DOKK Library

The Idris Tutorial v1.3.3


License CC0-1.0

The Idris Tutorial

    Version 1.3.3

1 Introduction                                                                                          2

2 Getting Started                                                                                       3

3 Types and Functions                                                                                   5

4 Interfaces                                                                                           22

5 Modules and Namespaces                                                                               31

6 Packages                                                                                             35

7 Example: The Well-Typed Interpreter                                                                  37

8 Views and the “with” rule                                                                            41

9 Theorem Proving                                                                                      43

10 Provisional Definitions                                                                             48

11 Interactive Editing                                                                                 52

12 Syntax Extensions                                                                                   55

13 Miscellany                                                                                          58

14 Further Reading                                                                                     66

This is the Idris Tutorial. It provides a brief introduction to programming in the Idris Language. It
covers the core language features, and assumes some familiarity with an existing functional programming
language such as Haskell or OCaml.

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:      

1     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 length1 , 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

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 general pur-
pose 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

1.1        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.

For a more in-depth introduction to Idris, which proceeds at a much slower pace, covering interactive
program development, with many more examples, see Type-Driven Development with Idris by Edwin
Brady, available from Manning.

1.2        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.

2        Getting Started

2.1        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 version of GHC. The earliest version we currently test with is 7.10.3.

    • The GNU Multiple Precision Arithmetic Library (GMP) is available from MacPorts/Homebrew
      and all major Linux distributions.

2.2        Downloading and Installing

The easiest way to install Idris, if you have all of the prerequisites, is to type:
    1   Typically, and perhaps confusingly, referred to in the dependently typed programming literature as “vectors”

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:

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

  • -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

  • --help display usage summary and command line options.

2.3      The Interactive Environment

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

$ idris
    ____    __    _
   / _/___/ /____(_)____
   / // __ / ___/ / ___/        Version 1.3.3
 _/ // /_/ / / / (__ )
/___/\__,_/_/ /_/____/          Type :? for help


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 1.3.3
  _/ // /_/ / / / (__ ) 
 /___/\__,_/_/ /_/____/           Type :? for help

Type checking ./hello.idr
*hello> :t main
Main.main : IO ()
*hello> :c hello
*hello> :q
Bye bye
$ ./hello
Hello world

3     Types and Functions

3.1    Primitive Types

Idris defines several primitive types: Int, Integer and Double for numeric operations, Char and String
for text manipulation, and Ptr which represents foreign pointers. There are also several data types
declared in the library, including Bool, with values True and False. We can declare some constants with
these types. Enter the following into a file Prims.idr and load it into the Idris interactive environment
by typing 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 (page 31). 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

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 (page 22) 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

3.2   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).

Data types may also be declared by giving just the types of the constructors. These definitions are
equivalent to those above:

data Nat : Type where
    Z : Nat
    S : Nat -> Nat

data List : Type -> Type where
    Nil : List a
    (::) : a -> List a -> List a

This syntax is more verbose, but more flexible, and is used for types that can’t be described with the
simpler syntax.

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 !.

3.3    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. By convention,
however, data types and constructor names typically begin with a capital letter. 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

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
       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


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

main : IO ()
main = putStrLn ?greeting

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

*Hello> :t greeting
greeting : String

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

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

We can check the type of even_rhs and see the expected return type, and the type of the variable k:

*Even> :t even_rhs
  k : Nat
even_rhs : Bool

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

3.4     Dependent Types

First Class Types

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

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

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

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

Or it can be used to have varying input types. The following function calculates either the sum of a list
of Nat, or returns the given Nat, depending on whether the singleton flag is true:

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


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
(++) : 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
When checking right hand side of Vect.++ with expected type
        Vect (S k + m) a

When checking an application of constructor Vect.:::
        Type mismatch between
                Vect (k + k) a (Type of xs ++ xs)
                Vect (plus k m) a (Expected type)

                  Type mismatch between
                          plus k k
                          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+1th 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.

