27 Sep 22:37 2010

## ANNOUNCE: Agda 2.2.8

Hi, Agda 2.2.8 has now been released. 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 proj₁ map : {A B : Set} {P : A → Set} {Q : B → Set} (f : A → B) → (∀ {x} → P x → Q (f x)) → Σ A P → Σ B Q map f g (x , y) = (f x , g y) The clause above is internally translated into the following one: map f g p = (f (Σ.proj₁ p) , g (Σ.proj₂ p)) Record patterns containing data type patterns are not translated. Example: add : ℕ × ℕ → ℕ add (zero , n) = n add (suc m , n) = suc (add (m , n)) Record patterns which do not contain data type patterns, but which do contain dot patterns, are currently rejected. Example: Foo : {A : Set} (p₁ p₂ : A × A) → proj₁ p₁ ≡ proj₁ p₂ → Set₁ Foo (x , y) (.x , y′) refl = Set * Proof irrelevant function types. Agda now supports irrelevant non-dependent function types: f : .A → B This type implies that f does not depend computationally on its argument. One intended use case is data structures with embedded proofs, like sorted lists: postulate _≤_ : ℕ → ℕ → Set p₁ : 0 ≤ 1 p₂ : 0 ≤ 1 data SList (bound : ℕ) : Set where [] : SList bound scons : (head : ℕ) → .(head ≤ bound) → (tail : SList head) → SList bound The effect of the irrelevant type in the signature of scons is that scons's second argument is never inspected after Agda has ensured that it has the right type. It is even thrown away, leading to smaller term sizes and hopefully some gain in efficiency. The type-checker ignores irrelevant arguments when checking equality, so two lists can be equal even if they contain different proofs: l₁ : SList 1 l₁ = scons 0 p₁ [] l₂ : SList 1 l₂ = scons 0 p₂ [] l₁≡l₂ : l₁ ≡ l₂ l₁≡l₂ = refl Irrelevant arguments can only be used in irrelevant contexts. Consider the following subset type: data Subset (A : Set) (P : A → Set) : Set where _#_ : (elem : A) → .(P elem) → Subset A P The following two uses are fine: elimSubset : ∀ {A C : Set} {P} → Subset A P → ((a : A) → .(P a) → C) → C elimSubset (a # p) k = k a p elem : {A : Set} {P : A → Set} → Subset A P → A elem (x # p) = x However, if we try to project out the proof component, then Agda complains that "variable p is declared irrelevant, so it cannot be used here": prjProof : ∀ {A P} (x : Subset A P) → P (elem x) prjProof (a # p) = p Matching against irrelevant arguments is also forbidden, except in the case of irrefutable matches (record constructor patterns which have been translated away). For instance, the match against the pattern (p , q) here is accepted: elim₂ : ∀ {A C : Set} {P Q : A → Set} → Subset A (λ x → Σ (P x) (λ _ → Q x)) → ((a : A) → .(P a) → .(Q a) → C) → C elim₂ (a # (p , q)) k = k a p q Absurd matches () are also allowed. Note that record fields can also be irrelevant. Example: record Subset (A : Set) (P : A → Set) : Set where constructor _#_ field elem : A .proof : P elem Irrelevant fields are never in scope, neither inside nor outside the record. This means that no record field can depend on an irrelevant field, and furthermore projections are not defined for such fields. Irrelevant fields can only be accessed using pattern matching, as in elimSubset above. Irrelevant function types were added very recently, and have not been subjected to much experimentation yet, so do not be surprised if something is changed before the next release. For instance, dependent irrelevant function spaces (.(x : A) → B) might be added in the future. * Mixfix binders. It is now possible to declare user-defined syntax that binds identifiers. Example: postulate State : Set → Set → Set put : ∀ {S} → S → State S ⊤ get : ∀ {S} → State S S return : ∀ {A S} → A → State S A bind : ∀ {A B S} → State S B → (B → State S A) → State S A syntax bind e₁ (λ x → e₂) = x ← e₁ , e₂ increment : State ℕ ⊤ increment = x ← get , put (1 + x) The syntax declaration for bind implies that x is in scope in e₂, but not in e₁. You can give fixity declarations along with syntax declarations: infixr 40 bind syntax bind e₁ (λ x → e₂) = x ← e₁ , e₂ The fixity applies to the syntax, not the name; syntax declarations are also restricted to ordinary, non-operator names. The following declaration is disallowed: syntax _==_ x y = x === y Syntax 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 B syntax Σ A (λ x → B) = Σ[ x ∶ A ] B -- Corecursive definition of the W-type. W : (A : Set) → (A → Set) → Set W A B = Rec (♯ (Σ[ x ∶ A ] (B x → W A B))) syntax W A (λ x → B) = W[ x ∶ A ] B sup : {A : Set} {B : A → Set} (x : A) (f : B x → W A B) → W A B sup x f = fold (x , f) W-rec : {A : Set} {B : A → Set} (P : W A B → Set) → (∀ {x} {f : B x → W A B} → (∀ y → P (f y)) → P (sup x f)) → ∀ x → P x W-rec P h (fold (x , f)) = h (λ y → W-rec P h (f y)) -- Induction-recursion encoded as corecursion-recursion. data