Release notes for Agda 2 version 2.3.0
======================================

Language
--------

* New more liberal syntax for mutually recursive definitions.

  It is no longer necessary to use the `mutual` keyword to define
  mutually recursive functions or datatypes. Instead, it is enough to
  declare things before they are used. Instead of

  ```agda
  mutual
    f : A
    f = a[f, g]

    g : B[f]
    g = b[f, g]
  ```

  you can now write

  ```agda
  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:

  ```agda
  -- Declaration.
  data Vec (A : Set) : Nat → Set  -- Note the absence of 'where'.

  -- Definition.
  data Vec A where
    []   : Vec A zero
    _::_ : {n : Nat} → A → Vec A n → Vec A (suc n)

  -- Declaration.
  record Sigma (A : Set) (B : A → Set) : Set

  -- Definition.
  record Sigma A B where
    constructor _,_
    field fst : A
          snd : B fst
  ```

  When making separated declarations/definitions private or abstract
  you should attach the `private` keyword to the declaration and the
  `abstract` keyword to the definition. For instance, a private,
  abstract function can be defined as

  ```agda
  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

  ```agda
  (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:

  ```agda
  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:

  ```agda
  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:

  ```agda
  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

  ```agda
  record old { upd-fields }
  ```

  is checked against a record type `R`, it is expanded to

  ```agda
  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:

  ```agda
  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:

  ```agda
  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](https://github.com/agda/agda/issues/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:

  ```agda
  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,

  ```agda
  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:

  ```agda
  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]:

  ```agda
  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:

  ```agda
  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:

  ```agda
  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:

  ```agda
    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

  ```agda
  record Eq (A : Set) : Set where
    field equal : A → A → Bool,
  ```

  then we can define the following projection:

  ```agda
  equal : {A : Set} {{eq : Eq A}} → A → A → Bool
  equal {{eq}} = Eq.equal eq
  ```

  Now consider the following expression:

  ```agda
  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:

  ```agda
  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:

  ```agda
  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:

  ```agda
  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:

  ```agda
  open Eq {{...}}
  ```

  Instance argument resolution is not recursive. As an example,
  consider the following "parametrised instance":

  ```agda
  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:

  ```agda
  test = equal (1 ∷ 2 ∷ []) (3 ∷ 4 ∷ [])
  ```

  However, we can make the code work by constructing a suitable
  instance manually:

  ```agda
  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:

  ```agda
  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

  ```agda
  (ih : (a : A) -> P (squash a)).
  ```

* Records with only irrelevant fields.

  The following now works:

  ```agda
  record IsEquivalence {A : Set} (_≈_ : A → A → Set) : Set where
    field
      .refl  : Reflexive _≈_
      .sym   : Symmetric _≈_
      .trans : Transitive _≈_

  record Setoid : Set₁ where
    infix 4 _≈_
    field
      Carrier        : Set
      _≈_            : Carrier → Carrier → Set
      .isEquivalence : IsEquivalence _≈_

    open IsEquivalence isEquivalence public
  ```

  Previously Agda complained about the application
  `IsEquivalence isEquivalence`, because `isEquivalence` is irrelevant
  and the `IsEquivalence` module expected a relevant argument. Now,
  when record modules are generated for records consisting solely of
  irrelevant arguments, the record parameter is made irrelevant:

   ```agda
  module IsEquivalence {A : Set} {_≈_ : A → A → Set}
                       .(r : IsEquivalence {A = A} _≈_) where
    …
  ```

* Irrelevant things are no longer erased internally. This means that
  they are printed as ordinary terms, not as `_` as before.

* The new flag `--experimental-irrelevance` enables irrelevant
  universe levels and matching on irrelevant data when only one
  constructor is available. These features are very experimental and
  likely to change or disappear.

### Reflection

* The reflection API has been extended to mirror features like
  irrelevance, instance arguments and universe polymorphism, and to
  give (limited) access to definitions. For completeness all the
  builtins and primitives are listed below:

  ```agda
  -- 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

  ```agda
  primQNameType (quote zero)
  ```

  is definitionally equal to

  ```agda
  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:

  1. The argument `t` must have type `Term` (see the reflection API above).

  2. The argument is normalised.

  3. 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:

  ```agda
  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:

  1. The type of `t` is inferred. The term `t` must be type-correct.

  2. The term `t` is normalised.

  3. 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:

  ```agda
  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

  ```haskell
    type AgdaIO a b = IO b,
  ```

  we can set up universe-polymorphic `IO` in the following way:

  ```agda
  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 `()`)

  ```haskell
  (\_ _ -> return)
    : forall a. () -> forall b. () -> b -> AgdaIO a b
    = T [[ ∀ {a} {A : Set a} → A → IO A ]]
  ```

  and

  ```haskell
  (\_ _ _ _ -> (>>=))
    : 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:

  ```agda
  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:

  ```agda
  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:

  ```agda
  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:

  ```agda
  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:

  ```agda
  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:

  ```agda
  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:

  ```javascript
  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:

  ```agda
  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" `OPTIONS`
  pragmas, some of which are known to make Agda inconsistent.

  Rejected pragmas:

  ```
  --allow-unsolved-metas
  --experimental-irrelevance
  --guardedness-preserving-type-construtors
  --injective-type-constructors
  --no-coverage-check
  --no-positivity-check
  --no-termination-check
  --sized-types
  --type-in-type
  ```

  Note that, at the moment, it is not possible to define the universe
  level or coinduction primitives when `--safe` is used (because they
  must be introduced as postulates). This can be worked around by
  type-checking trusted files in a first pass, without using `--safe`,
  and then using `--saf`e in a second pass. Modules which have already
  been type-checked are not re-type-checked just because `--safe` is
  used.

* Dependency graphs.

  The new flag `--dependency-graph=FILE` can be used to generate a DOT
  file containing a module dependency graph. The generated file (FILE)
  can be rendered using a tool like dot.

* The `--no-unreachable-check` flag has been removed.

* Projection functions are highlighted as functions instead of as
  fields. Field names (in record definitions and record values) are
  still highlighted as fields.

* Support for jumping to positions mentioned in the information
  buffer has been added.

* The `make install` command no longer installs Agda globally (by
  default).