# Agda

A dependently typed functional programming language and proof assistant http://wiki.portal.chalmers.se/agda/

Version on this page: | 2.5.1 |

LTS Haskell 8.19: | 2.5.2 |

Stackage Nightly 2017-06-22: | 2.5.2 |

Latest on Hackage: | 2.5.2 |

**Ulf Norell, Andreas Abel, Nils Anders Danielsson, Makoto Takeyama, Catarina Coquand, with contributions by Stevan Andjelkovic, Marcin Benke, Jean-Philippe Bernardy, James Chapman, Jesper Cockx, Dominique Devriese, Peter Divanski, Fredrik Nordvall Forsberg, Olle Fredriksson, Daniel Gustafsson, Philipp Hausmann, Patrik Jansson, Alan Jeffrey, Wolfram Kahl, Fredrik Lindblad, Francesco Mazzoli, Stefan Monnier, Darin Morrison, Guilhem Moulin, Nicolas Pouillard, Andrés Sicard-Ramírez, Andrea Vezzosi and many more.**

**Ulf Norell**

#### Module documentation for 2.5.1

- Agda
- Agda.Auto
- Agda.Benchmarking
- Agda.Compiler
- Agda.Compiler.CallCompiler
- Agda.Compiler.Common
- Agda.Compiler.Epic
- Agda.Compiler.Epic.AuxAST
- Agda.Compiler.Epic.CaseOpts
- Agda.Compiler.Epic.CompileState
- Agda.Compiler.Epic.Compiler
- Agda.Compiler.Epic.Epic
- Agda.Compiler.Epic.Erasure
- Agda.Compiler.Epic.ForceConstrs
- Agda.Compiler.Epic.Forcing
- Agda.Compiler.Epic.FromAgda
- Agda.Compiler.Epic.Injection
- Agda.Compiler.Epic.Interface
- Agda.Compiler.Epic.NatDetection
- Agda.Compiler.Epic.Primitive
- Agda.Compiler.Epic.Smashing
- Agda.Compiler.Epic.Static

- Agda.Compiler.HaskellTypes
- Agda.Compiler.JS
- Agda.Compiler.MAlonzo
- Agda.Compiler.ToTreeless
- Agda.Compiler.Treeless
- Agda.Compiler.Treeless.Builtin
- Agda.Compiler.Treeless.Compare
- Agda.Compiler.Treeless.Erase
- Agda.Compiler.Treeless.GuardsToPrims
- Agda.Compiler.Treeless.NormalizeNames
- Agda.Compiler.Treeless.Pretty
- Agda.Compiler.Treeless.Simplify
- Agda.Compiler.Treeless.Subst
- Agda.Compiler.Treeless.Uncase
- Agda.Compiler.Treeless.Unused

- Agda.Compiler.UHC

- Agda.ImpossibleTest
- Agda.Interaction
- Agda.Interaction.BasicOps
- Agda.Interaction.CommandLine
- Agda.Interaction.EmacsCommand
- Agda.Interaction.EmacsTop
- Agda.Interaction.Exceptions
- Agda.Interaction.FindFile
- Agda.Interaction.Highlighting
- Agda.Interaction.Imports
- Agda.Interaction.InteractionTop
- Agda.Interaction.Library
- Agda.Interaction.MakeCase
- Agda.Interaction.Monad
- Agda.Interaction.Options
- Agda.Interaction.Response
- Agda.Interaction.SearchAbout

- Agda.Main
- Agda.Syntax
- Agda.Termination
- Agda.Tests
- Agda.TheTypeChecker
- Agda.TypeChecking
- Agda.TypeChecking.Abstract
- Agda.TypeChecking.CheckInternal
- Agda.TypeChecking.CompiledClause
- Agda.TypeChecking.Constraints
- Agda.TypeChecking.Conversion
- Agda.TypeChecking.Coverage
- Agda.TypeChecking.Datatypes
- Agda.TypeChecking.DeadCode
- Agda.TypeChecking.DisplayForm
- Agda.TypeChecking.DropArgs
- Agda.TypeChecking.Empty
- Agda.TypeChecking.Errors
- Agda.TypeChecking.EtaContract
- Agda.TypeChecking.Forcing
- Agda.TypeChecking.Free
- Agda.TypeChecking.Implicit
- Agda.TypeChecking.Injectivity
- Agda.TypeChecking.InstanceArguments
- Agda.TypeChecking.Irrelevance
- Agda.TypeChecking.Level
- Agda.TypeChecking.LevelConstraints
- Agda.TypeChecking.MetaVars
- Agda.TypeChecking.Monad
- Agda.TypeChecking.Monad.Base
- Agda.TypeChecking.Monad.Benchmark
- Agda.TypeChecking.Monad.Builtin
- Agda.TypeChecking.Monad.Caching
- Agda.TypeChecking.Monad.Closure
- Agda.TypeChecking.Monad.Constraints
- Agda.TypeChecking.Monad.Context
- Agda.TypeChecking.Monad.Env
- Agda.TypeChecking.Monad.Exception
- Agda.TypeChecking.Monad.Imports
- Agda.TypeChecking.Monad.MetaVars
- Agda.TypeChecking.Monad.Mutual
- Agda.TypeChecking.Monad.Open
- Agda.TypeChecking.Monad.Options
- Agda.TypeChecking.Monad.Sharing
- Agda.TypeChecking.Monad.Signature
- Agda.TypeChecking.Monad.SizedTypes
- Agda.TypeChecking.Monad.State
- Agda.TypeChecking.Monad.Statistics
- Agda.TypeChecking.Monad.Trace

- Agda.TypeChecking.Patterns
- Agda.TypeChecking.Polarity
- Agda.TypeChecking.Positivity
- Agda.TypeChecking.Pretty
- Agda.TypeChecking.Primitive
- Agda.TypeChecking.ProjectionLike
- Agda.TypeChecking.Quote
- Agda.TypeChecking.ReconstructParameters
- Agda.TypeChecking.RecordPatterns
- Agda.TypeChecking.Records
- Agda.TypeChecking.Reduce
- Agda.TypeChecking.Rewriting
- Agda.TypeChecking.Rules
- Agda.TypeChecking.Serialise
- Agda.TypeChecking.SizedTypes
- Agda.TypeChecking.Substitute
- Agda.TypeChecking.SyntacticEquality
- Agda.TypeChecking.Telescope
- Agda.TypeChecking.Test
- Agda.TypeChecking.Tests
- Agda.TypeChecking.Unquote
- Agda.TypeChecking.With

- Agda.Utils
- Agda.Utils.AssocList
- Agda.Utils.Bag
- Agda.Utils.Benchmark
- Agda.Utils.BiMap
- Agda.Utils.Char
- Agda.Utils.Cluster
- Agda.Utils.Either
- Agda.Utils.Empty
- Agda.Utils.Environment
- Agda.Utils.Except
- Agda.Utils.Favorites
- Agda.Utils.FileName
- Agda.Utils.Function
- Agda.Utils.Functor
- Agda.Utils.Geniplate
- Agda.Utils.Graph
- Agda.Utils.Graph.AdjacencyMap

- Agda.Utils.Hash
- Agda.Utils.HashMap
- Agda.Utils.IO
- Agda.Utils.IORef
- Agda.Utils.Impossible
- Agda.Utils.Lens
- Agda.Utils.List
- Agda.Utils.ListT
- Agda.Utils.Map
- Agda.Utils.Maybe
- Agda.Utils.Memo
- Agda.Utils.Monad
- Agda.Utils.Null
- Agda.Utils.Parser
- Agda.Utils.PartialOrd
- Agda.Utils.Permutation
- Agda.Utils.Pointer
- Agda.Utils.Pretty
- Agda.Utils.QuickCheck
- Agda.Utils.SemiRing
- Agda.Utils.Singleton
- Agda.Utils.Size
- Agda.Utils.String
- Agda.Utils.Suffix
- Agda.Utils.TestHelpers
- Agda.Utils.Time
- Agda.Utils.Trie
- Agda.Utils.Tuple
- Agda.Utils.Update
- Agda.Utils.VarSet
- Agda.Utils.Warshall

- Agda.Version

# Agda 2

Table of contents:

Note that this README only discusses installation of Agda, not its standard library. See the Agda Wiki for information about the library.

## Documentation

## Prerequisites

You need recent versions of the following programs/libraries:

- GHC: http://www.haskell.org/ghc/
- cabal-install: http://www.haskell.org/cabal/
- Alex: http://www.haskell.org/alex/
- Happy: http://www.haskell.org/happy/
- cpphs: http://projects.haskell.org/cpphs/
- GNU Emacs: http://www.gnu.org/software/emacs/

You should also make sure that programs installed by cabal-install are on your shell's search path.

For instructions on installing a suitable version of Emacs under Windows, see [below]((#installing-emacs-under-windows).

Non-Windows users need to ensure that the development files for the C libraries zlib and ncurses are installed (see http://zlib.net and http://www.gnu.org/software/ncurses/). Your package manager may be able to install these files for you. For instance, on Debian or Ubuntu it should suffice to run

`apt-get install zlib1g-dev libncurses5-dev`

as root to get the correct files installed.

### Note on GHC's CPP language extension

Recent versions of Clang's preprocessor don't work well with Haskell. In order to get some dependencies to build, you may need to set up Cabal to have GHC use cpphs by default. You can do this by adding

```
program-default-options
ghc-options: -pgmPcpphs -optP--cpp
```

to your .cabal/config file. (You must be using cabal >= 1.18. Note that some packages may not compile with this option set.)

You don't need to set this option to install *Agda* from the current
development source; Agda.cabal now uses cpphs.

## Installing Agda

There are several ways to install Agda:

### Using a binary package prepared for your platform

Recommended if such a package exists. See the Agda Wiki.

### Using a released source package from Hackage

Install the prerequisites mentioned below, then run the following commands:

```
cabal update
cabal install Agda
agda-mode setup
```

The last command tries to set up Emacs for use with Agda. As an alternative you can copy the following text to your .emacs file:

```
(load-file (let ((coding-system-for-read 'utf-8))
(shell-command-to-string "agda-mode locate")))
```

It is also possible (but not necessary) to compile the Emacs mode's files:

`agda-mode compile`

This can, in some cases, give a noticeable speedup.

**WARNING**: If you reinstall the Agda mode without recompiling the Emacs
Lisp files, then Emacs may continue using the old, compiled files.

### Using the development version of the code

You can obtain tarballs of the development version from the Agda Wiki, or clone the repository.

Install the prerequisites discussed in Prerequisites.

Then, either:

*(1a)* Run the following commands in the top-level directory of the Agda source
tree to install Agda:

```
cabal update
cabal install
```

*(1b)* Run `agda-mode setup`

to set up Emacs for use with Agda. Alternatively,
add the following text to your .emacs file:

```
(load-file (let ((coding-system-for-read 'utf-8))
(shell-command-to-string "agda-mode locate")))
```

It is also possible (but not necessary) to compile the Emacs mode's files:

`agda-mode compile`

This can, in some cases, give a noticeable speedup.

**WARNING**: If you reinstall the Agda mode without recompiling the Emacs
Lisp files, then Emacs may continue using the old compiled files.

*(2)* Or, you can try to install Agda (including a compiled Emacs mode) by
running the following command:

`make install`

## Configuring the Emacs mode

If you want to you can customise the Emacs mode. Just start Emacs and type the following:

```
M-x load-library RET agda2-mode RET
M-x customize-group RET agda2 RET
```

This is useful if you want to change the Agda search path, in which case you should change the agda2-include-dirs variable.

If you want some specific settings for the Emacs mode you can add them to agda2-mode-hook. For instance, if you do not want to use the Agda input method (for writing various symbols like ∀≥ℕ→π⟦⟧) you can add the following to your .emacs:

```
(add-hook 'agda2-mode-hook
'(lambda ()
; If you do not want to use any input method:
(deactivate-input-method)
; (In some versions of Emacs you should use
; inactivate-input-method instead of
; deactivate-input-method.)
; If you want to use the X input method:
(set-input-method "X")))
```

Note that, on some systems, the Emacs mode changes the default font of the current frame in order to enable many Unicode symbols to be displayed. This only works if the right fonts are available, though. If you want to turn off this feature, then you should customise the agda2-fontset-name variable.

## Installing Emacs under Windows

A precompiled version of Emacs 24.3, with the necessary mathematical fonts, is available at http://homepage.cs.uiowa.edu/~astump/agda/

## Hacking on Agda

Head to `HACKING`

## Changes

# Release notes for Agda version 2.5.2

## Installation and infrastructure

Modular support for literate programming

Literate programming support has been moved out of the lexer and into the

`Agda.Syntax.Parser.Literate`

module.Files ending in

`.lagda`

are still interpreted as literate TeX. The extension`.lagda.tex`

may now also be used for literate TeX files.Support for more literate code formats and extensions can be added modularly.

By default,

`.lagda.*`

files are opened in the Emacs mode corresponding to their last extension. One may switch to and from Agda mode manually.reStructuredText

Literate Agda code can now be written in reStructuredText format, using the

`.lagda.rst`

extension.As a general rule, Agda will parse code following a line ending in

`::`

, as long as that line does not start with`..`

. The module name must match the path of the file in the documentation, and must be given explicitly. Several files have been converted already, for instance:`language/mixfix-operators.lagda.rst`

`tools/compilers.lagda.rst`

Note that:

- Code blocks inside an rST comment block will be type-checked by Agda, but not rendered in the documentation.
- Code blocks delimited by
`.. code-block:: agda`

will be rendered in the final documenation, but not type-checked by Agda. - All lines inside a codeblock must be further indented than the first line of the code block.
- Indentation must be consistent between code blocks. In other words, the file as a whole must be a valid Agda file if all the literate text is replaced by white space.

Documentation testing

All documentation files in the

`doc/user-manual`

directory that end in`.lagda.rst`

can be typechecked by running`make user-manual-test`

, and also as part of the general test suite.Support installation through Stack

The Agda sources now also include a configuration for the stack install tool (tested through continuous integration).

It should hence be possible to repeatably build any future Agda version (including unreleased commits) from source by checking out that version and running

`stack install`

from the checkout directory. By using repeatable builds, this should keep selecting the same dependencies in the face of new releases on Hackage.For further motivation, see Issue #2005.

Removed the

`--test`

command-line optionThis option ran the internal test-suite. This test-suite was implemented using Cabal supports for test-suites. [Issue #2083].

The

`--no-default-libraries`

flag has been split into two flags [Issue #1937]`--no-default-libraries`

: Ignore the defaults file but still look for local`.agda-lib`

files`--no-libraries`

: Don't use any`.agda-lib`

files (the previous behaviour of`--no-default-libraries`

).

If

`agda`

was built inside`git`

repository, then the`--version`

flag will display the hash of the commit used, and whether the tree was`-dirty`

(i.e. there were uncommited changes in the working directory). Otherwise, only the version number is shown.

## Language

Dot patterns are now optional

Consider the following program

`data Vec (A : Set) : Nat → Set where [] : Vec A zero cons : ∀ n → A → Vec A n → Vec A (suc n) vmap : ∀ {A B} n → (A → B) → Vec A n → Vec B n vmap .zero f [] = [] vmap .(suc m) f (cons m x xs) = cons m (f x) (vmap m f xs)`

If we don't care about the dot patterns they can (and could previously) be replaced by wildcards:

`vmap : ∀ {A B} n → (A → B) → Vec A n → Vec B n vmap _ f [] = [] vmap _ f (cons m x xs) = cons m (f x) (vmap m f xs)`

Now it is also allowed to give a variable pattern in place of the dot pattern. In this case the variable will be bound to the value of the dot pattern. For our example:

`vmap : ∀ {A B} n → (A → B) → Vec A n → Vec B n vmap n f [] = [] vmap n f (cons m x xs) = cons m (f x) (vmap m f xs)`

In the first clause

`n`

reduces to`zero`

and in the second clause`n`

reduces to`suc m`

.Module parameters can now be refined by pattern matching

Previously, pattern matches that would refine a variable outside the current left-hand side was disallowed. For instance, the following would give an error, since matching on the vector would instantiate

`n`

.`module _ {A : Set} {n : Nat} where f : Vec A n → Vec A n f [] = [] f (x ∷ xs) = x ∷ xs`

Now this is no longer disallowed. Instead

`n`

is bound to the appropriate value in each clause.With-abstraction now abstracts also in module parameters

The change that allows pattern matching to refine module parameters also allows with-abstraction to abstract in them. For instance,

`module _ (n : Nat) (xs : Vec Nat (n + n)) where f : Nat f with n + n f | nn = ? -- xs : Vec Nat nn`

Note: Any function argument or lambda-bound variable bound outside a given function counts as a module parameter.

To prevent abstraction in a parameter you can hide it inside a definition. In the above example,

`module _ (n : Nat) (xs : Vec Nat (n + n)) where ys : Vec Nat (n + n) ys = xs f : Nat f with n + n f | nn = ? -- xs : Vec Nat nn, ys : Vec Nat (n + n)`

As-patterns [Issue #78].

As-patterns (

`@`

-patterns) are finally working and can be used to name a pattern. The name has the same scope as normal pattern variables (i.e. the right-hand side, where clause, and dot patterns). The name reduces to the value of the named pattern. For example::`module _ {A : Set} (_<_ : A → A → Bool) where merge : List A → List A → List A merge xs [] = xs merge [] ys = ys merge xs@(x ∷ xs₁) ys@(y ∷ ys₁) = if x < y then x ∷ merge xs₁ ys else y ∷ merge xs ys₁`

Idiom brackets.

There is new syntactic sugar for idiom brackets:

`(| e a1 .. an |)`

expands to`pure e <*> a1 <*> .. <*> an`

The desugaring takes place before scope checking and only requires names

`pure`

and`_<*>_`

in scope. Idiom brackets work well with operators, for instance`(| if a then b else c |)`

desugars to`pure if_then_else_ <*> a <*> b <*> c`

Limitations:

The top-level application inside idiom brackets cannot include implicit applications, so

`(| foo {x = e} a b |)`

is illegal. In the case`e`

is pure you can write`(| (foo {x = e}) a b |)`

which desugars to`pure (foo {x = e}) <*> a <*> b`

Binding syntax and operator sections cannot appear immediately inside idiom brackets.

Layout for pattern matching lambdas.

You can now write pattern matching lambdas using the syntax

`λ where false → true true → false`

avoiding the need for explicit curly braces and semicolons.

Overloaded projections [Issue #1944].

Ambiguous projections are no longer a scope error. Instead they get resolved based on the type of the record value they are eliminating. This corresponds to constructors, which can be overloaded and get disambiguated based on the type they are introducing. Example:

`module _ (A : Set) (a : A) where record R B : Set where field f : B open R public record S B : Set where field f : B open S public`

Exporting

`f`

twice from both`R`

and`S`

is now allowed. Then,`r : R A f r = a s : S A f s = f r`

disambiguates to:

`r : R A R.f r = a s : S A S.f s = R.f r`

If the type of the projection is known, it can also be disambiguated unapplied.

`unapplied : R A -> A unapplied = f`

Postfix projections [Issue #1963].

Agda now supports a postfix syntax for projection application. This style is more in harmony with copatterns. For example:

`record Stream (A : Set) : Set where coinductive field head : A tail : Stream A open Stream repeat : ∀{A} (a : A) → Stream A repeat a .head = a repeat a .tail = repeat a zipWith : ∀{A B C} (f : A → B → C) (s : Stream A) (t : Stream B) → Stream C zipWith f s t .head = f (s .head) (t .head) zipWith f s t .tail = zipWith f (s .tail) (t .tail) module Fib (Nat : Set) (zero one : Nat) (plus : Nat → Nat → Nat) where {-# TERMINATING #-} fib : Stream Nat fib .head = zero fib .tail .head = one fib .tail .tail = zipWith plus fib (fib .tail)`

The thing we eliminate with projection now is visibly the head, i.e., the left-most expression of the sequence (e.g.

`repeat`

in`repeat a .tail`

).The syntax overlaps with dot patterns, but for type correct left hand sides there is no confusion: Dot patterns eliminate function types, while (postfix) projection patterns eliminate record types.

By default, Agda prints system-generated projections (such as by eta-expansion or case splitting) prefix. This can be changed with the new option:

`{-# OPTIONS --postfix-projections #-}`

Result splitting in extended lambdas (aka pattern lambdas) always produces postfix projections, as prefix projection pattern do not work here: a prefix projection needs to go left of the head, but the head is omitted in extended lambdas.

`dup : ∀{A : Set}(a : A) → A × A dup = λ{ a → ? }`

Result splitting (

`C-c C-c RET`

) here will yield:`dup = λ{ a .proj₁ → ? ; a .proj₂ → ? }`

Projection parameters [Issue #1954].

When copying a module, projection parameters will now stay hidden arguments, even if the module parameters are visible. This matches the situation we had for constructors since long. Example:

`module P (A : Set) where record R : Set where field f : A open module Q A = P A`

Parameter

`A`

is now hidden in`R.f`

:`test : ∀{A} → R A → A test r = R.f r`

Note that a module parameter that corresponds to the record value argument of a projection will not be hidden.

`module M (A : Set) (r : R A) where open R A r public test' : ∀{A} → R A → A test' r = M.f r`

Eager insertion of implicit arguments [Issue #2001]

Implicit arguments are now (again) eagerly inserted in left-hand sides. The previous behaviour of inserting implicits for where blocks, but not right-hand sides was not type safe.

Module applications can now be eta expanded/contracted without changing their behaviour [Issue #1985]

Previously definitions exported using

`open public`

got the incorrect type for underapplied module applications.Example:

`module A where postulate A : Set module B (X : Set) where open A public module C₁ = B module C₂ (X : Set) = B X`

Here both

`C₁.A`

and`C₂.A`

have type`(X : Set) → Set`

.Polarity pragmas.

Polarity pragmas can be attached to postulates. The polarities express how the postulate's arguments are used. The following polarities are available:

`_`

: Unused.`++`

: Strictly positive.`+`

: Positive.`-`

: Negative.`*`

: Unknown/mixed.Polarity pragmas have the form

`{-# POLARITY name <zero or more polarities> #-}`

and can be given wherever fixity declarations can be given. The listed polarities apply to the given postulate's arguments (explicit/implicit/instance), from left to right. Polarities currently cannot be given for module parameters. If the postulate takes n arguments (excluding module parameters), then the number of polarities given must be between 0 and n (inclusive).

Polarity pragmas make it possible to use postulated type formers in recursive types in the following way:

`postulate ∥_∥ : Set → Set {-# POLARITY ∥_∥ ++ #-} data D : Set where c : ∥ D ∥ → D`

Note that one can use postulates that may seem benign, together with polarity pragmas, to prove that the empty type is inhabited:

`postulate _⇒_ : Set → Set → Set lambda : {A B : Set} → (A → B) → A ⇒ B apply : {A B : Set} → A ⇒ B → A → B {-# POLARITY _⇒_ ++ #-} data ⊥ : Set where data D : Set where c : D ⇒ ⊥ → D not-inhabited : D → ⊥ not-inhabited (c f) = apply f (c f) inhabited : D inhabited = c (lambda not-inhabited) bad : ⊥ bad = not-inhabited inhabited`

Polarity pragmas are not allowed in safe mode.

Declarations in a

`where`

-block are now private. [Issue #2101] This means that`f ps = body where decls`

is now equivalent to

`f ps = body where private decls`

This changes little, since the

`decls`

were anyway not in scope outside`body`

. However, it makes a difference for abstract definitions, because private type signatures can see through abstract definitions. Consider:`record Wrap (A : Set) : Set where field unwrap : A postulate P : ∀{A : Set} → A → Set abstract unnamedWhere : (A : Set) → Set unnamedWhere A = A where -- the following definitions are private! B : Set B = Wrap A postulate b : B test : P (Wrap.unwrap b) -- succeeds`

The

`abstract`

is inherited in`where`

-blocks from the parent (here: function`unnamedWhere`

). Thus, the definition of`B`

is opaque and the type equation`B = Wrap A`

cannot be used to check type signatures, not even of abstract definitions. Thus, checking the type`P (Wrap.unwrap b)`

would fail. However, if`test`

is private, abstract definitions are translucent in its type, and checking succeeds. With the implemented change, all`where`

-definitions are private, in this case`B`

,`b`

, and`test`

, and the example succeeds.Nothing changes for the named forms of

`where`

,`module M where module _ where`

For instance, this still fails:

`abstract unnamedWhere : (A : Set) → Set unnamedWhere A = A module M where B : Set B = Wrap A postulate b : B test : P (Wrap.unwrap b) -- fails`

Private anonymous modules now work as expected [Issue #2199]

Previously the

`private`

was ignored for anonymous modules causing its definitions to be visible outside the module containing the anonymous module. This is no longer the case. For instance,`module M where private module _ (A : Set) where Id : Set Id = A foo : Set → Set foo = Id open M bar : Set → Set bar = Id -- Id is no longer in scope here`

Pattern synonyms are now expanded on left hand sides of DISPLAY pragmas [Issue #2132]. Example:

`data D : Set where C c : D g : D → D pattern C′ = C {-# DISPLAY C′ = C′ #-} {-# DISPLAY g C′ = c #-}`

This now behaves as:

`{-# DISPLAY C = C′ #-} {-# DISPLAY g C = c #-}`

Expected error for

`test : C ≡ g C test = refl`

is thus:

`C′ != c of type D`

The built-in floats have new semantics to fix inconsistencies and to improve cross-platform portability.

Float equality has been split into two primitives.

`primFloatEquality`

is designed to establish decidable propositional equality while`primFloatNumericalEquality`

is intended for numerical computations. They behave as follows:`primFloatEquality NaN NaN = True primFloatEquality 0.0 -0.0 = False primFloatNumericalEquality NaN NaN = False primFloatNumericalEquality 0.0 -0.0 = True`

This change fixes an inconsistency, see [Issue #2169]. For further detail see the user manual.

Floats now have only one

`NaN`

value. This is necessary for proper Float support in the JavaScript backend, as JavaScript (and some other platforms) only support one`NaN`

value.The primitive function

`primFloatLess`

was renamed`primFloatNumericalLess`

.

Added new primitives to built-in floats:

Anonymous declarations [Issue #1465].

A module can contain an arbitrary number of declarations named

`_`

which will scoped-checked and type-checked but won't be made available in the scope (nor exported). They cannot introduce arguments on the LHS (but one can use lambda-abstractions on the RHS) and they cannot be defined by recursion.`_ : Set → Set _ = λ x → x`

### Rewriting

The REWRITE pragma can now handle several names. E.g.:

`{-# REWRITE eq1 eq2 #-}`

### Reflection

You can now use macros in reflected terms [Issue #2130].

For instance, given a macro

`macro some-tactic : Term → TC ⊤ some-tactic = ...`

the term

`def (quote some-tactic) []`

represents a call to the macro. This makes it a lot easier to compose tactics.The reflection machinery now uses normalisation less often:

Macros no longer normalise the (automatically quoted) term arguments.

The TC primitives

`inferType`

,`checkType`

and`quoteTC`

no longer normalise their arguments.The following deprecated constructions may also have been changed:

`quoteGoal`

,`quoteTerm`

,`quoteContext`

and`tactic`

.

New TC primitive:

`withNormalisation`

.To recover the old normalising behaviour of

`inferType`

,`checkType`

,`quoteTC`

and`getContext`

, you can wrap them inside a call to`withNormalisation true`

:`withNormalisation : ∀ {a} {A : Set a} → Bool → TC A → TC A`

New TC primitive:

`reduce`

.`reduce : Term → TC Term`

Reduces its argument to weak head normal form.

