (plus m n) p`

.

We can use these properties in an Idris program, but in order to do so
we must *prove* them.

#### Equality Proofs¶

Idris has a built-in propositional equality type, conceptually defined as follows:

```
data (=) : a -> b -> Type where
Refl : x = x
```

Note that this must be built-in, rather than defined in the library,
because `=`

is a reserved operator — you cannot define this directly
in your own code.

It is *propositional* equality, where the type states that any two
values in different types `a`

and `b`

may be proposed to be equal.
There is only one way to *prove* equality, however, which is by
reflexivity (`Refl`

).

We have a *type* for propositional equality here, and correspondingly a
*program* inhabiting an instance of this type can be seen as a proof of
the corresponding proposition [1]. So, trivially, we can prove that
`4`

equals `4`

:

```
four_eq : 4 = 4
four_eq = Refl
```

However, trying to prove that `4 = 5`

results in failure:

```
four_eq_five : 4 = 5
four_eq_five = Refl
```

The type `4 = 5`

is a perfectly valid type, but is uninhabited, so
when trying to type check this definition, Idris gives the following
error:

```
When elaborating right hand side of four_eq_five:
Type mismatch between
x = x (Type of Refl)
and
4 = 5 (Expected type)
```

##### Type checking equality proofs¶

An important step in type checking Idris programs is *unification*,
which attempts to resolve implicit arguments such as the implicit
argument `x`

in `Refl`

. As far as our understanding of type checking
proofs is concerned, it suffices to know that unifying two terms
involves reducing both to normal form then trying to find an assignment
to implicit arguments which will make those normal forms equal.

When type checking `Refl`

, Idris requires that the type is of the form
`x = x`

, as we see from the type of `Refl`

. In the case of
`four_eq_five`

, Idris will try to unify the expected type `4 = 5`

with the type of `Refl`

, `x = x`

, notice that a solution requires
that `x`

be both `4`

and `5`

, and therefore fail.

Since type checking involves reduction to normal form, we can write the following equalities directly:

```
twoplustwo_eq_four : 2 + 2 = 4
twoplustwo_eq_four = Refl
plus_reduces_Z : (m : Nat) -> plus Z m = m
plus_reduces_Z m = Refl
plus_reduces_Sk : (k, m : Nat) -> plus (S k) m = S (plus k m)
plus_reduces_Sk k m = Refl
```

#### Heterogeneous Equality¶

Equality in Idris is *heterogeneous*, meaning that we can even propose
equalities between values in different types:

```
idris_not_php : 2 = "2"
```

Obviously, in Idris the type `2 = "2"`

is uninhabited, and one might
wonder why it is useful to be able to propose equalities between values
in different types. However, with dependent types, such equalities can
arise naturally. For example, if two vectors are equal, their lengths
must be equal:

```
vect_eq_length : (xs : Vect n a) -> (ys : Vect m a) ->
(xs = ys) -> n = m
```

In the above declaration, `xs`

and `ys`

have different types because
their lengths are different, but we would still like to draw a
conclusion about the lengths if they happen to be equal. We can define
`vect_eq_length`

as follows:

```
vect_eq_length xs xs Refl = Refl
```

By matching on `Refl`

for the third argument, we know that the only
valid value for `ys`

is `xs`

, because they must be equal, and
therefore their types must be equal, so the lengths must be equal.

Alternatively, we can put an underscore for the second `xs`

, since
there is only one value which will type check:

```
vect_eq_length xs _ Refl = Refl
```

#### Properties of `plus`

¶

Using the `(=)`

type, we can now state the properties of `plus`

given above as Idris type declarations:

```
plus_commutes : (n, m : Nat) -> plus n m = plus m n
plus_assoc : (n, m, p : Nat) -> plus n (plus m p) = plus (plus n m) p
```

Both of these properties (and many others) are proved for natural number
addition in the Idris standard library, using `(+)`

from the `Num`

interface rather than using `plus`

directly. They have the names
`plusCommutative`

and `plusAssociative`

respectively.

In the remainder of this tutorial, we will explore several different
ways of proving `plus_commutes`

(or, to put it another way, writing
the function.) We will also discuss how to use such equality proofs, and
see where the need for them arises in practice.

[1] | This is known as the Curry-Howard correspondence. |

### Inductive Proofs¶

Before embarking on proving `plus_commutes`

in Idris itself, let us
consider the overall structure of a proof of some property of natural
numbers. Recall that they are defined recursively, as follows:

```
data Nat : Type where
Z : Nat
S : Nat -> Nat
```

A *total* function over natural numbers must both terminate, and cover
all possible inputs. Idris checks functions for totality by checking that
all inputs are covered, and that all recursive calls are on
*structurally smaller* values (so recursion will always reach a base
case). Recalling `plus`

:

```
plus : Nat -> Nat -> Nat
plus Z m = m
plus (S k) m = S (plus k m)
```

This is total because it covers all possible inputs (the first argument
can only be `Z`

or `S k`

for some `k`

, and the second argument
`m`

covers all possible `Nat`

) and in the recursive call, `k`

is structurally smaller than `S k`

so the first argument will always
reach the base case `Z`

in any sequence of recursive calls.

In some sense, this resembles a mathematical proof by induction (and
this is no coincidence!). For some property `P`

of a natural number
`x`

, we can show that `P`

holds for all `x`

if:

`P`

holds for zero (the base case).- Assuming that
`P`

holds for`k`

, we can show`P`

also holds for`S k`

(the inductive step).

In `plus`

, the property we are trying to show is somewhat trivial (for
all natural numbers `x`

, there is a `Nat`

which need not have any
relation to `x`

). However, it still takes the form of a base case and
an inductive step. In the base case, we show that there is a `Nat`

arising from `plus n m`

when `n = Z`

, and in the inductive step we
show that there is a `Nat`

arising when `n = S k`

and we know we can
get a `Nat`

inductively from `plus k m`

. We could even write a
function capturing all such inductive definitions:

```
nat_induction : (P : Nat -> Type) -> -- Property to show
(P Z) -> -- Base case
((k : Nat) -> P k -> P (S k)) -> -- Inductive step
(x : Nat) -> -- Show for all x
P x
nat_induction P p_Z p_S Z = p_Z
nat_induction P p_Z p_S (S k) = p_S k (nat_induction P p_Z p_S k)
```

Using `nat_induction`

, we can implement an equivalent inductive
version of `plus`

:

```
plus_ind : Nat -> Nat -> Nat
plus_ind n m
= nat_induction (\x => Nat)
m -- Base case, plus_ind Z m
(\k, k_rec => S k_rec) -- Inductive step plus_ind (S k) m
-- where k_rec = plus_ind k m
n
```

To prove that `plus n m = plus m n`

for all natural numbers `n`

and
`m`

, we can also use induction. Either we can fix `m`

and perform
induction on `n`

, or vice versa. We can sketch an outline of a proof;
performing induction on `n`

, we have:

Property

`P`

is`\x => plus x m = plus m x`

.Show that

`P`

holds in the base case and inductive step:- Base case:
`P Z`

, i.e.`plus Z m = plus m Z`

, which reduces to`m = plus m Z`

due to the definition of`plus`

. - Inductive step: Inductively, we know that
`P k`

holds for a specific, fixed`k`

, i.e.`plus k m = plus m k`

(the induction hypothesis). Given this, show`P (S k)`

, i.e.`plus (S k) m = plus m (S k)`

, which reduces to`S (plus k m) = plus m (S k)`

. From the induction hypothesis, we can rewrite this to`S (plus m k) = plus m (S k)`

.

To complete the proof we therefore need to show that `m = plus m Z`

for all natural numbers `m`

, and that `S (plus m k) = plus m (S k)`

for all natural numbers `m`

and `k`

. Each of these can also be
proved by induction, this time on `m`

.

We are now ready to embark on a proof of commutativity of `plus`

formally in Idris.

### Pattern Matching Proofs¶

In this section, we will provide a proof of `plus_commutes`

directly,
by writing a pattern matching definition. We will use interactive
editing features extensively, since it is significantly easier to
produce a proof when the machine can give the types of intermediate
values and construct components of the proof itself. The commands we
will use are summarised below. Where we refer to commands
directly, we will use the Vim version, but these commands have a direct
mapping to Emacs commands.

Command | Vim binding | Emacs binding | Explanation |

Check type | `\t` |
`C-c C-t` |
Show type of identifier or hole under the cursor. |

Proof search | `\o` |
`C-c C-a` |
Attempt to solve hole under the cursor by applying simple proof search. |

Make new definition | `\d` |
`C-c C-s` |
Add a template definition for the type defined under the cursor. |

Make lemma | `\l` |
`C-c C-e` |
Add a top level function with a type which solves the hole under the cursor. |

Split cases | `\c` |
`C-c C-c` |
Create new constructor patterns for each possible case of the variable under the cursor. |

#### Creating a Definition¶

To begin, create a file `pluscomm.idr`

containing the following type
declaration:

```
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
```

To create a template definition for the proof, press `\d`

(or the
equivalent in your editor of choice) on the line with the type
declaration. You should see:

```
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes n m = ?plus_commutes_rhs
```

To prove this by induction on `n`

, as we sketched in Section
sect-inductive, we begin with a case split on `n`

(press
`\c`

with the cursor over the `n`

in the definition.) You
should see:

```
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = ?plus_commutes_rhs_1
plus_commutes (S k) m = ?plus_commutes_rhs_2
```

If we inspect the types of the newly created holes,
`plus_commutes_rhs_1`

and `plus_commutes_rhs_2`

, we see that the
type of each reflects that `n`

has been refined to `Z`

and `S k`

in each respective case. Pressing `\t`

over
`plus_commutes_rhs_1`

shows:

```
m : Nat
--------------------------------------
plus_commutes_rhs_1 : m = plus m 0
```

Note that `Z`

renders as `0`

because the pretty printer renders
natural numbers as integer literals for readability. Similarly, for
`plus_commutes_rhs_2`

:

```
k : Nat
m : Nat
--------------------------------------
plus_commutes_rhs_2 : S (plus k m) = plus m (S k)
```

It is a good idea to give these slightly more meaningful names:

```
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = ?plus_commutes_Z
plus_commutes (S k) m = ?plus_commutes_S
```

#### Base Case¶

We can create a separate lemma for the base case interactively, by
pressing `\l`

with the cursor over `plus_commutes_Z`

. This
yields:

```
plus_commutes_Z : m = plus m 0
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = plus_commutes_Z
plus_commutes (S k) m = ?plus_commutes_S
```

That is, the hole has been filled with a call to a top level
function `plus_commutes_Z`

. The argument `m`

has been made implicit
because it can be inferred from context when it is applied.

Unfortunately, we cannot prove this lemma directly, since `plus`

is
defined by matching on its *first* argument, and here `plus m 0`

has a
specific value for its *second argument* (in fact, the left hand side of
the equality has been reduced from `plus 0 m`

.) Again, we can prove
this by induction, this time on `m`

.

First, create a template definition with `\d`

:

```
plus_commutes_Z : m = plus m 0
plus_commutes_Z = ?plus_commutes_Z_rhs
```

Since we are going to write this by induction on `m`

, which is
implicit, we will need to bring `m`

into scope manually:

```
plus_commutes_Z : m = plus m 0
plus_commutes_Z {m} = ?plus_commutes_Z_rhs
```

Now, case split on `m`

with `\c`

:

```
plus_commutes_Z : m = plus m 0
plus_commutes_Z {m = Z} = ?plus_commutes_Z_rhs_1
plus_commutes_Z {m = (S k)} = ?plus_commutes_Z_rhs_2
```

Checking the type of `plus_commutes_Z_rhs_1`

shows the following,
which is easily proved by reflection:

```
--------------------------------------
plus_commutes_Z_rhs_1 : 0 = 0
```

For such trivial proofs, we can let write the proof automatically by
pressing `\o`

with the cursor over `plus_commutes_Z_rhs_1`

.
This yields:

```
plus_commutes_Z : m = plus m 0
plus_commutes_Z {m = Z} = Refl
plus_commutes_Z {m = (S k)} = ?plus_commutes_Z_rhs_2
```

For `plus_commutes_Z_rhs_2`

, we are not so lucky:

```
k : Nat
--------------------------------------
plus_commutes_Z_rhs_2 : S k = S (plus k 0)
```

Inductively, we should know that `k = plus k 0`

, and we can get access
to this inductive hypothesis by making a recursive call on k, as
follows:

```
plus_commutes_Z : m = plus m 0
plus_commutes_Z {m = Z} = Refl
plus_commutes_Z {m = (S k)} = let rec = plus_commutes_Z {m=k} in
?plus_commutes_Z_rhs_2
```

For `plus_commutes_Z_rhs_2`

, we now see:

```
k : Nat
rec : k = plus k (fromInteger 0)
--------------------------------------
plus_commutes_Z_rhs_2 : S k = S (plus k 0)
```

Again, the `fromInteger 0`

is merely due to `Nat`

having an implementation
of the `Num`

interface. So we know that `k = plus k 0`

, but how do
we use this to update the goal to `S k = S k`

?

To achieve this, Idris provides a `replace`

function as part of the
prelude:

```
*pluscomm> :t replace
replace : (x = y) -> P x -> P y
```

Given a proof that `x = y`

, and a property `P`

which holds for
`x`

, we can get a proof of the same property for `y`

, because we
know `x`

and `y`

must be the same. In practice, this function can be
a little tricky to use because in general the implicit argument `P`

can be hard to infer by unification, so Idris provides a high level
syntax which calculates the property and applies `replace`

:

```
rewrite prf in expr
```

If we have `prf : x = y`

, and the required type for `expr`

is some
property of `x`

, the `rewrite ... in`

syntax will search for `x`

in the required type of `expr`

and replace it with `y`

. Concretely,
in our example, we can say:

```
plus_commutes_Z {m = (S k)} = let rec = plus_commutes_Z {m=k} in
rewrite rec in ?plus_commutes_Z_rhs_2
```

Checking the type of `plus_commutes_Z_rhs_2`

now gives:

```
k : Nat
rec : k = plus k (fromInteger 0)
_rewrite_rule : plus k 0 = k
--------------------------------------
plus_commutes_Z_rhs_2 : S (plus k 0) = S (plus k 0)
```

Using the rewrite rule `rec`

(which we can see in the context here as
`_rewrite_rule`

[1], the goal type has been updated with `k`

replaced by `plus k 0`

.

Alternatively, we could have applied the rewrite in the other direction
using the `sym`

function:

```
*pluscomm> :t sym
sym : (l = r) -> r = l
```

```
plus_commutes_Z {m = (S k)} = let rec = plus_commutes_Z {m=k} in
rewrite sym rec in ?plus_commutes_Z_rhs_2
```

In this case, inspecting the type of the hole gives:

```
k : Nat
rec : k = plus k (fromInteger 0)
_rewrite_rule : k = plus k 0
--------------------------------------
plus_commutes_Z_rhs_2 : S k = S k
```

Either way, we can use proof search (`\o`

) to complete the
proof, giving:

```
plus_commutes_Z : m = plus m 0
plus_commutes_Z {m = Z} = Refl
plus_commutes_Z {m = (S k)} = let rec = plus_commutes_Z {m=k} in
rewrite rec in Refl
```

The base case is now complete.

