As with all known Haskell systems, GHC implements some extensions to the standard Haskell language. They can all be enabled or disabled by command line flags or language pragmas. By default GHC understands the most recent Haskell version it supports, plus a handful of extensions.

Some of the Glasgow extensions serve to give you access to the underlying facilities with which we implement Haskell. Thus, you can get at the Raw Iron, if you are willing to write some non-portable code at a more primitive level. You need not be “stuck” on performance because of the implementation costs of Haskell’s “high-level” features—you can always code “under” them. In an extreme case, you can write all your time-critical code in C, and then just glue it together with Haskell!

Before you get too carried away working at the lowest level (e.g., sloshing MutableByteArray#s around your program), you may wish to check if there are libraries that provide a “Haskellised veneer” over the features you want. The separate libraries documentation describes all the libraries that come with GHC.

9.1. Language options

The language option flags control what variation of the language are permitted.

Language options can be controlled in two ways:

• Every language option can switched on by a command-line flag “-X...” (e.g. -XTemplateHaskell), and switched off by the flag “-XNo...”; (e.g. -XNoTemplateHaskell).
• Language options recognised by Cabal can also be enabled using the LANGUAGE pragma, thus {-# LANGUAGE TemplateHaskell #-} (see LANGUAGE pragma).

The flag -fglasgow-exts -fglasgow-exts is equivalent to enabling the following extensions:

-XConstrainedClassMethods -XDeriveDataTypeable -XDeriveFoldable -XDeriveFunctor -XDeriveGeneric -XDeriveTraversable -XEmptyDataDecls -XExistentialQuantification -XExplicitNamespaces -XFlexibleContexts -XFlexibleInstances -XForeignFunctionInterface -XFunctionalDependencies -XGeneralizedNewtypeDeriving -XImplicitParams -XKindSignatures

-XLiberalTypeSynonyms -XMagicHash -XMultiParamTypeClasses -XParallelListComp -XPatternGuards -XPostfixOperators -XRankNTypes -XRecursiveDo -XScopedTypeVariables -XStandaloneDeriving -XTypeOperators -XTypeSynonymInstances -XUnboxedTuples -XUnicodeSyntax -XUnliftedFFITypes

Enabling these options is the only effect of -fglasgow-exts. We are trying to move away from this portmanteau flag, and towards enabling features individually.

9.2. Unboxed types and primitive operations

GHC is built on a raft of primitive data types and operations; “primitive” in the sense that they cannot be defined in Haskell itself. While you really can use this stuff to write fast code, we generally find it a lot less painful, and more satisfying in the long run, to use higher-level language features and libraries. With any luck, the code you write will be optimised to the efficient unboxed version in any case. And if it isn’t, we’d like to know about it.

All these primitive data types and operations are exported by the library GHC.Prim, for which there is detailed online documentation. (This documentation is generated from the file compiler/prelude/primops.txt.pp.)

If you want to mention any of the primitive data types or operations in your program, you must first import GHC.Prim to bring them into scope. Many of them have names ending in #, and to mention such names you need the -XMagicHash extension (The magic hash).

The primops make extensive use of unboxed types and unboxed tuples, which we briefly summarise here.

9.2.1. Unboxed types

Unboxed types (Glasgow extension)

Most types in GHC are boxed, which means that values of that type are represented by a pointer to a heap object. The representation of a Haskell Int, for example, is a two-word heap object. An unboxed type, however, is represented by the value itself, no pointers or heap allocation are involved.

Unboxed types correspond to the “raw machine” types you would use in C: Int# (long int), Double# (double), Addr# (void *), etc. The primitive operations (PrimOps) on these types are what you might expect; e.g., (+#) is addition on Int#s, and is the machine-addition that we all know and love—usually one instruction.

Primitive (unboxed) types cannot be defined in Haskell, and are therefore built into the language and compiler. Primitive types are always unlifted; that is, a value of a primitive type cannot be bottom. We use the convention (but it is only a convention) that primitive types, values, and operations have a # suffix (see The magic hash). For some primitive types we have special syntax for literals, also described in the same section.

Primitive values are often represented by a simple bit-pattern, such as Int#, Float#, Double#. But this is not necessarily the case: a primitive value might be represented by a pointer to a heap-allocated object. Examples include Array#, the type of primitive arrays. A primitive array is heap-allocated because it is too big a value to fit in a register, and would be too expensive to copy around; in a sense, it is accidental that it is represented by a pointer. If a pointer represents a primitive value, then it really does point to that value: no unevaluated thunks, no indirections…nothing can be at the other end of the pointer than the primitive value. A numerically-intensive program using unboxed types can go a lot faster than its “standard” counterpart—we saw a threefold speedup on one example.

There are some restrictions on the use of primitive types:

• The main restriction is that you can’t pass a primitive value to a polymorphic function or store one in a polymorphic data type. This rules out things like [Int#] (i.e. lists of primitive integers). The reason for this restriction is that polymorphic arguments and constructor fields are assumed to be pointers: if an unboxed integer is stored in one of these, the garbage collector would attempt to follow it, leading to unpredictable space leaks. Or a seq operation on the polymorphic component may attempt to dereference the pointer, with disastrous results. Even worse, the unboxed value might be larger than a pointer (Double# for instance).

• You cannot define a newtype whose representation type (the argument type of the data constructor) is an unboxed type. Thus, this is illegal:

  newtype A = MkA Int#
  

• You cannot bind a variable with an unboxed type in a top-level binding.

• You cannot bind a variable with an unboxed type in a recursive binding.

• You may bind unboxed variables in a (non-recursive, non-top-level) pattern binding, but you must make any such pattern-match strict. For example, rather than:

  data Foo = Foo Int Int#
  
  f x = let (Foo a b, w) = ..rhs.. in ..body..
  

  you must write:

  data Foo = Foo Int Int#
  
  f x = let !(Foo a b, w) = ..rhs.. in ..body..
  

  since b has type Int#.

9.2.2. Unboxed tuples

Unboxed tuples aren’t really exported by GHC.Exts; they are a syntactic extension enabled by the language flag -XUnboxedTuples. An unboxed tuple looks like this:

(# e_1, ..., e_n #)

where e_1..e_n are expressions of any type (primitive or non-primitive). The type of an unboxed tuple looks the same.

Note that when unboxed tuples are enabled, (# is a single lexeme, so for example when using operators like # and #- you need to write ( # ) and ( #- ) rather than (#) and (#-).

Unboxed tuples are used for functions that need to return multiple values, but they avoid the heap allocation normally associated with using fully-fledged tuples. When an unboxed tuple is returned, the components are put directly into registers or on the stack; the unboxed tuple itself does not have a composite representation. Many of the primitive operations listed in primops.txt.pp return unboxed tuples. In particular, the IO and ST monads use unboxed tuples to avoid unnecessary allocation during sequences of operations.

There are some restrictions on the use of unboxed tuples:

• Values of unboxed tuple types are subject to the same restrictions as other unboxed types; i.e. they may not be stored in polymorphic data structures or passed to polymorphic functions.

• The typical use of unboxed tuples is simply to return multiple values, binding those multiple results with a case expression, thus:

  f x y = (# x+1, y-1 #)
  g x = case f x x of { (# a, b #) -> a + b }
  

  You can have an unboxed tuple in a pattern binding, thus

  f x = let (# p,q #) = h x in ..body..
  

  If the types of p and q are not unboxed, the resulting binding is lazy like any other Haskell pattern binding. The above example desugars like this:

  f x = let t = case h x of { (# p,q #) -> (p,q) }
            p = fst t
            q = snd t
        in ..body..
  

  Indeed, the bindings can even be recursive.

9.3. Syntactic extensions

9.3.1. Unicode syntax

The language extension -XUnicodeSyntax-XUnicodeSyntax enables Unicode characters to be used to stand for certain ASCII character sequences. The following alternatives are provided:

ASCII	Unicode alternative	Code point	Name
::	∷	0x2237	PROPORTION
=>	⇒	0x21D2	RIGHTWARDS DOUBLE ARROW
->	→	0x2192	RIGHTWARDS ARROW
<-	←	0x2190	LEFTWARDS ARROW
>-	⤚	0x291a	RIGHTWARDS ARROW-TAIL
-<	⤙	0x2919	LEFTWARDS ARROW-TAIL
>>-	⤜	0x291C	RIGHTWARDS DOUBLE ARROW-TAIL
-<<	⤛	0x291B	LEFTWARDS DOUBLE ARROW-TAIL
*	★	0x2605	BLACK STAR
forall	∀	0x2200	FOR ALL

9.3.2. The magic hash

The language extension -XMagicHash allows “#” as a postfix modifier to identifiers. Thus, “x#” is a valid variable, and “T#” is a valid type constructor or data constructor.

The hash sign does not change semantics at all. We tend to use variable names ending in “#” for unboxed values or types (e.g. Int#), but there is no requirement to do so; they are just plain ordinary variables. Nor does the -XMagicHash extension bring anything into scope. For example, to bring Int# into scope you must import GHC.Prim (see Unboxed types and primitive operations); the -XMagicHash extension then allows you to refer to the Int# that is now in scope. Note that with this option, the meaning of x#y = 0 is changed: it defines a function x# taking a single argument y; to define the operator #, put a space: x # y = 0.

The -XMagicHash also enables some new forms of literals (see Unboxed types):

• 'x'# has type Char#
• "foo"# has type Addr#
• 3# has type Int#. In general, any Haskell integer lexeme followed by a # is an Int# literal, e.g. -0x3A# as well as 32#.
• 3## has type Word#. In general, any non-negative Haskell integer lexeme followed by ## is a Word#.
• 3.2# has type Float#.
• 3.2## has type Double#

9.3.3. Negative literals

The literal -123 is, according to Haskell98 and Haskell 2010, desugared as negate (fromInteger 123). The language extension -XNegativeLiterals means that it is instead desugared as fromInteger (-123).

This can make a difference when the positive and negative range of a numeric data type don’t match up. For example, in 8-bit arithmetic -128 is representable, but +128 is not. So negate (fromInteger 128) will elicit an unexpected integer-literal-overflow message.

9.3.4. Fractional looking integer literals

Haskell 2010 and Haskell 98 define floating literals with the syntax 1.2e6. These literals have the type Fractional a => a.

The language extension -XNumDecimals allows you to also use the floating literal syntax for instances of Integral, and have values like (1.2e6 :: Num a => a)

9.3.5. Binary integer literals

Haskell 2010 and Haskell 98 allows for integer literals to be given in decimal, octal (prefixed by 0o or 0O), or hexadecimal notation (prefixed by 0x or 0X).

The language extension -XBinaryLiterals adds support for expressing integer literals in binary notation with the prefix 0b or 0B. For instance, the binary integer literal 0b11001001 will be desugared into fromInteger 201 when -XBinaryLiterals is enabled.

9.3.6. Hierarchical Modules

GHC supports a small extension to the syntax of module names: a module name is allowed to contain a dot ‘.’. This is also known as the “hierarchical module namespace” extension, because it extends the normally flat Haskell module namespace into a more flexible hierarchy of modules.

This extension has very little impact on the language itself; modules names are always fully qualified, so you can just think of the fully qualified module name as “the module name”. In particular, this means that the full module name must be given after the module keyword at the beginning of the module; for example, the module A.B.C must begin

module A.B.C

It is a common strategy to use the as keyword to save some typing when using qualified names with hierarchical modules. For example:

import qualified Control.Monad.ST.Strict as ST

For details on how GHC searches for source and interface files in the presence of hierarchical modules, see The search path.

GHC comes with a large collection of libraries arranged hierarchically; see the accompanying library documentation. More libraries to install are available from HackageDB.

9.3.7. Pattern guards

Pattern guards (Glasgow extension) The discussion that follows is an abbreviated version of Simon Peyton Jones’s original proposal. (Note that the proposal was written before pattern guards were implemented, so refers to them as unimplemented.)

Suppose we have an abstract data type of finite maps, with a lookup operation:

lookup :: FiniteMap -> Int -> Maybe Int

The lookup returns Nothing if the supplied key is not in the domain of the mapping, and (Just v) otherwise, where v is the value that the key maps to. Now consider the following definition:

clunky env var1 var2 | ok1 && ok2 = val1 + val2
| otherwise  = var1 + var2
where
  m1 = lookup env var1
  m2 = lookup env var2
  ok1 = maybeToBool m1
  ok2 = maybeToBool m2
  val1 = expectJust m1
  val2 = expectJust m2

The auxiliary functions are

maybeToBool :: Maybe a -> Bool
maybeToBool (Just x) = True
maybeToBool Nothing  = False

expectJust :: Maybe a -> a
expectJust (Just x) = x
expectJust Nothing  = error "Unexpected Nothing"

What is clunky doing? The guard ok1 && ok2 checks that both lookups succeed, using maybeToBool to convert the Maybe types to booleans. The (lazily evaluated) expectJust calls extract the values from the results of the lookups, and binds the returned values to val1 and val2 respectively. If either lookup fails, then clunky takes the otherwise case and returns the sum of its arguments.

This is certainly legal Haskell, but it is a tremendously verbose and un-obvious way to achieve the desired effect. Arguably, a more direct way to write clunky would be to use case expressions:

clunky env var1 var2 = case lookup env var1 of
  Nothing -> fail
  Just val1 -> case lookup env var2 of
    Nothing -> fail
    Just val2 -> val1 + val2
where
  fail = var1 + var2

This is a bit shorter, but hardly better. Of course, we can rewrite any set of pattern-matching, guarded equations as case expressions; that is precisely what the compiler does when compiling equations! The reason that Haskell provides guarded equations is because they allow us to write down the cases we want to consider, one at a time, independently of each other. This structure is hidden in the case version. Two of the right-hand sides are really the same (fail), and the whole expression tends to become more and more indented.

Here is how I would write clunky:

clunky env var1 var2
  | Just val1 <- lookup env var1
  , Just val2 <- lookup env var2
  = val1 + val2
...other equations for clunky...

The semantics should be clear enough. The qualifiers are matched in order. For a <- qualifier, which I call a pattern guard, the right hand side is evaluated and matched against the pattern on the left. If the match fails then the whole guard fails and the next equation is tried. If it succeeds, then the appropriate binding takes place, and the next qualifier is matched, in the augmented environment. Unlike list comprehensions, however, the type of the expression to the right of the <- is the same as the type of the pattern to its left. The bindings introduced by pattern guards scope over all the remaining guard qualifiers, and over the right hand side of the equation.

Just as with list comprehensions, boolean expressions can be freely mixed with among the pattern guards. For example:

f x | [y] <- x
    , y > 3
    , Just z <- h y
    = ...

Haskell’s current guards therefore emerge as a special case, in which the qualifier list has just one element, a boolean expression.

9.3.8. View patterns

View patterns are enabled by the flag -XViewPatterns. More information and examples of view patterns can be found on the Wiki page.

View patterns are somewhat like pattern guards that can be nested inside of other patterns. They are a convenient way of pattern-matching against values of abstract types. For example, in a programming language implementation, we might represent the syntax of the types of the language as follows:

type Typ

data TypView = Unit
             | Arrow Typ Typ

view :: Typ -> TypView

-- additional operations for constructing Typ's ...

The representation of Typ is held abstract, permitting implementations to use a fancy representation (e.g., hash-consing to manage sharing). Without view patterns, using this signature a little inconvenient:

size :: Typ -> Integer
size t = case view t of
  Unit -> 1
  Arrow t1 t2 -> size t1 + size t2

It is necessary to iterate the case, rather than using an equational function definition. And the situation is even worse when the matching against t is buried deep inside another pattern.

View patterns permit calling the view function inside the pattern and matching against the result:

size (view -> Unit) = 1
size (view -> Arrow t1 t2) = size t1 + size t2

That is, we add a new form of pattern, written ⟨expression⟩ -> ⟨pattern⟩ that means “apply the expression to whatever we’re trying to match against, and then match the result of that application against the pattern”. The expression can be any Haskell expression of function type, and view patterns can be used wherever patterns are used.

The semantics of a pattern ( ⟨exp⟩ -> ⟨pat⟩ ) are as follows:

• Scoping: The variables bound by the view pattern are the variables bound by ⟨pat⟩.

  Any variables in ⟨exp⟩ are bound occurrences, but variables bound “to the left” in a pattern are in scope. This feature permits, for example, one argument to a function to be used in the view of another argument. For example, the function clunky from Pattern guards can be written using view patterns as follows:

  clunky env (lookup env -> Just val1) (lookup env -> Just val2) = val1 + val2
  ...other equations for clunky...
  

  More precisely, the scoping rules are:

  • In a single pattern, variables bound by patterns to the left of a view pattern expression are in scope. For example:

    example :: Maybe ((String -> Integer,Integer), String) -> Bool
    example Just ((f,_), f -> 4) = True
    

    Additionally, in function definitions, variables bound by matching earlier curried arguments may be used in view pattern expressions in later arguments:

    example :: (String -> Integer) -> String -> Bool
    example f (f -> 4) = True
    

    That is, the scoping is the same as it would be if the curried arguments were collected into a tuple.

  • In mutually recursive bindings, such as let, where, or the top level, view patterns in one declaration may not mention variables bound by other declarations. That is, each declaration must be self-contained. For example, the following program is not allowed:

    let {(x -> y) = e1 ;
         (y -> x) = e2 } in x
    

  (For some amplification on this design choice see Trac #4061.

• Typing: If ⟨exp⟩ has type ⟨T1⟩ -> ⟨T2⟩ and ⟨pat⟩ matches a ⟨T2⟩, then the whole view pattern matches a ⟨T1⟩.

• Matching: To the equations in Section 3.17.3 of the Haskell 98 Report, add the following:

  case v of { (e -> p) -> e1 ; _ -> e2 }
   =
  case (e v) of { p -> e1 ; _ -> e2 }
  

  That is, to match a variable ⟨v⟩ against a pattern ( ⟨exp⟩ -> ⟨pat⟩ ), evaluate ( ⟨exp⟩ ⟨v⟩ ) and match the result against ⟨pat⟩.

• Efficiency: When the same view function is applied in multiple branches of a function definition or a case expression (e.g., in size above), GHC makes an attempt to collect these applications into a single nested case expression, so that the view function is only applied once. Pattern compilation in GHC follows the matrix algorithm described in Chapter 4 of The Implementation of Functional Programming Languages. When the top rows of the first column of a matrix are all view patterns with the “same” expression, these patterns are transformed into a single nested case. This includes, for example, adjacent view patterns that line up in a tuple, as in

  f ((view -> A, p1), p2) = e1
  f ((view -> B, p3), p4) = e2
  

  The current notion of when two view pattern expressions are “the same” is very restricted: it is not even full syntactic equality. However, it does include variables, literals, applications, and tuples; e.g., two instances of view ("hi", "there") will be collected. However, the current implementation does not compare up to alpha-equivalence, so two instances of (x, view x -> y) will not be coalesced.

9.3.9. Pattern synonyms

Pattern synonyms are enabled by the flag -XPatternSynonyms, which is required for defining them, but not for using them. More information and examples of view patterns can be found on the Wiki page <PatternSynonyms>.

Pattern synonyms enable giving names to parametrized pattern schemes. They can also be thought of as abstract constructors that don’t have a bearing on data representation. For example, in a programming language implementation, we might represent types of the language as follows:

data Type = App String [Type]

Here are some examples of using said representation. Consider a few types of the Type universe encoded like this:

App "->" [t1, t2]          -- t1 -> t2
App "Int" []               -- Int
App "Maybe" [App "Int" []] -- Maybe Int

This representation is very generic in that no types are given special treatment. However, some functions might need to handle some known types specially, for example the following two functions collect all argument types of (nested) arrow types, and recognize the Int type, respectively:

collectArgs :: Type -> [Type]
collectArgs (App "->" [t1, t2]) = t1 : collectArgs t2
collectArgs _                   = []

isInt :: Type -> Bool
isInt (App "Int" []) = True
isInt _              = False

Matching on App directly is both hard to read and error prone to write. And the situation is even worse when the matching is nested:

isIntEndo :: Type -> Bool
isIntEndo (App "->" [App "Int" [], App "Int" []]) = True
isIntEndo _                                       = False

Pattern synonyms permit abstracting from the representation to expose matchers that behave in a constructor-like manner with respect to pattern matching. We can create pattern synonyms for the known types we care about, without committing the representation to them (note that these don’t have to be defined in the same module as the Type type):

pattern Arrow t1 t2 = App "->"    [t1, t2]
pattern Int         = App "Int"   []
pattern Maybe t     = App "Maybe" [t]

Which enables us to rewrite our functions in a much cleaner style:

collectArgs :: Type -> [Type]
collectArgs (Arrow t1 t2) = t1 : collectArgs t2
collectArgs _             = []

isInt :: Type -> Bool
isInt Int = True
isInt _   = False

isIntEndo :: Type -> Bool
isIntEndo (Arrow Int Int) = True
isIntEndo _               = False

In general there are three kinds of pattern synonyms. Unidirectional, bidirectional and explicitly bidirectional. The examples given so far are examples of bidirectional pattern synonyms. A bidirectional synonym behaves the same as an ordinary data constructor. We can use it in a pattern context to deconstruct values and in an expression context to construct values. For example, we can construct the value intEndo using the pattern synonyms Arrow and Int as defined previously.

intEndo :: Type
intEndo = Arrow Int Int

This example is equivalent to the much more complicated construction if we had directly used the Type constructors.

intEndo :: Type
intEndo = App "->" [App "Int" [], App "Int" []]

Unidirectional synonyms can only be used in a pattern context and are defined as follows:

pattern Head x <- x:xs

In this case, Head ⟨x⟩ cannot be used in expressions, only patterns, since it wouldn’t specify a value for the ⟨xs⟩ on the right-hand side. However, we can define an explicitly bidirectional pattern synonym by separately specifying how to construct and deconstruct a type. The syntax for doing this is as follows:

pattern HeadC x <- x:xs where
  HeadC x = [x]

We can then use HeadC in both expression and pattern contexts. In a pattern context it will match the head of any list with length at least one. In an expression context it will construct a singleton list.

The table below summarises where each kind of pattern synonym can be used.

Context	Unidirectional	Bidirectional	Explicitly Bidirectional
Pattern	Yes	Yes	Yes
Expression	No	Yes (Inferred)	Yes (Explicit)

9.3.9.1. Record Pattern Synonyms

It is also possible to define pattern synonyms which behave just like record constructors. The syntax for doing this is as follows:

pattern Point :: (Int, Int)
pattern Point{x, y} = (x, y)

The idea is that we can then use Point just as if we had defined a new datatype MyPoint with two fields x and y.

data MyPoint = Point { x :: Int, y :: Int }

Whilst a normal pattern synonym can be used in two ways, there are then seven ways in which to use Point. Precisely the ways in which a normal record constructor can be used.

Usage	Example
As a constructor	zero = Point 0 0
As a constructor with record syntax	zero = Point { x = 0, y = 0}
In a pattern context	isZero (Point 0 0) = True
In a pattern context with record syntax	isZero (Point { x = 0, y = 0 }
In a pattern context with field puns	getX (Point {x}) = x
In a record update	(0, 0) { x = 1 } == (1,0)
Using record selectors	x (0,0) == 0

For a unidirectional record pattern synonym we define record selectors but do not allow record updates or construction.

The syntax and semantics of pattern synonyms are elaborated in the following subsections. See the Wiki page for more details.

9.3.9.2. Syntax and scoping of pattern synonyms

A pattern synonym declaration can be either unidirectional, bidirectional or explicitly bidirectional. The syntax for unidirectional pattern synonyms is:

pattern pat_lhs <- pat

the syntax for bidirectional pattern synonyms is:

pattern pat_lhs = pat

and the syntax for explicitly bidirectional pattern synonyms is:

pattern pat_lhs <- pat where
  pat_lhs = expr

We can define either prefix, infix or record pattern synonyms by modifying the form of pat_lhs. The syntax for these is as follows:

Prefix	Name args
Infix	arg1 `Name` arg2 or arg1 op arg2
Record	Name{arg1,arg2,...,argn}

Pattern synonym declarations can only occur in the top level of a module. In particular, they are not allowed as local definitions.

The variables in the left-hand side of the definition are bound by the pattern on the right-hand side. For bidirectional pattern synonyms, all the variables of the right-hand side must also occur on the left-hand side; also, wildcard patterns and view patterns are not allowed. For unidirectional and explicitly bidirectional pattern synonyms, there is no restriction on the right-hand side pattern.

Pattern synonyms cannot be defined recursively.

9.3.9.3. Import and export of pattern synonyms

The name of the pattern synonym is in the same namespace as proper data constructors. Like normal data constructors, pattern synonyms can be imported and exported through association with a type constructor or independently.

To export them on their own, in an export or import specification, you must prefix pattern names with the pattern keyword, e.g.:

module Example (pattern Zero) where

data MyNum = MkNum Int

pattern Zero :: MyNum
pattern Zero = MkNum 0

Without the pattern prefix, Zero would be interpreted as a type constructor in the export list.

You may also use the pattern keyword in an import/export specification to import or export an ordinary data constructor. For example:

import Data.Maybe( pattern Just )

would bring into scope the data constructor Just from the Maybe type, without also bringing the type constructor Maybe into scope.

To bundle a pattern synonym with a type constructor, we list the pattern synonym in the export list of a module which exports the type constructor. For example, to bundle Zero with MyNum we could write the following:

module Example ( MyNum(Zero) ) where

If a module was then to import MyNum from Example, it would also import the pattern synonym Zero.

It is also possible to use the special token .. in an export list to mean all currently bundled constructors. For example, we could write:

module Example ( MyNum(.., Zero) ) where

in which case, Example would export the type constructor MyNum with the data constructor MkNum and also the pattern synonym Zero.

Bundled patterns synoyms are type checked to ensure that they are of the same type as the type constructor which they are bundled with. A pattern synonym P can not be bundled with a type constructor T if P‘s type is visibly incompatible with T.

A module which imports MyNum(..) from Example and then re-exports MyNum(..) will also export any pattern synonyms bundled with MyNum in Example. A more complete specification can be found on the wiki.

9.3.9.4. Typing of pattern synonyms

Given a pattern synonym definition of the form

pattern P var1 var2 ... varN <- pat

it is assigned a pattern type of the form

pattern P :: CReq => CProf => t1 -> t2 -> ... -> tN -> t

where ⟨CProv⟩ and ⟨CReq⟩ are type contexts, and ⟨t1⟩, ⟨t2⟩, ..., ⟨tN⟩ and ⟨t⟩ are types. Notice the unusual form of the type, with two contexts ⟨CProv⟩ and ⟨CReq⟩:

• ⟨CProv⟩ are the constraints made available (provided) by a successful pattern match.
• ⟨CReq⟩ are the constraints required to match the pattern.

For example, consider

data T a where
  MkT :: (Show b) => a -> b -> T a

f1 :: (Eq a, Num a) => T a -> String
f1 (MkT 42 x) = show x

pattern ExNumPat :: (Num a, Eq a) => (Show b) => b -> T a
pattern ExNumPat x = MkT 42 x

f2 :: (Eq a, Num a) => T a -> String
f2 (ExNumPat x) = show x

Here f1 does not use pattern synonyms. To match against the numeric pattern 42 requires the caller to satisfy the constraints (Num a, Eq a), so they appear in f1‘s type. The call to show generates a (Show b) constraint, where b is an existentially type variable bound by the pattern match on MkT. But the same pattern match also provides the constraint (Show b) (see MkT‘s type), and so all is well.

Exactly the same reasoning applies to ExNumPat: matching against ExNumPat requires the constraints (Num a, Eq a), and provides the constraint (Show b).

Note also the following points

• In the common case where Prov is empty, (), it can be omitted altogether.

• You may specify an explicit pattern signature, as we did for ExNumPat above, to specify the type of a pattern, just as you can for a function. As usual, the type signature can be less polymorphic than the inferred type. For example

  -- Inferred type would be 'a -> [a]'
  pattern SinglePair :: (a, a) -> [(a, a)]
  pattern SinglePair x = [x]
  

• The GHCi :info command shows pattern types in this format.

• For a bidirectional pattern synonym, a use of the pattern synonym as an expression has the type

  (CReq, CProv) => t1 -> t2 -> ... -> tN -> t
  

  So in the previous example, when used in an expression, ExNumPat has type

  ExNumPat :: (Num a, Eq a, Show b) => b -> T t
  

  Notice that this is a tiny bit more restrictive than the expression MkT 42 x which would not require (Eq a).

• Consider these two pattern synonyms:

  data S a where
     S1 :: Bool -> S Bool
  
  pattern P1 :: Bool -> Maybe Bool
  pattern P1 b = Just b
  
  pattern P2 :: () => (b ~ Bool) => Bool -> S b
  pattern P2 b = S1 b
  
  f :: Maybe a -> String
  f (P1 x) = "no no no"     -- Type-incorrect
  
  g :: S a -> String
  g (P2 b) = "yes yes yes"  -- Fine
  

  Pattern P1 can only match against a value of type Maybe Bool, so function f is rejected because the type signature is Maybe a. (To see this, imagine expanding the pattern synonym.)

  On the other hand, function g works fine, because matching against P2 (which wraps the GADT S) provides the local equality (a~Bool). If you were to give an explicit pattern signature P2 :: Bool -> S Bool, then P2 would become less polymorphic, and would behave exactly like P1 so that g would then be rejected.

  In short, if you want GADT-like behaviour for pattern synonyms, then (unlike unlike concrete data constructors like S1) you must write its type with explicit provided equalities. For a concrete data constructor like S1 you can write its type signature as either S1 :: Bool -> S Bool or S1 :: (b~Bool) => Bool -> S b; the two are equivalent. Not so for pattern synonyms: the two forms are different, in order to distinguish the two cases above. (See Trac #9953 for discussion of this choice.)

9.3.9.5. Matching of pattern synonyms

A pattern synonym occurrence in a pattern is evaluated by first matching against the pattern synonym itself, and then on the argument patterns. For example, in the following program, f and f' are equivalent:

pattern Pair x y <- [x, y]

f (Pair True True) = True
f _                = False

f' [x, y] | True <- x, True <- y = True
f' _                              = False

Note that the strictness of f differs from that of g defined below:

g [True, True] = True
g _            = False

*Main> f (False:undefined)
*** Exception: Prelude.undefined
*Main> g (False:undefined)
False

9.3.10. n+k patterns

n+k pattern support is disabled by default. To enable it, you can use the -XNPlusKPatterns flag.

9.3.11. The recursive do-notation

The do-notation of Haskell 98 does not allow recursive bindings, that is, the variables bound in a do-expression are visible only in the textually following code block. Compare this to a let-expression, where bound variables are visible in the entire binding group.

It turns out that such recursive bindings do indeed make sense for a variety of monads, but not all. In particular, recursion in this sense requires a fixed-point operator for the underlying monad, captured by the mfix method of the MonadFix class, defined in Control.Monad.Fix as follows:

class Monad m => MonadFix m where
   mfix :: (a -> m a) -> m a

Haskell’s Maybe, [] (list), ST (both strict and lazy versions), IO, and many other monads have MonadFix instances. On the negative side, the continuation monad, with the signature (a -> r) -> r, does not.

For monads that do belong to the MonadFix class, GHC provides an extended version of the do-notation that allows recursive bindings. The -XRecursiveDo (language pragma: RecursiveDo) provides the necessary syntactic support, introducing the keywords mdo and rec for higher and lower levels of the notation respectively. Unlike bindings in a do expression, those introduced by mdo and rec are recursively defined, much like in an ordinary let-expression. Due to the new keyword mdo, we also call this notation the mdo-notation.

Here is a simple (albeit contrived) example:

{-# LANGUAGE RecursiveDo #-}
justOnes = mdo { xs <- Just (1:xs)
               ; return (map negate xs) }

or equivalently

{-# LANGUAGE RecursiveDo #-}
justOnes = do { rec { xs <- Just (1:xs) }
              ; return (map negate xs) }

As you can guess justOnes will evaluate to Just [-1,-1,-1,....

GHC’s implementation the mdo-notation closely follows the original translation as described in the paper A recursive do for Haskell, which in turn is based on the work Value Recursion in Monadic Computations. Furthermore, GHC extends the syntax described in the former paper with a lower level syntax flagged by the rec keyword, as we describe next.

9.3.11.1. Recursive binding groups

The flag -XRecursiveDo also introduces a new keyword rec, which wraps a mutually-recursive group of monadic statements inside a do expression, producing a single statement. Similar to a let statement inside a do, variables bound in the rec are visible throughout the rec group, and below it. For example, compare

do { a <- getChar            do { a <- getChar
   ; let { r1 = f a r2          ; rec { r1 <- f a r2
   ;     ; r2 = g r1 }          ;     ; r2 <- g r1 }
   ; return (r1 ++ r2) }        ; return (r1 ++ r2) }

In both cases, r1 and r2 are available both throughout the let or rec block, and in the statements that follow it. The difference is that let is non-monadic, while rec is monadic. (In Haskell let is really letrec, of course.)

The semantics of rec is fairly straightforward. Whenever GHC finds a rec group, it will compute its set of bound variables, and will introduce an appropriate call to the underlying monadic value-recursion operator mfix, belonging to the MonadFix class. Here is an example:

rec { b <- f a c     ===>    (b,c) <- mfix (\ ~(b,c) -> do { b <- f a c
    ; c <- f b a }                                         ; c <- f b a
                                                           ; return (b,c) })

As usual, the meta-variables b, c etc., can be arbitrary patterns. In general, the statement rec ss is desugared to the statement

vs <- mfix (\ ~vs -> do { ss; return vs })

where vs is a tuple of the variables bound by ss.

Note in particular that the translation for a rec block only involves wrapping a call to mfix: it performs no other analysis on the bindings. The latter is the task for the mdo notation, which is described next.

9.3.11.2. The mdo notation

A rec-block tells the compiler where precisely the recursive knot should be tied. It turns out that the placement of the recursive knots can be rather delicate: in particular, we would like the knots to be wrapped around as minimal groups as possible. This process is known as segmentation, and is described in detail in Section 3.2 of A recursive do for Haskell. Segmentation improves polymorphism and reduces the size of the recursive knot. Most importantly, it avoids unnecessary interference caused by a fundamental issue with the so-called right-shrinking axiom for monadic recursion. In brief, most monads of interest (IO, strict state, etc.) do not have recursion operators that satisfy this axiom, and thus not performing segmentation can cause unnecessary interference, changing the termination behavior of the resulting translation. (Details can be found in Sections 3.1 and 7.2.2 of Value Recursion in Monadic Computations.)

The mdo notation removes the burden of placing explicit rec blocks in the code. Unlike an ordinary do expression, in which variables bound by statements are only in scope for later statements, variables bound in an mdo expression are in scope for all statements of the expression. The compiler then automatically identifies minimal mutually recursively dependent segments of statements, treating them as if the user had wrapped a rec qualifier around them.

The definition is syntactic:

• A generator ⟨g⟩ depends on a textually following generator ⟨g’⟩, if
  • ⟨g’⟩ defines a variable that is used by ⟨g⟩, or
  • ⟨g’⟩ textually appears between ⟨g⟩ and ⟨g’‘⟩, where ⟨g⟩ depends on ⟨g’‘⟩.
• A segment of a given mdo-expression is a minimal sequence of generators such that no generator of the sequence depends on an outside generator. As a special case, although it is not a generator, the final expression in an mdo-expression is considered to form a segment by itself.

Segments in this sense are related to strongly-connected components analysis, with the exception that bindings in a segment cannot be reordered and must be contiguous.

Here is an example mdo-expression, and its translation to rec blocks:

mdo { a <- getChar      ===> do { a <- getChar
    ; b <- f a c                ; rec { b <- f a c
    ; c <- f b a                ;     ; c <- f b a }
    ; z <- h a b                ; z <- h a b
    ; d <- g d e                ; rec { d <- g d e
    ; e <- g a z                ;     ; e <- g a z }
    ; putChar c }               ; putChar c }

Note that a given mdo expression can cause the creation of multiple rec blocks. If there are no recursive dependencies, mdo will introduce no rec blocks. In this latter case an mdo expression is precisely the same as a do expression, as one would expect.

In summary, given an mdo expression, GHC first performs segmentation, introducing rec blocks to wrap over minimal recursive groups. Then, each resulting rec is desugared, using a call to Control.Monad.Fix.mfix as described in the previous section. The original mdo-expression typechecks exactly when the desugared version would do so.

Here are some other important points in using the recursive-do notation:

• It is enabled with the flag -XRecursiveDo, or the LANGUAGE RecursiveDo pragma. (The same flag enables both mdo-notation, and the use of rec blocks inside do expressions.)
• rec blocks can also be used inside mdo-expressions, which will be treated as a single statement. However, it is good style to either use mdo or rec blocks in a single expression.
• If recursive bindings are required for a monad, then that monad must be declared an instance of the MonadFix class.
• The following instances of MonadFix are automatically provided: List, Maybe, IO. Furthermore, the Control.Monad.ST and Control.Monad.ST.Lazy modules provide the instances of the MonadFix class for Haskell’s internal state monad (strict and lazy, respectively).
• Like let and where bindings, name shadowing is not allowed within an mdo-expression or a rec-block; that is, all the names bound in a single rec must be distinct. (GHC will complain if this is not the case.)

9.3.12. Applicative do-notation

The language option -XApplicativeDo enables an alternative translation for the do-notation, which uses the operators <$>, <*>, along with join as far as possible. There are two main reasons for wanting to do this:

• We can use do-notation with types that are an instance of Applicative and Functor, but not Monad
• In some monads, using the applicative operators is more efficient than monadic bind. For example, it may enable more parallelism.

Applicative do-notation desugaring preserves the original semantics, provided that the Applicative instance satisfies <*> = ap and pure = return (these are true of all the common monadic types). Thus, you can normally turn on -XApplicativeDo without fear of breaking your program. There is one pitfall to watch out for; see Things to watch out for.

