## Changelog for Agda-2.4.2

If this renders badly as markdown, see the plain text version## -- Release notes for Agda version 2.4.2

Important changes since 2.4.0.2:

# 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 -> Aopen 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 : Brecord 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 AIf 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

Important changes since 2.4.0.1:

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 https://code.google.com/p/agda/issues): 1203 1205 1209 1213 1214 1216 1225 1226 1231 1233 1239 1241 1243

## -- Release notes for Agda 2 version 2.4.0.1

Important changes since 2.4.0:

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

Important changes since 2.3.2.2:

# 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.

`mutual {-# NO_TERMINATION_CHECK #-} c : A c = d d : 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 Sigmatest : (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) ≡ zDependent 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 = reflWhich 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.

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

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. [Issue207]

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.

- a constructor Issue207.C._.Foo.mkFoo brought into scope by
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/latex-backend/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/latex-backend/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

Important changes since 2.3.2.1:

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

Important changes since 2.3.2:

# 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

Important changes since 2.3.0:

# 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 !! iIn 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 fsttest : (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} = BNote 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:

\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

Important changes since 2.2.10:

# 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 fstWhen 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 otherwise
- if x is an explicit field then x = R.x r is included in new-fields, and
- if 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:

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 --safe 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

Important changes since 2.2.8:

## 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 ≡ xThen 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 Pelem : ∀ {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 elemNow 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

Important changes since 2.2.6:

## 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 proj₁map : {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 ≤ 1data 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 PThe 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 elemIrrelevant 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 === ySyntax declarations must also be linear; the following declaration is disallowed:

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 parameter

This behavior, which is still the default (N = 1), will not recognise the following functions as terminating.

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

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.

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 Bsyntax 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 ≡ mto 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:

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

Important changes since 2.2.4:

## 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 proj₁In 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 ≡ xcan 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 writef 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

Important changes since 2.2.0:

## 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.

The tokens forall, -> and \ can be written as ∀, → and λ.

Absurd lambdas: λ () and λ {}.

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.

## 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.

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.

Reload times when working on large projects are now considerably better.

## 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.

## 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.