#### Inductive Step¶

Our main theorem, `plus_commutes`

should currently be in the following
state:

```
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = plus_commutes_Z
plus_commutes (S k) m = ?plus_commutes_S
```

Looking again at the type of `plus_commutes_S`

, we have:

```
k : Nat
m : Nat
--------------------------------------
plus_commutes_S : S (plus k m) = plus m (S k)
```

Conveniently, by induction we can immediately tell that
`plus k m = plus m k`

, so let us rewrite directly by making a
recursive call to `plus_commutes`

. We add this directly, by hand, as
follows:

```
plus_commutes : (n : Nat) -> (m : Nat) -> n + m = m + n
plus_commutes Z m = plus_commutes_Z
plus_commutes (S k) m = rewrite plus_commutes k m in ?plus_commutes_S
```

Checking the type of `plus_commutes_S`

now gives:

```
k : Nat
m : Nat
_rewrite_rule : plus m k = plus k m
--------------------------------------
plus_commutes_S : S (plus m k) = plus m (S k)
```

The good news is that `m`

and `k`

now appear in the correct order.
However, we still have to show that the successor symbol `S`

can be
moved to the front in the right hand side of this equality. This
remaining lemma takes a similar form to the `plus_commutes_Z`

; we
begin by making a new top level lemma with `\l`

. This gives:

```
plus_commutes_S : (k : Nat) -> (m : Nat) -> S (plus m k) = plus m (S k)
```

Unlike the previous case, `k`

and `m`

are not made implicit because
we cannot in general infer arguments to a function from its result.
Again, we make a template definition with `\d`

:

```
plus_commutes_S : (k : Nat) -> (m : Nat) -> S (plus m k) = plus m (S k)
plus_commutes_S k m = ?plus_commutes_S_rhs
```

Again, this is defined by induction over `m`

, since `plus`

is
defined by matching on its first argument. The complete definition is:

```
total
plus_commutes_S : (k : Nat) -> (m : Nat) -> S (plus m k) = plus m (S k)
plus_commutes_S k Z = Refl
plus_commutes_S k (S j) = rewrite plus_commutes_S k j in Refl
```

All holes have now been solved.

The `total`

annotation means that we require the final function to
pass the totality checker; i.e. it will terminate on all possible
well-typed inputs. This is important for proofs, since it provides a
guarantee that the proof is valid in *all* cases, not just those for
which it happens to be well-defined.

Now that `plus_commutes`

has a `total`

annotation, we have completed the
proof of commutativity of addition on natural numbers.

[1] | Note that the left and right hand sides of the equality have been
swapped, because `replace` takes a proof of `x=y` and the
property for `x` , not `y` . |

### DEPRECATED: Interactive Theorem Proving¶

Warning

The interactive theorem-proving interface documented here has been deprecated in favor of Elaborator Reflection.

Idris also supports interactive theorem proving via tactics. This is generally not recommended to be used directly, but rather used as a mechanism for building proof automation which is beyond the scope of this tutorial. In this section, we briefly discus tactics.

One way to write proofs interactively is to write the general *structure* of
the proof, and use the interactive mode to complete the details.
Consider the following definition, proved in Theorem Proving:

```
plusReduces : (n:Nat) -> plus Z n = n
```

We’ll be constructing the proof by *induction*, so we write the cases for `Z`

and `S`

, with a recursive call in the `S`

case giving the inductive
hypothesis, and insert *holes* for the rest of the definition:

```
plusReducesZ' : (n:Nat) -> n = plus n Z
plusReducesZ' Z = ?plusredZ_Z
plusReducesZ' (S k) = let ih = plusReducesZ' k in
?plusredZ_S
```

On running , two global names are created, `plusredZ_Z`

and
`plusredZ_S`

, with no definition. We can use the `:m`

command at the
prompt to find out which holes are still to be solved (or, more
precisely, which functions exist but have no definitions), then the
`:t`

command to see their types:

```
*theorems> :m
Global holes:
[plusredZ_S,plusredZ_Z]
```

```
*theorems> :t plusredZ_Z
plusredZ_Z : Z = plus Z Z
*theorems> :t plusredZ_S
plusredZ_S : (k : Nat) -> (k = plus k Z) -> S k = plus (S k) Z
```

The `:p`

command enters interactive proof mode, which can be used to
complete the missing definitions.

```
*theorems> :p plusredZ_Z
---------------------------------- (plusredZ_Z) --------
{hole0} : Z = plus Z Z
```

This gives us a list of premises (above the line; there are none here)
and the current goal (below the line; named `{hole0}`

here). At the
prompt we can enter tactics to direct the construction of the proof. In
this case, we can normalise the goal with the `compute`

tactic:

```
-plusredZ_Z> compute
---------------------------------- (plusredZ_Z) --------
{hole0} : Z = Z
```

Now we have to prove that `Z`

equals `Z`

, which is easy to prove by
`Refl`

. To apply a function, such as `Refl`

, we use `refine`

which
introduces subgoals for each of the function’s explicit arguments
(`Refl`

has none):

```
-plusredZ_Z> refine Refl
plusredZ_Z: no more goals
```

Here, we could also have used the `trivial`

tactic, which tries to
refine by `Refl`

, and if that fails, tries to refine by each name in
the local context. When a proof is complete, we use the `qed`

tactic
to add the proof to the global context, and remove the hole from the
unsolved holes list. This also outputs a trace of the proof:

```
-plusredZ_Z> qed
plusredZ_Z = proof
compute
refine Refl
```

```
*theorems> :m
Global holes:
[plusredZ_S]
```

The `:addproof`

command, at the interactive prompt, will add the proof
to the source file (effectively in an appendix). Let us now prove the
other required lemma, `plusredZ_S`

:

```
*theorems> :p plusredZ_S
---------------------------------- (plusredZ_S) --------
{hole0} : (k : Nat) -> (k = plus k Z) -> S k = plus (S k) Z
```

In this case, the goal is a function type, using `k`

(the argument
accessible by pattern matching) and `ih`

— the local variable
containing the result of the recursive call. We can introduce these as
premises using the `intro`

tactic twice (or `intros`

, which
introduces all arguments as premises). This gives:

```
k : Nat
ih : k = plus k Z
---------------------------------- (plusredZ_S) --------
{hole2} : S k = plus (S k) Z
```

Since plus is defined by recursion on its first argument, the term
`plus (S k) Z`

in the goal can be simplified, so we use `compute`

.

```
k : Nat
ih : k = plus k Z
---------------------------------- (plusredZ_S) --------
{hole2} : S k = S (plus k Z)
```

We know, from the type of `ih`

, that `k = plus k Z`

, so we would
like to use this knowledge to replace `plus k Z`

in the goal with
`k`

. We can achieve this with the `rewrite`

tactic:

```
-plusredZ_S> rewrite ih
k : Nat
ih : k = plus k Z
---------------------------------- (plusredZ_S) --------
{hole3} : S k = S k
-plusredZ_S>
```

The `rewrite`

tactic takes an equality proof as an argument, and tries
to rewrite the goal using that proof. Here, it results in an equality
which is trivially provable:

```
-plusredZ_S> trivial
plusredZ_S: no more goals
-plusredZ_S> qed
plusredZ_S = proof {
intros;
rewrite ih;
trivial;
}
```

Again, we can add this proof to the end of our source file using the
`:addproof`

command at the interactive prompt.

## Language Reference¶

This is the reference guide for the Idris Language. It documents the language specification and internals. This will tell you how Idris works, for using it you should read the Idris Tutorial.

Note

*The
Idris Community* has waived all copyright and related or neighboring
rights to Documentation for Idris.

### Documenting Idris Code¶

Idris documentation comes in two major forms: comments, which exist
for a reader’s edification and are ignored by the compiler, and inline
API documentation, which the compiler parses and stores for future
reference. To consult the documentation for a declaration `f`

, write
`:doc f`

at the REPL or use the appropriate command in your editor
(`C-c C-d`

in Emacs, `<LocalLeader>h`

in Vim).

#### Comments¶

Use comments to explain why code is written the way that it
is. Idris’s comment syntax is the same as that of Haskell: lines
beginning with `--`

are comments, and regions bracketed by `{-`

and `-}`

are comments even if they extend across multiple
lines. These can be used to comment out lines of code or provide
simple documentation for the readers of Idris code.

#### Inline Documentation¶

Idris also supports a comprehensive and rich inline syntax for Idris code to be generated. This syntax also allows for named parameters and variables within type signatures to be individually annotated using a syntax similar to Javadoc parameter annotations.

Documentation always comes before the declaration being documented. Inline documentation applies to either top-level declarations or to constructors. Documentation for specific arguments to constructors, type constructors, or functions can be associated with these arguments using their names.

The inline documentation for a declaration is an unbroken string of
lines, each of which begins with `|||`

(three pipe symbols). The
first paragraph of the documentation is taken to be an overview, and
in some contexts, only this overview will be shown. After the
documentation for the declaration as a whole, it is possible to
associate documentation with specific named parameters, which can
either be explicitly name or the results of converting free variables
to implicit parameters. Annotations are the same as with Javadoc
annotations, that is for the named parameter `(n : T)`

, the
corresponding annotation is `||| @ n Some description`

that is
placed before the declaration.

Documentation is written in Markdown, though not all contexts will display all possible formatting (for example, images are not displayed when viewing documentation in the REPL, and only some terminals render italics correctly). A comprehensive set of examples is given below.

```
||| Modules can also be documented.
module Docs
||| Add some numbers.
|||
||| Addition is really great. This paragraph is not part of the overview.
||| Still the same paragraph. Lists are also nifty:
||| * Really nifty!
||| * Yep!
||| * The name `add` is a **bold** choice
||| @ n is the recursive param
||| @ m is not
add : (n, m : Nat) -> Nat
add Z m = m
add (S n) m = S (add n m)
||| Append some vectors
||| @ a the contents of the vectors
||| @ xs the first vector (recursive param)
||| @ ys the second vector (not analysed)
appendV : (xs : Vect n a) -> (ys : Vect m a) -> Vect (add n m) a
appendV [] ys = ys
appendV (x::xs) ys = x :: appendV xs ys
||| Here's a simple datatype
data Ty =
||| Unit
UNIT |
||| Functions
ARR Ty Ty
||| Points to a place in a typing context
data Elem : Vect n Ty -> Ty -> Type where
Here : {ts : Vect n Ty} -> Elem (t::ts) t
There : {ts : Vect n Ty} -> Elem ts t -> Elem (t'::ts) t
||| A more interesting datatype
||| @ n the number of free variables
||| @ ctxt a typing context for the free variables
||| @ ty the type of the term
data Term : (ctxt : Vect n Ty) -> (ty : Ty) -> Type where
||| The constructor of the unit type
||| More comment
||| @ ctxt the typing context
UnitCon : {ctxt : Vect n Ty} -> Term ctxt UNIT
||| Function application
||| @ f the function to apply
||| @ x the argument
App : {ctxt : Vect n Ty} -> (f : Term ctxt (ARR t1 t2)) -> (x : Term ctxt t1) -> Term ctxt t2
||| Lambda
||| @ body the function body
Lam : {ctxt : Vect n Ty} -> (body : Term (t1::ctxt) t2) -> Term ctxt (ARR t1 t2)
||| Variables
||| @ i de Bruijn index
Var : {ctxt : Vect n Ty} -> (i : Elem ctxt t) -> Term ctxt t
||| A computation that may someday finish
codata Partial : Type -> Type where
||| A finished computation
||| @ value the result
Now : (value : a) -> Partial a
||| A not-yet-finished computation
||| @ rest the remaining work
Later : (rest : Partial a) -> Partial a
||| We can document records, including their fields and constructors
record Yummy where
||| Make a yummy
constructor MkYummy
||| What to eat
food : String
```

### Packages¶

Idris includes a simple system for building packages from a package description file. These files can be used with the Idris compiler to manage the development process of your Idris programmes and packages.

#### Package Descriptions¶

A package description includes the following:

- A header, consisting of the keyword package followed by the package name.
- Fields describing package contents,
`<field> = <value>`

At least one field must be the modules field, where the value is a
comma separated list of modules. For example, a library test which
has two modules `foo.idr`

and `bar.idr`

as source files would be
written as follows:

```
package foo
modules = foo, bar
```

Other examples of package files can be found in the `libs`

directory
of the main Idris repository, and in third-party libraries.

##### Common Fields¶

Other common fields which may be present in an `ipkg`

file are:

`sourcedir = <dir>`

, which takes the directory (relative to the current directory) which contains the source. Default is the current directory.`executable = <output>`

, which takes the name of the executable file to generate.`main = <module>`

, which takes the name of the main module, and must be present if the executable field is present.`opts = "<idris options>"`

, which allows options to be passed to Idris.`pkgs = <pkg name> (',' <pkg name>)+`

, a comma separated list of package names that the Idris package requires.

##### Binding to C¶

In more advanced cases, particularly to support creating bindings to
external `C`

libraries, the following options are available:

`makefile = <file>`

, which specifies a`Makefile`

, to be built before the Idris modules, for example to support linking with a`C`

library.`libs = <libs>`

, which takes a comma separated list of libraries which must be present for the package to be usable.`objs = <objs>`

, which takes a comma separated list of additional object files to be installed, perhaps generated by the`Makefile`

.

##### Testing¶

For testing Idris packages there is a rudimentary testing harness, run in the `IO`

context.
The `iPKG`

file is used to specify the functions used for testing.
The following option is available:

`tests = <test functions>`

, which takes the qualified names of all test functions to be run.

Important

The modules containing the test functions must also be added to the list of modules.

#### Using Package files¶

Given an Idris package file `text.ipkg`

it can be used with the Idris compiler as follows:

`idris --build test.ipkg`

will build all modules in the package`idris --install test.ipkg`

will install the package, making it accessible by other Idris libraries and programs.`idris --clean test.ipkg`

will delete all intermediate code and executable files generated when building.`idris --mkdoc test.ipkg`

will build HTML documentation for your package in the folder`test_doc`

in your project’s root directory.`idris --checkpkg test.ipkg`

will type check all modules in the package only. This differs from build that type checks**and**generates code.`idris --testpkg test.ipkg`

will compile and run any embedded tests you have specified in the`tests`

parameter.

Once the test package has been installed, the command line option
`--package test`

makes it accessible (abbreviated to `-p test`

).
For example:

```
idris -p test Main.idr
```

### Uniqueness Types¶

Uniqueness Types are an experimental feature available from Idris
0.9.15. A value with a unique type is guaranteed to have *at most one*
reference to it at run-time, which means that it can safely be updated
in-place, reducing the need for memory allocation and garbage
collection. The motivation is that we would like to be able to write
reactive systems, programs which run in limited memory environments,
device drivers, and any other system with hard real-time requirements,
ideally while giving up as little high level conveniences as possible.

They are inspired by linear types, Uniqueness Types in the Clean programming language, and ownership types and borrowed pointers in the Rust programming language.

Some things we hope to be able to do eventually with uniqueness types include:

- Safe, pure, in-place update of arrays, lists, etc
- Provide guarantees of correct resource usage, state transitions, etc
- Provide guarantees that critical program fragments will
*never*allocate

#### Using Uniqueness¶

If `x : T`

and `T : UniqueType`

, then there is at most one reference
to `x`

at any time during run-time execution. For example, we can
declare the type of unique lists as follows:

```
data UList : Type -> UniqueType
Nil : UList a
(::) : a -> UList a -> UList a
```

If we have a value `xs : UList a`

, then there is at most one
reference to `xs`

at run-time. The type checker preserves this
guarantee by ensuring that there is at most one reference to any value
of a unique type in a pattern clause. For example, the following
function definition would be valid:

```
umap : (a -> b) -> UList a -> UList b
umap f [] = []
umap f (x :: xs) = f x :: umap f xs
```

In the second clause, `xs`

is a value of a unique type, and only
appears once on the right hand side, so this clause is valid. Not only
that, since we know there can be no other reference to the `UList a`

argument, we can reuse its space for building the result! The compiler
is aware of this, and compiles this definition to an in-place update
of the list.