Furthermore, {} can be used to pattern match on the left hand side, i.e. {var = pat} gets an implicit
variable and attempts to pattern match on “pat”; For example:

isEmpty : Vect n a -> Bool
isEmpty {n = Z} _   = True
isEmpty {n = S k} _ = False

“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 Main.testVec
inVect = There (There Here)

Important: Implicit Arguments and Scope

Within the type signature the typechecker will treat all variables that start with an lowercase letter and
are not applied to something else as an implicit variable. To get the above code example to compile you
will need to provide a qualified name for testVec. In the example above, we have assumed that the code
lives within the Main module.

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 type checking can rely on how particular functions are defined (though this
is only true of total functions; see Section Totality Checking (page 46)). However, this restriction can be
relaxed by using a mutual block, which allows data types and functions to be defined simultaneously:

  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.

3.5    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,

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.

3.6    “do” notation

I/O programs will typically need to sequence actions, feeding the output of one computation into the
input of the next. IO is an abstract type, however, so we can’t access the result of a computation directly.
Instead, we sequence operations with do notation:

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

The syntax x <- iovalue executes the I/O operation iovalue, of type IO a, and puts the result, of type
a into the variable x. In this case, getLine returns an IO String, so name has type String. Indentation
is significant — each statement in the do block must begin in the same column. The pure operation
allows us to inject a value directly into an IO operation:

pure : a -> IO a

As we will see later, do notation is more general than this, and can be overloaded.

3.7    Laziness

Normally, arguments to functions are evaluated before the function itself (that is, Idris uses eager eval-
uation). 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

3.8    Codata Types

Codata types allow us to define infinite data structures by marking recursive arguments as potentially
infinite. For a codata type T, each of its constructor arguments of type T are transformed into an
argument of type Inf T. This makes each of the T arguments lazy, and allows infinite data structures of
type T to be built. One example of a codata type is Stream, which is defined as follows.

codata Stream : Type -> Type where
  (::) : (e : a) -> Stream a -> Stream a

This gets translated into the following by the compiler.
data Stream : Type -> Type where
  (::) : (e : a) -> Inf (Stream a) -> Stream a

The following is an example of how the codata type Stream can be used to form an infinite data structure.
In this case we are creating an infinite stream of ones.
ones : Stream Nat
ones = 1 :: ones

It is important to note that codata does not allow the creation of infinite mutually recursive data
structures. For example the following will create an infinite loop and cause a stack overflow.
  codata Blue a = B a (Red a)
  codata Red a = R a (Blue a)

  blue : Blue Nat
  blue = B 1 red

  red : Red Nat
  red = R 1 blue

  findB : (a -> Bool) -> Blue a -> a
  findB f (B x r) = if f x then x else findR f r

  findR : (a -> Bool) -> Red a -> a
  findR f (R x b) = if f x then x else findB f b

main : IO ()
main = do printLn $ findB (== 1) blue

To fix this we must add explicit Inf declarations to the constructor parameter types, since codata will
not add it to constructor parameters of a different type from the one being defined. For example, the
following outputs 1.
  data Blue : Type -> Type where
   B : a -> Inf (Red a) -> Blue a

  data Red : Type -> Type where
   R : a -> Inf (Blue a) -> Red a

  blue : Blue Nat
  blue = B 1 red

  red : Red Nat
  red = R 1 blue

  findB : (a -> Bool) -> Blue a -> a
  findB f (B x r) = if f x then x else findR f r

                                                                                   (continues on next page)

                                                                               (continued from previous page)
  findR : (a -> Bool) -> Red a -> a
  findR f (R x b) = if f x then x else findB f b

main : IO ()
main = do printLn $ findB (== 1) blue

3.9    Useful Data Types

Idris includes a number of useful data types and library functions (see the libs/ directory in the distri-
bution, and the documentation). This section 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 (page 31)), 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.