There are no syntactic changes with -XApplicativeDo. The only way it shows up at the source level is that you can have a do expression that doesn’t require a Monad constraint. For example, in GHCi:

Prelude> :set -XApplicativeDo
Prelude> :t \m -> do { x <- m; return (not x) }
\m -> do { x <- m; return (not x) }
  :: Functor f => f Bool -> f Bool

This example only requires Functor, because it is translated into (\x -> not x) <$> m. A more complex example requires Applicative,

Prelude> :t \m -> do { x <- m 'a'; y <- m 'b'; return (x || y) }
\m -> do { x <- m 'a'; y <- m 'b'; return (x || y) }
  :: Applicative f => (Char -> f Bool) -> f Bool

Here GHC has translated the expression into

(\x y -> x || y) <$> m 'a' <*> m 'b'

It is possible to see the actual translation by using -ddump-ds, but be warned, the output is quite verbose.

Note that if the expression can’t be translated into uses of <$>, <*> only, then it will incur a Monad constraint as usual. This happens when there is a dependency on a value produced by an earlier statement in the do-block:

Prelude> :t \m -> do { x <- m True; y <- m x; return (x || y) }
\m -> do { x <- m True; y <- m x; return (x || y) }
  :: Monad m => (Bool -> m Bool) -> m Bool

Here, m x depends on the value of x produced by the first statement, so the expression cannot be translated using <*>.

In general, the rule for when a do statement incurs a Monad constraint is as follows. If the do-expression has the following form:

do p1 <- E1; ...; pn <- En; return E

where none of the variables defined by p1...pn are mentioned in E1...En, then the expression will only require Applicative. Otherwise, the expression will require Monad.

9.3.12.1. Things to watch out for

Your code should just work as before when -XApplicativeDo is enabled, provided you use conventional Applicative instances. However, if you define a Functor or Applicative instance using do-notation, then it will likely get turned into an infinite loop by GHC. For example, if you do this:

instance Functor MyType where
    fmap f m = do x <- m; return (f x)

Then applicative desugaring will turn it into

instance Functor MyType where
    fmap f m = fmap (\x -> f x) m

And the program will loop at runtime. Similarly, an Applicative instance like this

instance Applicative MyType where
    pure = return
    x <*> y = do f <- x; a <- y; return (f a)

will result in an infinte loop when <*> is called.

Just as you wouldn’t define a Monad instance using the do-notation, you shouldn’t define Functor or Applicative instance using do-notation (when using ApplicativeDo) either. The correct way to define these instances in terms of Monad is to use the Monad operations directly, e.g.

instance Functor MyType where
    fmap f m = m >>= return . f

instance Applicative MyType where
    pure = return
    (<*>) = ap

9.3.13. Parallel List Comprehensions

Parallel list comprehensions are a natural extension to list comprehensions. List comprehensions can be thought of as a nice syntax for writing maps and filters. Parallel comprehensions extend this to include the zipWith family.

A parallel list comprehension has multiple independent branches of qualifier lists, each separated by a | symbol. For example, the following zips together two lists:

[ (x, y) | x <- xs | y <- ys ]

The behaviour of parallel list comprehensions follows that of zip, in that the resulting list will have the same length as the shortest branch.

We can define parallel list comprehensions by translation to regular comprehensions. Here’s the basic idea:

Given a parallel comprehension of the form:

[ e | p1 <- e11, p2 <- e12, ...
    | q1 <- e21, q2 <- e22, ...
    ...
]

This will be translated to:

[ e | ((p1,p2), (q1,q2), ...) <- zipN [(p1,p2) | p1 <- e11, p2 <- e12, ...]
                                      [(q1,q2) | q1 <- e21, q2 <- e22, ...]
                                      ...
]

where zipN is the appropriate zip for the given number of branches.

9.3.14. Generalised (SQL-Like) List Comprehensions

Generalised list comprehensions are a further enhancement to the list comprehension syntactic sugar to allow operations such as sorting and grouping which are familiar from SQL. They are fully described in the paper Comprehensive comprehensions: comprehensions with “order by” and “group by”, except that the syntax we use differs slightly from the paper.

The extension is enabled with the flag -XTransformListComp.

Here is an example:

employees = [ ("Simon", "MS", 80)
, ("Erik", "MS", 100)
, ("Phil", "Ed", 40)
, ("Gordon", "Ed", 45)
, ("Paul", "Yale", 60)]

output = [ (the dept, sum salary)
| (name, dept, salary) <- employees
, then group by dept using groupWith
, then sortWith by (sum salary)
, then take 5 ]

In this example, the list output would take on the value:

[("Yale", 60), ("Ed", 85), ("MS", 180)]

There are three new keywords: group, by, and using. (The functions sortWith and groupWith are not keywords; they are ordinary functions that are exported by GHC.Exts.)

There are five new forms of comprehension qualifier, all introduced by the (existing) keyword then:

• then f
  

  This statement requires that f have the type forall a. [a] -> [a] . You can see an example of its use in the motivating example, as this form is used to apply take 5 .

• then f by e
  

  This form is similar to the previous one, but allows you to create a function which will be passed as the first argument to f. As a consequence f must have the type forall a. (a -> t) -> [a] -> [a]. As you can see from the type, this function lets f “project out” some information from the elements of the list it is transforming.

  An example is shown in the opening example, where sortWith is supplied with a function that lets it find out the sum salary for any item in the list comprehension it transforms.

• then group by e using f
  

  This is the most general of the grouping-type statements. In this form, f is required to have type forall a. (a -> t) -> [a] -> [[a]]. As with the then f by e case above, the first argument is a function supplied to f by the compiler which lets it compute e on every element of the list being transformed. However, unlike the non-grouping case, f additionally partitions the list into a number of sublists: this means that at every point after this statement, binders occurring before it in the comprehension refer to lists of possible values, not single values. To help understand this, let’s look at an example:

  -- This works similarly to groupWith in GHC.Exts, but doesn't sort its input first
  groupRuns :: Eq b => (a -> b) -> [a] -> [[a]]
  groupRuns f = groupBy (\x y -> f x == f y)
  
  output = [ (the x, y)
  | x <- ([1..3] ++ [1..2])
  , y <- [4..6]
  , then group by x using groupRuns ]
  

  This results in the variable output taking on the value below:

  [(1, [4, 5, 6]), (2, [4, 5, 6]), (3, [4, 5, 6]), (1, [4, 5, 6]), (2, [4, 5, 6])]
  

  Note that we have used the the function to change the type of x from a list to its original numeric type. The variable y, in contrast, is left unchanged from the list form introduced by the grouping.

• then group using f
  

  With this form of the group statement, f is required to simply have the type forall a. [a] -> [[a]], which will be used to group up the comprehension so far directly. An example of this form is as follows:

  output = [ x
  | y <- [1..5]
  , x <- "hello"
  , then group using inits]
  

  This will yield a list containing every prefix of the word “hello” written out 5 times:

  ["","h","he","hel","hell","hello","helloh","hellohe","hellohel","hellohell","hellohello","hellohelloh",...]
  

9.3.15. Monad comprehensions

Monad comprehensions generalise the list comprehension notation, including parallel comprehensions (Parallel List Comprehensions) and transform comprehensions (Generalised (SQL-Like) List Comprehensions) to work for any monad.

Monad comprehensions support:

• Bindings:

  [ x + y | x <- Just 1, y <- Just 2 ]
  

  Bindings are translated with the (>>=) and return functions to the usual do-notation:

  do x <- Just 1
     y <- Just 2
     return (x+y)
  

• Guards:

  [ x | x <- [1..10], x <= 5 ]
  

  Guards are translated with the guard function, which requires a MonadPlus instance:

  do x <- [1..10]
     guard (x <= 5)
     return x
  

• Transform statements (as with -XTransformListComp):

  [ x+y | x <- [1..10], y <- [1..x], then take 2 ]
  

  This translates to:

  do (x,y) <- take 2 (do x <- [1..10]
                         y <- [1..x]
                         return (x,y))
     return (x+y)
  

• Group statements (as with -XTransformListComp):

  [ x | x <- [1,1,2,2,3], then group by x using GHC.Exts.groupWith ]
  [ x | x <- [1,1,2,2,3], then group using myGroup ]
  

• Parallel statements (as with -XParallelListComp):

  [ (x+y) | x <- [1..10]
          | y <- [11..20]
          ]
  

  Parallel statements are translated using the mzip function, which requires a MonadZip instance defined in Control.Monad.Zip:

  do (x,y) <- mzip (do x <- [1..10]
                       return x)
                   (do y <- [11..20]
                       return y)
     return (x+y)
  

All these features are enabled by default if the MonadComprehensions extension is enabled. The types and more detailed examples on how to use comprehensions are explained in the previous chapters Generalised (SQL-Like) List Comprehensions and Parallel List Comprehensions. In general you just have to replace the type [a] with the type Monad m => m a for monad comprehensions.

Note

Even though most of these examples are using the list monad, monad comprehensions work for any monad. The base package offers all necessary instances for lists, which make MonadComprehensions backward compatible to built-in, transform and parallel list comprehensions.

More formally, the desugaring is as follows. We write D[ e | Q] to mean the desugaring of the monad comprehension [ e | Q]:

Expressions: e
Declarations: d
Lists of qualifiers: Q,R,S

-- Basic forms
D[ e | ]               = return e
D[ e | p <- e, Q ]  = e >>= \p -> D[ e | Q ]
D[ e | e, Q ]          = guard e >> \p -> D[ e | Q ]
D[ e | let d, Q ]      = let d in D[ e | Q ]

-- Parallel comprehensions (iterate for multiple parallel branches)
D[ e | (Q | R), S ]    = mzip D[ Qv | Q ] D[ Rv | R ] >>= \(Qv,Rv) -> D[ e | S ]

-- Transform comprehensions
D[ e | Q then f, R ]                  = f D[ Qv | Q ] >>= \Qv -> D[ e | R ]

D[ e | Q then f by b, R ]             = f (\Qv -> b) D[ Qv | Q ] >>= \Qv -> D[ e | R ]

D[ e | Q then group using f, R ]      = f D[ Qv | Q ] >>= \ys ->
                                        case (fmap selQv1 ys, ..., fmap selQvn ys) of
                                         Qv -> D[ e | R ]

D[ e | Q then group by b using f, R ] = f (\Qv -> b) D[ Qv | Q ] >>= \ys ->
                                        case (fmap selQv1 ys, ..., fmap selQvn ys) of
                                           Qv -> D[ e | R ]

where  Qv is the tuple of variables bound by Q (and used subsequently)
       selQvi is a selector mapping Qv to the ith component of Qv

Operator     Standard binding       Expected type
--------------------------------------------------------------------
return       GHC.Base               t1 -> m t2
(>>=)        GHC.Base               m1 t1 -> (t2 -> m2 t3) -> m3 t3
(>>)         GHC.Base               m1 t1 -> m2 t2         -> m3 t3
guard        Control.Monad          t1 -> m t2
fmap         GHC.Base               forall a b. (a->b) -> n a -> n b
mzip         Control.Monad.Zip      forall a b. m a -> m b -> m (a,b)

The comprehension should typecheck when its desugaring would typecheck, except that (as discussed in Generalised (SQL-Like) List Comprehensions) in the “then f” and “then group using f” clauses, when the “by b” qualifier is omitted, argument f should have a polymorphic type. In particular, “then Data.List.sort” and “then group using Data.List.group” are insufficiently polymorphic.

Monad comprehensions support rebindable syntax (Rebindable syntax and the implicit Prelude import). Without rebindable syntax, the operators from the “standard binding” module are used; with rebindable syntax, the operators are looked up in the current lexical scope. For example, parallel comprehensions will be typechecked and desugared using whatever “mzip” is in scope.

The rebindable operators must have the “Expected type” given in the table above. These types are surprisingly general. For example, you can use a bind operator with the type

(>>=) :: T x y a -> (a -> T y z b) -> T x z b

In the case of transform comprehensions, notice that the groups are parameterised over some arbitrary type n (provided it has an fmap, as well as the comprehension being over an arbitrary monad.

9.3.16. New monadic failure desugaring mechanism

The -XMonadFailDesugaring extension switches the desugaring of do-blocks to use MonadFail.fail instead of Monad.fail. This will eventually be the default behaviour in a future GHC release, under the MonadFail Proposal (MFP).

This extension is temporary, and will be deprecated in a future release. It is included so that library authors have a hard check for whether their code will work with future GHC versions.

9.3.17. Rebindable syntax and the implicit Prelude import

GHC normally imports Prelude.hi files for you. If you’d rather it didn’t, then give it a -XNoImplicitPrelude option. The idea is that you can then import a Prelude of your own. (But don’t call it Prelude; the Haskell module namespace is flat, and you must not conflict with any Prelude module.)

Suppose you are importing a Prelude of your own in order to define your own numeric class hierarchy. It completely defeats that purpose if the literal “1” means “Prelude.fromInteger 1”, which is what the Haskell Report specifies. So the -XRebindableSyntax flag causes the following pieces of built-in syntax to refer to whatever is in scope, not the Prelude versions:

• An integer literal 368 means “fromInteger (368::Integer)”, rather than “Prelude.fromInteger (368::Integer)”.
• Fractional literals are handed in just the same way, except that the translation is fromRational (3.68::Rational).
• The equality test in an overloaded numeric pattern uses whatever (==) is in scope.
• The subtraction operation, and the greater-than-or-equal test, in n+k patterns use whatever (-) and (>=) are in scope.
• Negation (e.g. “- (f x)”) means “negate (f x)”, both in numeric patterns, and expressions.
• Conditionals (e.g. “if e1 then e2 else e3”) means “ifThenElse e1 e2 e3”. However case expressions are unaffected.
• “Do” notation is translated using whatever functions (>>=), (>>), and fail, are in scope (not the Prelude versions). List comprehensions, mdo (The recursive do-notation), and parallel array comprehensions, are unaffected.
• Arrow notation (see Arrow notation) uses whatever arr, (>>>), first, app, (|||) and loop functions are in scope. But unlike the other constructs, the types of these functions must match the Prelude types very closely. Details are in flux; if you want to use this, ask!

-XRebindableSyntax implies -XNoImplicitPrelude.

In all cases (apart from arrow notation), the static semantics should be that of the desugared form, even if that is a little unexpected. For example, the static semantics of the literal 368 is exactly that of fromInteger (368::Integer); it’s fine for fromInteger to have any of the types:

fromInteger :: Integer -> Integer
fromInteger :: forall a. Foo a => Integer -> a
fromInteger :: Num a => a -> Integer
fromInteger :: Integer -> Bool -> Bool

Be warned: this is an experimental facility, with fewer checks than usual. Use -dcore-lint to typecheck the desugared program. If Core Lint is happy you should be all right.

9.3.18. Postfix operators

The -XPostfixOperators flag enables a small extension to the syntax of left operator sections, which allows you to define postfix operators. The extension is this: the left section

(e !)

is equivalent (from the point of view of both type checking and execution) to the expression

((!) e)

(for any expression e and operator (!). The strict Haskell 98 interpretation is that the section is equivalent to

(\y -> (!) e y)

That is, the operator must be a function of two arguments. GHC allows it to take only one argument, and that in turn allows you to write the function postfix.

The extension does not extend to the left-hand side of function definitions; you must define such a function in prefix form.

9.3.19. Tuple sections

The -XTupleSections flag enables Python-style partially applied tuple constructors. For example, the following program

(, True)

is considered to be an alternative notation for the more unwieldy alternative

\x -> (x, True)

You can omit any combination of arguments to the tuple, as in the following

(, "I", , , "Love", , 1337)

which translates to

\a b c d -> (a, "I", b, c, "Love", d, 1337)

If you have unboxed tuples enabled, tuple sections will also be available for them, like so

(# , True #)

Because there is no unboxed unit tuple, the following expression

(# #)

continues to stand for the unboxed singleton tuple data constructor.

9.3.20. Lambda-case

The -XLambdaCase flag enables expressions of the form

\case { p1 -> e1; ...; pN -> eN }

which is equivalent to

\freshName -> case freshName of { p1 -> e1; ...; pN -> eN }

Note that \case starts a layout, so you can write

\case
  p1 -> e1
  ...
  pN -> eN

9.3.21. Empty case alternatives

The -XEmptyCase flag enables case expressions, or lambda-case expressions, that have no alternatives, thus:

    case e of { }   -- No alternatives
or
    \case { }       -- -XLambdaCase is also required

This can be useful when you know that the expression being scrutinised has no non-bottom values. For example:

data Void
f :: Void -> Int
f x = case x of { }

With dependently-typed features it is more useful (see Trac #2431`). For example, consider these two candidate definitions of absurd:

data a :==: b where
  Refl :: a :==: a

absurd :: True :~: False -> a
absurd x = error "absurd"    -- (A)
absurd x = case x of {}      -- (B)

We much prefer (B). Why? Because GHC can figure out that (True :~: False) is an empty type. So (B) has no partiality and GHC should be able to compile with -fwarn-incomplete-patterns. (Though the pattern match checking is not yet clever enough to do that.) On the other hand (A) looks dangerous, and GHC doesn’t check to make sure that, in fact, the function can never get called.

9.3.22. Multi-way if-expressions

With -XMultiWayIf flag GHC accepts conditional expressions with multiple branches:

if | guard1 -> expr1
   | ...
   | guardN -> exprN

which is roughly equivalent to

case () of
  _ | guard1 -> expr1
  ...
  _ | guardN -> exprN

Multi-way if expressions introduce a new layout context. So the example above is equivalent to:

if { | guard1 -> expr1
   ; | ...
   ; | guardN -> exprN
   }

The following behaves as expected:

if | guard1 -> if | guard2 -> expr2
                  | guard3 -> expr3
   | guard4 -> expr4

because layout translates it as

if { | guard1 -> if { | guard2 -> expr2
                    ; | guard3 -> expr3
                    }
   ; | guard4 -> expr4
   }

Layout with multi-way if works in the same way as other layout contexts, except that the semi-colons between guards in a multi-way if are optional. So it is not necessary to line up all the guards at the same column; this is consistent with the way guards work in function definitions and case expressions.

9.3.23. Local Fixity Declarations

A careful reading of the Haskell 98 Report reveals that fixity declarations (infix, infixl, and infixr) are permitted to appear inside local bindings such those introduced by let and where. However, the Haskell Report does not specify the semantics of such bindings very precisely.

In GHC, a fixity declaration may accompany a local binding:

let f = ...
    infixr 3 `f`
in
    ...

and the fixity declaration applies wherever the binding is in scope. For example, in a let, it applies in the right-hand sides of other let-bindings and the body of the letC. Or, in recursive do expressions (The recursive do-notation), the local fixity declarations of a let statement scope over other statements in the group, just as the bound name does.

Moreover, a local fixity declaration must accompany a local binding of that name: it is not possible to revise the fixity of name bound elsewhere, as in

let infixr 9 $ in ...

Because local fixity declarations are technically Haskell 98, no flag is necessary to enable them.

9.3.24. Import and export extensions

9.3.24.1. Hiding things the imported module doesn’t export

Technically in Haskell 2010 this is illegal:

module A( f ) where
  f = True

module B where
  import A hiding( g )  -- A does not export g
  g = f

The import A hiding( g ) in module B is technically an error (Haskell Report, 5.3.1) because A does not export g. However GHC allows it, in the interests of supporting backward compatibility; for example, a newer version of A might export g, and you want B to work in either case.

The warning -fwarn-dodgy-imports, which is off by default but included with -W, warns if you hide something that the imported module does not export.

9.3.24.2. Package-qualified imports

With the -XPackageImports flag, GHC allows import declarations to be qualified by the package name that the module is intended to be imported from. For example:

import "network" Network.Socket

would import the module Network.Socket from the package network (any version). This may be used to disambiguate an import when the same module is available from multiple packages, or is present in both the current package being built and an external package.

The special package name this can be used to refer to the current package being built.

Note

You probably don’t need to use this feature, it was added mainly so that we can build backwards-compatible versions of packages when APIs change. It can lead to fragile dependencies in the common case: modules occasionally move from one package to another, rendering any package-qualified imports broken. See also Thinning and renaming modules for an alternative way of disambiguating between module names.

9.3.24.3. Safe imports

With the -XSafe, -XTrustworthy and -XUnsafe language flags, GHC extends the import declaration syntax to take an optional safe keyword after the import keyword. This feature is part of the Safe Haskell GHC extension. For example:

import safe qualified Network.Socket as NS

would import the module Network.Socket with compilation only succeeding if Network.Socket can be safely imported. For a description of when a import is considered safe see Safe Haskell.

9.3.24.4. Explicit namespaces in import/export

In an import or export list, such as

module M( f, (++) ) where ...
  import N( f, (++) )
  ...

the entities f and (++) are values. However, with type operators (Type operators) it becomes possible to declare (++) as a type constructor. In that case, how would you export or import it?

The -XExplicitNamespaces extension allows you to prefix the name of a type constructor in an import or export list with “type” to disambiguate this case, thus:

module M( f, type (++) ) where ...
  import N( f, type (++) )
  ...
module N( f, type (++) ) where
  data family a ++ b = L a | R b

The extension -XExplicitNamespaces is implied by -XTypeOperators and (for some reason) by -XTypeFamilies.

In addition, with -XPatternSynonyms you can prefix the name of a data constructor in an import or export list with the keyword pattern, to allow the import or export of a data constructor without its parent type constructor (see Import and export of pattern synonyms).

9.3.25. Summary of stolen syntax

Turning on an option that enables special syntax might cause working Haskell 98 code to fail to compile, perhaps because it uses a variable name which has become a reserved word. This section lists the syntax that is “stolen” by language extensions. We use notation and nonterminal names from the Haskell 98 lexical syntax (see the Haskell 98 Report). We only list syntax changes here that might affect existing working programs (i.e. “stolen” syntax). Many of these extensions will also enable new context-free syntax, but in all cases programs written to use the new syntax would not be compilable without the option enabled.

There are two classes of special syntax:

• New reserved words and symbols: character sequences which are no longer available for use as identifiers in the program.
• Other special syntax: sequences of characters that have a different meaning when this particular option is turned on.

The following syntax is stolen:

forall

Stolen (in types) by: -XExplicitForAll, and hence by -XScopedTypeVariables, -XLiberalTypeSynonyms, -XRankNTypes, -XExistentialQuantification

mdo

Stolen by: -XRecursiveDo

foreign

Stolen by: -XForeignFunctionInterface

rec, proc, -<, >-, -<<, >>-, (|, |)

Stolen by: -XArrows

?varid

Stolen by: -XImplicitParams

[|, [e|, [p|, [d|, [t|, $(, $$(, [||, [e||, $varid, $$varid

Stolen by: -XTemplateHaskell

[varid|

Stolen by: -XQuasiQuotes

⟨varid⟩, #⟨char⟩, #, ⟨string⟩, #, ⟨integer⟩, #, ⟨float⟩, #, ⟨float⟩, ##

Stolen by: -XMagicHash

(#, #)

Stolen by: -XUnboxedTuples

⟨varid⟩, !, ⟨varid⟩

Stolen by: -XBangPatterns

pattern

Stolen by: -XPatternSynonyms

9.4. Extensions to data types and type synonyms

9.4.1. Data types with no constructors

With the -XEmptyDataDecls flag (or equivalent LANGUAGE pragma), GHC lets you declare a data type with no constructors. For example:

data S      -- S :: *
data T a    -- T :: * -> *

Syntactically, the declaration lacks the “= constrs” part. The type can be parameterised over types of any kind, but if the kind is not * then an explicit kind annotation must be used (see Explicitly-kinded quantification).

Such data types have only one value, namely bottom. Nevertheless, they can be useful when defining “phantom types”.

9.4.2. Data type contexts

Haskell allows datatypes to be given contexts, e.g.

data Eq a => Set a = NilSet | ConsSet a (Set a)

give constructors with types:

NilSet :: Set a
ConsSet :: Eq a => a -> Set a -> Set a

This is widely considered a misfeature, and is going to be removed from the language. In GHC, it is controlled by the deprecated extension DatatypeContexts.

9.4.3. Infix type constructors, classes, and type variables

GHC allows type constructors, classes, and type variables to be operators, and to be written infix, very much like expressions. More specifically:

• A type constructor or class can be any non-reserved operator. Symbols used in types are always like capitalized identifiers; they are never variables. Note that this is different from the lexical syntax of data constructors, which are required to begin with a :.

• Data type and type-synonym declarations can be written infix, parenthesised if you want further arguments. E.g.

  data a :*: b = Foo a b
  type a :+: b = Either a b
  class a :=: b where ...
  
  data (a :**: b) x = Baz a b x
  type (a :++: b) y = Either (a,b) y
  

• Types, and class constraints, can be written infix. For example

  x :: Int :*: Bool
  f :: (a :=: b) => a -> b
  

• Back-quotes work as for expressions, both for type constructors and type variables; e.g. Int `Either` Bool, or Int `a` Bool. Similarly, parentheses work the same; e.g. (:*:) Int Bool.

• Fixities may be declared for type constructors, or classes, just as for data constructors. However, one cannot distinguish between the two in a fixity declaration; a fixity declaration sets the fixity for a data constructor and the corresponding type constructor. For example:

  infixl 7 T, :*:
  

  sets the fixity for both type constructor T and data constructor T, and similarly for :*:. Int `a` Bool.

• Function arrow is infixr with fixity 0 (this might change; it’s not clear what it should be).

9.4.4. Type operators

In types, an operator symbol like (+) is normally treated as a type variable, just like a. Thus in Haskell 98 you can say

type T (+) = ((+), (+))
-- Just like: type T a = (a,a)

f :: T Int -> Int
f (x,y)= x

As you can see, using operators in this way is not very useful, and Haskell 98 does not even allow you to write them infix.

The language -XTypeOperators changes this behaviour:

• Operator symbols become type constructors rather than type variables.

• Operator symbols in types can be written infix, both in definitions and uses. For example:

  data a + b = Plus a b
  type Foo = Int + Bool
  

• There is now some potential ambiguity in import and export lists; for example if you write import M( (+) ) do you mean the function (+) or the type constructor (+)? The default is the former, but with -XExplicitNamespaces (which is implied by -XTypeOperators) GHC allows you to specify the latter by preceding it with the keyword type, thus:

  import M( type (+) )
  

  See Explicit namespaces in import/export.

• The fixity of a type operator may be set using the usual fixity declarations but, as in Infix type constructors, classes, and type variables, the function and type constructor share a single fixity.

9.4.5. Liberalised type synonyms

Type synonyms are like macros at the type level, but Haskell 98 imposes many rules on individual synonym declarations. With the -XLiberalTypeSynonyms extension, GHC does validity checking on types only after expanding type synonyms. That means that GHC can be very much more liberal about type synonyms than Haskell 98.

• You can write a forall (including overloading) in a type synonym, thus:

  type Discard a = forall b. Show b => a -> b -> (a, String)
  
  f :: Discard a
  f x y = (x, show y)
  
  g :: Discard Int -> (Int,String)    -- A rank-2 type
  g f = f 3 True
  

• If you also use -XUnboxedTuples, you can write an unboxed tuple in a type synonym:

  type Pr = (# Int, Int #)
  
  h :: Int -> Pr
  h x = (# x, x #)
  

• You can apply a type synonym to a forall type:

  type Foo a = a -> a -> Bool
  
  f :: Foo (forall b. b->b)
  

  After expanding the synonym, f has the legal (in GHC) type:

  f :: (forall b. b->b) -> (forall b. b->b) -> Bool
  

• You can apply a type synonym to a partially applied type synonym:

  type Generic i o = forall x. i x -> o x
  type Id x = x
  
  foo :: Generic Id []
  

  After expanding the synonym, foo has the legal (in GHC) type:

  foo :: forall x. x -> [x]
  

GHC currently does kind checking before expanding synonyms (though even that could be changed)..

After expanding type synonyms, GHC does validity checking on types, looking for the following mal-formedness which isn’t detected simply by kind checking:

• Type constructor applied to a type involving for-alls (if XImpredicativeTypes is off)
• Partially-applied type synonym.

So, for example, this will be rejected:

type Pr = forall a. a

h :: [Pr]
h = ...

because GHC does not allow type constructors applied to for-all types.

9.4.6. Existentially quantified data constructors

The idea of using existential quantification in data type declarations was suggested by Perry, and implemented in Hope+ (Nigel Perry, The Implementation of Practical Functional Programming Languages, PhD Thesis, University of London, 1991). It was later formalised by Laufer and Odersky (Polymorphic type inference and abstract data types, TOPLAS, 16(5), pp. 1411-1430, 1994). It’s been in Lennart Augustsson’s hbc Haskell compiler for several years, and proved very useful. Here’s the idea. Consider the declaration:

data Foo = forall a. MkFoo a (a -> Bool)
         | Nil

The data type Foo has two constructors with types:

MkFoo :: forall a. a -> (a -> Bool) -> Foo
Nil   :: Foo

Notice that the type variable a in the type of MkFoo does not appear in the data type itself, which is plain Foo. For example, the following expression is fine:

[MkFoo 3 even, MkFoo 'c' isUpper] :: [Foo]

Here, (MkFoo 3 even) packages an integer with a function even that maps an integer to Bool; and MkFoo 'c' isUpper packages a character with a compatible function. These two things are each of type Foo and can be put in a list.

What can we do with a value of type Foo? In particular, what happens when we pattern-match on MkFoo?

f (MkFoo val fn) = ???

Since all we know about val and fn is that they are compatible, the only (useful) thing we can do with them is to apply fn to val to get a boolean. For example:

f :: Foo -> Bool
f (MkFoo val fn) = fn val

What this allows us to do is to package heterogeneous values together with a bunch of functions that manipulate them, and then treat that collection of packages in a uniform manner. You can express quite a bit of object-oriented-like programming this way.

9.4.6.1. Why existential?

What has this to do with existential quantification? Simply that MkFoo has the (nearly) isomorphic type

MkFoo :: (exists a . (a, a -> Bool)) -> Foo

But Haskell programmers can safely think of the ordinary universally quantified type given above, thereby avoiding adding a new existential quantification construct.

9.4.6.2. Existentials and type classes

An easy extension is to allow arbitrary contexts before the constructor. For example:

data Baz = forall a. Eq a => Baz1 a a
         | forall b. Show b => Baz2 b (b -> b)

The two constructors have the types you’d expect:

Baz1 :: forall a. Eq a => a -> a -> Baz
Baz2 :: forall b. Show b => b -> (b -> b) -> Baz

But when pattern matching on Baz1 the matched values can be compared for equality, and when pattern matching on Baz2 the first matched value can be converted to a string (as well as applying the function to it). So this program is legal:

f :: Baz -> String
f (Baz1 p q) | p == q    = "Yes"
             | otherwise = "No"
f (Baz2 v fn)            = show (fn v)

Operationally, in a dictionary-passing implementation, the constructors Baz1 and Baz2 must store the dictionaries for Eq and Show respectively, and extract it on pattern matching.

9.4.6.3. Record Constructors

GHC allows existentials to be used with records syntax as well. For example:

data Counter a = forall self. NewCounter
    { _this    :: self
    , _inc     :: self -> self
    , _display :: self -> IO ()
    , tag      :: a
    }

Here tag is a public field, with a well-typed selector function tag :: Counter a -> a. The self type is hidden from the outside; any attempt to apply _this, _inc or _display as functions will raise a compile-time error. In other words, GHC defines a record selector function only for fields whose type does not mention the existentially-quantified variables. (This example used an underscore in the fields for which record selectors will not be defined, but that is only programming style; GHC ignores them.)

To make use of these hidden fields, we need to create some helper functions:

inc :: Counter a -> Counter a
inc (NewCounter x i d t) = NewCounter
    { _this = i x, _inc = i, _display = d, tag = t }

display :: Counter a -> IO ()
display NewCounter{ _this = x, _display = d } = d x

Now we can define counters with different underlying implementations:

counterA :: Counter String
counterA = NewCounter
    { _this = 0, _inc = (1+), _display = print, tag = "A" }

counterB :: Counter String
counterB = NewCounter
    { _this = "", _inc = ('#':), _display = putStrLn, tag = "B" }

main = do
    display (inc counterA)         -- prints "1"
    display (inc (inc counterB))   -- prints "##"

Record update syntax is supported for existentials (and GADTs):

setTag :: Counter a -> a -> Counter a
setTag obj t = obj{ tag = t }

The rule for record update is this:

the types of the updated fields may mention only the universally-quantified type variables of the data constructor. For GADTs, the field may mention only types that appear as a simple type-variable argument in the constructor’s result type.

For example:

data T a b where { T1 { f1::a, f2::b, f3::(b,c) } :: T a b } -- c is existential
upd1 t x = t { f1=x }   -- OK:   upd1 :: T a b -> a' -> T a' b
upd2 t x = t { f3=x }   -- BAD   (f3's type mentions c, which is
                        --        existentially quantified)

data G a b where { G1 { g1::a, g2::c } :: G a [c] }
upd3 g x = g { g1=x }   -- OK:   upd3 :: G a b -> c -> G c b
upd4 g x = g { g2=x }   -- BAD (f2's type mentions c, which is not a simple
                        --      type-variable argument in G1's result type)

9.4.6.4. Restrictions

There are several restrictions on the ways in which existentially-quantified constructors can be used.

• When pattern matching, each pattern match introduces a new, distinct, type for each existential type variable. These types cannot be unified with any other type, nor can they escape from the scope of the pattern match. For example, these fragments are incorrect:

  f1 (MkFoo a f) = a
  

  Here, the type bound by MkFoo “escapes”, because a is the result of f1. One way to see why this is wrong is to ask what type f1 has:

  f1 :: Foo -> a             -- Weird!
  

  What is this “a” in the result type? Clearly we don’t mean this:

  f1 :: forall a. Foo -> a   -- Wrong!
  

  The original program is just plain wrong. Here’s another sort of error

  f2 (Baz1 a b) (Baz1 p q) = a==q
  

  It’s ok to say a==b or p==q, but a==q is wrong because it equates the two distinct types arising from the two Baz1 constructors.

• You can’t pattern-match on an existentially quantified constructor in a let or where group of bindings. So this is illegal:

  f3 x = a==b where { Baz1 a b = x }
  

  Instead, use a case expression:

  f3 x = case x of Baz1 a b -> a==b
  

  In general, you can only pattern-match on an existentially-quantified constructor in a case expression or in the patterns of a function definition. The reason for this restriction is really an implementation one. Type-checking binding groups is already a nightmare without existentials complicating the picture. Also an existential pattern binding at the top level of a module doesn’t make sense, because it’s not clear how to prevent the existentially-quantified type “escaping”. So for now, there’s a simple-to-state restriction. We’ll see how annoying it is.

• You can’t use existential quantification for newtype declarations. So this is illegal:

  newtype T = forall a. Ord a => MkT a
  

  Reason: a value of type T must be represented as a pair of a dictionary for Ord t and a value of type t. That contradicts the idea that newtype should have no concrete representation. You can get just the same efficiency and effect by using data instead of newtype. If there is no overloading involved, then there is more of a case for allowing an existentially-quantified newtype, because the data version does carry an implementation cost, but single-field existentially quantified constructors aren’t much use. So the simple restriction (no existential stuff on newtype) stands, unless there are convincing reasons to change it.

• You can’t use deriving to define instances of a data type with existentially quantified data constructors. Reason: in most cases it would not make sense. For example:;

  data T = forall a. MkT [a] deriving( Eq )
  

  To derive Eq in the standard way we would need to have equality between the single component of two MkT constructors:

  instance Eq T where
    (MkT a) == (MkT b) = ???
  

  But a and b have distinct types, and so can’t be compared. It’s just about possible to imagine examples in which the derived instance would make sense, but it seems altogether simpler simply to prohibit such declarations. Define your own instances!

9.4.7. Declaring data types with explicit constructor signatures

When the GADTSyntax extension is enabled, GHC allows you to declare an algebraic data type by giving the type signatures of constructors explicitly. For example:

data Maybe a where
    Nothing :: Maybe a
    Just    :: a -> Maybe a

The form is called a “GADT-style declaration” because Generalised Algebraic Data Types, described in Generalised Algebraic Data Types (GADTs), can only be declared using this form.

Notice that GADT-style syntax generalises existential types (Existentially quantified data constructors). For example, these two declarations are equivalent:

data Foo = forall a. MkFoo a (a -> Bool)
data Foo' where { MKFoo :: a -> (a->Bool) -> Foo' }

Any data type that can be declared in standard Haskell 98 syntax can also be declared using GADT-style syntax. The choice is largely stylistic, but GADT-style declarations differ in one important respect: they treat class constraints on the data constructors differently. Specifically, if the constructor is given a type-class context, that context is made available by pattern matching. For example:

data Set a where
  MkSet :: Eq a => [a] -> Set a

makeSet :: Eq a => [a] -> Set a
makeSet xs = MkSet (nub xs)

insert :: a -> Set a -> Set a
insert a (MkSet as) | a `elem` as = MkSet as
                    | otherwise   = MkSet (a:as)

A use of MkSet as a constructor (e.g. in the definition of makeSet) gives rise to a (Eq a) constraint, as you would expect. The new feature is that pattern-matching on MkSet (as in the definition of insert) makes available an (Eq a) context. In implementation terms, the MkSet constructor has a hidden field that stores the (Eq a) dictionary that is passed to MkSet; so when pattern-matching that dictionary becomes available for the right-hand side of the match. In the example, the equality dictionary is used to satisfy the equality constraint generated by the call to elem, so that the type of insert itself has no Eq constraint.

For example, one possible application is to reify dictionaries:

data NumInst a where
  MkNumInst :: Num a => NumInst a

intInst :: NumInst Int
intInst = MkNumInst

plus :: NumInst a -> a -> a -> a
plus MkNumInst p q = p + q

Here, a value of type NumInst a is equivalent to an explicit (Num a) dictionary.