The following function definition would not be valid (even assuming an
implementation of `++`

), however, since `xs`

appears twice:

```
dupList : UList a -> UList a
dupList xs = xs ++ xs
```

This would result in a shared pointer to `xs`

, so the typechecker
reports:

```
unique.idr:12:5:Unique name xs is used more than once
```

If we explicitly copy, however, the typechecker is happy:

```
dup : UList a -> UList a
dup [] = []
dup (x :: xs) = x :: x :: dup xs
```

Note that it’s fine to use `x`

twice, because `a`

is a `Type`

,
rather than a `UniqueType`

.

There are some other restrictions on where a `UniqueType`

can
appear, so that the uniqueness property is preserved. In particular,
the type of the function type, `(x : a) -> b`

depends on the type of
`a`

or `b`

- if either is a `UniqueType`

, then the function type
is also a `UniqueType`

. Then, in a data declaration, if the type
constructor builds a `Type`

, then no constructor can have a
`UniqueType`

. For example, the following definition is invalid,
since it would embed a unique value in a possible non-unique value:

```
data BadList : UniqueType -> Type
Nil : {a : UniqueType} -> BadList a
(::) : {a : UniqueType} -> a -> BadList a -> BadList a
```

Finally, types may be polymorphic in their uniqueness, to a limited
extent. Since `Type`

and `UniqueType`

are different types, we are
limited in how much we can use polymorphic functions on unique types.
For example, if we have function composition defined as follows:

```
(.) : {a, b, c : Type} -> (b -> c) -> (a -> b) -> a -> c
(.) f g x = f (g x)
```

And we have some functions over unique types:

```
foo : UList a -> UList b
bar : UList b -> UList c
```

Then we cannot compose `foo`

and `bar`

as `bar . foo`

, because
`UList`

does not compute a `Type`

! Instead, we can define
composition as follows:

```
(.) : {a, b, c : Type*} -> (b -> c) -> (a -> b) -> a -> c
(.) f g x = f (g x)
```

The `Type*`

type stands for either unique or non-unique types. Since
such a function may be passed a `UniqueType`

, any value of type
`Type*`

must also satisfy the requirement that it appears at most
once on the right hand side.

##### Borrowed Types¶

It quickly becomes obvious when working with uniqueness types that having only one reference at a time can be painful. For example, what if we want to display a list before updating it?

```
showU : Show a => UList a -> String
showU xs = "[" ++ showU' xs ++ "]" where
showU' : UList a -> String
showU' [] = ""
showU' [x] = show x
showU' (x :: xs) = show x ++ ", " ++ showU' xs
```

This is a valid definition of `showU`

, but unfortunately it consumes
the list! So the following function would be invalid:

```
printAndUpdate : UList Int -> IO ()
printAndUpdate xs = do putStrLn (showU xs)
let xs' = umap (*2) xs -- xs no longer available!
putStrLn (showU xs')
```

Still, one would hope to be able to display a unique list without
problem, since it merely *inspects* the list; there are no updates. We
can achieve this, using the notion of *borrowing*. A Borrowed type is
a Unique type which can be inspected at the top level (by pattern
matching, or by *lending* to another function) but no further. This
ensures that the internals (i.e. the arguments to top level patterns)
will not be passed to any function which will update them.

`Borrowed`

converts a `UniqueType`

to a `BorrowedType`

. It is
defined as follows (along with some additional rules in the
typechecker):

```
data Borrowed : UniqueType -> BorrowedType where
Read : {a : UniqueType} -> a -> Borrowed a
implicit
lend : {a : UniqueType} -> a -> Borrowed a
lend x = Read x
```

A value can be “lent” to another function using `lend`

. Arguments to
`lend`

are not counted by the type checker as a reference to a unique
value, therefore a value can be lent as many times as desired. Using
this, we can write `showU`

as follows:

```
showU : Show a => Borrowed (UList a) -> String
showU xs = "[" ++ showU' xs ++ "]" where
showU' : Borrowed (UList a) -> String
showU' [] = ""
showU' [x] = show x
showU' (Read (x :: xs)) = show x ++ ", " ++ showU' (lend xs)
```

Unlike a unique value, a borrowed value may be referred to as many times as desired. However, there is a restriction on how a borrowed value can be used. After all, much like a library book or your neighbour’s lawnmower, if a function borrows a value it is expected to return it in exactly the condition in which it was received!

The restriction is that when a `Borrowed`

type is matched, any
pattern variables under the `Read`

which have a unique type may not
be referred to at all on the right hand side (unless they are
themselves `lent`

to another function).

Uniqueness information is stored in the type, and in particular in function types. Once we’re in a unique context, any new function which is constructed will be required to have unique type, which prevents the following sort of bad program being implemented:

```
foo : UList Int -> IO ()
foo xs = do let f = \x : Int => showU xs
putStrLn $ free xs
putStrLn $ f 42
return ()
```

Since `lend`

is implicit, in practice for functions to lend and borrow
values merely requires the argument to be marked as `Borrowed`

. We can
therefore write `showU`

as follows:

```
showU : Show a => Borrowed (UList a) -> String
showU xs = "[" ++ showU' xs ++ "]" where
showU' : Borrowed (UList a) -> String
showU' [] = ""
showU' [x] = show x
showU' (x :: xs) = show x ++ ", " ++ showU' xs
```

##### Problems/Disadvantages/Still to do...¶

This is a work in progress, there is lots to do. The most obvious
problem is the loss of abstraction. On the one hand, we have more
precise control over memory usage with `UniqueType`

and
`BorrowedType`

, but they are not in general compatible with
functions polymorphic over `Type`

. In the short term, we can start
to write reactive and low memory systems with this, but longer term it
would be nice to support more abstraction.

We also haven’t checked any of the metatheory, so this could all be fatally flawed! The implementation is based to a large extent on Uniqueness Typing Simplified, by de Vries et al, so there is reason to believe things should be fine, but we still have to do the work.

Much as there are with linear types, there are some annoyances when
trying to prove properties of functions with unique types (for
example, what counts as a use of a value). Since we require *at most*
one use of a value, rather than *exactly* one, this seems to be less
of an issue in practice, but still needs thought.

### New Foreign Function Interface¶

Ever since Idris has had multiple backends compiling to different target languages on potentially different platforms, we have had the problem that the foreign function interface (FFI) was written under the assumption of compiling to C. As a result, it has been hard to write generic code for multiple targets, or even to be sure that if code compiles that it will run on the expected target.

As of 0.9.17, Idris will have a new foreign function interface (FFI) which is aware of multiple targets. Users who are working with the default code generator can happily continue writing programs as before with no changes, but if you are writing bindings for an external library, writing a back end, or working with a non-C back end, there are some things you will need to be aware of, which this page describes.

#### The `IO'`

monad, and `main`

¶

The `IO`

monad exists as before, but is now specific to the C
backend (or, more precisely, any backend whose foreign function calls
are compatible with C.) Additionally, there is now an `IO'`

monad,
which is parameterised over a FFI descriptor:

```
data IO' : (lang : FFI) -> Type -> Type
```

The Prelude defines two FFI descriptors which are imported
automatically, for C and JavaScript/Node, and defines `IO`

to use
the C FFI and `JS_IO`

to use the JavaScript FFI:

```
FFI_C : FFI
FFI_JS : FFI
IO : Type -> Type
IO a = IO' FFI_C a
JS_IO : Type -> Type
JS_IO a = IO' FFI_JS a
```

As before, the entry point to an Idris program is `main`

, but the
type of `main`

can now be any instance of `IO'`

, e.g. the
following are both valid:

```
main : IO ()
main : JS_IO ()
```

The FFI descriptor includes details about which types can be marshalled between the foreign language and Idris, and the “target” of a foreign function call (typically just a String representation of the function’s name, but potentially something more complicated such as an external library file or even a URL).

#### FFI descriptors¶

An FFI descriptor is a record containing a predicate which holds when a type can be marshalled, and the type of the target of a foreign call:

```
record FFI where
constructor MkFFI
ffi_types : Type -> Type
ffi_fn : Type
```

For C, this is:

```
||| Supported C integer types
data C_IntTypes : Type -> Type where
C_IntChar : C_IntTypes Char
C_IntNative : C_IntTypes Int
... -- more integer types
||| Supported C foreign types
data C_Types : Type -> Type where
C_Str : C_Types String
C_Float : C_Types Float
C_Ptr : C_Types Ptr
C_MPtr : C_Types ManagedPtr
C_Unit : C_Types ()
C_Any : C_Types (Raw a)
C_IntT : C_IntTypes i -> C_Types i
FFI_C : FFI
FFI_C = MkFFI C_Types
String -- the name of the C function
```

#### Foreign calls¶

To call a foreign function, the `foreign`

function is used. For
example:

```
do_fopen : String -> String -> IO Ptr
do_fopen f m
= foreign FFI_C "fileOpen" (String -> String -> IO Ptr) f m
```

The `foreign`

function takes an FFI description, a function name (the
type is given by the `ffi_fn`

field of `FFI_C`

here), and a function
type, which gives the expected types of the remaining arguments. Here,
we’re calling an external function `fileOpen`

which takes, in the C, a
`char*`

file name, a `char*`

mode, and returns a file pointer. It is
the job of the C back end to convert Idris `String`

to C `char*`

and vice versa.

The argument types and return type given here must be present in the
`fn_types`

predicate of the `FFI_C`

description for the foreign
call to be valid.

**Note** The arguments to `foreign`

*must* be known at compile time,
because the foreign calls are generated statically. The `%inline`

directive on a function can be used to give hints to help this, for
example a shorthand for calling external JavaScript functions:

```
%inline
jscall : (fname : String) -> (ty : Type) ->
{auto fty : FTy FFI_JS [] ty} -> ty
jscall fname ty = foreign FFI_JS fname ty
```

##### FFI implementation¶

In order to write bindings to external libraries, the details of how
`foreign`

works are unnecessary — you simply need to know that
`foreign`

takes an FFI descriptor, the function name, and its
type. It is instructive to look a little deeper, however:

The type of `foreign`

is as follows:

```
foreign : (ffi : FFI)
-> (fname : ffi_fn f)
-> (ty : Type)
-> {auto fty : FTy ffi [] ty}
-> ty
```

The important argument here is the implicit `fty`

, which contains a
proof (`FTy`

) that the given type is valid according to the FFI
description `ffi`

:

```
data FTy : FFI -> List Type -> Type -> Type where
FRet : ffi_types f t -> FTy f xs (IO' f t)
FFun : ffi_types f s -> FTy f (s :: xs) t -> FTy f xs (s -> t)
```

Notice that this uses the `ffi_types`

field of the FFI descriptor
— these arguments to `FRet`

and `FFun`

give explicit proofs that
the type is valid in this FFI. For example, the above `do_fopen`

builds the following implicit proof as the `fty`

argument to
`foreign`

:

```
FFun C_Str (FFun C_Str (FRet C_Ptr))
```

#### Compiling foreign calls¶

(This section assumes some knowledge of the Idris internals.)

When writing a back end, we now need to know how to compile
`foreign`

. We’ll skip the details here of how a `foreign`

call
reaches the intermediate representation (the IR), though you can look
in `IO.idr`

in the `prelude`

package to see a bit more detail —
a `foreign`

call is implemented by the primitive function
`mkForeignPrim`

. The important part of the IR as defined in
`Lang.hs`

is the following constructor:

```
data LExp = ...
| LForeign FDesc -- Function descriptor
FDesc -- Return type descriptor
[(FDesc, LExp)]
```

So, a `foreign`

call appears in the IR as the `LForeign`

constructor, which takes a function descriptor (of a type given by the
`ffi_fn`

field in the FFI descriptor), a return type descriptor
(given by an application of `FTy`

), and a list of arguments with
type descriptors (also given by an application of `FTy`

).

An `FDesc`

describes an application of a name to some arguments, and
is really just a simplified subset of an `LExp`

:

```
data FDesc = FCon Name
| FStr String
| FUnknown
| FApp Name [FDesc]
```

There are corresponding structures in the lower level IRs, such as the defunctionalised, simplified and bytecode forms.

Our `do_fopen`

example above arrives in the `LExp`

form as:

```
LForeign (FStr "fileOpen") (FCon (sUN "C_Ptr"))
[(FCon (sUN "C_Str"), f), (FCon (sUN "C_Str"), m)]
```

(Assuming that `f`

and `m`

stand for the `LExp`

representations
of the arguments.) This information should be enough for any back end
to marshal the arguments and return value appropriately.

Note

When processing `FDesc`

, be aware that there may be implicit
arguments, which have not been erased. For example, `C_IntT`

has
an implicit argument `i`

, so will appear in an `FDesc`

as
something of the form `FApp (sUN "C_IntT") [i, t]`

where `i`

is
the implicit argument (which can be ignored) and `t`

is the
descriptor of the integer type. See `CodegenC.hs`

, specifically
the function `toFType`

, to see how this works in practice.

#### JavaScript FFI descriptor¶

The JavaScript FFI descriptor is a little more complex, because the JavaScript FFI supports marshalling functions. It is defined as follows:

```
mutual
data JsFn t = MkJsFn t
data JS_IntTypes : Type -> Type where
JS_IntChar : JS_IntTypes Char
JS_IntNative : JS_IntTypes Int
data JS_FnTypes : Type -> Type where
JS_Fn : JS_Types s -> JS_FnTypes t -> JS_FnTypes (s -> t)
JS_FnIO : JS_Types t -> JS_FnTypes (IO' l t)
JS_FnBase : JS_Types t -> JS_FnTypes t
data JS_Types : Type -> Type where
JS_Str : JS_Types String
JS_Float : JS_Types Float
JS_Ptr : JS_Types Ptr
JS_Unit : JS_Types ()
JS_FnT : JS_FnTypes a -> JS_Types (JsFn a)
JS_IntT : JS_IntTypes i -> JS_Types i
```

The reason for wrapping function types in a `JsFn`

is to help the
proof search when building `FTy`

. We hope to improve proof search
eventually, but for the moment it works much more reliably if the
indices are disjoint! An example of using this appears in IdrisScript when setting
timeouts:

```
setTimeout : (() -> JS_IO ()) -> (millis : Int) -> JS_IO Timeout
setTimeout f millis = do
timeout <- jscall "setTimeout(%0, %1)"
(JsFn (() -> JS_IO ()) -> Int -> JS_IO Ptr)
(MkJsFn f) millis
return $ MkTimeout timeout
```

### Syntax Guide¶

Examples are mostly adapted from the Idris tutorial.

#### Source File Structure¶

Source files consist of:

- An optional Module Header.
- Zero or more Imports.
- Zero or more declarations, e.g. Variables, Data types, etc.

For example:

```
module MyModule -- module header
import Data.Vect -- an import
%default total -- a directive
foo : Nat -- a declaration
foo = 5
```

##### Module Header¶

A file can start with a module header, introduced by the `module`

keyword:

```
module Semantics
```

Module names can be hierarchical, with parts separated by `.`

:

```
module Semantics.Transform
```

Each file can define only a single module, which includes everything defined in that file.

Like with declarations, a docstring can be used to provide documentation for a module:

```
||| Implementation of predicate transformer semantics.
module Semantics.Transform
```

##### Imports¶

An `import`

makes the names in another module available for use by the current
module:

```
import Data.Vect
```

All the declarations in an imported module are available for use in the file. In a case where a name is ambiguous — e.g. because it is imported from multiple modules, or appears in multiple visible namespaces — the ambiguity can be resolved using Qualified Names. (Often, the compiler can resolve the ambiguity for you, using the types involved.)

