agda/doc/release-notes/2.4.0.md

Release notes for Agda 2 version 2.4.0

Installation and infrastructure

• A new module called Agda.Primitive has been introduced. This module is available to all users, even if the standard library is not used. Currently the module contains level primitives and their representation in Haskell when compiling with MAlonzo:

  infixl 6 _⊔_

  postulate
    Level : Set
    lzero : Level
    lsuc  : (ℓ : Level) → Level
    _⊔_   : (ℓ₁ ℓ₂ : Level) → Level

  {-# COMPILED_TYPE Level ()      #-}
  {-# COMPILED lzero ()           #-}
  {-# COMPILED lsuc  (\_ -> ())   #-}
  {-# COMPILED _⊔_   (\_ _ -> ()) #-}

  {-# BUILTIN LEVEL     Level  #-}
  {-# BUILTIN LEVELZERO lzero  #-}
  {-# BUILTIN LEVELSUC  lsuc   #-}
  {-# BUILTIN LEVELMAX  _⊔_    #-}

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

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

The standard library reexports these primitives (using the names zero and suc instead of lzero and lsuc) from the Level module.

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

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

To fix this problem, please remove the duplicate bindings.

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

The include path now always contains a directory <DATADIR>/lib/prim, and this directory is supposed to contain a subdirectory Agda containing a file Primitive.agda.

The standard location of <DATADIR> is system- and installation-specific. E.g., in a Cabal --user installation of Agda-2.3.4 on a standard single-ghc Linux system it would be $HOME/.cabal/share/Agda-2.3.4 or something similar.

The location of the <DATADIR> directory can be configured at compile-time using Cabal flags (--datadir and --datasubdir). The location can also be set at run-time, using the Agda_datadir environment variable.

Pragmas and options

• Pragma NO_TERMINATION_CHECK placed within a mutual block is now applied to the whole mutual block (rather than being discarded silently). Adding to the uses 1.-4. outlined in the release notes for 2.3.2 we allow:

3a. Skipping an old-style mutual block: Somewhere within mutual

  block before a type signature or first function clause.

   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 Sigma

  test : (A : Set) (B : A -> Set) ->
    let X : (x : A) (y : B x) -> Sigma A B
        X = _
    in  (z : Sigma A B) -> X (fst z) (snd z) ≡ z
  test A B z = refl

  test' : (A : Set) (B : A -> Set) ->
    let X : A -> A
        X = _
    in  (z : Sigma A B) -> X (fst z) ≡ fst z
  test' A B z = refl

The fresh bound variables are named fst(z) and snd(z) and can appear in error messages, e.g.:

  fail : (A : Set) (B : A -> Set) ->
    let X : A -> Sigma A B
        X = _
    in  (z : Sigma A B) -> X (fst z) ≡ z
  fail A B z = refl

results in error:

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

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

For a simple example, consider

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

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

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

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

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

• The implementation of the primitive primTrustMe has changed. It now only reduces to REFL if the two arguments x and y have the same computational normal form. Before, it reduced when x and y were definitionally equal, which included type-directed equality laws such as eta-equality. Yet because reduction is untyped, calling conversion from reduction lead to Agda crashes [Issue #882].

The amended description of primTrustMe is (cf. release notes for 2.2.6):

  primTrustMe : {A : Set} {x y : A} → x ≡ y

Here _≡_ is the builtin equality (see BUILTIN hooks for equality, above).

If x and y have the same computational normal form, then primTrustMe {x = x} {y = y} reduces to refl.

A note on primTrustMe's runtime behavior: The MAlonzo compiler replaces all uses of primTrustMe with the REFL builtin, without any check for definitional equality. Incorrect uses of primTrustMe can potentially lead to segfaults or similar problems of the compiled code.

• Implicit patterns of record type are now only eta-expanded if there is a record constructor. [Issues #473, #635]

  data D : Set where
    d : D

  data P : D → Set where
    p : P d

  record Rc : Set where
    constructor c
    field f : D

  works : {r : Rc} → P (Rc.f r) → Set
  works p = D

This works since the implicit pattern r is eta-expanded to c x which allows the type of p to reduce to P x and x to be unified with d. The corresponding explicit version is:

  works' : (r : Rc) → P (Rc.f r) → Set
  works' (c .d) p = D

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

  record R : Set where
    field f : D

  fails : {r : R} → P (R.f r) → Set
  fails p = D

  -- d != R.f r of type D
  -- when checking that the pattern p has type P (R.f r)

The error is justified since there is no pattern we could write down for r. It would have to look like

  record { f = .d }

but anonymous record patterns are not part of the language.

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

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

Which is a good thing!

• Printing of named implicit function types.