Added new TC primitive:

`isMacro`

[Issue #2182]`isMacro : Name → TC Bool`

Returns

`true`

if the name refers to a macro, otherwise`false`

.

## Type checking

Files with open metas can be imported now [Issue #964]. This should make simultaneous interactive development on several modules more pleasant.

Requires option:

`--allow-unsolved-metas`

Internally, before serialization, open metas are turned into postulates named

`unsolved#meta.<nnn>`

where

`<nnn>`

is the internal meta variable number.The performance of the compile-time evaluator has been greatly improved.

Fixed a memory leak in evaluator (Issue #2147).

Reduction speed improved by an order of magnitude and is now comparable to the performance of GHCi. Still call-by-name though.

The detection of types that satisfy K added in Agda 2.5.1 has been rolled back (see Issue #2003).

Eta-equality for record types is now only on after the positivity checker has confirmed it is safe to have it. Eta-equality for unguarded inductive records previously lead to looping of the type checker. [See Issue #2197]

`record R : Set where inductive field r : R loops : R loops = ?`

As a consequence of this change, the following example does not type-check any more:

`mutual record ⊤ : Set where test : ∀ {x y : ⊤} → x ≡ y test = refl`

It fails because the positivity checker is only run after the mutual block, thus, eta-equality for

`⊤`

is not available when checking test.One can declare eta-equality explicitly, though, to make this example work.

`mutual record ⊤ : Set where eta-equality test : ∀ {x y : ⊤} → x ≡ y test = refl`

Records with instance fields are now eta expanded before instance search.

For instance, assuming

`Eq`

and`Ord`

with boolean functions`_==_`

and`_<_`

respectively,`record EqAndOrd (A : Set) : Set where field {{eq}} : Eq A {{ord}} : Ord A leq : {A : Set} {{_ : EqAndOrd A}} → A → A → Bool leq x y = x == y || x < y`

Here the

`EqAndOrd`

record is automatically unpacked before instance search, revealing the component`Eq`

and`Ord`

instances.This can be used to simulate superclass dependencies.

Overlappable record instance fields.

Instance fields in records can be marked as overlappable using the new

`overlap`

keyword:`record Ord (A : Set) : Set where field _<_ : A → A → Bool overlap {{eqA}} : Eq A`

When instance search finds multiple candidates for a given instance goal and they are

**all**overlappable it will pick the left-most candidate instead of refusing to solve the instance goal.This can be use to solve the problem arising from shared "superclass" dependencies. For instance, if you have, in addition to

`Ord`

above, a`Num`

record that also has an`Eq`

field and want to write a function requiring both`Ord`

and`Num`

, any`Eq`

constraint will be solved by the`Eq`

instance from whichever argument that comes first.`record Num (A : Set) : Set where field fromNat : Nat → A overlap {{eqA}} : Eq A lessOrEqualFive : {A : Set} {{NumA : Num A}} {{OrdA : Ord A}} → A → Bool lessOrEqualFive x = x == fromNat 5 || x < fromNat 5`

In this example the call to

`_==_`

will use the`eqA`

field from`NumA`

rather than the one from`OrdA`

. Note that these may well be different.Instance fields can be left out of copattern matches [Issue #2288]

Missing cases for instance fields (marked

`{{`

`}}`

) in copattern matches will be solved using instance search. This makes defining instances with superclass fields much nicer. For instance, we can define`Nat`

instances of`Eq`

,`Ord`

and`Num`

from above as follows:`instance EqNat : Eq Nat _==_ {{EqNat}} n m = eqNat n m OrdNat : Ord Nat _<_ {{OrdNat}} n m = lessNat n m NumNat : Num Nat fromNat {{NumNat}} n = n`

The

`eqA`

fields of`Ord`

and`Num`

are filled in using instance search (with`EqNat`

in this case).Limited instance search depth [Issue #2269]

To prevent instance search from looping on bad instances (see Issue #1743) the search depth of instance search is now limited. The maximum depth can be set with the

`--instance-search-depth`

flag and the default value is`500`

.

## Emacs mode

New command

`C-u C-u C-c C-n`

: Use`show`

to display the result of normalisation.Calling

`C-u C-u C-c C-n`

on an expression`e`

(in a hole or at top level) normalises`show e`

and prints the resulting string, or an error message if the expression does not normalise to a literal string.This is useful when working with complex data structures for which you have defined a nice

`Show`

instance.Note that the name

`show`

is hardwired into the command.Changed feature: Interactively split result.

Make-case (

`C-c C-c`

) with no variables will now*either*introduce function arguments*or*do a copattern split (or fail).This is as before:

`test : {A B : Set} (a : A) (b : B) → A × B test a b = ? -- expected: -- proj₁ (test a b) = {!!} -- proj₂ (test a b) = {!!} testFun : {A B : Set} (a : A) (b : B) → A × B testFun = ? -- expected: -- testFun a b = {!!}`

This is has changed:

`record FunRec A : Set where field funField : A → A open FunRec testFunRec : ∀{A} → FunRec A testFunRec = ? -- expected (since 2016-05-03): -- funField testFunRec = {!!} -- used to be: -- funField testFunRec x = {!!}`

Changed feature: Split on hidden variables.

Make-case (

`C-c C-c`

) will no longer split on the given hidden variables, but only make them visible. (Splitting can then be performed in a second go.)`test : ∀{N M : Nat} → Nat → Nat → Nat test N M = {!.N N .M!}`

Invoking splitting will result in:

`test {N} {M} zero M₁ = ? test {N} {M} (suc N₁) M₁ = ?`

The hidden

`.N`

and`.M`

have been brought into scope, the visible`N`

has been split upon.Non-fatal errors/warnings.

Non-fatal errors and warnings are now displayed in the info buffer and do not interrupt the typechecking of the file.

Currently termination errors, unsolved metavariables, unsolved constraints, positivity errors, deprecated BUILTINs, and empty REWRITING pragmas are non-fatal errors.

Highlighting for positivity check failures

Negative occurences of a datatype in its definition are now highlighted in a way similar to termination errors.

The abbrev for codata was replaced by an abbrev for code environments.

If you type

`c C-x '`

(on a suitably standard setup), then Emacs will insert the following text:`\begin{code}<newline> <cursor><newline>\end{code}<newline>.`

The LaTeX backend can now be invoked from the Emacs mode.

Using the compilation command (

`C-c C-x C-c`

).The flag

`--latex-dir`

can be used to set the output directory (by default:`latex`

). Note that if this directory is a relative path, then it is interpreted relative to the "project root". (When the LaTeX backend is invoked from the command line the path is interpreted relative to the current working directory.) Example: If the module`A.B.C`

is located in the file`/foo/A/B/C.agda`

, then the project root is`/foo/`

, and the default output directory is`/foo/latex/`

.The compilation command (

`C-c C-x C-c`

) now by default asks for a backend.To avoid this question, set the customisation variable

`agda2-backend`

to an appropriate value.The command

`agda2-measure-load-time`

no longer "touches" the file, and the optional argument`DONT-TOUCH`

has been removed.New command

`C-u (C-u) C-c C-s`

: Simplify or normalise the solution`C-c C-s`

producesWhen writing examples, it is nice to have the hole filled in with a normalised version of the solution. Calling

`C-c C-s`

on`_ : reverse (0 ∷ 1 ∷ []) ≡ ? _ = refl`

used to yield the non informative

`reverse (0 ∷ 1 ∷ [])`

when we would have hopped to get`1 ∷ 0 ∷ []`

instead. We can now control finely the degree to which the solution is simplified.Changed feature: Solving the hole at point

Calling

`C-c C-s`

inside a specific goal does not solve*all*the goals already instantiated internally anymore: it only solves the one at hand (if possible).New bindings: All the blackboard bold letters are now available [Pull Request #2305]

The Agda input method only bound a handful of the blackboard bold letters but programmers were actually using more than these. They are now all available: lowercase and uppercase. Some previous bindings had to be modified for consistency. The naming scheme is as follows:

- \bx for lowercase blackboard bold
- \bX for uppercase blackboard bold
- \bGx for lowercase greek blackboard bold (similar to \Gx for greeks)
- \bGX for uppercase greek blackboard bold (similar to \GX for uppercase greeks)

Replaced binding for go back

Use

`M-,`

(instead of`M-*`

) for go back in Emacs ≥ 25.1 (and continue using`M-*`

with previous versions of Emacs).

## Compiler backends

JS compiler backend

The JavaScript backend has been (partially) rewritten. The JavaScript backend now supports most Agda features, notably copatterns can now be compiled to JavaScript. Furthermore, the existing optimizations from the other backends now apply to the JavaScript backend as well.

GHC, JS and UHC compiler backends

Added new primitives to built-in floats [Issues #2194 and #2200]:

`primFloatNegate : Float → Float primCos : Float → Float primTan : Float → Float primASin : Float → Float primACos : Float → Float primATan : Float → Float primATan2 : Float → Float → Float`

## LaTeX backend

Code blocks are now (by default) surrounded by vertical space. [Issue #2198]

Use

`\AgdaNoSpaceAroundCode{}`

to avoid this vertical space, and`\AgdaSpaceAroundCode{}`

to reenable it.Note that, if

`\AgdaNoSpaceAroundCode{}`

is used, then empty lines before or after a code block will not necessarily lead to empty lines in the generated document. However, empty lines*inside*the code block do (by default) lead to empty lines in the output.If you prefer the previous behaviour, then you can use the

`agda.sty`

file that came with the previous version of Agda.`\AgdaHide{...}`

now eats trailing spaces (using`\ignorespaces`

).New environments:

`AgdaAlign`

,`AgdaSuppressSpace`

and`AgdaMultiCode`

.Sometimes one might want to break up a code block into multiple pieces, but keep code in different blocks aligned with respect to each other. Then one can use the

`AgdaAlign`

environment. Example usage:`\begin{AgdaAlign} \begin{code} code code (more code) \end{code} Explanation... \begin{code} aligned with "code" code (aligned with (more code)) \end{code} \end{AgdaAlign}`

Note that

`AgdaAlign`

environments should not be nested.Sometimes one might also want to hide code in the middle of a code block. This can be accomplished in the following way:

`\begin{AgdaAlign} \begin{code} visible \end{code} \AgdaHide{ \begin{code} hidden \end{code}} \begin{code} visible \end{code} \end{AgdaAlign}`

However, the result may be ugly: extra space is perhaps inserted around the code blocks.

The

`AgdaSuppressSpace`

environment ensures that extra space is only inserted before the first code block, and after the last one (but not if`\AgdaNoSpaceAroundCode{}`

is used).The environment takes one argument, the number of wrapped code blocks (excluding hidden ones). Example usage:

`\begin{AgdaAlign} \begin{code} code more code \end{code} Explanation... \begin{AgdaSuppressSpace}{2} \begin{code} aligned with "code" aligned with "more code" \end{code} \AgdaHide{ \begin{code} hidden code \end{code}} \begin{code} also aligned with "more code" \end{code} \end{AgdaSuppressSpace} \end{AgdaAlign}`

Note that

`AgdaSuppressSpace`

environments should not be nested.There is also a combined environment,

`AgdaMultiCode`

, that combines the effects of`AgdaAlign`

and`AgdaSuppressSpace`

.

## Tools

### agda-ghc-names

The `agda-ghc-names`

now has its own repository at

https://github.com/agda/agda-ghc-names

and is no longer distributed with Agda.

# Release notes for Agda version 2.5.1.2

Fixed broken type signatures that were incorrectly accepted due to GHC #12784.

# Release notes for Agda version 2.5.1.1

## Installation and infrastructure

Added support for GHC 8.0.1.

Documentation is now built with Python >=3.3, as done by readthedocs.org.

## Bug fixes

Fixed a serious performance problem with instance search

Interactively splitting variable with

`C-c C-c`

no longer introduces new trailing patterns. This fixes Issue #1950.`data Ty : Set where _⇒_ : Ty → Ty → Ty ⟦_⟧ : Ty → Set ⟦ A ⇒ B ⟧ = ⟦ A ⟧ → ⟦ B ⟧ data Term : Ty → Set where K : (A B : Ty) → Term (A ⇒ (B ⇒ A)) test : (A : Ty) (a : Term A) → ⟦ A ⟧ test A a = {!a!}`

Before change, case splitting on

`a`

would give`test .(A ⇒ (B ⇒ A)) (K A B) x x₁ = ?`

Now, it yields

`test .(A ⇒ (B ⇒ A)) (K A B) = ?`

In literate TeX files,

`\begin{code}`

and`\end{code}`

can be preceded (resp. followed) by TeX code on the same line. This fixes Issue #2077.Other issues fixed (see bug tracker):

#1951 (mixfix binders not working in 'syntax')

#1967 (too eager insteance search error)

#1974 (lost constraint dependencies)

#1982 (internal error in unifier)

#2034 (function type instance goals)

## Compiler backends

UHC compiler backend

Added support for UHC 1.1.9.4.

# Release notes for Agda version 2.5.1

## Documentation

There is now an official Agda User Manual: http://agda.readthedocs.org/en/stable/

## Installation and infrastructure

Builtins and primitives are now defined in a new set of modules available to all users, independent of any particular library. The modules are

`Agda.Builtin.Bool Agda.Builtin.Char Agda.Builtin.Coinduction Agda.Builtin.Equality Agda.Builtin.Float Agda.Builtin.FromNat Agda.Builtin.FromNeg Agda.Builtin.FromString Agda.Builtin.IO Agda.Builtin.Int Agda.Builtin.List Agda.Builtin.Nat Agda.Builtin.Reflection Agda.Builtin.Size Agda.Builtin.Strict Agda.Builtin.String Agda.Builtin.TrustMe Agda.Builtin.Unit`

The standard library reexports the primitives from the new modules.

The

`Agda.Builtin`

modules are installed in the same way as`Agda.Primitive`

, but unlike`Agda.Primitive`

they are not loaded automatically.

## Pragmas and options

Library management

There is a new 'library' concept for managing include paths. A library consists of - a name, - a set of libraries it depends on, and - a set of include paths.

A library is defined in a

`.agda-lib`

file using the following format:`name: LIBRARY-NAME -- Comment depend: LIB1 LIB2 LIB3 LIB4 include: PATH1 PATH2 PATH3`

Dependencies are library names, not paths to

`.agda-lib`

files, and include paths are relative to the location of the library-file.To be useable, a library file has to be listed (with its full path) in

`AGDA_DIR/libraries`

(or`AGDA_DIR/libraries-VERSION`

, for a given Agda version).`AGDA_DIR`

defaults to`~/.agda`

on Unix-like systems and`C:/Users/USERNAME/AppData/Roaming/agda`

or similar on Windows, and can be overridden by setting the`AGDA_DIR`

environment variable.Environment variables in the paths (of the form

`$VAR`

or`${VAR}`

) are expanded. The location of the libraries file used can be overridden using the`--library-file=FILE`

flag, although this is not expected to be very useful.You can find out the precise location of the 'libraries' file by calling

`agda -l fjdsk Dummy.agda`

and looking at the error message (assuming you don't have a library called fjdsk installed).There are three ways a library gets used:

You supply the

`--library=LIB`

(or`-l LIB`

) option to Agda. This is equivalent to adding a`-iPATH`

for each of the include paths of`LIB`

and its (transitive) dependencies.No explicit

`--library`

flag is given, and the current project root (of the Agda file that is being loaded) or one of its parent directories contains a`.agda-lib`

file defining a library`LIB`

. This library is used as if a`--librarary=LIB`

option had been given, except that it is not necessary for the library to be listed in the`AGDA_DIR/libraries`

file.No explicit

`--library`

flag, and no`.agda-lib`

file in the project root. In this case the file`AGDA_DIR/defaults`

is read and all libraries listed are added to the path. The defaults file should contain a list of library names, each on a separate line. In this case the current directory is also added to the path.To disable default libraries, you can give the flag

`--no-default-libraries`

.

Library names can end with a version number (for instance,

`mylib-1.2.3`

). When resolving a library name (given in a`--library`

flag, or listed as a default library or library dependency) the following rules are followed:If you don't give a version number, any version will do.

If you give a version number an exact match is required.

When there are multiple matches an exact match is preferred, and otherwise the latest matching version is chosen.

For example, suppose you have the following libraries installed:

`mylib`

,`mylib-1.0`

,`otherlib-2.1`

, and`otherlib-2.3`

. In this case, aside from the exact matches you can also say`--library=otherlib`

to get`otherlib-2.3`

.New Pragma

`COMPILED_DECLARE_DATA`

for binding recursively defined Haskell data types to recursively defined Agda data types.If you have a Haskell type like

`{-# LANGUAGE GADTs #-} module Issue223 where data A where BA :: B -> A data B where AB :: A -> B BB :: B`

You can now bind it to corresponding mutual Agda inductive data types as follows:

`{-# IMPORT Issue223 #-} data A : Set {-# COMPILED_DECLARE_DATA A Issue223.A #-} data B : Set {-# COMPILED_DECLARE_DATA B Issue223.B #-} data A where BA : B → A {-# COMPILED_DATA A Issue223.A Issue223.BA #-} data B where AB : A → B BB : B {-# COMPILED_DATA B Issue223.B Issue223.AB Issue223.BB #-}`

This fixes Issue #223.

New pragma

`HASKELL`

for adding inline Haskell code (GHC backend only)Arbitrary Haskell code can be added to a module using the

`HASKELL`

pragma. For instance,`{-# HASKELL echo :: IO () echo = getLine >>= putStrLn #-} postulate echo : IO ⊤ {-# COMPILED echo echo #-}`

New option

`--exact-split`

.The

`--exact-split`

flag causes Agda to raise an error whenever a clause in a definition by pattern matching cannot be made to hold definitionally (i.e. as a reduction rule). Specific clauses can be excluded from this check by means of the`{-# CATCHALL #-}`

pragma.For instance, the following definition will be rejected as the second clause cannot be made to hold definitionally:

`min : Nat → Nat → Nat min zero y = zero min x zero = zero min (suc x) (suc y) = suc (min x y`

Catchall clauses have to be marked as such, for instance:

`eq : Nat → Nat → Bool eq zero zero = true eq (suc m) (suc n) = eq m n {-# CATCHALL #-} eq _ _ = false`

New option:

`--no-exact-split`

.This option can be used to override a global

`--exact-split`

in a file, by adding a pragma`{-# OPTIONS --no-exact-split #-}`

.New options:

`--sharing`

and`--no-sharing`

.These options are used to enable/disable sharing and call-by-need evaluation. The default is

`--no-sharing`

.Note that they cannot appear in an OPTIONS pragma, but have to be given as command line arguments or added to the Agda Program Args from Emacs with

`M-x customize-group agda2`

.New pragma

`DISPLAY`

.`{-# DISPLAY f e1 .. en = e #-}`

This causes

`f e1 .. en`

to be printed in the same way as`e`

, where`ei`

can bind variables used in`e`

. The expressions`ei`

and`e`

are scope checked, but not type checked.For example this can be used to print overloaded (instance) functions with the overloaded name:

`instance NumNat : Num Nat NumNat = record { ..; _+_ = natPlus } {-# DISPLAY natPlus a b = a + b #-}`

Limitations

Left-hand sides are restricted to variables, constructors, defined functions or types, and literals. In particular, lambdas are not allowed in left-hand sides.

Since

`DISPLAY`

pragmas are not type checked implicit argument insertion may not work properly if the type of`f`

computes to an implicit function space after pattern matching.

Removed pragma

`{-# ETA R #-}`

The pragma

`{-# ETA R #-}`

is replaced by the`eta-equality`

directive inside record declarations.New option

`--no-eta-equality`

.The

`--no-eta-equality`

flag disables eta rules for declared record types. It has the same effect as`no-eta-equality`

inside each declaration of a record type`R`

.If used with the OPTIONS pragma it will not affect records defined in other modules.

The semantics of

`{-# REWRITE r #-}`

pragmas in parametrized modules has changed (see Issue #1652).Rewrite rules are no longer lifted to the top context. Instead, they now only apply to terms in (extensions of) the module context. If you want the old behaviour, you should put the

`{-# REWRITE r #-}`

pragma outside of the module (i.e. unindent it).New pragma

`{-# INLINE f #-}`

causes`f`

to be inlined during compilation.The

`STATIC`

pragma is now taken into account during compilation.Calls to a function marked

`STATIC`

are normalised before compilation. The typical use case for this is to mark the interpreter of an embedded language as`STATIC`

.Option

`--type-in-type`

no longer implies`--no-universe-polymorphism`

, thus, it can be used with explicit universe levels. [Issue #1764] It simply turns off error reporting for any level mismatch now. Examples:`{-# OPTIONS --type-in-type #-} Type : Set Type = Set data D {α} (A : Set α) : Set where d : A → D A data E α β : Set β where e : Set α → E α β`

New

`NO_POSITIVITY_CHECK`

pragma to switch off the positivity checker for data/record definitions and mutual blocks.The pragma must precede a data/record definition or a mutual block.

The pragma cannot be used in

`--safe`

mode.Examples (see

`Issue1614*.agda`

and`Issue1760*.agda`

in`test/Succeed/`

):Skipping a single data definition.

`{-# NO_POSITIVITY_CHECK #-} data D : Set where lam : (D → D) → D`

Skipping a single record definition.

`{-# NO_POSITIVITY_CHECK #-} record U : Set where field ap : U → U`

Skipping an old-style mutual block: Somewhere within a

`mutual`

block before a data/record definition.`mutual data D : Set where lam : (D → D) → D {-# NO_POSITIVITY_CHECK #-} record U : Set where field ap : U → U`

Skipping an old-style mutual block: Before the

`mutual`

keyword.`{-# NO_POSITIVITY_CHECK #-} mutual data D : Set where lam : (D → D) → D record U : Set where field ap : U → U`

Skipping a new-style mutual block: Anywhere before the declaration or the definition of data/record in the block.

`record U : Set data D : Set record U where field ap : U → U {-# NO_POSITIVITY_CHECK #-} data D where lam : (D → D) → D`

Removed

`--no-coverage-check`

option. [Issue #1918]

## Language

### Operator syntax

The default fixity for syntax declarations has changed from -666 to 20.

Sections.

Operators can be sectioned by replacing arguments with underscores. There must not be any whitespace between these underscores and the adjacent nameparts. Examples:

`pred : ℕ → ℕ pred = _∸ 1 T : Bool → Set T = if_then ⊤ else ⊥ if : {A : Set} (b : Bool) → A → A → A if b = if b then_else_`

Sections are translated into lambda expressions. Examples:

`_∸ 1 ↦ λ section → section ∸ 1 if_then ⊤ else ⊥ ↦ λ section → if section then ⊤ else ⊥ if b then_else_ ↦ λ section section₁ → if b then section else section₁`

Operator sections have the same fixity as the underlying operator (except in cases like

`if b then_else_`

, in which the section is "closed", but the operator is not).Operator sections are not supported in patterns (with the exception of dot patterns), and notations coming from syntax declarations cannot be sectioned.

A long-standing operator fixity bug has been fixed. As a consequence some programs that used to parse no longer do.

Previously each precedence level was (incorrectly) split up into five separate ones, ordered as follows, with the earlier ones binding less tightly than the later ones:

Non-associative operators.

Left associative operators.

Right associative operators.

Prefix operators.

Postfix operators.

Now this problem has been addressed. It is no longer possible to mix operators of a given precedence level but different associativity. However, prefix and right associative operators are seen as having the same associativity, and similarly for postfix and left associative operators.

## Examples

The following code is no longer accepted:

`infixl 6 _+_ infix 6 _∸_ rejected : ℕ rejected = 1 + 0 ∸ 1`

However, the following previously rejected code is accepted:

`infixr 4 _,_ infix 4 ,_ ,_ : {A : Set} {B : A → Set} {x : A} → B x → Σ A B , y = _ , y accepted : Σ ℕ λ i → Σ ℕ λ j → Σ (i ≡ j) λ _ → Σ ℕ λ k → j ≡ k accepted = 5 , , refl , , refl`

The classification of notations with binders into the categories infix, prefix, postfix or closed has changed. [Issue #1450]

The difference is that, when classifying the notation, only

*regular*holes are taken into account, not*binding*ones.Example: The notation

`syntax m >>= (λ x → f) = x <- m , f`

was previously treated as infix, but is now treated as prefix.

Notation can now include wildcard binders.

Example:

`syntax Σ A (λ _ → B) = A × B`

If an overloaded operator is in scope with several distinct precedence levels, then several instances of this operator will be included in the operator grammar, possibly leading to ambiguity. Previously the operator was given the default fixity [Issue #1436].

There is an exception to this rule: If there are multiple precedences, but at most one is explicitly declared, then only one instance will be included in the grammar. If there are no explicitly declared precedences, then this instance will get the default precedence, and otherwise it will get the declared precedence.

If multiple occurrences of an operator are "merged" in the grammar, and they have distinct associativities, then they are treated as being non-associative.

The three paragraphs above also apply to identical notations (coming from syntax declarations) for a given overloaded name.

Examples:

`module A where infixr 5 _∷_ infixr 5 _∙_ infixl 3 _+_ infix 1 bind syntax bind c (λ x → d) = x ← c , d module B where infix 5 _∷_ infixr 4 _∙_ -- No fixity declaration for _+_. infixl 2 bind syntax bind c d = c ∙ d module C where infixr 2 bind syntax bind c d = c ∙ d open A open B open C -- _∷_ is infix 5. -- _∙_ has two fixities: infixr 4 and infixr 5. -- _+_ is infixl 3. -- A.bind's notation is infix 1. -- B.bind and C.bind's notations are infix 2. -- There is one instance of "_ ∷ _" in the grammar, and one -- instance of "_ + _". -- There are three instances of "_ ∙ _" in the grammar, one -- corresponding to A._∙_, one corresponding to B._∙_, and one -- corresponding to both B.bind and C.bind.`

### Reflection

The reflection framework has received a massive overhaul.

A new type of reflected type checking computations supplants most of the old reflection primitives. The

`quoteGoal`

,`quoteContext`

and tactic primitives are deprecated and will be removed in the future, and the`unquoteDecl`

and`unquote`

primitives have changed behaviour. Furthermore the following primitive functions have been replaced by builtin type checking computations:`- primQNameType --> AGDATCMGETTYPE - primQNameDefinition --> AGDATCMGETDEFINITION - primDataConstructors --> subsumed by AGDATCMGETDEFINITION - primDataNumberOfParameters --> subsumed by AGDATCMGETDEFINITION`

See below for details.

Types are no longer packaged with a sort.

The

`AGDATYPE`

and`AGDATYPEEL`

built-ins have been removed. Reflected types are now simply terms.Reflected definitions have more information.

The type for reflected definitions has changed to

`data Definition : Set where fun-def : List Clause → Definition data-type : Nat → List Name → Definition -- parameters and constructors record-type : Name → Definition -- name of the data/record type data-con : Name → Definition -- name of the constructor axiom : Definition prim-fun : Definition`

Correspondingly the built-ins for function, data and record definitions (

`AGDAFUNDEF`

,`AGDAFUNDEFCON`

,`AGDADATADEF`

,`AGDARECORDDEF`

) have been removed.Reflected type checking computations.