Imported modules can be given aliases to make qualified names more compact:

```
import Data.Vect as V
```

Note that names made visible by import are not, by default, re-exported to
users of the module being written. This can be done using `import public`

:

```
import public Data.Vect
```

#### Variables¶

A variable is always defined by defining its type on one line, and its value on the next line, using the syntax

```
<id> : <type>
<id> = <value>
```

Examples

```
x : Int
x = 100
hello : String
hello = "hello"
```

#### Types¶

In Idris, types are first class values. So a type declaration is the
same as just declaration of a variable whose type is `Type`

. In Idris,
variables that denote a type need not be capitalised. Example:

```
MyIntType : Type
MyIntType = Int
```

a more interesting example:

```
MyListType : Type
MyListType = List Int
```

While capitalising types is not required, the rules for generating implicit arguments mean it is often a good idea.

##### Data types¶

Idris provides two kinds of syntax for defining data types. The first, Haskell style syntax, defines a regular algebraic data type. For example

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

or

```
data List a = Nil | (::) a (List a)
```

The second, more general kind of data type, is defined using Agda or
GADT style syntax. This syntax defines a data type that is parameterised
by some values (in the `Vect`

example, a value of type `Nat`

and a
value of type `Type`

).

```
data Vect : Nat -> Type -> Type where
Nil : Vect Z a
(::) : (x : a) -> (xs : Vect n a) -> Vect (S n) a
```

#### Operators¶

##### Arithmetic¶

```
x + y
x - y
x * y
x / y
(x * y) + (a / b)
```

##### Equality and Relational¶

```
x == y
x /= y
x >= y
x > y
x <= y
x < y
```

##### Conditional¶

```
x && y
x || y
not x
```

#### Conditionals¶

##### If Then Else¶

```
if <test> then <true> else <false>
```

##### Case Expressions¶

```
case <test> of
<case 1> => <expr>
<case 2> => <expr>
...
otherwise => <expr>
```

#### Functions¶

##### Named¶

Named functions are defined in the same way as variables, with the type followed by the definition.

```
<id> : <argument type> -> <return type>
<id> arg = <expr>
```

Example

```
plusOne : Int -> Int
plusOne x = x + 1
```

Functions can also have multiple inputs, for example

```
makeHello : String -> String -> String
makeHello first last = "hello, my name is " ++ first ++ " " ++ last
```

Functions can also have named arguments. This is required if you want to
annotate parameters in a docstring. The following shows the same
`makeHello`

function as above, but with named parameters which are
also annotated in the docstring

```
||| Makes a string introducing a person
||| @first The person's first name
||| @last The person's last name
makeHello : (first : String) -> (last : String) -> String
makeHello first last = "hello, my name is " ++ first ++ " " ++ last
```

Like Haskell, Idris functions can be defined by pattern matching. For example

```
sum : List Int -> Int
sum [] = 0
sum (x :: xs) = x + (sum xs)
```

Similarly case analysis looks like

```
answerString : Bool -> String
answerString False = "Wrong answer"
answerString True = "Correct answer"
```

##### Dependent Functions¶

Dependent functions are functions where the type of the return value depends on the input value. In order to define a dependent function, named parameters must be used, since the parameter will appear in the return type. For example, consider

```
zeros : (n : Nat) -> Vect n Int
zeros Z = []
zeros (S k) = 0 :: (zeros k)
```

In this example, the return type is `Vect n Int`

which is an
expression which depends on the input parameter `n`

. ### Anonymous
Arguments in anonymous functions are separated by comma.

```
(\x => <expr>)
(\x, y => <expr>)
```

#### Misc¶

##### Qualified Names¶

If multiple declarations with the same name are visible, using the name can
result in an ambiguous situation. The compiler will attempt to resolve the
ambiguity using the types involved. If it’s unable — for example, because
the declarations with the same name also have the same type signatures — the
situation can be cleared up using a *qualified name*.

A qualified name has the symbol’s namespace prefixed, separated by a `.`

:

```
Data.Vect.length
```

This would specifically reference a `length`

declaration from `Data.Vect`

.

Qualified names can be written using two different shorthands:

- Names in modules that are imported using an alias can be qualified by the alias.
- The name can be qualified by the
*shortest unique suffix*of the namespace in question. For example, the`length`

case above can likely be shortened to`Vect.length`

.

##### Comments¶

```
-- Single Line
{- Multiline -}
||| Docstring (goes before definition)
```

##### Multi line String literals¶

```
foo = """
this is a
string literal"""
```

#### Directives¶

```
%lib <path>
%link <path>
%flag <path>
%include <path>
%hide <function>
%freeze <name>
%access <accessibility>
%default <totality>
%logging <level 0--11>
%dynamic <list of libs>
%name <list of names>
%error_handlers <list of names>
%language <extension>
```

### Erasure By Usage Analysis¶

This work stems from this feature proposal (obsoleted by this page). Beware that the information in the proposal is out of date — and sometimes even in direct contradiction with the eventual implementation.

#### Motivation¶

Traditional dependently typed languages (Agda, Coq) are good at
erasing *proofs* (either via irrelevance or an extra universe).

```
half : (n : Nat) -> Even n -> Nat
half Z EZ = Z
half (S (S n)) (ES pf) = S (half n pf)
```

For example, in the above snippet, the second argument is a proof, which is used only to convince the compiler that the function is total. This proof is never inspected at runtime and thus can be erased. In this case, the mere existence of the proof is sufficient and we can use irrelevance-related methods to achieve erasure.

However, sometimes we want to erase *indices* and this is where the
traditional approaches stop being useful, mainly for reasons described
in the original proposal.

```
uninterleave : {n : Nat} -> Vect (n * 2) a -> (Vect n a, Vect n a)
uninterleave [] = ([] , [])
uninterleave (x :: y :: rest) with (unzipPairs rest)
| (xs, ys) = (x :: xs, y :: ys)
```

Notice that in this case, the second argument is the important one and
we would like to get rid of the `n`

instead, although the shape of
the program is generally the same as in the previous case.

There are methods described by Brady, McBride and McKinna in [BMM04] to remove the indices from data structures, exploiting the fact that functions operating on them either already have a copy of the appropriate index or the index can be quickly reconstructed if needed. However, we often want to erase the indices altogether, from the whole program, even in those cases where reconstruction is not possible.

The following two sections describe two cases where doing so improves the runtime performance asymptotically.

##### Binary numbers¶

- O(n) instead of O(log n)

Consider the following `Nat`

-indexed type family representing binary
numbers:

```
data Bin : Nat -> Type where
N : Bin 0
O : {n : Nat} -> Bin n -> Bin (0 + 2*n)
I : {n : Nat} -> Bin n -> Bin (1 + 2*n)
```

These are supposed to be (at least asymptotically) fast and memory-efficient because their size is logarithmic compared to the numbers they represent.

Unfortunately this is not the case. The problem is that these binary
numbers still carry the *unary* indices with them, performing
arithmetic on the indices whenever arithmetic is done on the binary
numbers themselves. Hence the real representation of the number 15
looks like this:

```
I -> I -> I -> I -> N
S S S Z
S S Z
S S
S Z
S
S
S
Z
```

The used memory is actually *linear*, not logarithmic and therefore we
cannot get below O(n) with time complexities.

One could argue that Idris in fact compiles `Nat`

via GMP but
that’s a moot point for two reasons:

- First, whenever we try to index our data structures with anything
else than
`Nat`

, the compiler is not going to come to the rescue. - Second, even with
`Nat`

, the GMP integers are*still*there and they slow the runtime down.

This ought not to be the case since the `Nat`

are never used at
runtime and they are only there for typechecking purposes. Hence we
should get rid of them and get runtime code similar to what a idris
programmer would write.

##### U-views of lists¶

- O(n^2) instead of O(n)

Consider the type of U-views of lists:

```
data U : List a -> Type where
nil : U []
one : (z : a) -> U [z]
two : {xs : List a} -> (x : a) -> (u : U xs) -> (y : a) -> U (x :: xs ++ [y])
```

For better intuition, the shape of the U-view of
`[x0,x1,x2,z,y2,y1,y0]`

looks like this:

```
x0 y0 (two)
x1 y1 (two)
x2 y1 (two)
z (one)
```

When recursing over this structure, the values of `xs`

range over
`[x0,x1,x2,z,y2,y1,y0]`

, `[x1,x2,z,y2,y1]`

, `[x2,z,y2]`

,
`[z]`

. No matter whether these lists are stored or built on demand,
they take up a quadratic amount of memory (because they cannot share
nodes), and hence it takes a quadratic amount of time just to build
values of this index alone.

But the reasonable expectation is that operations with U-views take
linear time — so we need to erase the index `xs`

if we want to
achieve this goal.

#### Changes to Idris¶

Usage analysis is run at every compilation and its outputs are used for various purposes. This is actually invisible to the user but it’s a relatively big and important change, which enables the new features.

Everything that is found to be unused is erased. No annotations are needed, just don’t use the thing and it will vanish from the generated code. However, if you wish, you can use the dot annotations to get a warning if the thing is accidentally used.

“Being used” in this context means that the value of the “thing” may influence run-time behaviour of the program. (More precisely, it is not found to be irrelevant to the run-time behaviour by the usage analysis algorithm.)

“Things” considered for removal by erasure include:

- function arguments
- data constructor fields (including record fields and dictionary fields of class instances)

For example, `Either`

often compiles to the same runtime
representation as `Bool`

. Constructor field removal sometimes
combines with the newtype optimisation to have quite a strong effect.

There is a new compiler option `--warnreach`

, which will enable
warnings coming from erasure. Since we have full usage analysis, we
can compile even those programs that violate erasure annotations –
it’s just that the binaries may run slower than expected. The warnings
will be enabled by default in future versions of Idris (and possibly
turned to errors). However, in this transitional period, we chose to
keep them on-demand to avoid confusion until better documentation is
written.

Case-tree elaboration tries to avoid using dotted “things” whenever possible. (NB. This is not yet perfect and it’s being worked on: https://gist.github.com/ziman/10458331)

Postulates are no longer required to be collapsible. They are now
required to be *unused* instead.

#### Changes to the language¶

You can use dots to mark fields that are not intended to be used at runtime.

```
data Bin : Nat -> Type where
N : Bin 0
O : .{n : Nat} -> Bin n -> Bin (0 + 2*n)
I : .{n : Nat} -> Bin n -> Bin (1 + 2*n)
```

If these fields are found to be used at runtime, the dots will trigger
a warning (with `--warnreach`

).

Note that free (unbound) implicits are dotted by default so, for
example, the constructor `O`

can be defined as:

```
O : Bin n -> Bin (0 + 2*n)
```

and this is actually the preferred form.

If you have a free implicit which is meant to be used at runtime, you
have to change it into an (undotted) `{bound : implicit}`

.

You can also put dots in types of functions to get more guarantees.

```
half : (n : Nat) -> .(pf : Even n) -> Nat
```

and free implicits are automatically dotted here, too.

#### What it means¶

Dot annotations serve two purposes:

- influence case-tree elaboration to avoid dotted variables
- trigger warnings when a dotted variable is used

However, there’s no direct connection between being dotted and being erased. The compiler erases everything it can, dotted or not. The dots are there mainly to help the programmer (and the compiler) refrain from using the values they want to erase.

#### How to use it¶

Ideally, few or no extra annotations are needed – in practice, it turns out that having free implicits automatically dotted is enough to get good erasure.

Therefore, just compile with `--warnreach`

to see warnings if
erasure cannot remove parts of the program.

However, those programs that have been written without runtime behaviour in mind, will need some help to get in the form that compiles to a reasonable binary. Generally, it’s sufficient to follow erasure warnings (which may be sometimes unhelpful at the moment).

#### Benchmarks¶

It can be clearly seen that asymptotics are improved by erasure.

#### Shortcomings¶

You can’t get warnings in libraries because usage analysis starts from
`Main.main`

. This will be solved by the planned `%default_usage`

pragma.

Usage warnings are quite bad and unhelpful at the moment. We should include more information and at least translate argument numbers to their names.

There is no decent documentation yet. This wiki page is the first one.

There is no generally accepted terminology. We switch between “dotted”, “unused”, “erased”, “irrelevant”, “inaccessible”, while each has a slightly different meaning. We need more consistent and understandable naming.

If the same type is used in both erased and non-erased context, it
will retain its fields to accommodate the least common denominator –
the non-erased context. This is particularly troublesome in the case
of the type of (dependent) pairs, where it actually means that no
erasure would be performed. We should probably locate disjoint uses of
data types and split them into “sub-types”. There are three different
flavours of dependent types now: `Sigma`

(nothing erased),
`Exists`

(first component erased), `Subset`

(second component
erased).

Case-tree building does not avoid dotted values coming from pattern-matched constructors (https://gist.github.com/ziman/10458331). This is to be fixed soon. (Fixed.)

Higher-order function arguments and opaque functional variables are
considered to be using all their arguments. To work around this, you
can force erasure via the type system, using the `Erased`

wrapper:
https://github.com/idris-lang/Idris-dev/blob/master/libs/base/Data/Erased.idr

Typeclass methods are considered to be using the union of all their implementations. In other words, an argument of a method is unused only if it is unused in every implementation of the method that occurs in the program.

#### Planned features¶

Fixes to the above shortcomings in general.

- Improvements to the case-tree elaborator so that it properly avoids
dotted fields of data constructors. Done.

- Compiler pragma
`%default_usage used/unused`

and per-function overrides

`used`

and`unused`

, which allow the programmer to mark the return value of a function as used, even if the function is not used in`main`

(which is the case when writing library code). These annotations will help library writers discover usage violations in their code before it is actually published and used in compiled programs.

- Compiler pragma

#### Troubleshooting¶

##### My program is slower¶

The patch introducing erasure by usage analysis also disabled some optimisations that were in place before; these are subsumed by the new erasure. However, in some erasure-unaware programs, where erasure by usage analysis does not exercise its full potential (but the old optimisations would have worked), certain slowdown may be observed (up to ~10% according to preliminary benchmarking), due to retention and computation of information that should not be necessary at runtime.

A simple check whether this is the case is to compile with
`--warnreach`

. If you see warnings, there is some unnecessary code
getting compiled into the binary.

The solution is to change the code so that there are no warnings.

##### Usage warnings are unhelpful¶

This is a known issue and we are working on it. For now, see the section How to read and resolve erasure warnings.

##### There should be no warnings in this function¶

A possible cause is non-totality of the function (more precisely, non-coverage). If a function is non-covering, the program needs to inspect all arguments in order to detect coverage failures at runtime. Since the function inspects all its arguments, nothing can be erased and this may transitively cause usage violations. The solution is to make the function total or accept the fact that it will use its arguments and remove some dots from the appropriate constructor fields and function arguments. (Please note that this is not a shortcoming of erasure and there is nothing we can do about it.)

Another possible cause is the currently imperfect case-tree elaboration, which does not avoid dotted constructor fields (see https://gist.github.com/ziman/10458331). You can either rephrase the function or wait until this is fixed, hopefully soon. Fixed.