All this applies to constructors declared using the syntax of Existentials and type classes. For example, the NumInst data type above could equivalently be declared like this:

data NumInst a
   = Num a => MkNumInst (NumInst a)

Notice that, unlike the situation when declaring an existential, there is no forall, because the Num constrains the data type’s universally quantified type variable a. A constructor may have both universal and existential type variables: for example, the following two declarations are equivalent:

data T1 a
 = forall b. (Num a, Eq b) => MkT1 a b
data T2 a where
 MkT2 :: (Num a, Eq b) => a -> b -> T2 a

All this behaviour contrasts with Haskell 98’s peculiar treatment of contexts on a data type declaration (Section 4.2.1 of the Haskell 98 Report). In Haskell 98 the definition

data Eq a => Set' a = MkSet' [a]

gives MkSet' the same type as MkSet above. But instead of making available an (Eq a) constraint, pattern-matching on MkSet' requires an (Eq a) constraint! GHC faithfully implements this behaviour, odd though it is. But for GADT-style declarations, GHC’s behaviour is much more useful, as well as much more intuitive.

The rest of this section gives further details about GADT-style data type declarations.

• The result type of each data constructor must begin with the type constructor being defined. If the result type of all constructors has the form T a1 ... an, where a1 ... an are distinct type variables, then the data type is ordinary; otherwise is a generalised data type (Generalised Algebraic Data Types (GADTs)).

• As with other type signatures, you can give a single signature for several data constructors. In this example we give a single signature for T1 and T2:

  data T a where
    T1,T2 :: a -> T a
    T3 :: T a
  

• The type signature of each constructor is independent, and is implicitly universally quantified as usual. In particular, the type variable(s) in the “data T a where” header have no scope, and different constructors may have different universally-quantified type variables:

  data T a where        -- The 'a' has no scope
    T1,T2 :: b -> T b   -- Means forall b. b -> T b
    T3 :: T a           -- Means forall a. T a
  

• A constructor signature may mention type class constraints, which can differ for different constructors. For example, this is fine:

  data T a where
    T1 :: Eq b => b -> b -> T b
    T2 :: (Show c, Ix c) => c -> [c] -> T c
  

  When pattern matching, these constraints are made available to discharge constraints in the body of the match. For example:

  f :: T a -> String
  f (T1 x y) | x==y      = "yes"
             | otherwise = "no"
  f (T2 a b)             = show a
  

  Note that f is not overloaded; the Eq constraint arising from the use of == is discharged by the pattern match on T1 and similarly the Show constraint arising from the use of show.

• Unlike a Haskell-98-style data type declaration, the type variable(s) in the “data Set a where” header have no scope. Indeed, one can write a kind signature instead:

  data Set :: * -> * where ...
  

  or even a mixture of the two:

  data Bar a :: (* -> *) -> * where ...
  

  The type variables (if given) may be explicitly kinded, so we could also write the header for Foo like this:

  data Bar a (b :: * -> *) where ...
  

• You can use strictness annotations, in the obvious places in the constructor type:

  data Term a where
      Lit    :: !Int -> Term Int
      If     :: Term Bool -> !(Term a) -> !(Term a) -> Term a
      Pair   :: Term a -> Term b -> Term (a,b)
  

• You can use a deriving clause on a GADT-style data type declaration. For example, these two declarations are equivalent

  data Maybe1 a where {
      Nothing1 :: Maybe1 a ;
      Just1    :: a -> Maybe1 a
    } deriving( Eq, Ord )
  
  data Maybe2 a = Nothing2 | Just2 a
       deriving( Eq, Ord )
  

• The type signature may have quantified type variables that do not appear in the result type:

  data Foo where
     MkFoo :: a -> (a->Bool) -> Foo
     Nil   :: Foo
  

  Here the type variable a does not appear in the result type of either constructor. Although it is universally quantified in the type of the constructor, such a type variable is often called “existential”. Indeed, the above declaration declares precisely the same type as the data Foo in Existentially quantified data constructors.

  The type may contain a class context too, of course:

  data Showable where
    MkShowable :: Show a => a -> Showable
  

• You can use record syntax on a GADT-style data type declaration:

  data Person where
      Adult :: { name :: String, children :: [Person] } -> Person
      Child :: Show a => { name :: !String, funny :: a } -> Person
  

  As usual, for every constructor that has a field f, the type of field f must be the same (modulo alpha conversion). The Child constructor above shows that the signature may have a context, existentially-quantified variables, and strictness annotations, just as in the non-record case. (NB: the “type” that follows the double-colon is not really a type, because of the record syntax and strictness annotations. A “type” of this form can appear only in a constructor signature.)

• Record updates are allowed with GADT-style declarations, only fields that have the following property: the type of the field mentions no existential type variables.

• As in the case of existentials declared using the Haskell-98-like record syntax (Record Constructors), record-selector functions are generated only for those fields that have well-typed selectors. Here is the example of that section, in GADT-style syntax:

  data Counter a where
      NewCounter :: { _this    :: self
                    , _inc     :: self -> self
                    , _display :: self -> IO ()
                    , tag      :: a
                    } -> Counter a
  

  As before, only one selector function is generated here, that for tag. Nevertheless, you can still use all the field names in pattern matching and record construction.

• In a GADT-style data type declaration there is no obvious way to specify that a data constructor should be infix, which makes a difference if you derive Show for the type. (Data constructors declared infix are displayed infix by the derived show.) So GHC implements the following design: a data constructor declared in a GADT-style data type declaration is displayed infix by Show iff (a) it is an operator symbol, (b) it has two arguments, (c) it has a programmer-supplied fixity declaration. For example

  infix 6 (:--:)
  data T a where
    (:--:) :: Int -> Bool -> T Int
  

9.4.8. Generalised Algebraic Data Types (GADTs)

Generalised Algebraic Data Types generalise ordinary algebraic data types by allowing constructors to have richer return types. Here is an example:

data Term a where
    Lit    :: Int -> Term Int
    Succ   :: Term Int -> Term Int
    IsZero :: Term Int -> Term Bool
    If     :: Term Bool -> Term a -> Term a -> Term a
    Pair   :: Term a -> Term b -> Term (a,b)

Notice that the return type of the constructors is not always Term a, as is the case with ordinary data types. This generality allows us to write a well-typed eval function for these Terms:

eval :: Term a -> a
eval (Lit i)      = i
eval (Succ t)     = 1 + eval t
eval (IsZero t)   = eval t == 0
eval (If b e1 e2) = if eval b then eval e1 else eval e2
eval (Pair e1 e2) = (eval e1, eval e2)

The key point about GADTs is that pattern matching causes type refinement. For example, in the right hand side of the equation

eval :: Term a -> a
eval (Lit i) =  ...

the type a is refined to Int. That’s the whole point! A precise specification of the type rules is beyond what this user manual aspires to, but the design closely follows that described in the paper Simple unification-based type inference for GADTs, (ICFP 2006). The general principle is this: type refinement is only carried out based on user-supplied type annotations. So if no type signature is supplied for eval, no type refinement happens, and lots of obscure error messages will occur. However, the refinement is quite general. For example, if we had:

eval :: Term a -> a -> a
eval (Lit i) j =  i+j

the pattern match causes the type a to be refined to Int (because of the type of the constructor Lit), and that refinement also applies to the type of j, and the result type of the case expression. Hence the addition i+j is legal.

These and many other examples are given in papers by Hongwei Xi, and Tim Sheard. There is a longer introduction on the wiki, and Ralf Hinze’s Fun with phantom types also has a number of examples. Note that papers may use different notation to that implemented in GHC.

The rest of this section outlines the extensions to GHC that support GADTs. The extension is enabled with -XGADTs. The -XGADTs flag also sets -XGADTSyntax and -XMonoLocalBinds.

• A GADT can only be declared using GADT-style syntax (Declaring data types with explicit constructor signatures); the old Haskell 98 syntax for data declarations always declares an ordinary data type. The result type of each constructor must begin with the type constructor being defined, but for a GADT the arguments to the type constructor can be arbitrary monotypes. For example, in the Term data type above, the type of each constructor must end with Term ty, but the ty need not be a type variable (e.g. the Lit constructor).

• It is permitted to declare an ordinary algebraic data type using GADT-style syntax. What makes a GADT into a GADT is not the syntax, but rather the presence of data constructors whose result type is not just T a b.

• You cannot use a deriving clause for a GADT; only for an ordinary data type.

• As mentioned in Declaring data types with explicit constructor signatures, record syntax is supported. For example:

  data Term a where
      Lit    :: { val  :: Int }      -> Term Int
      Succ   :: { num  :: Term Int } -> Term Int
      Pred   :: { num  :: Term Int } -> Term Int
      IsZero :: { arg  :: Term Int } -> Term Bool
      Pair   :: { arg1 :: Term a
                , arg2 :: Term b
                }                    -> Term (a,b)
      If     :: { cnd  :: Term Bool
                , tru  :: Term a
                , fls  :: Term a
                }                    -> Term a
  

  However, for GADTs there is the following additional constraint: every constructor that has a field f must have the same result type (modulo alpha conversion) Hence, in the above example, we cannot merge the num and arg fields above into a single name. Although their field types are both Term Int, their selector functions actually have different types:

  num :: Term Int -> Term Int
  arg :: Term Bool -> Term Int
  

• When pattern-matching against data constructors drawn from a GADT, for example in a case expression, the following rules apply:

  • The type of the scrutinee must be rigid.
  • The type of the entire case expression must be rigid.
  • The type of any free variable mentioned in any of the case alternatives must be rigid.

  A type is “rigid” if it is completely known to the compiler at its binding site. The easiest way to ensure that a variable a rigid type is to give it a type signature. For more precise details see Simple unification-based type inference for GADTs. The criteria implemented by GHC are given in the Appendix.

9.5. Extensions to the record system

9.5.1. Traditional record syntax

Traditional record syntax, such as C {f = x}, is enabled by default. To disable it, you can use the -XNoTraditionalRecordSyntax flag.

9.5.2. Record field disambiguation

In record construction and record pattern matching it is entirely unambiguous which field is referred to, even if there are two different data types in scope with a common field name. For example:

module M where
  data S = MkS { x :: Int, y :: Bool }

module Foo where
  import M

  data T = MkT { x :: Int }

  ok1 (MkS { x = n }) = n+1   -- Unambiguous
  ok2 n = MkT { x = n+1 }     -- Unambiguous

  bad1 k = k { x = 3 }  -- Ambiguous
  bad2 k = x k          -- Ambiguous

Even though there are two x‘s in scope, it is clear that the x in the pattern in the definition of ok1 can only mean the field x from type S. Similarly for the function ok2. However, in the record update in bad1 and the record selection in bad2 it is not clear which of the two types is intended.

Haskell 98 regards all four as ambiguous, but with the -XDisambiguateRecordFields flag, GHC will accept the former two. The rules are precisely the same as those for instance declarations in Haskell 98, where the method names on the left-hand side of the method bindings in an instance declaration refer unambiguously to the method of that class (provided they are in scope at all), even if there are other variables in scope with the same name. This reduces the clutter of qualified names when you import two records from different modules that use the same field name.

Some details:

• Field disambiguation can be combined with punning (see Record puns). For example:

  module Foo where
    import M
    x=True
    ok3 (MkS { x }) = x+1   -- Uses both disambiguation and punning
  

• With -XDisambiguateRecordFields you can use unqualified field names even if the corresponding selector is only in scope qualified For example, assuming the same module M as in our earlier example, this is legal:

  module Foo where
    import qualified M    -- Note qualified
  
    ok4 (M.MkS { x = n }) = n+1   -- Unambiguous
  

  Since the constructor MkS is only in scope qualified, you must name it M.MkS, but the field x does not need to be qualified even though M.x is in scope but x is not (In effect, it is qualified by the constructor).

9.5.3. Duplicate record fields

Going beyond -XDisambiguateRecordFields (see Record field disambiguation), the -XDuplicateRecordFields extension allows multiple datatypes to be declared using the same field names in a single module. For example, it allows this:

module M where
  data S = MkS { x :: Int }
  data T = MkT { x :: Bool }

Uses of fields that are always unambiguous because they mention the constructor, including construction and pattern-matching, may freely use duplicated field names. For example, the following are permitted (just as with -XDisambiguateRecordFields):

s = MkS { x = 3 }

f (MkT { x = b }) = b

Field names used as selector functions or in record updates must be unambiguous, either because there is only one such field in scope, or because a type signature is supplied, as described in the following sections.

9.5.3.1. Selector functions

Fields may be used as selector functions only if they are unambiguous, so this is still not allowed if both S(x) and T(x) are in scope:

bad r = x r

An ambiguous selector may be disambiguated by the type being “pushed down” to the occurrence of the selector (see Type inference for more details on what “pushed down” means). For example, the following are permitted:

ok1 = x :: S -> Int

ok2 :: S -> Int
ok2 = x

ok3 = k x -- assuming we already have k :: (S -> Int) -> _

In addition, the datatype that is meant may be given as a type signature on the argument to the selector:

ok4 s = x (s :: S)

However, we do not infer the type of the argument to determine the datatype, or have any way of deferring the choice to the constraint solver. Thus the following is ambiguous:

bad :: S -> Int
bad s = x s

Even though a field label is duplicated in its defining module, it may be possible to use the selector unambiguously elsewhere. For example, another module could import S(x) but not T(x), and then use x unambiguously.

9.5.3.2. Record updates

In a record update such as e { x = 1 }, if there are multiple x fields in scope, then the type of the context must fix which record datatype is intended, or a type annotation must be supplied. Consider the following definitions:

data S = MkS { foo :: Int }
data T = MkT { foo :: Int, bar :: Int }
data U = MkU { bar :: Int, baz :: Int }

Without -XDuplicateRecordFields, an update mentioning foo will always be ambiguous if all these definitions were in scope. When the extension is enabled, there are several options for disambiguating updates:

• Check for types that have all the fields being updated. For example:

  f x = x { foo = 3, bar = 2 }
  

  Here f must be updating T because neither S nor U have both fields.

• Use the type being pushed in to the record update, as in the following:

  g1 :: T -> T
  g1 x = x { foo = 3 }
  
  g2 x = x { foo = 3 } :: T
  
  g3 = k (x { foo = 3 }) -- assuming we already have k :: T -> _
  

• Use an explicit type signature on the record expression, as in:

  h x = (x :: T) { foo = 3 }
  

The type of the expression being updated will not be inferred, and no constraint-solving will be performed, so the following will be rejected as ambiguous:

let x :: T
    x = blah
in x { foo = 3 }

\x -> [x { foo = 3 },  blah :: T ]

\ (x :: T) -> x { foo = 3 }

9.5.3.3. Import and export of record fields

When -XDuplicateRecordFields is enabled, an ambiguous field must be exported as part of its datatype, rather than at the top level. For example, the following is legal:

module M (S(x), T(..)) where
  data S = MkS { x :: Int }
  data T = MkT { x :: Bool }

However, this would not be permitted, because x is ambiguous:

module M (x) where ...

Similar restrictions apply on import.

9.5.4. Record puns

Record puns are enabled by the flag -XNamedFieldPuns.

When using records, it is common to write a pattern that binds a variable with the same name as a record field, such as:

data C = C {a :: Int}
f (C {a = a}) = a

Record punning permits the variable name to be elided, so one can simply write

f (C {a}) = a

to mean the same pattern as above. That is, in a record pattern, the pattern a expands into the pattern a = a for the same name a.

Note that:

• Record punning can also be used in an expression, writing, for example,

  let a = 1 in C {a}
  

  instead of

  let a = 1 in C {a = a}
  

  The expansion is purely syntactic, so the expanded right-hand side expression refers to the nearest enclosing variable that is spelled the same as the field name.

• Puns and other patterns can be mixed in the same record:

  data C = C {a :: Int, b :: Int}
  f (C {a, b = 4}) = a
  

• Puns can be used wherever record patterns occur (e.g. in let bindings or at the top-level).

• A pun on a qualified field name is expanded by stripping off the module qualifier. For example:

  f (C {M.a}) = a
  

  means

  f (M.C {M.a = a}) = a
  

  (This is useful if the field selector a for constructor M.C is only in scope in qualified form.)

9.5.5. Record wildcards

Record wildcards are enabled by the flag -XRecordWildCards. This flag implies -XDisambiguateRecordFields.

For records with many fields, it can be tiresome to write out each field individually in a record pattern, as in

data C = C {a :: Int, b :: Int, c :: Int, d :: Int}
f (C {a = 1, b = b, c = c, d = d}) = b + c + d

Record wildcard syntax permits a “..” in a record pattern, where each elided field f is replaced by the pattern f = f. For example, the above pattern can be written as

f (C {a = 1, ..}) = b + c + d

More details:

• Record wildcards in patterns can be mixed with other patterns, including puns (Record puns); for example, in a pattern (C {a = 1, b, ..}). Additionally, record wildcards can be used wherever record patterns occur, including in let bindings and at the top-level. For example, the top-level binding

  C {a = 1, ..} = e
  

  defines b, c, and d.

• Record wildcards can also be used in an expression, when constructing a record. For example,

  let {a = 1; b = 2; c = 3; d = 4} in C {..}
  

  in place of

  let {a = 1; b = 2; c = 3; d = 4} in C {a=a, b=b, c=c, d=d}
  

  The expansion is purely syntactic, so the record wildcard expression refers to the nearest enclosing variables that are spelled the same as the omitted field names.

• Record wildcards may not be used in record updates. For example this is illegal:

  f r = r { x = 3, .. }
  

• For both pattern and expression wildcards, the “..” expands to the missing in-scope record fields. Specifically the expansion of “C {..}” includes f if and only if:

  • f is a record field of constructor C.
  • The record field f is in scope somehow (either qualified or unqualified).
  • In the case of expressions (but not patterns), the variable f is in scope unqualified, apart from the binding of the record selector itself.

  These rules restrict record wildcards to the situations in which the user could have written the expanded version. For example

  module M where
    data R = R { a,b,c :: Int }
  module X where
    import M( R(a,c) )
    f b = R { .. }
  

  The R{..} expands to R{M.a=a}, omitting b since the record field is not in scope, and omitting c since the variable c is not in scope (apart from the binding of the record selector c, of course).

• Record wildcards cannot be used (a) in a record update construct, and (b) for data constructors that are not declared with record fields. For example:

  f x = x { v=True, .. }   -- Illegal (a)
  
  data T = MkT Int Bool
  g = MkT { .. }           -- Illegal (b)
  h (MkT { .. }) = True    -- Illegal (b)
  

9.6. Extensions to the “deriving” mechanism

9.6.1. Inferred context for deriving clauses

The Haskell Report is vague about exactly when a deriving clause is legal. For example:

data T0 f a = MkT0 a         deriving( Eq )
data T1 f a = MkT1 (f a)     deriving( Eq )
data T2 f a = MkT2 (f (f a)) deriving( Eq )

The natural generated Eq code would result in these instance declarations:

instance Eq a         => Eq (T0 f a) where ...
instance Eq (f a)     => Eq (T1 f a) where ...
instance Eq (f (f a)) => Eq (T2 f a) where ...

The first of these is obviously fine. The second is still fine, although less obviously. The third is not Haskell 98, and risks losing termination of instances.

GHC takes a conservative position: it accepts the first two, but not the third. The rule is this: each constraint in the inferred instance context must consist only of type variables, with no repetitions.

This rule is applied regardless of flags. If you want a more exotic context, you can write it yourself, using the standalone deriving mechanism.

9.6.2. Stand-alone deriving declarations

GHC allows stand-alone deriving declarations, enabled by -XStandaloneDeriving:

data Foo a = Bar a | Baz String

deriving instance Eq a => Eq (Foo a)

The syntax is identical to that of an ordinary instance declaration apart from (a) the keyword deriving, and (b) the absence of the where part.

However, standalone deriving differs from a deriving clause in a number of important ways:

• The standalone deriving declaration does not need to be in the same module as the data type declaration. (But be aware of the dangers of orphan instances (Orphan modules and instance declarations).

• You must supply an explicit context (in the example the context is (Eq a)), exactly as you would in an ordinary instance declaration. (In contrast, in a deriving clause attached to a data type declaration, the context is inferred.)

• Unlike a deriving declaration attached to a data declaration, the instance can be more specific than the data type (assuming you also use -XFlexibleInstances, Relaxed rules for instance contexts). Consider for example

  data Foo a = Bar a | Baz String
  
  deriving instance Eq a => Eq (Foo [a])
  deriving instance Eq a => Eq (Foo (Maybe a))
  

  This will generate a derived instance for (Foo [a]) and (Foo (Maybe a)), but other types such as (Foo (Int,Bool)) will not be an instance of Eq.

• Unlike a deriving declaration attached to a data declaration, GHC does not restrict the form of the data type. Instead, GHC simply generates the appropriate boilerplate code for the specified class, and typechecks it. If there is a type error, it is your problem. (GHC will show you the offending code if it has a type error.)

  The merit of this is that you can derive instances for GADTs and other exotic data types, providing only that the boilerplate code does indeed typecheck. For example:

  data T a where
     T1 :: T Int
     T2 :: T Bool
  
  deriving instance Show (T a)
  

  In this example, you cannot say ... deriving( Show ) on the data type declaration for T, because T is a GADT, but you can generate the instance declaration using stand-alone deriving.

  The down-side is that, if the boilerplate code fails to typecheck, you will get an error message about that code, which you did not write. Whereas, with a deriving clause the side-conditions are necessarily more conservative, but any error message may be more comprehensible.

In other ways, however, a standalone deriving obeys the same rules as ordinary deriving:

• A deriving instance declaration must obey the same rules concerning form and termination as ordinary instance declarations, controlled by the same flags; see Instance declarations.

• The stand-alone syntax is generalised for newtypes in exactly the same way that ordinary deriving clauses are generalised (Generalised derived instances for newtypes). For example:

  newtype Foo a = MkFoo (State Int a)
  
  deriving instance MonadState Int Foo
  

  GHC always treats the last parameter of the instance (Foo in this example) as the type whose instance is being derived.

9.6.3. Deriving instances of extra classes (Data, etc.)

Haskell 98 allows the programmer to add “deriving( Eq, Ord )” to a data type declaration, to generate a standard instance declaration for classes specified in the deriving clause. In Haskell 98, the only classes that may appear in the deriving clause are the standard classes Eq, Ord, Enum, Ix, Bounded, Read, and Show.

GHC extends this list with several more classes that may be automatically derived:

• With -XDeriveGeneric, you can derive instances of the classes Generic and Generic1, defined in GHC.Generics. You can use these to define generic functions, as described in Generic programming.
• With -XDeriveFunctor, you can derive instances of the class Functor, defined in GHC.Base. See Deriving Functor instances.
• With -XDeriveDataTypeable, you can derive instances of the class Data, defined in Data.Data. See Deriving Typeable instances for deriving Typeable.
• With -XDeriveFoldable, you can derive instances of the class Foldable, defined in Data.Foldable. See Deriving Foldable instances.
• With -XDeriveTraversable, you can derive instances of the class Traversable, defined in Data.Traversable. Since the Traversable instance dictates the instances of Functor and Foldable, you’ll probably want to derive them too, so -XDeriveTraversable implies -XDeriveFunctor and -XDeriveFoldable. See Deriving Traversable instances.
• With -XDeriveLift, you can derive instances of the class Lift, defined in the Language.Haskell.TH.Syntax module of the template-haskell package. See Deriving Lift instances.

You can also use a standalone deriving declaration instead (see Stand-alone deriving declarations).

In each case the appropriate class must be in scope before it can be mentioned in the deriving clause.

9.6.4. Deriving Functor instances

With -XDeriveFunctor, one can derive Functor instances for data types of kind * -> *. For example, this declaration:

data Example a = Ex a Char (Example a) (Example Char)
  deriving Functor

would generate the following instance:

instance Functor Example where
  fmap f (Ex a1 a2 a3 a4) = Ex (f a1) a2 (fmap f a3) a4

The basic algorithm for -XDeriveFunctor walks the arguments of each constructor of a data type, applying a mapping function depending on the type of each argument. If a plain type variable is found that is syntactically equivalent to the last type parameter of the data type (a in the above example), then we apply the function f directly to it. If a type is encountered that is not syntactically equivalent to the last type parameter but does mention the last type parameter somewhere in it, then a recursive call to fmap is made. If a type is found which doesn’t mention the last type paramter at all, then it is left alone.

The second of those cases, in which a type is unequal to the type parameter but does contain the type parameter, can be surprisingly tricky. For example, the following example compiles:

newtype Right a = Right (Either Int a) deriving Functor

Modifying the code slightly, however, produces code which will not compile:

newtype Wrong a = Wrong (Either a Int) deriving Functor

The difference involves the placement of the last type parameter, a. In the Right case, a occurs within the type Either Int a, and moreover, it appears as the last type argument of Either. In the Wrong case, however, a is not the last type argument to Either; rather, Int is.

This distinction is important because of the way -XDeriveFunctor works. The derived Functor Right instance would be:

instance Functor Right where
  fmap f (Right a) = Right (fmap f a)

Given a value of type Right a, GHC must produce a value of type Right b. Since the argument to the Right constructor has type Either Int a, the code recursively calls fmap on it to produce a value of type Either Int b, which is used in turn to construct a final value of type Right b.

The generated code for the Functor Wrong instance would look exactly the same, except with Wrong replacing every occurrence of Right. The problem is now that fmap is being applied recursively to a value of type Either a Int. This cannot possibly produce a value of type Either b Int, as fmap can only change the last type parameter! This causes the generated code to be ill-typed.

As a general rule, if a data type has a derived Functor instance and its last type parameter occurs on the right-hand side of the data declaration, then either it must (1) occur bare (e.g., newtype Id a = a), or (2) occur as the last argument of a type constructor (as in Right above).

There are two exceptions to this rule:

1. Tuple types. When a non-unit tuple is used on the right-hand side of a data declaration, -XDeriveFunctor treats it as a product of distinct types. In other words, the following code:

   newtype Triple a = Triple (a, Int, [a]) deriving Functor
   

   Would result in a generated Functor instance like so:

   instance Functor Triple where
     fmap f (Triple a) =
       Triple (case a of
                    (a1, a2, a3) -> (f a1, a2, fmap f a3))
   

   That is, -XDeriveFunctor pattern-matches its way into tuples and maps over each type that constitutes the tuple. The generated code is reminiscient of what would be generated from data Triple a = Triple a Int [a], except with extra machinery to handle the tuple.

2. Function types. The last type parameter can appear anywhere in a function type as long as it occurs in a covariant position. To illustrate what this means, consider the following three examples:

   newtype CovFun1 a = CovFun1 (Int -> a) deriving Functor
   newtype CovFun2 a = CovFun2 ((a -> Int) -> a) deriving Functor
   newtype CovFun3 a = CovFun3 (((Int -> a) -> Int) -> a) deriving Functor
   

   All three of these examples would compile without issue. On the other hand:

   newtype ContraFun1 a = ContraFun1 (a -> Int) deriving Functor
   newtype ContraFun2 a = ContraFun2 ((Int -> a) -> Int) deriving Functor
   newtype ContraFun3 a = ContraFun3 (((a -> Int) -> a) -> Int) deriving Functor
   

   While these examples look similar, none of them would successfully compile. This is because all occurrences of the last type parameter a occur in contravariant positions, not covariant ones.

   Intuitively, a covariant type is produced, and a contravariant type is consumed. Most types in Haskell are covariant, but the function type is special in that the lefthand side of a function arrow reverses variance. If a function type a -> b appears in a covariant position (e.g., CovFun1 above), then a is in a contravariant position and b is in a covariant position. Similarly, if a -> b appears in a contravariant position (e.g., CovFun2 above), then a is in a covariant position and b is in a contravariant position.

   To see why a data type with a contravariant occurrence of its last type parameter cannot have a derived Functor instance, let’s suppose that a Functor ContraFun1 instance exists. The implementation would look something like this:

   instance Functor ContraFun1 where
     fmap f (ContraFun g) = ContraFun (\x -> _)
   

   We have f :: a -> b, g :: a -> Int, and x :: b. Using these, we must somehow fill in the hole (denoted with an underscore) with a value of type Int. What are our options?

   We could try applying g to x. This won’t work though, as g expects an argument of type a, and x :: b. Even worse, we can’t turn x into something of type a, since f also needs an argument of type a! In short, there’s no good way to make this work.

   On the other hand, a derived Functor instances for the CovFuns are within the realm of possibility:

   instance Functor CovFun1 where
     fmap f (CovFun1 g) = CovFun1 (\x -> f (g x))
   
   instance Functor CovFun2 where
     fmap f (CovFun2 g) = CovFun2 (\h -> f (g (\x -> h (f x))))
   
   instance Functor CovFun3 where
     fmap f (CovFun3 g) = CovFun3 (\h -> f (g (\k -> h (\x -> f (k x)))))
   

There are some other scenarios in which a derived Functor instance will fail to compile:

1. A data type has no type parameters (e.g., data Nothing = Nothing).

2. A data type’s last type variable is used in a -XDatatypeContexts constraint (e.g., data Ord a => O a = O a).

3. A data type’s last type variable is used in an -XExistentialQuantification constraint, or is refined in a GADT. For example,

   data T a b where
       T4 :: Ord b => b -> T a b
       T5 :: b -> T b b
       T6 :: T a (b,b)
   
   deriving instance Functor (T a)
   

   would not compile successfully due to the way in which b is constrained.

9.6.5. Deriving Foldable instances

With -XDeriveFoldable, one can derive Foldable instances for data types of kind * -> *. For example, this declaration:

data Example a = Ex a Char (Example a) (Example Char)
  deriving Foldable

would generate the following instance:

instance Foldable Example where
  foldr f z (Ex a1 a2 a3 a4) = f a1 (foldr f z a3)
  foldMap f (Ex a1 a2 a3 a4) = mappend (f a1) (foldMap f a3)

The algorithm for -XDeriveFoldable is adapted from the -XDeriveFunctor algorithm, but it generates definitions for foldMap and foldr instead of fmap. Here are the differences between the generated code in each extension:

1. When a bare type variable a is encountered, -XDeriveFunctor would generate f a for an fmap definition. -XDeriveFoldable would generate f a z for foldr, and f a for foldMap.
2. When a type that is not syntactically equivalent to a, but which does contain a, is encountered, -XDeriveFunctor recursively calls fmap on it. Similarly, -XDeriveFoldable would recursively call foldr and foldMap.
3. When a type that does not mention a is encountered, -XDeriveFunctor leaves it alone. On the other hand, -XDeriveFoldable would generate z (the state value) for foldr and mempty for foldMap.
4. -XDeriveFunctor puts everything back together again at the end by invoking the constructor. -XDeriveFoldable, however, builds up a value of some type. For foldr, this is accomplished by chaining applications of f and recursive foldr calls on the state value z. For foldMap, this happens by combining all values with mappend.

There are some other differences regarding what data types can have derived Foldable instances:

1. Data types containing function types on the right-hand side cannot have derived Foldable instances.

2. Foldable instances can be derived for data types in which the last type parameter is existentially constrained or refined in a GADT. For example, this data type:

   data E a where
       E1 :: (a ~ Int) => a   -> E a
       E2 ::              Int -> E Int
       E3 :: (a ~ Int) => a   -> E Int
       E4 :: (a ~ Int) => Int -> E a
   
   deriving instance Foldable E
   

   would have the following generated Foldable instance:

   instance Foldable E where
       foldr f z (E1 e) = f e z
       foldr f z (E2 e) = z
       foldr f z (E3 e) = z
       foldr f z (E4 e) = z
   
       foldMap f (E1 e) = f e
       foldMap f (E2 e) = mempty
       foldMap f (E3 e) = mempty
       foldMap f (E4 e) = mempty
   

   Notice how every constructor of E utilizes some sort of existential quantification, but only the argument of E1 is actually “folded over”. This is because we make a deliberate choice to only fold over universally polymorphic types that are syntactically equivalent to the last type parameter. In particular:

• We don’t fold over the arguments of E1 or E4 beacause even though (a ~ Int), Int is not syntactically equivalent to a.
• We don’t fold over the argument of E3 because a is not universally polymorphic. The a in E3 is (implicitly) existentially quantified, so it is not the same as the last type parameter of E.

9.6.6. Deriving Traversable instances

With -XDeriveTraversable, one can derive Traversable instances for data types of kind * -> *. For example, this declaration:

data Example a = Ex a Char (Example a) (Example Char)
  deriving (Functor, Foldable, Traversable)

would generate the following Traversable instance:

instance Traversable Example where
  traverse f (Ex a1 a2 a3 a4)
    = fmap Ex (f a1) <*> traverse f a3

The algorithm for -XDeriveTraversable is adapted from the -XDeriveFunctor algorithm, but it generates a definition for traverse instead of fmap. Here are the differences between the generated code in each extension:

1. When a bare type variable a is encountered, both -XDeriveFunctor and -XDeriveTraversable would generate f a for an fmap and traverse definition, respectively.
2. When a type that is not syntactically equivalent to a, but which does contain a, is encountered, -XDeriveFunctor recursively calls fmap on it. Similarly, -XDeriveTraversable would recursively call traverse.
3. When a type that does not mention a is encountered, -XDeriveFunctor leaves it alone. On the other hand, -XDeriveTravserable would call pure on the value of that type.
4. -XDeriveFunctor puts everything back together again at the end by invoking the constructor. -XDeriveTraversable does something similar, but it works in an Applicative context by chaining everything together with (<*>).

Unlike -XDeriveFunctor, -XDeriveTraversable cannot be used on data types containing a function type on the right-hand side.

For a full specification of the algorithms used in -XDeriveFunctor, -XDeriveFoldable, and -XDeriveTraversable, see this wiki page.

9.6.7. Deriving Typeable instances

The class Typeable is very special:

• Typeable is kind-polymorphic (see Kind polymorphism).

• GHC has a custom solver for discharging constraints that involve class Typeable, and handwritten instances are forbidden. This ensures that the programmer cannot subvert the type system by writing bogus instances.

• Derived instances of Typeable are ignored, and may be reported as an error in a later version of the compiler.

• The rules for solving `Typeable` constraints are as follows:

  • A concrete type constructor applied to some types.

    instance (Typeable t1, .., Typeable t_n) =>
      Typeable (T t1 .. t_n)
    

    This rule works for any concrete type constructor, including type constructors with polymorphic kinds. The only restriction is that if the type constructor has a polymorphic kind, then it has to be applied to all of its kinds parameters, and these kinds need to be concrete (i.e., they cannot mention kind variables).

  • A type variable applied to some types.
    instance (Typeable f, Typeable t1, .., Typeable t_n) =>
      Typeable (f t1 .. t_n)
    

  • A concrete type literal.
    instance Typeable 0       -- Type natural literals
    instance Typeable "Hello" -- Type-level symbols
    

9.6.8. Deriving Lift instances

The class Lift, unlike other derivable classes, lives in template-haskell instead of base. Having a data type be an instance of Lift permits its values to be promoted to Template Haskell expressions (of type ExpQ), which can then be spliced into Haskell source code.

Here is an example of how one can derive Lift:

{-# LANGUAGE DeriveLift #-}
module Bar where

import Language.Haskell.TH.Syntax

data Foo a = Foo a | a :^: a deriving Lift

{-
instance (Lift a) => Lift (Foo a) where
    lift (Foo a)
    = appE
        (conE
            (mkNameG_d "package-name" "Bar" "Foo"))
        (lift a)
    lift (u :^: v)
    = infixApp
        (lift u)
        (conE
            (mkNameG_d "package-name" "Bar" ":^:"))
        (lift v)
-}

-----
{-# LANGUAGE TemplateHaskell #-}
module Baz where

import Bar
import Language.Haskell.TH.Lift

foo :: Foo String
foo = $(lift $ Foo "foo")

fooExp :: Lift a => Foo a -> Q Exp
fooExp f = [| f |]

-XDeriveLift also works for certain unboxed types (Addr#, Char#, Double#, Float#, Int#, and Word#):

{-# LANGUAGE DeriveLift, MagicHash #-}
module Unboxed where

import GHC.Exts
import Language.Haskell.TH.Syntax

data IntHash = IntHash Int# deriving Lift

{-
instance Lift IntHash where
    lift (IntHash i)
    = appE
        (conE
            (mkNameG_d "package-name" "Unboxed" "IntHash"))
        (litE
            (intPrimL (toInteger (I# i))))
-}

9.6.9. Generalised derived instances for newtypes

When you define an abstract type using newtype, you may want the new type to inherit some instances from its representation. In Haskell 98, you can inherit instances of Eq, Ord, Enum and Bounded by deriving them, but for any other classes you have to write an explicit instance declaration. For example, if you define

newtype Dollars = Dollars Int

and you want to use arithmetic on Dollars, you have to explicitly define an instance of Num:

instance Num Dollars where
  Dollars a + Dollars b = Dollars (a+b)
  ...

All the instance does is apply and remove the newtype constructor. It is particularly galling that, since the constructor doesn’t appear at run-time, this instance declaration defines a dictionary which is wholly equivalent to the Int dictionary, only slower!

9.6.9.1. Generalising the deriving clause

GHC now permits such instances to be derived instead, using the flag -XGeneralizedNewtypeDeriving, so one can write

newtype Dollars = Dollars Int deriving (Eq,Show,Num)

and the implementation uses the same Num dictionary for Dollars as for Int. Notionally, the compiler derives an instance declaration of the form

instance Num Int => Num Dollars

which just adds or removes the newtype constructor according to the type.

We can also derive instances of constructor classes in a similar way. For example, suppose we have implemented state and failure monad transformers, such that

instance Monad m => Monad (State s m)
instance Monad m => Monad (Failure m)

In Haskell 98, we can define a parsing monad by

type Parser tok m a = State [tok] (Failure m) a

which is automatically a monad thanks to the instance declarations above. With the extension, we can make the parser type abstract, without needing to write an instance of class Monad, via

newtype Parser tok m a = Parser (State [tok] (Failure m) a)
                       deriving Monad

In this case the derived instance declaration is of the form

instance Monad (State [tok] (Failure m)) => Monad (Parser tok m)

Notice that, since Monad is a constructor class, the instance is a partial application of the new type, not the entire left hand side. We can imagine that the type declaration is “eta-converted” to generate the context of the instance declaration.