Label : Set where ′0 ′1 ′2 ′σ ′π ′w : Label mutual U : Set U = Σ Label U′ U′ : Label → Set U′ ′0 = ⊤ U′ ′1 = ⊤ U′ ′2 = ⊤ U′ ′σ = Rec (♯ (Σ[ a ∶ U ] (El a → U))) U′ ′π = Rec (♯ (Σ[ a ∶ U ] (El a → U))) U′ ′w = Rec (♯ (Σ[ a ∶ U ] (El a → U))) El : U → Set El (′0 , _) = ⊥ El (′1 , _) = ⊤ El (′2 , _) = Bool El (′σ , fold (a , b)) = Σ[ x ∶ El a ] El (b x) El (′π , fold (a , b)) = (x : El a) → El (b x) El (′w , fold (a , b)) = W[ x ∶ El a ] El (b x) U-rec : (P : ∀ u → El u → Set) → P (′1 , _) tt → P (′2 , _) true → P (′2 , _) false → (∀ {a b x y} → P a x → P (b x) y → P (′σ , fold (a , b)) (x , y)) → (∀ {a b f} → (∀ x → P (b x) (f x)) → P (′π , fold (a , b)) f) → (∀ {a b x f} → (∀ y → P (′w , fold (a , b)) (f y)) → P (′w , fold (a , b)) (sup x f)) → ∀ u (x : El u) → P u x U-rec P P1 P2t P2f Pσ Pπ Pw = rec where rec : ∀ u (x : El u) → P u x rec (′0 , _) () rec (′1 , _) _ = P1 rec (′2 , _) true = P2t rec (′2 , _) false = P2f rec (′σ , fold (a , b)) (x , y) = Pσ (rec _ x) (rec _ y) rec (′π , fold (a , b)) f = Pπ (λ x → rec _ (f x)) rec (′w , fold (a , b)) (fold (x , f)) = Pw (λ y → rec _ (f y)) The --guardedness-preserving-type-constructors extension is based on a rather operational understanding of ∞/♯_; it's not yet clear if this extension is consistent. * Qualified constructors. Constructors can now be referred to qualified by their data type. For instance, given data Nat : Set where zero : Nat suc : Nat → Nat data Fin : Nat → Set where zero : ∀ {n} → Fin (suc n) suc : ∀ {n} → Fin n → Fin (suc n) you can refer to the constructors unambiguously as Nat.zero, Nat.suc, Fin.zero, and Fin.suc (Nat and Fin are modules containing the respective constructors). Example: inj : (n m : Nat) → Nat.suc n ≡ suc m → n ≡ m inj .m m refl = refl Previously you had to write something like inj : (n m : Nat) → _≡_ {Nat} (suc n) (suc m) → n ≡ m to make the type checker able to figure out that you wanted the natural number suc in this case. * Reflection. There are two new constructs for reflection: - quoteGoal x in e In e the value of x will be a representation of the goal type (the type expected of the whole expression) as an element in a datatype of Agda terms (see below). For instance, example : ℕ example = quoteGoal x in {! at this point x = def (quote ℕ) [] !} - quote x : Name If x is the name of a definition (function, datatype, record, or a constructor), quote x gives you the representation of x as a value in the primitive type Name (see below). Quoted terms use the following BUILTINs and primitives (available from the standard library module Reflection): -- The type of Agda names. postulate Name : Set {-# BUILTIN QNAME Name #-} primitive primQNameEquality : Name → Name → Bool -- Arguments. Explicit? = Bool data Arg A : Set where arg : Explicit? → A → Arg A {-# BUILTIN ARG Arg #-} {-# BUILTIN ARGARG arg #-} -- The type of Agda terms. data Term : Set where var : ℕ → List (Arg Term) → Term con : Name → List (Arg Term) → Term def : Name → List (Arg Term) → Term lam : Explicit? → Term → Term pi : Arg Term → Term → Term sort : Term unknown : Term {-# BUILTIN AGDATERM Term #-} {-# BUILTIN AGDATERMVAR var #-} {-# BUILTIN AGDATERMCON con #-} {-# BUILTIN AGDATERMDEF def #-} {-# BUILTIN AGDATERMLAM lam #-} {-# BUILTIN AGDATERMPI pi #-} {-# BUILTIN AGDATERMSORT sort #-} {-# BUILTIN AGDATERMUNSUPPORTED unknown #-} Reflection may be useful when working with internal decision procedures, such as the standard library's ring solver. * Minor record definition improvement. The definition of a record type is now available when type checking record module definitions. This means that you can define things like the following: record Cat : Set₁ where field Obj : Set _=>_ : Obj → Obj → Set -- ... -- not possible before: op : Cat op = record { Obj = Obj; _=>_ = λ A B → B => A } Tools ----- * The "Goal type and context" command now shows the goal type before the context, and the context is shown in reverse order. The "Goal type, context and inferred type" command has been modified in a similar way. * Show module contents command. Given a module name M the Emacs mode can now display all the top-level modules and names inside M, along with types for the names. The command is activated using C-c C-o or the menus. * Auto command. A command which searches for type inhabitants has been added. The command is invoked by pressing C-C C-a (or using the goal menu). There are several flags and parameters, e.g. '-c' which enables case-splitting in the search. For further information, see the Agda wiki: http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Main.Auto * HTML generation is now possible for a module with unsolved meta-variables, provided that the --allow-unsolved-metas flag is used. -- /NAD