There is a primitive

`TC`

monad representing type checking computations. The`unquote`

,`unquoteDecl`

, and the new`unquoteDef`

all expect computations in this monad (see below). The interface to the monad is the following`-- Error messages can contain embedded names and terms. data ErrorPart : Set where strErr : String → ErrorPart termErr : Term → ErrorPart nameErr : Name → ErrorPart {-# BUILTIN AGDAERRORPART ErrorPart #-} {-# BUILTIN AGDAERRORPARTSTRING strErr #-} {-# BUILTIN AGDAERRORPARTTERM termErr #-} {-# BUILTIN AGDAERRORPARTNAME nameErr #-} postulate TC : ∀ {a} → Set a → Set a returnTC : ∀ {a} {A : Set a} → A → TC A bindTC : ∀ {a b} {A : Set a} {B : Set b} → TC A → (A → TC B) → TC B -- Unify two terms, potentially solving metavariables in the process. unify : Term → Term → TC ⊤ -- Throw a type error. Can be caught by catchTC. typeError : ∀ {a} {A : Set a} → List ErrorPart → TC A -- Block a type checking computation on a metavariable. This will abort -- the computation and restart it (from the beginning) when the -- metavariable is solved. blockOnMeta : ∀ {a} {A : Set a} → Meta → TC A -- Backtrack and try the second argument if the first argument throws a -- type error. catchTC : ∀ {a} {A : Set a} → TC A → TC A → TC A -- Infer the type of a given term inferType : Term → TC Type -- Check a term against a given type. This may resolve implicit arguments -- in the term, so a new refined term is returned. Can be used to create -- new metavariables: newMeta t = checkType unknown t checkType : Term → Type → TC Term -- Compute the normal form of a term. normalise : Term → TC Term -- Get the current context. getContext : TC (List (Arg Type)) -- Extend the current context with a variable of the given type. extendContext : ∀ {a} {A : Set a} → Arg Type → TC A → TC A -- Set the current context. inContext : ∀ {a} {A : Set a} → List (Arg Type) → TC A → TC A -- Quote a value, returning the corresponding Term. quoteTC : ∀ {a} {A : Set a} → A → TC Term -- Unquote a Term, returning the corresponding value. unquoteTC : ∀ {a} {A : Set a} → Term → TC A -- Create a fresh name. freshName : String → TC QName -- Declare a new function of the given type. The function must be defined -- later using 'defineFun'. Takes an Arg Name to allow declaring instances -- and irrelevant functions. The Visibility of the Arg must not be hidden. declareDef : Arg QName → Type → TC ⊤ -- Define a declared function. The function may have been declared using -- 'declareDef' or with an explicit type signature in the program. defineFun : QName → List Clause → TC ⊤ -- Get the type of a defined name. Replaces 'primQNameType'. getType : QName → TC Type -- Get the definition of a defined name. Replaces 'primQNameDefinition'. getDefinition : QName → TC Definition {-# BUILTIN AGDATCM TC #-} {-# BUILTIN AGDATCMRETURN returnTC #-} {-# BUILTIN AGDATCMBIND bindTC #-} {-# BUILTIN AGDATCMUNIFY unify #-} {-# BUILTIN AGDATCMNEWMETA newMeta #-} {-# BUILTIN AGDATCMTYPEERROR typeError #-} {-# BUILTIN AGDATCMBLOCKONMETA blockOnMeta #-} {-# BUILTIN AGDATCMCATCHERROR catchTC #-} {-# BUILTIN AGDATCMINFERTYPE inferType #-} {-# BUILTIN AGDATCMCHECKTYPE checkType #-} {-# BUILTIN AGDATCMNORMALISE normalise #-} {-# BUILTIN AGDATCMGETCONTEXT getContext #-} {-# BUILTIN AGDATCMEXTENDCONTEXT extendContext #-} {-# BUILTIN AGDATCMINCONTEXT inContext #-} {-# BUILTIN AGDATCMQUOTETERM quoteTC #-} {-# BUILTIN AGDATCMUNQUOTETERM unquoteTC #-} {-# BUILTIN AGDATCMFRESHNAME freshName #-} {-# BUILTIN AGDATCMDECLAREDEF declareDef #-} {-# BUILTIN AGDATCMDEFINEFUN defineFun #-} {-# BUILTIN AGDATCMGETTYPE getType #-} {-# BUILTIN AGDATCMGETDEFINITION getDefinition #-}`

Builtin type for metavariables

There is a new builtin type for metavariables used by the new reflection framework. It is declared as follows and comes with primitive equality, ordering and show.

`postulate Meta : Set {-# BUILTIN AGDAMETA Meta #-} primitive primMetaEquality : Meta → Meta → Bool primitive primMetaLess : Meta → Meta → Bool primitive primShowMeta : Meta → String`

There are corresponding new constructors in the

`Term`

and`Literal`

data types:`data Term : Set where ... meta : Meta → List (Arg Term) → Term {-# BUILTIN AGDATERMMETA meta #-} data Literal : Set where ... meta : Meta → Literal {-# BUILTIN AGDALITMETA meta #-}`

Builtin unit type

The type checker needs to know about the unit type, which you can allow by

`record ⊤ : Set where {-# BUILTIN UNIT ⊤ #-}`

Changed behaviour of

`unquote`

The

`unquote`

primitive now expects a type checking computation instead of a pure term. In particular`unquote e`

requires`e : Term → TC ⊤`

where the argument is the representation of the hole in which the result should go. The old

`unquote`

behaviour (where`unquote`

expected a`Term`

argument) can be recovered by`OLD: unquote v NEW: unquote λ hole → unify hole v`

Changed behaviour of

`unquoteDecl`

The

`unquoteDecl`

primitive now expects a type checking computation instead of a pure function definition. It is possible to define multiple (mutually recursive) functions at the same time. More specifically`unquoteDecl x₁ .. xₙ = m`

requires

`m : TC ⊤`

and that`x₁ .. xₙ`

are defined (using`declareDef`

and`defineFun`

) after executing`m`

. As before`x₁ .. xₙ : QName`

in`m`

, but have their declared types outside the`unquoteDecl`

.New primitive

`unquoteDef`

There is a new declaration

`unquoteDef x₁ .. xₙ = m`

This works exactly as

`unquoteDecl`

(see above) with the exception that`x₁ .. xₙ`

are required to already be declared.The main advantage of

`unquoteDef`

over`unquoteDecl`

is that`unquoteDef`

is allowed in mutual blocks, allowing mutually recursion between generated definitions and hand-written definitions.The reflection interface now exposes the name hint (as a string) for variables. As before, the actual binding structure is with de Bruijn indices. The String value is just a hint used as a prefix to help display the variable. The type

`Abs`

is a new builtin type used for the constructors`Term.lam`

,`Term.pi`

,`Pattern.var`

(bultins`AGDATERMLAM`

,`AGDATERMPI`

and`AGDAPATVAR`

).`data Abs (A : Set) : Set where abs : (s : String) (x : A) → Abs A {-# BUILTIN ABS Abs #-} {-# BUILTIN ABSABS abs #-}`

Updated constructor types:

`Term.lam : Hiding → Abs Term → Term Term.pi : Arg Type → Abs Type → Term Pattern.var : String → Pattern`

Reflection-based macros

Macros are functions of type

`t1 → t2 → .. → Term → TC ⊤`

that are defined in a`macro`

block. Macro application is guided by the type of the macro, where`Term`

arguments desugar into the`quoteTerm`

syntax and`Name`

arguments into the`quote`

syntax. Arguments of any other type are preserved as-is. The last`Term`

argument is the hole term given to`unquote`

computation (see above).For example, the macro application

`f u v w`

where the macro`f`

has the type`Term → Name → Bool → Term → TC ⊤`

desugars into`unquote (f (quoteTerm u) (quote v) w)`

Limitations:

Macros cannot be recursive. This can be worked around by defining the recursive function outside the macro block and have the macro call the recursive function.

Silly example:

`macro plus-to-times : Term → Term → TC ⊤ plus-to-times (def (quote _+_) (a ∷ b ∷ [])) hole = unify hole (def (quote _*_) (a ∷ b ∷ [])) plus-to-times v hole = unify hole v thm : (a b : Nat) → plus-to-times (a + b) ≡ a * b thm a b = refl`

Macros are most useful when writing tactics, since they let you hide the reflection machinery. For instance, suppose you have a solver

`magic : Type → Term`

that takes a reflected goal and outputs a proof (when successful). You can then define the following macro

`macro by-magic : Term → TC ⊤ by-magic hole = bindTC (inferType hole) λ goal → unify hole (magic goal)`

This lets you apply the magic tactic without any syntactic noise at all:

`thm : ¬ P ≡ NP thm = by-magic`

### Literals and built-ins

Overloaded number literals.

You can now overload natural number literals using the new builtin

`FROMNAT`

:`{-# BUILTIN FROMNAT fromNat #-}`

The target of the builtin should be a defined name. Typically you would do something like

`record Number (A : Set) : Set where field fromNat : Nat → A open Number {{...}} public {-# BUILTIN FROMNAT fromNat #-}`

This will cause number literals

`n`

to be desugared to`fromNat n`

before type checking.Negative number literals.

Number literals can now be negative. For floating point literals it works as expected. For integer literals there is a new builtin

`FROMNEG`

that enables negative integer literals:`{-# BUILTIN FROMNEG fromNeg #-}`

This causes negative literals

`-n`

to be desugared to`fromNeg n`

.Overloaded string literals.

String literals can be overladed using the

`FROMSTRING`

builtin:`{-# BUILTIN FROMSTRING fromString #-}`

The will cause string literals

`s`

to be desugared to`fromString s`

before type checking.Change to builtin integers.

The

`INTEGER`

builtin now needs to be bound to a datatype with two constructors that should be bound to the new builtins`INTEGERPOS`

and`INTEGERNEGSUC`

as follows:`data Int : Set where pos : Nat -> Int negsuc : Nat -> Int {-# BUILTIN INTEGER Int #-} {-# BUILTIN INTEGERPOS pos #-} {-# BUILTIN INTEGERNEGSUC negsuc #-}`

where

`negsuc n`

represents the integer`-n - 1`

. For instance,`-5`

is represented as`negsuc 4`

. All primitive functions on integers except`primShowInteger`

have been removed, since these can be defined without too much trouble on the above representation using the corresponding functions on natural numbers.The primitives that have been removed are

`primIntegerPlus primIntegerMinus primIntegerTimes primIntegerDiv primIntegerMod primIntegerEquality primIntegerLess primIntegerAbs primNatToInteger`

New primitives for strict evaluation

`primitive primForce : ∀ {a b} {A : Set a} {B : A → Set b} (x : A) → (∀ x → B x) → B x primForceLemma : ∀ {a b} {A : Set a} {B : A → Set b} (x : A) (f : ∀ x → B x) → primForce x f ≡ f x`

`primForce x f`

evaluates to`f x`

if x is in weak head normal form, and`primForceLemma x f`

evaluates to`refl`

in the same situation. The following values are considered to be in weak head normal form:- constructor applications
- literals
- lambda abstractions
- type constructor (data/record types) applications
- function types
- Set a

### Modules

Modules in import directives

When you use

`using`

/`hiding`

/`renaming`

on a name it now automatically applies to any module of the same name, unless you explicitly mention the module. For instance,`open M using (D)`

is equivalent to

`open M using (D; module D)`

if

`M`

defines a module`D`

. This is most useful for record and data types where you always get a module of the same name as the type.With this feature there is no longer useful to be able to qualify a constructor (or field) by the name of the data type even when it differs from the name of the corresponding module. The follow (weird) code used to work, but doesn't work anymore:

`module M where data D where c : D open M using (D) renaming (module D to MD) foo : D foo = D.c`

If you want to import only the type name and not the module you have to hide it explicitly:

`open M using (D) hiding (module D)`

See discussion on Issue #836.

Private definitions of a module are no longer in scope at the Emacs mode top-level.

The reason for this change is that

`.agdai-files`

are stripped of unused private definitions (which can yield significant performance improvements for module-heavy code).To test private definitions you can create a hole at the bottom of the module, in which private definitions will be visible.

### Records

New record directives

`eta-equality`

/`no-eta-equality`

The keywords

`eta-equality`

/`no-eta-equality`

enable/disable eta rules for the (inductive) record type being declared.`record Σ (A : Set) (B : A -> Set) : Set where no-eta-equality constructor _,_ field fst : A snd : B fst open Σ -- fail : ∀ {A : Set}{B : A -> Set} → (x : Σ A B) → x ≡ (fst x , snd x) -- fail x = refl -- -- x != fst x , snd x of type Σ .A .B -- when checking that the expression refl has type x ≡ (fst x , snd x)`

Building records from modules.

The

`record { <fields> }`

syntax is now extended to accept module names as well. Fields are thus defined using the corresponding definitions from the given module.For instance assuming this record type

`R`

and module`M`

:`record R : Set where field x : X y : Y z : Z module M where x = {! ... !} y = {! ... !} r : R r = record { M; z = {! ... !} }`

Previously one had to write

`record { x = M.x; y = M.y; z = {! ... !} }`

.More precisely this construction now supports any combination of explicit field definitions and applied modules.

If a field is both given explicitly and available in one of the modules, then the explicit one takes precedence.

If a field is available in more than one module then this is ambiguous and therefore rejected. As a consequence the order of assignments does not matter.

The modules can be both applied to arguments and have import directives such as

`hiding`

,`using`

, and`renaming`

. In particular this construct subsumes the record update construction.Here is an example of record update:

`-- Record update. Same as: record r { y = {! ... !} } r2 : R r2 = record { R r; y = {! ... !} }`

A contrived example showing the use of

`hiding`

/`renaming`

:`module M2 (a : A) where w = {! ... !} z = {! ... !} r3 : A → R r3 a = record { M hiding (y); M2 a renaming (w to y) }`

Record patterns are now accepted.

Examples:

`swap : {A B : Set} (p : A × B) → B × A swap record{ proj₁ = a; proj₂ = b } = record{ proj₁ = b; proj₂ = a } thd3 : ... thd3 record{ proj₂ = record { proj₂ = c }} = c`

Record modules now properly hide all their parameters [Issue #1759]

Previously parameters to parent modules were not hidden in the record module, resulting in different behaviour between

`module M (A : Set) where record R (B : Set) : Set where`

and

`module M where record R (A B : Set) : Set where`

where in the former case,

`A`

would be an explicit argument to the module`M.R`

, but implicit in the latter case. Now`A`

is implicit in both cases.

### Instance search

Performance has been improved, recursive instance search which was previously exponential in the depth is now only quadratic.

Constructors of records and datatypes are not anymore automatically considered as instances, you have to do so explicitely, for instance:

`-- only [b] is an instance of D data D : Set where a : D instance b : D c : D -- the constructor is now an instance record tt : Set where instance constructor tt`

Lambda-bound variables are no longer automatically considered instances.

Lambda-bound variables need to be bound as instance arguments to be considered for instance search. For example,

`_==_ : {A : Set} {{_ : Eq A}} → A → A → Bool fails : {A : Set} → Eq A → A → Bool fails eqA x = x == x works : {A : Set} {{_ : Eq A}} → A → Bool works x = x == x`

Let-bound variables are no longer automatically considered instances.

To make a let-bound variable available as an instance it needs to be declared with the

`instance`

keyword, just like top-level instances. For example,`mkEq : {A : Set} → (A → A → Bool) → Eq A fails : {A : Set} → (A → A → Bool) → A → Bool fails eq x = let eqA = mkEq eq in x == x works : {A : Set} → (A → A → Bool) → A → Bool works eq x = let instance eqA = mkEq eq in x == x`

Record fields can be declared instances.

For example,

`record EqSet : Set₁ where field set : Set instance eq : Eq set`

This causes the projection function

`eq : (E : EqSet) → Eq (set E)`

to be considered for instance search.Instance search can now find arguments in variable types (but such candidates can only be lambda-bound variables, they can’t be declared as instances)

`module _ {A : Set} (P : A → Set) where postulate bla : {x : A} {{_ : P x}} → Set → Set -- Works, the instance argument is found in the context test : {x : A} {{_ : P x}} → Set → Set test B = bla B -- Still forbidden, because [P] could be instantiated later to anything instance postulate forbidden : {x : A} → P x`

Instance search now refuses to solve constraints with unconstrained metavariables, since this can lead to non-termination.

See [Issue #1532] for an example.

Top-level instances are now only considered if they are in scope. [Issue #1913]

Note that lambda-bound instances need not be in scope.

### Other changes

Unicode ellipsis character is allowed for the ellipsis token

`...`

in`with`

expressions.`Prop`

is no longer a reserved word.

## Type checking

Large indices.

Force constructor arguments no longer count towards the size of a datatype. For instance, the definition of equality below is accepted.

`data _≡_ {a} {A : Set a} : A → A → Set where refl : ∀ x → x ≡ x`

This gets rid of the asymmetry that the version of equality which indexes only on the second argument could be small, but not the version above which indexes on both arguments.

Detection of datatypes that satisfy K (i.e. sets)

Agda will now try to detect datatypes that satisfy K when

`--without-K`

is enabled. A datatype satisfies K when it follows these three rules:The types of all non-recursive constructor arguments should satisfy K.

All recursive constructor arguments should be first-order.

The types of all indices should satisfy K.

For example, the types

`Nat`

,`List Nat`

, and`x ≡ x`

(where`x : Nat`

) are all recognized by Agda as satisfying K.New unifier for case splitting

The unifier used by Agda for case splitting has been completely rewritten. The new unifier takes a much more type-directed approach in order to avoid the problems in issues #1406, #1408, #1427, and #1435.

The new unifier also has eta-equality for record types built-in. This should avoid unnecessary case splitting on record constructors and improve the performance of Agda on code that contains deeply nested record patterns (see issues #473, #635, #1575, #1603, #1613, and #1645).

In some cases, the locations of the dot patterns computed by the unifier did not correspond to the locations given by the user (see Issue #1608). This has now been fixed by adding an extra step after case splitting that checks whether the user-written patterns are compatible with the computed ones.

In some rare cases, the new unifier is still too restrictive when

`--without-K`

is enabled because it cannot generalize over the datatype indices (yet). For example, the following code is rejected:`data Bar : Set₁ where bar : Bar baz : (A : Set) → Bar data Foo : Bar → Set where foo : Foo bar test : foo ≡ foo → Set₁ test refl = Set`

The aggressive behaviour of

`with`

introduced in 2.4.2.5 has been rolled back [Issue #1692]. With no longer abstracts in the types of variables appearing in the with-expressions. [Issue #745]This means that the following example no longer works:

`fails : (f : (x : A) → a ≡ x) (b : A) → b ≡ a fails f b with a | f b fails f b | .b | refl = f b`

The

`with`

no longer abstracts the type of`f`

over`a`

, since`f`

appears in the second with-expression`f b`

. You can use a nested`with`

to make this example work.This example does work again:

`test : ∀{A : Set}{a : A}{f : A → A} (p : f a ≡ a) → f (f a) ≡ a test p rewrite p = p`

After

`rewrite p`

the goal has changed to`f a ≡ a`

, but the type of`p`

has not been rewritten, thus, the final`p`

solves the goal.The following, which worked in 2.4.2.5, no longer works:

`fails : (f : (x : A) → a ≡ x) (b : A) → b ≡ a fails f b rewrite f b = f b`

The rewrite with

`f b : a ≡ b`

is not applied to`f`

as the latter is part of the rewrite expression`f b`

. Thus, the type of`f`

remains untouched, and the changed goal`b ≡ b`

is not solved by`f b`

.When using

`rewrite`

on a term`eq`

of type`lhs ≡ rhs`

, the`lhs`

is no longer abstracted in`rhs`

[Issue #520]. This means that`f pats rewrite eq = body`

is more than syntactic sugar for

`f pats with lhs | eq f pats | _ | refl = body`

In particular, the following application of

`rewrite`

is now possible`id : Bool → Bool id true = true id false = false is-id : ∀ x → x ≡ id x is-id true = refl is-id false = refl postulate P : Bool → Set b : Bool p : P (id b) proof : P b proof rewrite is-id b = p`

Previously, this was desugared to

`proof with b | is-id b proof | _ | refl = p`

which did not type check as

`refl`

does not have type`b ≡ id b`

. Now, Agda gets the task of checking`refl : _ ≡ id b`

leading to instantiation of`_`

to`id b`

.

## Compiler backends

Major Bug Fixes:

- Function clauses with different arities are now always compiled correctly by the GHC/UHC backends. (Issue #727)

Co-patterns

Optimizations

- Builtin naturals are now represented as arbitrary-precision Integers. See the user manual, section "Agda Compilers -> Optimizations" for details.

GHC Haskell backend (MAlonzo)

Pragmas

Since builtin naturals are compiled to

`Integer`

you can no longer give a`{-# COMPILED_DATA #-}`

pragma for`Nat`

. The same goes for builtin booleans, integers, floats, characters and strings which are now hard-wired to appropriate Haskell types.

UHC compiler backend

A new backend targeting the Utrecht Haskell Compiler (UHC) is available. It targets the UHC Core language, and it's design is inspired by the Epic backend. See the user manual, section "Agda Compilers -> UHC Backend" for installation instructions.

FFI

The UHC backend has a FFI to Haskell similar to MAlonzo's. The target Haskell code also needs to be compilable using UHC, which does not support the Haskell base library version 4.*.

FFI pragmas for the UHC backend are not checked in any way. If the pragmas are wrong, bad things will happen.

Imports

Additional Haskell modules can be brought into scope with the

`IMPORT_UHC`

pragma:`{-# IMPORT_UHC Data.Char #-}`

The Haskell modules

`UHC.Base`

and`UHC.Agda.Builtins`

are always in scope and don't need to be imported explicitly.Datatypes

Agda datatypes can be bound to Haskell datatypes as follows:

Haskell:

`data HsData a = HsCon1 | HsCon2 (HsData a)`

Agda:

`data AgdaData (A : Set) : Set where AgdaCon1 : AgdaData A AgdaCon2 : AgdaData A -> AgdaData A {-# COMPILED_DATA_UHC AgdaData HsData HsCon1 HsCon2 #-}`

The mapping has to cover all constructors of the used Haskell datatype, else runtime behavior is undefined!

There are special reserved names to bind Agda datatypes to certain Haskell datatypes. For example, this binds an Agda datatype to Haskell's list datatype:

Agda:

`data AgdaList (A : Set) : Set where Nil : AgdaList A Cons : A -> AgdaList A -> AgdaList A {-# COMPILED_DATA_UHC AgdaList __LIST__ __NIL__ __CONS__ #-}`

The following "magic" datatypes are available:

`HS Datatype | Datatype Pragma | HS Constructor | Constructor Pragma () __UNIT__ () __UNIT__ List __LIST__ (:) __CONS__ [] __NIL__ Bool __BOOL__ True __TRUE__ False __FALSE__`

Functions

Agda postulates can be bound to Haskell functions. Similar as in MAlonzo, all arguments of type

`Set`

need to be dropped before calling Haskell functions. An example calling the return function:Agda:

`postulate hs-return : {A : Set} -> A -> IO A {-# COMPILED_UHC hs-return (\_ -> UHC.Agda.Builtins.primReturn) #-}`

## Emacs mode and interaction

Module contents (

`C-c C-o`

) now also works for records. [See Issue #1926 ] If you have an inferable expression of record type in an interaction point, you can invoke`C-c C-o`

to see its fields and types. Example`record R : Set where field f : A test : R → R test r = {!r!} -- C-c C-o here`

Less aggressive error notification.

Previously Emacs could jump to the position of an error even if the type-checking process was not initiated in the current buffer. Now this no longer happens: If the type-checking process was initiated in another buffer, then the cursor is moved to the position of the error in the buffer visiting the file (if any) and in every window displaying the file, but focus should not change from one file to another.

In the cases where focus does change from one file to another, one can now use the go-back functionality to return to the previous position.

Removed the

`agda-include-dirs`

customization parameter.Use

`agda-program-args`

with`-iDIR`

or`-lLIB`

instead, or add libraries to`~/.agda/defaults`

(`C:/Users/USERNAME/AppData/Roaming/agda/defaults`

or similar on Windows). See Library management, above, for more information.

## Tools

### LaTeX-backend

The default font has been changed to XITS (which is part of TeX Live):

http://www.ctan.org/tex-archive/fonts/xits/

This font is more complete with respect to Unicode.

### agda-ghc-names

New tool: The command

`agda-ghc-names fixprof <compile-dir> <ProgName>.prof`

converts

`*.prof`

files obtained from profiling runs of MAlonzo-compiled code to`*.agdaIdents.prof`

, with the original Agda identifiers replacing the MAlonzo-generated Haskell identifiers.For usage and more details, see

`src/agda-ghc-names/README.txt`

.

## Highlighting and textual backends

Names in import directives are now highlighted and are clickable. [Issue #1714] This leads also to nicer printing in the LaTeX and html backends.

## Fixed issues

See bug tracker (milestone 2.5.1)

# Release notes for Agda version 2.4.2.5

## Installation and infrastructure

Added support for GHC 7.10.3.

Added

`cpphs`

Cabal flagTurn on/off this flag to choose cpphs/cpp as the C preprocessor.

This flag is turn on by default.

(This flag was added in Agda 2.4.2.1 but it was not documented)

## Pragmas and options

Termination pragmas are no longer allowed inside

`where`

clauses [Issue #1137].

## Type checking

`with`

-abstraction is more aggressive, abstracts also in types of variables that are used in the`with`

-expressions, unless they are also used in the types of the`with`

-expressions. [Issue #1692]Example:

`test : (f : (x : A) → a ≡ x) (b : A) → b ≡ a test f b with a | f b test f b | .b | refl = f b`

Previously,

`with`

would not abstract in types of variables that appear in the`with`

-expressions, in this case, both`f`

and`b`

, leaving their types unchanged. Now, it tries to abstract in`f`

, as only`b`

appears in the types of the`with`

-expressions which are`A`

(of`a`

) and`a ≡ b`

(of`f b`

). As a result, the type of`f`

changes to`(x : A) → b ≡ x`

and the type of the goal to`b ≡ b`

(as previously).This also affects

`rewrite`

, which is implemented in terms of`with`

.`test : (f : (x : A) → a ≡ x) (b : A) → b ≡ a test f b rewrite f b = f b`

As the new

`with`

is not fully backwards-compatible, some parts of your Agda developments using`with`

or`rewrite`

might need maintenance.

## Fixed issues

See bug tracker

# Release notes for Agda version 2.4.2.4

## Installation and infrastructure

Removed support for GHC 7.4.2.

## Pragmas and options

Option

`--copatterns`

is now on by default. To switch off parsing of copatterns, use:`{-# OPTIONS --no-copatterns #-}`

Option

`--rewriting`

is now needed to use`REWRITE`

pragmas and rewriting during reduction. Rewriting is not`--safe`

.To use rewriting, first specify a relation symbol

`R`

that will later be used to add rewrite rules. A canonical candidate would be propositional equality`{-# BUILTIN REWRITE _≡_ #-}`

but any symbol

`R`

of type`Δ → A → A → Set i`

for some`A`

and`i`

is accepted. Then symbols`q`

can be added to rewriting provided their type is of the form`Γ → R ds l r`

. This will add a rewrite rule`Γ ⊢ l ↦ r : A[ds/Δ]`

to the signature, which fires whenever a term is an instance of

`l`

. For example, if`plus0 : ∀ x → x + 0 ≡ x`

(ideally, there is a proof for

`plus0`

, but it could be a postulate), then`{-# REWRITE plus0 #-}`

will prompt Agda to rewrite any well-typed term of the form

`t + 0`

to`t`

.Some caveats: Agda accepts and applies rewrite rules naively, it is very easy to break consistency and termination of type checking. Some examples of rewrite rules that should

*not*be added:`refl : ∀ x → x ≡ x -- Agda loops plus-sym : ∀ x y → x + y ≡ y + x -- Agda loops absurd : true ≡ false -- Breaks consistency`

Adding only proven equations should at least preserve consistency, but this is only a conjecture, so know what you are doing! Using rewriting, you are entering into the wilderness, where you are on your own!