We can even derive instances of multi-parameter classes, provided the newtype is the last class parameter. In this case, a “partial application” of the class appears in the deriving clause. For example, given the class

class StateMonad s m | m -> s where ...
instance Monad m => StateMonad s (State s m) where ...

then we can derive an instance of StateMonad for Parser by

newtype Parser tok m a = Parser (State [tok] (Failure m) a)
                       deriving (Monad, StateMonad [tok])

The derived instance is obtained by completing the application of the class to the new type:

instance StateMonad [tok] (State [tok] (Failure m)) =>
         StateMonad [tok] (Parser tok m)

As a result of this extension, all derived instances in newtype declarations are treated uniformly (and implemented just by reusing the dictionary for the representation type), except Show and Read, which really behave differently for the newtype and its representation.

9.6.9.2. A more precise specification

A derived instance is derived only for declarations of these forms (after expansion of any type synonyms)

newtype T v1..vn                   = MkT (t vk+1..vn) deriving (C t1..tj)
newtype instance T s1..sk vk+1..vn = MkT (t vk+1..vn) deriving (C t1..tj)

where

• v1..vn are type variables, and t, s1..sk, t1..tj are types.
• The (C t1..tj) is a partial applications of the class C, where the arity of C is exactly j+1. That is, C lacks exactly one type argument.
• k is chosen so that C t1..tj (T v1...vk) is well-kinded. (Or, in the case of a data instance, so that C t1..tj (T s1..sk) is well kinded.)
• The type t is an arbitrary type.
• The type variables vk+1...vn do not occur in the types t, s1..sk, or t1..tj.
• C is not Read, Show, Typeable, or Data. These classes should not “look through” the type or its constructor. You can still derive these classes for a newtype, but it happens in the usual way, not via this new mechanism.
• It is safe to coerce each of the methods of C. That is, the missing last argument to C is not used at a nominal role in any of the C‘s methods. (See Roles.)

Then the derived instance declaration is of the form

instance C t1..tj t => C t1..tj (T v1...vk)

As an example which does not work, consider

newtype NonMonad m s = NonMonad (State s m s) deriving Monad

Here we cannot derive the instance

instance Monad (State s m) => Monad (NonMonad m)

because the type variable s occurs in State s m, and so cannot be “eta-converted” away. It is a good thing that this deriving clause is rejected, because NonMonad m is not, in fact, a monad — for the same reason. Try defining >>= with the correct type: you won’t be able to.

Notice also that the order of class parameters becomes important, since we can only derive instances for the last one. If the StateMonad class above were instead defined as

class StateMonad m s | m -> s where ...

then we would not have been able to derive an instance for the Parser type above. We hypothesise that multi-parameter classes usually have one “main” parameter for which deriving new instances is most interesting.

Lastly, all of this applies only for classes other than Read, Show, Typeable, and Data, for which the built-in derivation applies (section 4.3.3. of the Haskell Report). (For the standard classes Eq, Ord, Ix, and Bounded it is immaterial whether the standard method is used or the one described here.)

9.6.10. Deriving any other class

With -XDeriveAnyClass you can derive any other class. The compiler will simply generate an instance declaration with no explicitly-defined methods. This is mostly useful in classes whose minimal set is empty, and especially when writing generic functions.

As an example, consider a simple pretty-printer class SPretty, which outputs pretty strings:

{-# LANGUAGE DefaultSignatures, DeriveAnyClass #-}

class SPretty a where
  sPpr :: a -> String
  default sPpr :: Show a => a -> String
  sPpr = show

If a user does not provide a manual implementation for sPpr, then it will default to show. Now we can leverage the -XDeriveAnyClass extension to easily implement a SPretty instance for a new data type:

data Foo = Foo deriving (Show, SPretty)

The above code is equivalent to:

data Foo = Foo deriving Show
instance SPretty Foo

That is, an SPretty Foo instance will be created with empty implementations for all methods. Since we are using -XDefaultSignatures in this example, a default implementation of sPpr is filled in automatically.

Note the following details

• In case you try to derive some class on a newtype, and -XGeneralizedNewtypeDeriving is also on, -XDeriveAnyClass takes precedence.

• -XDeriveAnyClass is allowed only when the last argument of the class has kind * or (* -> *). So this is not allowed:

  data T a b = MkT a b deriving( Bifunctor )
  

  because the last argument of Bifunctor :: (* -> * -> *) -> Constraint has the wrong kind.

• The instance context will be generated according to the same rules used when deriving Eq (if the kind of the type is *), or the rules for Functor (if the kind of the type is (* -> *)). For example

  instance C a => C (a,b) where ...
  
  data T a b = MkT a (a,b) deriving( C )
  

  The deriving clause will generate

  instance C a => C (T a b) where {}
  

  The constraints C a and C (a,b) are generated from the data constructor arguments, but the latter simplifies to C a.

• -XDeriveAnyClass can be used with partially applied classes, such as

  data T a = MKT a deriving( D Int )
  

  which generates

  instance D Int a => D Int (T a) where {}
  

• -XDeriveAnyClass can be used to fill in default instances for associated type families:

  {-# LANGUAGE DeriveAnyClass, TypeFamilies #-}
  
  class Sizable a where
    type Size a
    type Size a = Int
  
  data Bar = Bar deriving Sizable
  
  doubleBarSize :: Size Bar -> Size Bar
  doubleBarSize s = 2*s
  

  The deriving( Sizable ) is equivalent to saying

  instance Sizeable Bar where {}
  

  and then the normal rules for filling in associated types from the default will apply, making Size Bar equal to Int.

9.7. Class and instances declarations

9.7.1. Class declarations

This section, and the next one, documents GHC’s type-class extensions. There’s lots of background in the paper Type classes: exploring the design space (Simon Peyton Jones, Mark Jones, Erik Meijer).

9.7.1.1. Multi-parameter type classes

Multi-parameter type classes are permitted, with flag -XMultiParamTypeClasses. For example:

class Collection c a where
  union :: c a -> c a -> c a
  ...etc.

9.7.1.2. The superclasses of a class declaration

In Haskell 98 the context of a class declaration (which introduces superclasses) must be simple; that is, each predicate must consist of a class applied to type variables. The flag -XFlexibleContexts (The context of a type signature) lifts this restriction, so that the only restriction on the context in a class declaration is that the class hierarchy must be acyclic. So these class declarations are OK:

class Functor (m k) => FiniteMap m k where
  ...

class (Monad m, Monad (t m)) => Transform t m where
  lift :: m a -> (t m) a

As in Haskell 98, The class hierarchy must be acyclic. However, the definition of “acyclic” involves only the superclass relationships. For example, this is OK:

class C a where {
  op :: D b => a -> b -> b
}

class C a => D a where { ... }

Here, C is a superclass of D, but it’s OK for a class operation op of C to mention D. (It would not be OK for D to be a superclass of C.)

With the extension that adds a kind of constraints, you can write more exotic superclass definitions. The superclass cycle check is even more liberal in these case. For example, this is OK:

class A cls c where
  meth :: cls c => c -> c

class A B c => B c where

A superclass context for a class C is allowed if, after expanding type synonyms to their right-hand-sides, and uses of classes (other than C) to their superclasses, C does not occur syntactically in the context.

9.7.1.3. Class method types

Haskell 98 prohibits class method types to mention constraints on the class type variable, thus:

class Seq s a where
  fromList :: [a] -> s a
  elem     :: Eq a => a -> s a -> Bool

The type of elem is illegal in Haskell 98, because it contains the constraint Eq a, which constrains only the class type variable (in this case a).

GHC lifts this restriction with language extension -XConstrainedClassMethods. The restriction is a pretty stupid one in the first place, so -XConstrainedClassMethods is implied by -XMultiParamTypeClasses.

9.7.1.4. Default method signatures

Haskell 98 allows you to define a default implementation when declaring a class:

class Enum a where
  enum :: [a]
  enum = []

The type of the enum method is [a], and this is also the type of the default method. You can lift this restriction and give another type to the default method using the flag -XDefaultSignatures. For instance, if you have written a generic implementation of enumeration in a class GEnum with method genum in terms of GHC.Generics, you can specify a default method that uses that generic implementation:

class Enum a where
  enum :: [a]
  default enum :: (Generic a, GEnum (Rep a)) => [a]
  enum = map to genum

We reuse the keyword default to signal that a signature applies to the default method only; when defining instances of the Enum class, the original type [a] of enum still applies. When giving an empty instance, however, the default implementation map to genum is filled-in, and type-checked with the type (Generic a, GEnum (Rep a)) => [a].

We use default signatures to simplify generic programming in GHC (Generic programming).

9.7.1.5. Nullary type classes

Nullary (no parameter) type classes are enabled with -XMultiTypeClasses; historically, they were enabled with the (now deprecated) -XNullaryTypeClasses. Since there are no available parameters, there can be at most one instance of a nullary class. A nullary type class might be used to document some assumption in a type signature (such as reliance on the Riemann hypothesis) or add some globally configurable settings in a program. For example,

class RiemannHypothesis where
  assumeRH :: a -> a

-- Deterministic version of the Miller test
-- correctness depends on the generalised Riemann hypothesis
isPrime :: RiemannHypothesis => Integer -> Bool
isPrime n = assumeRH (...)

The type signature of isPrime informs users that its correctness depends on an unproven conjecture. If the function is used, the user has to acknowledge the dependence with:

instance RiemannHypothesis where
  assumeRH = id

9.7.2. Functional dependencies

Functional dependencies are implemented as described by Mark Jones in “Type Classes with Functional Dependencies”, Mark P. Jones, In Proceedings of the 9th European Symposium on Programming, ESOP 2000, Berlin, Germany, March 2000, Springer-Verlag LNCS 1782, .

Functional dependencies are introduced by a vertical bar in the syntax of a class declaration; e.g.

class (Monad m) => MonadState s m | m -> s where ...

class Foo a b c | a b -> c where ...

There should be more documentation, but there isn’t (yet). Yell if you need it.

9.7.2.1. Rules for functional dependencies

In a class declaration, all of the class type variables must be reachable (in the sense mentioned in The context of a type signature) from the free variables of each method type. For example:

class Coll s a where
  empty  :: s
  insert :: s -> a -> s

is not OK, because the type of empty doesn’t mention a. Functional dependencies can make the type variable reachable:

class Coll s a | s -> a where
  empty  :: s
  insert :: s -> a -> s

Alternatively Coll might be rewritten

class Coll s a where
  empty  :: s a
  insert :: s a -> a -> s a

which makes the connection between the type of a collection of a‘s (namely (s a)) and the element type a. Occasionally this really doesn’t work, in which case you can split the class like this:

class CollE s where
  empty  :: s

class CollE s => Coll s a where
  insert :: s -> a -> s

9.7.2.2. Background on functional dependencies

The following description of the motivation and use of functional dependencies is taken from the Hugs user manual, reproduced here (with minor changes) by kind permission of Mark Jones.

Consider the following class, intended as part of a library for collection types:

class Collects e ce where
    empty  :: ce
    insert :: e -> ce -> ce
    member :: e -> ce -> Bool

The type variable e used here represents the element type, while ce is the type of the container itself. Within this framework, we might want to define instances of this class for lists or characteristic functions (both of which can be used to represent collections of any equality type), bit sets (which can be used to represent collections of characters), or hash tables (which can be used to represent any collection whose elements have a hash function). Omitting standard implementation details, this would lead to the following declarations:

instance Eq e => Collects e [e] where ...
instance Eq e => Collects e (e -> Bool) where ...
instance Collects Char BitSet where ...
instance (Hashable e, Collects a ce)
           => Collects e (Array Int ce) where ...

All this looks quite promising; we have a class and a range of interesting implementations. Unfortunately, there are some serious problems with the class declaration. First, the empty function has an ambiguous type:

empty :: Collects e ce => ce

By “ambiguous” we mean that there is a type variable e that appears on the left of the => symbol, but not on the right. The problem with this is that, according to the theoretical foundations of Haskell overloading, we cannot guarantee a well-defined semantics for any term with an ambiguous type.

We can sidestep this specific problem by removing the empty member from the class declaration. However, although the remaining members, insert and member, do not have ambiguous types, we still run into problems when we try to use them. For example, consider the following two functions:

f x y = insert x . insert y
g     = f True 'a'

for which GHC infers the following types:

f :: (Collects a c, Collects b c) => a -> b -> c -> c
g :: (Collects Bool c, Collects Char c) => c -> c

Notice that the type for f allows the two parameters x and y to be assigned different types, even though it attempts to insert each of the two values, one after the other, into the same collection. If we’re trying to model collections that contain only one type of value, then this is clearly an inaccurate type. Worse still, the definition for g is accepted, without causing a type error. As a result, the error in this code will not be flagged at the point where it appears. Instead, it will show up only when we try to use g, which might even be in a different module.

9.7.2.2.1. An attempt to use constructor classes

Faced with the problems described above, some Haskell programmers might be tempted to use something like the following version of the class declaration:

class Collects e c where
   empty  :: c e
   insert :: e -> c e -> c e
   member :: e -> c e -> Bool

The key difference here is that we abstract over the type constructor c that is used to form the collection type c e, and not over that collection type itself, represented by ce in the original class declaration. This avoids the immediate problems that we mentioned above: empty has type Collects e c => c e, which is not ambiguous.

The function f from the previous section has a more accurate type:

f :: (Collects e c) => e -> e -> c e -> c e

The function g from the previous section is now rejected with a type error as we would hope because the type of f does not allow the two arguments to have different types. This, then, is an example of a multiple parameter class that does actually work quite well in practice, without ambiguity problems. There is, however, a catch. This version of the Collects class is nowhere near as general as the original class seemed to be: only one of the four instances for Collects given above can be used with this version of Collects because only one of them—the instance for lists—has a collection type that can be written in the form c e, for some type constructor c, and element type e.

9.7.2.2.2. Adding functional dependencies

To get a more useful version of the Collects class, GHC provides a mechanism that allows programmers to specify dependencies between the parameters of a multiple parameter class (For readers with an interest in theoretical foundations and previous work: The use of dependency information can be seen both as a generalisation of the proposal for “parametric type classes” that was put forward by Chen, Hudak, and Odersky, or as a special case of Mark Jones’s later framework for “improvement” of qualified types. The underlying ideas are also discussed in a more theoretical and abstract setting in a manuscript [implparam], where they are identified as one point in a general design space for systems of implicit parameterisation). To start with an abstract example, consider a declaration such as:

class C a b where ...

which tells us simply that C can be thought of as a binary relation on types (or type constructors, depending on the kinds of a and b). Extra clauses can be included in the definition of classes to add information about dependencies between parameters, as in the following examples:

class D a b | a -> b where ...
class E a b | a -> b, b -> a where ...

The notation a -> b used here between the | and where symbols — not to be confused with a function type — indicates that the a parameter uniquely determines the b parameter, and might be read as “a determines b.” Thus D is not just a relation, but actually a (partial) function. Similarly, from the two dependencies that are included in the definition of E, we can see that E represents a (partial) one-to-one mapping between types.

More generally, dependencies take the form x1 ... xn -> y1 ... ym, where x1, ..., xn, and y1, ..., yn are type variables with n>0 and m>=0, meaning that the y parameters are uniquely determined by the x parameters. Spaces can be used as separators if more than one variable appears on any single side of a dependency, as in t -> a b. Note that a class may be annotated with multiple dependencies using commas as separators, as in the definition of E above. Some dependencies that we can write in this notation are redundant, and will be rejected because they don’t serve any useful purpose, and may instead indicate an error in the program. Examples of dependencies like this include a -> a, a -> a a, a ->, etc. There can also be some redundancy if multiple dependencies are given, as in a->b, b->c, a->c, and in which some subset implies the remaining dependencies. Examples like this are not treated as errors. Note that dependencies appear only in class declarations, and not in any other part of the language. In particular, the syntax for instance declarations, class constraints, and types is completely unchanged.

By including dependencies in a class declaration, we provide a mechanism for the programmer to specify each multiple parameter class more precisely. The compiler, on the other hand, is responsible for ensuring that the set of instances that are in scope at any given point in the program is consistent with any declared dependencies. For example, the following pair of instance declarations cannot appear together in the same scope because they violate the dependency for D, even though either one on its own would be acceptable:

instance D Bool Int where ...
instance D Bool Char where ...

Note also that the following declaration is not allowed, even by itself:

instance D [a] b where ...

The problem here is that this instance would allow one particular choice of [a] to be associated with more than one choice for b, which contradicts the dependency specified in the definition of D. More generally, this means that, in any instance of the form:

instance D t s where ...

for some particular types t and s, the only variables that can appear in s are the ones that appear in t, and hence, if the type t is known, then s will be uniquely determined.

The benefit of including dependency information is that it allows us to define more general multiple parameter classes, without ambiguity problems, and with the benefit of more accurate types. To illustrate this, we return to the collection class example, and annotate the original definition of Collects with a simple dependency:

class Collects e ce | ce -> e where
   empty  :: ce
   insert :: e -> ce -> ce
   member :: e -> ce -> Bool

The dependency ce -> e here specifies that the type e of elements is uniquely determined by the type of the collection ce. Note that both parameters of Collects are of kind *; there are no constructor classes here. Note too that all of the instances of Collects that we gave earlier can be used together with this new definition.

What about the ambiguity problems that we encountered with the original definition? The empty function still has type Collects e ce => ce, but it is no longer necessary to regard that as an ambiguous type: Although the variable e does not appear on the right of the => symbol, the dependency for class Collects tells us that it is uniquely determined by ce, which does appear on the right of the => symbol. Hence the context in which empty is used can still give enough information to determine types for both ce and e, without ambiguity. More generally, we need only regard a type as ambiguous if it contains a variable on the left of the => that is not uniquely determined (either directly or indirectly) by the variables on the right.

Dependencies also help to produce more accurate types for user defined functions, and hence to provide earlier detection of errors, and less cluttered types for programmers to work with. Recall the previous definition for a function f:

f x y = insert x y = insert x . insert y

for which we originally obtained a type:

f :: (Collects a c, Collects b c) => a -> b -> c -> c

Given the dependency information that we have for Collects, however, we can deduce that a and b must be equal because they both appear as the second parameter in a Collects constraint with the same first parameter c. Hence we can infer a shorter and more accurate type for f:

f :: (Collects a c) => a -> a -> c -> c

In a similar way, the earlier definition of g will now be flagged as a type error.

Although we have given only a few examples here, it should be clear that the addition of dependency information can help to make multiple parameter classes more useful in practice, avoiding ambiguity problems, and allowing more general sets of instance declarations.

9.7.3. Instance declarations

An instance declaration has the form

instance ( assertion1, ..., assertionn) => class type1 ... typem where ...

The part before the “=>” is the context, while the part after the “=>” is the head of the instance declaration.

9.7.3.1. Instance resolution

When GHC tries to resolve, say, the constraint C Int Bool, it tries to match every instance declaration against the constraint, by instantiating the head of the instance declaration. Consider these declarations:

instance context1 => C Int a     where ...  -- (A)
instance context2 => C a   Bool  where ...  -- (B)

GHC’s default behaviour is that exactly one instance must match the constraint it is trying to resolve. For example, the constraint C Int Bool matches instances (A) and (B), and hence would be rejected; while C Int Char matches only (A) and hence (A) is chosen.

Notice that

• When matching, GHC takes no account of the context of the instance declaration (context1 etc).
• It is fine for there to be a potential of overlap (by including both declarations (A) and (B), say); an error is only reported if a particular constraint matches more than one.

See also Overlapping instances for flags that loosen the instance resolution rules.

9.7.3.2. Relaxed rules for the instance head

In Haskell 98 the head of an instance declaration must be of the form C (T a1 ... an), where C is the class, T is a data type constructor, and the a1 ... an are distinct type variables. In the case of multi-parameter type classes, this rule applies to each parameter of the instance head (Arguably it should be okay if just one has this form and the others are type variables, but that’s the rules at the moment).

GHC relaxes this rule in two ways:

• With the -XTypeSynonymInstances flag, instance heads may use type synonyms. As always, using a type synonym is just shorthand for writing the RHS of the type synonym definition. For example:

  type Point a = (a,a)
  instance C (Point a)   where ...
  

  is legal. The instance declaration is equivalent to

  instance C (a,a) where ...
  

  As always, type synonyms must be fully applied. You cannot, for example, write:

  instance Monad Point where ...
  

• The -XFlexibleInstances flag allows the head of the instance declaration to mention arbitrary nested types. For example, this becomes a legal instance declaration

  instance C (Maybe Int) where ...
  

  See also the rules on overlap.

  The -XFlexibleInstances flag implies -XTypeSynonymInstances.

However, the instance declaration must still conform to the rules for instance termination: see Instance termination rules.

9.7.3.3. Relaxed rules for instance contexts

In Haskell 98, the class constraints in the context of the instance declaration must be of the form C a where a is a type variable that occurs in the head.

The -XFlexibleContexts flag relaxes this rule, as well as relaxing the corresponding rule for type signatures (see The context of a type signature). Specifically, -XFlexibleContexts, allows (well-kinded) class constraints of form (C t1 ... tn) in the context of an instance declaration.

Notice that the flag does not affect equality constraints in an instance context; they are permitted by -XTypeFamilies or -XGADTs.

However, the instance declaration must still conform to the rules for instance termination: see Instance termination rules.

9.7.3.4. Instance termination rules

Regardless of -XFlexibleInstances and -XFlexibleContexts, instance declarations must conform to some rules that ensure that instance resolution will terminate. The restrictions can be lifted with -XUndecidableInstances (see Undecidable instances).

The rules are these:

1. The Paterson Conditions: for each class constraint (C t1 ... tn) in the context
   1. No type variable has more occurrences in the constraint than in the head
   2. The constraint has fewer constructors and variables (taken together and counting repetitions) than the head
   3. The constraint mentions no type functions. A type function application can in principle expand to a type of arbitrary size, and so are rejected out of hand
2. The Coverage Condition. For each functional dependency, ⟨tvs⟩_left -> ⟨tvs⟩_right, of the class, every type variable in S(⟨tvs⟩_right) must appear in S(⟨tvs⟩_left), where S is the substitution mapping each type variable in the class declaration to the corresponding type in the instance head.

These restrictions ensure that instance resolution terminates: each reduction step makes the problem smaller by at least one constructor. You can find lots of background material about the reason for these restrictions in the paper Understanding functional dependencies via Constraint Handling Rules.

For example, these are okay:

instance C Int [a]          -- Multiple parameters
instance Eq (S [a])         -- Structured type in head

    -- Repeated type variable in head
instance C4 a a => C4 [a] [a]
instance Stateful (ST s) (MutVar s)

    -- Head can consist of type variables only
instance C a
instance (Eq a, Show b) => C2 a b

    -- Non-type variables in context
instance Show (s a) => Show (Sized s a)
instance C2 Int a => C3 Bool [a]
instance C2 Int a => C3 [a] b

But these are not:

    -- Context assertion no smaller than head
instance C a => C a where ...
    -- (C b b) has more occurrences of b than the head
instance C b b => Foo [b] where ...

The same restrictions apply to instances generated by deriving clauses. Thus the following is accepted:

data MinHeap h a = H a (h a)
  deriving (Show)

because the derived instance

instance (Show a, Show (h a)) => Show (MinHeap h a)

conforms to the above rules.

A useful idiom permitted by the above rules is as follows. If one allows overlapping instance declarations then it’s quite convenient to have a “default instance” declaration that applies if something more specific does not:

instance C a where
  op = ... -- Default

9.7.3.5. Undecidable instances

Sometimes even the termination rules of Instance termination rules are too onerous. So GHC allows you to experiment with more liberal rules: if you use the experimental flag -XUndecidableInstances, both the Paterson Conditions and the Coverage Condition (described in Instance termination rules) are lifted. Termination is still ensured by having a fixed-depth recursion stack. If you exceed the stack depth you get a sort of backtrace, and the opportunity to increase the stack depth with -freduction-depth=N. However, if you should exceed the default reduction depth limit, it is probably best just to disable depth checking, with -freduction-depth=0. The exact depth your program requires depends on minutiae of your code, and it may change between minor GHC releases. The safest bet for released code – if you’re sure that it should compile in finite time – is just to disable the check.

For example, sometimes you might want to use the following to get the effect of a “class synonym”:

class (C1 a, C2 a, C3 a) => C a where { }

instance (C1 a, C2 a, C3 a) => C a where { }

This allows you to write shorter signatures:

f :: C a => ...

instead of

f :: (C1 a, C2 a, C3 a) => ...

The restrictions on functional dependencies (Functional dependencies) are particularly troublesome. It is tempting to introduce type variables in the context that do not appear in the head, something that is excluded by the normal rules. For example:

class HasConverter a b | a -> b where
   convert :: a -> b

data Foo a = MkFoo a

instance (HasConverter a b,Show b) => Show (Foo a) where
   show (MkFoo value) = show (convert value)

This is dangerous territory, however. Here, for example, is a program that would make the typechecker loop:

class D a
class F a b | a->b
instance F [a] [[a]]
instance (D c, F a c) => D [a]   -- 'c' is not mentioned in the head

Similarly, it can be tempting to lift the coverage condition:

class Mul a b c | a b -> c where
  (.*.) :: a -> b -> c

instance Mul Int Int Int where (.*.) = (*)
instance Mul Int Float Float where x .*. y = fromIntegral x * y
instance Mul a b c => Mul a [b] [c] where x .*. v = map (x.*.) v

The third instance declaration does not obey the coverage condition; and indeed the (somewhat strange) definition:

f = \ b x y -> if b then x .*. [y] else y

makes instance inference go into a loop, because it requires the constraint (Mul a [b] b).

The -XUndecidableInstances flag is also used to lift some of the restrictions imposed on type family instances. See Decidability of type synonym instances.

9.7.3.6. Overlapping instances

In general, as discussed in Instance resolution, GHC requires that it be unambiguous which instance declaration should be used to resolve a type-class constraint. GHC also provides a way to to loosen the instance resolution, by allowing more than one instance to match, provided there is a most specific one. Moreover, it can be loosened further, by allowing more than one instance to match irrespective of whether there is a most specific one. This section gives the details.

To control the choice of instance, it is possible to specify the overlap behavior for individual instances with a pragma, written immediately after the instance keyword. The pragma may be one of: {-# OVERLAPPING #-}, {-# OVERLAPPABLE #-}, {-# OVERLAPS #-}, or {-# INCOHERENT #-}.

The matching behaviour is also influenced by two module-level language extension flags: -XOverlappingInstances -XOverlappingInstances and -XIncoherentInstances -XIncoherentInstances. These flags are now deprecated (since GHC 7.10) in favour of the fine-grained per-instance pragmas.

A more precise specification is as follows. The willingness to be overlapped or incoherent is a property of the instance declaration itself, controlled as follows:

• An instance is incoherent if: it has an INCOHERENT pragma; or if the instance has no pragma and it appears in a module compiled with -XIncoherentInstances.
• An instance is overlappable if: it has an OVERLAPPABLE or OVERLAPS pragma; or if the instance has no pragma and it appears in a module compiled with -XOverlappingInstances; or if the instance is incoherent.
• An instance is overlapping if: it has an OVERLAPPING or OVERLAPS pragma; or if the instance has no pragma and it appears in a module compiled with -XOverlappingInstances; or if the instance is incoherent.

Now suppose that, in some client module, we are searching for an instance of the target constraint (C ty1 .. tyn). The search works like this:

• Find all instances I that match the target constraint; that is, the target constraint is a substitution instance of I. These instance declarations are the candidates.
• Eliminate any candidate IX for which both of the following hold:
  • There is another candidate IY that is strictly more specific; that is, IY is a substitution instance of IX but not vice versa.
  • Either IX is overlappable, or IY is overlapping. (This “either/or” design, rather than a “both/and” design, allow a client to deliberately override an instance from a library, without requiring a change to the library.)
• If exactly one non-incoherent candidate remains, select it. If all remaining candidates are incoherent, select an arbitrary one. Otherwise the search fails (i.e. when more than one surviving candidate is not incoherent).
• If the selected candidate (from the previous step) is incoherent, the search succeeds, returning that candidate.
• If not, find all instances that unify with the target constraint, but do not match it. Such non-candidate instances might match when the target constraint is further instantiated. If all of them are incoherent, the search succeeds, returning the selected candidate; if not, the search fails.

Notice that these rules are not influenced by flag settings in the client module, where the instances are used. These rules make it possible for a library author to design a library that relies on overlapping instances without the client having to know.

Errors are reported lazily (when attempting to solve a constraint), rather than eagerly (when the instances themselves are defined). Consider, for example

instance C Int  b where ..
instance C a Bool where ..

These potentially overlap, but GHC will not complain about the instance declarations themselves, regardless of flag settings. If we later try to solve the constraint (C Int Char) then only the first instance matches, and all is well. Similarly with (C Bool Bool). But if we try to solve (C Int Bool), both instances match and an error is reported.

As a more substantial example of the rules in action, consider

instance {-# OVERLAPPABLE #-} context1 => C Int b     where ...  -- (A)
instance {-# OVERLAPPABLE #-} context2 => C a   Bool  where ...  -- (B)
instance {-# OVERLAPPABLE #-} context3 => C a   [b]   where ...  -- (C)
instance {-# OVERLAPPING  #-} context4 => C Int [Int] where ...  -- (D)

Now suppose that the type inference engine needs to solve the constraint C Int [Int]. This constraint matches instances (A), (C) and (D), but the last is more specific, and hence is chosen.

If (D) did not exist then (A) and (C) would still be matched, but neither is most specific. In that case, the program would be rejected, unless -XIncoherentInstances is enabled, in which case it would be accepted and (A) or (C) would be chosen arbitrarily.

An instance declaration is more specific than another iff the head of former is a substitution instance of the latter. For example (D) is “more specific” than (C) because you can get from (C) to (D) by substituting a := Int.

GHC is conservative about committing to an overlapping instance. For example:

f :: [b] -> [b]
f x = ...

Suppose that from the RHS of f we get the constraint C b [b]. But GHC does not commit to instance (C), because in a particular call of f, b might be instantiate to Int, in which case instance (D) would be more specific still. So GHC rejects the program.

If, however, you add the flag -XIncoherentInstances when compiling the module that contains (D), GHC will instead pick (C), without complaining about the problem of subsequent instantiations.

Notice that we gave a type signature to f, so GHC had to check that f has the specified type. Suppose instead we do not give a type signature, asking GHC to infer it instead. In this case, GHC will refrain from simplifying the constraint C Int [b] (for the same reason as before) but, rather than rejecting the program, it will infer the type

f :: C b [b] => [b] -> [b]

That postpones the question of which instance to pick to the call site for f by which time more is known about the type b. You can write this type signature yourself if you use the -XFlexibleContexts flag.

Exactly the same situation can arise in instance declarations themselves. Suppose we have

class Foo a where
   f :: a -> a
instance Foo [b] where
   f x = ...

and, as before, the constraint C Int [b] arises from f‘s right hand side. GHC will reject the instance, complaining as before that it does not know how to resolve the constraint C Int [b], because it matches more than one instance declaration. The solution is to postpone the choice by adding the constraint to the context of the instance declaration, thus:

instance C Int [b] => Foo [b] where
   f x = ...

(You need -XFlexibleInstances to do this.)

Warning

Overlapping instances must be used with care. They can give rise to incoherence (i.e. different instance choices are made in different parts of the program) even without -XIncoherentInstances. Consider:

{-# LANGUAGE OverlappingInstances #-}
module Help where

    class MyShow a where
    myshow :: a -> String

    instance MyShow a => MyShow [a] where
    myshow xs = concatMap myshow xs

    showHelp :: MyShow a => [a] -> String
    showHelp xs = myshow xs

{-# LANGUAGE FlexibleInstances, OverlappingInstances #-}
module Main where
    import Help

    data T = MkT

    instance MyShow T where
    myshow x = "Used generic instance"

    instance MyShow [T] where
    myshow xs = "Used more specific instance"

    main = do { print (myshow [MkT]); print (showHelp [MkT]) }

In function showHelp GHC sees no overlapping instances, and so uses the MyShow [a] instance without complaint. In the call to myshow in main, GHC resolves the MyShow [T] constraint using the overlapping instance declaration in module Main. As a result, the program prints

"Used more specific instance"
"Used generic instance"

(An alternative possible behaviour, not currently implemented, would be to reject module Help on the grounds that a later instance declaration might overlap the local one.)

9.7.3.7. Instance signatures: type signatures in instance declarations

In Haskell, you can’t write a type signature in an instance declaration, but it is sometimes convenient to do so, and the language extension -XInstanceSigs allows you to do so. For example:

data T a = MkT a a
instance Eq a => Eq (T a) where
  (==) :: T a -> T a -> Bool   -- The signature
  (==) (MkT x1 x2) (MkTy y1 y2) = x1==y1 && x2==y2

Some details

• The type signature in the instance declaration must be more polymorphic than (or the same as) the one in the class declaration, instantiated with the instance type. For example, this is fine:

  instance Eq a => Eq (T a) where
     (==) :: forall b. b -> b -> Bool
     (==) x y = True
  

  Here the signature in the instance declaration is more polymorphic than that required by the instantiated class method.

• The code for the method in the instance declaration is typechecked against the type signature supplied in the instance declaration, as you would expect. So if the instance signature is more polymorphic than required, the code must be too.

• One stylistic reason for wanting to write a type signature is simple documentation. Another is that you may want to bring scoped type variables into scope. For example:

  class C a where
    foo :: b -> a -> (a, [b])
  
  instance C a => C (T a) where
    foo :: forall b. b -> T a -> (T a, [b])
    foo x (T y) = (T y, xs)
       where
         xs :: [b]
         xs = [x,x,x]
  

  Provided that you also specify -XScopedTypeVariables (Lexically scoped type variables), the forall b scopes over the definition of foo, and in particular over the type signature for xs.

9.7.4. Overloaded string literals

GHC supports overloaded string literals. Normally a string literal has type String, but with overloaded string literals enabled (with -XOverloadedStrings) a string literal has type (IsString a) => a.

This means that the usual string syntax can be used, e.g., for ByteString, Text, and other variations of string like types. String literals behave very much like integer literals, i.e., they can be used in both expressions and patterns. If used in a pattern the literal with be replaced by an equality test, in the same way as an integer literal is.

The class IsString is defined as:

class IsString a where
    fromString :: String -> a

The only predefined instance is the obvious one to make strings work as usual:

instance IsString [Char] where
    fromString cs = cs

The class IsString is not in scope by default. If you want to mention it explicitly (for example, to give an instance declaration for it), you can import it from module GHC.Exts.

Haskell’s defaulting mechanism (Haskell Report, Section 4.3.4) is extended to cover string literals, when -XOverloadedStrings is specified. Specifically:

• Each type in a default declaration must be an instance of Num or of IsString.
• If no default declaration is given, then it is just as if the module contained the declaration default( Integer, Double, String).
• The standard defaulting rule is extended thus: defaulting applies when all the unresolved constraints involve standard classes or IsString; and at least one is a numeric class or IsString.

So, for example, the expression length "foo" will give rise to an ambiguous use of IsString a0 which, because of the above rules, will default to String.

A small example:

module Main where

import GHC.Exts( IsString(..) )

newtype MyString = MyString String deriving (Eq, Show)
instance IsString MyString where
    fromString = MyString

greet :: MyString -> MyString
greet "hello" = "world"
greet other = other

main = do
    print $ greet "hello"
    print $ greet "fool"

Note that deriving Eq is necessary for the pattern matching to work since it gets translated into an equality comparison.

9.7.5. Overloaded lists

GHC supports overloading of the list notation. Let us recap the notation for constructing lists. In Haskell, the list notation can be be used in the following seven ways:

[]          -- Empty list
[x]         -- x : []
[x,y,z]     -- x : y : z : []
[x .. ]     -- enumFrom x
[x,y ..]    -- enumFromThen x y
[x .. y]    -- enumFromTo x y
[x,y .. z]  -- enumFromThenTo x y z

When the OverloadedLists extension is turned on, the aforementioned seven notations are desugared as follows:

[]          -- fromListN 0 []
[x]         -- fromListN 1 (x : [])
[x,y,z]     -- fromListN 3 (x : y : z : [])
[x .. ]     -- fromList (enumFrom x)
[x,y ..]    -- fromList (enumFromThen x y)
[x .. y]    -- fromList (enumFromTo x y)
[x,y .. z]  -- fromList (enumFromThenTo x y z)

This extension allows programmers to use the list notation for construction of structures like: Set, Map, IntMap, Vector, Text and Array. The following code listing gives a few examples:

['0' .. '9']             :: Set Char
[1 .. 10]                :: Vector Int
[("default",0), (k1,v1)] :: Map String Int
['a' .. 'z']             :: Text

List patterns are also overloaded. When the OverloadedLists extension is turned on, these definitions are desugared as follows

f [] = ...          -- f (toList -> []) = ...
g [x,y,z] = ...     -- g (toList -> [x,y,z]) = ...

(Here we are using view-pattern syntax for the translation, see View patterns.)

9.7.5.1. The IsList class

In the above desugarings, the functions toList, fromList and fromListN are all methods of the IsList class, which is itself exported from the GHC.Exts module. The type class is defined as follows:

class IsList l where
  type Item l

  fromList :: [Item l] -> l
  toList   :: l -> [Item l]

  fromListN :: Int -> [Item l] -> l
  fromListN _ = fromList

The IsList class and its methods are intended to be used in conjunction with the OverloadedLists extension.