When printing terms in a context with bound variables Agda renames new bindings to avoid clashes with the previously bound names. For instance, if A is in scope, the type (A : Set) → A is printed as (A₁ : Set) → A₁. However, for implicit function types the name of the binding matters, since it can be used when giving implicit arguments.

For this situation, the following new syntax has been introduced: {x = y : A} → B is an implicit function type whose bound variable (in scope in B) is y, but where the name of the argument is x for the purposes of giving it explicitly. For instance, with A in scope, the type {A : Set} → A is now printed as {A = A₁ : Set} → A₁.

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

• Changed the semantics of --without-K. [Issue #712, Issue #865, Issue #1025]

New specification of --without-K:

When --without-K is enabled, the unification of indices for pattern matching is restricted in two ways:

1. Reflexive equations of the form x == x are no longer solved, instead Agda gives an error when such an equation is encountered.

2. When unifying two same-headed constructor forms c us and c vs of type D pars ixs, the datatype indices ixs (but not the parameters) have to be self-unifiable, i.e. unification of ixs with itself should succeed positively. This is a nontrivial requirement because of point 1.

Examples:

• The J rule is accepted.

  ```agda J : {A : Set} (P : {x y : A} → x ≡ y → Set) →

  (∀ x → P (refl x)) →
  ∀ {x y} (x≡y : x ≡ y) → P x≡y
  

  J P p (refl x) = p x ```agda

  This definition is accepted since unification of x with y doesn't require deletion or injectivity.

• The K rule is rejected.

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

  Definition is rejected with the following error:

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

• Symmetry of the new criterion.

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

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

• Handling of parameters.

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

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

• A larger example: antisymmetry of ≤.

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

• [ Issue #1025 ]

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

  When applying injectivity to the equation foo ≡ foo of type Foo refl, it is checked that the index refl of type myPoint ≡ myPoint is self-unifiable. The equation refl ≡ refl again requires injectivity, so now the index myPoint is checked for self-unifiability, hence the error:

  Cannot eliminate reflexive equation myPoint = myPoint of type
  mySpace because K has been disabled.
  when checking that the pattern refl has type foo ≡ foo
  

Termination checking

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

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

• Termination checking of functions defined by with has been improved.

Cases which previously required --termination-depth to pass the termination checker (due to use of with) no longer need the flag. For example

  merge : List A → List A → List A
  merge [] ys = ys
  merge xs [] = xs
  merge (x ∷ xs) (y ∷ ys) with x ≤ y
  merge (x ∷ xs) (y ∷ ys)    | false = y ∷ merge (x ∷ xs) ys
  merge (x ∷ xs) (y ∷ ys)    | true  = x ∷ merge xs (y ∷ ys)

This failed to termination check previously, since the with expands to an auxiliary function merge-aux:

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

This function makes a call to merge in which the size of one of the arguments is increasing. To make this pass the termination checker now inlines the definition of merge-aux before checking, thus effectively termination checking the original source program.

As a result of this transformation doing with on a variable no longer preserves termination. For instance, this does not termination check:

  bad : Nat → Nat
  bad n with n
  ... | zero  = zero
  ... | suc m = bad m

• The performance of the termination checker has been improved. For higher --termination-depth the improvement is significant. While the default --termination-depth is still 1, checking with higher --termination-depth should now be feasible.

Compiler backends

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

  • A new command-line option --compile-no-main: the command

  agda --compile-no-main Test.agda
  

  will compile Test.agda and all its dependencies to Haskell and compile the resulting Haskell files with --make, but (unlike --compile) not tell GHC to treat Test.hs as the main module. This type of compilation can be invoked from Emacs by customizing the agda2-backend variable to value MAlonzoNoMain and then calling C-c C-x C-c as before.

  • A new pragma COMPILED_EXPORT was added as part of the MAlonzo FFI. If we have an Agda file containing the following:

   module A.B where
  
   test : SomeType
   test = someImplementation
  
   {-# COMPILED_EXPORT test someHaskellId #-}
  

  then test will be compiled to a Haskell function called someHaskellId in module MAlonzo.Code.A.B that can be invoked from other Haskell code. Its type will be translated according to the normal MAlonzo rules.

Tools

Emacs mode

• A new goal command Helper Function Type (C-c C-h) has been added.