## Language

`forall`

/`∀`

now parses like`λ`

, i.e., the following parses now [Issue #1583]:`⊤ × ∀ (B : Set) → B → B`

The underscore pattern

`_`

can now also stand for an inaccessible pattern (dot pattern). This alleviates the need for writing`._`

. [Issue #1605] Instead of`transVOld : ∀{A : Set} (a b c : A) → a ≡ b → b ≡ c → a ≡ c transVOld _ ._ ._ refl refl = refl`

one can now write

`transVNew : ∀{A : Set} (a b c : A) → a ≡ b → b ≡ c → a ≡ c transVNew _ _ _ refl refl = refl`

and let Agda decide where to put the dots. This was always possible by using hidden arguments

`transH : ∀{A : Set}{a b c : A} → a ≡ b → b ≡ c → a ≡ c transH refl refl = refl`

which is now equivalent to

`transHNew : ∀{A : Set}{a b c : A} → a ≡ b → b ≡ c → a ≡ c transHNew {a = _}{b = _}{c = _} refl refl = refl`

Before, underscore

`_`

stood for an unnamed variable that could not be instantiated by an inaccessible pattern. If one no wants to prevent Agda from instantiating, one needs to use a variable name other than underscore (however, in practice this situation seems unlikely).

## Type checking

Polarity of phantom arguments to data and record types has changed. [Issue #1596] Polarity of size arguments is Nonvariant (both monotone and antitone). Polarity of other arguments is Covariant (monotone). Both were Invariant before (neither monotone nor antitone).

The following example type-checks now:

`open import Common.Size -- List should be monotone in both arguments -- (even when `cons' is missing). data List (i : Size) (A : Set) : Set where [] : List i A castLL : ∀{i A} → List i (List i A) → List ∞ (List ∞ A) castLL x = x -- Stream should be antitone in the first and monotone in the second argument -- (even with field `tail' missing). record Stream (i : Size) (A : Set) : Set where coinductive field head : A castSS : ∀{i A} → Stream ∞ (Stream ∞ A) → Stream i (Stream i A) castSS x = x`

`SIZELT`

lambdas must be consistent [Issue #1523, see Abel and Pientka, ICFP 2013]. When lambda-abstracting over type (`Size< size`

) then`size`

must be non-zero, for any valid instantiation of size variables.The good:

`data Nat (i : Size) : Set where zero : ∀ (j : Size< i) → Nat i suc : ∀ (j : Size< i) → Nat j → Nat i {-# TERMINATING #-} -- This definition is fine, the termination checker is too strict at the moment. fix : ∀ {C : Size → Set} → (∀ i → (∀ (j : Size< i) → Nat j -> C j) → Nat i → C i) → ∀ i → Nat i → C i fix t i (zero j) = t i (λ (k : Size< i) → fix t k) (zero j) fix t i (suc j n) = t i (λ (k : Size< i) → fix t k) (suc j n)`

The

`λ (k : Size< i)`

is fine in both cases, as context`i : Size, j : Size< i`

guarantees that

`i`

is non-zero.The bad:

`record Stream {i : Size} (A : Set) : Set where coinductive constructor _∷ˢ_ field head : A tail : ∀ {j : Size< i} → Stream {j} A open Stream public _++ˢ_ : ∀ {i A} → List A → Stream {i} A → Stream {i} A [] ++ˢ s = s (a ∷ as) ++ˢ s = a ∷ˢ (as ++ˢ s)`

This fails, maybe unjustified, at

`i : Size, s : Stream {i} A ⊢ a ∷ˢ (λ {j : Size< i} → as ++ˢ s)`

Fixed by defining the constructor by copattern matching:

`record Stream {i : Size} (A : Set) : Set where coinductive field head : A tail : ∀ {j : Size< i} → Stream {j} A open Stream public _∷ˢ_ : ∀ {i A} → A → Stream {i} A → Stream {↑ i} A head (a ∷ˢ as) = a tail (a ∷ˢ as) = as _++ˢ_ : ∀ {i A} → List A → Stream {i} A → Stream {i} A [] ++ˢ s = s (a ∷ as) ++ˢ s = a ∷ˢ (as ++ˢ s)`

The ugly:

`fix : ∀ {C : Size → Set} → (∀ i → (∀ (j : Size< i) → C j) → C i) → ∀ i → C i fix t i = t i λ (j : Size< i) → fix t j`

For

`i=0`

, there is no such`j`

at runtime, leading to looping behavior.

## Interaction

Issue #635 has been fixed. Case splitting does not spit out implicit record patterns any more.

`record Cont : Set₁ where constructor _◃_ field Sh : Set Pos : Sh → Set open Cont data W (C : Cont) : Set where sup : (s : Sh C) (k : Pos C s → W C) → W C bogus : {C : Cont} → W C → Set bogus w = {!w!}`

Case splitting on

`w`

yielded, since the fix of Issue #473,`bogus {Sh ◃ Pos} (sup s k) = ?`

Now it gives, as expected,

`bogus (sup s k) = ?`

## Performance

As one result of the 21st Agda Implementor's Meeting (AIM XXI), serialization of the standard library is 50% faster (time reduced by a third), without using additional disk space for the interface files.

## Bug fixes

Issues fixed (see bug tracker):

#1546 (copattern matching and with-clauses)

#1560 (positivity checker inefficiency)

#1584 (let pattern with trailing implicit)

# Release notes for Agda version 2.4.2.3

## Installation and infrastructure

Added support for GHC 7.10.1.

Removed support for GHC 7.0.4.

## Language

`_`

is no longer a valid name for a definition. The following fails now: [Issue #1465]`postulate _ : Set`

Typed bindings can now contain hiding information [Issue #1391]. This means you can now write

`assoc : (xs {ys zs} : List A) → ((xs ++ ys) ++ zs) ≡ (xs ++ (ys ++ zs))`

instead of the longer

`assoc : (xs : List A) {ys zs : List A} → ...`

It also works with irrelevance

`.(xs {ys zs} : List A) → ...`

but of course does not make sense if there is hiding information already. Thus, this is (still) a parse error:

`{xs {ys zs} : List A} → ...`

The builtins for sized types no longer need accompanying postulates. The BUILTIN pragmas for size stuff now also declare the identifiers they bind to.

`{-# BUILTIN SIZEUNIV SizeUniv #-} -- SizeUniv : SizeUniv {-# BUILTIN SIZE Size #-} -- Size : SizeUniv {-# BUILTIN SIZELT Size<_ #-} -- Size<_ : ..Size → SizeUniv {-# BUILTIN SIZESUC ↑_ #-} -- ↑_ : Size → Size {-# BUILTIN SIZEINF ∞ #-} -- ∞ : Size`

`Size`

and`Size<`

now live in the new universe`SizeUniv`

. It is forbidden to build function spaces in this universe, in order to prevent the malicious assumption of a size predecessor`pred : (i : Size) → Size< i`

[Issue #1428].

Unambiguous notations (coming from syntax declarations) that resolve to ambiguous names are now parsed unambiguously [Issue #1194].

If only some instances of an overloaded name have a given associated notation (coming from syntax declarations), then this name can only be resolved to the given instances of the name, not to other instances [Issue #1194].

Previously, if different instances of an overloaded name had

*different*associated notations, then none of the notations could be used. Now all of them can be used.Note that notation identity does not only involve the right-hand side of the syntax declaration. For instance, the following notations are not seen as identical, because the implicit argument names are different:

`module A where data D : Set where c : {x y : D} → D syntax c {x = a} {y = b} = a ∙ b module B where data D : Set where c : {y x : D} → D syntax c {y = a} {x = b} = a ∙ b`

If an overloaded operator is in scope with at least two distinct fixities, then it gets the default fixity [Issue #1436].

Similarly, if two or more identical notations for a given overloaded name are in scope, and these notations do not all have the same fixity, then they get the default fixity.

## Type checking

Functions of varying arity can now have with-clauses and use rewrite.

Example:

`NPred : Nat → Set NPred 0 = Bool NPred (suc n) = Nat → NPred n const : Bool → ∀{n} → NPred n const b {0} = b const b {suc n} m = const b {n} allOdd : ∀ n → NPred n allOdd 0 = true allOdd (suc n) m with even m ... | true = const false ... | false = allOdd n`

Function defined by copattern matching can now have

`with`

-clauses and use`rewrite`

.Example:

`{-# OPTIONS --copatterns #-} record Stream (A : Set) : Set where coinductive constructor delay field force : A × Stream A open Stream map : ∀{A B} → (A → B) → Stream A → Stream B force (map f s) with force s ... | a , as = f a , map f as record Bisim {A B} (R : A → B → Set) (s : Stream A) (t : Stream B) : Set where coinductive constructor ~delay field ~force : let a , as = force s b , bs = force t in R a b × Bisim R as bs open Bisim SEq : ∀{A} (s t : Stream A) → Set SEq = Bisim (_≡_) -- Slightly weird definition of symmetry to demonstrate rewrite. ~sym' : ∀{A} {s t : Stream A} → SEq s t → SEq t s ~force (~sym' {s = s} {t} p) with force s | force t | ~force p ... | a , as | b , bs | r , q rewrite r = refl , ~sym' q`

Instances can now be defined by copattern matching. [Issue #1413] The following example extends the one in [Abel, Pientka, Thibodeau, Setzer, POPL 2013, Section 2.2]:

`{-# OPTIONS --copatterns #-} -- The Monad type class record Monad (M : Set → Set) : Set1 where field return : {A : Set} → A → M A _>>=_ : {A B : Set} → M A → (A → M B) → M B open Monad {{...}} -- The State newtype record State (S A : Set) : Set where field runState : S → A × S open State -- State is an instance of Monad instance stateMonad : {S : Set} → Monad (State S) runState (return {{stateMonad}} a ) s = a , s -- NEW runState (_>>=_ {{stateMonad}} m k) s₀ = -- NEW let a , s₁ = runState m s₀ in runState (k a) s₁ -- stateMonad fulfills the monad laws leftId : {A B S : Set}(a : A)(k : A → State S B) → (return a >>= k) ≡ k a leftId a k = refl rightId : {A B S : Set}(m : State S A) → (m >>= return) ≡ m rightId m = refl assoc : {A B C S : Set}(m : State S A)(k : A → State S B)(l : B → State S C) → ((m >>= k) >>= l) ≡ (m >>= λ a → k a >>= l) assoc m k l = refl`

## Emacs mode

The new menu option

`Switch to another version of Agda`

tries to do what it says.Changed feature: Interactively split result.

[ This is as before: ] Make-case (

`C-c C-c`

) with no variables given tries to split on the result to introduce projection patterns. The hole needs to be of record type, of course.`test : {A B : Set} (a : A) (b : B) → A × B test a b = ?`

Result-splitting

`?`

will produce the new clauses:`proj₁ (test a b) = ? proj₂ (test a b) = ?`

[ This has changed: ] If hole is of function type,

`make-case`

will introduce only pattern variables (as much as it can).`testFun : {A B : Set} (a : A) (b : B) → A × B testFun = ?`

Result-splitting

`?`

will produce the new clause:`testFun a b = ?`

A second invocation of

`make-case`

will then introduce projection patterns.

## Error messages

Agda now suggests corrections of misspelled options, e.g.

`{-# OPTIONS --dont-termination-check --without-k --senf-gurke #-}`

Unrecognized options:

`--dont-termination-check (did you mean --no-termination-check ?) --without-k (did you mean --without-K ?) --senf-gurke`

Nothing close to

`--senf-gurke`

, I am afraid.

## Compiler backends

The Epic backend has been removed [Issue #1481].

## Bug fixes

Fixed bug with

`unquoteDecl`

not working in instance blocks [Issue #1491].Other issues fixed (see bug tracker

# Release notes for Agda version 2.4.2.2

## Bug fixes

# Release notes for Agda version 2.4.2.1

## Pragmas and options

New pragma

`{-# TERMINATING #-}`

replacing`{-# NO_TERMINATION_CHECK #-}`

Complements the existing pragma

`{-# NON_TERMINATING #-}`

. Skips termination check for the associated definitions and marks them as terminating. Thus, it is a replacement for`{-# NO_TERMINATION_CHECK #-}`

with the same semantics.You can no longer use pragma

`{-# NO_TERMINATION_CHECK #-}`

to skip the termination check, but must label your definitions as either`{-# TERMINATING #-}`

or`{-# NON_TERMINATING #-}`

instead.Note:

`{-# OPTION --no-termination-check #-}`

labels all your definitions as`{-# TERMINATING #-}`

, putting you in the danger zone of a loop in the type checker.

## Language

Referring to a local variable shadowed by module opening is now an error. Previous behavior was preferring the local over the imported definitions. [Issue #1266]

Note that module parameters are locals as well as variables bound by λ, dependent function type, patterns, and let.

Example:

`module M where A = Set1 test : (A : Set) → let open M in A`

The last

`A`

produces an error, since it could refer to the local variable`A`

or to the definition imported from module`M`

.`with`

on a variable bound by a module telescope or a pattern of a parent function is now forbidden. [Issue #1342]`data Unit : Set where unit : Unit id : (A : Set) → A → A id A a = a module M (x : Unit) where dx : Unit → Unit dx unit = x g : ∀ u → x ≡ dx u g with x g | unit = id (∀ u → unit ≡ dx u) ?`

Even though this code looks right, Agda complains about the type expression

`∀ u → unit ≡ dx u`

. If you ask Agda what should go there instead, it happily tells you that it wants`∀ u → unit ≡ dx u`

. In fact what you do not see and Agda will never show you is that the two expressions actually differ in the invisible first argument to`dx`

, which is visible only outside module`M`

. What Agda wants is an invisible`unit`

after`dx`

, but all you can write is an invisible`x`

(which is inserted behind the scenes).To avoid those kinds of paradoxes,

`with`

is now outlawed on module parameters. This should ensure that the invisible arguments are always exactly the module parameters.Since a

`where`

block is desugared as module with pattern variables of the parent clause as module parameters, the same strikes you for uses of`with`

on pattern variables of the parent function.`f : Unit → Unit f x = unit where dx : Unit → Unit dx unit = x g : ∀ u → x ≡ dx u g with x g | unit = id ((u : Unit) → unit ≡ dx u) ?`

The

`with`

on pattern variable`x`

of the parent clause`f x = unit`

is outlawed now.

## Type checking

Termination check failure is now a proper error.

We no longer continue type checking after termination check failures. Use pragmas

`{-# NON_TERMINATING #-}`

and`{-# NO_TERMINATION_CHECK #-}`

near the offending definitions if you want to do so. Or switch off the termination checker altogether with`{-# OPTIONS --no-termination-check #-}`

(at your own risk!).(Since Agda 2.4.2): Termination checking

`--without-K`

restricts structural descent to arguments ending in data types or`Size`

. Likewise, guardedness is only tracked when result type is data or record type.`mutual data WOne : Set where wrap : FOne → WOne FOne = ⊥ → WOne noo : (X : Set) → (WOne ≡ X) → X → ⊥ noo .WOne refl (wrap f) = noo FOne iso f`

`noo`

is rejected since at type`X`

the structural descent`f < wrap f`

is discounted`--without-K`

.`data Pandora : Set where C : ∞ ⊥ → Pandora loop : (A : Set) → A ≡ Pandora → A loop .Pandora refl = C (♯ (loop ⊥ foo))`

`loop`

is rejected since guardedness is not tracked at type`A`

`--without-K`

.

## Termination checking

The termination checker can now recognize simple subterms in dot patterns.

`data Subst : (d : Nat) → Set where c₁ : ∀ {d} → Subst d → Subst d c₂ : ∀ {d₁ d₂} → Subst d₁ → Subst d₂ → Subst (suc d₁ + d₂) postulate comp : ∀ {d₁ d₂} → Subst d₁ → Subst d₂ → Subst (d₁ + d₂) lookup : ∀ d → Nat → Subst d → Set₁ lookup d zero (c₁ ρ) = Set lookup d (suc v) (c₁ ρ) = lookup d v ρ lookup .(suc d₁ + d₂) v (c₂ {d₁} {d₂} ρ σ) = lookup (d₁ + d₂) v (comp ρ σ)`

The dot pattern here is actually normalized, so it is

`suc (d₁ + d₂)`

and the corresponding recursive call argument is

`(d₁ + d₂)`

. In such simple cases, Agda can now recognize that the pattern is constructor applied to call argument, which is valid descent.Note however, that Agda only looks for syntactic equality when identifying subterms, since it is not allowed to normalize terms on the rhs during termination checking.

Actually writing the dot pattern has no effect, this works as well, and looks pretty magical... ;-)

`hidden : ∀{d} → Nat → Subst d → Set₁ hidden zero (c₁ ρ) = Set hidden (suc v) (c₁ ρ) = hidden v ρ hidden v (c₂ ρ σ) = hidden v (comp ρ σ)`

## Tools

### LaTeX-backend

Fixed the issue of identifiers containing operators being typeset with excessive math spacing.

## Bug fixes

Issue #1194

Issue #836: Fields and constructors can be qualified by the record/data

*type*as well as by their record/data module. This now works also for record/data type imported from parametrized modules:`module M (_ : Set₁) where record R : Set₁ where field X : Set open M Set using (R) -- rather than using (module R) X : R → Set X = R.X`

# Release notes for Agda version 2.4.2

## Pragmas and options

New option:

`--with-K`

This can be used to override a global

`--without-K`

in a file, by adding a pragma`{-# OPTIONS --with-K #-}`

.New pragma

`{-# NON_TERMINATING #-}`

This is a safer version of

`NO_TERMINATION_CHECK`

which doesn't treat the affected functions as terminating. This means that`NON_TERMINATING`

functions do not reduce during type checking. They do reduce at run-time and when invoking`C-c C-n`

at top-level (but not in a hole).

## Language

Instance search is now more efficient and recursive (see Issue #938) (but without termination check yet).

A new keyword

`instance`

has been introduced (in the style of`abstract`

and`private`

) which must now be used for every definition/postulate that has to be taken into account during instance resolution. For example:`record RawMonoid (A : Set) : Set where field nil : A _++_ : A -> A -> A open RawMonoid {{...}} instance rawMonoidList : {A : Set} -> RawMonoid (List A) rawMonoidList = record { nil = []; _++_ = List._++_ } rawMonoidMaybe : {A : Set} {{m : RawMonoid A}} -> RawMonoid (Maybe A) rawMonoidMaybe {A} = record { nil = nothing ; _++_ = catMaybe } where catMaybe : Maybe A -> Maybe A -> Maybe A catMaybe nothing mb = mb catMaybe ma nothing = ma catMaybe (just a) (just b) = just (a ++ b)`

Moreover, each type of an instance must end in (something that reduces to) a named type (e.g. a record, a datatype or a postulate). This allows us to build a simple index structure

`data/record name --> possible instances`

that speeds up instance search.

Instance search takes into account all local bindings and all global

`instance`

bindings and the search is recursive. For instance, searching for`? : RawMonoid (Maybe (List A))`

will consider the candidates {

`rawMonoidList`

,`rawMonoidMaybe`

}, fail to unify the first one, succeeding with the second one`? = rawMonoidMaybe {A = List A} {{m = ?m}} : RawMonoid (Maybe (List A))`

and continue with goal

`?m : RawMonoid (List A)`

This will then find

`?m = rawMonoidList {A = A}`

and putting together we have the solution.

Be careful that there is no termination check for now, you can easily make Agda loop by declaring the identity function as an instance. But it shouldn’t be possible to make Agda loop by only declaring structurally recursive instances (whatever that means).

Additionally:

Uniqueness of instances is up to definitional equality (see Issue #899).

Instances of the following form are allowed:

`EqSigma : {A : Set} {B : A → Set} {{EqA : Eq A}} {{EqB : {a : A} → Eq (B a)}} → Eq (Σ A B)`

When searching recursively for an instance of type

`{a : A} → Eq (B a)`

, a lambda will automatically be introduced and instance search will search for something of type`Eq (B a)`

in the context extended by`a : A`

. When searching for an instance, the`a`

argument does not have to be implicit, but in the definition of`EqSigma`

, instance search will only be able to use`EqB`

if`a`

is implicit.There is no longer any attempt to solve irrelevant metas by instance search.

Constructors of records and datatypes are automatically added to the instance table.

You can now use

`quote`

in patterns.For instance, here is a function that unquotes a (closed) natural number term.

`unquoteNat : Term → Maybe Nat unquoteNat (con (quote Nat.zero) []) = just zero unquoteNat (con (quote Nat.suc) (arg _ n ∷ [])) = fmap suc (unquoteNat n) unquoteNat _ = nothing`

The builtin constructors

`AGDATERMUNSUPPORTED`

and`AGDASORTUNSUPPORTED`

are now translated to meta variables when unquoting.New syntactic sugar

`tactic e`

and`tactic e | e1 | .. | en`

.It desugars as follows and makes it less unwieldy to call reflection-based tactics.

`tactic e --> quoteGoal g in unquote (e g) tactic e | e1 | .. | en --> quoteGoal g in unquote (e g) e1 .. en`

Note that in the second form the tactic function should generate a function from a number of new subgoals to the original goal. The type of

`e`

should be`Term -> Term`

in both cases.New reflection builtins for literals.

The term data type

`AGDATERM`

now needs an additional constructor`AGDATERMLIT`

taking a reflected literal defined as follows (with appropriate builtin bindings for the types`Nat`

,`Float`

, etc).`data Literal : Set where nat : Nat → Literal float : Float → Literal char : Char → Literal string : String → Literal qname : QName → Literal {-# BUILTIN AGDALITERAL Literal #-} {-# BUILTIN AGDALITNAT nat #-} {-# BUILTIN AGDALITFLOAT float #-} {-# BUILTIN AGDALITCHAR char #-} {-# BUILTIN AGDALITSTRING string #-} {-# BUILTIN AGDALITQNAME qname #-}`

When quoting (

`quoteGoal`

or`quoteTerm`

) literals will be mapped to the`AGDATERMLIT`

constructor. Previously natural number literals were quoted to`suc`

/`zero`

application and other literals were quoted to`AGDATERMUNSUPPORTED`

.New reflection builtins for function definitions.

`AGDAFUNDEF`

should now map to a data type defined as follows(with

`{-# BUILTIN QNAME QName #-} {-# BUILTIN ARG Arg #-} {-# BUILTIN AGDATERM Term #-} {-# BUILTIN AGDATYPE Type #-} {-# BUILTIN AGDALITERAL Literal #-}`

).

`data Pattern : Set where con : QName → List (Arg Pattern) → Pattern dot : Pattern var : Pattern lit : Literal → Pattern proj : QName → Pattern absurd : Pattern {-# BUILTIN AGDAPATTERN Pattern #-} {-# BUILTIN AGDAPATCON con #-} {-# BUILTIN AGDAPATDOT dot #-} {-# BUILTIN AGDAPATVAR var #-} {-# BUILTIN AGDAPATLIT lit #-} {-# BUILTIN AGDAPATPROJ proj #-} {-# BUILTIN AGDAPATABSURD absurd #-} data Clause : Set where clause : List (Arg Pattern) → Term → Clause absurd-clause : List (Arg Pattern) → Clause {-# BUILTIN AGDACLAUSE Clause #-} {-# BUILTIN AGDACLAUSECLAUSE clause #-} {-# BUILTIN AGDACLAUSEABSURD absurd-clause #-} data FunDef : Set where fun-def : Type → List Clause → FunDef {-# BUILTIN AGDAFUNDEF FunDef #-} {-# BUILTIN AGDAFUNDEFCON fun-def #-}`

New reflection builtins for extended (pattern-matching) lambda.

The

`AGDATERM`

data type has been augmented with a constructor`AGDATERMEXTLAM : List AGDACLAUSE → List (ARG AGDATERM) → AGDATERM`

Absurd lambdas (

`λ ()`

) are quoted to extended lambdas with an absurd clause.Unquoting declarations.

You can now define (recursive) functions by reflection using the new

`unquoteDecl`

declaration`unquoteDecl x = e`

Here e should have type

`AGDAFUNDEF`

and evaluate to a closed value. This value is then spliced in as the definition of`x`

. In the body`e`

,`x`

has type`QNAME`

which lets you splice in recursive definitions.Standard modifiers, such as fixity declarations, can be applied to

`x`

as expected.Quoted levels

Universe levels are now quoted properly instead of being quoted to

`AGDASORTUNSUPPORTED`

.`Setω`

still gets an unsupported sort, however.Module applicants can now be operator applications.

Example:

`postulate [_] : A -> B module M (b : B) where module N (a : A) = M [ a ]`

[See Issue #1245]

Minor change in module application semantics. [Issue #892]

Previously re-exported functions were not redefined when instantiating a module. For instance

`module A where f = ... module B (X : Set) where open A public module C = B Nat`

In this example

`C.f`

would be an alias for`A.f`

, so if both`A`

and`C`

were opened`f`

would not be ambiguous. However, this behaviour is not correct when`A`

and`B`

share some module parameters (Issue #892). To fix this`C`

now defines its own copy of`f`

(which evaluates to`A.f`

), which means that opening`A`

and`C`

results in an ambiguous`f`

.

## Type checking

Recursive records need to be declared as either

`inductive`

or`coinductive`

.`inductive`

is no longer default for recursive records. Examples:`record _×_ (A B : Set) : Set where constructor _,_ field fst : A snd : B record Tree (A : Set) : Set where inductive constructor tree field elem : A subtrees : List (Tree A) record Stream (A : Set) : Set where coinductive constructor _::_ field head : A tail : Stream A`

If you are using old-style (musical) coinduction, a record may have to be declared as inductive, paradoxically.

`record Stream (A : Set) : Set where inductive -- YES, THIS IS INTENDED ! constructor _∷_ field head : A tail : ∞ (Stream A)`

This is because the "coinduction" happens in the use of

`∞`

and not in the use of`record`

.

## Tools

### Emacs mode

A new menu option

`Display`

can be used to display the version of the running Agda process.

### LaTeX-backend

New experimental option

`references`

has been added. When specified, i.e.:`\usepackage[references]{agda}`

a new command called

`\AgdaRef`

is provided, which lets you reference previously typeset commands, e.g.:Let us postulate

`\AgdaRef{apa}`

.`\begin{code} postulate apa : Set \end{code}`

Above

`apa`

will be typeset (highlighted) the same in the text as in the code, provided that the LaTeX output is post-processed using`src/data/postprocess-latex.pl`

, e.g.:`cp $(dirname $(dirname $(agda-mode locate)))/postprocess-latex.pl . agda -i. --latex Example.lagda cd latex/ perl ../postprocess-latex.pl Example.tex > Example.processed mv Example.processed Example.tex xelatex Example.tex`

Mix-fix and Unicode should work as expected (Unicode requires XeLaTeX/LuaLaTeX), but there are limitations:

Overloading identifiers should be avoided, if multiples exist

`\AgdaRef`

will typeset according to the first it finds.Only the current module is used, should you need to reference identifiers in other modules then you need to specify which other module manually, i.e.

`\AgdaRef[module]{identifier}`

.

# Release notes for Agda 2 version 2.4.0.2

The Agda input mode now supports alphabetical super and subscripts, in addition to the numerical ones that were already present. [Issue #1240]

New feature: Interactively split result.