Aside: Anonymous functions and operator sections

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


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

data Pair a b = MkPair a b

As syntactic sugar, we can write (a, b) which, according to context, means either Pair a b or MkPair
a b. Tuples can contain an arbitrary number of values, represented as nested pairs:

fred : (String, Int)
fred = ("Fred", 42)

jim : (String, Int, String)
jim = ("Jim", 25, "Cambridge")

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

Dependent Pairs

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

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

Again, there is syntactic sugar for this. (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 : DPair Nat (\n => Vect n Int)
vec = MkDPair 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

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.

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


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 $= (+ 1) } fred
MkPerson   "Jim" "Joe" "Bloggs"   31 : Person

The syntax record { field = val, ... } generates a function which updates the given fields in a
record. = assigns a new value to a field, and $= applies a function to update its value.

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

Records, 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

We could also use $= to define addStudent more concisely:
addStudent' : Person -> Class -> Class
addStudent' p c = record { students $= (p ::) } c

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

The $= notation is also valid for nested record updates.

Dependent Records

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

Using the Class record from earlier, the size of the class can be restricted using a Vect and the size
included in the type by parameterising the record with the size. For example:
record SizedClass (size : Nat) where
    constructor SizedClassInfo
    students : Vect size Person
    className : String

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)

3.10    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

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

The [a..b] notation is another shorthand which builds a list of numbers between a and b. Alternatively
[a,b..c] builds a list of numbers between a and c with the increment specified by the difference between
a and b. This works for type Nat, Int and Integer, using the enumFromTo and enumFromThenTo function
from the prelude.

case expressions

Another way of inspecting intermediate values 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.

3.11     Totality

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

    • Terminates for all possible inputs, or

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

If a function is total, we can consider its type a precise description of what that function will do. For
example, if we have a function with a return type of String we know something different, depending on
whether or not it’s total:

    • If it’s total, it will return a value of type String in finite time;

    • If it’s partial, then as long as it doesn’t crash or enter an infinite loop, it will return a String.

Idris makes this distinction so that it knows which functions are safe to evaluate while type checking
(as we’ve seen with First Class Types (page 9)). After all, if it tries to evaluate a function during type
checking which doesn’t terminate, then type checking won’t terminate! Therefore, only total functions
will be evaluated during type checking. Partial functions can still be used in types, but will not be
evaluated further.

4      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 Double 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 unnamed implementation of an interface can be given for a type, and implementations may
not overlap. But see Named Implementations (page 29) below.

Implementation declarations can themselves have constraints. To help with resolution, the arguments of
an implementation must be constructors (either data or type constructors) or variables (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

4.1    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
                                                                                      (continues on next page)

                                                                               (continued from previous page)

      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

4.2     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 round brackets (parentheses) in a comma separated list, for example:

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

Note: Interfaces and mutual blocks

Idris is strictly “define before use”, except in mutual blocks. In a mutual block, Idris elaborates in
two passes: types on the first pass and definitions on the second. When the mutual block contains an

interface declaration, it elaborates the interface header but none of the method types on the first pass,
and elaborates the method types and any default definitions on the second pass.

4.3   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

4.4   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 (page 14). 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 implemen-
tation 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
               pure (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.
*Interfaces> m_add (Just 20) (Just 22)
Just 42 : Maybe Int
*Interfaces> 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
                                                                                   (continues on next page)

                                                                                 (continued from previous page)
      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.


