Pragmas

See the official user manual for the most up-to-date version of the information on this page.

TOC

Index of pragmas

Index of options

The table below summarize the options available in Agda 2.3.2.1. You can also use agda --help for the latest list. We only list those options that can be given in .agda files in the {-# OPTIONS ... #-} form.

OPTIONS

 {-# OPTIONS -opt1 --opt2 ... #-}

Passes command-line options to Agda.

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 B

  syntax Σ A (λ x → B) = Σ[ x ∶ A ] B

  -- Corecursive definition of the W-type.

  W : (A : Set) → (A → Set) → Set
  W A B = Rec (♯ (Σ[ x ∶ A ] (B x → W A B)))

  syntax W A (λ x → B) = W[ x ∶ A ] B

  sup : {A : Set} {B : A → Set} (x : A) (f : B x → W A B) → W A B
  sup x f = fold (x , f)

  W-rec : {A : Set} {B : A → Set}
          (P : W A B → Set) →
          (∀ {x} {f : B x → W A B} → (∀ y → P (f y)) → P (sup x f)) →
          ∀ x → P x
  W-rec P h (fold (x , f)) = h (λ y → W-rec P h (f y))

  -- Induction-recursion encoded as corecursion-recursion.

  data Label : Set where
    ′0 ′1 ′2 ′σ ′π ′w : Label

  mutual

    U : Set
    U = Σ Label U′

    U′ : Label → Set
    U′ ′0 = ⊤
    U′ ′1 = ⊤
    U′ ′2 = ⊤
    U′ ′σ = Rec (♯ (Σ[ a ∶ U ] (El a → U)))
    U′ ′π = Rec (♯ (Σ[ a ∶ U ] (El a → U)))
    U′ ′w = Rec (♯ (Σ[ a ∶ U ] (El a → U)))

    El : U → Set
    El (′0 , _)            = ⊥
    El (′1 , _)            = ⊤
    El (′2 , _)            = Bool
    El (′σ , fold (a , b)) = Σ[ x ∶ El a ]  El (b x)
    El (′π , fold (a , b)) =   (x : El a) → El (b x)
    El (′w , fold (a , b)) = W[ x ∶ El a ]  El (b x)

  U-rec : (P : ∀ u → El u → Set) →
          P (′1 , _) tt →
          P (′2 , _) true →
          P (′2 , _) false →
          (∀ {a b x y} →
           P a x → P (b x) y → P (′σ , fold (a , b)) (x , y)) →
          (∀ {a b f} →
           (∀ x → P (b x) (f x)) → P (′π , fold (a , b)) f) →
          (∀ {a b x f} →
           (∀ y → P (′w , fold (a , b)) (f y)) →
           P (′w , fold (a , b)) (sup x f)) →
          ∀ u (x : El u) → P u x
  U-rec P P1 P2t P2f Pσ Pπ Pw = rec
    where
    rec : ∀ u (x : El u) → P u x
    rec (′0 , _)            ()
    rec (′1 , _)            _              = P1
    rec (′2 , _)            true           = P2t
    rec (′2 , _)            false          = P2f
    rec (′σ , fold (a , b)) (x , y)        = Pσ (rec _ x) (rec _ y)
    rec (′π , fold (a , b)) f              = Pπ (λ x → rec _ (f x))
    rec (′w , fold (a , b)) (fold (x , f)) = Pw (λ y → rec _ (f y))

The --guardedness-preserving-type-constructors extension is based on a rather operational understanding of ∞/♯_; it's not yet clear if this extension is consistent.

others

TODO