Make case (

`C-c C-c`

) with no variables given tries to split on the result to introduce projection patterns. The hole needs to be of record type, of course.`test : {A B : Set} (a : A) (b : B) → A × B test a b = ?`

Result-splitting

`?`

will produce the new clauses:`proj₁ (test a b) = ? proj₂ (test a b) = ?`

If hole is of function type ending in a record type, the necessary pattern variables will be introduced before the split. Thus, the same result can be obtained by starting from:

`test : {A B : Set} (a : A) (b : B) → A × B test = ?`

The so far undocumented

`ETA`

pragma now throws an error if applied to definitions that are not records.`ETA`

can be used to force eta-equality at recursive record types, for which eta is not enabled automatically by Agda. Here is such an example:`mutual data Colist (A : Set) : Set where [] : Colist A _∷_ : A → ∞Colist A → Colist A record ∞Colist (A : Set) : Set where coinductive constructor delay field force : Colist A open ∞Colist {-# ETA ∞Colist #-} test : {A : Set} (x : ∞Colist A) → x ≡ delay (force x) test x = refl`

Note: Unsafe use of

`ETA`

can make Agda loop, e.g. by triggering infinite eta expansion!Bugs fixed (see bug tracker):

# Release notes for Agda 2 version 2.4.0.1

The option

`--compile-no-main`

has been renamed to`--no-main`

.`COMPILED_DATA`

pragmas can now be given for records.Various bug fixes.

# Release notes for Agda 2 version 2.4.0

## Installation and infrastructure

A new module called

`Agda.Primitive`

has been introduced. This module is available to all users, even if the standard library is not used. Currently the module contains level primitives and their representation in Haskell when compiling with MAlonzo:`infixl 6 _⊔_ postulate Level : Set lzero : Level lsuc : (ℓ : Level) → Level _⊔_ : (ℓ₁ ℓ₂ : Level) → Level {-# COMPILED_TYPE Level () #-} {-# COMPILED lzero () #-} {-# COMPILED lsuc (\_ -> ()) #-} {-# COMPILED _⊔_ (\_ _ -> ()) #-} {-# BUILTIN LEVEL Level #-} {-# BUILTIN LEVELZERO lzero #-} {-# BUILTIN LEVELSUC lsuc #-} {-# BUILTIN LEVELMAX _⊔_ #-}`

To bring these declarations into scope you can use a declaration like the following one:

`open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)`

The standard library reexports these primitives (using the names

`zero`

and`suc`

instead of`lzero`

and`lsuc`

) from the`Level`

module.Existing developments using universe polymorphism might now trigger the following error message:

`Duplicate binding for built-in thing LEVEL, previous binding to .Agda.Primitive.Level`

To fix this problem, please remove the duplicate bindings.

Technical details (perhaps relevant to those who build Agda packages):

The include path now always contains a directory

`<DATADIR>/lib/prim`

, and this directory is supposed to contain a subdirectory Agda containing a file`Primitive.agda`

.The standard location of

`<DATADIR>`

is system- and installation-specific. E.g., in a Cabal`--user`

installation of Agda-2.3.4 on a standard single-ghc Linux system it would be`$HOME/.cabal/share/Agda-2.3.4`

or something similar.The location of the

`<DATADIR>`

directory can be configured at compile-time using Cabal flags (`--datadir`

and`--datasubdir`

). The location can also be set at run-time, using the`Agda_datadir`

environment variable.

## Pragmas and options

Pragma

`NO_TERMINATION_CHECK`

placed within a mutual block is now applied to the whole mutual block (rather than being discarded silently). Adding to the uses 1.-4. outlined in the release notes for 2.3.2 we allow:3a. Skipping an old-style mutual block: Somewhere within

`mutual`

block before a type signature or first function clause.```agda mutual {-# NO

*TERMINATION*CHECK #-} c : A c = dd : A d = c ```

New option

`--no-pattern-matching`

Disables all forms of pattern matching (for the current file). You can still import files that use pattern matching.

New option

`-v profile:7`

Prints some stats on which phases Agda spends how much time. (Number might not be very reliable, due to garbage collection interruptions, and maybe due to laziness of Haskell.)

New option

`--no-sized-types`

Option

`--sized-types`

is now default.`--no-sized-types`

will turn off an extra (inexpensive) analysis on data types used for subtyping of sized types.

## Language

Experimental feature:

`quoteContext`

There is a new keyword

`quoteContext`

that gives users access to the list of names in the current local context. For instance:`open import Data.Nat open import Data.List open import Reflection foo : ℕ → ℕ → ℕ foo 0 m = 0 foo (suc n) m = quoteContext xs in ?`

In the remaining goal, the list

`xs`

will consist of two names,`n`

and`m`

, corresponding to the two local variables. At the moment it is not possible to access let bound variables (this feature may be added in the future).Experimental feature: Varying arity. Function clauses may now have different arity, e.g.,

`Sum : ℕ → Set Sum 0 = ℕ Sum (suc n) = ℕ → Sum n sum : (n : ℕ) → ℕ → Sum n sum 0 acc = acc sum (suc n) acc m = sum n (m + acc)`

or,

`T : Bool → Set T true = Bool T false = Bool → Bool f : (b : Bool) → T b f false true = false f false false = true f true = true`

This feature is experimental. Yet unsupported: - Varying arity and

`with`

.- Compilation of functions with varying arity to Haskell, JS, or Epic.

Experimental feature: copatterns. (Activated with option

`--copatterns`

)We can now define a record by explaining what happens if you project the record. For instance:

`{-# OPTIONS --copatterns #-} record _×_ (A B : Set) : Set where constructor _,_ field fst : A snd : B open _×_ pair : {A B : Set} → A → B → A × B fst (pair a b) = a snd (pair a b) = b swap : {A B : Set} → A × B → B × A fst (swap p) = snd p snd (swap p) = fst p swap3 : {A B C : Set} → A × (B × C) → C × (B × A) fst (swap3 t) = snd (snd t) fst (snd (swap3 t)) = fst (snd t) snd (snd (swap3 t)) = fst t`

Taking a projection on the left hand side (lhs) is called a projection pattern, applying to a pattern is called an application pattern. (Alternative terms: projection/application copattern.)

In the first example, the symbol

`pair`

, if applied to variable patterns`a`

and`b`

and then projected via`fst`

, reduces to`a`

.`pair`

by itself does not reduce.A typical application are coinductive records such as streams:

`record Stream (A : Set) : Set where coinductive field head : A tail : Stream A open Stream repeat : {A : Set} (a : A) -> Stream A head (repeat a) = a tail (repeat a) = repeat a`

Again,

`repeat a`

by itself will not reduce, but you can take a projection (head or tail) and then it will reduce to the respective rhs. This way, we get the lazy reduction behavior necessary to avoid looping corecursive programs.Application patterns do not need to be trivial (i.e., variable patterns), if we mix with projection patterns. E.g., we can have

`nats : Nat -> Stream Nat head (nats zero) = zero tail (nats zero) = nats zero head (nats (suc x)) = x tail (nats (suc x)) = nats x`

Here is an example (not involving coinduction) which demostrates records with fields of function type:

`-- The State monad record State (S A : Set) : Set where constructor state field runState : S → A × S open State -- The Monad type class record Monad (M : Set → Set) : Set1 where constructor monad field return : {A : Set} → A → M A _>>=_ : {A B : Set} → M A → (A → M B) → M B -- State is an instance of Monad -- Demonstrates the interleaving of projection and application patterns stateMonad : {S : Set} → Monad (State S) runState (Monad.return stateMonad a ) s = a , s runState (Monad._>>=_ stateMonad m k) s₀ = let a , s₁ = runState m s₀ in runState (k a) s₁ module MonadLawsForState {S : Set} where open Monad (stateMonad {S}) leftId : {A B : Set}(a : A)(k : A → State S B) → (return a >>= k) ≡ k a leftId a k = refl rightId : {A B : Set}(m : State S A) → (m >>= return) ≡ m rightId m = refl assoc : {A B C : Set}(m : State S A)(k : A → State S B)(l : B → State S C) → ((m >>= k) >>= l) ≡ (m >>= λ a → (k a >>= l)) assoc m k l = refl`

Copatterns are yet experimental and the following does not work:

Copatterns and

`with`

clauses.Compilation of copatterns to Haskell, JS, or Epic.

Projections generated by

`open R {{...}}`

are not handled properly on lhss yet.

Conversion checking is slower in the presence of copatterns, since stuck definitions of record type do no longer count as neutral, since they can become unstuck by applying a projection. Thus, comparing two neutrals currently requires comparing all they projections, which repeats a lot of work.

Top-level module no longer required.

The top-level module can be omitted from an Agda file. The module name is then inferred from the file name by dropping the path and the

`.agda`

extension. So, a module defined in`/A/B/C.agda`

would get the name`C`

.You can also suppress only the module name of the top-level module by writing

`module _ where`

This works also for parameterised modules.

Module parameters are now always hidden arguments in projections. For instance:

`module M (A : Set) where record Prod (B : Set) : Set where constructor _,_ field fst : A snd : B open Prod public open M`

Now, the types of

`fst`

and`snd`

are`fst : {A : Set}{B : Set} → Prod A B → A snd : {A : Set}{B : Set} → Prod A B → B`

Until 2.3.2, they were

`fst : (A : Set){B : Set} → Prod A B → A snd : (A : Set){B : Set} → Prod A B → B`

This change is a step towards symmetry of constructors and projections. (Constructors always took the module parameters as hidden arguments).

Telescoping lets: Local bindings are now accepted in telescopes of modules, function types, and lambda-abstractions.

The syntax of telescopes as been extended to support

`let`

:`id : (let ★ = Set) (A : ★) → A → A id A x = x`

In particular one can now

`open`

modules inside telescopes:`module Star where ★ : Set₁ ★ = Set module MEndo (let open Star) (A : ★) where Endo : ★ Endo = A → A`

Finally a shortcut is provided for opening modules:

`module N (open Star) (A : ★) (open MEndo A) (f : Endo) where ...`

The semantics of the latter is

`module _ where open Star module _ (A : ★) where open MEndo A module N (f : Endo) where ...`

The semantics of telescoping lets in function types and lambda abstractions is just expanding them into ordinary lets.

More liberal left-hand sides in lets [Issue #1028]:

You can now write left-hand sides with arguments also for let bindings without a type signature. For instance,

`let f x = suc x in f zero`

Let bound functions still can't do pattern matching though.

Ambiguous names in patterns are now optimistically resolved in favor of constructors. [Issue #822] In particular, the following succeeds now:

`module M where data D : Set₁ where [_] : Set → D postulate [_] : Set → Set open M Foo : _ → Set Foo [ A ] = A`

Anonymous

`where`

-modules are opened public. [Issue #848]`<clauses> f args = rhs module _ telescope where body <more clauses>`

means the following (not proper Agda code, since you cannot put a module in-between clauses)

`<clauses> module _ {arg-telescope} telescope where body f args = rhs <more clauses>`

Example:

`A : Set1 A = B module _ where B : Set1 B = Set C : Set1 C = B`

Builtin

`ZERO`

and`SUC`

have been merged with`NATURAL`

.When binding the

`NATURAL`

builtin,`ZERO`

and`SUC`

are bound to the appropriate constructors automatically. This means that instead of writing`{-# BUILTIN NATURAL Nat #-} {-# BUILTIN ZERO zero #-} {-# BUILTIN SUC suc #-}`

you just write

`{-# BUILTIN NATURAL Nat #-}`

Pattern synonym can now have implicit arguments. [Issue #860]

For example,

`pattern tail=_ {x} xs = x ∷ xs len : ∀ {A} → List A → Nat len [] = 0 len (tail= xs) = 1 + len xs`

Syntax declarations can now have implicit arguments. [Issue #400]

For example

`id : ∀ {a}{A : Set a} -> A -> A id x = x syntax id {A} x = x ∈ A`

Minor syntax changes

`-}`

is now parsed as end-comment even if no comment was begun. As a consequence, the following definition gives a parse error`f : {A- : Set} -> Set f {A-} = A-`

because Agda now sees

`ID(f) LBRACE ID(A) END-COMMENT`

, and no longer`ID(f) LBRACE ID(A-) RBRACE`

.The rational is that the previous lexing was to context-sensitive, attempting to comment-out

`f`

using`{-`

and`-}`

lead to a parse error.Fixities (binding strengths) can now be negative numbers as well. [Issue #1109]

`infix -1 _myop_`

Postulates are now allowed in mutual blocks. [Issue #977]

Empty where blocks are now allowed. [Issue #947]

Pattern synonyms are now allowed in parameterised modules. [Issue #941]

Empty hiding and renaming lists in module directives are now allowed.

Module directives

`using`

,`hiding`

,`renaming`

and`public`

can now appear in arbitrary order. Multiple`using`

/`hiding`

/`renaming`

directives are allowed, but you still cannot have both using and`hiding`

(because that doesn't make sense). [Issue #493]

## Goal and error display

The error message

`Refuse to construct infinite term`

has been removed, instead one gets unsolved meta variables. Reason: the error was thrown over-eagerly. [Issue #795]If an interactive case split fails with message

`Since goal is solved, further case distinction is not supported; try `Solve constraints' instead`

then the associated interaction meta is assigned to a solution. Press

`C-c C-=`

(Show constraints) to view the solution and`C-c C-s`

(Solve constraints) to apply it. [Issue #289]

## Type checking

[ Issue #376 ] Implemented expansion of bound record variables during meta assignment. Now Agda can solve for metas X that are applied to projected variables, e.g.:

`X (fst z) (snd z) = z X (fst z) = fst z`

Technically, this is realized by substituting

`(x , y)`

for`z`

with fresh bound variables`x`

and`y`

. Here the full code for the examples:`record Sigma (A : Set)(B : A -> Set) : Set where constructor _,_ field fst : A snd : B fst open Sigma test : (A : Set) (B : A -> Set) -> let X : (x : A) (y : B x) -> Sigma A B X = _ in (z : Sigma A B) -> X (fst z) (snd z) ≡ z test A B z = refl test' : (A : Set) (B : A -> Set) -> let X : A -> A X = _ in (z : Sigma A B) -> X (fst z) ≡ fst z test' A B z = refl`

The fresh bound variables are named

`fst(z)`

and`snd(z)`

and can appear in error messages, e.g.:`fail : (A : Set) (B : A -> Set) -> let X : A -> Sigma A B X = _ in (z : Sigma A B) -> X (fst z) ≡ z fail A B z = refl`

results in error:

`Cannot instantiate the metavariable _7 to solution fst(z) , snd(z) since it contains the variable snd(z) which is not in scope of the metavariable or irrelevant in the metavariable but relevant in the solution when checking that the expression refl has type _7 A B (fst z) ≡ z`

Dependent record types and definitions by copatterns require reduction with previous function clauses while checking the current clause. [Issue #907]

For a simple example, consider

`test : ∀ {A} → Σ Nat λ n → Vec A n proj₁ test = zero proj₂ test = []`

For the second clause, the lhs and rhs are typed as

`proj₂ test : Vec A (proj₁ test) [] : Vec A zero`

In order for these types to match, we have to reduce the lhs type with the first function clause.

Note that termination checking comes after type checking, so be careful to avoid non-termination! Otherwise, the type checker might get into an infinite loop.

The implementation of the primitive

`primTrustMe`

has changed. It now only reduces to`REFL`

if the two arguments`x`

and`y`

have the same computational normal form. Before, it reduced when`x`

and`y`

were definitionally equal, which included type-directed equality laws such as eta-equality. Yet because reduction is untyped, calling conversion from reduction lead to Agda crashes [Issue #882].The amended description of

`primTrustMe`

is (cf. release notes for 2.2.6):`primTrustMe : {A : Set} {x y : A} → x ≡ y`

Here

`_≡_`

is the builtin equality (see BUILTIN hooks for equality, above).If

`x`

and`y`

have the same computational normal form, then`primTrustMe {x = x} {y = y}`

reduces to`refl`

.A note on

`primTrustMe`

's runtime behavior: The MAlonzo compiler replaces all uses of`primTrustMe`

with the`REFL`

builtin, without any check for definitional equality. Incorrect uses of`primTrustMe`

can potentially lead to segfaults or similar problems of the compiled code.Implicit patterns of record type are now only eta-expanded if there is a record constructor. [Issues #473, #635]

`data D : Set where d : D data P : D → Set where p : P d record Rc : Set where constructor c field f : D works : {r : Rc} → P (Rc.f r) → Set works p = D`

This works since the implicit pattern

`r`

is eta-expanded to`c x`

which allows the type of`p`

to reduce to`P x`

and`x`

to be unified with`d`

. The corresponding explicit version is:`works' : (r : Rc) → P (Rc.f r) → Set works' (c .d) p = D`

However, if the record constructor is removed, the same example will fail:

`record R : Set where field f : D fails : {r : R} → P (R.f r) → Set fails p = D -- d != R.f r of type D -- when checking that the pattern p has type P (R.f r)`

The error is justified since there is no pattern we could write down for

`r`

. It would have to look like`record { f = .d }`

but anonymous record patterns are not part of the language.

Absurd lambdas at different source locations are no longer different. [Issue #857] In particular, the following code type-checks now:

`absurd-equality : _≡_ {A = ⊥ → ⊥} (λ()) λ() absurd-equality = refl`

Which is a good thing!

Printing of named implicit function types.

When printing terms in a context with bound variables Agda renames new bindings to avoid clashes with the previously bound names. For instance, if

`A`

is in scope, the type`(A : Set) → A`

is printed as`(A₁ : Set) → A₁`

. However, for implicit function types the name of the binding matters, since it can be used when giving implicit arguments.For this situation, the following new syntax has been introduced:

`{x = y : A} → B`

is an implicit function type whose bound variable (in scope in`B`

) is`y`

, but where the name of the argument is`x`

for the purposes of giving it explicitly. For instance, with`A`

in scope, the type`{A : Set} → A`

is now printed as`{A = A₁ : Set} → A₁`

.This syntax is only used when printing and is currently not being parsed.

Changed the semantics of

`--without-K`

. [Issue #712, Issue #865, Issue #1025]New specification of

`--without-K`

:When

`--without-K`

is enabled, the unification of indices for pattern matching is restricted in two ways:Reflexive equations of the form

`x == x`

are no longer solved, instead Agda gives an error when such an equation is encountered.When unifying two same-headed constructor forms

`c us`

and`c vs`

of type`D pars ixs`

, the datatype indices`ixs`

(but not the parameters) have to be*self-unifiable*, i.e. unification of`ixs`

with itself should succeed positively. This is a nontrivial requirement because of point 1.

Examples:

The J rule is accepted.

```agda J : {A : Set} (P : {x y : A} → x ≡ y → Set) → (∀ x → P (refl x)) → ∀ {x y} (x≡y : x ≡ y) → P x≡y J P p (refl x) = p x

```agda

This definition is accepted since unification of

`x`

with`y`

doesn't require deletion or injectivity.The K rule is rejected.

`K : {A : Set} (P : {x : A} → x ≡ x → Set) → (∀ x → P (refl {x = x})) → ∀ {x} (x≡x : x ≡ x) → P x≡x K P p refl = p _`

Definition is rejected with the following error:

`Cannot eliminate reflexive equation x = x of type A because K has been disabled. when checking that the pattern refl has type x ≡ x`

Symmetry of the new criterion.

`test₁ : {k l m : ℕ} → k + l ≡ m → ℕ test₁ refl = zero test₂ : {k l m : ℕ} → k ≡ l + m → ℕ test₂ refl = zero`

Both versions are now accepted (previously only the first one was).

Handling of parameters.

`cons-injective : {A : Set} (x y : A) → (x ∷ []) ≡ (y ∷ []) → x ≡ y cons-injective x .x refl = refl`

Parameters are not unified, so they are ignored by the new criterion.

A larger example: antisymmetry of ≤.

`data _≤_ : ℕ → ℕ → Set where lz : (n : ℕ) → zero ≤ n ls : (m n : ℕ) → m ≤ n → suc m ≤ suc n ≤-antisym : (m n : ℕ) → m ≤ n → n ≤ m → m ≡ n ≤-antisym .zero .zero (lz .zero) (lz .zero) = refl ≤-antisym .(suc m) .(suc n) (ls m n p) (ls .n .m q) = cong suc (≤-antisym m n p q)`

[ Issue #1025 ]

`postulate mySpace : Set postulate myPoint : mySpace data Foo : myPoint ≡ myPoint → Set where foo : Foo refl test : (i : foo ≡ foo) → i ≡ refl test refl = {!!}`

When applying injectivity to the equation

`foo ≡ foo`

of type`Foo refl`

, it is checked that the index`refl`

of type`myPoint ≡ myPoint`

is self-unifiable. The equation`refl ≡ refl`

again requires injectivity, so now the index`myPoint`

is checked for self-unifiability, hence the error:`Cannot eliminate reflexive equation myPoint = myPoint of type mySpace because K has been disabled. when checking that the pattern refl has type foo ≡ foo`

## Termination checking

A buggy facility coined "matrix-shaped orders" that supported uncurried functions (which take tuples of arguments instead of one argument after another) has been removed from the termination checker. [Issue #787]

Definitions which fail the termination checker are not unfolded any longer to avoid loops or stack overflows in Agda. However, the termination checker for a mutual block is only invoked after type-checking, so there can still be loops if you define a non-terminating function. But termination checking now happens before the other supplementary checks: positivity, polarity, injectivity and projection-likeness. Note that with the pragma

`{-# NO_TERMINATION_CHECK #-}`

you can make Agda treat any function as terminating.Termination checking of functions defined by

`with`

has been improved.Cases which previously required

`--termination-depth`

to pass the termination checker (due to use of`with`

) no longer need the flag. For example`merge : List A → List A → List A merge [] ys = ys merge xs [] = xs merge (x ∷ xs) (y ∷ ys) with x ≤ y merge (x ∷ xs) (y ∷ ys) | false = y ∷ merge (x ∷ xs) ys merge (x ∷ xs) (y ∷ ys) | true = x ∷ merge xs (y ∷ ys)`

This failed to termination check previously, since the

`with`

expands to an auxiliary function`merge-aux`

:`merge-aux x y xs ys false = y ∷ merge (x ∷ xs) ys merge-aux x y xs ys true = x ∷ merge xs (y ∷ ys)`

This function makes a call to

`merge`

in which the size of one of the arguments is increasing. To make this pass the termination checker now inlines the definition of`merge-aux`

before checking, thus effectively termination checking the original source program.As a result of this transformation doing

`with`

on a variable no longer preserves termination. For instance, this does not termination check:`bad : Nat → Nat bad n with n ... | zero = zero ... | suc m = bad m`

The performance of the termination checker has been improved. For higher

`--termination-depth`

the improvement is significant. While the default`--termination-depth`

is still 1, checking with higher`--termination-depth`

should now be feasible.

## Compiler backends

The MAlonzo compiler backend now has support for compiling modules that are not full programs (i.e. don't have a main function). The goal is that you can write part of a program in Agda and the rest in Haskell, and invoke the Agda functions from the Haskell code. The following features were added for this reason:

A new command-line option

`--compile-no-main`

: the command`agda --compile-no-main Test.agda`

will compile

`Test.agda`

and all its dependencies to Haskell and compile the resulting Haskell files with`--make`

, but (unlike`--compile`

) not tell GHC to treat`Test.hs`

as the main module. This type of compilation can be invoked from Emacs by customizing the`agda2-backend`

variable to value`MAlonzoNoMain`

and then calling`C-c C-x C-c`

as before.A new pragma

`COMPILED_EXPORT`

was added as part of the MAlonzo FFI. If we have an Agda file containing the following:`module A.B where test : SomeType test = someImplementation {-# COMPILED_EXPORT test someHaskellId #-}`

then test will be compiled to a Haskell function called

`someHaskellId`

in module`MAlonzo.Code.A.B`

that can be invoked from other Haskell code. Its type will be translated according to the normal MAlonzo rules.

## Tools

### Emacs mode

A new goal command

`Helper Function Type`

(`C-c C-h`

) has been added.If you write an application of an undefined function in a goal, the

`Helper Function Type`

command will print the type that the function needs to have in order for it to fit the goal. The type is also added to the Emacs kill-ring and can be pasted into the buffer using`C-y`

.The application must be of the form

`f args`

where`f`

is the name of the helper function you want to create. The arguments can use all the normal features like named implicits or instance arguments.Example:

Here's a start on a naive reverse on vectors:

`reverse : ∀ {A n} → Vec A n → Vec A n reverse [] = [] reverse (x ∷ xs) = {!snoc (reverse xs) x!}`

Calling

`C-c C-h`

in the goal prints`snoc : ∀ {A} {n} → Vec A n → A → Vec A (suc n)`

A new command

`Explain why a particular name is in scope`

(`C-c C-w`

) has been added. [Issue #207]This command can be called from a goal or from the top-level and will as the name suggests explain why a particular name is in scope.

For each definition or module that the given name can refer to a trace is printed of all open statements and module applications leading back to the original definition of the name.

For example, given

`module A (X : Set₁) where data Foo : Set where mkFoo : Foo module B (Y : Set₁) where open A Y public module C = B Set open C`

Calling

`C-c C-w`

on`mkFoo`

at the top-level prints`mkFoo is in scope as * a constructor Issue207.C._.Foo.mkFoo brought into scope by - the opening of C at Issue207.agda:13,6-7 - the application of B at Issue207.agda:11,12-13 - the application of A at Issue207.agda:9,8-9 - its definition at Issue207.agda:6,5-10`

This command is useful if Agda complains about an ambiguous name and you need to figure out how to hide the undesired interpretations.

Improvements to the

`make case`

command (`C-c C-c`

)One can now also split on hidden variables, using the name (starting with

`.`

) with which they are printed. Use`C-c C-`

, to see all variables in context.Concerning the printing of generated clauses:

Uses named implicit arguments to improve readability.

Picks explicit occurrences over implicit ones when there is a choice of binding site for a variable.

Avoids binding variables in implicit positions by replacing dot patterns that uses them by wildcards (

`._`

).

Key bindings for lots of "mathematical" characters (examples: 𝐴𝑨𝒜𝓐𝔄) have been added to the Agda input method. Example: type

`\MiA\MIA\McA\MCA\MfA`

to get 𝐴𝑨𝒜𝓐𝔄.Note:

`\McB`

does not exist in Unicode (as well as others in that style), but the`\MC`

(bold) alphabet is complete.Key bindings for "blackboard bold" B (𝔹) and 0-9 (𝟘-𝟡) have been added to the Agda input method (

`\bb`

and`\b[0-9]`

).Key bindings for controlling simplification/normalisation:

[TODO: Simplification should be explained somewhere.]

Commands like

`Goal type and context`

(`C-c C-,`

) could previously be invoked in two ways. By default the output was normalised, but if a prefix argument was used (for instance via`C-u C-c C-,`

), then no explicit normalisation was performed. Now there are three options:By default (

`C-c C-,`

) the output is simplified.If

`C-u`

is used exactly once (`C-u C-c C-,`

), then the result is neither (explicitly) normalised nor simplified.If

`C-u`

is used twice (`C-u C-u C-c C-,`

), then the result is normalised.

[TODO: As part of the release of Agda 2.3.4 the key binding page on the wiki should be updated.]

### LaTeX-backend

Two new color scheme options were added to

`agda.sty`

:`\usepackage[bw]{agda}`

, which highlights in black and white;`\usepackage[conor]{agda}`

, which highlights using Conor's colors.The default (no options passed) is to use the standard colors.