##### The compiler refuses to recognise this thing as erased¶

You can force anything to be erased by wrapping it in the `Erased`

monad. While this program triggers usage warnings,

```
f : (g : Nat -> Nat) -> .(x : Nat) -> Nat
f g x = g x -- WARNING: g uses x
```

the following program does not:

```
f : (g : Erased Nat -> Nat) -> .(x : Nat) -> Nat
f g x = g (Erase x) -- OK
```

#### How to read and resolve erasure warnings¶

##### Example 1¶

Consider the following program:

```
vlen : Vect n a -> Nat
vlen {n = n} xs = n
sumLengths : List (Vect n a) -> Nat
sumLengths [] = 0
sumLengths (v :: vs) = vlen v + sumLengths vs
main : IO ()
main = print . sumLengths $ [[0,1],[2,3]]
```

When you compile it using `--warnreach`

, there is one warning:

```
Main.sumLengths: inaccessible arguments reachable:
n (no more information available)
```

The warning does not contain much detail at this point so we can try
compiling with `--dumpcases cases.txt`

and look up the compiled
definition in `cases.txt`

:

```
Main.sumLengths {e0} {e1} {e2} =
case {e2} of
| Prelude.List.::({e6}) => LPlus (ATInt ITBig)({e0}, Main.sumLengths({e0}, ____, {e6}))
| Prelude.List.Nil() => 0
```

The reason for the warning is that `sumLengths`

calls `vlen`

, which
gets inlined. The second clause of `sumLengths`

then accesses the
variable `n`

, compiled as `{e0}`

. Since `n`

is a free implicit, it
is automatically considered dotted and this triggers the warning.

A solution would be either making the argument `n`

a bound implicit
parameter to indicate that we wish to keep it at runtime,

```
sumLengths : {n : Nat} -> List (Vect n a) -> Nat
```

or fixing `vlen`

to not use the index:

```
vlen : Vect n a -> Nat
vlen [] = Z
vlen (x :: xs) = S (vlen xs)
```

Which solution is appropriate depends on the usecase.

##### Example 2¶

Consider the following program manipulating value-indexed binary numbers.

```
data Bin : Nat -> Type where
N : Bin Z
O : Bin n -> Bin (0 + n + n)
I : Bin n -> Bin (1 + n + n)
toN : (b : Bin n) -> Nat
toN N = Z
toN (O {n} bs) = 0 + n + n
toN (I {n} bs) = 1 + n + n
main : IO ()
main = print . toN $ I (I (O (O (I N))))
```

In the function `toN`

, we attempted to “cheat” and instead of
traversing the whole structure, we just projected the value index `n`

out of constructors `I`

and `O`

. However, this index is a free
implicit, therefore it is considered dotted.

Inspecting it then produces the following warnings when compiling with
`--warnreach`

:

```
Main.I: inaccessible arguments reachable:
n from Main.toN arg# 1
Main.O: inaccessible arguments reachable:
n from Main.toN arg# 1
```

We can see that the argument `n`

of both `I`

and `O`

is used in
the function `toN`

, argument 1.

At this stage of development, warnings only contain argument numbers,
not names; this will hopefully be fixed. When numbering arguments, we
go from 0, taking free implicits first, left-to-right; then the bound
arguments. The function `toN`

has therefore in fact two arguments:
`n`

(argument 0) and `b`

(argument 1). And indeed, as the warning
says, we project the dotted field from `b`

.

Again, one solution is to fix the function `toN`

to calculate its
result honestly; the other one is to accept that we carry a `Nat`

with every constructor of `Bin`

and make it a bound implicit:

```
O : {n : Nat} -> Bin n -> Bin (0 + n + n)
I : {n : Nat} -> bin n -> Bin (1 + n + n)
```

#### References¶

[BMM04] | Edwin Brady, Conor McBride, James McKinna: Inductive families need not store their indices |

### The IDE Protocol¶

The Idris REPL has two modes of interaction: a human-readable syntax designed for direct use in a terminal, and a machine-readable syntax designed for using Idris as a backend for external tools.

#### Protocol Overview¶

The communication protocol is of asynchronous request-reply style: a single request from the client is handled by Idris at a time. Idris waits for a request on its standard input stream, and outputs the answer or answers to standard output. The result of a request can be either success, failure, or intermediate output; and furthermore, before the result is delivered, there might be additional meta-messages.

A reply can consist of multiple messages: any number of messages to inform the user about the progress of the request or other informational output, and finally a result, either `ok`

or `error`

.

The wire format is the length of the message in characters, encoded in 6 characters hexadecimal, followed by the message encoded as S-expression (sexp). Additionally, each request includes a unique integer (counting upwards), which is repeated in all messages corresponding to that request.

An example interaction from loading the file `/home/hannes/empty.idr`

looks as follows on the wire::

```
00002a((:load-file "/home/hannes/empty.idr") 1)
000039(:write-string "Type checking /home/hannes/empty.idr" 1)
000025(:set-prompt "/home/hannes/empty" 1)
000032(:return (:ok "Loaded /home/hannes/empty.idr") 1)
```

The first message is the request from idris-mode to load the specific file, which length is hex 2a, decimal 42 (including the newline at the end).
The request identifier is set to 1.
The first message from Idris is to write the string `Type checking /home/hannes/empty.idr`

, another is to set the prompt to `*/home/hannes/empty`

.
The answer, starting with `:return`

is `ok`

, and additional information is that the file was loaded.

There are three atoms in the wire language: numbers, strings, and symbols. The only compound object is a list, which is surrounded by parenthesis. The syntax is:

```
A ::= NUM | '"' STR '"' | ':' ALPHA+
S ::= A | '(' S* ')' | nil
```

where `NUM`

is either 0 or a positive integer, `ALPHA`

is an alphabetical character, and `STR`

is the contents of a string, with `"`

escaped by a backslash.
The atom `nil`

is accepted instead of `()`

for compatibility with some regexp pretty-printing routines.

The state of the Idris process is mainly the active file, which needs to be kept synchronised between the editor and Idris.
This is achieved by the already seen `:load-file`

command.

The available commands include:

`(:load-file FILENAME)`

- Load the named file
`(:interpret STRING)`

- Interpret
`STRING`

at the Idris REPL, returning a highlighted result`(:repl-completions STRING)`

- Return the result of tab-completing
`STRING`

as a REPL command`(:type-of STRING)`

- Return the type of the name, written with Idris syntax in the
`STRING`

. The reply may contain highlighting information.`(:case-split LINE NAME)`

- Generate a case-split for the pattern variable
`NAME`

on program line`LINE`

. The pattern-match cases to be substituted are returned as a string with no highlighting.`(:add-clause LINE NAME)`

- Generate an initial pattern-match clause for the function declared as
`NAME`

on program line`LINE`

. The initial clause is returned as a string with no highlighting.`(:add-proof-clause LINE NAME)`

- Add a clause driven by the
`<==`

syntax.`(:add-missing LINE NAME)`

- Add the missing cases discovered by totality checking the function declared as
`NAME`

on program line`LINE`

. The missing clauses are returned as a string with no highlighting.`(:make-with LINE NAME)`

- Create a with-rule pattern match template for the clause of function
`NAME`

on line`LINE`

. The new code is returned with no highlighting.`(:proof-search LINE NAME HINTS)`

- Attempt to fill out the holes on
`LINE```named ``NAME`

by proof search.`HINTS`

is a possibly-empty list of additional things to try while searching.`(:docs-for NAME)`

- Look up the documentation for
`NAME`

, and return it as a highlighted string.`(:metavariables WIDTH)`

- List the currently-active holes, with their types pretty-printed with
`WIDTH`

columns.`(:who-calls NAME)`

- Get a list of callers of
`NAME`

`(:calls-who NAME)`

- Get a list of callees of
`NAME`

`(:browse-namespace NAMESPACE)`

- Return the contents of
`NAMESPACE`

, like`:browse`

at the command-line REPL`(:normalise-term TM)`

- Return a highlighted string consisting of the results of normalising the serialised term
`TM`

(which would previously have been sent as the`tt-term`

property of a string)`(:show-term-implicits TM)`

- Return a highlighted string consisting of the results of making all arguments in serialised term
`TM`

(which would previously have been sent as the`tt-term`

property of a string) explicit.`(:hide-term-implicits TM)`

- Return a highlighted string consisting of the results of making all arguments in serialised term
`TM`

(which would previously have been sent as the`tt-term`

property of a string) follow their usual implicitness setting.`(:elaborate-term TM)`

- Return a highlighted string consisting of the the core language term corresponding to serialised term
`TM`

(which would previously have been sent as the`tt-term`

property of a string).`(:print-definition NAME)`

- Return the definition of
`NAME`

as a highlighted string

Possible replies include a normal final reply::

```
(:return (:ok SEXP [HIGHLIGHTING]))
(:return (:error String [HIGHLIGHTING]))
```

A normal intermediate reply::

```
(:output (:ok SEXP [HIGHLIGHTING]))
(:output (:error String [HIGHLIGHTING]))
```

Informational and/or abnormal replies::

```
(:write-string String)
(:set-prompt String)
(:warning (FilePath (LINE COL) (LINE COL) String [HIGHLIGHTING]))
```

Proof mode replies::

```
(:start-proof-mode)
(:write-proof-state [String] [HIGHLIGHTING])
(:end-proof-mode)
(:write-goal String)
```

#### Output Highlighting¶

Idris mode supports highlighting the output from Idris. In reality, this highlighting is controlled by the Idris compiler. Some of the return forms from Idris support an optional extra parameter: a list mapping spans of text to metadata about that text. Clients can then use this list both to highlight the displayed output and to enable richer interaction by having more metadata present. For example, the Emacs mode allows right-clicking identifiers to get a menu with access to documentation and type signatures.

A particular semantic span is a three element list. The first element of the list is the index at which the span begins, the second element is the number of characters included in the span, and the third is the semantic data itself. The semantic data is a list of lists. The head of each list is a key that denotes what kind of metadata is in the list, and the tail is the metadata itself.

- The following keys are available:
`name`

- gives a reference to the fully-qualified Idris name
`implicit`

- provides a Boolean value that is True if the region is the name of an implicit argument
`decor`

- describes the category of a token, which can be
`type`

,`function`

,`data`

,`keyword`

, or`bound`

. `source-loc`

states that the region refers to a source code location. Its body is a collection of key-value pairs, with the following possibilities:

`filename`

- provides the filename
`start`

- provides the line and column that the source location starts at as a two-element tail
`end`

- provides the line and column that the source location ends at as a two-element tail

`text-formatting`

- provides an attribute of formatted text. This is for use with natural-language text, not code, and is presently emitted only from inline documentation. The potential values are
`bold`

,`italic`

, and`underline`

. `link-href`

- provides a URL that the corresponding text is a link to.
`quasiquotation`

- states that the region is quasiquoted.
`antiquotation`

- states that the region is antiquoted.
`tt-term`

- A serialised representation of the Idris core term corresponding to the region of text.

#### Source Code Highlighting¶

Idris supports instructing editors how to colour their code. When elaborating source code or REPL input, Idris will locate regions of the source code corresponding to names, and emit information about these names using the same metadata as output highlighting.

These messages will arrive as replies to the command that caused elaboration to occur, such as `:load-file`

or `:interpret`

.
They have the format::

```
(:output (:ok (:highlight-source POSNS)))
```

where `POSNS`

is a list of positions to highlight. Each of these is a two-element list whose first element is a position (encoded as for the `source-loc`

property above) and whose second element is highlighting metadata in the same format used for output.

### Semantic Highlighting & Pretty Printing¶

Since `v0.9.18`

Idris comes with support for semantic highlighting.
When using the `REPL`

or IDE support, Idris will highlight your code accordingly to its meaning within the Idris structure. A precursor to semantic highlighting support is the pretty printing of definitions to console, LaTeX, or HTML.

The default styling scheme used was inspired by Conor McBride’s own set of stylings, informally known as *Conor Colours*.

#### Legend¶

The concepts and their default stylings are as follows:

Idris Term | HTML | LaTeX | IDE/REPL |
---|---|---|---|

Bound Variable | Purple | Magenta | |

Keyword | Bold | Underlined | |

Function | Green | Green | |

Type | Blue | Blue | |

Data | Red | Red | |

Implicit | Italic Purple | Italic Magenta |

#### Pretty Printing¶

Idris also supports the pretty printing of code to HTML and LaTeX using the commands:

`:pp <latex|html> <width> <function name>`

`:pprint <latex|html> <width> <function name>`

#### Customisation¶

If you are not happy with the colours used, the VIM and Emacs editor support allows for customisation of the colours. When pretty printing Idris code as LaTeX and HTML, commands and a CSS style are provided. The colours used by the REPL can be customised through the initialisation script.

#### Further Information¶

Please also see the Idris Extras project for links to editor support, and pre-made style files for LaTeX and HTML.

### DEPRECATED: Tactics and Theorem Proving¶

Warning

The interactive theorem-proving interface documented here has been deprecated in favor of Elaborator Reflection.

Idris supports interactive theorem proving, and the analyse of context
through holes. To list all unproven holes, use the command `:m`

.
This will display their qualified names and the expected types. To
interactively prove a holes, use the command `:p name`

where `name`

is the hole. Once the proof is complete, the command `:a`

will append
it to the current module.

Once in the interactive prover, the following commands are available:

#### Basic commands¶

`:q`

- Quits the prover (gives up on proving current lemma).`:abandon`

- Same as :q`:state`

- Displays the current state of the proof.`:term`

- Displays the current proof term complete with its yet-to-be-filled holes (is only really useful for debugging).`:undo`

- Undoes the last tactic.`:qed`

- Once the interactive theorem prover tells you “No more goals,” you get to type this in celebration! (Completes the proof and exits the prover)

#### Commonly Used Tactics¶

##### Compute¶

`compute`

- Normalises all terms in the goal (note: does not normalise assumptions)

```
---------- Goal: ----------
(Vect (S (S Z + (S Z) + (S n))) Nat) -> Vect (S (S (S (S n)))) Nat
-lemma> compute
---------- Goal: ----------
(Vect (S (S (S (S n)))) Nat) -> Vect (S (S (S (S n)))) Nat
-lemma>
```

##### Exact¶

`exact`

- Provide a term of the goal type directly.

```
---------- Goal: ----------
Nat
-lemma> exact Z
lemma: No more goals.
-lemma>
```

##### Refine¶

`refine`

- Use a name to refine the goal. If the name needs arguments, introduce them as new goals.

##### Trivial¶

`trivial`

- Satisfies the goal using an assumption that matches its type.

```
---------- Assumptions: ----------
value : Nat
---------- Goal: ----------
Nat
-lemma> trivial
lemma: No more goals.
-lemma>
```

##### Intro¶

`intro`

- If your goal is an arrow, turns the left term into an assumption.

```
---------- Goal: ----------
Nat -> Nat -> Nat
-lemma> intro
---------- Assumptions: ----------
n : Nat
---------- Goal: ----------
Nat -> Nat
-lemma>
```

You can also supply your own name for the assumption:

```
---------- Goal: ----------
Nat -> Nat -> Nat
-lemma> intro number
---------- Assumptions: ----------
number : Nat
---------- Goal: ----------
Nat -> Nat
```

##### Intros¶

`intros`

- Exactly like intro, but it operates on all left terms at once.

```
---------- Goal: ----------
Nat -> Nat -> Nat
-lemma> intros
---------- Assumptions: ----------
n : Nat
m : Nat
---------- Goal: ----------
Nat
-lemma>
```

##### let¶

`let`

- Introduces a new assumption; you may use current assumptions to define the new one.

```
---------- Assumptions: ----------
n : Nat
---------- Goal: ----------
BigInt
-lemma> let x = toIntegerNat n
---------- Assumptions: ----------
n : Nat
x = toIntegerNat n: BigInt
---------- Goal: ----------
BigInt
-lemma>
```

##### rewrite¶

`rewrite`

- Takes an expression with an equality type (x = y), and replaces all instances of x in the goal with y. Is often useful in combination with ‘sym’.

```
---------- Assumptions: ----------
n : Nat
a : Type
value : Vect Z a
---------- Goal: ----------
Vect (mult n Z) a
-lemma> rewrite sym (multZeroRightZero n)
---------- Assumptions: ----------
n : Nat
a : Type
value : Vect Z a
---------- Goal: ----------
Vect Z a
-lemma>
```

##### induction¶

`induction`

- (`Note that this is still experimental`

and you may get strange results and error messages. We are aware of these and will finish the implementation eventually!) Prove the goal by induction. Each constructor of the datatype becomes a goal. Constructors with recursive arguments become induction steps, while simple constructors become base cases. Note that this only works for datatypes that have eliminators: a datatype definition must have the`%elim`

modifier.