• The type function Item returns the type of items of the structure l.
• The function fromList constructs the structure l from the given list of Item l.
• The function fromListN takes the input list’s length as a hint. Its behaviour should be equivalent to fromList. The hint can be used for more efficient construction of the structure l compared to fromList. If the given hint is not equal to the input list’s length the behaviour of fromListN is not specified.
• The function toList should be the inverse of fromList.

It is perfectly fine to declare new instances of IsList, so that list notation becomes useful for completely new data types. Here are several example instances:

instance IsList [a] where
  type Item [a] = a
  fromList = id
  toList = id

instance (Ord a) => IsList (Set a) where
  type Item (Set a) = a
  fromList = Set.fromList
  toList = Set.toList

instance (Ord k) => IsList (Map k v) where
  type Item (Map k v) = (k,v)
  fromList = Map.fromList
  toList = Map.toList

instance IsList (IntMap v) where
  type Item (IntMap v) = (Int,v)
  fromList = IntMap.fromList
  toList = IntMap.toList

instance IsList Text where
  type Item Text = Char
  fromList = Text.pack
  toList = Text.unpack

instance IsList (Vector a) where
  type Item (Vector a) = a
  fromList  = Vector.fromList
  fromListN = Vector.fromListN
  toList = Vector.toList

9.7.5.2. Rebindable syntax

When desugaring list notation with -XOverloadedLists GHC uses the fromList (etc) methods from module GHC.Exts. You do not need to import GHC.Exts for this to happen.

However if you use -XRebindableSyntax, then GHC instead uses whatever is in scope with the names of toList, fromList and fromListN. That is, these functions are rebindable; c.f. Rebindable syntax and the implicit Prelude import.

9.7.5.3. Defaulting

Currently, the IsList class is not accompanied with defaulting rules. Although feasible, not much thought has gone into how to specify the meaning of the default declarations like:

default ([a])

9.7.5.4. Speculation about the future

The current implementation of the OverloadedLists extension can be improved by handling the lists that are only populated with literals in a special way. More specifically, the compiler could allocate such lists statically using a compact representation and allow IsList instances to take advantage of the compact representation. Equipped with this capability the OverloadedLists extension will be in a good position to subsume the OverloadedStrings extension (currently, as a special case, string literals benefit from statically allocated compact representation).

9.8. Type families

Indexed type families form an extension to facilitate type-level programming. Type families are a generalisation of associated data types [AssocDataTypes2005] and associated type synonyms [AssocTypeSyn2005] Type families themselves are described in Schrijvers 2008 [TypeFamilies2008]. Type families essentially provide type-indexed data types and named functions on types, which are useful for generic programming and highly parameterised library interfaces as well as interfaces with enhanced static information, much like dependent types. They might also be regarded as an alternative to functional dependencies, but provide a more functional style of type-level programming than the relational style of functional dependencies.

Indexed type families, or type families for short, are type constructors that represent sets of types. Set members are denoted by supplying the type family constructor with type parameters, which are called type indices. The difference between vanilla parametrised type constructors and family constructors is much like between parametrically polymorphic functions and (ad-hoc polymorphic) methods of type classes. Parametric polymorphic functions behave the same at all type instances, whereas class methods can change their behaviour in dependence on the class type parameters. Similarly, vanilla type constructors imply the same data representation for all type instances, but family constructors can have varying representation types for varying type indices.

Indexed type families come in three flavours: data families, open type synonym families, and closed type synonym families. They are the indexed family variants of algebraic data types and type synonyms, respectively. The instances of data families can be data types and newtypes.

Type families are enabled by the flag -XTypeFamilies. Additional information on the use of type families in GHC is available on the Haskell wiki page on type families.

[AssocDataTypes2005]

“Associated Types with Class”, M. Chakravarty, G. Keller, S. Peyton Jones, and S. Marlow. In Proceedings of “The 32nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL‘05)”, pages 1-13, ACM Press, 2005)

[AssocTypeSyn2005]

“Type Associated Type Synonyms”. M. Chakravarty, G. Keller, and S. Peyton Jones. In Proceedings of “The Tenth ACM SIGPLAN International Conference on Functional Programming”, ACM Press, pages 241-253, 2005).

[TypeFamilies2008]

“Type Checking with Open Type Functions”, T. Schrijvers, S. Peyton-Jones, M. Chakravarty, and M. Sulzmann, in Proceedings of “ICFP 2008: The 13th ACM SIGPLAN International Conference on Functional Programming”, ACM Press, pages 51-62, 2008.

9.8.1. Data families

Data families appear in two flavours: (1) they can be defined on the toplevel or (2) they can appear inside type classes (in which case they are known as associated types). The former is the more general variant, as it lacks the requirement for the type-indexes to coincide with the class parameters. However, the latter can lead to more clearly structured code and compiler warnings if some type instances were - possibly accidentally - omitted. In the following, we always discuss the general toplevel form first and then cover the additional constraints placed on associated types.

9.8.1.1. Data family declarations

Indexed data families are introduced by a signature, such as

data family GMap k :: * -> *

The special family distinguishes family from standard data declarations. The result kind annotation is optional and, as usual, defaults to * if omitted. An example is

data family Array e

Named arguments can also be given explicit kind signatures if needed. Just as with GADT declarations named arguments are entirely optional, so that we can declare Array alternatively with

data family Array :: * -> *

9.8.1.2. Data instance declarations

Instance declarations of data and newtype families are very similar to standard data and newtype declarations. The only two differences are that the keyword data or newtype is followed by instance and that some or all of the type arguments can be non-variable types, but may not contain forall types or type synonym families. However, data families are generally allowed in type parameters, and type synonyms are allowed as long as they are fully applied and expand to a type that is itself admissible - exactly as this is required for occurrences of type synonyms in class instance parameters. For example, the Either instance for GMap is

data instance GMap (Either a b) v = GMapEither (GMap a v) (GMap b v)

In this example, the declaration has only one variant. In general, it can be any number.

When the name of a type argument of a data or newtype instance declaration doesn’t matter, it can be replaced with an underscore (_). This is the same as writing a type variable with a unique name.

data family F a b :: *
data instance F Int _ = Int
-- Equivalent to
data instance F Int b = Int

This resembles the wildcards that can be used in Partial Type Signatures. However, there are some differences. Only anonymous wildcards are allowed in these instance declarations, named and extra-constraints wildcards are not. No error messages reporting the inferred types are generated, nor does the flag -XPartialTypeSignatures have any effect.

Data and newtype instance declarations are only permitted when an appropriate family declaration is in scope - just as a class instance declaration requires the class declaration to be visible. Moreover, each instance declaration has to conform to the kind determined by its family declaration. This implies that the number of parameters of an instance declaration matches the arity determined by the kind of the family.

A data family instance declaration can use the full expressiveness of ordinary data or newtype declarations:

• Although, a data family is introduced with the keyword “data”, a data family instance can use either data or newtype. For example:

  data family T a
  data    instance T Int  = T1 Int | T2 Bool
  newtype instance T Char = TC Bool
  

• A data instance can use GADT syntax for the data constructors, and indeed can define a GADT. For example:

  data family G a b
  data instance G [a] b where
     G1 :: c -> G [Int] b
     G2 :: G [a] Bool
  

• You can use a deriving clause on a data instance or newtype instance declaration.

Even if data families are defined as toplevel declarations, functions that perform different computations for different family instances may still need to be defined as methods of type classes. In particular, the following is not possible:

data family T a
data instance T Int  = A
data instance T Char = B
foo :: T a -> Int
foo A = 1             -- WRONG: These two equations together...
foo B = 2             -- ...will produce a type error.

Instead, you would have to write foo as a class operation, thus:

class Foo a where
  foo :: T a -> Int
instance Foo Int where
  foo A = 1
instance Foo Char where
  foo B = 2

Given the functionality provided by GADTs (Generalised Algebraic Data Types), it might seem as if a definition, such as the above, should be feasible. However, type families are - in contrast to GADTs - are open; i.e., new instances can always be added, possibly in other modules. Supporting pattern matching across different data instances would require a form of extensible case construct.

9.8.1.3. Overlap of data instances

The instance declarations of a data family used in a single program may not overlap at all, independent of whether they are associated or not. In contrast to type class instances, this is not only a matter of consistency, but one of type safety.

9.8.2. Synonym families

Type families appear in three flavours: (1) they can be defined as open families on the toplevel, (2) they can be defined as closed families on the toplevel, or (3) they can appear inside type classes (in which case they are known as associated type synonyms). Toplevel families are more general, as they lack the requirement for the type-indexes to coincide with the class parameters. However, associated type synonyms can lead to more clearly structured code and compiler warnings if some type instances were - possibly accidentally - omitted. In the following, we always discuss the general toplevel forms first and then cover the additional constraints placed on associated types. Note that closed associated type synonyms do not exist.

9.8.2.1. Type family declarations

Open indexed type families are introduced by a signature, such as

type family Elem c :: *

The special family distinguishes family from standard type declarations. The result kind annotation is optional and, as usual, defaults to * if omitted. An example is

type family Elem c

Parameters can also be given explicit kind signatures if needed. We call the number of parameters in a type family declaration, the family’s arity, and all applications of a type family must be fully saturated with respect to to that arity. This requirement is unlike ordinary type synonyms and it implies that the kind of a type family is not sufficient to determine a family’s arity, and hence in general, also insufficient to determine whether a type family application is well formed. As an example, consider the following declaration:

type family F a b :: * -> *   -- F's arity is 2,
                              -- although its overall kind is * -> * -> * -> *

Given this declaration the following are examples of well-formed and malformed types:

F Char [Int]       -- OK!  Kind: * -> *
F Char [Int] Bool  -- OK!  Kind: *
F IO Bool          -- WRONG: kind mismatch in the first argument
F Bool             -- WRONG: unsaturated application

The result kind annotation is optional and defaults to * (like argument kinds) if omitted. Polykinded type families can be declared using a parameter in the kind annotation:

type family F a :: k

In this case the kind parameter k is actually an implicit parameter of the type family.

9.8.2.2. Type instance declarations

Instance declarations of type families are very similar to standard type synonym declarations. The only two differences are that the keyword type is followed by instance and that some or all of the type arguments can be non-variable types, but may not contain forall types or type synonym families. However, data families are generally allowed, and type synonyms are allowed as long as they are fully applied and expand to a type that is admissible - these are the exact same requirements as for data instances. For example, the [e] instance for Elem is

type instance Elem [e] = e

Type arguments can be replaced with underscores (_) if the names of the arguments don’t matter. This is the same as writing type variables with unique names. The same rules apply as for Data instance declarations.

Type family instance declarations are only legitimate when an appropriate family declaration is in scope - just like class instances require the class declaration to be visible. Moreover, each instance declaration has to conform to the kind determined by its family declaration, and the number of type parameters in an instance declaration must match the number of type parameters in the family declaration. Finally, the right-hand side of a type instance must be a monotype (i.e., it may not include foralls) and after the expansion of all saturated vanilla type synonyms, no synonyms, except family synonyms may remain.

9.8.2.3. Closed type families

A type family can also be declared with a where clause, defining the full set of equations for that family. For example:

type family F a where
  F Int  = Double
  F Bool = Char
  F a    = String

A closed type family’s equations are tried in order, from top to bottom, when simplifying a type family application. In this example, we declare an instance for F such that F Int simplifies to Double, F Bool simplifies to Char, and for any other type a that is known not to be Int or Bool, F a simplifies to String. Note that GHC must be sure that a cannot unify with Int or Bool in that last case; if a programmer specifies just F a in their code, GHC will not be able to simplify the type. After all, a might later be instantiated with Int.

A closed type family’s equations have the same restrictions as the equations for open type family instances.

A closed type family may be declared with no equations. Such closed type families are opaque type-level definitions that will never reduce, are not necessarily injective (unlike empty data types), and cannot be given any instances. This is different from omitting the equations of a closed type family in a hs-boot file, which uses the syntax where .., as in that case there may or may not be equations given in the hs file.

9.8.2.4. Type family examples

Here are some examples of admissible and illegal type instances:

type family F a :: *
type instance F [Int]   = Int   -- OK!
type instance F String  = Char  -- OK!
type instance F (F a)   = a     -- WRONG: type parameter mentions a type family
type instance
  F (forall a. (a, b))  = b     -- WRONG: a forall type appears in a type parameter
type instance
  F Float = forall a.a          -- WRONG: right-hand side may not be a forall type
type family H a where          -- OK!
  H Int  = Int
  H Bool = Bool
  H a    = String
type instance H Char = Char    -- WRONG: cannot have instances of closed family
type family K a where          -- OK!

type family G a b :: * -> *
type instance G Int            = (,)     -- WRONG: must be two type parameters
type instance G Int Char Float = Double  -- WRONG: must be two type parameters

9.8.2.5. Compatibility and apartness of type family equations

There must be some restrictions on the equations of type families, lest we define an ambiguous rewrite system. So, equations of open type families are restricted to be compatible. Two type patterns are compatible if

1. all corresponding types and implicit kinds in the patterns are apart, or
2. the two patterns unify producing a substitution, and the right-hand sides are equal under that substitution.

Two types are considered apart if, for all possible substitutions, the types cannot reduce to a common reduct.

The first clause of “compatible” is the more straightforward one. It says that the patterns of two distinct type family instances cannot overlap. For example, the following is disallowed:

type instance F Int = Bool
type instance F Int = Char

The second clause is a little more interesting. It says that two overlapping type family instances are allowed if the right-hand sides coincide in the region of overlap. Some examples help here:

type instance F (a, Int) = [a]
type instance F (Int, b) = [b]   -- overlap permitted

type instance G (a, Int)  = [a]
type instance G (Char, a) = [a]  -- ILLEGAL overlap, as [Char] /= [Int]

Note that this compatibility condition is independent of whether the type family is associated or not, and it is not only a matter of consistency, but one of type safety.

For a polykinded type family, the kinds are checked for apartness just like types. For example, the following is accepted:

type family J a :: k
type instance J Int = Bool
type instance J Int = Maybe

These instances are compatible because they differ in their implicit kind parameter; the first uses * while the second uses * -> *.

The definition for “compatible” uses a notion of “apart”, whose definition in turn relies on type family reduction. This condition of “apartness”, as stated, is impossible to check, so we use this conservative approximation: two types are considered to be apart when the two types cannot be unified, even by a potentially infinite unifier. Allowing the unifier to be infinite disallows the following pair of instances:

type instance H x   x = Int
type instance H [x] x = Bool

The type patterns in this pair equal if x is replaced by an infinite nesting of lists. Rejecting instances such as these is necessary for type soundness.

Compatibility also affects closed type families. When simplifying an application of a closed type family, GHC will select an equation only when it is sure that no incompatible previous equation will ever apply. Here are some examples:

type family F a where
  F Int = Bool
  F a   = Char

type family G a where
  G Int = Int
  G a   = a

In the definition for F, the two equations are incompatible – their patterns are not apart, and yet their right-hand sides do not coincide. Thus, before GHC selects the second equation, it must be sure that the first can never apply. So, the type F a does not simplify; only a type such as F Double will simplify to Char. In G, on the other hand, the two equations are compatible. Thus, GHC can ignore the first equation when looking at the second. So, G a will simplify to a.

However see Type, class and other declarations for the overlap rules in GHCi.

9.8.2.6. Decidability of type synonym instances

In order to guarantee that type inference in the presence of type families decidable, we need to place a number of additional restrictions on the formation of type instance declarations (c.f., Definition 5 (Relaxed Conditions) of “Type Checking with Open Type Functions”). Instance declarations have the general form

type instance F t1 .. tn = t

where we require that for every type family application (G s1 .. sm) in t,

1. s1 .. sm do not contain any type family constructors,
2. the total number of symbols (data type constructors and type variables) in s1 .. sm is strictly smaller than in t1 .. tn, and
3. for every type variable a, a occurs in s1 .. sm at most as often as in t1 .. tn.

These restrictions are easily verified and ensure termination of type inference. However, they are not sufficient to guarantee completeness of type inference in the presence of, so called, ‘’loopy equalities’‘, such as a ~ [F a], where a recursive occurrence of a type variable is underneath a family application and data constructor application - see the above mentioned paper for details.

If the option -XUndecidableInstances is passed to the compiler, the above restrictions are not enforced and it is on the programmer to ensure termination of the normalisation of type families during type inference.

9.8.3. Associated data and type families

A data or type synonym family can be declared as part of a type class, thus:

class GMapKey k where
  data GMap k :: * -> *
  ...

class Collects ce where
  type Elem ce :: *
  ...

When doing so, we (optionally) may drop the “family” keyword.

The type parameters must all be type variables, of course, and some (but not necessarily all) of then can be the class parameters. Each class parameter may only be used at most once per associated type, but some may be omitted and they may be in an order other than in the class head. Hence, the following contrived example is admissible:

class C a b c where
  type T c a x :: *

Here c and a are class parameters, but the type is also indexed on a third parameter x.

9.8.3.1. Associated instances

When an associated data or type synonym family instance is declared within a type class instance, we (optionally) may drop the instance keyword in the family instance:

instance (GMapKey a, GMapKey b) => GMapKey (Either a b) where
  data GMap (Either a b) v = GMapEither (GMap a v) (GMap b v)
  ...

instance Eq (Elem [e]) => Collects [e] where
  type Elem [e] = e
  ...

Note the following points:

• The type indexes corresponding to class parameters must have precisely the same shape the type given in the instance head. To have the same “shape” means that the two types are identical modulo renaming of type variables. For example:

  instance Eq (Elem [e]) => Collects [e] where
    -- Choose one of the following alternatives:
    type Elem [e] = e       -- OK
    type Elem [x] = x       -- OK
    type Elem x   = x       -- BAD; shape of 'x' is different to '[e]'
    type Elem [Maybe x] = x -- BAD: shape of '[Maybe x]' is different to '[e]'
  

• An instances for an associated family can only appear as part of an instance declarations of the class in which the family was declared, just as with the equations of the methods of a class.

• The instance for an associated type can be omitted in class instances. In that case, unless there is a default instance (see Associated type synonym defaults), the corresponding instance type is not inhabited; i.e., only diverging expressions, such as undefined, can assume the type.

• Although it is unusual, there (currently) can be multiple instances for an associated family in a single instance declaration. For example, this is legitimate:

  instance GMapKey Flob where
    data GMap Flob [v] = G1 v
    data GMap Flob Int = G2 Int
    ...
  

  Here we give two data instance declarations, one in which the last parameter is [v], and one for which it is Int. Since you cannot give any subsequent instances for (GMap Flob ...), this facility is most useful when the free indexed parameter is of a kind with a finite number of alternatives (unlike *). WARNING: this facility may be withdrawn in the future.

9.8.3.2. Associated type synonym defaults

It is possible for the class defining the associated type to specify a default for associated type instances. So for example, this is OK:

class IsBoolMap v where
  type Key v
  type instance Key v = Int

  lookupKey :: Key v -> v -> Maybe Bool

instance IsBoolMap [(Int, Bool)] where
  lookupKey = lookup

In an instance declaration for the class, if no explicit type instance declaration is given for the associated type, the default declaration is used instead, just as with default class methods.

Note the following points:

• The instance keyword is optional.
• There can be at most one default declaration for an associated type synonym.
• A default declaration is not permitted for an associated data type.
• The default declaration must mention only type variables on the left hand side, and the right hand side must mention only type variables bound on the left hand side. However, unlike the associated type family declaration itself, the type variables of the default instance are independent of those of the parent class.

Here are some examples:

class C a where
  type F1 a :: *
  type instance F1 a = [a]     -- OK
  type instance F1 a = a->a    -- BAD; only one default instance is allowed

  type F2 b a                  -- OK; note the family has more type
                               --     variables than the class
  type instance F2 c d = c->d  -- OK; you don't have to use 'a' in the type instance

  type F3 a
  type F3 [b] = b              -- BAD; only type variables allowed on the LHS

  type F4 a
  type F4 b = a                -- BAD; 'a' is not in scope  in the RHS

9.8.3.3. Scoping of class parameters

The visibility of class parameters in the right-hand side of associated family instances depends solely on the parameters of the family. As an example, consider the simple class declaration

class C a b where
  data T a

Only one of the two class parameters is a parameter to the data family. Hence, the following instance declaration is invalid:

instance C [c] d where
  data T [c] = MkT (c, d)    -- WRONG!!  'd' is not in scope

Here, the right-hand side of the data instance mentions the type variable d that does not occur in its left-hand side. We cannot admit such data instances as they would compromise type safety.

9.8.3.4. Instance contexts and associated type and data instances

Associated type and data instance declarations do not inherit any context specified on the enclosing instance. For type instance declarations, it is unclear what the context would mean. For data instance declarations, it is unlikely a user would want the context repeated for every data constructor. The only place where the context might likely be useful is in a deriving clause of an associated data instance. However, even here, the role of the outer instance context is murky. So, for clarity, we just stick to the rule above: the enclosing instance context is ignored. If you need to use a non-trivial context on a derived instance, use a standalone deriving clause (at the top level).

9.8.4. Import and export

The rules for export lists (Haskell Report Section 5.2) needs adjustment for type families:

• The form T(..), where T is a data family, names the family T and all the in-scope constructors (whether in scope qualified or unqualified) that are data instances of T.
• The form T(.., ci, .., fj, ..), where T is a data family, names T and the specified constructors ci and fields fj as usual. The constructors and field names must belong to some data instance of T, but are not required to belong to the same instance.
• The form C(..), where C is a class, names the class C and all its methods and associated types.
• The form C(.., mi, .., type Tj, ..), where C is a class, names the class C, and the specified methods mi and associated types Tj. The types need a keyword “type” to distinguish them from data constructors.
• Whenever there is no export list and a data instance is defined, the corresponding data family type constructor is exported along with the new data constructors, regardless of whether the data family is defined locally or in another module.

9.8.4.1. Examples

Recall our running GMapKey class example:

class GMapKey k where
  data GMap k :: * -> *
  insert :: GMap k v -> k -> v -> GMap k v
  lookup :: GMap k v -> k -> Maybe v
  empty  :: GMap k v

instance (GMapKey a, GMapKey b) => GMapKey (Either a b) where
  data GMap (Either a b) v = GMapEither (GMap a v) (GMap b v)
  ...method declarations...

Here are some export lists and their meaning:

• module GMap( GMapKey )
  

  Exports just the class name.

• module GMap( GMapKey(..) )
  

  Exports the class, the associated type GMap and the member functions empty, lookup, and insert. The data constructors of GMap (in this case GMapEither) are not exported.

• module GMap( GMapKey( type GMap, empty, lookup, insert ) )
  

  Same as the previous item. Note the “type” keyword.

• module GMap( GMapKey(..), GMap(..) )
  

  Same as previous item, but also exports all the data constructors for GMap, namely GMapEither.

• module GMap ( GMapKey( empty, lookup, insert), GMap(..) )
  

  Same as previous item.

• module GMap ( GMapKey, empty, lookup, insert, GMap(..) )
  

  Same as previous item.

Two things to watch out for:

• You cannot write GMapKey(type GMap(..)) — i.e., sub-component specifications cannot be nested. To specify GMap‘s data constructors, you have to list it separately.

• Consider this example:

  module X where
    data family D
  
  module Y where
    import X
    data instance D Int = D1 | D2
  

  Module Y exports all the entities defined in Y, namely the data constructors D1 and D2, and implicitly the data family D, even though it’s defined in X. This means you can write import Y( D(D1,D2) ) without giving an explicit export list like this:

       module Y( D(..) ) where ...
  or   module Y( module Y, D ) where ...
  

9.8.4.2. Instances

Family instances are implicitly exported, just like class instances. However, this applies only to the heads of instances, not to the data constructors an instance defines.

9.8.5. Type families and instance declarations

Type families require us to extend the rules for the form of instance heads, which are given in Relaxed rules for the instance head. Specifically:

• Data type families may appear in an instance head
• Type synonym families may not appear (at all) in an instance head

The reason for the latter restriction is that there is no way to check for instance matching. Consider

type family F a
type instance F Bool = Int

class C a

instance C Int
instance C (F a)

Now a constraint (C (F Bool)) would match both instances. The situation is especially bad because the type instance for F Bool might be in another module, or even in a module that is not yet written.

However, type class instances of instances of data families can be defined much like any other data type. For example, we can say

data instance T Int = T1 Int | T2 Bool
instance Eq (T Int) where
  (T1 i) == (T1 j) = i==j
  (T2 i) == (T2 j) = i==j
  _      == _      = False

Note that class instances are always for particular instances of a data family and never for an entire family as a whole. This is for essentially the same reasons that we cannot define a toplevel function that performs pattern matching on the data constructors of different instances of a single type family. It would require a form of extensible case construct.

Data instance declarations can also have deriving clauses. For example, we can write

data GMap () v = GMapUnit (Maybe v)
               deriving Show

which implicitly defines an instance of the form

instance Show v => Show (GMap () v) where ...

9.8.6. Injective type families

Starting with GHC 7.12 type families can be annotated with injectivity information. This information is then used by GHC during type checking to resolve type ambiguities in situations where a type variable appears only under type family applications.

For full details on injective type families refer to Haskell Symposium 2015 paper Injective type families for Haskell.

9.8.6.1. Syntax of injectivity annotation

Injectivity annotation is added after type family head and consists of two parts:

• a type variable that names the result of a type family. Syntax: = tyvar or = (tyvar :: kind). Type variable must be fresh.
• an injectivity annotation of the form | A -> B, where A is the result type variable (see previous bullet) and B is a list of argument type and kind variables in which type family is injective. It is possible to omit some variables if type family is not injective in them.

Examples:

type family Id a = result | result -> a where
type family F a b c = d | d -> a c b
type family G (a :: k) b c = foo | foo -> k b where

For open and closed type families it is OK to name the result but skip the injectivity annotation. This is not the case for associated type synonyms, where the named result without injectivity annotation will be interpreted as associated type synonym default.

9.8.6.2. Verifying injectivity annotation against type family equations

Once the user declares type family to be injective GHC must verify that this declaration is correct, ie. type family equations don’t violate the injectivity annotation. A general idea is that if at least one equation (bullets (1), (2) and (3) below) or a pair of equations (bullets (4) and (5) below) violates the injectivity annotation then a type family is not injective in a way user claims and an error is reported. In the bullets below RHS refers to the right-hand side of the type family equation being checked for injectivity. LHS refers to the arguments of that type family equation. Below are the rules followed when checking injectivity of a type family:

1. If a RHS of a type family equation is a type family application GHC reports that the type family is not injective.

2. If a RHS of a type family equation is a bare type variable we require that all LHS variables (including implicit kind variables) are also bare. In other words, this has to be a sole equation of that type family and it has to cover all possible patterns. If the patterns are not covering GHC reports that the type family is not injective.

3. If a LHS type variable that is declared as injective is not mentioned on injective position in the RHS GHC reports that the type family is not injective. Injective position means either argument to a type constructor or injective argument to a type family.

4. Open type familiesOpen type families are typechecked incrementally. This means that when a module is imported type family instances contained in that module are checked against instances present in already imported modules.

   A pair of an open type family equations is checked by attempting to unify their RHSs. If the RHSs don’t unify this pair does not violate injectivity annotation. If unification succeeds with a substitution then LHSs of unified equations must be identical under that substitution. If they are not identical then GHC reports that the type family is not injective.

5. In a closed type family all equations are ordered and in one place. Equations are also checked pair-wise but this time an equation has to be paired with all the preceeding equations. Of course a single-equation closed type family is trivially injective (unless (1), (2) or (3) above holds).

   When checking a pair of closed type family equations GHC tried to unify their RHSs. If they don’t unify this pair of equations does not violate injectivity annotation. If the RHSs can be unified under some substitution (possibly empty) then either the LHSs unify under the same substitution or the LHS of the latter equation is subsumed by earlier equations. If neither condition is met GHC reports that a type family is not injective.

Note that for the purpose of injectivity check in bullets (4) and (5) GHC uses a special variant of unification algorithm that treats type family applications as possibly unifying with anything.

9.9. Kind polymorphism

This section describes kind polymorphism, and extension enabled by -XPolyKinds. It is described in more detail in the paper Giving Haskell a Promotion, which appeared at TLDI 2012.

9.9.1. Overview of kind polymorphism

Currently there is a lot of code duplication in the way Typeable is implemented (Deriving Typeable instances):

class Typeable (t :: *) where
  typeOf :: t -> TypeRep

class Typeable1 (t :: * -> *) where
  typeOf1 :: t a -> TypeRep

class Typeable2 (t :: * -> * -> *) where
  typeOf2 :: t a b -> TypeRep

Kind polymorphism (with -XPolyKinds) allows us to merge all these classes into one:

data Proxy t = Proxy

class Typeable t where
  typeOf :: Proxy t -> TypeRep

instance Typeable Int  where typeOf _ = TypeRep
instance Typeable []   where typeOf _ = TypeRep

Note that the datatype Proxy has kind forall k. k -> * (inferred by GHC), and the new Typeable class has kind forall k. k -> Constraint.

Note the following specific points:

• Generally speaking, with -XPolyKinds, GHC will infer a polymorphic kind for un-decorated declarations, whenever possible. For example, in GHCi

  ghci> :set -XPolyKinds
  ghci> data T m a = MkT (m a)
  ghci> :k T
  T :: (k -> *) -> k -> *
  

• GHC does not usually print explicit foralls, including kind foralls. You can make GHC show them explicitly with -fprint-explicit-foralls (see Verbosity options):

  ghci> :set -XPolyKinds
  ghci> :set -fprint-explicit-foralls
  ghci> data T m a = MkT (m a)
  ghci> :k T
  T :: forall (k :: BOX). (k -> *) -> k -> *
  

• Just as in the world of terms, you can restrict polymorphism using a kind signature (sometimes called a kind annotation)

  data T m (a :: *) = MkT (m a)
  -- GHC now infers kind   T :: (* -> *) -> * -> *
  

  NB: -XPolyKinds implies -XKindSignatures (see Explicitly-kinded quantification).

• The source language does not support an explicit forall for kind variables. Instead, when binding a type variable, you can simply mention a kind variable in a kind annotation for that type-variable binding, thus:

  data T (m :: k -> *) a = MkT (m a)
  -- GHC now infers kind   T :: forall k. (k -> *) -> k -> *
  

• The (implicit) kind “forall” is placed just outside the outermost type-variable binding whose kind annotation mentions the kind variable. For example

  f1 :: (forall a m. m a -> Int) -> Int
           -- f1 :: forall (k::BOX).
           --       (forall (a::k) (m::k->*). m a -> Int)
           --       -> Int
  
  f2 :: (forall (a::k) m. m a -> Int) -> Int
           -- f2 :: (forall (k::BOX) (a::k) (m::k->*). m a -> Int)
           --       -> Int
  

  Here in f1 there is no kind annotation mentioning the polymorphic kind variable, so k is generalised at the top level of the signature for f1. But in the case of of f2 we give a kind annotation in the forall (a:k) binding, and GHC therefore puts the kind forall right there too. This design decision makes default case (f1) as polymorphic as possible; remember that a more polymorphic argument type (as in f2 makes the overall function less polymorphic, because there are fewer acceptable arguments.

Note

These rules are a bit indirect and clumsy. Perhaps GHC should allow explicit kind quantification. But the implicit quantification (e.g. in the declaration for data type T above) is certainly very convenient, and it is not clear what the syntax for explicit quantification should be.

9.9.2. Principles of kind inference

Generally speaking, when -XPolyKinds is on, GHC tries to infer the most general kind for a declaration. For example:

data T f a = MkT (f a)   -- GHC infers:
                         -- T :: forall k. (k->*) -> k -> *

In this case the definition has a right-hand side to inform kind inference. But that is not always the case. Consider

type family F a

Type family declarations have no right-hand side, but GHC must still infer a kind for F. Since there are no constraints, it could infer F :: forall k1 k2. k1 -> k2, but that seems too polymorphic. So GHC defaults those entirely-unconstrained kind variables to * and we get F :: * -> *. You can still declare F to be kind-polymorphic using kind signatures:

type family F1 a               -- F1 :: * -> *
type family F2 (a :: k)        -- F2 :: forall k. k -> *
type family F3 a :: k          -- F3 :: forall k. * -> k
type family F4 (a :: k1) :: k  -- F4 :: forall k1 k2. k1 -> k2

The general principle is this:

• When there is a right-hand side, GHC infers the most polymorphic kind consistent with the right-hand side. Examples: ordinary data type and GADT declarations, class declarations. In the case of a class declaration the role of “right hand side” is played by the class method signatures.
• When there is no right hand side, GHC defaults argument and result kinds to ``*``, except when directed otherwise by a kind signature. Examples: data and type family declarations.

This rule has occasionally-surprising consequences (see Trac #10132.

class C a where    -- Class declarations are generalised
                   -- so C :: forall k. k -> Constraint
  data D1 a        -- No right hand side for these two family
  type F1 a        -- declarations, but the class forces (a :: k)
                   -- so   D1, F1 :: forall k. k -> *

data D2 a   -- No right-hand side so D2 :: * -> *
type F2 a   -- No right-hand side so F2 :: * -> *

The kind-polymorphism from the class declaration makes D1 kind-polymorphic, but not so D2; and similarly F1, F1.

9.9.3. Polymorphic kind recursion and complete kind signatures

Just as in type inference, kind inference for recursive types can only use monomorphic recursion. Consider this (contrived) example:

data T m a = MkT (m a) (T Maybe (m a))
-- GHC infers kind  T :: (* -> *) -> * -> *

The recursive use of T forced the second argument to have kind *. However, just as in type inference, you can achieve polymorphic recursion by giving a complete kind signature for T. A complete kind signature is present when all argument kinds and the result kind are known, without any need for inference. For example:

data T (m :: k -> *) :: k -> * where
  MkT :: m a -> T Maybe (m a) -> T m a

The complete user-supplied kind signature specifies the polymorphic kind for T, and this signature is used for all the calls to T including the recursive ones. In particular, the recursive use of T is at kind *.

What exactly is considered to be a “complete user-supplied kind signature” for a type constructor? These are the forms:

• For a datatype, every type variable must be annotated with a kind. In a GADT-style declaration, there may also be a kind signature (with a top-level :: in the header), but the presence or absence of this annotation does not affect whether or not the declaration has a complete signature.

  data T1 :: (k -> *) -> k -> *       where ...
  -- Yes  T1 :: forall k. (k->*) -> k -> *
  
  data T2 (a :: k -> *) :: k -> *     where ...
  -- Yes  T2 :: forall k. (k->*) -> k -> *
  
  data T3 (a :: k -> *) (b :: k) :: * where ...
  -- Yes  T3 :: forall k. (k->*) -> k -> *
  
  data T4 (a :: k -> *) (b :: k)      where ...
  -- Yes  T4 :: forall k. (k->*) -> k -> *
  
  data T5 a (b :: k) :: *             where ...
  -- No  kind is inferred
  
  data T6 a b                         where ...
  -- No  kind is inferred
  

• For a class, every type variable must be annotated with a kind.

• For a type synonym, every type variable and the result type must all be annotated with kinds.

  type S1 (a :: k) = (a :: k)    -- Yes   S1 :: forall k. k -> k
  type S2 (a :: k) = a           -- No    kind is inferred
  type S3 (a :: k) = Proxy a     -- No    kind is inferred
  

  Note that in S2 and S3, the kind of the right-hand side is rather apparent, but it is still not considered to have a complete signature – no inference can be done before detecting the signature.

• An open type or data family declaration always has a complete user-specified kind signature; un-annotated type variables default to kind *.

  data family D1 a               -- D1 :: * -> *
  data family D2 (a :: k)        -- D2 :: forall k. k -> *
  data family D3 (a :: k) :: *   -- D3 :: forall k. k -> *
  type family S1 a :: k -> *     -- S1 :: forall k. * -> k -> *
  
  class C a where                -- C  :: k -> Constraint
    type AT a b                  -- AT :: k -> * -> *
  

  In the last example, the variable a has an implicit kind variable annotation from the class declaration. It keeps its polymorphic kind in the associated type declaration. The variable b, however, gets defaulted to *.

• A closed type family has a complete signature when all of its type variables are annotated and a return kind (with a top-level ::) is supplied.

9.9.4. Kind inference in closed type families

Although all open type families are considered to have a complete user-specified kind signature, we can relax this condition for closed type families, where we have equations on which to perform kind inference. GHC will infer kinds for the arguments and result types of a closed type family.

GHC supports kind-indexed type families, where the family matches both on the kind and type. GHC will not infer this behaviour without a complete user-supplied kind signature, as doing so would sometimes infer non-principal types.

For example:

type family F1 a where
  F1 True  = False
  F1 False = True
  F1 x     = x
-- F1 fails to compile: kind-indexing is not inferred

type family F2 (a :: k) where
  F2 True  = False
  F2 False = True
  F2 x     = x
-- F2 fails to compile: no complete signature

type family F3 (a :: k) :: k where
  F3 True  = False
  F3 False = True
  F3 x     = x
-- OK

9.9.5. Kind inference in class instance declarations

Consider the following example of a poly-kinded class and an instance for it:

class C a where
  type F a

instance C b where
  type F b = b -> b

In the class declaration, nothing constrains the kind of the type a, so it becomes a poly-kinded type variable (a :: k). Yet, in the instance declaration, the right-hand side of the associated type instance b -> b says that b must be of kind *. GHC could theoretically propagate this information back into the instance head, and make that instance declaration apply only to type of kind *, as opposed to types of any kind. However, GHC does not do this.

In short: GHC does not propagate kind information from the members of a class instance declaration into the instance declaration head.

This lack of kind inference is simply an engineering problem within GHC, but getting it to work would make a substantial change to the inference infrastructure, and it’s not clear the payoff is worth it. If you want to restrict b‘s kind in the instance above, just use a kind signature in the instance head.

9.10. Datatype promotion

This section describes data type promotion, an extension to the kind system that complements kind polymorphism. It is enabled by -XDataKinds, and described in more detail in the paper Giving Haskell a Promotion, which appeared at TLDI 2012.