If

`agda.sty`

cannot be found by the LateX environment, it is now copied into the LateX output directory (`latex`

by default) instead of the working directory. This means that the commands needed to produce a PDF now is`agda --latex -i . <file>.lagda cd latex pdflatex <file>.tex`

The LaTeX-backend has been made more tool agnostic, in particular XeLaTeX and LuaLaTeX should now work. Here is a small example (

`test/LaTeXAndHTML/succeed/UnicodeInput.lagda`

):`\documentclass{article} \usepackage{agda} \begin{document} \begin{code} data αβγδεζθικλμνξρστυφχψω : Set₁ where postulate →⇒⇛⇉⇄↦⇨↠⇀⇁ : Set \end{code} \[ ∀X [ ∅ ∉ X ⇒ ∃f:X ⟶ ⋃ X\ ∀A ∈ X (f(A) ∈ A) ] \] \end{document}`

Compiled as follows, it should produce a nice looking PDF (tested with TeX Live 2012):

`agda --latex <file>.lagda cd latex xelatex <file>.tex (or lualatex <file>.tex)`

If symbols are missing or XeLaTeX/LuaLaTeX complains about the font missing, try setting a different font using:

`\setmathfont{<math-font>}`

Use the

`fc-list`

tool to list available fonts.Add experimental support for hyperlinks to identifiers

If the

`hyperref`

LateX package is loaded before the Agda package and the links option is passed to the Agda package, then the Agda package provides a function called`\AgdaTarget`

. Identifiers which have been declared targets, by the user, will become clickable hyperlinks in the rest of the document. Here is a small example (`test/LaTeXAndHTML/succeed/Links.lagda`

):`\documentclass{article} \usepackage{hyperref} \usepackage[links]{agda} \begin{document} \AgdaTarget{ℕ} \AgdaTarget{zero} \begin{code} data ℕ : Set where zero : ℕ suc : ℕ → ℕ \end{code} See next page for how to define \AgdaFunction{two} (doesn't turn into a link because the target hasn't been defined yet). We could do it manually though; \hyperlink{two}{\AgdaDatatype{two}}. \newpage \AgdaTarget{two} \hypertarget{two}{} \begin{code} two : ℕ two = suc (suc zero) \end{code} \AgdaInductiveConstructor{zero} is of type \AgdaDatatype{ℕ}. \AgdaInductiveConstructor{suc} has not been defined to be a target so it doesn't turn into a link. \newpage Now that the target for \AgdaFunction{two} has been defined the link works automatically. \begin{code} data Bool : Set where true false : Bool \end{code} The AgdaTarget command takes a list as input, enabling several targets to be specified as follows: \AgdaTarget{if, then, else, if\_then\_else\_} \begin{code} if_then_else_ : {A : Set} → Bool → A → A → A if true then t else f = t if false then t else f = f \end{code} \newpage Mixfix identifier need their underscores escaped: \AgdaFunction{if\_then\_else\_}. \end{document}`

The boarders around the links can be suppressed using hyperref's hidelinks option:

`\usepackage[hidelinks]{hyperref}`

Note that the current approach to links does not keep track of scoping or types, and hence overloaded names might create links which point to the wrong place. Therefore it is recommended to not overload names when using the links option at the moment, this might get fixed in the future.

# Release notes for Agda 2 version 2.3.2.2

Fixed a bug that sometimes made it tricky to use the Emacs mode on Windows [Issue #757].

Made Agda build with newer versions of some libraries.

Fixed a bug that caused ambiguous parse error messages [Issue #147].

# Release notes for Agda 2 version 2.3.2.1

## Installation

Made it possible to compile Agda with more recent versions of hashable, QuickCheck and Win32.

Excluded mtl-2.1.

## Type checking

Fixed bug in the termination checker (Issue #754).

# Release notes for Agda 2 version 2.3.2

## Installation

The Agda-executable package has been removed.

The executable is now provided as part of the Agda package.

The Emacs mode no longer depends on haskell-mode or GHCi.

Compilation of Emacs mode Lisp files.

You can now compile the Emacs mode Lisp files by running

`agda-mode compile`

. This command is run by`make install`

.Compilation can, in some cases, give a noticeable speedup.

WARNING: If you reinstall the Agda mode without recompiling the Emacs Lisp files, then Emacs may continue using the old, compiled files.

## Pragmas and options

The

`--without-K`

check now reconstructs constructor parameters.New specification of

`--without-K`

:If the flag is activated, then Agda only accepts certain case-splits. If the type of the variable to be split is

`D pars ixs`

, where`D`

is a data (or record) type,`pars`

stands for the parameters, and`ixs`

the indices, then the following requirements must be satisfied:The indices

`ixs`

must be applications of constructors (or literals) to distinct variables. Constructors are usually not applied to parameters, but for the purposes of this check constructor parameters are treated as other arguments.These distinct variables must not be free in pars.

Irrelevant arguments are printed as

`_`

by default now. To turn on printing of irrelevant arguments, use option`--show-irrelevant`

New: Pragma

`NO_TERMINATION_CHECK`

to switch off termination checker for individual function definitions and mutual blocks.The pragma must precede a function definition or a mutual block. Examples (see

`test/Succeed/NoTerminationCheck.agda`

):Skipping a single definition: before type signature.

`{-# NO_TERMINATION_CHECK #-} a : A a = a`

Skipping a single definition: before first clause.

`b : A {-# NO_TERMINATION_CHECK #-} b = b`

Skipping an old-style mutual block: Before

`mutual`

keyword.`{-# NO_TERMINATION_CHECK #-} mutual c : A c = d d : A d = c`

Skipping a new-style mutual block: Anywhere before a type signature or first function clause in the block

`i : A j : A i = j {-# NO_TERMINATION_CHECK #-} j = i`

The pragma cannot be used in

`--safe`

mode.

## Language

Let binding record patterns

`record _×_ (A B : Set) : Set where constructor _,_ field fst : A snd : B open _×_ let (x , (y , z)) = t in u`

will now be interpreted as

`let x = fst t y = fst (snd t) z = snd (snd t) in u`

Note that the type of

`t`

needs to be inferable. If you need to provide a type signature, you can write the following:`let a : ... a = t (x , (y , z)) = a in u`

Pattern synonyms

A pattern synonym is a declaration that can be used on the left hand side (when pattern matching) as well as the right hand side (in expressions). For example:

`pattern z = zero pattern ss x = suc (suc x) f : ℕ -> ℕ f z = z f (suc z) = ss z f (ss n) = n`

Pattern synonyms are implemented by substitution on the abstract syntax, so definitions are scope-checked but not type-checked. They are particularly useful for universe constructions.

Qualified mixfix operators

It is now possible to use a qualified mixfix operator by qualifying the first part of the name. For instance

`import Data.Nat as Nat import Data.Bool as Bool two = Bool.if true then 1 Nat.+ 1 else 0`

Sections [Issue #735]. Agda now parses anonymous modules as sections:

`module _ {a} (A : Set a) where data List : Set a where [] : List _∷_ : (x : A) (xs : List) → List module _ {a} {A : Set a} where _++_ : List A → List A → List A [] ++ ys = ys (x ∷ xs) ++ ys = x ∷ (xs ++ ys) test : List Nat test = (5 ∷ []) ++ (3 ∷ [])`

In general, now the syntax

`module _ parameters where declarations`

is accepted and has the same effect as

`private module M parameters where declarations open M public`

for a fresh name

`M`

.Instantiating a module in an open import statement [Issue #481]. Now accepted:

`open import Path.Module args [using/hiding/renaming (...)]`

This only brings the imported identifiers from

`Path.Module`

into scope, not the module itself! Consequently, the following is pointless, and raises an error:`import Path.Module args [using/hiding/renaming (...)]`

You can give a private name

`M`

to the instantiated module via`import Path.Module args as M [using/hiding/renaming (...)] open import Path.Module args as M [using/hiding/renaming (...)]`

Try to avoid

`as`

as part of the arguments.`as`

is not a keyword; the following can be legal, although slightly obfuscated Agda code:`open import as as as as as as`

Implicit module parameters can be given by name. E.g.

`open M {namedArg = bla}`

This feature has been introduced in Agda 2.3.0 already.

Multiple type signatures sharing a same type can now be written as a single type signature.

`one two : ℕ one = suc zero two = suc one`

## Goal and error display

Meta-variables that were introduced by hidden argument

`arg`

are now printed as`_arg_number`

instead of just`_number`

. [Issue #526]Agda expands identifiers in anonymous modules when printing. Should make some goals nicer to read. [Issue #721]

When a module identifier is ambiguous, Agda tells you if one of them is a data type module. [Issues #318, #705]

## Type checking

Improved coverage checker. The coverage checker splits on arguments that have constructor or literal pattern, committing to the left-most split that makes progress. Consider the lookup function for vectors:

`data Fin : Nat → Set where zero : {n : Nat} → Fin (suc n) suc : {n : Nat} → Fin n → Fin (suc n) data Vec (A : Set) : Nat → Set where [] : Vec A zero _∷_ : {n : Nat} → A → Vec A n → Vec A (suc n) _!!_ : {A : Set}{n : Nat} → Vec A n → Fin n → A (x ∷ xs) !! zero = x (x ∷ xs) !! suc i = xs !! i`

In Agda up to 2.3.0, this definition is rejected unless we add an absurd clause

`[] !! ()`

This is because the coverage checker committed on splitting on the vector argument, even though this inevitably lead to failed coverage, because a case for the empty vector

`[]`

is missing.The improvement to the coverage checker consists on committing only on splits that have a chance of covering, since all possible constructor patterns are present. Thus, Agda will now split first on the

`Fin`

argument, since cases for both`zero`

and`suc`

are present. Then, it can split on the`Vec`

argument, since the empty vector is already ruled out by instantiating`n`

to a`suc _`

.Instance arguments resolution will now consider candidates which still expect hidden arguments. For example:

`record Eq (A : Set) : Set where field eq : A → A → Bool open Eq {{...}} eqFin : {n : ℕ} → Eq (Fin n) eqFin = record { eq = primEqFin } testFin : Bool testFin = eq fin1 fin2`

The type-checker will now resolve the instance argument of the

`eq`

function to`eqFin {_}`

. This is only done for hidden arguments, not instance arguments, so that the instance search stays non-recursive.Constraint solving: Upgraded Miller patterns to record patterns. [Issue #456]

Agda now solves meta-variables that are applied to record patterns. A typical (but here, artificial) case is:

`record Sigma (A : Set)(B : A -> Set) : Set where constructor _,_ field fst : A snd : B fst test : (A : Set)(B : A -> Set) -> let X : Sigma A B -> Sigma A B X = _ in (x : A)(y : B x) -> X (x , y) ≡ (x , y) test A B x y = refl`

This yields a constraint of the form

`_X A B (x , y) := t[x,y]`

(with

`t[x,y] = (x, y)`

) which is not a Miller pattern. However, Agda now solves this as`_X A B z := t[fst z,snd z].`

Changed: solving recursive constraints. [Issue #585]

Until 2.3.0, Agda sometimes inferred values that did not pass the termination checker later, or would even make Agda loop. To prevent this, the occurs check now also looks into the definitions of the current mutual block, to avoid constructing recursive solutions. As a consequence, also terminating recursive solutions are no longer found automatically.

This effects a programming pattern where the recursively computed type of a recursive function is left to Agda to solve.

`mutual T : D -> Set T pattern1 = _ T pattern2 = _ f : (d : D) -> T d f pattern1 = rhs1 f pattern2 = rhs2`

This might no longer work from now on. See examples

`test/Fail/Issue585*.agda`

.Less eager introduction of implicit parameters. [Issue #679]

Until Agda 2.3.0, trailing hidden parameters were introduced eagerly on the left hand side of a definition. For instance, one could not write

`test : {A : Set} -> Set test = \ {A} -> A`

because internally, the hidden argument

`{A : Set}`

was added to the left-hand side, yielding`test {_} = \ {A} -> A`

which raised a type error. Now, Agda only introduces the trailing implicit parameters it has to, in order to maintain uniform function arity. For instance, in

`test : Bool -> {A B C : Set} -> Set test true {A} = A test false {B = B} = B`

Agda will introduce parameters

`A`

and`B`

in all clauses, but not`C`

, resulting in`test : Bool -> {A B C : Set} -> Set test true {A} {_} = A test false {_} {B = B} = B`

Note that for checking

`where`

-clauses, still all hidden trailing parameters are in scope. For instance:`id : {i : Level}{A : Set i} -> A -> A id = myId where myId : forall {A} -> A -> A myId x = x`

To be able to fill in the meta variable

`_1`

in`myId : {A : Set _1} -> A -> A`

the hidden parameter

`{i : Level}`

needs to be in scope.As a result of this more lazy introduction of implicit parameters, the following code now passes.

`data Unit : Set where unit : Unit T : Unit → Set T unit = {u : Unit} → Unit test : (u : Unit) → T u test unit with unit ... | _ = λ {v} → v`

Before, Agda would eagerly introduce the hidden parameter

`{v}`

as unnamed left-hand side parameter, leaving no way to refer to it.The related Issue #655 has also been addressed. It is now possible to make `synonym' definitions

`name = expression`

even when the type of expression begins with a hidden quantifier. Simple example:

`id2 = id`

That resulted in unsolved metas until 2.3.0.

Agda detects unused arguments and ignores them during equality checking. [Issue #691, solves also Issue #44]

Agda's polarity checker now assigns 'Nonvariant' to arguments that are not actually used (except for absurd matches). If

`f`

's first argument is Nonvariant, then`f x`

is definitionally equal to`f y`

regardless of`x`

and`y`

. It is similar to irrelevance, but does not require user annotation.For instance, unused module parameters do no longer get in the way:

`module M (x : Bool) where not : Bool → Bool not true = false not false = true open M true open M false renaming (not to not′) test : (y : Bool) → not y ≡ not′ y test y = refl`

Matching against record or absurd patterns does not count as `use', so we get some form of proof irrelevance:

`data ⊥ : Set where record ⊤ : Set where constructor trivial data Bool : Set where true false : Bool True : Bool → Set True true = ⊤ True false = ⊥ fun : (b : Bool) → True b → Bool fun true trivial = true fun false () test : (b : Bool) → (x y : True b) → fun b x ≡ fun b y test b x y = refl`

More examples in

`test/Succeed/NonvariantPolarity.agda`

.Phantom arguments: Parameters of record and data types are considered `used' even if they are not actually used. Consider:

`False : Nat → Set False zero = ⊥ False (suc n) = False n module Invariant where record Bla (n : Nat)(p : False n) : Set where module Nonvariant where Bla : (n : Nat) → False n → Set Bla n p = ⊤`

Even though record

`Bla`

does not use its parameters`n`

and`p`

, they are considered as used, allowing "phantom type" techniques.In contrast, the arguments of function

`Bla`

are recognized as unused. The following code type-checks if we open`Invariant`

but leaves unsolved metas if we open`Nonvariant`

.`drop-suc : {n : Nat}{p : False n} → Bla (suc n) p → Bla n p drop-suc _ = _ bla : (n : Nat) → {p : False n} → Bla n p → ⊥ bla zero {()} b bla (suc n) b = bla n (drop-suc b)`

If

`Bla`

is considered invariant, the hidden argument in the recursive call can be inferred to be`p`

. If it is considered non-variant, then`Bla n X = Bla n p`

does not entail`X = p`

and the hidden argument remains unsolved. Since`bla`

does not actually use its hidden argument, its value is not important and it could be searched for. Unfortunately, polarity analysis of`bla`

happens only after type checking, thus, the information that`bla`

is non-variant in`p`

is not available yet when meta-variables are solved. (See`test/Fail/BrokenInferenceDueToNonvariantPolarity.agda`

)Agda now expands simple definitions (one clause, terminating) to check whether a function is constructor headed. [Issue #747] For instance, the following now also works:

`MyPair : Set -> Set -> Set MyPair A B = Pair A B Vec : Set -> Nat -> Set Vec A zero = Unit Vec A (suc n) = MyPair A (Vec A n)`

Here,

`Unit`

and`Pair`

are data or record types.

## Compiler backends

`-Werror`

is now overridable.To enable compilation of Haskell modules containing warnings, the

`-Werror`

flag for the MAlonzo backend has been made overridable. If, for example,`--ghc-flag=-Wwarn`

is passed when compiling, one can get away with things like:`data PartialBool : Set where true : PartialBool {-# COMPILED_DATA PartialBool Bool True #-}`

The default behavior remains as it used to be and rejects the above program.

## Tools

### Emacs mode

Asynchronous Emacs mode.

One can now use Emacs while a buffer is type-checked. If the buffer is edited while the type-checker runs, then syntax highlighting will not be updated when type-checking is complete.

Interactive syntax highlighting.

The syntax highlighting is updated while a buffer is type-checked:

At first the buffer is highlighted in a somewhat crude way (without go-to-definition information for overloaded constructors).

If the highlighting level is "interactive", then the piece of code that is currently being type-checked is highlighted as such. (The default is "non-interactive".)

When a mutual block has been type-checked it is highlighted properly (this highlighting includes warnings for potential non-termination).

The highlighting level can be controlled via the new configuration variable

`agda2-highlight-level`

.Multiple case-splits can now be performed in one go.

Consider the following example:

`_==_ : Bool → Bool → Bool b₁ == b₂ = {!!}`

If you split on

`b₁ b₂`

, then you get the following code:`_==_ : Bool → Bool → Bool true == true = {!!} true == false = {!!} false == true = {!!} false == false = {!!}`

The order of the variables matters. Consider the following code:

`lookup : ∀ {a n} {A : Set a} → Vec A n → Fin n → A lookup xs i = {!!}`

If you split on

`xs i`

, then you get the following code:`lookup : ∀ {a n} {A : Set a} → Vec A n → Fin n → A lookup [] () lookup (x ∷ xs) zero = {!!} lookup (x ∷ xs) (suc i) = {!!}`

However, if you split on

`i xs`

, then you get the following code instead:`lookup : ∀ {a n} {A : Set a} → Vec A n → Fin n → A lookup (x ∷ xs) zero = ? lookup (x ∷ xs) (suc i) = ?`

This code is rejected by Agda 2.3.0, but accepted by 2.3.2 thanks to improved coverage checking (see above).

The Emacs mode now presents information about which module is currently being type-checked.

New global menu entry:

`Information about the character at point`

.If this entry is selected, then information about the character at point is displayed, including (in many cases) information about how to type the character.

Commenting/uncommenting the rest of the buffer.

One can now comment or uncomment the rest of the buffer by typing

`C-c C-x M-;`

or by selecting the menu entry`Comment/uncomment`

the rest of the buffer".The Emacs mode now uses the Agda executable instead of GHCi.

The

`*ghci*`

buffer has been renamed to`*agda2*`

.A new configuration variable has been introduced:

`agda2-program-name`

, the name of the Agda executable (by default`agda`

).The variable

`agda2-ghci-options`

has been replaced by`agda2-program-args`

: extra arguments given to the Agda executable (by default`none`

).If you want to limit Agda's memory consumption you can add some arguments to

`agda2-program-args`

, for instance`+RTS -M1.5G -RTS`

.The Emacs mode no longer depends on haskell-mode.

Users who have customised certain haskell-mode variables (such as

`haskell-ghci-program-args`

) may want to update their configuration.

### LaTeX-backend

An experimental LaTeX-backend which does precise highlighting a la the HTML-backend and code alignment a la lhs2TeX has been added.

Here is a sample input literate Agda file:

```latex \documentclass{article}

\usepackage{agda}

\begin{document}

The following module declaration will be hidden in the output.

\AgdaHide{ \begin{code} module M where \end{code} }

Two or more spaces can be used to make the backend align stuff.

\begin{code} data ℕ : Set where zero : ℕ suc : ℕ → ℕ

*+* : ℕ → ℕ → ℕ
zero + n = n
suc m + n = suc (m + n)
\end{code}

\end{document} ```

To produce an output PDF issue the following commands:

```
`
agda --latex -i . <file>.lagda
pdflatex latex/<file>.tex
```

`

Only the top-most module is processed, like with lhs2tex and unlike
with the HTML-backend. If you want to process imported modules you
have to call `agda --latex`

manually on each of those modules.

There are still issues related to formatting, see the bug tracker for more information:

https://code.google.com/p/agda/issues/detail?id=697

The default `agda.sty`

might therefore change in backwards-incompatible
ways, as work proceeds in trying to resolve those problems.

Implemented features:

Two or more spaces can be used to force alignment of things, like with lhs2tex. See example above.

The highlighting information produced by the type checker is used to generate the output. For example, the data declaration in the example above, produces:

`\AgdaKeyword{data} \AgdaDatatype{ℕ} \AgdaSymbol{:} \AgdaPrimitiveType{Set} \AgdaKeyword{where}`

These LaTeX commands are defined in

`agda.sty`

(which is imported by`\usepackage{agda}`

) and cause the highlighting.The LaTeX-backend checks if

`agda.sty`

is found by the LaTeX environment, if it isn't a default`agda.sty`

is copied from Agda's`data-dir`

into the working directory (and thus made available to the LaTeX environment).If the default

`agda.sty`

isn't satisfactory (colors, fonts, spacing, etc) then the user can modify it and make put it somewhere where the LaTeX environment can find it. Hopefully most aspects should be modifiable via`agda.sty`

rather than having to tweak the implementation.`--latex-dir`

can be used to change the default output directory.

# Release notes for Agda 2 version 2.3.0

## Language

New more liberal syntax for mutually recursive definitions.

It is no longer necessary to use the

`mutual`

keyword to define mutually recursive functions or datatypes. Instead, it is enough to declare things before they are used. Instead of`mutual f : A f = a[f, g] g : B[f] g = b[f, g]`

you can now write

`f : A g : B[f] f = a[f, g] g = b[f, g].`

With the new style you have more freedom in choosing the order in which things are type checked (previously type signatures were always checked before definitions). Furthermore you can mix arbitrary declarations, such as modules and postulates, with mutually recursive definitions.

For data types and records the following new syntax is used to separate the declaration from the definition:

`-- Declaration. data Vec (A : Set) : Nat → Set -- Note the absence of 'where'. -- Definition. data Vec A where [] : Vec A zero _::_ : {n : Nat} → A → Vec A n → Vec A (suc n) -- Declaration. record Sigma (A : Set) (B : A → Set) : Set -- Definition. record Sigma A B where constructor _,_ field fst : A snd : B fst`

When making separated declarations/definitions private or abstract you should attach the

`private`

keyword to the declaration and the`abstract`

keyword to the definition. For instance, a private, abstract function can be defined as`private f : A abstract f = e`

Finally it may be worth noting that the old style of mutually recursive definitions is still supported (it basically desugars into the new style).

Pattern matching lambdas.

Anonymous pattern matching functions can be defined using the syntax

`\ { p11 .. p1n -> e1 ; ... ; pm1 .. pmn -> em }`

(where, as usual,

`\`

and`->`

can be replaced by`λ`

and`→`

). Internally this is translated into a function definition of the following form:`.extlam p11 .. p1n = e1 ... .extlam pm1 .. pmn = em`

This means that anonymous pattern matching functions are generative. For instance,

`refl`

will not be accepted as an inhabitant of the type`(λ { true → true ; false → false }) ≡ (λ { true → true ; false → false }),`

because this is equivalent to

`extlam1 ≡ extlam2`

for some distinct fresh names`extlam1`

and`extlam2`

.Currently the

`where`

and`with`

constructions are not allowed in (the top-level clauses of) anonymous pattern matching functions.Examples:

`and : Bool → Bool → Bool and = λ { true x → x ; false _ → false } xor : Bool → Bool → Bool xor = λ { true true → false ; false false → false ; _ _ → true } fst : {A : Set} {B : A → Set} → Σ A B → A fst = λ { (a , b) → a } snd : {A : Set} {B : A → Set} (p : Σ A B) → B (fst p) snd = λ { (a , b) → b }`

Record update syntax.

Assume that we have a record type and a corresponding value:

`record MyRecord : Set where field a b c : ℕ old : MyRecord old = record { a = 1; b = 2; c = 3 }`

Then we can update (some of) the record value's fields in the following way:

`new : MyRecord new = record old { a = 0; c = 5 }`

Here new normalises to

`record { a = 0; b = 2; c = 5 }`

. Any expression yielding a value of type`MyRecord`

can be used instead of old.Record updating is not allowed to change types: the resulting value must have the same type as the original one, including the record parameters. Thus, the type of a record update can be inferred if the type of the original record can be inferred.

The record update syntax is expanded before type checking. When the expression

`record old { upd-fields }`

is checked against a record type

`R`

, it is expanded to`let r = old in record { new-fields },`

where old is required to have type

`R`

and new-fields is defined as follows: for each field`x`

in`R`

,if

`x = e`

is contained in`upd-fields`

then`x = e`

is included in`new-fields`

, and otherwiseif

`x`

is an explicit field then`x = R.x r`

is included in`new-fields`

, andif

`x`

is an implicit or instance field, then it is omitted from`new-fields`

.

(Instance arguments are explained below.) The reason for treating implicit and instance fields specially is to allow code like the following:

`record R : Set where field {length} : ℕ vec : Vec ℕ length -- More fields… xs : R xs = record { vec = 0 ∷ 1 ∷ 2 ∷ [] } ys = record xs { vec = 0 ∷ [] }`

Without the special treatment the last expression would need to include a new binding for length (for instance

`length = _`

).Record patterns which do not contain data type patterns, but which do contain dot patterns, are no longer rejected.

When the

`--without-K`

flag is used literals are now treated as constructors.Under-applied functions can now reduce.

Consider the following definition:

`id : {A : Set} → A → A id x = x`

Previously the expression

`id`

would not reduce. This has been changed so that it now reduces to`λ x → x`

. Usually this makes little difference, but it can be important in conjunction with`with`

. See Issue #365 for an example.Unused AgdaLight legacy syntax

`(x y : A; z v : B)`

for telescopes has been removed.

### Universe polymorphism

Universe polymorphism is now enabled by default. Use

`--no-universe-polymorphism`

to disable it.Universe levels are no longer defined as a data type.

The basic level combinators can be introduced in the following way:

`postulate Level : Set zero : Level suc : Level → Level max : Level → Level → Level {-# BUILTIN LEVEL Level #-} {-# BUILTIN LEVELZERO zero #-} {-# BUILTIN LEVELSUC suc #-} {-# BUILTIN LEVELMAX max #-}`

The BUILTIN equality is now required to be universe-polymorphic.

`trustMe`

is now universe-polymorphic.

### Meta-variables and unification

Unsolved meta-variables are now frozen after every mutual block. This means that they cannot be instantiated by subsequent code. For instance,

`one : Nat one = _ bla : one ≡ suc zero bla = refl`

leads to an error now, whereas previously it lead to the instantiation of

`_`

with`suc zero`

. If you want to make use of the old behaviour, put the two definitions in a mutual block.All meta-variables are unfrozen during interactive editing, so that the user can fill holes interactively. Note that type-checking of interactively given terms is not perfect: Agda sometimes refuses to load a file, even though no complaints were raised during the interactive construction of the file. This is because certain checks (for instance, positivity) are only invoked when a file is loaded.

Record types can now be inferred.

If there is a unique known record type with fields matching the fields in a record expression, then the type of the expression will be inferred to be the record type applied to unknown parameters.

If there is no known record type with the given fields the type checker will give an error instead of producing lots of unsolved meta-variables.

Note that "known record type" refers to any record type in any imported module, not just types which are in scope.

The occurrence checker distinguishes rigid and strongly rigid occurrences [Reed, LFMTP 2009; Abel & Pientka, TLCA 2011].

The completeness checker now accepts the following code:

`h : (n : Nat) → n ≡ suc n → Nat h n ()`

Internally this generates a constraint

`_n = suc _n`

where the meta-variable`_n`

occurs strongly rigidly, i.e. on a constructor path from the root, in its own defining term tree. This is never solvable.Weakly rigid recursive occurrences may have a solution [Jason Reed's PhD thesis, page 106]:

`test : (k : Nat) → let X : (Nat → Nat) → Nat X = _ in (f : Nat → Nat) → X f ≡ suc (f (X (λ x → k))) test k f = refl`

The constraint

`_X k f = suc (f (_X k (λ x → k)))`

has the solution`_X k f = suc (f (suc k))`

, despite the recursive occurrence of`_X`

. Here`_X`

is not strongly rigid, because it occurs under the bound variable`f`

. Previously Agda rejected this code; now it instead complains about an unsolved meta-variable.Equation constraints involving the same meta-variable in the head now trigger pruning [Pientka, PhD, Sec. 3.1.2; Abel & Pientka, TLCA 2011]. Example:

`same : let X : A → A → A → A × A X = _ in {x y z : A} → X x y y ≡ (x , y) × X x x y ≡ X x y y same = refl , refl`

The second equation implies that

`X`

cannot depend on its second argument. After pruning the first equation is linear and can be solved.Instance arguments.

A new type of hidden function arguments has been added: instance arguments. This new feature is based on influences from Scala's implicits and Agda's existing implicit arguments.

Plain implicit arguments are marked by single braces:

`{…}`

. Instance arguments are instead marked by double braces:`{{…}}`

. Example:`postulate A : Set B : A → Set a : A f : {{a : A}} → B a`

Instead of the double braces you can use the symbols

`⦃`

and`⦄`

, but these symbols must in many cases be surrounded by whitespace. (If you are using Emacs and the Agda input method, then you can conjure up the symbols by typing`\{{`

and`\}}`

, respectively.)Instance arguments behave as ordinary implicit arguments, except for one important aspect: resolution of arguments which are not provided explicitly. For instance, consider the following code:

`test = f`

Here Agda will notice that

`f`

's instance argument was not provided explicitly, and try to infer it. All definitions in scope at`f`

's call site, as well as all variables in the context, are considered. If exactly one of these names has the required type`A`

, then the instance argument will be instantiated to this name.This feature can be used as an alternative to Haskell type classes. If we define

`record Eq (A : Set) : Set where field equal : A → A → Bool,`

then we can define the following projection:

`equal : {A : Set} {{eq : Eq A}} → A → A → Bool equal {{eq}} = Eq.equal eq`

Now consider the following expression:

`equal false false ∨ equal 3 4`

If the following

`Eq`

"instances" for`Bool`

and`ℕ`

are in scope, and no others, then the expression is accepted:`eq-Bool : Eq Bool eq-Bool = record { equal = … } eq-ℕ : Eq ℕ eq-ℕ = record { equal = … }`

A shorthand notation is provided to avoid the need to define projection functions manually:

`module Eq-with-implicits = Eq {{...}}`

This notation creates a variant of

`Eq`

's record module, where the main`Eq`

argument is an instance argument instead of an explicit one. It is equivalent to the following definition:`module Eq-with-implicits {A : Set} {{eq : Eq A}} = Eq eq`

Note that the short-hand notation allows you to avoid naming the "-with-implicits" module:

`open Eq {{...}}`

Instance argument resolution is not recursive. As an example, consider the following "parametrised instance":

`eq-List : {A : Set} → Eq A → Eq (List A) eq-List {A} eq = record { equal = eq-List-A } where eq-List-A : List A → List A → Bool eq-List-A [] [] = true eq-List-A (a ∷ as) (b ∷ bs) = equal a b ∧ eq-List-A as bs eq-List-A _ _ = false`

Assume that the only

`Eq`

instances in scope are`eq-List`

and`eq-ℕ`

. Then the following code does not type-check:`test = equal (1 ∷ 2 ∷ []) (3 ∷ 4 ∷ [])`

However, we can make the code work by constructing a suitable instance manually:

`test′ = equal (1 ∷ 2 ∷ []) (3 ∷ 4 ∷ []) where eq-List-ℕ = eq-List eq-ℕ`

By restricting the "instance search" to be non-recursive we avoid introducing a new, compile-time-only evaluation model to Agda.

For more information about instance arguments, see Devriese & Piessens [ICFP 2011]. Some examples are also available in the examples/instance-arguments subdirectory of the Agda distribution.

### Irrelevance

Dependent irrelevant function types.

Some examples illustrating the syntax of dependent irrelevant function types:

`.(x y : A) → B .{x y z : A} → B ∀ x .y → B ∀ x .{y} {z} .v → B`

The declaration

`f : .(x : A) → B[x] f x = t[x]`

requires that

`x`

is irrelevant both in`t[x]`

and in`B[x]`

. This is possible if, for instance,`B[x] = B′ x`

, with`B′ : .A → Set`

.Dependent irrelevance allows us to define the eliminator for the

`Squash`

type:`record Squash (A : Set) : Set where constructor squash field .proof : A elim-Squash : {A : Set} (P : Squash A → Set) (ih : .(a : A) → P (squash a)) → (a⁻ : Squash A) → P a⁻ elim-Squash P ih (squash a) = ih a`

Note that this would not type-check with

`(ih : (a : A) -> P (squash a)).`

Records with only irrelevant fields.

The following now works:

`record IsEquivalence {A : Set} (_≈_ : A → A → Set) : Set where field .refl : Reflexive _≈_ .sym : Symmetric _≈_ .trans : Transitive _≈_ record Setoid : Set₁ where infix 4 _≈_ field Carrier : Set _≈_ : Carrier → Carrier → Set .isEquivalence : IsEquivalence _≈_ open IsEquivalence isEquivalence public`

Previously Agda complained about the application

`IsEquivalence isEquivalence`

, because`isEquivalence`

is irrelevant and the`IsEquivalence`

module expected a relevant argument. Now, when record modules are generated for records consisting solely of irrelevant arguments, the record parameter is made irrelevant:```agda module IsEquivalence {A : Set} {

*≈*: A → A → Set} .(r : IsEquivalence {A = A}*≈*) where …```

Irrelevant things are no longer erased internally. This means that they are printed as ordinary terms, not as

`_`

as before.The new flag

`--experimental-irrelevance`

enables irrelevant universe levels and matching on irrelevant data when only one constructor is available. These features are very experimental and likely to change or disappear.

### Reflection

The reflection API has been extended to mirror features like irrelevance, instance arguments and universe polymorphism, and to give (limited) access to definitions. For completeness all the builtins and primitives are listed below:

`-- Names. postulate Name : Set {-# BUILTIN QNAME Name #-} primitive -- Equality of names. primQNameEquality : Name → Name → Bool -- Is the argument visible (explicit), hidden (implicit), or an -- instance argument? data Visibility : Set where visible hidden instance : Visibility {-# BUILTIN HIDING Visibility #-} {-# BUILTIN VISIBLE visible #-} {-# BUILTIN HIDDEN hidden #-} {-# BUILTIN INSTANCE instance #-} -- Arguments can be relevant or irrelevant. data Relevance : Set where relevant irrelevant : Relevance {-# BUILTIN RELEVANCE Relevance #-} {-# BUILTIN RELEVANT relevant #-} {-# BUILTIN IRRELEVANT irrelevant #-} -- Arguments. data Arg A : Set where arg : (v : Visibility) (r : Relevance) (x : A) → Arg A {-# BUILTIN ARG Arg #-} {-# BUILTIN ARGARG arg #-} -- Terms. mutual data Term : Set where -- Variable applied to arguments. var : (x : ℕ) (args : List (Arg Term)) → Term -- Constructor applied to arguments. con : (c : Name) (args : List (Arg Term)) → Term -- Identifier applied to arguments. def : (f : Name) (args : List (Arg Term)) → Term -- Different kinds of λ-abstraction. lam : (v : Visibility) (t : Term) → Term -- Pi-type. pi : (t₁ : Arg Type) (t₂ : Type) → Term -- A sort. sort : Sort → Term -- Anything else. unknown : Term data Type : Set where el : (s : Sort) (t : Term) → Type data Sort : Set where -- A Set of a given (possibly neutral) level. set : (t : Term) → Sort -- A Set of a given concrete level. lit : (n : ℕ) → Sort -- Anything else. unknown : Sort {-# BUILTIN AGDASORT Sort #-} {-# BUILTIN AGDATYPE Type #-} {-# BUILTIN AGDATERM Term #-} {-# BUILTIN AGDATERMVAR var #-} {-# BUILTIN AGDATERMCON con #-} {-# BUILTIN AGDATERMDEF def #-} {-# BUILTIN AGDATERMLAM lam #-} {-# BUILTIN AGDATERMPI pi #-} {-# BUILTIN AGDATERMSORT sort #-} {-# BUILTIN AGDATERMUNSUPPORTED unknown #-} {-# BUILTIN AGDATYPEEL el #-} {-# BUILTIN AGDASORTSET set #-} {-# BUILTIN AGDASORTLIT lit #-} {-# BUILTIN AGDASORTUNSUPPORTED unknown #-} postulate -- Function definition. Function : Set -- Data type definition. Data-type : Set -- Record type definition. Record : Set {-# BUILTIN AGDAFUNDEF Function #-} {-# BUILTIN AGDADATADEF Data-type #-} {-# BUILTIN AGDARECORDDEF Record #-} -- Definitions. data Definition : Set where function : Function → Definition data-type : Data-type → Definition record′ : Record → Definition constructor′ : Definition axiom : Definition primitive′ : Definition {-# BUILTIN AGDADEFINITION Definition #-} {-# BUILTIN AGDADEFINITIONFUNDEF function #-} {-# BUILTIN AGDADEFINITIONDATADEF data-type #-} {-# BUILTIN AGDADEFINITIONRECORDDEF record′ #-} {-# BUILTIN AGDADEFINITIONDATACONSTRUCTOR constructor′ #-} {-# BUILTIN AGDADEFINITIONPOSTULATE axiom #-} {-# BUILTIN AGDADEFINITIONPRIMITIVE primitive′ #-} primitive -- The type of the thing with the given name. primQNameType : Name → Type -- The definition of the thing with the given name. primQNameDefinition : Name → Definition -- The constructors of the given data type. primDataConstructors : Data-type → List Name`

As an example the expression

`primQNameType (quote zero)`

is definitionally equal to

`el (lit 0) (def (quote ℕ) [])`

(if

`zero`

is a constructor of the data type`ℕ`

).New keyword:

`unquote`

.The construction

`unquote t`

converts a representation of an Agda term to actual Agda code in the following way:The argument

`t`

must have type`Term`

(see the reflection API above).The argument is normalised.

The entire construction is replaced by the normal form, which is treated as syntax written by the user and type-checked in the usual way.

Examples:

`test : unquote (def (quote ℕ) []) ≡ ℕ test = refl id : (A : Set) → A → A id = unquote (lam visible (lam visible (var 0 []))) id-ok : id ≡ (λ A (x : A) → x) id-ok = refl`

New keyword:

`quoteTerm`

.The construction

`quoteTerm t`

is similar to`quote n`

, but whereas`quote`

is restricted to names`n`

,`quoteTerm`

accepts terms`t`

. The construction is handled in the following way:The type of

`t`

is inferred. The term`t`

must be type-correct.The term

`t`

is normalised.The construction is replaced by the Term representation (see the reflection API above) of the normal form. Any unsolved metavariables in the term are represented by the

`unknown`

term constructor.

Examples:

`test₁ : quoteTerm (λ {A : Set} (x : A) → x) ≡ lam hidden (lam visible (var 0 [])) test₁ = refl -- Local variables are represented as de Bruijn indices. test₂ : (λ {A : Set} (x : A) → quoteTerm x) ≡ (λ x → var 0 []) test₂ = refl -- Terms are normalised before being quoted. test₃ : quoteTerm (0 + 0) ≡ con (quote zero) [] test₃ = refl`

## Compiler backends

### MAlonzo

The MAlonzo backend's FFI now handles universe polymorphism in a better way.

The translation of Agda types and kinds into Haskell now supports universe-polymorphic postulates. The core changes are that the translation of function types has been changed from

`T[[ Pi (x : A) B ]] = if A has a Haskell kind then forall x. () -> T[[ B ]] else if x in fv B then undef else T[[ A ]] -> T[[ B ]]`

into

`T[[ Pi (x : A) B ]] = if x in fv B then forall x. T[[ A ]] -> T[[ B ]] -- Note: T[[A]] not Unit. else T[[ A ]] -> T[[ B ]],`

and that the translation of constants (postulates, constructors and literals) has been changed from

`T[[ k As ]] = if COMPILED_TYPE k T then T T[[ As ]] else undef`

into

`T[[ k As ]] = if COMPILED_TYPE k T then T T[[ As ]] else if COMPILED k E then () else undef.`

For instance, assuming a Haskell definition

`type AgdaIO a b = IO b,`

we can set up universe-polymorphic

`IO`

in the following way:`postulate IO : ∀ {ℓ} → Set ℓ → Set ℓ return : ∀ {a} {A : Set a} → A → IO A _>>=_ : ∀ {a b} {A : Set a} {B : Set b} → IO A → (A → IO B) → IO B {-# COMPILED_TYPE IO AgdaIO #-} {-# COMPILED return (\_ _ -> return) #-} {-# COMPILED _>>=_ (\_ _ _ _ -> (>>=)) #-}`

This is accepted because (assuming that the universe level type is translated to the Haskell unit type

`()`

)`(\_ _ -> return) : forall a. () -> forall b. () -> b -> AgdaIO a b = T [[ ∀ {a} {A : Set a} → A → IO A ]]`

and

`(\_ _ _ _ -> (>>=)) : forall a. () -> forall b. () -> forall c. () -> forall d. () -> AgdaIO a c -> (c -> AgdaIO b d) -> AgdaIO b d = T [[ ∀ {a b} {A : Set a} {B : Set b} → IO A → (A → IO B) → IO B ]].`

### Epic

New Epic backend pragma:

`STATIC`

.In the Epic backend, functions marked with the

`STATIC`

pragma will be normalised before compilation. Example usage:`{-# STATIC power #-} power : ℕ → ℕ → ℕ power 0 x = 1 power 1 x = x power (suc n) x = power n x * x`

Occurrences of

`power 4 x`

will be replaced by`((x * x) * x) * x`

.Some new optimisations have been implemented in the Epic backend:

Removal of unused arguments.

A worker/wrapper transformation is performed so that unused arguments can be removed by Epic's inliner. For instance, the map function is transformed in the following way:

`map_wrap : (A B : Set) → (A → B) → List A → List B map_wrap A B f xs = map_work f xs map_work f [] = [] map_work f (x ∷ xs) = f x ∷ map_work f xs`

If

`map_wrap`

is inlined (which it will be in any saturated call), then`A`

and`B`

disappear in the generated code.Unused arguments are found using abstract interpretation. The bodies of all functions in a module are inspected to decide which variables are used. The behaviour of postulates is approximated based on their types. Consider

`return`

, for instance:`postulate return : {A : Set} → A → IO A`

The first argument of

`return`

can be removed, because it is of type Set and thus cannot affect the outcome of a program at runtime.Injection detection.

At runtime many functions may turn out to be inefficient variants of the identity function. This is especially true after forcing. Injection detection replaces some of these functions with more efficient versions. Example:

`inject : {n : ℕ} → Fin n → Fin (1 + n) inject {suc n} zero = zero inject {suc n} (suc i) = suc (inject {n} i)`

Forcing removes the

`Fin`

constructors'`ℕ`

arguments, so this function is an inefficient identity function that can be replaced by the following one:`inject {_} x = x`

To actually find this function, we make the induction hypothesis that inject is an identity function in its second argument and look at the branches of the function to decide if this holds.

Injection detection also works over data type barriers. Example:

`forget : {A : Set} {n : ℕ} → Vec A n → List A forget [] = [] forget (x ∷ xs) = x ∷ forget xs`

Given that the constructor tags (in the compiled Epic code) for

`Vec.[]`

and`List.[]`

are the same, and that the tags for`Vec._∷_`

and`List._∷_`

are also the same, this is also an identity function. We can hence replace the definition with the following one:`forget {_} xs = xs`

To get this to apply as often as possible, constructor tags are chosen

*after*injection detection has been run, in a way to make as many functions as possible injections.Constructor tags are chosen once per source file, so it may be advantageous to define conversion functions like forget in the same module as one of the data types. For instance, if

`Vec.agda`

imports`List.agda`

, then the forget function should be put in`Vec.agda`

to ensure that vectors and lists get the same tags (unless some other injection function, which puts different constraints on the tags, is prioritised).Smashing.

This optimisation finds types whose values are inferable at runtime:

A data type with only one constructor where all fields are inferable is itself inferable.

`Set ℓ`

is inferable (as it has no runtime representation).

A function returning an inferable data type can be smashed, which means that it is replaced by a function which simply returns the inferred value.

An important example of an inferable type is the usual propositional equality type (

`_≡_`

). Any function returning a propositional equality can simply return the reflexivity constructor directly without computing anything.This optimisation makes more arguments unused. It also makes the Epic code size smaller, which in turn speeds up compilation.

### JavaScript

ECMAScript compiler backend.

A new compiler backend is being implemented, targetting ECMAScript (also known as JavaScript), with the goal of allowing Agda programs to be run in browsers or other ECMAScript environments.

The backend is still at an experimental stage: the core language is implemented, but many features are still missing.

The ECMAScript compiler can be invoked from the command line using the flag

`--js`

:`agda --js --compile-dir=<DIR> <FILE>.agda`

Each source

`<FILE>.agda`

is compiled into an ECMAScript target`<DIR>/jAgda.<TOP-LEVEL MODULE NAME>.js`

. The compiler can also be invoked using the Emacs mode (the variable`agda2-backend`

controls which backend is used).Note that ECMAScript is a strict rather than lazy language. Since Agda programs are total, this should not impact program semantics, but it may impact their space or time usage.

ECMAScript does not support algebraic datatypes or pattern-matching. These features are translated to a use of the visitor pattern. For instance, the standard library's

`List`

data type and`null`

function are translated into the following code:`exports["List"] = {}; exports["List"]["[]"] = function (x0) { return x0["[]"](); }; exports["List"]["_∷_"] = function (x0) { return function (x1) { return function (x2) { return x2["_∷_"](x0, x1); }; }; }; exports["null"] = function (x0) { return function (x1) { return function (x2) { return x2({ "[]": function () { return jAgda_Data_Bool["Bool"]["true"]; }, "_∷_": function (x3, x4) { return jAgda_Data_Bool["Bool"]["false"]; } }); }; }; };`

Agda records are translated to ECMAScript objects, preserving field names.

Top-level Agda modules are translated to ECMAScript modules, following the

`common.js`

module specification. A top-level Agda module`Foo.Bar`

is translated to an ECMAScript module`jAgda.Foo.Bar`

.The ECMAScript compiler does not compile to Haskell, so the pragmas related to the Haskell FFI (

`IMPORT`

,`COMPILED_DATA`

and`COMPILED`

) are not used by the ECMAScript backend. Instead, there is a`COMPILED_JS`

pragma which may be applied to any declaration. For postulates, primitives, functions and values, it gives the ECMAScript code to be emitted by the compiler. For data types, it gives a function which is applied to a value of that type, and a visitor object. For instance, a binding of natural numbers to ECMAScript integers (ignoring overflow errors) is:`data ℕ : Set where zero : ℕ suc : ℕ → ℕ {-# COMPILED_JS ℕ function (x,v) { if (x < 1) { return v.zero(); } else { return v.suc(x-1); } } #-} {-# COMPILED_JS zero 0 #-} {-# COMPILED_JS suc function (x) { return x+1; } #-} _+_ : ℕ → ℕ → ℕ zero + n = n suc m + n = suc (m + n) {-# COMPILED_JS _+_ function (x) { return function (y) { return x+y; }; } #-}`

To allow FFI code to be optimised, the ECMAScript in a

`COMPILED_JS`

declaration is parsed, using a simple parser that recognises a pure functional subset of ECMAScript, consisting of functions, function applications, return, if-statements, if-expressions, side-effect-free binary operators (no precedence, left associative), side-effect-free prefix operators, objects (where all member names are quoted), field accesses, and string and integer literals. Modules may be imported using the require (`<module-id>`

) syntax: any impure code, or code outside the supported fragment, can be placed in a module and imported.

## Tools

New flag

`--safe`

, which can be used to type-check untrusted code.This flag disables postulates,

`primTrustMe`

, and "unsafe" OPTION pragmas, some of which are known to make Agda inconsistent.Rejected pragmas:

`--allow-unsolved-metas --experimental-irrelevance --guardedness-preserving-type-construtors --injective-type-constructors --no-coverage-check --no-positivity-check --no-termination-check --sized-types --type-in-type`

Note that, at the moment, it is not possible to define the universe level or coinduction primitives when

`--safe`

is used (because they must be introduced as postulates). This can be worked around by type-checking trusted files in a first pass, without using`--safe`

, and then using`--saf`

e in a second pass. Modules which have already been type-checked are not re-type-checked just because`--safe`

is used.Dependency graphs.

The new flag

`--dependency-graph=FILE`

can be used to generate a DOT file containing a module dependency graph. The generated file (FILE) can be rendered using a tool like dot.The

`--no-unreachable-check`

flag has been removed.Projection functions are highlighted as functions instead of as fields. Field names (in record definitions and record values) are still highlighted as fields.

Support for jumping to positions mentioned in the information buffer has been added.

The

`make install`

command no longer installs Agda globally (by default).

# Release notes for Agda 2 version 2.2.10

## Language

New flag:

`--without-K`

.This flag makes pattern matching more restricted. If the flag is activated, then Agda only accepts certain case-splits. If the type of the variable to be split is

`D pars ixs`

, where`D`

is a data (or record) type, pars stands for the parameters, and`ixs`

the indices, then the following requirements must be satisfied:The indices

`ixs`

must be applications of constructors to distinct variables.These variables must not be free in pars.

The intended purpose of

`--without-K`

is to enable experiments with a propositional equality without the K rule. Let us define propositional equality as follows:`data _≡_ {A : Set} : A → A → Set where refl : ∀ x → x ≡ x`

Then the obvious implementation of the J rule is accepted:

`J : {A : Set} (P : {x y : A} → x ≡ y → Set) → (∀ x → P (refl x)) → ∀ {x y} (x≡y : x ≡ y) → P x≡y J P p (refl x) = p x`

The same applies to Christine Paulin-Mohring's version of the J rule:

`J′ : {A : Set} {x : A} (P : {y : A} → x ≡ y → Set) → P (refl x) → ∀ {y} (x≡y : x ≡ y) → P x≡y J′ P p (refl x) = p`

On the other hand, the obvious implementation of the K rule is not accepted:

`K : {A : Set} (P : {x : A} → x ≡ x → Set) → (∀ x → P (refl x)) → ∀ {x} (x≡x : x ≡ x) → P x≡x K P p (refl x) = p x`

However, we have

*not*proved that activation of`--without-K`

ensures that the K rule cannot be proved in some other way.Irrelevant declarations.

Postulates and functions can be marked as irrelevant by prefixing the name with a dot when the name is declared. Example:

`postulate .irrelevant : {A : Set} → .A → A`

Irrelevant names may only be used in irrelevant positions or in definitions of things which have been declared irrelevant.

The axiom irrelevant above can be used to define a projection from an irrelevant record field:

`data Subset (A : Set) (P : A → Set) : Set where _#_ : (a : A) → .(P a) → Subset A P elem : ∀ {A P} → Subset A P → A elem (a # p) = a .certificate : ∀ {A P} (x : Subset A P) → P (elem x) certificate (a # p) = irrelevant p`

The right-hand side of certificate is relevant, so we cannot define

`certificate (a # p) = p`

(because

`p`

is irrelevant). However, certificate is declared to be irrelevant, so it can use the axiom irrelevant. Furthermore the first argument of the axiom is irrelevant, which means that irrelevant`p`