##### sourceLocation¶

`sourceLocation`

- Solve the current goal with information about the location in the source code where the tactic was invoked. This is mostly for embedded DSLs and programmer tools like assertions that need to know where they are called. See`Language.Reflection.SourceLocation`

for more information.

#### Less commonly-used tactics¶

`applyTactic`

- Apply a user-defined tactic. This should be a function of type`List (TTName, Binder TT) -> TT -> Tactic`

, where the first argument represents the proof context and the second represents the goal. If your tactic will produce a proof term directly, use the`Exact`

constructor from`Tactic`

.`attack`

- ?`equiv`

- Replaces the goal with a new one that is convertible with the old one`fill`

- ?`focus`

- ?`mrefine`

- Refining by matching against a type`reflect`

- ?`solve`

- Takes a guess with the correct type and fills a hole with it, closing a proof obligation. This happens automatically in the interactive prover, so`solve`

is really only relevant in tactic scripts used for helping implicit argument resolution.`try`

- ?

### The Idris REPL¶

Idris comes with a `REPL`

.

#### Evaluation¶

Being a fully dependently typed language, Idris has two phases where it evaluates things, compile-time and run-time. At compile-time it will only evaluate things which it knows to be total (i.e. terminating and covering all possible inputs) in order to keep type checking decidable. The compile-time evaluator is part of the Idris kernel, and is implemented in Haskell using a HOAS (higher order abstract syntax) style representation of values. Since everything is known to have a normal form here, the evaluation strategy doesn’t actually matter because either way it will get the same answer, and in practice it will do whatever the Haskell run-time system chooses to do.

The REPL, for convenience, uses the compile-time notion of evaluation. As well as being easier to implement (because we have the evaluator available) this can be very useful to show how terms evaluate in the type checker. So you can see the difference between:

```
Idris> \n, m => (S n) + m
\n => \m => S (plus n m) : Nat -> Nat -> Nat
Idris> \n, m => n + (S m)
\n => \m => plus n (S m) : Nat -> Nat -> Nat
```

#### Customisation¶

Idris supports initialisation scripts.

##### Initialisation scripts¶

When the Idris REPL starts up, it will attempt to open the file
repl/init in Idris’s application data directory. The application data
directory is the result of the Haskell function call
`getAppUserDataDirectory "idris"`

, which on most Unix-like systems
will return $HOME/.idris and on various versions of Windows will return
paths such as `C:/Documents And Settings/user/Application Data/appName`

.

The file repl/init is a newline-separate list of REPL commands. Not all commands are supported in initialisation scripts — only the subset that will not interfere with the normal operation of the REPL. In particular, setting colours, display options such as showing implicits, and log levels are supported.

##### Example initialisation script¶

```
:colour prompt white italic bold
:colour implicit magenta italic
```

#### The `REPL`

Commands¶

The current set of supported commands are:

Command | Arguments | Purpose |
---|---|---|

<expr> | Evaluate an expression | |

:t :type | <expr> | Check the type of an expression |

:core | <expr> | View the core language representation of a term |

:miss :missing | <name> | Show missing clauses |

:doc | <name> | Show internal documentation |

:mkdoc | <namespace> | Generate IdrisDoc for namespace(s) and dependencies |

:apropos | [<package list>] <name> | Search names, types, and documentation |

:s :search | [<package list>] <expr> | Search for values by type |

:wc :whocalls | <name> | List the callers of some name |

:cw :callswho | <name> | List the callees of some name |

:browse | <namespace> | List the contents of some namespace |

:total | <name> | Check the totality of a name |

:r :reload | Reload current file | |

:l :load | <filename> | Load a new file |

:cd | <filename> | Change working directory |

:module | <module> | Import an extra module |

:e :edit | Edit current file using $EDITOR or $VISUAL | |

:m :metavars | Show remaining proof obligations (holes) | |

:p :prove | <hole> | Prove a hole |

:a :addproof | <name> | Add proof to source file |

:rmproof | <name> | Remove proof from proof stack |

:showproof | <name> | Show proof |

:proofs | Show available proofs | |

:x | <expr> | Execute IO actions resulting from an expression using the interpreter |

:c :compile | <filename> | Compile to an executable [codegen] <filename> |

:exec :execute | [<expr>] | Compile to an executable and run |

:dynamic | <filename> | Dynamically load a C library (similar to %dynamic) |

:dynamic | List dynamically loaded C libraries | |

:? :h :help | Display this help text | |

:set | <option> | Set an option (errorcontext, showimplicits, originalerrors, autosolve, nobanner, warnreach, evaltypes, desugarnats) |

:unset | <option> | Unset an option |

:color :colour | <option> | Turn REPL colours on or off; set a specific colour |

:consolewidth | auto|infinite|<number> | Set the width of the console |

:printerdepth | <number-or-blank> | Set the maximum pretty-printing depth, or infinite if nothing specified |

:q :quit | Exit the Idris system | |

:w :warranty | Displays warranty information | |

:let | (<top-level-declaration>)... | Evaluate a declaration, such as a function definition, instance implementation, or fixity declaration |

:unlet :undefine | (<name>)... | Remove the listed repl definitions, or all repl definitions if no names given |

:printdef | <name> | Show the definition of a function |

:pp :pprint | <option> <number> <name> | Pretty prints an Idris function in either LaTeX or HTML and for a specified width. |

#### Using the REPL¶

##### Getting help¶

The command `:help`

(or `:h`

or `:?`

) prints a short summary of
the available commands.

##### Quitting Idris¶

If you would like to leave Idris, simply use `:q`

or `:quit`

.

##### Evaluating expressions¶

To evaluate an expression, simply type it. If Idris is unable to infer
the type, it can be helpful to use the operator `the`

to manually
provide one, as Idris’s syntax does not allow for direct type
annotations. Examples of `the`

include:

```
Idris> the Nat 4
4 : Nat
Idris> the Int 4
4 : Int
Idris> the (List Nat) [1,2]
[1,2] : List Nat
Idris> the (Vect _ Nat) [1,2]
[1,2] : Vect 2 Nat
```

This may not work in cases where the expression still involves ambiguous
names. The name can be disambiguated by using the `with`

keyword:

```
Idris> sum [1,2,3]
When elaborating an application of function Prelude.Foldable.sum:
Can't disambiguate name: Prelude.List.::,
Prelude.Stream.::,
Prelude.Vect.::
Idris> with List sum [1,2,3]
6 : Integer
```

##### Adding let bindings¶

To add a let binding to the REPL, use `:let`

. It’s likely you’ll also
need to provide a type annotation. `:let`

also works for other
declarations as well, such as `data`

.

```
Idris> :let x : String; x = "hello"
Idris> x
"hello" : String
Idris> :let y = 10
Idris> y
10 : Integer
Idris> :let data Foo : Type where Bar : Foo
Idris> Bar
Bar : Foo
```

##### Getting type information¶

To ask Idris for the type of some expression, use the `:t`

command.
Additionally, if used with an overloaded name, Idris will provide all
overloadings and their types. To ask for the type of an infix operator,
surround it in parentheses.

```
Idris> :t "foo"
"foo" : String
Idris> :t plus
Prelude.Nat.plus : Nat -> Nat -> Nat
Idris> :t (++)
Builtins.++ : String -> String -> String
Prelude.List.++ : (List a) -> (List a) -> List a
Prelude.Vect.++ : (Vect m a) -> (Vect n a) -> Vect (m + n) a
Idris> :t plus 4
plus (Builtins.fromInteger 4) : Nat -> Nat
```

You can also ask for basic information about typeclasses with `:doc`

:

```
Idris> :doc Monad
Type class Monad
Parameters:
m
Methods:
(>>=) : Monad m => m a -> (a -> m b) -> m b
infixl 5
Instances:
Monad id
Monad PrimIO
Monad IO
Monad Maybe
...
```

Other documentation is also available from `:doc`

:

```
Idris> :doc (+)
Prelude.Classes.+ : (a : Type) -> (Num a) -> a -> a -> a
infixl 8
Arguments:
Class constraint Prelude.Classes.Num a
__pi_arg : a
__pi_arg1 : a
```

```
Idris> :doc Vect
Data type Prelude.Vect.Vect : Nat -> Type -> Type
Arguments:
Nat
Type
Constructors:
Prelude.Vect.Nil : (a : Type) -> Vect 0 a
Prelude.Vect.:: : (a : Type) -> (n : Nat) -> a -> (Vect n a) -> Vect (S n) a
infixr 7
Arguments:
a
Vect n a
```

```
Idris> :doc Monad
Type class Prelude.Monad.Monad
Methods:
Prelude.Monad.>>= : (m : Type -> Type) -> (a : Type) -> (b : Type) -> (Monad m) -> (m a) -> (a -> m b) -> m b
infixl 5
Arguments:
Class constraint Prelude.Monad.Monad m
__pi_arg : m a
__pi_arg1 : a -> m b
```

##### Finding things¶

The command `:apropos`

searches names, types, and documentation for
some string, and prints the results. For example:

```
Idris> :apropos eq
eqPtr : Ptr -> Ptr -> IO Bool
eqSucc : (left : Nat) -> (right : Nat) -> (left = right) -> S left = S right
S preserves equality
lemma_both_neq : ((x = x') -> _|_) -> ((y = y') -> _|_) -> ((x, y) = (x', y')) -> _|_
lemma_fst_neq_snd_eq : ((x = x') -> _|_) -> (y = y') -> ((x, y) = (x', y)) -> _|_
lemma_snd_neq : (x = x) -> ((y = y') -> _|_) -> ((x, y) = (x, y')) -> _|_
lemma_x_eq_xs_neq : (x = y) -> ((xs = ys) -> _|_) -> (x :: xs = y :: ys) -> _|_
lemma_x_neq_xs_eq : ((x = y) -> _|_) -> (xs = ys) -> (x :: xs = y :: ys) -> _|_
lemma_x_neq_xs_neq : ((x = y) -> _|_) -> ((xs = ys) -> _|_) -> (x :: xs = y :: ys) -> _|_
prim__eqB16 : Bits16 -> Bits16 -> Int
prim__eqB16x8 : Bits16x8 -> Bits16x8 -> Bits16x8
prim__eqB32 : Bits32 -> Bits32 -> Int
prim__eqB32x4 : Bits32x4 -> Bits32x4 -> Bits32x4
prim__eqB64 : Bits64 -> Bits64 -> Int
prim__eqB64x2 : Bits64x2 -> Bits64x2 -> Bits64x2
prim__eqB8 : Bits8 -> Bits8 -> Int
prim__eqB8x16 : Bits8x16 -> Bits8x16 -> Bits8x16
prim__eqBigInt : Integer -> Integer -> Int
prim__eqChar : Char -> Char -> Int
prim__eqFloat : Float -> Float -> Int
prim__eqInt : Int -> Int -> Int
prim__eqString : String -> String -> Int
prim__syntactic_eq : (a : Type) -> (b : Type) -> (x : a) -> (y : b) -> Maybe (x = y)
sequence : Traversable t => Applicative f => (t (f a)) -> f (t a)
sequence_ : Foldable t => Applicative f => (t (f a)) -> f ()
Eq : Type -> Type
The Eq class defines inequality and equality.
GTE : Nat -> Nat -> Type
Greater than or equal to
LTE : Nat -> Nat -> Type
Proofs that n is less than or equal to m
gte : Nat -> Nat -> Bool
Boolean test than one Nat is greater than or equal to another
lte : Nat -> Nat -> Bool
Boolean test than one Nat is less than or equal to another
ord : Char -> Int
Convert the number to its ASCII equivalent.
replace : (x = y) -> (P x) -> P y
Perform substitution in a term according to some equality.
sym : (l = r) -> r = l
Symmetry of propositional equality
trans : (a = b) -> (b = c) -> a = c
Transitivity of propositional equality
```

`:search`

does a type-based search, in the spirit of Hoogle. See Type-directed search (:search) for more details. Here is an example:

```
Idris> :search a -> b -> a
= Prelude.Basics.const : a -> b -> a
Constant function. Ignores its second argument.
= assert_smaller : a -> b -> b
Assert to the totality checker than y is always structurally
smaller than x (which is typically a pattern argument)
> malloc : Int -> a -> a
> Prelude.pow : Num a => a -> Nat -> a
> Prelude.Classes.(*) : Num a => a -> a -> a
> Prelude.Classes.(+) : Num a => a -> a -> a
... (More results)
```

`:search`

can also look for dependent types:

```
Idris> :search plus (S n) n = plus n (S n)
< Prelude.Nat.plusSuccRightSucc : (left : Nat) ->
(right : Nat) ->
S (left + right) = left + S right
```

##### Loading and reloading Idris code¶

The command `:l File.idr`

will load File.idr into the
currently-running REPL, and `:r`

will reload the last file that was
loaded.

##### Totality¶

All Idris definitions are checked for totality. The command
`:total <NAME>`

will display the result of that check. If a definition
is not total, this may be due to an incomplete pattern match. If that is
the case, `:missing`

or `:miss`

will display the missing cases.

##### Editing files¶

The command `:e`

launches your default editor on the current module.
After control returns to Idris, the file is reloaded.

##### Invoking the compiler¶

The current module can be compiled to an executable using the command
`:c <FILENAME>`

or `:compile <FILENAME>`

. This command allows to
specify codegen, so for example JavaScript can be generated using
`:c javascript <FILENAME>`

. The `:exec`

command will compile the
program to a temporary file and run the resulting executable.

##### IO actions¶

Unlike GHCI, the Idris REPL is not inside of an implicit IO monad. This
means that a special command must be used to execute IO actions.
`:x tm`

will execute the IO action `tm`

in an Idris interpreter.

##### Dynamically loading C libraries¶

Sometimes, an Idris program will depend on external libraries written in
C. In order to use these libraries from the Idris interpreter, they must
first be dynamically loaded. This is achieved through the
`%dynamic <LIB>`

directive in Idris source files or through the
`:dynamic <LIB>`

command at the REPL. The current set of dynamically
loaded libraries can be viewed by executing `:dynamic`

with no
arguments. These libraries are available through the Idris FFI in type
providers and `:exec`

.

#### Colours¶

Idris terms are available in amazing colour! By default, the Idris REPL uses colour to distinguish between data constructors, types or type constructors, operators, bound variables, and implicit arguments. This feature is available on all POSIX-like systems, and there are plans to allow it to work on Windows as well.

If you do not like the default colours, they can be turned off using the command

```
:colour off
```

and, when boredom strikes, they can be re-enabled using the command

```
:colour on
```

To modify a colour, use the command

```
:colour <CATEGORY> <OPTIONS>
```

where `<CATEGORY`

is one of `keyword`

, `boundvar`

, `implicit`

,
`function`

, `type`

, `data`

, or `prompt`

, and is a
space-separated list drawn from the colours and the font options. The
available colours are `default`

, `black`

, `yellow`

, `cyan`

,
`red`

, `blue`

, `white`

, `green`

, and `magenta`

. If more than
one colour is specified, the last one takes precedence. The available
options are `dull`

and `vivid`

, `bold`

and `nobold`

, `italic`

and `noitalic`

, `underline`

and `nounderline`

, forming pairs of
opposites. The colour `default`

refers to your terminal’s default
colour.

The colours used at startup can be changed using REPL initialisation scripts.

Colour can be disabled at startup by the `--nocolour`

command-line
option.

### Compilation and Logging¶

This section provides highlights of the Idris compilation process, and provides details over how you can follow the process through logging.