9.10.1. Motivation

Standard Haskell has a rich type language. Types classify terms and serve to avoid many common programming mistakes. The kind language, however, is relatively simple, distinguishing only lifted types (kind *), type constructors (e.g. kind * -> * -> *), and unlifted types (Unboxed types). In particular when using advanced type system features, such as type families (Type families) or GADTs (Generalised Algebraic Data Types (GADTs)), this simple kind system is insufficient, and fails to prevent simple errors. Consider the example of type-level natural numbers, and length-indexed vectors:

data Ze
data Su n

data Vec :: * -> * -> * where
  Nil  :: Vec a Ze
  Cons :: a -> Vec a n -> Vec a (Su n)

The kind of Vec is * -> * -> *. This means that eg. Vec Int Char is a well-kinded type, even though this is not what we intend when defining length-indexed vectors.

With -XDataKinds, the example above can then be rewritten to:

data Nat = Ze | Su Nat

data Vec :: * -> Nat -> * where
  Nil  :: Vec a Ze
  Cons :: a -> Vec a n -> Vec a (Su n)

With the improved kind of Vec, things like Vec Int Char are now ill-kinded, and GHC will report an error.

9.10.2. Overview

With -XDataKinds, GHC automatically promotes every suitable datatype to be a kind, and its (value) constructors to be type constructors. The following types

data Nat = Ze | Su Nat

data List a = Nil | Cons a (List a)

data Pair a b = Pair a b

data Sum a b = L a | R b

give rise to the following kinds and type constructors:

Nat :: BOX
Ze :: Nat
Su :: Nat -> Nat

List k :: BOX
Nil  :: List k
Cons :: k -> List k -> List k

Pair k1 k2 :: BOX
Pair :: k1 -> k2 -> Pair k1 k2

Sum k1 k2 :: BOX
L :: k1 -> Sum k1 k2
R :: k2 -> Sum k1 k2

where BOX is the (unique) sort that classifies kinds. Note that List, for instance, does not get sort BOX -> BOX, because we do not further classify kinds; all kinds have sort BOX.

The following restrictions apply to promotion:

• We promote data types and newtypes, but not type synonyms, or type/data families (Type families).
• We only promote types whose kinds are of the form * -> ... -> * -> *. In particular, we do not promote higher-kinded datatypes such as data Fix f = In (f (Fix f)), or datatypes whose kinds involve promoted types such as Vec :: * -> Nat -> *.
• We do not promote data constructors that are kind polymorphic, involve constraints, mention type or data families, or involve types that are not promotable.

9.10.3. Distinguishing between types and constructors

Since constructors and types share the same namespace, with promotion you can get ambiguous type names:

data P          -- 1

data Prom = P   -- 2

type T = P      -- 1 or promoted 2?

In these cases, if you want to refer to the promoted constructor, you should prefix its name with a quote:

type T1 = P     -- 1

type T2 = 'P    -- promoted 2

Note that promoted datatypes give rise to named kinds. Since these can never be ambiguous, we do not allow quotes in kind names.

Just as in the case of Template Haskell (Syntax), there is no way to quote a data constructor or type constructor whose second character is a single quote.

9.10.4. Promoted list and tuple types

With -XDataKinds, Haskell’s list and tuple types are natively promoted to kinds, and enjoy the same convenient syntax at the type level, albeit prefixed with a quote:

data HList :: [*] -> * where
  HNil  :: HList '[]
  HCons :: a -> HList t -> HList (a ': t)

data Tuple :: (*,*) -> * where
  Tuple :: a -> b -> Tuple '(a,b)

foo0 :: HList '[]
foo0 = HNil

foo1 :: HList '[Int]
foo1 = HCons (3::Int) HNil

foo2 :: HList [Int, Bool]
foo2 = ...

(Note: the declaration for HCons also requires -XTypeOperators because of infix type operator (:').) For type-level lists of two or more elements, such as the signature of foo2 above, the quote may be omitted because the meaning is unambiguous. But for lists of one or zero elements (as in foo0 and foo1), the quote is required, because the types [] and [Int] have existing meanings in Haskell.

9.10.5. Promoting existential data constructors

Note that we do promote existential data constructors that are otherwise suitable. For example, consider the following:

data Ex :: * where
  MkEx :: forall a. a -> Ex

Both the type Ex and the data constructor MkEx get promoted, with the polymorphic kind 'MkEx :: forall k. k -> Ex. Somewhat surprisingly, you can write a type family to extract the member of a type-level existential:

type family UnEx (ex :: Ex) :: k
type instance UnEx (MkEx x) = x

At first blush, UnEx seems poorly-kinded. The return kind k is not mentioned in the arguments, and thus it would seem that an instance would have to return a member of k for any k. However, this is not the case. The type family UnEx is a kind-indexed type family. The return kind k is an implicit parameter to UnEx. The elaborated definitions are as follows:

type family UnEx (k :: BOX) (ex :: Ex) :: k
type instance UnEx k (MkEx k x) = x

Thus, the instance triggers only when the implicit parameter to UnEx matches the implicit parameter to MkEx. Because k is actually a parameter to UnEx, the kind is not escaping the existential, and the above code is valid.

See also Trac #7347.

9.10.6. Promoting type operators

Type operators are not promoted to the kind level. Why not? Because * is a kind, parsed the way identifiers are. Thus, if a programmer tried to write Either * Bool, would it be Either applied to * and Bool? Or would it be * applied to Either and Bool. To avoid this quagmire, we simply forbid promoting type operators to the kind level.

9.11. Type-Level Literals

GHC supports numeric and string literals at the type level, giving convenient access to a large number of predefined type-level constants. Numeric literals are of kind Nat, while string literals are of kind Symbol. This feature is enabled by the XDataKinds language extension.

The kinds of the literals and all other low-level operations for this feature are defined in module GHC.TypeLits. Note that the module defines some type-level operators that clash with their value-level counterparts (e.g. (+)). Import and export declarations referring to these operators require an explicit namespace annotation (see Explicit namespaces in import/export).

Here is an example of using type-level numeric literals to provide a safe interface to a low-level function:

import GHC.TypeLits
import Data.Word
import Foreign

newtype ArrPtr (n :: Nat) a = ArrPtr (Ptr a)

clearPage :: ArrPtr 4096 Word8 -> IO ()
clearPage (ArrPtr p) = ...

Here is an example of using type-level string literals to simulate simple record operations:

data Label (l :: Symbol) = Get

class Has a l b | a l -> b where
  from :: a -> Label l -> b

data Point = Point Int Int deriving Show

instance Has Point "x" Int where from (Point x _) _ = x
instance Has Point "y" Int where from (Point _ y) _ = y

example = from (Point 1 2) (Get :: Label "x")

9.11.1. Runtime Values for Type-Level Literals

Sometimes it is useful to access the value-level literal associated with a type-level literal. This is done with the functions natVal and symbolVal. For example:

GHC.TypeLits> natVal (Proxy :: Proxy 2)
2

These functions are overloaded because they need to return a different result, depending on the type at which they are instantiated.

natVal :: KnownNat n => proxy n -> Integer

-- instance KnownNat 0
-- instance KnownNat 1
-- instance KnownNat 2
-- ...

GHC discharges the constraint as soon as it knows what concrete type-level literal is being used in the program. Note that this works only for literals and not arbitrary type expressions. For example, a constraint of the form KnownNat (a + b) will not be simplified to (KnownNat a, KnownNat b); instead, GHC will keep the constraint as is, until it can simplify a + b to a constant value.

It is also possible to convert a run-time integer or string value to the corresponding type-level literal. Of course, the resulting type literal will be unknown at compile-time, so it is hidden in an existential type. The conversion may be performed using someNatVal for integers and someSymbolVal for strings:

someNatVal :: Integer -> Maybe SomeNat
SomeNat    :: KnownNat n => Proxy n -> SomeNat

The operations on strings are similar.

9.11.2. Computing With Type-Level Naturals

GHC 7.8 can evaluate arithmetic expressions involving type-level natural numbers. Such expressions may be constructed using the type-families (+), (*), (^) for addition, multiplication, and exponentiation. Numbers may be compared using (<=?), which returns a promoted boolean value, or (<=), which compares numbers as a constraint. For example:

GHC.TypeLits> natVal (Proxy :: Proxy (2 + 3))
5

At present, GHC is quite limited in its reasoning about arithmetic: it will only evaluate the arithmetic type functions and compare the results— in the same way that it does for any other type function. In particular, it does not know more general facts about arithmetic, such as the commutativity and associativity of (+), for example.

However, it is possible to perform a bit of “backwards” evaluation. For example, here is how we could get GHC to compute arbitrary logarithms at the type level:

lg :: Proxy base -> Proxy (base ^ pow) -> Proxy pow
lg _ _ = Proxy

GHC.TypeLits> natVal (lg (Proxy :: Proxy 2) (Proxy :: Proxy 8))
3

9.12. Equality constraints

A type context can include equality constraints of the form t1 ~ t2, which denote that the types t1 and t2 need to be the same. In the presence of type families, whether two types are equal cannot generally be decided locally. Hence, the contexts of function signatures may include equality constraints, as in the following example:

sumCollects :: (Collects c1, Collects c2, Elem c1 ~ Elem c2) => c1 -> c2 -> c2

where we require that the element type of c1 and c2 are the same. In general, the types t1 and t2 of an equality constraint may be arbitrary monotypes; i.e., they may not contain any quantifiers, independent of whether higher-rank types are otherwise enabled.

Equality constraints can also appear in class and instance contexts. The former enable a simple translation of programs using functional dependencies into programs using family synonyms instead. The general idea is to rewrite a class declaration of the form

class C a b | a -> b

to

class (F a ~ b) => C a b where
  type F a

That is, we represent every functional dependency (FD) a1 .. an -> b by an FD type family F a1 .. an and a superclass context equality F a1 .. an ~ b, essentially giving a name to the functional dependency. In class instances, we define the type instances of FD families in accordance with the class head. Method signatures are not affected by that process.

9.12.1. The Coercible constraint

The constraint Coercible t1 t2 is similar to t1 ~ t2, but denotes representational equality between t1 and t2 in the sense of Roles (Roles). It is exported by Data.Coerce, which also contains the documentation. More details and discussion can be found in the paper “Safe Coercions”.

9.13. The Constraint kind

Normally, constraints (which appear in types to the left of the => arrow) have a very restricted syntax. They can only be:

• Class constraints, e.g. Show a
• Implicit parameter constraints, e.g. ?x::Int (with the -XImplicitParams flag)
• Equality constraints, e.g. a ~ Int (with the -XTypeFamilies or -XGADTs flag)

With the -XConstraintKinds flag, GHC becomes more liberal in what it accepts as constraints in your program. To be precise, with this flag any type of the new kind Constraint can be used as a constraint. The following things have kind Constraint:

• Anything which is already valid as a constraint without the flag: saturated applications to type classes, implicit parameter and equality constraints.

• Tuples, all of whose component types have kind Constraint. So for example the type (Show a, Ord a) is of kind Constraint.

• Anything whose form is not yet known, but the user has declared to have kind Constraint (for which they need to import it from GHC.Exts). So for example type Foo (f :: \* -> Constraint) = forall b. f b => b -> b is allowed, as well as examples involving type families:

  type family Typ a b :: Constraint
  type instance Typ Int  b = Show b
  type instance Typ Bool b = Num b
  
  func :: Typ a b => a -> b -> b
  func = ...
  

Note that because constraints are just handled as types of a particular kind, this extension allows type constraint synonyms:

type Stringy a = (Read a, Show a)
foo :: Stringy a => a -> (String, String -> a)
foo x = (show x, read)

Presently, only standard constraints, tuples and type synonyms for those two sorts of constraint are permitted in instance contexts and superclasses (without extra flags). The reason is that permitting more general constraints can cause type checking to loop, as it would with these two programs:

type family Clsish u a
type instance Clsish () a = Cls a
class Clsish () a => Cls a where

class OkCls a where

type family OkClsish u a
type instance OkClsish () a = OkCls a
instance OkClsish () a => OkCls a where

You may write programs that use exotic sorts of constraints in instance contexts and superclasses, but to do so you must use -XUndecidableInstances to signal that you don’t mind if the type checker fails to terminate.

9.14. Other type system extensions

9.14.1. Explicit universal quantification (forall)

Haskell type signatures are implicitly quantified. When the language option -XExplicitForAll is used, the keyword forall allows us to say exactly what this means. For example:

g :: b -> b

means this:

g :: forall b. (b -> b)

The two are treated identically.

Of course forall becomes a keyword; you can’t use forall as a type variable any more!

9.14.2. The context of a type signature

The -XFlexibleContexts flag lifts the Haskell 98 restriction that the type-class constraints in a type signature must have the form (class type-variable) or (class (type-variable type1 type2 ... typen)). With -XFlexibleContexts these type signatures are perfectly okay

g :: Eq [a] => ...
g :: Ord (T a ()) => ...

The flag -XFlexibleContexts also lifts the corresponding restriction on class declarations (The superclasses of a class declaration) and instance declarations (Relaxed rules for instance contexts).

9.14.3. Ambiguous types and the ambiguity check

Each user-written type signature is subjected to an ambiguity check. The ambiguity check rejects functions that can never be called; for example:

f :: C a => Int

The idea is there can be no legal calls to f because every call will give rise to an ambiguous constraint. Indeed, the only purpose of the ambiguity check is to report functions that cannot possibly be called. We could soundly omit the ambiguity check on type signatures entirely, at the expense of delaying ambiguity errors to call sites. Indeed, the language extension -XAllowAmbiguousTypes switches off the ambiguity check.

Ambiguity can be subtle. Consider this example which uses functional dependencies:

class D a b | a -> b where ..
h :: D Int b => Int

The Int may well fix b at the call site, so that signature should not be rejected. Moreover, the dependencies might be hidden. Consider

class X a b where ...
class D a b | a -> b where ...
instance D a b => X [a] b where...
h :: X a b => a -> a

Here h‘s type looks ambiguous in b, but here’s a legal call:

...(h [True])...

That gives rise to a (X [Bool] beta) constraint, and using the instance means we need (D Bool beta) and that fixes beta via D‘s fundep!

Behind all these special cases there is a simple guiding principle. Consider

f :: type
f = ...blah...

g :: type
g = f

You would think that the definition of g would surely typecheck! After all f has exactly the same type, and g=f. But in fact f‘s type is instantiated and the instantiated constraints are solved against the constraints bound by g‘s signature. So, in the case an ambiguous type, solving will fail. For example, consider the earlier definition f :: C a => Int:

f :: C a => Int
f = ...blah...

g :: C a => Int
g = f

In g‘s definition, we’ll instantiate to (C alpha) and try to deduce (C alpha) from (C a), and fail.

So in fact we use this as our definition of ambiguity: a type ty is ambiguous if and only if ((undefined :: ty) :: ty) would fail to typecheck. We use a very similar test for inferred types, to ensure that they too are unambiguous.

Switching off the ambiguity check. Even if a function is has an ambiguous type according the “guiding principle”, it is possible that the function is callable. For example:

class D a b where ...
instance D Bool b where ...

strange :: D a b => a -> a
strange = ...blah...

foo = strange True

Here strange‘s type is ambiguous, but the call in foo is OK because it gives rise to a constraint (D Bool beta), which is soluble by the (D Bool b) instance. So the language extension -XAllowAmbiguousTypes allows you to switch off the ambiguity check. But even with ambiguity checking switched off, GHC will complain about a function that can never be called, such as this one:

f :: (Int ~ Bool) => a -> a

Note

A historical note. GHC used to impose some more restrictive and less principled conditions on type signatures. For type type forall tv1..tvn (c1, ...,cn) => type GHC used to require (a) that each universally quantified type variable tvi must be “reachable” from type, and (b) that every constraint ci mentions at least one of the universally quantified type variables tvi. These ad-hoc restrictions are completely subsumed by the new ambiguity check.

9.14.4. Implicit parameters

Implicit parameters are implemented as described in “Implicit parameters: dynamic scoping with static types”, J Lewis, MB Shields, E Meijer, J Launchbury, 27th ACM Symposium on Principles of Programming Languages (POPL‘00), Boston, Jan 2000. (Most of the following, still rather incomplete, documentation is due to Jeff Lewis.)

Implicit parameter support is enabled with the option -XImplicitParams.

A variable is called dynamically bound when it is bound by the calling context of a function and statically bound when bound by the callee’s context. In Haskell, all variables are statically bound. Dynamic binding of variables is a notion that goes back to Lisp, but was later discarded in more modern incarnations, such as Scheme. Dynamic binding can be very confusing in an untyped language, and unfortunately, typed languages, in particular Hindley-Milner typed languages like Haskell, only support static scoping of variables.

However, by a simple extension to the type class system of Haskell, we can support dynamic binding. Basically, we express the use of a dynamically bound variable as a constraint on the type. These constraints lead to types of the form (?x::t') => t, which says “this function uses a dynamically-bound variable ?x of type t'”. For example, the following expresses the type of a sort function, implicitly parameterised by a comparison function named cmp.

sort :: (?cmp :: a -> a -> Bool) => [a] -> [a]

The dynamic binding constraints are just a new form of predicate in the type class system.

An implicit parameter occurs in an expression using the special form ?x, where x is any valid identifier (e.g. ord ?x is a valid expression). Use of this construct also introduces a new dynamic-binding constraint in the type of the expression. For example, the following definition shows how we can define an implicitly parameterised sort function in terms of an explicitly parameterised sortBy function:

sortBy :: (a -> a -> Bool) -> [a] -> [a]

sort   :: (?cmp :: a -> a -> Bool) => [a] -> [a]
sort    = sortBy ?cmp

9.14.4.1. Implicit-parameter type constraints

Dynamic binding constraints behave just like other type class constraints in that they are automatically propagated. Thus, when a function is used, its implicit parameters are inherited by the function that called it. For example, our sort function might be used to pick out the least value in a list:

least   :: (?cmp :: a -> a -> Bool) => [a] -> a
least xs = head (sort xs)

Without lifting a finger, the ?cmp parameter is propagated to become a parameter of least as well. With explicit parameters, the default is that parameters must always be explicit propagated. With implicit parameters, the default is to always propagate them.

An implicit-parameter type constraint differs from other type class constraints in the following way: All uses of a particular implicit parameter must have the same type. This means that the type of (?x, ?x) is (?x::a) => (a,a), and not (?x::a, ?x::b) => (a, b), as would be the case for type class constraints.

You can’t have an implicit parameter in the context of a class or instance declaration. For example, both these declarations are illegal:

class (?x::Int) => C a where ...
instance (?x::a) => Foo [a] where ...

Reason: exactly which implicit parameter you pick up depends on exactly where you invoke a function. But the “invocation” of instance declarations is done behind the scenes by the compiler, so it’s hard to figure out exactly where it is done. Easiest thing is to outlaw the offending types.

Implicit-parameter constraints do not cause ambiguity. For example, consider:

f :: (?x :: [a]) => Int -> Int
f n = n + length ?x

g :: (Read a, Show a) => String -> String
g s = show (read s)

Here, g has an ambiguous type, and is rejected, but f is fine. The binding for ?x at f‘s call site is quite unambiguous, and fixes the type a.

9.14.4.2. Implicit-parameter bindings

An implicit parameter is bound using the standard let or where binding forms. For example, we define the min function by binding cmp.

min :: [a] -> a
min  = let ?cmp = (<=) in least

A group of implicit-parameter bindings may occur anywhere a normal group of Haskell bindings can occur, except at top level. That is, they can occur in a let (including in a list comprehension, or do-notation, or pattern guards), or a where clause. Note the following points:

• An implicit-parameter binding group must be a collection of simple bindings to implicit-style variables (no function-style bindings, and no type signatures); these bindings are neither polymorphic or recursive.

• You may not mix implicit-parameter bindings with ordinary bindings in a single let expression; use two nested lets instead. (In the case of where you are stuck, since you can’t nest where clauses.)

• You may put multiple implicit-parameter bindings in a single binding group; but they are not treated as a mutually recursive group (as ordinary let bindings are). Instead they are treated as a non-recursive group, simultaneously binding all the implicit parameter. The bindings are not nested, and may be re-ordered without changing the meaning of the program. For example, consider:

  f t = let { ?x = t; ?y = ?x+(1::Int) } in ?x + ?y
  

  The use of ?x in the binding for ?y does not “see” the binding for ?x, so the type of f is

  f :: (?x::Int) => Int -> Int
  

9.14.4.3. Implicit parameters and polymorphic recursion

Consider these two definitions:

len1 :: [a] -> Int
len1 xs = let ?acc = 0 in len_acc1 xs

len_acc1 [] = ?acc
len_acc1 (x:xs) = let ?acc = ?acc + (1::Int) in len_acc1 xs

------------

len2 :: [a] -> Int
len2 xs = let ?acc = 0 in len_acc2 xs

len_acc2 :: (?acc :: Int) => [a] -> Int
len_acc2 [] = ?acc
len_acc2 (x:xs) = let ?acc = ?acc + (1::Int) in len_acc2 xs

The only difference between the two groups is that in the second group len_acc is given a type signature. In the former case, len_acc1 is monomorphic in its own right-hand side, so the implicit parameter ?acc is not passed to the recursive call. In the latter case, because len_acc2 has a type signature, the recursive call is made to the polymorphic version, which takes ?acc as an implicit parameter. So we get the following results in GHCi:

Prog> len1 "hello"
0
Prog> len2 "hello"
5

Adding a type signature dramatically changes the result! This is a rather counter-intuitive phenomenon, worth watching out for.

9.14.4.4. Implicit parameters and monomorphism

GHC applies the dreaded Monomorphism Restriction (section 4.5.5 of the Haskell Report) to implicit parameters. For example, consider:

f :: Int -> Int
f v = let ?x = 0     in
      let y = ?x + v in
      let ?x = 5     in
      y

Since the binding for y falls under the Monomorphism Restriction it is not generalised, so the type of y is simply Int, not (?x::Int) => Int. Hence, (f 9) returns result 9. If you add a type signature for y, then y will get type (?x::Int) => Int, so the occurrence of y in the body of the let will see the inner binding of ?x, so (f 9) will return 14.

9.14.4.5. Special implicit parameters

GHC treats implicit parameters of type GHC.Types.CallStack specially, by resolving them to the current location in the program. Consider:

f :: String
f = show (?loc :: CallStack)

GHC will automatically resolve ?loc to its source location. If another implicit parameter with type CallStack is in scope, GHC will append the two locations, creating an explicit call-stack. For example:

f :: (?stk :: CallStack) => String
f = show (?stk :: CallStack)

will produce the location of ?stk, followed by f‘s call-site. Note that the name of the implicit parameter does not matter (we used ?loc above), GHC will solve any implicit parameter with the right type. The name does, however, matter when pushing new locations onto existing stacks. Consider:

f :: (?stk :: CallStack) => String
f = show (?loc :: CallStack)

When we call f, the stack will include the use of ?loc, but not the call to f; in this case the names must match.

CallStack is kept abstract, but GHC provides a function

getCallStack :: CallStack -> [(String, SrcLoc)]

to access the individual call-sites in the stack. The String is the name of the function that was called, and the SrcLoc provides the package, module, and file name, as well as the line and column numbers. The stack will never be empty, as the first call-site will be the location at which the implicit parameter was used. GHC will also never infer ?loc :: CallStack as a type constraint, which means that functions must explicitly ask to be told about their call-sites.

A potential “gotcha” when using implicit CallStacks is that the :type command in GHCi will not report the ?loc :: CallStack constraint, as the typechecker will immediately solve it. Use :info instead to print the unsolved type.

9.14.5. Explicitly-kinded quantification

Haskell infers the kind of each type variable. Sometimes it is nice to be able to give the kind explicitly as (machine-checked) documentation, just as it is nice to give a type signature for a function. On some occasions, it is essential to do so. For example, in his paper “Restricted Data Types in Haskell” (Haskell Workshop 1999) John Hughes had to define the data type:

data Set cxt a = Set [a]
               | Unused (cxt a -> ())

The only use for the Unused constructor was to force the correct kind for the type variable cxt.

GHC now instead allows you to specify the kind of a type variable directly, wherever a type variable is explicitly bound, with the flag -XKindSignatures.

This flag enables kind signatures in the following places:

• data declarations:

  data Set (cxt :: * -> *) a = Set [a]
  

• type declarations:

  type T (f :: * -> *) = f Int
  

• class declarations:

  class (Eq a) => C (f :: * -> *) a where ...
  

• forall‘s in type signatures:

  f :: forall (cxt :: * -> *). Set cxt Int
  

The parentheses are required. Some of the spaces are required too, to separate the lexemes. If you write “(f::*->*)” you will get a parse error, because “::*->*” is a single lexeme in Haskell.

As part of the same extension, you can put kind annotations in types as well. Thus:

f :: (Int :: *) -> Int
g :: forall a. a -> (a :: *)

The syntax is

atype ::= '(' ctype '::' kind ')

The parentheses are required.

9.14.6. Arbitrary-rank polymorphism

GHC’s type system supports arbitrary-rank explicit universal quantification in types. For example, all the following types are legal:

f1 :: forall a b. a -> b -> a
g1 :: forall a b. (Ord a, Eq  b) => a -> b -> a

f2 :: (forall a. a->a) -> Int -> Int
g2 :: (forall a. Eq a => [a] -> a -> Bool) -> Int -> Int

f3 :: ((forall a. a->a) -> Int) -> Bool -> Bool

f4 :: Int -> (forall a. a -> a)

Here, f1 and g1 are rank-1 types, and can be written in standard Haskell (e.g. f1 :: a->b->a). The forall makes explicit the universal quantification that is implicitly added by Haskell.

The functions f2 and g2 have rank-2 types; the forall is on the left of a function arrow. As g2 shows, the polymorphic type on the left of the function arrow can be overloaded.

The function f3 has a rank-3 type; it has rank-2 types on the left of a function arrow.

The language option -XRankNTypes (which implies -XExplicitForAll, Explicit universal quantification (forall)) enables higher-rank types. That is, you can nest foralls arbitrarily deep in function arrows. For example, a forall-type (also called a “type scheme”), including a type-class context, is legal:

• On the left or right (see f4, for example) of a function arrow
• As the argument of a constructor, or type of a field, in a data type declaration. For example, any of the f1, f2, f3, g1, g2 above would be valid field type signatures.
• As the type of an implicit parameter
• In a pattern type signature (see Lexically scoped type variables)

The -XRankNTypes option is also required for any type with a forall or context to the right of an arrow (e.g. f :: Int -> forall a. a->a, or g :: Int -> Ord a => a -> a). Such types are technically rank 1, but are clearly not Haskell-98, and an extra flag did not seem worth the bother.

In particular, in data and newtype declarations the constructor arguments may be polymorphic types of any rank; see examples in Examples. Note that the declared types are nevertheless always monomorphic. This is important because by default GHC will not instantiate type variables to a polymorphic type (Impredicative polymorphism).

The obsolete language options -XPolymorphicComponents and -XRank2Types are synonyms for -XRankNTypes. They used to specify finer distinctions that GHC no longer makes. (They should really elicit a deprecation warning, but they don’t, purely to avoid the need to library authors to change their old flags specifications.)

9.14.6.1. Examples

These are examples of data and newtype declarations whose data constructors have polymorphic argument types:

data T a = T1 (forall b. b -> b -> b) a

data MonadT m = MkMonad { return :: forall a. a -> m a,
                          bind   :: forall a b. m a -> (a -> m b) -> m b
                        }

newtype Swizzle = MkSwizzle (forall a. Ord a => [a] -> [a])

The constructors have rank-2 types:

T1 :: forall a. (forall b. b -> b -> b) -> a -> T a
MkMonad :: forall m. (forall a. a -> m a)
                  -> (forall a b. m a -> (a -> m b) -> m b)
                  -> MonadT m
MkSwizzle :: (forall a. Ord a => [a] -> [a]) -> Swizzle

In earlier versions of GHC, it was possible to omit the forall in the type of the constructor if there was an explicit context. For example:

newtype Swizzle' = MkSwizzle' (Ord a => [a] -> [a])

As of GHC 7.10, this is deprecated. The -fwarn-context-quantification flag detects this situation and issues a warning. In GHC 7.12, declarations such as MkSwizzle' will cause an out-of-scope error.

As for type signatures, implicit quantification happens for non-overloaded types too. So if you write this:

f :: (a -> a) -> a

it’s just as if you had written this:

f :: forall a. (a -> a) -> a

That is, since the type variable a isn’t in scope, it’s implicitly universally quantified.

You construct values of types T1, MonadT, Swizzle by applying the constructor to suitable values, just as usual. For example,

a1 :: T Int
a1 = T1 (\xy->x) 3

a2, a3 :: Swizzle
a2 = MkSwizzle sort
a3 = MkSwizzle reverse

a4 :: MonadT Maybe
a4 = let r x = Just x
     b m k = case m of
           Just y -> k y
           Nothing -> Nothing
     in
     MkMonad r b

mkTs :: (forall b. b -> b -> b) -> a -> [T a]
mkTs f x y = [T1 f x, T1 f y]

The type of the argument can, as usual, be more general than the type required, as (MkSwizzle reverse) shows. (reverse does not need the Ord constraint.)

When you use pattern matching, the bound variables may now have polymorphic types. For example:

f :: T a -> a -> (a, Char)
f (T1 w k) x = (w k x, w 'c' 'd')

g :: (Ord a, Ord b) => Swizzle -> [a] -> (a -> b) -> [b]
g (MkSwizzle s) xs f = s (map f (s xs))

h :: MonadT m -> [m a] -> m [a]
h m [] = return m []
h m (x:xs) = bind m x          $ \y ->
             bind m (h m xs)   $ \ys ->
             return m (y:ys)

In the function h we use the record selectors return and bind to extract the polymorphic bind and return functions from the MonadT data structure, rather than using pattern matching.

9.14.6.2. Type inference

In general, type inference for arbitrary-rank types is undecidable. GHC uses an algorithm proposed by Odersky and Laufer (“Putting type annotations to work”, POPL‘96) to get a decidable algorithm by requiring some help from the programmer. We do not yet have a formal specification of “some help” but the rule is this:

For a lambda-bound or case-bound variable, x, either the programmer provides an explicit polymorphic type for x, or GHC’s type inference will assume that x’s type has no foralls in it.

What does it mean to “provide” an explicit type for x? You can do that by giving a type signature for x directly, using a pattern type signature (Lexically scoped type variables), thus:

\ f :: (forall a. a->a) -> (f True, f 'c')

Alternatively, you can give a type signature to the enclosing context, which GHC can “push down” to find the type for the variable:

(\ f -> (f True, f 'c')) :: (forall a. a->a) -> (Bool,Char)

Here the type signature on the expression can be pushed inwards to give a type signature for f. Similarly, and more commonly, one can give a type signature for the function itself:

h :: (forall a. a->a) -> (Bool,Char)
h f = (f True, f 'c')

You don’t need to give a type signature if the lambda bound variable is a constructor argument. Here is an example we saw earlier:

f :: T a -> a -> (a, Char)
f (T1 w k) x = (w k x, w 'c' 'd')

Here we do not need to give a type signature to w, because it is an argument of constructor T1 and that tells GHC all it needs to know.

9.14.6.3. Implicit quantification

GHC performs implicit quantification as follows. At the top level (only) of user-written types, if and only if there is no explicit forall, GHC finds all the type variables mentioned in the type that are not already in scope, and universally quantifies them. For example, the following pairs are equivalent:

f :: a -> a
f :: forall a. a -> a

g (x::a) = let
              h :: a -> b -> b
              h x y = y
           in ...
g (x::a) = let
              h :: forall b. a -> b -> b
              h x y = y
           in ...

Notice that GHC does not find the inner-most possible quantification point. For example:

f :: (a -> a) -> Int
         -- MEANS
f :: forall a. (a -> a) -> Int
         -- NOT
f :: (forall a. a -> a) -> Int


g :: (Ord a => a -> a) -> Int
         -- MEANS the illegal type
g :: forall a. (Ord a => a -> a) -> Int
         -- NOT
g :: (forall a. Ord a => a -> a) -> Int

The latter produces an illegal type, which you might think is silly, but at least the rule is simple. If you want the latter type, you can write your foralls explicitly. Indeed, doing so is strongly advised for rank-2 types.

9.14.7. Impredicative polymorphism

In general, GHC will only instantiate a polymorphic function at a monomorphic type (one with no foralls). For example,

runST :: (forall s. ST s a) -> a
id :: forall b. b -> b

foo = id runST   -- Rejected

The definition of foo is rejected because one would have to instantiate id‘s type with b := (forall s. ST s a) -> a, and that is not allowed. Instanting polymorpic type variables with polymorphic types is called impredicative polymorphism.

GHC has extremely flaky support for impredicative polymorphism, enabled with -XImpredicativeTypes. If it worked, this would mean that you could call a polymorphic function at a polymorphic type, and parameterise data structures over polymorphic types. For example:

f :: Maybe (forall a. [a] -> [a]) -> Maybe ([Int], [Char])
f (Just g) = Just (g [3], g "hello")
f Nothing  = Nothing

Notice here that the Maybe type is parameterised by the polymorphic type (forall a. [a] -> [a]). However the extension should be considered highly experimental, and certainly un-supported. You are welcome to try it, but please don’t rely on it working consistently, or working the same in subsequent releases. See this wiki page for more details.

If you want impredicative polymorphism, the main workaround is to use a newtype wrapper. The id runST example can be written using theis workaround like this:

runST :: (forall s. ST s a) -> a
id :: forall b. b -> b

nwetype Wrap a = Wrap { unWrap :: (forall s. ST s a) -> a }

foo :: (forall s. ST s a) -> a
foo = unWrap (id (Wrap runST))
      -- Here id is called at monomorphic type (Wrap a)

9.14.8. Lexically scoped type variables

GHC supports lexically scoped type variables, without which some type signatures are simply impossible to write. For example:

f :: forall a. [a] -> [a]
f xs = ys ++ ys
     where
       ys :: [a]
       ys = reverse xs

The type signature for f brings the type variable a into scope, because of the explicit forall (Declaration type signatures). The type variables bound by a forall scope over the entire definition of the accompanying value declaration. In this example, the type variable a scopes over the whole definition of f, including over the type signature for ys. In Haskell 98 it is not possible to declare a type for ys; a major benefit of scoped type variables is that it becomes possible to do so.

Lexically-scoped type variables are enabled by -XScopedTypeVariables. This flag implies -XRelaxedPolyRec.

9.14.8.1. Overview

The design follows the following principles

• A scoped type variable stands for a type variable, and not for a type. (This is a change from GHC’s earlier design.)
• Furthermore, distinct lexical type variables stand for distinct type variables. This means that every programmer-written type signature (including one that contains free scoped type variables) denotes a rigid type; that is, the type is fully known to the type checker, and no inference is involved.
• Lexical type variables may be alpha-renamed freely, without changing the program.

A lexically scoped type variable can be bound by:

• A declaration type signature (Declaration type signatures)
• An expression type signature (Expression type signatures)
• A pattern type signature (Pattern type signatures)
• Class and instance declarations (Class and instance declarations)

In Haskell, a programmer-written type signature is implicitly quantified over its free type variables (Section 4.1.2 of the Haskell Report). Lexically scoped type variables affect this implicit quantification rules as follows: any type variable that is in scope is not universally quantified. For example, if type variable a is in scope, then

(e :: a -> a)     means     (e :: a -> a)
(e :: b -> b)     means     (e :: forall b. b->b)
(e :: a -> b)     means     (e :: forall b. a->b)

9.14.8.2. Declaration type signatures

A declaration type signature that has explicit quantification (using forall) brings into scope the explicitly-quantified type variables, in the definition of the named function. For example:

f :: forall a. [a] -> [a]
f (x:xs) = xs ++ [ x :: a ]

The “forall a” brings “a” into scope in the definition of “f”.

This only happens if:

• The quantification in f‘s type signature is explicit. For example:

  g :: [a] -> [a]
  g (x:xs) = xs ++ [ x :: a ]
  

  This program will be rejected, because “a” does not scope over the definition of “g”, so “x::a” means “x::forall a. a” by Haskell’s usual implicit quantification rules.

• The signature gives a type for a function binding or a bare variable binding, not a pattern binding. For example:

  f1 :: forall a. [a] -> [a]
  f1 (x:xs) = xs ++ [ x :: a ]   -- OK
  
  f2 :: forall a. [a] -> [a]
  f2 = \(x:xs) -> xs ++ [ x :: a ]   -- OK
  
  f3 :: forall a. [a] -> [a]
  Just f3 = Just (\(x:xs) -> xs ++ [ x :: a ])   -- Not OK!
  

  The binding for f3 is a pattern binding, and so its type signature does not bring a into scope. However f1 is a function binding, and f2 binds a bare variable; in both cases the type signature brings a into scope.

9.14.8.3. Expression type signatures

An expression type signature that has explicit quantification (using forall) brings into scope the explicitly-quantified type variables, in the annotated expression. For example:

f = runST ( (op >>= \(x :: STRef s Int) -> g x) :: forall s. ST s Bool )

Here, the type signature forall s. ST s Bool brings the type variable s into scope, in the annotated expression (op >>= \(x :: STRef s Int) -> g x).

9.14.8.4. Pattern type signatures

A type signature may occur in any pattern; this is a pattern type signature. For example:

-- f and g assume that 'a' is already in scope
f = \(x::Int, y::a) -> x
g (x::a) = x
h ((x,y) :: (Int,Bool)) = (y,x)

In the case where all the type variables in the pattern type signature are already in scope (i.e. bound by the enclosing context), matters are simple: the signature simply constrains the type of the pattern in the obvious way.

Unlike expression and declaration type signatures, pattern type signatures are not implicitly generalised. The pattern in a pattern binding may only mention type variables that are already in scope. For example:

f :: forall a. [a] -> (Int, [a])
f xs = (n, zs)
  where
    (ys::[a], n) = (reverse xs, length xs) -- OK
    zs::[a] = xs ++ ys                     -- OK

    Just (v::b) = ...  -- Not OK; b is not in scope

Here, the pattern signatures for ys and zs are fine, but the one for v is not because b is not in scope.