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 = pure (!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 (page 21) 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; pure 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 ]

4.5     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 |]
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
evaluator1 , for a language similar to the following:
   1 Conor McBride and Ross Paterson. 2008. Applicative programming with effects. J. Funct. Program. 18, 1 (January

2008), 1-13. DOI=10.1017/S0956796807006326

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

Evaluation will take place relative to a context mapping variables (represented as Strings) 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

4.6     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

Sometimes, we also need access to a named parent implementation. For example, the prelude defines
the following Semigroup interface:
interface Semigroup ty where
  (<+>) : ty -> ty -> ty

Then it defines Monoid, which extends Semigroup with a “neutral” value:
interface Semigroup ty => Monoid ty where
  neutral : ty

We can define two different implementations of Semigroup and Monoid for Nat, one based on addition
and one on multiplication:
[PlusNatSemi] Semigroup Nat where
  (<+>) x y = x + y

[MultNatSemi] Semigroup Nat where
  (<+>) x y = x * y

The neutral value for addition is 0, but the neutral value for multiplication is 1. It’s important, therefore,
that when we define implementations of Monoid they extend the correct Semigroup implementation. We
can do this with a using clause in the implementation as follows:
[PlusNatMonoid] Monoid Nat using PlusNatSemi where
  neutral = 0

[MultNatMonoid] Monoid Nat using MultNatSemi where
  neutral = 1

The using PlusNatSemi clause indicates that PlusNatMonoid should extend PlusNatSemi specifically.

4.7    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.

5     Modules and Namespaces

An Idris program consists of a collection of modules. Each module includes an optional module decla-
ration 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

public export
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)

The modifiers export and public export say which names are visible from other modules. These are
explained further below.

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: 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 would import the file foo/bar.idr, which would conventionally have the
module declaration module 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.

5.1   Export Modifiers

Idris allows for fine-grained control over the visibility of a module’s contents. By default, all names
defined in a module are kept private. This aides in specification of a minimal interface and for internal
details to be left hidden. Idris allows for functions, types, and interfaces to be marked as: private,
export, or public export. Their general meaning is as follows:

  • private meaning that it’s not exported at all. This is the default.

  • export meaning that its top level type is exported.

  • public export meaning that the entire definition is exported.

A further restriction in modifying the visibility is that definitions must not refer to anything within a
lower level of visibility. For example, public export definitions cannot use private names, and export
types cannot use private names. This is to prevent private names leaking into a module’s interface.

Meaning for Functions

  • export the type is exported

  • public export the type and definition are exported, and the definition can be used after it is
    imported. In other words, the definition itself is considered part of the module’s interface. The
    long name public export is intended to make you think twice about doing this.

Note: Type synonyms in Idris are created by writing a function. When setting the visibility for a
module, it might be a good idea to public export all type synonyms if they are to be used outside the
module. Otherwise, Idris won’t know what the synonym is a synonym for.

Since public export means that a function’s definition is exported, this effectively makes the function
definition part of the module’s API. Therefore, it’s generally a good idea to avoid using public export
for functions unless you really mean to export the full definition.

Meaning for Data Types

For data types, the meanings are:

  • export the type constructor is exported

  • public export the type constructor and data constructors are exported

Meaning for Interfaces

For interfaces, the meanings are:

  • export the interface name is exported

  • public export the interface name, method names and default definitions are exported

%access Directive

The default export mode can be changed with the %access directive, for example:

module Btree

%access export

public export
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 export, rather than left private.

Propagating Inner Module API’s

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

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.

5.2   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

                                                                                  (continues on next page)

                                                                           (continued from previous page)
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

5.3    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

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

6     Packages

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

6.1     Package Descriptions

A package description includes the following:

    • A header, consisting of the keyword package followed by a package name. Package names can
      be any valid Idris identifier. The iPKG format also takes a quoted version that accepts any valid

    • 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.

6.2     Using Package files

Idris itself is aware about packages, and special commands are available to help with, for example,
building packages, installing packages, and cleaning packages. For instance, given the maths package
from earlier we can use Idris 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

Once the maths package has been installed, the command line option --package maths makes it acces-
sible (abbreviated to -p maths). For example:

idris -p maths Main.idr

6.3     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

%access export -- to make functions under test visible

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

%access export   -- to make the test functions visible

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
                                                                                  (continues on next page)

                                                                              (continued from previous page)
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.