If you write an application of an undefined function in a goal, the Helper Function Type command will print the type that the function needs to have in order for it to fit the goal. The type is also added to the Emacs kill-ring and can be pasted into the buffer using C-y.

The application must be of the form f args where f is the name of the helper function you want to create. The arguments can use all the normal features like named implicits or instance arguments.

Example:

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

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

Calling C-c C-h in the goal prints

  snoc : ∀ {A} {n} → Vec A n → A → Vec A (suc n)

• A new command Explain why a particular name is in scope (C-c C-w) has been added. [Issue #207]

This command can be called from a goal or from the top-level and will as the name suggests explain why a particular name is in scope.

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

For example, given

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

Calling C-c C-w on mkFoo at the top-level prints

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

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

• Improvements to the make case command (C-c C-c)

  • One can now also split on hidden variables, using the name (starting with .) with which they are printed. Use C-c C-, to see all variables in context.

  • Concerning the printing of generated clauses:

  • Uses named implicit arguments to improve readability.

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

  • Avoids binding variables in implicit positions by replacing dot patterns that uses them by wildcards (._).

• Key bindings for lots of "mathematical" characters (examples: 𝐴𝑨𝒜𝓐𝔄) have been added to the Agda input method. Example: type \MiA\MIA\McA\MCA\MfA to get 𝐴𝑨𝒜𝓐𝔄.

Note: \McB does not exist in Unicode (as well as others in that style), but the \MC (bold) alphabet is complete.

• Key bindings for "blackboard bold" B (𝔹) and 0-9 (𝟘-𝟡) have been added to the Agda input method (\bb and \b[0-9]).

• Key bindings for controlling simplification/normalisation:

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.

LaTeX-backend

• Two new color scheme options were added to agda.sty:

\usepackage[bw]{agda}, which highlights in black and white; \usepackage[conor]{agda}, which highlights using Conor's colors.

The default (no options passed) is to use the standard colors.

• If agda.sty cannot be found by the LateX environment, it is now copied into the LateX output directory (latex by default) instead of the working directory. This means that the commands needed to produce a PDF now is

  agda --latex -i . <file>.lagda
  cd latex
  pdflatex <file>.tex

• The LaTeX-backend has been made more tool agnostic, in particular XeLaTeX and LuaLaTeX should now work. Here is a small example (test/LaTeXAndHTML/succeed/UnicodeInput.lagda):

  \documentclass{article}
  \usepackage{agda}
  \begin{document}

  \begin{code}
  data αβγδεζθικλμνξρστυφχψω : Set₁ where

  postulate
    →⇒⇛⇉⇄↦⇨↠⇀⇁ : Set
  \end{code}

  \[
  ∀X [ ∅ ∉ X ⇒ ∃f:X ⟶  ⋃ X\ ∀A ∈ X (f(A) ∈ A) ]
  \]
  \end{document}

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

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

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

  \setmathfont{<math-font>}

Use the fc-list tool to list available fonts.

• Add experimental support for hyperlinks to identifiers

If the hyperref LateX package is loaded before the Agda package and the links option is passed to the Agda package, then the Agda package provides a function called \AgdaTarget. Identifiers which have been declared targets, by the user, will become clickable hyperlinks in the rest of the document. Here is a small example (test/LaTeXAndHTML/succeed/Links.lagda):

  \documentclass{article}
  \usepackage{hyperref}
  \usepackage[links]{agda}
  \begin{document}

  \AgdaTarget{ℕ}
  \AgdaTarget{zero}
  \begin{code}
  data ℕ : Set where
    zero  : ℕ
    suc   : ℕ → ℕ
  \end{code}

  See next page for how to define \AgdaFunction{two} (doesn't turn into a
  link because the target hasn't been defined yet). We could do it
  manually though; \hyperlink{two}{\AgdaDatatype{two}}.

  \newpage

  \AgdaTarget{two}
  \hypertarget{two}{}
  \begin{code}
  two : ℕ
  two = suc (suc zero)
  \end{code}

  \AgdaInductiveConstructor{zero} is of type
  \AgdaDatatype{ℕ}. \AgdaInductiveConstructor{suc} has not been defined to
  be a target so it doesn't turn into a link.

  \newpage

  Now that the target for \AgdaFunction{two} has been defined the link
  works automatically.

  \begin{code}
  data Bool : Set where
    true false : Bool
  \end{code}

  The AgdaTarget command takes a list as input, enabling several
  targets to be specified as follows:

  \AgdaTarget{if, then, else, if\_then\_else\_}
  \begin{code}
  if_then_else_ : {A : Set} → Bool → A → A → A
  if true  then t else f = t
  if false then t else f = f
  \end{code}

  \newpage

  Mixfix identifier need their underscores escaped:
  \AgdaFunction{if\_then\_else\_}.

  \end{document}

The boarders around the links can be suppressed using hyperref's hidelinks option:

    \usepackage[hidelinks]{hyperref}

Note that the current approach to links does not keep track of scoping or types, and hence overloaded names might create links which point to the wrong place. Therefore it is recommended to not overload names when using the links option at the moment, this might get fixed in the future.