However, in all patterns other than pattern bindings, a pattern type signature may mention a type variable that is not in scope; in this case, the signature brings that type variable into scope. This is particularly important for existential data constructors. For example:

data T = forall a. MkT [a]

k :: T -> T
k (MkT [t::a]) = MkT t3
               where
                 t3::[a] = [t,t,t]

Here, the pattern type signature (t::a) mentions a lexical type variable that is not already in scope. Indeed, it cannot already be in scope, because it is bound by the pattern match. GHC’s rule is that in this situation (and only then), a pattern type signature can mention a type variable that is not already in scope; the effect is to bring it into scope, standing for the existentially-bound type variable.

When a pattern type signature binds a type variable in this way, GHC insists that the type variable is bound to a rigid, or fully-known, type variable. This means that any user-written type signature always stands for a completely known type.

If all this seems a little odd, we think so too. But we must have some way to bring such type variables into scope, else we could not name existentially-bound type variables in subsequent type signatures.

This is (now) the only situation in which a pattern type signature is allowed to mention a lexical variable that is not already in scope. For example, both f and g would be illegal if a was not already in scope.

9.14.8.5. Class and instance declarations

The type variables in the head of a class or instance declaration scope over the methods defined in the where part. You do not even need an explicit forall. For example:

class C a where
  op :: [a] -> a

  op xs = let ys::[a]
              ys = reverse xs
          in
          head ys

instance C b => C [b] where
  op xs = reverse (head (xs :: [[b]]))

9.14.9. Bindings and generalisation

9.14.9.1. Switching off the dreaded Monomorphism Restriction

Haskell’s monomorphism restriction (see Section 4.5.5 of the Haskell Report) can be completely switched off by -XNoMonomorphismRestriction. Since GHC 7.8.1, the monomorphism restriction is switched off by default in GHCi’s interactive options (see Setting options for interactive evaluation only).

9.14.9.2. Generalised typing of mutually recursive bindings

The Haskell Report specifies that a group of bindings (at top level, or in a let or where) should be sorted into strongly-connected components, and then type-checked in dependency order (Haskell Report, Section 4.5.1). As each group is type-checked, any binders of the group that have an explicit type signature are put in the type environment with the specified polymorphic type, and all others are monomorphic until the group is generalised (Haskell Report, Section 4.5.2).

Following a suggestion of Mark Jones, in his paper Typing Haskell in Haskell, GHC implements a more general scheme. If -XRelaxedPolyRec is specified: the dependency analysis ignores references to variables that have an explicit type signature. As a result of this refined dependency analysis, the dependency groups are smaller, and more bindings will typecheck. For example, consider:

f :: Eq a => a -> Bool
f x = (x == x) || g True || g "Yes"

g y = (y <= y) || f True

This is rejected by Haskell 98, but under Jones’s scheme the definition for g is typechecked first, separately from that for f, because the reference to f in g‘s right hand side is ignored by the dependency analysis. Then g‘s type is generalised, to get

g :: Ord a => a -> Bool

Now, the definition for f is typechecked, with this type for g in the type environment.

The same refined dependency analysis also allows the type signatures of mutually-recursive functions to have different contexts, something that is illegal in Haskell 98 (Section 4.5.2, last sentence). With -XRelaxedPolyRec GHC only insists that the type signatures of a refined group have identical type signatures; in practice this means that only variables bound by the same pattern binding must have the same context. For example, this is fine:

f :: Eq a => a -> Bool
f x = (x == x) || g True

g :: Ord a => a -> Bool
g y = (y <= y) || f True

9.14.9.3. Let-generalisation

An ML-style language usually generalises the type of any let-bound or where-bound variable, so that it is as polymorphic as possible. With the flag -XMonoLocalBinds GHC implements a slightly more conservative policy, using the following rules:

• A variable is closed if and only if
  • the variable is let-bound
  • one of the following holds:
    • the variable has an explicit type signature that has no free type variables, or
    • its binding group is fully generalised (see next bullet)
• A binding group is fully generalised if and only if
  • each of its free variables is either imported or closed, and
  • the binding is not affected by the monomorphism restriction (Haskell Report, Section 4.5.5)

For example, consider

f x = x + 1
g x = let h y = f y * 2
          k z = z+x
      in  h x + k x

Here f is generalised because it has no free variables; and its binding group is unaffected by the monomorphism restriction; and hence f is closed. The same reasoning applies to g, except that it has one closed free variable, namely f. Similarly h is closed, even though it is not bound at top level, because its only free variable f is closed. But k is not closed, because it mentions x which is not closed (because it is not let-bound).

Notice that a top-level binding that is affected by the monomorphism restriction is not closed, and hence may in turn prevent generalisation of bindings that mention it.

The rationale for this more conservative strategy is given in the papers “Let should not be generalised” and “Modular type inference with local assumptions”, and a related blog post.

The flag -XMonoLocalBinds is implied by -XTypeFamilies and -XGADTs. You can switch it off again with -XNoMonoLocalBinds but type inference becomes less predicatable if you do so. (Read the papers!)

9.15. Typed Holes

Typed holes are a feature of GHC that allows special placeholders written with a leading underscore (e.g., “_”, “_foo”, “_bar”), to be used as expressions. During compilation these holes will generate an error message that describes which type is expected at the hole’s location, information about the origin of any free type variables, and a list of local bindings that might help fill the hole with actual code. Typed holes are always enabled in GHC.

The goal of typed holes is to help with writing Haskell code rather than to change the type system. Typed holes can be used to obtain extra information from the type checker, which might otherwise be hard to get. Normally, using GHCi, users can inspect the (inferred) type signatures of all top-level bindings. However, this method is less convenient with terms that are not defined on top-level or inside complex expressions. Holes allow the user to check the type of the term they are about to write.

For example, compiling the following module with GHC:

f :: a -> a
f x = _

will fail with the following error:

hole.hs:2:7:
    Found hole `_' with type: a
    Where: `a' is a rigid type variable bound by
               the type signature for f :: a -> a at hole.hs:1:6
    Relevant bindings include
      f :: a -> a (bound at hole.hs:2:1)
      x :: a (bound at hole.hs:2:3)
    In the expression: _
    In an equation for `f': f x = _

Here are some more details:

• A “Found hole” error usually terminates compilation, like any other type error. After all, you have omitted some code from your program. Nevertheless, you can run and test a piece of code containing holes, by using the -fdefer-typed-holes flag. This flag defers errors produced by typed holes until runtime, and converts them into compile-time warnings. These warnings can in turn be suppressed entirely by -fnowarn-typed-holes).

  The result is that a hole will behave like undefined, but with the added benefits that it shows a warning at compile time, and will show the same message if it gets evaluated at runtime. This behaviour follows that of the -fdefer-type-errors option, which implies -fdefer-typed-holes. See Deferring type errors to runtime.

• All unbound identifiers are treated as typed holes, whether or not they start with an underscore. The only difference is in the error message:

  cons z = z : True : _x : y
  

  yields the errors

  Foo.hs:5:15: error:
      Found hole: _x :: Bool
      Relevant bindings include
        p :: Bool (bound at Foo.hs:3:6)
        cons :: Bool -> [Bool] (bound at Foo.hs:3:1)
  
  Foo.hs:5:20: error:
      Variable not in scope: y :: [Bool]
  

  More information is given for explicit holes (i.e. ones that start with an underscore), than for out-of-scope variables, because the latter are often unintended typos, so the extra information is distracting. If you the detailed information, use a leading underscore to make explicit your intent to use a hole.

• Unbound identifiers with the same name are never unified, even within the same function, but shown individually. For example:

  cons = _x : _x
  

  results in the following errors:

  unbound.hs:1:8:
      Found hole '_x' with type: a
      Where: `a' is a rigid type variable bound by
                 the inferred type of cons :: [a] at unbound.hs:1:1
      Relevant bindings include cons :: [a] (bound at unbound.hs:1:1)
      In the first argument of `(:)', namely `_x'
      In the expression: _x : _x
      In an equation for `cons': cons = _x : _x
  
  unbound.hs:1:13:
      Found hole '_x' with type: [a]
      Arising from: an undeclared identifier `_x' at unbound.hs:1:13-14
      Where: `a' is a rigid type variable bound by
                 the inferred type of cons :: [a] at unbound.hs:1:1
      Relevant bindings include cons :: [a] (bound at unbound.hs:1:1)
      In the second argument of `(:)', namely `_x'
      In the expression: _x : _x
      In an equation for `cons': cons = _x : _x
  

  Notice the two different types reported for the two different occurrences of _x.

• No language extension is required to use typed holes. The lexeme “_” was previously illegal in Haskell, but now has a more informative error message. The lexeme “_x” is a perfectly legal variable, and its behaviour is unchanged when it is in scope. For example

  f _x = _x + 1
  

  does not elict any errors. Only a variable that is not in scope (whether or not it starts with an underscore) is treated as an error (which it always was), albeit now with a more informative error message.

• Unbound data constructors used in expressions behave exactly as above. However, unbound data constructors used in patterns cannot be deferred, and instead bring compilation to a halt. (In implementation terms, they are reported by the renamer rather than the type checker.)

9.16. Partial Type Signatures

A partial type signature is a type signature containing special placeholders written with a leading underscore (e.g., “_”, “_foo”, “_bar”) called wildcards. Partial type signatures are to type signatures what Typed Holes are to expressions. During compilation these wildcards or holes will generate an error message that describes which type was inferred at the hole’s location, and information about the origin of any free type variables. GHC reports such error messages by default.

Unlike Typed Holes, which make the program incomplete and will generate errors when they are evaluated, this needn’t be the case for holes in type signatures. The type checker is capable (in most cases) of type-checking a binding with or without a type signature. A partial type signature bridges the gap between the two extremes, the programmer can choose which parts of a type to annotate and which to leave over to the type-checker to infer.

By default, the type-checker will report an error message for each hole in a partial type signature, informing the programmer of the inferred type. When the -XPartialTypeSignatures flag is enabled, the type-checker will accept the inferred type for each hole, generating warnings instead of errors. Additionally, these warnings can be silenced with the -fno-warn-partial-type-signatures flag.

9.16.1. Syntax

A (partial) type signature has the following form: forall a b .. . (C1, C2, ..) => tau. It consists of three parts:

• The type variables: a b ..
• The constraints: (C1, C2, ..)
• The (mono)type: tau

We distinguish three kinds of wildcards.

9.16.1.1. Type Wildcards

Wildcards occurring within the monotype (tau) part of the type signature are type wildcards (“type” is often omitted as this is the default kind of wildcard). Type wildcards can be instantiated to any monotype like Bool or Maybe [Bool], including functions and higher-kinded types like (Int -> Bool) or Maybe.

not' :: Bool -> _
not' x = not x
-- Inferred: Bool -> Bool

maybools :: _
maybools = Just [True]
-- Inferred: Maybe [Bool]

just1 :: _ Int
just1 = Just 1
-- Inferred: Maybe Int

filterInt :: _ -> _ -> [Int]
filterInt = filter -- has type forall a. (a -> Bool) -> [a] -> [a]
-- Inferred: (Int -> Bool) -> [Int] -> [Int]

For instance, the first wildcard in the type signature not' would produce the following error message:

Test.hs:4:17:
    Found hole ‘_’ with type: Bool
    To use the inferred type, enable PartialTypeSignatures
    In the type signature for ‘not'’: Bool -> _

When a wildcard is not instantiated to a monotype, it will be generalised over, i.e. replaced by a fresh type variable (of which the name will often start with w_), e.g.

foo :: _ -> _
foo x = x
-- Inferred: forall w_. w_ -> w_

filter' :: _
filter' = filter -- has type forall a. (a -> Bool) -> [a] -> [a]
-- Inferred: (a -> Bool) -> [a] -> [a]

9.16.1.2. Named Wildcards

Type wildcards can also be named by giving the underscore an identifier as suffix, i.e. _a. These are called named wildcards. All occurrences of the same named wildcard within one type signature will unify to the same type. For example:

f :: _x -> _x
f ('c', y) = ('d', error "Urk")
-- Inferred: forall t. (Char, t) -> (Char, t)

The named wildcard forces the argument and result types to be the same. Lacking a signature, GHC would have inferred forall a b. (Char, a) -> (Char, b). A named wildcard can be mentioned in constraints, provided it also occurs in the monotype part of the type signature to make sure that it unifies with something:

somethingShowable :: Show _x => _x -> _
somethingShowable x = show x
-- Inferred type: Show w_x => w_x -> String

somethingShowable' :: Show _x => _x -> _
somethingShowable' x = show (not x)
-- Inferred type: Bool -> String

Besides an extra-constraints wildcard (see Extra-Constraints Wildcard), only named wildcards can occur in the constraints, e.g. the _x in Show _x.

Named wildcards should not be confused with type variables. Even though syntactically similar, named wildcards can unify with monotypes as well as be generalised over (and behave as type variables).

In the first example above, _x is generalised over (and is effectively replaced by a fresh type variable w_x). In the second example, _x is unified with the Bool type, and as Bool implements the Show type class, the constraint Show Bool can be simplified away.

By default, GHC (as the Haskell 2010 standard prescribes) parses identifiers starting with an underscore in a type as type variables. To treat them as named wildcards, the -XNamedWildCards flag should be enabled. The example below demonstrated the effect.

foo :: _a -> _a
foo _ = False

Compiling this program without enabling -XNamedWildCards produces the following error message complaining about the type variable _a no matching the actual type Bool.

Test.hs:5:9:
    Couldn't match expected type ‘_a’ with actual type ‘Bool’
      ‘_a’ is a rigid type variable bound by
           the type signature for foo :: _a -> _a at Test.hs:4:8
    Relevant bindings include foo :: _a -> _a (bound at Test.hs:4:1)
    In the expression: False
    In an equation for ‘foo’: foo _ = False

Compiling this program with -XNamedWildCards enabled produces the following error message reporting the inferred type of the named wildcard _a.

Test.hs:4:8: Warning:
    Found hole ‘_a’ with type: Bool
    In the type signature for ‘foo’: _a -> _a

9.16.1.3. Extra-Constraints Wildcard

The third kind of wildcard is the extra-constraints wildcard. The presence of an extra-constraints wildcard indicates that an arbitrary number of extra constraints may be inferred during type checking and will be added to the type signature. In the example below, the extra-constraints wildcard is used to infer three extra constraints.

arbitCs :: _ => a -> String
arbitCs x = show (succ x) ++ show (x == x)
-- Inferred:
--   forall a. (Enum a, Eq a, Show a) => a -> String
-- Error:
Test.hs:5:12:
    Found hole ‘_’ with inferred constraints: (Enum a, Eq a, Show a)
    To use the inferred type, enable PartialTypeSignatures
    In the type signature for ‘arbitCs’: _ => a -> String

An extra-constraints wildcard shouldn’t prevent the programmer from already listing the constraints he knows or wants to annotate, e.g.

-- Also a correct partial type signature:
arbitCs' :: (Enum a, _) => a -> String
arbitCs' x = arbitCs x
-- Inferred:
--   forall a. (Enum a, Show a, Eq a) => a -> String
-- Error:
Test.hs:9:22:
    Found hole ‘_’ with inferred constraints: (Eq a, Show a)
    To use the inferred type, enable PartialTypeSignatures
    In the type signature for ‘arbitCs'’: (Enum a, _) => a -> String

An extra-constraints wildcard can also lead to zero extra constraints to be inferred, e.g.

noCs :: _ => String
noCs = "noCs"
-- Inferred: String
-- Error:
Test.hs:13:9:
    Found hole ‘_’ with inferred constraints: ()
    To use the inferred type, enable PartialTypeSignatures
    In the type signature for ‘noCs’: _ => String

As a single extra-constraints wildcard is enough to infer any number of constraints, only one is allowed in a type signature and it should come last in the list of constraints.

Extra-constraints wildcards cannot be named.

9.16.2. Where can they occur?

Partial type signatures are allowed for bindings, pattern and expression signatures. In all other contexts, e.g. type class or type family declarations, they are disallowed. In the following example a wildcard is used in each of the three possible contexts. Extra-constraints wildcards are not supported in pattern or expression signatures.

{-# LANGUAGE ScopedTypeVariables #-}
foo :: _
foo (x :: _) = (x :: _)
-- Inferred: forall w_. w_ -> w_

Anonymous wildcards can occur in type or data instance declarations. However, these declarations are not partial type signatures and different rules apply. See Data instance declarations for more details.

Partial type signatures can also be used in Template Haskell splices.

• Declaration splices: partial type signature are fully supported.

  {-# LANGUAGE TemplateHaskell, NamedWildCards #-}
  $( [d| foo :: _ => _a -> _a -> _
         foo x y = x == y|] )
  

• Expression splices: anonymous and named wildcards can be used in expression signatures. Extra-constraints wildcards are not supported, just like in regular expression signatures.

  {-# LANGUAGE TemplateHaskell, NamedWildCards #-}
  $( [e| foo = (Just True :: _m _) |] )
  

• Typed expression splices: the same wildcards as in (untyped) expression splices are supported.

• Pattern splices: Template Haskell doesn’t support type signatures in pattern splices. Consequently, partial type signatures are not supported either.

• Type splices: only anonymous wildcards are supported in type splices. Named and extra-constraints wildcards are not.

  {-# LANGUAGE TemplateHaskell #-}
  foo :: $( [t| _ |] ) -> a
  foo x = x
  

9.17. Deferring type errors to runtime

While developing, sometimes it is desirable to allow compilation to succeed even if there are type errors in the code. Consider the following case:

module Main where

a :: Int
a = 'a'

main = print "b"

Even though a is ill-typed, it is not used in the end, so if all that we’re interested in is main it can be useful to be able to ignore the problems in a.

For more motivation and details please refer to the Wiki page or the original paper.

9.17.1. Enabling deferring of type errors

The flag -fdefer-type-errors controls whether type errors are deferred to runtime. Type errors will still be emitted as warnings, but will not prevent compilation. You can use -fno-warn-deferred-type-errors to suppress these warnings.

This flag implies the -fdefer-typed-holes flag, which enables this behaviour for typed holes. Should you so wish, it is possible to enable -fdefer-type-errors without enabling -fdefer-typed-holes, by explicitly specifying -fno-defer-typed-holes on the command-line after the -fdefer-type-errors flag.

At runtime, whenever a term containing a type error would need to be evaluated, the error is converted into a runtime exception of type TypeError. Note that type errors are deferred as much as possible during runtime, but invalid coercions are never performed, even when they would ultimately result in a value of the correct type. For example, given the following code:

x :: Int
x = 0

y :: Char
y = x

z :: Int
z = y

evaluating z will result in a runtime TypeError.

9.17.2. Deferred type errors in GHCi

The flag -fdefer-type-errors works in GHCi as well, with one exception: for “naked” expressions typed at the prompt, type errors don’t get delayed, so for example:

Prelude> fst (True, 1 == 'a')

<interactive>:2:12:
    No instance for (Num Char) arising from the literal `1'
    Possible fix: add an instance declaration for (Num Char)
    In the first argument of `(==)', namely `1'
    In the expression: 1 == 'a'
    In the first argument of `fst', namely `(True, 1 == 'a')'

Otherwise, in the common case of a simple type error such as typing reverse True at the prompt, you would get a warning and then an immediately-following type error when the expression is evaluated.

This exception doesn’t apply to statements, as the following example demonstrates:

Prelude> let x = (True, 1 == 'a')

<interactive>:3:16: Warning:
    No instance for (Num Char) arising from the literal `1'
    Possible fix: add an instance declaration for (Num Char)
    In the first argument of `(==)', namely `1'
    In the expression: 1 == 'a'
    In the expression: (True, 1 == 'a')
Prelude> fst x
True

9.18. Template Haskell

Template Haskell allows you to do compile-time meta-programming in Haskell. The background to the main technical innovations is discussed in “Template Meta-programming for Haskell” (Proc Haskell Workshop 2002).

There is a Wiki page about Template Haskell at http://www.haskell.org/haskellwiki/Template_Haskell, and that is the best place to look for further details. You may also consult the online Haskell library reference material (look for module Language.Haskell.TH). Many changes to the original design are described in Notes on Template Haskell version 2. Not all of these changes are in GHC, however.

The first example from that paper is set out below (A Template Haskell Worked Example) as a worked example to help get you started.

The documentation here describes the realisation of Template Haskell in GHC. It is not detailed enough to understand Template Haskell; see the Wiki page.

9.18.1. Syntax

Template Haskell has the following new syntactic constructions. You need to use the flag -XTemplateHaskell to switch these syntactic extensions on. Alternatively, the -XTemplateHaskellQuotes flag can be used to enable the quotation subset of Template Haskell (i.e. without splice syntax). The -XTemplateHaskellQuotes extension is considered safe under Safe Haskell while -XTemplateHaskell is not.

• A splice is written $x, where x is an identifier, or $(...), where the ”...” is an arbitrary expression. There must be no space between the “$” and the identifier or parenthesis. This use of “$” overrides its meaning as an infix operator, just as “M.x” overrides the meaning of ”.” as an infix operator. If you want the infix operator, put spaces around it.

  A splice can occur in place of

  • an expression; the spliced expression must have type Q Exp
  • a pattern; the spliced pattern must have type Q Pat
  • a type; the spliced expression must have type Q Type
  • a list of declarations at top level; the spliced expression must have type Q [Dec]

  Inside a splice you can only call functions defined in imported modules, not functions defined elsewhere in the same module. Note that declaration splices are not allowed anywhere except at top level (outside any other declarations).

• A expression quotation is written in Oxford brackets, thus:

  • [| ... |], or [e| ... |], where the ”...” is an expression; the quotation has type Q Exp.
  • [d| ... |], where the ”...” is a list of top-level declarations; the quotation has type Q [Dec].
  • [t| ... |], where the ”...” is a type; the quotation has type Q Type.
  • [p| ... |], where the ”...” is a pattern; the quotation has type Q Pat.

  See Where can they occur? for using partial type signatures in quotations.

• A typed expression splice is written $$x, where x is an identifier, or $$(...), where the ”...” is an arbitrary expression.

  A typed expression splice can occur in place of an expression; the spliced expression must have type Q (TExp a)

• A typed expression quotation is written as [|| ... ||], or [e|| ... ||], where the ”...” is an expression; if the ”...” expression has type a, then the quotation has type Q (TExp a).

  Values of type TExp a may be converted to values of type Exp using the function unType :: TExp a -> Exp.

• A quasi-quotation can appear in a pattern, type, expression, or declaration context and is also written in Oxford brackets:

  • [varid| ... |], where the ”...” is an arbitrary string; a full description of the quasi-quotation facility is given in Template Haskell Quasi-quotation.

• A name can be quoted with either one or two prefix single quotes:

  • 'f has type Name, and names the function f. Similarly 'C has type Name and names the data constructor C. In general '⟨thing⟩ interprets ⟨thing⟩ in an expression context.

    A name whose second character is a single quote (sadly) cannot be quoted in this way, because it will be parsed instead as a quoted character. For example, if the function is called f'7 (which is a legal Haskell identifier), an attempt to quote it as 'f'7 would be parsed as the character literal 'f' followed by the numeric literal 7. There is no current escape mechanism in this (unusual) situation.

  • ''T has type Name, and names the type constructor T. That is, ''⟨thing⟩ interprets ⟨thing⟩ in a type context.

  These Names can be used to construct Template Haskell expressions, patterns, declarations etc. They may also be given as an argument to the reify function.

• It is possible for a splice to expand to an expression that contain names which are not in scope at the site of the splice. As an example, consider the following code:

  module Bar where
  
  import Language.Haskell.TH
  
  add1 :: Int -> Q Exp
  add1 x = [| x + 1 |]
  

  Now consider a splice using <literal>add1</literal> in a separate module:

  module Foo where
  
  import Bar
  
  two :: Int
  two = $(add1 1)
  

  Template Haskell cannot know what the argument to add1 will be at the function’s definition site, so a lifting mechanism is used to promote x into a value of type Q Exp. This functionality is exposed to the user as the Lift typeclass in the Language.Haskell.TH.Syntax module. If a type has a Lift instance, then any of its values can be lifted to a Template Haskell expression:

  class Lift t where
      lift :: t -> Q Exp
  

  In general, if GHC sees an expression within Oxford brackets (e.g., [| foo bar |], then GHC looks up each name within the brackets. If a name is global (e.g., suppose foo comes from an import or a top-level declaration), then the fully qualified name is used directly in the quotation. If the name is local (e.g., suppose bar is bound locally in the function definition mkFoo bar = [| foo bar |]), then GHC uses lift on it (so GHC pretends [| foo bar |] actually contains [| foo $(lift bar) |]). Local names, which are not in scope at splice locations, are actually evaluated when the quotation is processed.

  The template-haskell library provides Lift instances for many common data types. Furthermore, it is possible to derive Lift instances automatically by using the -XDeriveLift language extension. See Deriving Lift instances for more information.

• You may omit the $(...) in a top-level declaration splice. Simply writing an expression (rather than a declaration) implies a splice. For example, you can write

  module Foo where
  import Bar
  
  f x = x
  
  $(deriveStuff 'f)   -- Uses the $(...) notation
  
  g y = y+1
  
  deriveStuff 'g      -- Omits the $(...)
  
  h z = z-1
  

  This abbreviation makes top-level declaration slices quieter and less intimidating.

• Pattern splices introduce variable binders but scoping of variables in expressions inside the pattern’s scope is only checked when a splice is run. Note that pattern splices that occur outside of any quotation brackets are run at compile time. Pattern splices occurring inside a quotation bracket are not run at compile time; they are run when the bracket is spliced in, sometime later. For example,

  mkPat :: Q Pat
  mkPat = [p| (x, y) |]
  
  -- in another module:
  foo :: (Char, String) -> String
  foo $(mkPat) = x : z
  
  bar :: Q Exp
  bar = [| \ $(mkPat) -> x : w |]
  

  will fail with z being out of scope in the definition of foo but it will not fail with w being out of scope in the definition of bar. That will only happen when bar is spliced.

• A pattern quasiquoter may generate binders that scope over the right-hand side of a definition because these binders are in scope lexically. For example, given a quasiquoter haskell that parses Haskell, in the following code, the y in the right-hand side of f refers to the y bound by the haskell pattern quasiquoter, not the top-level y = 7.

  y :: Int
  y = 7
  
  f :: Int -> Int -> Int
  f n = \ [haskell|y|] -> y+n
  

• Top-level declaration splices break up a source file into declaration groups. A declaration group is the group of declarations created by a top-level declaration splice, plus those following it, down to but not including the next top-level declaration splice. The first declaration group in a module includes all top-level definitions down to but not including the first top-level declaration splice.

  Each declaration group is mutually recursive only within the group. Declaration groups can refer to definitions within previous groups, but not later ones.

  Accordingly, the type environment seen by reify includes all the top-level declarations up to the end of the immediately preceding declaration group, but no more.

  Unlike normal declaration splices, declaration quasiquoters do not cause a break. These quasiquoters are expanded before the rest of the declaration group is processed, and the declarations they generate are merged into the surrounding declaration group. Consequently, the type environment seen by reify from a declaration quasiquoter will not include anything from the quasiquoter’s declaration group.

  Concretely, consider the following code

  module M where
     import ...
     f x = x
     $(th1 4)
     h y = k y y $(blah1)
     [qq|blah|]
     k x y = x + y
     $(th2 10)
     w z = $(blah2)
  

  In this example

  1. The body of h would be unable to refer to the function w.

     A reify inside the splice $(th1 ..) would see the definition of f.

  2. A reify inside the splice $(blah1) would see the definition of f, but would not see the definition of h.

  3. A reify inside the splice $(th2..) would see the definition of f, all the bindings created by $(th1..), and the definition of h.

  4. A reify inside the splice $(blah2) would see the same definitions as the splice $(th2...).

  5. The body of h is able to refer to the function k appearing on the other side of the declaration quasiquoter, as quasiquoters never cause a declaration group to be broken up.

     A reify inside the qq quasiquoter would be able to see the definition of f from the preceding declaration group, but not the definitions of h or k, or any definitions from subsequent declaration groups.

• Expression quotations accept most Haskell language constructs. However, there are some GHC-specific extensions which expression quotations currently do not support, including

  • Recursive do-statements (see Trac #1262)
  • Pattern synonyms (see Trac #8761)
  • Typed holes (see Trac #10267)

(Compared to the original paper, there are many differences of detail. The syntax for a declaration splice uses “$” not “splice”. The type of the enclosed expression must be Q [Dec], not [Q Dec]. Typed expression splices and quotations are supported.)

9.18.2. Using Template Haskell

• The data types and monadic constructor functions for Template Haskell are in the library Language.Haskell.THSyntax.

• You can only run a function at compile time if it is imported from another module. That is, you can’t define a function in a module, and call it from within a splice in the same module. (It would make sense to do so, but it’s hard to implement.)

• You can only run a function at compile time if it is imported from another module that is not part of a mutually-recursive group of modules that includes the module currently being compiled. Furthermore, all of the modules of the mutually-recursive group must be reachable by non-SOURCE imports from the module where the splice is to be run.

  For example, when compiling module A, you can only run Template Haskell functions imported from B if B does not import A (directly or indirectly). The reason should be clear: to run B we must compile and run A, but we are currently type-checking A.

• If you are building GHC from source, you need at least a stage-2 bootstrap compiler to run Template Haskell splices and quasi-quotes. A stage-1 compiler will only accept regular quotes of Haskell. Reason: TH splices and quasi-quotes compile and run a program, and then looks at the result. So it’s important that the program it compiles produces results whose representations are identical to those of the compiler itself.

Template Haskell works in any mode (--make, --interactive, or file-at-a-time). There used to be a restriction to the former two, but that restriction has been lifted.

9.18.3. Viewing Template Haskell generated code

The flag -ddump-splices shows the expansion of all top-level declaration splices, both typed and untyped, as they happen. As with all dump flags, the default is for this output to be sent to stdout. For a non-trivial program, you may be interested in combining this with the -ddump-to-file flag (see Dumping out compiler intermediate structures. For each file using Template Haskell, this will show the output in a .dump-splices file.

The flag -dth-dec-file shows the expansions of all top-level TH declaration splices, both typed and untyped, in the file M.th.hs where M is the name of the module being compiled. Note that other types of splices (expressions, types, and patterns) are not shown. Application developers can check this into their repository so that they can grep for identifiers that were defined in Template Haskell. This is similar to using -ddump-to-file with -ddump-splices but it always generates a file instead of being coupled to -ddump-to-file. The format is also different: it does not show code from the original file, instead it only shows generated code and has a comment for the splice location of the original file.

Below is a sample output of -ddump-splices

TH_pragma.hs:(6,4)-(8,26): Splicing declarations
  [d| foo :: Int -> Int
      foo x = x + 1 |]
======>
  foo :: Int -> Int
  foo x = (x + 1)

Below is the output of the same sample using -dth-dec-file

-- TH_pragma.hs:(6,4)-(8,26): Splicing declarations
foo :: Int -> Int
foo x = (x + 1)

9.18.4. A Template Haskell Worked Example

To help you get over the confidence barrier, try out this skeletal worked example. First cut and paste the two modules below into “Main.hs” and “Printf.hs”:

{- Main.hs -}
module Main where

-- Import our template "pr"
import Printf ( pr )

-- The splice operator $ takes the Haskell source code
-- generated at compile time by "pr" and splices it into
-- the argument of "putStrLn".
main = putStrLn ( $(pr "Hello") )


{- Printf.hs -}
module Printf where

-- Skeletal printf from the paper.
-- It needs to be in a separate module to the one where
-- you intend to use it.

-- Import some Template Haskell syntax
import Language.Haskell.TH

-- Describe a format string
data Format = D | S | L String

-- Parse a format string.  This is left largely to you
-- as we are here interested in building our first ever
-- Template Haskell program and not in building printf.
parse :: String -> [Format]
parse s   = [ L s ]

-- Generate Haskell source code from a parsed representation
-- of the format string.  This code will be spliced into
-- the module which calls "pr", at compile time.
gen :: [Format] -> Q Exp
gen [D]   = [| \n -> show n |]
gen [S]   = [| \s -> s |]
gen [L s] = stringE s

-- Here we generate the Haskell code for the splice
-- from an input format string.
pr :: String -> Q Exp
pr s = gen (parse s)

Now run the compiler (here we are a Cygwin prompt on Windows):

$ ghc --make -XTemplateHaskell main.hs -o main.exe

Run “main.exe” and here is your output:

$ ./main
Hello

9.18.5. Using Template Haskell with Profiling

Template Haskell relies on GHC’s built-in bytecode compiler and interpreter to run the splice expressions. The bytecode interpreter runs the compiled expression on top of the same runtime on which GHC itself is running; this means that the compiled code referred to by the interpreted expression must be compatible with this runtime, and in particular this means that object code that is compiled for profiling cannot be loaded and used by a splice expression, because profiled object code is only compatible with the profiling version of the runtime.

This causes difficulties if you have a multi-module program containing Template Haskell code and you need to compile it for profiling, because GHC cannot load the profiled object code and use it when executing the splices. Fortunately GHC provides a workaround. The basic idea is to compile the program twice:

1. Compile the program or library first the normal way, without -prof.

2. Then compile it again with -prof, and additionally use -osuf p_o to name the object files differently (you can choose any suffix that isn’t the normal object suffix here). GHC will automatically load the object files built in the first step when executing splice expressions. If you omit the -osuf flag when building with -prof and Template Haskell is used, GHC will emit an error message.

9.18.6. Template Haskell Quasi-quotation

Quasi-quotation allows patterns and expressions to be written using programmer-defined concrete syntax; the motivation behind the extension and several examples are documented in “Why It’s Nice to be Quoted: Quasiquoting for Haskell” (Proc Haskell Workshop 2007). The example below shows how to write a quasiquoter for a simple expression language.

Here are the salient features

• A quasi-quote has the form [quoter| string |].

  • The ⟨quoter⟩ must be the name of an imported quoter, either qualified or unqualified; it cannot be an arbitrary expression.
  • The ⟨quoter⟩ cannot be “e”, “t”, “d”, or “p”, since those overlap with Template Haskell quotations.
  • There must be no spaces in the token [quoter|.
  • The quoted ⟨string⟩ can be arbitrary, and may contain newlines.
  • The quoted ⟨string⟩ finishes at the first occurrence of the two-character sequence "|]". Absolutely no escaping is performed. If you want to embed that character sequence in the string, you must invent your own escape convention (such as, say, using the string "|~]" instead), and make your quoter function interpret "|~]" as "|]". One way to implement this is to compose your quoter with a pre-processing pass to perform your escape conversion. See the discussion in Trac #5348 for details.

• A quasiquote may appear in place of

  • An expression
  • A pattern
  • A type
  • A top-level declaration

  (Only the first two are described in the paper.)

• A quoter is a value of type Language.Haskell.TH.Quote.QuasiQuoter, which is defined thus:

  data QuasiQuoter = QuasiQuoter { quoteExp  :: String -> Q Exp,
                                   quotePat  :: String -> Q Pat,
                                   quoteType :: String -> Q Type,
                                   quoteDec  :: String -> Q [Dec] }
  

  That is, a quoter is a tuple of four parsers, one for each of the contexts in which a quasi-quote can occur.

• A quasi-quote is expanded by applying the appropriate parser to the string enclosed by the Oxford brackets. The context of the quasi-quote (expression, pattern, type, declaration) determines which of the parsers is called.

• Unlike normal declaration splices of the form $(...), declaration quasi-quotes do not cause a declaration group break. See Syntax for more information.

The example below shows quasi-quotation in action. The quoter expr is bound to a value of type QuasiQuoter defined in module Expr. The example makes use of an antiquoted variable n, indicated by the syntax 'int:n (this syntax for anti-quotation was defined by the parser’s author, not by GHC). This binds n to the integer value argument of the constructor IntExpr when pattern matching. Please see the referenced paper for further details regarding anti-quotation as well as the description of a technique that uses SYB to leverage a single parser of type String -> a to generate both an expression parser that returns a value of type Q Exp and a pattern parser that returns a value of type Q Pat.

Quasiquoters must obey the same stage restrictions as Template Haskell, e.g., in the example, expr cannot be defined in Main.hs where it is used, but must be imported.

{- ------------- file Main.hs --------------- -}
module Main where

import Expr

main :: IO ()
main = do { print $ eval [expr|1 + 2|]
          ; case IntExpr 1 of
              { [expr|'int:n|] -> print n
              ;  _              -> return ()
              }
          }


{- ------------- file Expr.hs --------------- -}
module Expr where

import qualified Language.Haskell.TH as TH
import Language.Haskell.TH.Quote

data Expr  =  IntExpr Integer
           |  AntiIntExpr String
           |  BinopExpr BinOp Expr Expr
           |  AntiExpr String
    deriving(Show, Typeable, Data)

data BinOp  =  AddOp
            |  SubOp
            |  MulOp
            |  DivOp
    deriving(Show, Typeable, Data)

eval :: Expr -> Integer
eval (IntExpr n)        = n
eval (BinopExpr op x y) = (opToFun op) (eval x) (eval y)
  where
    opToFun AddOp = (+)
    opToFun SubOp = (-)
    opToFun MulOp = (*)
    opToFun DivOp = div

expr = QuasiQuoter { quoteExp = parseExprExp, quotePat =  parseExprPat }

-- Parse an Expr, returning its representation as
-- either a Q Exp or a Q Pat. See the referenced paper
-- for how to use SYB to do this by writing a single
-- parser of type String -> Expr instead of two
-- separate parsers.

parseExprExp :: String -> Q Exp
parseExprExp ...

parseExprPat :: String -> Q Pat
parseExprPat ...