#### Compilation Process¶

Idris follows the following compilation process:

- Parsing
- Type Checking
- Elaboration
- Coverage
- Unification
- Totality Checking
- Erasure

- Code Generation
- Defunctionalisation
- Inlining
- Resolving variables
- Code Generation

#### Type Checking Only¶

With Idris you can ask it to terminate the compilation process after type checking has completed. This is achieved through use of either:

- The command line options
`--check`

for files`--checkpkg`

for packages

- The REPL command:
`:check`

Use of this option will still result in the generation of the Idris binary `.ibc`

files, and is suitable if you do not wish to generate code from one of the supported backends.

#### Logging Output¶

The internal operation of Idris is captured using a category based logger. Currently, the logging infrastructure has support for the following categories:

- Parser
- Elaborator
- Code generation
- Erasure
- Coverage Checking
- IBC generation

These categories are specified using the command-line option:
`--logging-categories CATS`

, where `CATS`

is a quoted colon
seperated string of the categories you want to see. By default if this
option is not specified all categories are allowed. Sub-categories
have yet to be defined but will be in the future, especially for the
elaborator.

Further, the verbosity of logging can be controlled by specifying a
logging level between: 1 to 10 using the command-line option: ```
--log
<level>
```

.

- Level 0: Show no logging output. Default level
- Level 1: High level details of the compilation process.
- Level 2: Provides details of the coverage checking, and further details the elaboration process specifically: Class, Clauses, Data, Term, and Types,
- Level 3: Provides details of compilation of the IRTS, erasure, parsing, case splitting, and further details elaboration of: Instances, Providers, and Values.
- Level 4: Provides further details on: Erasure, Coverage Checking, Case splitting, and elaboration of clauses.
- Level 5: Provides details on the prover, and further details elaboration (adding declarations) and compilation of the IRTS.
- Level 6: Further details elaboration and coverage checking.
- Level 7:
- Level 8:
- Level 9:
- Level 10: Further details elaboration.

### Core Language Features¶

- Full-spectrum dependent types
- Strict evaluation (plus
`Lazy : Type -> Type`

type constructor for explicit laziness) - Lambda, Pi (forall), Let bindings
- Pattern matching definitions
- Export modifiers
`public`

,`abstract`

,`private`

- Function options
`partial`

,`total`

`where`

clauses- “magic with”
- Implicit arguments (in top level types)
- “Bound” implicit arguments
`{n : Nat} -> {a : Type} -> Vect n a`

- “Unbound” implicit arguments —
`Vect n a`

is equivalent to the above in a type,`n`

and`a`

are implicitly bound. This applies to names beginning with a lower case letter in an argument position. - ‘Tactic’ implicit arguments, which are solved by running a tactic script or giving a default argument, rather than by unification.
- Unit type
`()`

, empty type`Void`

- Tuples (desugaring to nested pairs)
- Dependent pair syntax
`(x : T ** P x)`

(there exists an`x`

of type`T`

such that`P x`

) - Inline
`case`

expressions - Heterogeneous equality
`do`

notation- Idiom brackets
- Interfaces (like type classes), supporting default methods and dependencies between methods
`rewrite`

prf`in`

expr- Metavariables
- Inline proof/tactic scripts
- Implicit coercion
`codata`

- Also
`Inf : Type -> Type`

type constructor for mixed data/codata. In fact`codata`

is implemented by putting recursive arguments under`Inf`

. `syntax`

rules for defining pattern and term syntactic sugar- these are used in the standard library to define
`if ... then ... else`

expressions and an Agda-style preorder reasoning syntax. - Uniqueness
typing
using the
`UniqueType`

universe. - Partial
evaluation
by
`[static]`

argument annotations. - Error message reflection
- Eliminators
- Label types
`'name`

`%logging n`

`%unifyLog`

### Language Extensions¶

#### Type Providers¶

Idris type providers are a way to get the type system to reflect observations about the world outside of Idris. Similarly to F# type providers, they cause effectful computations to run during type checking, returning information that the type checker can use when checking the rest of the program. While F# type providers are based on code generation, Idris type providers use only the ordinary execution semantics of Idris to generate the information.

A type provider is simply a term of type `IO (Provider t)`

, where
`Provider`

is a data type with constructors for a successful result
and an error. The type `t`

can be either `Type`

(the type of types)
or a concrete type. Then, a type provider `p`

is invoked using the
syntax `%provide (x : t) with p`

. When the type checker encounters
this line, the IO action `p`

is executed. Then, the resulting term is
extracted from the IO monad. If it is `Provide y`

for some `y : t`

,
then `x`

is bound to `y`

for the remainder of typechecking and in
the compiled code. If execution fails, a generic error is reported and
type checking terminates. If the resulting term is `Error e`

for some
string `e`

, then type checking fails and the error `e`

is reported
to the user.

Example Idris type providers can be seen at this repository. More detailed descriptions are available in David Christiansen’s WGP ‘13 paper and M.Sc. thesis.

### Elaborator Reflection¶

The Idris elaborator is responsible for converting high-level Idris code into the core language.
It is implemented as a kind of embedded tactic language in Haskell, where tactic scripts are written in an *elaboration monad* that provides error handling and a proof state.
For details, see Edwin Brady’s 2013 paper in the Journal of Functional Programming.

Elaborator reflection makes the elaboration type as well as a selection of its tactics available to Idris code. This means that metaprograms written in Idris can have complete control over the elaboration process, generating arbitrary code, and they have access to all of the facilities available in the elaborator, such as higher-order unification, type checking, and emitting auxiliary definitions.

#### The Elaborator State¶

The elaborator state contains information about the ongoing elaboration process.
In particular, it contains a *goal type*, which is to be filled by an under-construction *proof term*.
The proof term can contain *holes*, each of which has a scope in which it is valid and a type.
Some holes may additionally contain *guesses*, which can be substituted in the scope of the hole.
The holes are tracked in a *hole queue*, and one of them is *focused*.
In addition to the goal type, proof term, and holes, the elaborator state contains a collection of unsolved unification problems that can affect elaboration.

The elaborator state is not directly available to Idris programs.
Instead, it is modified through the use of *tactics*, which are operations that affect the elaborator state.
A tactic that returns a value of type `a`

, potentially modifying the elaborator state, has type `Elab a`

.
The default tactics are all in the namespace `Language.Reflection.Elab.Tactics`

.

#### Running Elaborator Scripts¶

On their own, tactics have no effect.
The meta-operation `%runElab script`

runs `script`

in the current elaboration context.
For example, the following script constructs the identity function at type `Nat`

:

```
idNat : Nat -> Nat
idNat = %runElab (do intro (Just (UN "x"))
fill (Var (UN "x"))
solve)
```

On the right-hand side, the Idris elaborator has the goal `Nat -> Nat`

.
When it encounters the `%runElab`

directive, it fulfills this goal by running the provided script.
The first tactic, `intro`

, constructs a lambda that binds the name `x`

.
The name argument is optional because a default name can be taken from the function type.
Now, the proof term is of the form `\x : Nat => {hole}`

.
The second tactic, `fill`

, fills this hole with a guess, giving the term `\x : Nat => {hole≈x}`

.
Finally, the `solve`

tactic instantiates the guess, giving the result `\x : Nat => x`

.

Because elaborator scripts are ordinary Idris expressions, it is also possible to use them in multiple contexts.
Note that there is nothing `Nat`

-specific about the above script.
We can generate identity functions at any concrete type using the same script:

```
mkId : Elab ()
mkId = do intro (Just (UN "x"))
fill (Var (UN "x"))
solve
idNat : Nat -> Nat
idNat = %runElab mkId
idUnit : () -> ()
idUnit = %runElab mkId
idString : String -> String
idString = %runElab mkId
```

#### Interactively Building Elab Scripts¶

You can build an `Elab`

script interactively at the REPL.
Use the command `:metavars`

, or `:m`

for short, to list the available holes.
Then, issue the `:elab <hole>`

command at the REPL
to enter the elaboration shell.

At the shell, you can enter proof tactics to alter the proof state.
You can view the system-provided tactics prior to entering the shell
by issuing the REPL command `:browse Language.Reflection.Elab.Tactics`

.
When you have discharged all goals, you can complete the proof
using the `:qed`

command and receive in return an elaboration script
that fills the hole.

The interactive elaboration shell accepts a limited number of commands, including a subset of the commands understood by the normal Idris REPL as well as some elaboration-specific commands.

General-purpose commands:

`:eval <EXPR>`

, or`:e <EXPR>`

for short, evaluates the provided expression and prints the result.`:type <EXPR>`

, or`:t <EXPR>`

for short, prints the provided expression together with its type.`:search <TYPE>`

searches for definitions having the provided type.`:doc <NAME>`

searches for definitions with the provided name and prints their documentation.

Commands for viewing the proof state:

`:state`

displays the current state of the term being constructed. It lists both other goals and the current goal.`:term`

displays the current proof term as well as its yet-to-be-filled holes.

Commands for manipulating the proof state:

`:undo`

undoes the effects of the last tactic.`:abandon`

gives up on proving the current lemma and quits the elaboration shell.`:qed`

finishes the script and exits the elaboration shell. The shell will only accept this command once it reports, “No more goals.” On exit, it will print out the finished elaboration script for you to copy into your program.

#### Failure¶

Some tactics may *fail*.
For example, `intro`

will fail if the focused hole does not have a function type, `solve`

will fail if the current hole does not contain a guess, and `fill`

will fail if the term to be filled in has the wrong type.
Scripts can also fail explicitly using the `fail`

tactic.

To account for failure, there is an `Alternative`

instance for `Elab`

.
The `<|>`

operator first tries the script to its left.
If that script fails, any changes that it made to the state are undone and the right argument is executed.
If the first argument succeeds, then the second argument is not executed.

#### Querying the Elaboration State¶

`Elab`

includes operations to query the elaboration state, allowing scripts to use information about their environment to steer the elaboration process.
The ordinary Idris bind syntax can be used to propagate this information.
For example, a tactic that solves the current goal when it is the unit type might look like this:

```
triv : Elab ()
triv = do compute
g <- getGoal
case (snd g) of
`(() : Type) => do fill `(() : ())
solve
otherGoal => fail [ TermPart otherGoal
, TextPart "is not trivial"
]
```

The tactic `compute`

normalises the type of its goal with respect to the current context.
While not strictly necessary, this allows `triv`

to be used in contexts where the triviality of the goal is not immediately apparent.
Then, `getGoal`

is used, and its result is bound to `g`

.
Because it returns a pair consisting of the current goal’s name and type, we case-split on its second projection.
If the goal type turns out to have been the unit type, we fill using the unit constructor and solve the goal.
Otherwise, we fail with an error message informing the user that the current goal is not trivial.

Additionally, the elaboration state can be dumped into an error message with the `debug`

tactic.
A variant, `debugMessage`

, allows arbitrary messages to be included with the state, allowing for a kind of “`printf`

debugging” of elaboration scripts.
The message format used by `debugMessage`

is the same for errors produced by the error reflection mechanism, allowing the re-use of the Idris pretty-printer when rendering messages.

#### Changing the Global Context¶

`Elab`

scripts can modify the global context during execution.
Just as the Idris elaborator produces auxiliary definitions to implement features such as `where`

-blocks and `case`

expressions, user elaboration scripts may need to define functions.
Furthermore, this allows `Elab`

reflection to be used to implement features such as type class deriving.
The operations `declareType`

, `defineFunction`

, and `addInstance`

allow `Elab`

scripts to modify the global context.

#### Using Idris’s Features¶

The Idris compiler has a number of ways to automate the construction of terms.
On its own, the `Elab`

state and its interactions with the unifier allow implicits to be solved using unification.
Additional operations use further features of Idris.
In particular, `resolveTC`

solves the current goal using type class resolution, `search`

invokes the proof search mechanism, and `sourceLocation`

finds the context in the original file at which the elaboration script is invoked.

#### Recursive Elaboration¶

The elaboration mechanism can be invoked recursively using the `runElab`

tactic.
This tactic takes a goal type and an elaboration script as arguments and runs the script in a fresh lexical environment to create an inhabitant of the provided goal type.
This is primarily useful for code generation, particularly for generating pattern-matching clauses, where variable scope needs to be one that isn’t the present local context.

#### Learn More¶

While this documentation is still incomplete, elaboration reflection works in Idris today.
As you wait for the completion of the documentation, the list of built-in tactics can be obtained using the `:browse`

command in an Idris REPL or the corresponding feature in one of the graphical IDE clients to explore the `Language.Reflection.Elab.Tactics`

namespace.
All of the built-in tactics contain documentation strings.

### Miscellaneous¶

Things we have yet to classify, or are two small to justify their own page.

#### The Unifier Log¶

If you’re having a hard time debugging why the unifier won’t accept
something (often while debugging the compiler itself), try applying the
special operator `%unifyLog`

to the expression in question. This will
cause the type checker to spit out all sorts of informative messages.

#### Namespaces and type-directed disambiguation¶

Names can be defined in separate namespaces, and disambiguated by type.
An expression `with NAME EXPR`

will privilege the namespace `NAME`

in the expression `EXPR`

. For example:

```
Idris> with List [[1,2],[3,4],[5,6]]
[[1, 2], [3, 4], [5, 6]] : List (List Integer)
Idris> with Vect [[1,2],[3,4],[5,6]]
[[1, 2], [3, 4], [5, 6]] : Vect 3 (Vect 2 Integer)
Idris> [[1,2],[3,4],[5,6]]
Can't disambiguate name: Prelude.List.::, Prelude.Stream.::, Prelude.Vect.::
```

#### Alternatives¶

The syntax `(| option1, option2, option3, ... |)`

type checks each
of the options in turn until one of them works. This is used, for
example, when translating integer literals.

```
Idris> the Nat (| "foo", Z, (-3) |)
0 : Nat
```

This can also be used to give simple automated proofs, for example: trying some constructors of proofs.

```
syntax Trivial = (| oh, refl |)
```

#### Totality checking assertions¶

All definitions are checked for *coverage* (i.e. all well-typed
applications are handled) and either for *termination* (i.e. all
well-typed applications will eventually produce an answer) or, if
returning codata, for productivity (in practice, all recursive calls are
constructor guarded).

Obviously, termination checking is undecidable. In practice, the
termination checker looks for *size change* - every cycle of recursive
calls must have a decreasing argument, such as a recursive argument of a
strictly positive data type.

There are two built-in functions which can be used to give the totality checker a hint:

`assert_total x`

asserts that the expression`x`

is terminating and covering, even if the totality checker cannot tell. This can be used for example if`x`

uses a function which does not cover all inputs, but the caller knows that the specific input is covered.`assert_smaller p x`

asserts that the expression`x`

is structurally smaller than the pattern`p`

.

For example, the following function is not checked as total:

```
qsort : Ord a => List a -> List a
qsort [] = []
qsort (x :: xs) = qsort (filter (<= x) xs) ++ (x :: qsort (filter (>= x) xs)))
```

This is because the checker cannot tell that `filter`

will always
produce a value smaller than the pattern `x :: xs`

for the recursive
call to `qsort`

. We can assert that this will always be true as
follows:

```
total
qsort : Ord a => List a -> List a
qsort [] = []
qsort (x :: xs) = qsort (assert_smaller (x :: xs) (filter (<= x) xs)) ++
(x :: qsort (assert_smaller (x :: xs) (filter (>= x) xs))))
```

#### Preorder reasoning¶

This syntax is defined in the module `Syntax.PreorderReasoning`

in the
`base`

package. It provides a syntax for composing proofs of
reflexive-transitive relations, using overloadable functions called
`step`

and `qed`

. This module also defines `step`

and `qed`

functions allowing the syntax to be used for demonstrating equality.
Here is an example:

```
import Syntax.PreorderReasoning
multThree : (a, b, c : Nat) -> a * b * c = c * a * b
multThree a b c =
(a * b * c) ={ sym (multAssociative a b c) }=
(a * (b * c)) ={ cong (multCommutative b c) }=
(a * (c * b)) ={ multAssociative a c b }=
(a * c * b) ={ cong {f = (*b)} (multCommutative a c) }=
(c * a * b) QED
```

Note that the parentheses are required – only a simple expression can
be on the left of `={ }=`

or `QED`

. Also, when using preorder
reasoning syntax to prove things about equality, remember that you can
only relate the entire expression, not subexpressions. This might
occasionally require the use of `cong`

.

Finally, although equality is the most obvious application of preorder
reasoning, it can be used for any reflexive-transitive relation.
Something like `step1 ={ just1 }= step2 ={ just2 }= end QED`

is
translated to `(step step1 just1 (step step2 just2 (qed end)))`

,
selecting the appropriate definitions of `step`

and `qed`

through
the normal disambiguation process. The standard library, for example,
also contains an implementation of preorder reasoning on isomorphisms.

#### Pattern matching on Implicit Arguments¶

Pattern matching is only allowed on implicit arguments when they are referred by name, e.g.