6.4     Package Dependencies Using Atom

If you are using the Atom editor and have a dependency on another package, corresponding to for
instance import Lightyear or import Pruviloj, you need to let Atom know that it should be loaded.
The easiest way to accomplish that is with a .ipkg file. The general contents of an ipkg file will be
described in the next section of the tutorial, but for now here is a simple recipe for this trivial case:

    • Create a folder myProject.

    • Add a file myProject.ipkg containing just a couple of lines:

package myProject

pkgs = pruviloj, lightyear

    • In Atom, use the File menu to Open Folder myProject.

6.5     More information

More details, including a complete listing of available fields, can be found in the reference manual in

7     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.

7.1     Representing Languages

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

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        = Integer
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 : Integer) -> 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 nth most recently defined variable is well-typed, so is the n+1th. 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 : Integer) -> 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

7.2      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 oper-
ators, 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

7.3    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)))

7.4    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
          printLn (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 1.3.2
  _/ // /_/ / / / (__ )    
 /___/\__,_/_/ /_/____/              Type :? for help

Type checking ./interp.idr
*interp> :exec
Enter a number: 6

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.

8     Views and the “with” rule

8.1    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.

8.2    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 Epigram1 , 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.

We have already seen a vector filter function. This time, we define it using with as follows:
filter :   (a -> Bool) -> Vect n a -> (p ** Vect p a)
filter p   [] = ( _ ** [] )
filter p   (x :: xs) with (filter p xs)
  filter   p (x :: xs) | ( _ ** xs' ) = if (p x) then ( _ ** x :: xs' ) else ( _ ** xs' )

   1 Conor McBride and James McKinna. 2004. The view from the left. J. Funct. Program. 14, 1 (January 2004), 69-111.

Here, the with clause allows us to deconstruct the result of filter p xs. The view refined argument
pattern filter p (x :: xs) goes beneath the with clause, followed by a vertical bar |, followed by
the deconstructed intermediate result ( _ ** xs' ). If the view refined argument pattern is unchanged
from the original function argument pattern, then the left side of | is extraneous and may be omitted:

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

with clauses can also be nested:

foo : Int -> Int -> Bool
foo n m with (succ n)
  foo _ m | 2 with (succ m)
    foo _ _ | 2 | 3 = True
    foo _ _ | 2 | _ = False
  foo _ _ | _ = False

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. In these cases, view refined
argument patterns must be explicit. For example, a Nat is either even or odd. If it is 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 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 |. That is:

  • When parity k evaluates to Even, we can refine the original argument k to a refined pattern (j
    + j) according to Parity (n + n) from the Even constructor definition. So (j + j) replaces k
    on the left side of |, and the Even constructor appears on the right side. The natural number j in
    the refined pattern can be used on the right side of the = sign.

  • Otherwise, when parity k evaluates to Odd, the original argument k is refined to S (j + j)
    according to Parity (S (n + n)) from the Odd constructor definition, and Odd now appears on
    the right side of |, again with the natural number j used on the right side of the = sign.

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 the next section Theorems in Practice (page 44) to complete the
definition of parity.

8.3   With and proofs

To use a dependent pattern match for theorem proving, it is sometimes necessary to explicitly construct
the proof resulting from the pattern match. To do this, you can postfix the with clause with proof p
and the proof generated by the pattern match will be in scope and named p. For example:

data Foo = FInt Int | FBool Bool

optional : Foo -> Maybe Int
optional (FInt x) = Just x
optional (FBool b) = Nothing

isFInt : (foo:Foo) -> Maybe (x : Int ** (optional foo = Just x))
isFInt foo with (optional foo) proof p
  isFInt foo | Nothing = Nothing           -- here, p : Nothing = optional foo
  isFInt foo | (Just x) = Just (x ** Refl) -- here, p : Just x = optional foo

