------------------------------------------------------------------------
-- 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 -> A
open 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 : B
record 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 A
If 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 /lib/prim,
and this directory is supposed to contain a subdirectory Agda
containing a file Primitive.agda.
The standard location of 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 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]
f args = rhs
module _ telescope where
body
means the following (not proper Agda code, since you cannot put a
module in-between clauses)
module _ {arg-telescope} telescope where
body
f args = rhs
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.
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.
* 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 . .lagda
cd latex
pdflatex .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 .lagda
cd latex
xelatex .tex (or lualatex .tex)
If symbols are missing or xelatex/lualatex complains about the font
missing, try setting a different font using:
\setmathfont{}
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):
1. Skipping a single definition: before type signature.
{-# NO_TERMINATION_CHECK #-}
a : A
a = a
2. Skipping a single definition: before first clause.
b : A
{-# NO_TERMINATION_CHECK #-}
b = b
3. Skipping an old-style mutual block: Before 'mutual' keyword.
{-# NO_TERMINATION_CHECK #-}
mutual
c : A
c = d
d : A
d = c
4. 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 !! i
In 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 fst
test : (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} = B
Note 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 . .lagda
pdflatex latex/.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 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
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:
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:
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:
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= .agda
Each source .agda is compiled into an ECMAScript target
/jAgda..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("") 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 ≡ x
Then 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 P
elem : ∀ {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 elem
Now 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= --compile-dir= .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 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.
------------------------------------------------------------------------
-- 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 proj₁
In 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 ≡ x
can 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 write
f 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 #-}
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 /A/B/C.agda, then the root directory is
.)
* 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
now translates to
import A
open A
instead of
import A
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.
http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Docs.FFI
* The tokens forall, -> and \ can be written as ∀, → and λ.
* Absurd lambdas: λ () and λ {}.
http://thread.gmane.org/gmane.comp.lang.agda/440
* 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.
http://thread.gmane.org/gmane.comp.lang.agda/720
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.
http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode
* 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.
http://thread.gmane.org/gmane.comp.lang.agda/245
* Reload times when working on large projects are now considerably
better.
http://thread.gmane.org/gmane.comp.lang.agda/551
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.
http://wiki.portal.chalmers.se/agda/
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.
http://hackage.haskell.org/