```
foo : {n : Nat} -> Nat
foo {n = Z} = Z
foo {n = S k} = k
```

or

```
foo : {n : Nat} -> Nat
foo {n = n} = n
```

The latter could be shortened to the following:

```
foo : {n : Nat} -> Nat
foo {n} = n
```

That is, `{x}`

behaves like `{x=x}`

.

#### Existence of an instance¶

In order to show that an instance of some typeclass is defined for some
type, one could use the `%instance`

keyword:

```
foo : Num Nat
foo = %instance
```

#### ‘match’ application¶

`ty <== name`

applies the function `name`

in such a way that it has
the type `ty`

, by matching `ty`

against the function’s type. This
can be used in proofs, for example:

```
plus_comm : (n : Nat) -> (m : Nat) -> (n + m = m + n)
-- Base case
(Z + m = m + Z) <== plus_comm =
rewrite ((m + Z = m) <== plusZeroRightNeutral) ==>
(Z + m = m) in refl
-- Step case
(S k + m = m + S k) <== plus_comm =
rewrite ((k + m = m + k) <== plus_comm) in
rewrite ((S (m + k) = m + S k) <== plusSuccRightSucc) in
refl
```

#### Reflection¶

Including `%reflection`

functions and `quoteGoal x by fn in t`

,
which applies `fn`

to the expected type of the current expression, and
puts the result in `x`

which is in scope when elaborating `t`

.

## Tutorials on the Idris Language¶

Tutorials submitted by community members.

Note

*The
Idris Community* has waived all copyright and related or neighboring
rights to Documentation for Idris.

### Type Providers in Idris¶

Type providers in Idris are simple enough, but there are a few caveats to using them that it would be worthwhile to go through the basic steps. We also go over foreign functions, because these will often be used with type providers.

#### The use case¶

First, let’s talk about *why* we might want type providers. There are
a number of reasons to use them and there are other examples available
around the net, but in this tutorial we’ll be using them to port C’s
`struct stat`

to Idris.

Why do we need type providers? Well, Idris’s FFI needs to know the
types of the things it passes to and from C, but the fields of a
`struct stat`

are implementation-dependent types that cannot be
relied upon. We don’t just want to hard-code these types into our
program... so we’ll use a type provider to find them at compile time!

#### A simple example¶

First, let’s go over a basic usage of type providers, because foreign functions can be confusing but it’s important to remember that providers themselves are simple.

A type provider is simply an IO action that returns a value of this type:

```
data Provider a = Provide a | Error String
```

Looks familiar? `Provider`

is just `Either a String`

, given a
slightly more descriptive name.

Remember though, type providers we use in our program must be IO actions. Let’s write a simple one now:

```
module Provider
-- Asks nicely for the user to supply the size of C's size_t type on this
-- machine
getSizeT : IO (Provider Int)
getSizeT = do
putStrLn "I'm sorry, I don't know how big size_t is. Can you tell me, in bytes?"
resp <- getLine
case readInt resp of
Just sizeTSize => return (Provide sizeTSize)
Nothing => return (Error "I'm sorry, I don't understand.")
-- the readInt function is left as an exercise
```

We assume that whoever’s compiling the library knows the size of
`size_t`

, so we’ll just ask them! (Don’t worry, we’ll get it
ourselves later.) Then, if their response can be converted to an
integer, we present `Provide sizeTSize`

as the result of our IO
action; or if it can’t, we signal a failure. (This will then become a
compile-time error.)

Now we can use this IO action as a type provider:

```
module Main
-- to gain access to the IO action we're using as a provider
import Provider
-- TypeProviders is an extension, so we'll enable it
%language TypeProviders
-- And finally, use the provider!
-- Note that the parentheses are mandatory.
%provide (sizeTSize : Int) with getSizeT
-- From now on it's just a normal program where `sizeTSize` is available
-- as a top-level constant
main : IO ()
main = do
putStr "Look! I figured out how big size_t is! It's "
putStr (show sizeTSize)
putStr " bytes!"
```

Yay! We... asked the user something at compile time? That’s not very
good, actually. Our library is going to be difficult to compile! This
is hardly a step up from having them edit in the size of `size_t`

themselves!

Don’t worry, there’s a better way.

#### Foreign Functions¶

It’s actually pretty easy to write a C function that figures out the
size of `size_t`

:

```
int sizeof_size_t() { return sizeof(size_t); }
```

(Why an int and not a `size_t`

? The FFI needs to know how to receive
the return value of this function and translate it into an Idris
value. If we knew how to do this for values of C type `size_t`

, we
wouldn’t need to write this function at all! If we really wanted to be
safe from overflow, we could use an array of multiple integers, but
the SIZE of `size_t`

is never going to be a 65535 byte integer.)

So now we can get the size of `size_t`

as long as we’re in C code.
We’d like to be able to use this from Idris. Can we do this? It turns
out we can.

`mkForeign`

¶

With mkForeign, we can turn a C function into an IO action. It works like this:

```
getSizeT : IO Int
getSizeT = mkForeign (FFun "sizeof_size_t" [] FInt)
```

Pretty simple. `mkForeign`

takes a specification of what function it
needs to call, and we construct this specification with `FFun`

. And
`FFun`

just takes a name, a list of argument types (we have none),
and a return type.

One thing you might want to note: the return type we’ve specified is
`FInt`

, not `Int`

. That’s because `Int`

is an idris type and C
functions don’t return idris types. `FInt`

is not an idris type, but
is just the representation of the type of a C int. It tells the
compiler “Treat the return value of this C function like it’s a C int,
and when you pass it back into Idris, convert it to an Idris int.”

##### Caveats of mkForeign¶

First and foremost: `mkForeign`

is not actually a function. It is
treated specially by the compiler, and there are certain rules you
need to follow when using it.

- Rule 1: the name string must be a literal or constant

This does not work:

```
intIntToInt : String -> Int -> Int -> IO Int
intIntToInt name = mkForeign (FFun name [FInt, FInt] FInt)
```

You’ll just have to bite the bullet and write out the whole
`mkForeign`

and `FFun`

expression each time.

- Rule 2: the “call” to
`mkForeign`

must be fully applied

This just means that every argument appearing in the list of argument
types must be applied wherever you write `mkForeign`

. The arguments
don’t have to be literals or even known at compile time; they just
have to be there. For example, if we have ```
strlen : String -> IO
Int
```

, then this is fine:

```
strlen str = mkForeign (FFun "strlen" [FString] FInt) str
```

but this is not fine:

```
strlen = mkForeign (FFun "strlen" [FString] FInt)
```

Note that this only applies to places where you literally typed
`mkForeign`

. Once you’ve defined it, `strlen`

is just a normal
function returning an IO action, and it doesn’t need to be fully
applied. This is okay:

```
lengths : IO [Int]
lengths = mapM strlen listOfStrings
```

##### Running foreign functions¶

This is all well and good for writing code that will typecheck. To actually run the code, we’ll need to do just a bit more work. Exactly what we need to do depends on whether we want to interpret or compile our code.

##### In the interpreter¶

If we want to call our foreign functions from interpreted code (such
as the REPL or a type provider), we need to dynamically link a library
containing the symbols we need. This is pretty easy to do with the
`%dynamic`

directive:

```
%dynamic "./filename.so"
```

Note that the leading ”./” is important: currently, the string you
provide is interpreted as by `dlopen()`

, which on Unix does not search
in the current directory by default. If you use the ”./”, the library
will be searched for in the directory from which you run idris (*not*
the location of the file you’re running!). Of course, if you’re using
functions from an installed library rather than something you wrote
yourself, the ”./” is not necessary.

##### In an executable¶

If we want to run our code from an executable, we can statically link
instead. We’ll use the `%include`

and `%link`

directives:

```
%include C "filename.h"
%link C "filename.o"
```

Note the extra argument to the directive! We specify that we’re
linking a C header and library. Also, unlike `%dynamic`

, these
directives search in the current directory by default. (That is, the
directory from which we run idris.)

#### Putting it all together¶

So, at the beginning of this article I said we’d use type providers to
port `struct stat`

to Idris. The relevant part is just translating
all the mysterious typedef’d C types into Idris types, and that’s what
we’ll do here.

First, let’s write a C file containing functions that we’ll bind to.

```
/* stattypes.c */
int sizeof_dev_t() { return sizeof(dev_t); }
int sizeof_ino_t() { return sizeof(ino_t); }
/* lots more functions like this */
```

Next, an Idris file to define our providers:

```
-- Providers.idr
module Providers
%dynamic "./stattypes.so"
sizeOfDevT : IO Int
sizeOfDevT = mkForeign (FFun "sizeof_dev_t" [] FInt)
{- lots of similar functions -}
-- now we have an integer, but we want a Provider FTy
-- since our sizeOf* functions are ordinary IO actions, we
-- can just map over them.
bytesToType : Int -> Provider FTy
bytesToType 1 = Provide (FIntT IT8) -- "8 bit foreign integer"
bytesToType 2 = Provide (FIntT IT16)
bytesToType 4 = Provide (FIntT IT32)
bytesToType 8 = Provide (FIntT IT64)
bytesToType _ = Error "Unrecognised integral type."
getDevT : IO (Provider FTy)
getDevT = map bytesToType sizeOfDevT
{- lots of similar functions -}
```

Finally, we’ll write one more idris file where we use the type providers:

```
-- Main.idr
module Main
import Providers
%language TypeProviders
%provide (FDevT : FTy) with getDevT
-- interpFTy translates a foreign type to the corresponding idris type
DevT : Type
DevT = interpFTy FDevT -- on most systems, DevT = Bits64
-- We can now use DevT in our program and FDevT in our FFun expressions!
```

### The Interactive Theorem Prover¶

This short guide contributed by a community member illustrates how to prove associativity of addition on `Nat`

using the interactive theorem prover.

First we define a module `Foo.idr`

```
module Foo
plusAssoc : plus n (plus m o) = plus (plus n m) o
plusAssoc = ?rhs
```

We wish to perform induction on `n`

. First we load the file into the Idris `REPL`

as follows:

```
$ idris Foo.idr
```

We will be given the following prompt, in future releases the version string will differ:

```
____ __ _
/ _/___/ /____(_)____
/ // __ / ___/ / ___/ Version 0.9.18.1
_/ // /_/ / / / (__ ) http://www.idris-lang.org/
/___/\__,_/_/ /_/____/ Type :? for help
Idris is free software with ABSOLUTELY NO WARRANTY.
For details type :warranty.
Type checking ./Foo.idr
Metavariables: Foo.rhs
*Foo>
```

#### Explore the Context¶

We start the interactive session by asking Idris to prove the
hole `rhs`

using the command `:p rhs`

. Idris by default
will show us the initial context. This looks as follows:

```
*Foo> :p rhs
---------- Goal: ----------
{ hole 0 }:
(n : Nat) ->
(m : Nat) ->
(o : Nat) ->
plus n (plus m o) = plus (plus n m) o
```

#### Application of Intros¶

We first apply the `intros`

tactic:

```
-Foo.rhs> intros
---------- Other goals: ----------
{ hole 2 }
{ hole 1 }
{ hole 0 }
---------- Assumptions: ----------
n : Nat
m : Nat
o : Nat
---------- Goal: ----------
{ hole 3 }:
plus n (plus m o) = plus (plus n m) o
```

#### Induction on `n`

¶

Then apply `induction`

on to `n`

:

```
-Foo.rhs> induction n
---------- Other goals: ----------
elim_S0
{ hole 2 }
{ hole 1 }
{ hole 0 }
---------- Assumptions: ----------
n : Nat
m : Nat
o : Nat
---------- Goal: ----------
elim_Z0:
plus Z (plus m o) = plus (plus Z m) o
```

#### Compute¶

```
-Foo.rhs> compute
---------- Other goals: ----------
elim_S0
{ hole 2 }
{ hole 1 }
{ hole 0 }
---------- Assumptions: ----------
n : Nat
m : Nat
o : Nat
---------- Goal: ----------
elim_Z0:
plus m o = plus m o
```

#### Trivial¶

```
-Foo.rhs> trivial
---------- Other goals: ----------
{ hole 2 }
{ hole 1 }
{ hole 0 }
---------- Assumptions: ----------
n : Nat
m : Nat
o : Nat
---------- Goal: ----------
elim_S0:
(n__0 : Nat) ->
(plus n__0 (plus m o) = plus (plus n__0 m) o) ->
plus (S n__0) (plus m o) = plus (plus (S n__0) m) o
```

#### Intros¶

```
-Foo.rhs> intros
---------- Other goals: ----------
{ hole 4 }
elim_S0
{ hole 2 }
{ hole 1 }
{ hole 0 }
---------- Assumptions: ----------
n : Nat
m : Nat
o : Nat
n__0 : Nat
ihn__0 : plus n__0 (plus m o) = plus (plus n__0 m) o
---------- Goal: ----------
{ hole 5 }:
plus (S n__0) (plus m o) = plus (plus (S n__0) m) o
```

#### Compute¶

```
-Foo.rhs> compute
---------- Other goals: ----------
{ hole 4 }
elim_S0
{ hole 2 }
{ hole 1 }
{ hole 0 }
---------- Assumptions: ----------
n : Nat
m : Nat
o : Nat
n__0 : Nat
ihn__0 : plus n__0 (plus m o) = plus (plus n__0 m) o
---------- Goal: ----------
{ hole 5 }:
S (plus n__0 (plus m o)) = S (plus (plus n__0 m) o)
```

#### Rewrite¶

```
-Foo.rhs> rewrite ihn__0
---------- Other goals: ----------
{ hole 5 }
{ hole 4 }
elim_S0
{ hole 2 }
{ hole 1 }
{ hole 0 }
---------- Assumptions: ----------
n : Nat
m : Nat
o : Nat
n__0 : Nat
ihn__0 : plus n__0 (plus m o) = plus (plus n__0 m) o
---------- Goal: ----------
{ hole 6 }:
S (plus n__0 (plus m o)) = S (plus n__0 (plus m o))
```

#### Trivial¶

```
-Foo.rhs> trivial
rhs: No more goals.
-Foo.rhs> qed
Proof completed!
Foo.rhs = proof
intros
induction n
compute
trivial
intros
compute
rewrite ihn__0
trivial
```

Two goals were created: one for `Z`

and one for `S`

.
Here we have proven associativity, and assembled a tactic based proof script.
This proof script can be added to `Foo.idr`

.

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