9     Theorem Proving

9.1   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

9.2   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 (page 46) 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 ()
    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

9.3    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 correspondence1 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

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

9.4    Theorems in Practice

The need to prove theorems can arise naturally in practice. For example, previously (Views and the
“with” rule (page 41)) we implemented natToBin using a function parity:

   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.

parity : (n:Nat) -> Parity n

However, we didn’t provide a definition for parity. We might expect it to look something like the

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}

Unfortunately, this fails with a type error:

When checking right hand side of with block in views.parity with expected type
        Parity (S (S (j + j)))

Type mismatch between
        Parity (S j + S j) (Type of Even)
        Parity (S (S (plus j j))) (Expected type)

The problem is that normalising S j + S j, in the type of Even doesn’t result in what we need for the
type of the right hand side of Parity. We know that S (S (plus j j)) is going to be equal to S j
+ S j, but we need to explain it to Idris with a proof. We can begin by adding some holes (see Holes
(page 8)) to the definition:

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 = let result = Even {n=S j} in
  parity (S (S (S (j + j)))) | Odd = let result = Odd {n=S j} in

Checking the type of helpEven shows us what we need to prove for the Even case:

  j : Nat
  result : Parity (S (plus j (S j)))
helpEven : Parity (S (S (plus j j)))

We can therefore write a helper function to rewrite the type to the form we need:

helpEven : (j : Nat) -> Parity (S j + S j) -> Parity (S (S (plus j j)))
helpEven j p = rewrite plusSuccRightSucc j j in p

The rewrite ... in syntax allows you to change the required type of an expression by rewriting it
according to an equality proof. Here, we have used plusSuccRightSucc, which has the following type:

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

We can see the effect of rewrite by replacing the right hand side of helpEven with a hole, and working
step by step. Beginning with the following:

helpEven : (j : Nat) -> Parity (S j + S j) -> Parity (S (S (plus j j)))
helpEven j p = ?helpEven_rhs

We can look at the type of helpEven_rhs:
  j : Nat
  p : Parity (S (plus j (S j)))
helpEven_rhs : Parity (S (S (plus j j)))

Then we can rewrite by applying plusSuccRightSucc j j, which gives an equation S (j + j) = j +
S j, thus replacing S (j + j) (or, in this case, S (plus j j) since S (j + j) reduces to that) in the
type with j + S j:
helpEven : (j : Nat) -> Parity (S j + S j) -> Parity (S (S (plus j j)))
helpEven j p = rewrite plusSuccRightSucc j j in ?helpEven_rhs

Checking the type of helpEven_rhs now shows what has happened, including the type of the equation
we just used (as the type of _rewrite_rule):
  j : Nat
  p : Parity (S (plus j (S j)))
  _rewrite_rule : S (plus j j) = plus j (S j)
helpEven_rhs : Parity (S (plus j (S j)))

Using rewrite and another helper for the Odd case, we can complete parity as follows:
helpEven : (j : Nat) -> Parity (S j + S j) -> Parity (S (S (plus j j)))
helpEven j p = rewrite plusSuccRightSucc j j in p

helpOdd : (j : Nat) -> Parity (S (S (j + S j))) -> Parity (S (S (S (j + j))))
helpOdd j p = rewrite plusSuccRightSucc j j in p

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 = helpEven j (Even {n = S j})
  parity (S (S (S (j + j)))) | Odd = helpOdd j (Odd {n = S j})

Full details of rewrite are beyond the scope of this introductory tutorial, but it is covered in the theorem
proving tutorial (see proofs-index).

9.5    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 (page 43) that the zero and successor constructors
are disjoint is total:

*theorems> :total disjoint

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:
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

10     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}
                                                                                      (continues on next page)

                                                                                 (continued from previous page)
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))
    Parity (S (S (plus j j)))

    Type mismatch between
        plus (S j) (S j)
        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?