is well-formed.As shown above the axiom irrelevant justifies irrelevant projections. Previously no projections were generated for irrelevant record fields, such as the field certificate in the following record type:

`record Subset (A : Set) (P : A → Set) : Set where constructor _#_ field elem : A .certificate : P elem`

Now projections are generated automatically for irrelevant fields (unless the flag

`--no-irrelevant-projections`

is used). Note that irrelevant projections are highly experimental.Termination checker recognises projections.

Projections now preserve sizes, both in patterns and expressions. Example:

`record Wrap (A : Set) : Set where constructor wrap field unwrap : A open Wrap public data WNat : Set where zero : WNat suc : Wrap WNat → WNat id : WNat → WNat id zero = zero id (suc w) = suc (wrap (id (unwrap w)))`

In the structural ordering

`unwrap w`

≤`w`

. This means that`unwrap w ≤ w < suc w,`

and hence the recursive call to id is accepted.

Projections also preserve guardedness.

## Tools

Hyperlinks for top-level module names now point to the start of the module rather than to the declaration of the module name. This applies both to the Emacs mode and to the output of

`agda --html`

.Most occurrences of record field names are now highlighted as "fields". Previously many occurrences were highlighted as "functions".

Emacs mode: It is no longer possible to change the behaviour of the

`TAB`

key by customising`agda2-indentation`

.Epic compiler backend.

A new compiler backend is being implemented. This backend makes use of Edwin Brady's language Epic (http://www.cs.st-andrews.ac.uk/~eb/epic.php) and its compiler. The backend should handle most Agda code, but is still at an experimental stage: more testing is needed, and some things written below may not be entirely true.

The Epic compiler can be invoked from the command line using the flag

`--epic`

:`agda --epic --epic-flag=<EPIC-FLAG> --compile-dir=<DIR> <FILE>.agda`

The

`--epic-flag`

flag can be given multiple times; each flag is given verbatim to the Epic compiler (in the given order). The resulting executable is named after the main module and placed in the directory specified by the`--compile-dir`

flag (default: the project root). Intermediate files are placed in a subdirectory called`Epic`

.The backend requires that there is a definition named main. This definition should be a value of type

`IO Unit`

, but at the moment this is not checked (so it is easy to produce a program which segfaults). Currently the backend represents actions of type`IO A`

as functions from`Unit`

to`A`

, and main is applied to the unit value.The Epic compiler compiles via C, not Haskell, so the pragmas related to the Haskell FFI (

`IMPORT`

,`COMPILED_DATA`

and`COMPILED`

) are not used by the Epic backend. Instead there is a new pragma`COMPILED_EPIC`

. This pragma is used to give Epic code for postulated definitions (Epic code can in turn call C code). The form of the pragma is`{-# COMPILED_EPIC def code #-}`

, where`def`

is the name of an Agda postulate and`code`

is some Epic code which should include the function arguments, return type and function body. As an example the`IO`

monad can be defined as follows:`postulate IO : Set → Set return : ∀ {A} → A → IO A _>>=_ : ∀ {A B} → IO A → (A → IO B) → IO B {-# COMPILED_EPIC return (u : Unit, a : Any) -> Any = ioreturn(a) #-} {-# COMPILED_EPIC _>>=_ (u1 : Unit, u2 : Unit, x : Any, f : Any) -> Any = iobind(x,f) #-}`

Here

`ioreturn`

and`iobind`

are Epic functions which are defined in the file`AgdaPrelude.e`

which is always included.By default the backend will remove so-called forced constructor arguments (and case-splitting on forced variables will be rewritten). This optimisation can be disabled by using the flag

`--no-forcing`

.All data types which look like unary natural numbers after forced constructor arguments have been removed (i.e. types with two constructors, one nullary and one with a single recursive argument) will be represented as "BigInts". This applies to the standard

`Fin`

type, for instance.The backend supports Agda's primitive functions and the BUILTIN pragmas. If the BUILTIN pragmas for unary natural numbers are used, then some operations, like addition and multiplication, will use more efficient "BigInt" operations.

If you want to make use of the Epic backend you need to install some dependencies, see the README.

The Emacs mode can compile using either the MAlonzo or the Epic backend. The variable

`agda2-backend`

controls which backend is used.

# Release notes for Agda 2 version 2.2.8

## Language

Record pattern matching.

It is now possible to pattern match on named record constructors. Example:

`record Σ (A : Set) (B : A → Set) : Set where constructor _,_ field proj₁ : A proj₂ : B proj₁ map : {A B : Set} {P : A → Set} {Q : B → Set} (f : A → B) → (∀ {x} → P x → Q (f x)) → Σ A P → Σ B Q map f g (x , y) = (f x , g y)`

The clause above is internally translated into the following one:

`map f g p = (f (Σ.proj₁ p) , g (Σ.proj₂ p))`

Record patterns containing data type patterns are not translated. Example:

`add : ℕ × ℕ → ℕ add (zero , n) = n add (suc m , n) = suc (add (m , n))`

Record patterns which do not contain data type patterns, but which do contain dot patterns, are currently rejected. Example:

`Foo : {A : Set} (p₁ p₂ : A × A) → proj₁ p₁ ≡ proj₁ p₂ → Set₁ Foo (x , y) (.x , y′) refl = Set`

Proof irrelevant function types.

Agda now supports irrelevant non-dependent function types:

`f : .A → B`

This type implies that

`f`

does not depend computationally on its argument. One intended use case is data structures with embedded proofs, like sorted lists:`postulate _≤_ : ℕ → ℕ → Set p₁ : 0 ≤ 1 p₂ : 0 ≤ 1 data SList (bound : ℕ) : Set where [] : SList bound scons : (head : ℕ) → .(head ≤ bound) → (tail : SList head) → SList bound`

The effect of the irrelevant type in the signature of

`scons`

is that`scons`

's second argument is never inspected after Agda has ensured that it has the right type. It is even thrown away, leading to smaller term sizes and hopefully some gain in efficiency. The type-checker ignores irrelevant arguments when checking equality, so two lists can be equal even if they contain different proofs:`l₁ : SList 1 l₁ = scons 0 p₁ [] l₂ : SList 1 l₂ = scons 0 p₂ [] l₁≡l₂ : l₁ ≡ l₂ l₁≡l₂ = refl`

Irrelevant arguments can only be used in irrelevant contexts. Consider the following subset type:

`data Subset (A : Set) (P : A → Set) : Set where _#_ : (elem : A) → .(P elem) → Subset A P`

The following two uses are fine:

`elimSubset : ∀ {A C : Set} {P} → Subset A P → ((a : A) → .(P a) → C) → C elimSubset (a # p) k = k a p elem : {A : Set} {P : A → Set} → Subset A P → A elem (x # p) = x`

However, if we try to project out the proof component, then Agda complains that

`variable p is declared irrelevant, so it cannot be used here`

:`prjProof : ∀ {A P} (x : Subset A P) → P (elem x) prjProof (a # p) = p`

Matching against irrelevant arguments is also forbidden, except in the case of irrefutable matches (record constructor patterns which have been translated away). For instance, the match against the pattern

`(p , q)`

here is accepted:`elim₂ : ∀ {A C : Set} {P Q : A → Set} → Subset A (λ x → Σ (P x) (λ _ → Q x)) → ((a : A) → .(P a) → .(Q a) → C) → C elim₂ (a # (p , q)) k = k a p q`

Absurd matches

`()`

are also allowed.Note that record fields can also be irrelevant. Example:

`record Subset (A : Set) (P : A → Set) : Set where constructor _#_ field elem : A .proof : P elem`

Irrelevant fields are never in scope, neither inside nor outside the record. This means that no record field can depend on an irrelevant field, and furthermore projections are not defined for such fields. Irrelevant fields can only be accessed using pattern matching, as in

`elimSubset`

above.Irrelevant function types were added very recently, and have not been subjected to much experimentation yet, so do not be surprised if something is changed before the next release. For instance, dependent irrelevant function spaces (

`.(x : A) → B`

) might be added in the future.Mixfix binders.

It is now possible to declare user-defined syntax that binds identifiers. Example:

`postulate State : Set → Set → Set put : ∀ {S} → S → State S ⊤ get : ∀ {S} → State S S return : ∀ {A S} → A → State S A bind : ∀ {A B S} → State S B → (B → State S A) → State S A syntax bind e₁ (λ x → e₂) = x ← e₁ , e₂ increment : State ℕ ⊤ increment = x ← get , put (1 + x)`

The syntax declaration for

`bind`

implies that`x`

is in scope in`e₂`

, but not in`e₁`

.You can give fixity declarations along with syntax declarations:

`infixr 40 bind syntax bind e₁ (λ x → e₂) = x ← e₁ , e₂`

The fixity applies to the syntax, not the name; syntax declarations are also restricted to ordinary, non-operator names. The following declaration is disallowed:

`syntax _==_ x y = x === y ```agda Syntax declarations must also be linear; the following declaration is disallowed: ```agda syntax wrong x = x + x`

Syntax declarations were added very recently, and have not been subjected to much experimentation yet, so do not be surprised if something is changed before the next release.

`Prop`

has been removed from the language.The experimental sort

`Prop`

has been disabled. Any program using`Prop`

should typecheck if`Prop`

is replaced by`Set₀`

. Note that`Prop`

is still a keyword.Injective type constructors off by default.

Automatic injectivity of type constructors has been disabled (by default). To enable it, use the flag

`--injective-type-constructors`

, either on the command line or in an OPTIONS pragma. Note that this flag makes Agda anti-classical and possibly inconsistent:Agda with excluded middle is inconsistent http://thread.gmane.org/gmane.comp.lang.agda/1367

See

`test/Succeed/InjectiveTypeConstructors.agda`

for an example.Termination checker can count.

There is a new flag

`--termination-depth=N`

accepting values`N >= 1`

(with`N = 1`

being the default) which influences the behavior of the termination checker. So far, the termination checker has only distinguished three cases when comparing the argument of a recursive call with the formal parameter of the callee.`<`

: the argument is structurally smaller than the parameter`=`

: they are equal`?`

: the argument is bigger or unrelated to the parameterThis behavior, which is still the default (

`N = 1`

), will not recognise the following functions as terminating.```agda mutual

`f : ℕ → ℕ f zero = zero f (suc zero) = zero f (suc (suc n)) = aux n aux : ℕ → ℕ aux m = f (suc m)`

`The call graph`

f --(<)--> aux --(?)--> f

`yields a recursive call from `f` to `f` via `aux` where the relation of call argument to callee parameter is computed as "unrelated" (composition of `<` and `?`). Setting `N >= 2` allows a finer analysis: `n` has two constructors less than `suc (suc n)`, and `suc m` has one more than `m`, so we get the call graph:`

f --(-2)--> aux --(+1)--> f

`The indirect call `f --> f` is now labeled with `(-1)`, and the termination checker can recognise that the call argument is decreasing on this path. Setting the termination depth to `N` means that the termination checker counts decrease up to `N` and increase up to `N-1`. The default, `N=1`, means that no increase is counted, every increase turns to "unrelated". In practice, examples like the one above sometimes arise when `with` is used. As an example, the program ```agda f : ℕ → ℕ f zero = zero f (suc zero) = zero f (suc (suc n)) with zero ... | _ = f (suc n)`

is internally represented as

`mutual f : ℕ → ℕ f zero = zero f (suc zero) = zero f (suc (suc n)) = aux n zero aux : ℕ → ℕ → ℕ aux m k = f (suc m)`

Thus, by default, the definition of

`f`

using`with`

is not accepted by the termination checker, even though it looks structural (`suc n`

is a subterm of`suc suc n`

). Now, the termination checker is satisfied if the option`--termination-depth=2`

is used.Caveats:

This is an experimental feature, hopefully being replaced by something smarter in the near future.

Increasing the termination depth will quickly lead to very long termination checking times. So, use with care. Setting termination depth to

`100`

by habit, just to be on the safe side, is not a good idea!Increasing termination depth only makes sense for linear data types such as

`ℕ`

and`Size`

. For other types, increase cannot be recognised. For instance, consider a similar example with lists.```agda data List : Set where nil : List cons : ℕ → List → List

mutual f : List → List f nil = nil f (cons x nil) = nil f (cons x (cons y ys)) = aux y ys

aux : ℕ → List → List aux z zs = f (cons z zs) ```

Here the termination checker compares

`cons z zs`

to`z`

and also to`zs`

. In both cases, the result will be "unrelated", no matter how high we set the termination depth. This is because when comparing`cons z zs`

to`zs`

, for instance,`z`

is unrelated to`zs`

, thus,`cons z zs`

is also unrelated to`zs`

. We cannot say it is just "one larger" since`z`

could be a very large term. Note that this points to a weakness of untyped termination checking.To regain the benefit of increased termination depth, we need to index our lists by a linear type such as

`ℕ`

or`Size`

. With termination depth`2`

, the above example is accepted for vectors instead of lists.

The

`codata`

keyword has been removed. To use coinduction, use the following new builtins:`INFINITY`

,`SHARP`

and`FLAT`

. Example:`{-# OPTIONS --universe-polymorphism #-} module Coinduction where open import Level infix 1000 ♯_ postulate ∞ : ∀ {a} (A : Set a) → Set a ♯_ : ∀ {a} {A : Set a} → A → ∞ A ♭ : ∀ {a} {A : Set a} → ∞ A → A {-# BUILTIN INFINITY ∞ #-} {-# BUILTIN SHARP ♯_ #-} {-# BUILTIN FLAT ♭ #-}`

Note that (non-dependent) pattern matching on

`SHARP`

is no longer allowed.Note also that strange things might happen if you try to combine the pragmas above with

`COMPILED_TYPE`

,`COMPILED_DATA`

or`COMPILED`

pragmas, or if the pragmas do not occur right after the postulates.The compiler compiles the

`INFINITY`

builtin to nothing (more or less), so that the use of coinduction does not get in the way of FFI declarations:`data Colist (A : Set) : Set where [] : Colist A _∷_ : (x : A) (xs : ∞ (Colist A)) → Colist A {-# COMPILED_DATA Colist [] [] (:) #-}`

Infinite types.

If the new flag

`--guardedness-preserving-type-constructors`

is used, then type constructors are treated as inductive constructors when we check productivity (but only in parameters, and only if they are used strictly positively or not at all). This makes examples such as the following possible:`data Rec (A : ∞ Set) : Set where fold : ♭ A → Rec A -- Σ cannot be a record type below. data Σ (A : Set) (B : A → Set) : Set where _,_ : (x : A) → B x → Σ A B syntax Σ A (λ x → B) = Σ[ x ∶ A ] B -- Corecursive definition of the W-type. W : (A : Set) → (A → Set) → Set W A B = Rec (♯ (Σ[ x ∶ A ] (B x → W A B))) syntax W A (λ x → B) = W[ x ∶ A ] B sup : {A : Set} {B : A → Set} (x : A) (f : B x → W A B) → W A B sup x f = fold (x , f) W-rec : {A : Set} {B : A → Set} (P : W A B → Set) → (∀ {x} {f : B x → W A B} → (∀ y → P (f y)) → P (sup x f)) → ∀ x → P x W-rec P h (fold (x , f)) = h (λ y → W-rec P h (f y)) -- Induction-recursion encoded as corecursion-recursion. data Label : Set where ′0 ′1 ′2 ′σ ′π ′w : Label mutual U : Set U = Σ Label U′ U′ : Label → Set U′ ′0 = ⊤ U′ ′1 = ⊤ U′ ′2 = ⊤ U′ ′σ = Rec (♯ (Σ[ a ∶ U ] (El a → U))) U′ ′π = Rec (♯ (Σ[ a ∶ U ] (El a → U))) U′ ′w = Rec (♯ (Σ[ a ∶ U ] (El a → U))) El : U → Set El (′0 , _) = ⊥ El (′1 , _) = ⊤ El (′2 , _) = Bool El (′σ , fold (a , b)) = Σ[ x ∶ El a ] El (b x) El (′π , fold (a , b)) = (x : El a) → El (b x) El (′w , fold (a , b)) = W[ x ∶ El a ] El (b x) U-rec : (P : ∀ u → El u → Set) → P (′1 , _) tt → P (′2 , _) true → P (′2 , _) false → (∀ {a b x y} → P a x → P (b x) y → P (′σ , fold (a , b)) (x , y)) → (∀ {a b f} → (∀ x → P (b x) (f x)) → P (′π , fold (a , b)) f) → (∀ {a b x f} → (∀ y → P (′w , fold (a , b)) (f y)) → P (′w , fold (a , b)) (sup x f)) → ∀ u (x : El u) → P u x U-rec P P1 P2t P2f Pσ Pπ Pw = rec where rec : ∀ u (x : El u) → P u x rec (′0 , _) () rec (′1 , _) _ = P1 rec (′2 , _) true = P2t rec (′2 , _) false = P2f rec (′σ , fold (a , b)) (x , y) = Pσ (rec _ x) (rec _ y) rec (′π , fold (a , b)) f = Pπ (λ x → rec _ (f x)) rec (′w , fold (a , b)) (fold (x , f)) = Pw (λ y → rec _ (f y))`

The

`--guardedness-preserving-type-constructors`

extension is based on a rather operational understanding of`∞`

/`♯_`

; it's not yet clear if this extension is consistent.Qualified constructors.

Constructors can now be referred to qualified by their data type. For instance, given

`data Nat : Set where zero : Nat suc : Nat → Nat data Fin : Nat → Set where zero : ∀ {n} → Fin (suc n) suc : ∀ {n} → Fin n → Fin (suc n)`

you can refer to the constructors unambiguously as

`Nat.zero`

,`Nat.suc`

,`Fin.zero`

, and`Fin.suc`

(`Nat`

and`Fin`

are modules containing the respective constructors). Example:`inj : (n m : Nat) → Nat.suc n ≡ suc m → n ≡ m inj .m m refl = refl`

Previously you had to write something like

`inj : (n m : Nat) → _≡_ {Nat} (suc n) (suc m) → n ≡ m`

to make the type checker able to figure out that you wanted the natural number suc in this case.

Reflection.

There are two new constructs for reflection:

`quoteGoal x in e`

In

`e`

the value of`x`

will be a representation of the goal type (the type expected of the whole expression) as an element in a datatype of Agda terms (see below). For instance,`example : ℕ example = quoteGoal x in {! at this point x = def (quote ℕ) [] !}`

`quote x : Name`

If

`x`

is the name of a definition (function, datatype, record, or a constructor),`quote x`

gives you the representation of`x`

as a value in the primitive type`Name`

(see below).

Quoted terms use the following BUILTINs and primitives (available from the standard library module

`Reflection`

):`-- The type of Agda names. postulate Name : Set {-# BUILTIN QNAME Name #-} primitive primQNameEquality : Name → Name → Bool -- Arguments. Explicit? = Bool data Arg A : Set where arg : Explicit? → A → Arg A {-# BUILTIN ARG Arg #-} {-# BUILTIN ARGARG arg #-} -- The type of Agda terms. data Term : Set where var : ℕ → List (Arg Term) → Term con : Name → List (Arg Term) → Term def : Name → List (Arg Term) → Term lam : Explicit? → Term → Term pi : Arg Term → Term → Term sort : Term unknown : Term {-# BUILTIN AGDATERM Term #-} {-# BUILTIN AGDATERMVAR var #-} {-# BUILTIN AGDATERMCON con #-} {-# BUILTIN AGDATERMDEF def #-} {-# BUILTIN AGDATERMLAM lam #-} {-# BUILTIN AGDATERMPI pi #-} {-# BUILTIN AGDATERMSORT sort #-} {-# BUILTIN AGDATERMUNSUPPORTED unknown #-}`

Reflection may be useful when working with internal decision procedures, such as the standard library's ring solver.

Minor record definition improvement.

The definition of a record type is now available when type checking record module definitions. This means that you can define things like the following:

`record Cat : Set₁ where field Obj : Set _=>_ : Obj → Obj → Set -- ... -- not possible before: op : Cat op = record { Obj = Obj; _=>_ = λ A B → B => A }`

## Tools

The

`Goal type and context`

command now shows the goal type before the context, and the context is shown in reverse order. The`Goal type, context and inferred type`

command has been modified in a similar way.Show module contents command.

Given a module name

`M`

the Emacs mode can now display all the top-level modules and names inside`M`

, along with types for the names. The command is activated using`C-c C-o`

or the menus.Auto command.

A command which searches for type inhabitants has been added. The command is invoked by pressing

`C-C C-a`

(or using the goal menu). There are several flags and parameters, e.g.`-c`

which enables case-splitting in the search. For further information, see the Agda wiki:http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Main.Auto

HTML generation is now possible for a module with unsolved meta-variables, provided that the

`--allow-unsolved-metas`

flag is used.

# Release notes for Agda 2 version 2.2.6

## Language

Universe polymorphism (experimental extension).

To enable universe polymorphism give the flag

`--universe-polymorphism`

on the command line or (recommended) as an OPTIONS pragma.When universe polymorphism is enabled

`Set`

takes an argument which is the universe level. For instance, the type of universe polymorphic identity is`id : {a : Level} {A : Set a} → A → A.`

The type Level is isomorphic to the unary natural numbers and should be specified using the BUILTINs

`LEVEL`

,`LEVELZERO`

, and`LEVELSUC`

:`data Level : Set where zero : Level suc : Level → Level {-# BUILTIN LEVEL Level #-} {-# BUILTIN LEVELZERO zero #-} {-# BUILTIN LEVELSUC suc #-}`

There is an additional BUILTIN

`LEVELMAX`

for taking the maximum of two levels:`max : Level → Level → Level max zero m = m max (suc n) zero = suc n max (suc n) (suc m) = suc (max n m) {-# BUILTIN LEVELMAX max #-}`

The non-polymorphic universe levels

`Set`

,`Set₁`

and so on are sugar for`Set zero`

,`Set (suc zero)`

, etc.At present there is no automatic lifting of types from one level to another. It can still be done (rather clumsily) by defining types like the following one:

`data Lifted {a} (A : Set a) : Set (suc a) where lift : A → Lifted A`

However, it is likely that automatic lifting is introduced at some point in the future.

Multiple constructors, record fields, postulates or primitives can be declared using a single type signature:

`data Bool : Set where false true : Bool postulate A B : Set`

Record fields can be implicit:

`record R : Set₁ where field {A} : Set f : A → A {B C} D {E} : Set g : B → C → E`

By default implicit fields are not printed.

Record constructors can be defined:

`record Σ (A : Set) (B : A → Set) : Set where constructor _,_ field proj₁ : A proj₂ : B proj₁`

In this example

`_,_`

gets the type`(proj₁ : A) → B proj₁ → Σ A B.`

For implicit fields the corresponding constructor arguments become implicit.

Note that the constructor is defined in the

*outer*scope, so any fixity declaration has to be given outside the record definition. The constructor is not in scope inside the record module.Note also that pattern matching for records has not been implemented yet.

BUILTIN hooks for equality.

The data type

`data _≡_ {A : Set} (x : A) : A → Set where refl : x ≡ x`

can be specified as the builtin equality type using the following pragmas:

`{-# BUILTIN EQUALITY _≡_ #-} {-# BUILTIN REFL refl #-}`

The builtin equality is used for the new rewrite construct and the

`primTrustMe`

primitive described below.New

`rewrite`

construct.If

`eqn : a ≡ b`

, where`_≡_`

is the builtin equality (see above) you can now write`f ps rewrite eqn = rhs`

instead of

`f ps with a | eqn ... | ._ | refl = rhs`

The

`rewrite`

construct has the effect of rewriting the goal and the context by the given equation (left to right).You can rewrite using several equations (in sequence) by separating them with vertical bars (|):

`f ps rewrite eqn₁ | eqn₂ | … = rhs`

It is also possible to add

`with`

-clauses after rewriting:`f ps rewrite eqns with e ... | p = rhs`

Note that pattern matching happens before rewriting—if you want to rewrite and then do pattern matching you can use a with after the rewrite.

See

`test/Succeed/Rewrite.agda`

for some examples.A new primitive,

`primTrustMe`

, has been added:`primTrustMe : {A : Set} {x y : A} → x ≡ y`

Here

`_≡_`

is the builtin equality (see BUILTIN hooks for equality, above).If

`x`

and`y`

are definitionally equal, then`primTrustMe {x = x} {y = y}`

reduces to`refl`

.Note that the compiler replaces all uses of

`primTrustMe`

with the`REFL`

builtin, without any check for definitional equality. Incorrect uses of`primTrustMe`

can potentially lead to segfaults or similar problems.For an example of the use of

`primTrustMe`

, see`Data.String`

in version 0.3 of the standard library, where it is used to implement decidable equality on strings using the primitive boolean equality.Changes to the syntax and semantics of IMPORT pragmas, which are used by the Haskell FFI. Such pragmas must now have the following form:

`{-# IMPORT <module name> #-}`

These pragmas are interpreted as

*qualified*imports, so Haskell names need to be given qualified (unless they come from the Haskell prelude).The horizontal tab character (U+0009) is no longer treated as white space.

Line pragmas are no longer supported.

The

`--include-path`

flag can no longer be used as a pragma.The experimental and incomplete support for proof irrelevance has been disabled.

## Tools

New

`intro`

command in the Emacs mode. When there is a canonical way of building something of the goal type (for instance, if the goal type is a pair), the goal can be refined in this way. The command works for the following goal types:A data type where only one of its constructors can be used to construct an element of the goal type. (For instance, if the goal is a non-empty vector, a

`cons`

will be introduced.)A record type. A record value will be introduced. Implicit fields will not be included unless showing of implicit arguments is switched on.

A function type. A lambda binding as many variables as possible will be introduced. The variable names will be chosen from the goal type if its normal form is a dependent function type, otherwise they will be variations on

`x`

. Implicit lambdas will only be inserted if showing of implicit arguments is switched on.

This command can be invoked by using the

`refine`

command (`C-c C-r`

) when the goal is empty. (The old behaviour of the refine command in this situation was to ask for an expression using the minibuffer.)The Emacs mode displays

`Checked`

in the mode line if the current file type checked successfully without any warnings.If a file

`F`

is loaded, and this file defines the module`M`

, it is an error if`F`

is not the file which defines`M`

according to the include path.Note that the command-line tool and the Emacs mode define the meaning of relative include paths differently: the command-line tool interprets them relative to the current working directory, whereas the Emacs mode interprets them relative to the root directory of the current project. (As an example, if the module

`A.B.C`

is loaded from the file`<some-path>/A/B/C.agda`

, then the root directory is`<some-path>`

.)It is an error if there are several files on the include path which match a given module name.

Interface files are relocatable. You can move around source trees as long as the include path is updated in a corresponding way. Note that a module

`M`

may be re-typechecked if its time stamp is strictly newer than that of the corresponding interface file (`M.agdai`

).Type-checking is no longer done when an up-to-date interface exists. (Previously the initial module was always type-checked.)

Syntax highlighting files for Emacs (

`.agda.el`

) are no longer used. The`--emacs`

flag has been removed. (Syntax highlighting information is cached in the interface files.)The Agate and Alonzo compilers have been retired. The options

`--agate`

,`--alonzo`

and`--malonzo`

have been removed.The default directory for MAlonzo output is the project's root directory. The

`--malonzo-dir`

flag has been renamed to`--compile-dir`

.Emacs mode:

`C-c C-x C-d`

no longer resets the type checking state.`C-c C-x C-r`

can be used for a more complete reset.`C-c C-x C-s`

(which used to reload the syntax highlighting information) has been removed.`C-c C-l`

can be used instead.The Emacs mode used to define some "abbrevs", unless the user explicitly turned this feature off. The new default is

*not*to add any abbrevs. The old default can be obtained by customising`agda2-mode-abbrevs-use-defaults`

(a customisation buffer can be obtained by typing`M-x customize-group agda2 RET`

after an Agda file has been loaded).

# Release notes for Agda 2 version 2.2.4

Important changes since 2.2.2:

Change to the semantics of

`open import`

and`open module`

. The declaration`open import M <using/hiding/renaming>`

now translates to

`import A open A <using/hiding/renaming>`

instead of

`import A <using/hiding/renaming> open A`

The same translation is used for

`open module M = E …`

. Declarations involving the keywords as or public are changed in a corresponding way (`as`

always goes with import, and`public`

always with open).This change means that import directives do not affect the qualified names when open import/module is used. To get the old behaviour you can use the expanded version above.

Names opened publicly in parameterised modules no longer inherit the module parameters. Example:

`module A where postulate X : Set module B (Y : Set) where open A public`

In Agda 2.2.2

`B.X`

has type`(Y : Set) → Set`

, whereas in Agda 2.2.4`B.X`

has type Set.Previously it was not possible to export a given constructor name through two different

`open public`

statements in the same module. This is now possible.Unicode subscript digits are now allowed for the hierarchy of universes (

`Set₀`

,`Set₁`

, …):`Set₁`

is equivalent to`Set1`

.

# Release notes for Agda 2 version 2.2.2

## Tools

The

`--malonzodir`

option has been renamed to`--malonzo-dir`

.The output of

`agda --html`

is by default placed in a directory called`html`

.

## Infrastructure

The Emacs mode is included in the Agda Cabal package, and installed by

`cabal install`

. The recommended way to enable the Emacs mode is to include the following code in`.emacs`

:`(load-file (let ((coding-system-for-read 'utf-8)) (shell-command-to-string "agda-mode locate")))`

# Release notes for Agda 2 version 2.2.0

Important changes since 2.1.2 (which was released 2007-08-16):

## Language

Exhaustive pattern checking. Agda complains if there are missing clauses in a function definition.

Coinductive types are supported. This feature is under development/evaluation, and may change.

http://wiki.portal.chalmers.se/agda/agda.php?n=ReferenceManual.Codatatypes

Another experimental feature: Sized types, which can make it easier to explain why your code is terminating.

Improved constraint solving for functions with constructor headed right hand sides.

http://wiki.portal.chalmers.se/agda/agda.php?n=ReferenceManual.FindingTheValuesOfImplicitArguments

A simple, well-typed foreign function interface, which allows use of Haskell functions in Agda code.

http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Docs.FFI

The tokens

`forall`

,`->`

and`\`

can be written as`∀`

,`→`

and`λ`

.Absurd lambdas:

`λ ()`

and`λ {}`

.http://thread.gmane.org/gmane.comp.lang.agda/440

Record fields whose values can be inferred can be omitted.

Agda complains if it spots an unreachable clause, or if a pattern variable "shadows" a hidden constructor of matching type.

http://thread.gmane.org/gmane.comp.lang.agda/720

## Tools

Case-split: The user interface can replace a pattern variable with the corresponding constructor patterns. You get one new left-hand side for every possible constructor.

http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode

The MAlonzo compiler.

http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Docs.MAlonzo

A new Emacs input method, which contains bindings for many Unicode symbols, is by default activated in the Emacs mode.

http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Docs.UnicodeInput

Highlighted, hyperlinked HTML can be generated from Agda source code.

http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Main.HowToGenerateWebPagesFromSourceCode

The command-line interactive mode (

`agda -I`

) is no longer supported, but should still work.http://thread.gmane.org/gmane.comp.lang.agda/245

Reload times when working on large projects are now considerably better.

http://thread.gmane.org/gmane.comp.lang.agda/551

## Libraries

A standard library is under development.

http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Libraries.StandardLibrary

## Documentation

The Agda wiki is better organised. It should be easier for a newcomer to find relevant information now.

http://wiki.portal.chalmers.se/agda/

## Infrastructure

Easy-to-install packages for Windows and Debian/Ubuntu have been prepared.

http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Main.Download

Agda 2.2.0 is available from Hackage.

http://hackage.haskell.org/