10.1    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:

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)))


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
(page 41) 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

10.2    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

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 {
    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.

10.3    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 {
                                                                                    (continues on next page)

                                                                              (continued from previous page)
     rewrite sym (plusSuccRightSucc j j);

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 =   ""

11     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

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.

11.1    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' : 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.

11.2    Editing Commands


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.


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.


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 empty vectors, generates the required clauses, and eliminates
the clauses which would lead to unification errors.


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 pos-
sibility. Similarly, and perhaps surprisingly, there is only one possibility if we try to solve :ps 97

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.


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.

11.3     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 the current                             line       (using

   • \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 the cursor (using :proofsearch).

   • \p invokes a proof search with additional hints to solve the hole under the cursor (using

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

   • \t displays the type of the (globally visible) name under the cursor.                        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.

12      Syntax Extensions

Idris supports the implementation of Embedded Domain Specific Languages (EDSLs) in several ways1 .
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.

12.1     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 (page 14)):

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

   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-

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 arbi-
    trary 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 : Double) -> (upper : Double) ->
                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.

12.2     dsl notation

The well-typed interpreter in Section Example: The Well-Typed Interpreter (page 37) 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 data2 or running concurrent processes safely3 .

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 overloading1 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:

   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.
   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.

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

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)

13     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.

13.1    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

Auto implicit arguments

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}

Default implicit arguments

Besides having Idris automatically find a value of a given type, sometimes we want to have an implicit
argument with a specific default value. In Idris, we can do this using the default annotation. While
this is primarily intended to assist in automatically constructing a proof where auto fails, or finds an
unhelpful value, it might be easier to first consider a simpler case, not involving proofs.

If we want to compute the n’th fibonacci number (and defining the 0th fibonacci number as 0), we could

fibonacci : {default 0 lag : Nat} -> {default 1 lead : Nat} -> (n : Nat) -> Nat
fibonacci {lag} Z = lag
fibonacci {lag} {lead} (S n) = fibonacci {lag=lead} {lead=lag+lead} n

After this definition, fibonacci 5 is equivalent to fibonacci {lag=0} {lead=1} 5, and will return the
5th fibonacci number. Note that while this works, this is not the intended use of the default annotation.
It is included here for illustrative purposes only. Usually, default is used to provide things like a custom
proof search script.

13.2    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.

13.3    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).

13.4    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 = Double
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 pass-
    ing 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 — dynamically link the interpreter with the shared object

Testing foreign function calls

Normally, the Idris interpreter (used for typechecking and as 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
Idris> :x unsafePerformIO ((mkForeign (FFun "sin" [FFloat] FFloat)) 1.6)
0.9995736030415051 : Double

13.5    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 Right str <- readFile fname
                      | Left err => pure (Provide Void)
                    pure (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 collection1 .

13.6      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

To compile the generated C with debugging information e.g. to use gdb to debug segmentation faults in
Idris programs, use the %flag C pragma to include debugging symbols, as is shown below:
%flag C "-g"

13.7    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

                                                                                  (continues on next page)

                                                                             (continued from previous page)
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

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. For library packages you can also use the
ipkg objs option to include the js file in the installation, and use:

%include Node "package/external.js"

The JavaScript and NodeJS backends of Idris will also lookup for the file on that location.

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.

13.8    Cumulativity

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

*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.

14      Further Reading

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

   • The Idris web site ( and by asking questions on the mailing list.

   • The IRC channel #idris, on

   • The wiki ( has further user provided in-
         formation, in particular:



   • Examining the prelude and exploring the samples in the distribution.                          The Idris source
        can be found online at:

   • Existing projects on the Idris Hackers web space:

   • Various papers (e.g.1 ,2 , and3 ). Although these mostly describe older versions of Idris.

   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-
   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.
   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 confer-
ence on Functional programming (ICFP ‘10). ACM, New York, NY, USA, 297-308. DOI=10.1145/1863543.1863587