Now run the compiler:

$ ghc --make -XQuasiQuotes Main.hs -o main

Run “main” and here is your output:

$ ./main
3
1

9.19. Arrow notation

Arrows are a generalisation of monads introduced by John Hughes. For more details, see

• “Generalising Monads to Arrows”, John Hughes, in Science of Computer Programming 37, pp. 67–111, May 2000. The paper that introduced arrows: a friendly introduction, motivated with programming examples.
• “A New Notation for Arrows”, Ross Paterson, in ICFP, Sep 2001. Introduced the notation described here.
• “Arrows and Computation”, Ross Paterson, in The Fun of Programming, Palgrave, 2003.
• “Programming with Arrows”, John Hughes, in 5th International Summer School on Advanced Functional Programming, Lecture Notes in Computer Science vol. 3622, Springer, 2004. This paper includes another introduction to the notation, with practical examples.
• “Type and Translation Rules for Arrow Notation in GHC”, Ross Paterson and Simon Peyton Jones, September 16, 2004. A terse enumeration of the formal rules used (extracted from comments in the source code).
• The arrows web page at http://www.haskell.org/arrows/ <http://www.haskell.org/arrows/>`__.

With the -XArrows flag, GHC supports the arrow notation described in the second of these papers, translating it using combinators from the Control.Arrow module. What follows is a brief introduction to the notation; it won’t make much sense unless you’ve read Hughes’s paper.

The extension adds a new kind of expression for defining arrows:

exp10 ::= ...
       |  proc apat -> cmd

where proc is a new keyword. The variables of the pattern are bound in the body of the proc-expression, which is a new sort of thing called a command. The syntax of commands is as follows:

cmd   ::= exp10 -<  exp
       |  exp10 -<< exp
       |  cmd0

with ⟨cmd⟩⁰ up to ⟨cmd⟩⁹ defined using infix operators as for expressions, and

cmd10 ::= \ apat ... apat -> cmd
       |  let decls in cmd
       |  if exp then cmd else cmd
       |  case exp of { calts }
       |  do { cstmt ; ... cstmt ; cmd }
       |  fcmd

fcmd  ::= fcmd aexp
       |  ( cmd )
       |  (| aexp cmd ... cmd |)

cstmt ::= let decls
       |  pat <- cmd
       |  rec { cstmt ; ... cstmt [;] }
       |  cmd

where ⟨calts⟩ are like ⟨alts⟩ except that the bodies are commands instead of expressions.

Commands produce values, but (like monadic computations) may yield more than one value, or none, and may do other things as well. For the most part, familiarity with monadic notation is a good guide to using commands. However the values of expressions, even monadic ones, are determined by the values of the variables they contain; this is not necessarily the case for commands.

A simple example of the new notation is the expression

proc x -> f -< x+1

We call this a procedure or arrow abstraction. As with a lambda expression, the variable x is a new variable bound within the proc-expression. It refers to the input to the arrow. In the above example, -< is not an identifier but an new reserved symbol used for building commands from an expression of arrow type and an expression to be fed as input to that arrow. (The weird look will make more sense later.) It may be read as analogue of application for arrows. The above example is equivalent to the Haskell expression

arr (\ x -> x+1) >>> f

That would make no sense if the expression to the left of -< involves the bound variable x. More generally, the expression to the left of -< may not involve any local variable, i.e. a variable bound in the current arrow abstraction. For such a situation there is a variant -<<, as in

proc x -> f x -<< x+1

which is equivalent to

arr (\ x -> (f x, x+1)) >>> app

so in this case the arrow must belong to the ArrowApply class. Such an arrow is equivalent to a monad, so if you’re using this form you may find a monadic formulation more convenient.

9.19.1. do-notation for commands

Another form of command is a form of do-notation. For example, you can write

proc x -> do
        y <- f -< x+1
        g -< 2*y
        let z = x+y
        t <- h -< x*z
        returnA -< t+z

You can read this much like ordinary do-notation, but with commands in place of monadic expressions. The first line sends the value of x+1 as an input to the arrow f, and matches its output against y. In the next line, the output is discarded. The arrow returnA is defined in the Control.Arrow module as arr id. The above example is treated as an abbreviation for

arr (\ x -> (x, x)) >>>
        first (arr (\ x -> x+1) >>> f) >>>
        arr (\ (y, x) -> (y, (x, y))) >>>
        first (arr (\ y -> 2*y) >>> g) >>>
        arr snd >>>
        arr (\ (x, y) -> let z = x+y in ((x, z), z)) >>>
        first (arr (\ (x, z) -> x*z) >>> h) >>>
        arr (\ (t, z) -> t+z) >>>
        returnA

Note that variables not used later in the composition are projected out. After simplification using rewrite rules (see Rewrite rules) defined in the Control.Arrow module, this reduces to

arr (\ x -> (x+1, x)) >>>
        first f >>>
        arr (\ (y, x) -> (2*y, (x, y))) >>>
        first g >>>
        arr (\ (_, (x, y)) -> let z = x+y in (x*z, z)) >>>
        first h >>>
        arr (\ (t, z) -> t+z)

which is what you might have written by hand. With arrow notation, GHC keeps track of all those tuples of variables for you.

Note that although the above translation suggests that let-bound variables like z must be monomorphic, the actual translation produces Core, so polymorphic variables are allowed.

It’s also possible to have mutually recursive bindings, using the new rec keyword, as in the following example:

counter :: ArrowCircuit a => a Bool Int
counter = proc reset -> do
        rec     output <- returnA -< if reset then 0 else next
                next <- delay 0 -< output+1
        returnA -< output

The translation of such forms uses the loop combinator, so the arrow concerned must belong to the ArrowLoop class.

9.19.2. Conditional commands

In the previous example, we used a conditional expression to construct the input for an arrow. Sometimes we want to conditionally execute different commands, as in

proc (x,y) ->
        if f x y
        then g -< x+1
        else h -< y+2

which is translated to

arr (\ (x,y) -> if f x y then Left x else Right y) >>>
        (arr (\x -> x+1) >>> g) ||| (arr (\y -> y+2) >>> h)

Since the translation uses |||, the arrow concerned must belong to the ArrowChoice class.

There are also case commands, like

case input of
    [] -> f -< ()
    [x] -> g -< x+1
    x1:x2:xs -> do
        y <- h -< (x1, x2)
        ys <- k -< xs
        returnA -< y:ys

The syntax is the same as for case expressions, except that the bodies of the alternatives are commands rather than expressions. The translation is similar to that of if commands.

9.19.3. Defining your own control structures

As we’re seen, arrow notation provides constructs, modelled on those for expressions, for sequencing, value recursion and conditionals. But suitable combinators, which you can define in ordinary Haskell, may also be used to build new commands out of existing ones. The basic idea is that a command defines an arrow from environments to values. These environments assign values to the free local variables of the command. Thus combinators that produce arrows from arrows may also be used to build commands from commands. For example, the ArrowPlus class includes a combinator

ArrowPlus a => (<+>) :: a b c -> a b c -> a b c

so we can use it to build commands:

expr' = proc x -> do
                returnA -< x
        <+> do
                symbol Plus -< ()
                y <- term -< ()
                expr' -< x + y
        <+> do
                symbol Minus -< ()
                y <- term -< ()
                expr' -< x - y

(The do on the first line is needed to prevent the first <+> ... from being interpreted as part of the expression on the previous line.) This is equivalent to

expr' = (proc x -> returnA -< x)
        <+> (proc x -> do
                symbol Plus -< ()
                y <- term -< ()
                expr' -< x + y)
        <+> (proc x -> do
                symbol Minus -< ()
                y <- term -< ()
                expr' -< x - y)

We are actually using <+> here with the more specific type

ArrowPlus a => (<+>) :: a (e,()) c -> a (e,()) c -> a (e,()) c

It is essential that this operator be polymorphic in e (representing the environment input to the command and thence to its subcommands) and satisfy the corresponding naturality property

arr (first k) >>> (f <+> g) = (arr (first k) >>> f) <+> (arr (first k) >>> g)

at least for strict k. (This should be automatic if you’re not using seq.) This ensures that environments seen by the subcommands are environments of the whole command, and also allows the translation to safely trim these environments. (The second component of the input pairs can contain unnamed input values, as described in the next section.) The operator must also not use any variable defined within the current arrow abstraction.

We could define our own operator

untilA :: ArrowChoice a => a (e,s) () -> a (e,s) Bool -> a (e,s) ()
untilA body cond = proc x ->
        b <- cond -< x
        if b then returnA -< ()
        else do
                body -< x
                untilA body cond -< x

and use it in the same way. Of course this infix syntax only makes sense for binary operators; there is also a more general syntax involving special brackets:

proc x -> do
        y <- f -< x+1
        (|untilA (increment -< x+y) (within 0.5 -< x)|)

9.19.4. Primitive constructs

Some operators will need to pass additional inputs to their subcommands. For example, in an arrow type supporting exceptions, the operator that attaches an exception handler will wish to pass the exception that occurred to the handler. Such an operator might have a type

handleA :: ... => a (e,s) c -> a (e,(Ex,s)) c -> a (e,s) c

where Ex is the type of exceptions handled. You could then use this with arrow notation by writing a command

body `handleA` \ ex -> handler

so that if an exception is raised in the command body, the variable ex is bound to the value of the exception and the command handler, which typically refers to ex, is entered. Though the syntax here looks like a functional lambda, we are talking about commands, and something different is going on. The input to the arrow represented by a command consists of values for the free local variables in the command, plus a stack of anonymous values. In all the prior examples, we made no assumptions about this stack. In the second argument to handleA, the value of the exception has been added to the stack input to the handler. The command form of lambda merely gives this value a name.

More concretely, the input to a command consists of a pair of an environment and a stack. Each value on the stack is paired with the remainder of the stack, with an empty stack being (). So operators like handleA that pass extra inputs to their subcommands can be designed for use with the notation by placing the values on the stack paired with the environment in this way. More precisely, the type of each argument of the operator (and its result) should have the form

a (e, (t1, ... (tn, ())...)) t

where ⟨e⟩ is a polymorphic variable (representing the environment) and ⟨ti⟩ are the types of the values on the stack, with ⟨t1⟩ being the “top”. The polymorphic variable ⟨e⟩ must not occur in ⟨a⟩, ⟨ti⟩ or ⟨t⟩. However the arrows involved need not be the same. Here are some more examples of suitable operators:

bracketA :: ... => a (e,s) b -> a (e,(b,s)) c -> a (e,(c,s)) d -> a (e,s) d
runReader :: ... => a (e,s) c -> a' (e,(State,s)) c
runState :: ... => a (e,s) c -> a' (e,(State,s)) (c,State)

We can supply the extra input required by commands built with the last two by applying them to ordinary expressions, as in

proc x -> do
        s <- ...
        (|runReader (do { ... })|) s

which adds s to the stack of inputs to the command built using runReader.

The command versions of lambda abstraction and application are analogous to the expression versions. In particular, the beta and eta rules describe equivalences of commands. These three features (operators, lambda abstraction and application) are the core of the notation; everything else can be built using them, though the results would be somewhat clumsy. For example, we could simulate do-notation by defining

bind :: Arrow a => a (e,s) b -> a (e,(b,s)) c -> a (e,s) c
u `bind` f = returnA &&& u >>> f

bind_ :: Arrow a => a (e,s) b -> a (e,s) c -> a (e,s) c
u `bind_` f = u `bind` (arr fst >>> f)

We could simulate if by defining

cond :: ArrowChoice a => a (e,s) b -> a (e,s) b -> a (e,(Bool,s)) b
cond f g = arr (\ (e,(b,s)) -> if b then Left (e,s) else Right (e,s)) >>> f ||| g

9.19.5. Differences with the paper

• Instead of a single form of arrow application (arrow tail) with two translations, the implementation provides two forms “-<” (first-order) and “-<<” (higher-order).
• User-defined operators are flagged with banana brackets instead of a new form keyword.
• In the paper and the previous implementation, values on the stack were paired to the right of the environment in a single argument, but now the environment and stack are separate arguments.

9.19.6. Portability

Although only GHC implements arrow notation directly, there is also a preprocessor (available from the arrows web page) that translates arrow notation into Haskell 98 for use with other Haskell systems. You would still want to check arrow programs with GHC; tracing type errors in the preprocessor output is not easy. Modules intended for both GHC and the preprocessor must observe some additional restrictions:

• The module must import Control.Arrow.
• The preprocessor cannot cope with other Haskell extensions. These would have to go in separate modules.
• Because the preprocessor targets Haskell (rather than Core), let-bound variables are monomorphic.

9.20. Bang patterns

GHC supports an extension of pattern matching called bang patterns, written !pat. Bang patterns are under consideration for Haskell Prime. The Haskell prime feature description contains more discussion and examples than the material below.

The key change is the addition of a new rule to the semantics of pattern matching in the Haskell 98 report. Add new bullet 10, saying: Matching the pattern !⟨pat⟩ against a value ⟨v⟩ behaves as follows:

• if ⟨v⟩ is bottom, the match diverges
• otherwise, ⟨pat⟩ is matched against ⟨v⟩

Bang patterns are enabled by the flag -XBangPatterns.

9.20.1. Informal description of bang patterns

The main idea is to add a single new production to the syntax of patterns:

pat ::= !pat

Matching an expression e against a pattern !p is done by first evaluating e (to WHNF) and then matching the result against p. Example:

f1 !x = True

This definition makes f1 is strict in x, whereas without the bang it would be lazy. Bang patterns can be nested of course:

f2 (!x, y) = [x,y]

Here, f2 is strict in x but not in y. A bang only really has an effect if it precedes a variable or wild-card pattern:

f3 !(x,y) = [x,y]
f4 (x,y)  = [x,y]

Here, f3 and f4 are identical; putting a bang before a pattern that forces evaluation anyway does nothing.

There is one (apparent) exception to this general rule that a bang only makes a difference when it precedes a variable or wild-card: a bang at the top level of a let or where binding makes the binding strict, regardless of the pattern. (We say “apparent” exception because the Right Way to think of it is that the bang at the top of a binding is not part of the pattern; rather it is part of the syntax of the binding, creating a “bang-pattern binding”.) See Strict recursive and polymorphic let bindings for how bang-pattern bindings are compiled.

However, nested bangs in a pattern binding behave uniformly with all other forms of pattern matching. For example

let (!x,[y]) = e in b

is equivalent to this:

let { t = case e of (x,[y]) -> x `seq` (x,y)
      x = fst t
      y = snd t }
in b

The binding is lazy, but when either x or y is evaluated by b the entire pattern is matched, including forcing the evaluation of x.

Bang patterns work in case expressions too, of course:

g5 x = let y = f x in body
g6 x = case f x of { y -> body }
g7 x = case f x of { !y -> body }

The functions g5 and g6 mean exactly the same thing. But g7 evaluates (f x), binds y to the result, and then evaluates body.

9.20.2. Syntax and semantics

We add a single new production to the syntax of patterns:

pat ::= !pat

There is one problem with syntactic ambiguity. Consider:

f !x = 3

Is this a definition of the infix function “(!)”, or of the “f” with a bang pattern? GHC resolves this ambiguity in favour of the latter. If you want to define (!) with bang-patterns enabled, you have to do so using prefix notation:

(!) f x = 3

The semantics of Haskell pattern matching is described in Section 3.17.2 of the Haskell Report. To this description add one extra item 10, saying:

• Matching the pattern !pat against a value v behaves as follows:
  • if v is bottom, the match diverges
  • otherwise, pat is matched against v

Similarly, in Figure 4 of Section 3.17.3, add a new case (t):

case v of { !pat -> e; _ -> e' }
   = v `seq` case v of { pat -> e; _ -> e' }

That leaves let expressions, whose translation is given in Section 3.12 of the Haskell Report. In the translation box, first apply the following transformation: for each pattern pi that is of form !qi = ei, transform it to (xi,!qi) = ((),ei), and replace e0 by (xi `seq` e0). Then, when none of the left-hand-side patterns have a bang at the top, apply the rules in the existing box.

The effect of the let rule is to force complete matching of the pattern qi before evaluation of the body is begun. The bang is retained in the translated form in case qi is a variable, thus:

let !y = f x in b

The let-binding can be recursive. However, it is much more common for the let-binding to be non-recursive, in which case the following law holds: (let !p = rhs in body) is equivalent to (case rhs of !p -> body)

A pattern with a bang at the outermost level is not allowed at the top level of a module.

9.21. Assertions

If you want to make use of assertions in your standard Haskell code, you could define a function like the following:

assert :: Bool -> a -> a
assert False x = error "assertion failed!"
assert _     x = x

which works, but gives you back a less than useful error message – an assertion failed, but which and where?

One way out is to define an extended assert function which also takes a descriptive string to include in the error message and perhaps combine this with the use of a pre-processor which inserts the source location where assert was used.

GHC offers a helping hand here, doing all of this for you. For every use of assert in the user’s source:

kelvinToC :: Double -> Double
kelvinToC k = assert (k >= 0.0) (k+273.15)

GHC will rewrite this to also include the source location where the assertion was made,

assert pred val ==> assertError "Main.hs|15" pred val

The rewrite is only performed by the compiler when it spots applications of Control.Exception.assert, so you can still define and use your own versions of assert, should you so wish. If not, import Control.Exception to make use assert in your code.

GHC ignores assertions when optimisation is turned on with the -O flag. That is, expressions of the form assert pred e will be rewritten to e. You can also disable assertions using the -fignore-asserts option. The option -fno-ignore-asserts allows enabling assertions even when optimisation is turned on.

Assertion failures can be caught, see the documentation for the Control.Exception library for the details.

9.22. Static pointers

The language extension -XStaticPointers adds a new syntactic form static e, which stands for a reference to the closed expression ⟨e⟩. This reference is stable and portable, in the sense that it remains valid across different processes on possibly different machines. Thus, a process can create a reference and send it to another process that can resolve it to ⟨e⟩.

With this extension turned on, static is no longer a valid identifier.

Static pointers were first proposed in the paper Towards Haskell in the cloud, Jeff Epstein, Andrew P. Black and Simon Peyton-Jones, Proceedings of the 4th ACM Symposium on Haskell, pp. 118-129, ACM, 2011.

9.22.1. Using static pointers

Each reference is given a key which can be used to locate it at runtime with unsafeLookupStaticPtr which uses a global and immutable table called the Static Pointer Table. The compiler includes entries in this table for all static forms found in the linked modules. The value can be obtained from the reference via deRefStaticPtr.

The body e of a static e expression must be a closed expression. That is, there can be no free variables occurring in e, i.e. lambda- or let-bound variables bound locally in the context of the expression.

All of the following are permissible:

inc :: Int -> Int
inc x = x + 1

ref1 = static 1
ref2 = static inc
ref3 = static (inc 1)
ref4 = static ((\x -> x + 1) (1 :: Int))
ref5 y = static (let x = 1 in x)

While the following definitions are rejected:

ref6 = let x = 1 in static x
ref7 y = static (let x = 1 in y)

9.22.2. Static semantics of static pointers

Informally, if we have a closed expression

e :: forall a_1 ... a_n . t

the static form is of type

static e :: (Typeable a_1, ... , Typeable a_n) => StaticPtr t

Furthermore, type t is constrained to have a Typeable instance. The following are therefore illegal:

static show                    -- No Typeable instance for (Show a => a -> String)
static Control.Monad.ST.runST  -- No Typeable instance for ((forall s. ST s a) -> a)

That being said, with the appropriate use of wrapper datatypes, the above limitations induce no loss of generality:

{-# LANGUAGE ConstraintKinds           #-}
{-# LANGUAGE DeriveDataTypeable        #-}
{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE Rank2Types                #-}
{-# LANGUAGE StandaloneDeriving        #-}
{-# LANGUAGE StaticPointers            #-}

import Control.Monad.ST
import Data.Typeable
import GHC.StaticPtr

data Dict c = c => Dict
  deriving Typeable

g1 :: Typeable a => StaticPtr (Dict (Show a) -> a -> String)
g1 = static (\Dict -> show)

data Rank2Wrapper f = R2W (forall s. f s)
  deriving Typeable
newtype Flip f a s = Flip { unFlip :: f s a }
  deriving Typeable

g2 :: Typeable a => StaticPtr (Rank2Wrapper (Flip ST a) -> a)
g2 = static (\(R2W f) -> runST (unFlip f))

9.23. Pragmas

GHC supports several pragmas, or instructions to the compiler placed in the source code. Pragmas don’t normally affect the meaning of the program, but they might affect the efficiency of the generated code.

Pragmas all take the form {-# word ... #-} where ⟨word⟩ indicates the type of pragma, and is followed optionally by information specific to that type of pragma. Case is ignored in ⟨word⟩. The various values for ⟨word⟩ that GHC understands are described in the following sections; any pragma encountered with an unrecognised ⟨word⟩ is ignored. The layout rule applies in pragmas, so the closing #-} should start in a column to the right of the opening {-#.

Certain pragmas are file-header pragmas:

• A file-header pragma must precede the module keyword in the file.
• There can be as many file-header pragmas as you please, and they can be preceded or followed by comments.
• File-header pragmas are read once only, before pre-processing the file (e.g. with cpp).
• The file-header pragmas are: {-# LANGUAGE #-}, {-# OPTIONS_GHC #-}, and {-# INCLUDE #-}.

9.23.1. LANGUAGE pragma

The LANGUAGE pragma allows language extensions to be enabled in a portable way. It is the intention that all Haskell compilers support the LANGUAGE pragma with the same syntax, although not all extensions are supported by all compilers, of course. The LANGUAGE pragma should be used instead of OPTIONS_GHC, if possible.

For example, to enable the FFI and preprocessing with CPP:

{-# LANGUAGE ForeignFunctionInterface, CPP #-}

LANGUAGE is a file-header pragma (see Pragmas).

Every language extension can also be turned into a command-line flag by prefixing it with “-X”; for example -XForeignFunctionInterface. (Similarly, all “-X” flags can be written as LANGUAGE pragmas.)

A list of all supported language extensions can be obtained by invoking ghc --supported-extensions (see Modes of operation).

Any extension from the Extension type defined in Language.Haskell.Extension may be used. GHC will report an error if any of the requested extensions are not supported.

9.23.2. OPTIONS_GHC pragma

The OPTIONS_GHC pragma is used to specify additional options that are given to the compiler when compiling this source file. See Command line options in source files for details.

Previous versions of GHC accepted OPTIONS rather than OPTIONS_GHC, but that is now deprecated.

OPTIONS_GHC is a file-header pragma (see Pragmas).

9.23.3. INCLUDE pragma

The INCLUDE used to be necessary for specifying header files to be included when using the FFI and compiling via C. It is no longer required for GHC, but is accepted (and ignored) for compatibility with other compilers.

9.23.4. WARNING and DEPRECATED pragmas

The WARNING pragma allows you to attach an arbitrary warning to a particular function, class, or type. A DEPRECATED pragma lets you specify that a particular function, class, or type is deprecated. There are two ways of using these pragmas.

• You can work on an entire module thus:

  module Wibble {-# DEPRECATED "Use Wobble instead" #-} where
    ...
  

  Or:

  module Wibble {-# WARNING "This is an unstable interface." #-} where
    ...
  

  When you compile any module that import Wibble, GHC will print the specified message.

• You can attach a warning to a function, class, type, or data constructor, with the following top-level declarations:

  {-# DEPRECATED f, C, T "Don't use these" #-}
  {-# WARNING unsafePerformIO "This is unsafe; I hope you know what you're doing" #-}
  

  When you compile any module that imports and uses any of the specified entities, GHC will print the specified message.

  You can only attach to entities declared at top level in the module being compiled, and you can only use unqualified names in the list of entities. A capitalised name, such as T refers to either the type constructor T or the data constructor T, or both if both are in scope. If both are in scope, there is currently no way to specify one without the other (c.f. fixities Infix type constructors, classes, and type variables).

Warnings and deprecations are not reported for (a) uses within the defining module, (b) defining a method in a class instance, and (c) uses in an export list. The latter reduces spurious complaints within a library in which one module gathers together and re-exports the exports of several others.

You can suppress the warnings with the flag -fno-warn-warnings-deprecations.

9.23.5. MINIMAL pragma

The MINIMAL pragma is used to specify the minimal complete definition of a class, i.e. specify which methods must be implemented by all instances. If an instance does not satisfy the minimal complete definition, then a warning is generated. This can be useful when a class has methods with circular defaults. For example

class Eq a where
    (==) :: a -> a -> Bool
    (/=) :: a -> a -> Bool
    x == y = not (x /= y)
    x /= y = not (x == y)
    {-# MINIMAL (==) | (/=) #-}

Without the MINIMAL pragma no warning would be generated for an instance that implements neither method.

The syntax for minimal complete definition is:

mindef ::= name
        |  '(' mindef ')'
        |  mindef '|' mindef
        |  mindef ',' mindef

A vertical bar denotes disjunction, i.e. one of the two sides is required. A comma denotes conjunction, i.e. both sides are required. Conjunction binds stronger than disjunction.

If no MINIMAL pragma is given in the class declaration, it is just as if a pragma {-# MINIMAL op1, op2, ..., opn #-} was given, where the opi are the methods (a) that lack a default method in the class declaration, and (b) whose name that does not start with an underscore (c.f. -fwarn-missing-methods, Warnings and sanity-checking).

This warning can be turned off with the flag -fno-warn-missing-methods.

9.23.6. INLINE and NOINLINE pragmas

These pragmas control the inlining of function definitions.

9.23.6.1. INLINE pragma

GHC (with -O, as always) tries to inline (or “unfold”) functions/values that are “small enough,” thus avoiding the call overhead and possibly exposing other more-wonderful optimisations. GHC has a set of heuristics, tuned over a long period of time using many benchmarks, that decide when it is beneficial to inline a function at its call site. The heuristics are designed to inline functions when it appears to be beneficial to do so, but without incurring excessive code bloat. If a function looks too big, it won’t be inlined, and functions larger than a certain size will not even have their definition exported in the interface file. Some of the thresholds that govern these heuristic decisions can be changed using flags, see -f*: platform-independent flags.

Normally GHC will do a reasonable job of deciding by itself when it is a good idea to inline a function. However, sometimes you might want to override the default behaviour. For example, if you have a key function that is important to inline because it leads to further optimisations, but GHC judges it to be too big to inline.

The sledgehammer you can bring to bear is the INLINE pragma, used thusly:

key_function :: Int -> String -> (Bool, Double)
{-# INLINE key_function #-}

The major effect of an INLINE pragma is to declare a function’s “cost” to be very low. The normal unfolding machinery will then be very keen to inline it. However, an INLINE pragma for a function “f” has a number of other effects:

• While GHC is keen to inline the function, it does not do so blindly. For example, if you write

  map key_function xs
  

  there really isn’t any point in inlining key_function to get

  map (\x -> body) xs
  

  In general, GHC only inlines the function if there is some reason (no matter how slight) to suppose that it is useful to do so.

• Moreover, GHC will only inline the function if it is fully applied, where “fully applied” means applied to as many arguments as appear (syntactically) on the LHS of the function definition. For example:

  comp1 :: (b -> c) -> (a -> b) -> a -> c
  {-# INLINE comp1 #-}
  comp1 f g = \x -> f (g x)
  
  comp2 :: (b -> c) -> (a -> b) -> a -> c
  {-# INLINE comp2 #-}
  comp2 f g x = f (g x)
  

  The two functions comp1 and comp2 have the same semantics, but comp1 will be inlined when applied to two arguments, while comp2 requires three. This might make a big difference if you say

  map (not `comp1` not) xs
  

  which will optimise better than the corresponding use of comp2.

• It is useful for GHC to optimise the definition of an INLINE function f just like any other non-INLINE function, in case the non-inlined version of f is ultimately called. But we don’t want to inline the optimised version of f; a major reason for INLINE pragmas is to expose functions in f‘s RHS that have rewrite rules, and it’s no good if those functions have been optimised away.

  So GHC guarantees to inline precisely the code that you wrote, no more and no less. It does this by capturing a copy of the definition of the function to use for inlining (we call this the “inline-RHS”), which it leaves untouched, while optimising the ordinarily RHS as usual. For externally-visible functions the inline-RHS (not the optimised RHS) is recorded in the interface file.

• An INLINE function is not worker/wrappered by strictness analysis. It’s going to be inlined wholesale instead.

GHC ensures that inlining cannot go on forever: every mutually-recursive group is cut by one or more loop breakers that is never inlined (see Secrets of the GHC inliner, JFP 12(4) July 2002). GHC tries not to select a function with an INLINE pragma as a loop breaker, but when there is no choice even an INLINE function can be selected, in which case the INLINE pragma is ignored. For example, for a self-recursive function, the loop breaker can only be the function itself, so an INLINE pragma is always ignored.

Syntactically, an INLINE pragma for a function can be put anywhere its type signature could be put.

INLINE pragmas are a particularly good idea for the then/return (or bind/unit) functions in a monad. For example, in GHC’s own UniqueSupply monad code, we have:

{-# INLINE thenUs #-}
{-# INLINE returnUs #-}

See also the NOINLINE (NOINLINE pragma) and INLINABLE (INLINABLE pragma) pragmas.

9.23.6.2. INLINABLE pragma

An {-# INLINABLE f #-} pragma on a function f has the following behaviour:

• While INLINE says “please inline me”, the INLINABLE says “feel free to inline me; use your discretion”. In other words the choice is left to GHC, which uses the same rules as for pragma-free functions. Unlike INLINE, that decision is made at the call site, and will therefore be affected by the inlining threshold, optimisation level etc.
• Like INLINE, the INLINABLE pragma retains a copy of the original RHS for inlining purposes, and persists it in the interface file, regardless of the size of the RHS.
• One way to use INLINABLE is in conjunction with the special function inline (Special built-in functions). The call inline f tries very hard to inline f. To make sure that f can be inlined, it is a good idea to mark the definition of f as INLINABLE, so that GHC guarantees to expose an unfolding regardless of how big it is. Moreover, by annotating f as INLINABLE, you ensure that f‘s original RHS is inlined, rather than whatever random optimised version of f GHC’s optimiser has produced.
• The INLINABLE pragma also works with SPECIALISE: if you mark function f as INLINABLE, then you can subsequently SPECIALISE in another module (see SPECIALIZE pragma).
• Unlike INLINE, it is OK to use an INLINABLE pragma on a recursive function. The principal reason do to so to allow later use of SPECIALISE

9.23.6.3. NOINLINE pragma

The NOINLINE pragma does exactly what you’d expect: it stops the named function from being inlined by the compiler. You shouldn’t ever need to do this, unless you’re very cautious about code size.

NOTINLINE is a synonym for NOINLINE (NOINLINE is specified by Haskell 98 as the standard way to disable inlining, so it should be used if you want your code to be portable).

9.23.6.4. CONLIKE modifier

An INLINE or NOINLINE pragma may have a CONLIKE modifier, which affects matching in RULEs (only). See How rules interact with CONLIKE pragmas.

9.23.6.5. Phase control

Sometimes you want to control exactly when in GHC’s pipeline the INLINE pragma is switched on. Inlining happens only during runs of the simplifier. Each run of the simplifier has a different phase number; the phase number decreases towards zero. If you use -dverbose-core2core you’ll see the sequence of phase numbers for successive runs of the simplifier. In an INLINE pragma you can optionally specify a phase number, thus:

• “INLINE[k] f” means: do not inline f until phase k, but from phase k onwards be very keen to inline it.
• “INLINE[~k] f” means: be very keen to inline f until phase k, but from phase k onwards do not inline it.
• “NOINLINE[k] f” means: do not inline f until phase k, but from phase k onwards be willing to inline it (as if there was no pragma).
• “NOINLINE[~k] f” means: be willing to inline f until phase k, but from phase k onwards do not inline it.

The same information is summarised here:

                         -- Before phase 2     Phase 2 and later
{-# INLINE   [2]  f #-}  --      No                 Yes
{-# INLINE   [~2] f #-}  --      Yes                No
{-# NOINLINE [2]  f #-}  --      No                 Maybe
{-# NOINLINE [~2] f #-}  --      Maybe              No

{-# INLINE   f #-}       --      Yes                Yes
{-# NOINLINE f #-}       --      No                 No

By “Maybe” we mean that the usual heuristic inlining rules apply (if the function body is small, or it is applied to interesting-looking arguments etc). Another way to understand the semantics is this:

• For both INLINE and NOINLINE, the phase number says when inlining is allowed at all.
• The INLINE pragma has the additional effect of making the function body look small, so that when inlining is allowed it is very likely to happen.

The same phase-numbering control is available for RULES (Rewrite rules).

9.23.7. LINE pragma

This pragma is similar to C’s #line pragma, and is mainly for use in automatically generated Haskell code. It lets you specify the line number and filename of the original code; for example

{-# LINE 42 "Foo.vhs" #-}

if you’d generated the current file from something called Foo.vhs and this line corresponds to line 42 in the original. GHC will adjust its error messages to refer to the line/file named in the LINE pragma.

LINE pragmas generated from Template Haskell set the file and line position for the duration of the splice and are limited to the splice. Note that because Template Haskell splices abstract syntax, the file positions are not automatically advanced.

9.23.8. RULES pragma

The RULES pragma lets you specify rewrite rules. It is described in Rewrite rules.

9.23.9. SPECIALIZE pragma

(UK spelling also accepted.) For key overloaded functions, you can create extra versions (NB: more code space) specialised to particular types. Thus, if you have an overloaded function:

hammeredLookup :: Ord key => [(key, value)] -> key -> value

If it is heavily used on lists with Widget keys, you could specialise it as follows:

{-# SPECIALIZE hammeredLookup :: [(Widget, value)] -> Widget -> value #-}

• A SPECIALIZE pragma for a function can be put anywhere its type signature could be put. Moreover, you can also SPECIALIZE an imported function provided it was given an INLINABLE pragma at its definition site (INLINABLE pragma).

• A SPECIALIZE has the effect of generating (a) a specialised version of the function and (b) a rewrite rule (see Rewrite rules) that rewrites a call to the un-specialised function into a call to the specialised one. Moreover, given a SPECIALIZE pragma for a function f, GHC will automatically create specialisations for any type-class-overloaded functions called by f, if they are in the same module as the SPECIALIZE pragma, or if they are INLINABLE; and so on, transitively.

• You can add phase control (Phase control) to the RULE generated by a SPECIALIZE pragma, just as you can if you write a RULE directly. For example:

  {-# SPECIALIZE [0] hammeredLookup :: [(Widget, value)] -> Widget -> value #-}
  

  generates a specialisation rule that only fires in Phase 0 (the final phase). If you do not specify any phase control in the SPECIALIZE pragma, the phase control is inherited from the inline pragma (if any) of the function. For example:

  foo :: Num a => a -> a
  foo = ...blah...
  {-# NOINLINE [0] foo #-}
  {-# SPECIALIZE foo :: Int -> Int #-}
  

  The NOINLINE pragma tells GHC not to inline foo until Phase 0; and this property is inherited by the specialisation RULE, which will therefore only fire in Phase 0.

  The main reason for using phase control on specialisations is so that you can write optimisation RULES that fire early in the compilation pipeline, and only then specialise the calls to the function. If specialisation is done too early, the optimisation rules might fail to fire.

• The type in a SPECIALIZE pragma can be any type that is less polymorphic than the type of the original function. In concrete terms, if the original function is f then the pragma

  {-# SPECIALIZE f :: <type> #-}
  

  is valid if and only if the definition

  f_spec :: <type>
  f_spec = f
  

  is valid. Here are some examples (where we only give the type signature for the original function, not its code):

  f :: Eq a => a -> b -> b
  {-# SPECIALISE f :: Int -> b -> b #-}
  
  g :: (Eq a, Ix b) => a -> b -> b
  {-# SPECIALISE g :: (Eq a) => a -> Int -> Int #-}
  
  h :: Eq a => a -> a -> a
  {-# SPECIALISE h :: (Eq a) => [a] -> [a] -> [a] #-}
  

  The last of these examples will generate a RULE with a somewhat-complex left-hand side (try it yourself), so it might not fire very well. If you use this kind of specialisation, let us know how well it works.

9.23.9.1. SPECIALIZE INLINE

A SPECIALIZE pragma can optionally be followed with a INLINE or NOINLINE pragma, optionally followed by a phase, as described in INLINE and NOINLINE pragmas. The INLINE pragma affects the specialised version of the function (only), and applies even if the function is recursive. The motivating example is this:

-- A GADT for arrays with type-indexed representation
data Arr e where
  ArrInt :: !Int -> ByteArray# -> Arr Int
  ArrPair :: !Int -> Arr e1 -> Arr e2 -> Arr (e1, e2)

(!:) :: Arr e -> Int -> e
{-# SPECIALISE INLINE (!:) :: Arr Int -> Int -> Int #-}
{-# SPECIALISE INLINE (!:) :: Arr (a, b) -> Int -> (a, b) #-}
(ArrInt _ ba)     !: (I# i) = I# (indexIntArray# ba i)
(ArrPair _ a1 a2) !: i      = (a1 !: i, a2 !: i)

Here, (!:) is a recursive function that indexes arrays of type Arr e. Consider a call to (!:) at type (Int,Int). The second specialisation will fire, and the specialised function will be inlined. It has two calls to (!:), both at type Int. Both these calls fire the first specialisation, whose body is also inlined. The result is a type-based unrolling of the indexing function.

You can add explicit phase control (Phase control) to SPECIALISE INLINE pragma, just like on an INLINE pragma; if you do so, the same phase is used for the rewrite rule and the INLINE control of the specialised function.

Warning

You can make GHC diverge by using SPECIALISE INLINE on an ordinarily-recursive function.

9.23.9.2. SPECIALIZE for imported functions

Generally, you can only give a SPECIALIZE pragma for a function defined in the same module. However if a function f is given an INLINABLE pragma at its definition site, then it can subsequently be specialised by importing modules (see INLINABLE pragma). For example

module Map( lookup, blah blah ) where
  lookup :: Ord key => [(key,a)] -> key -> Maybe a
  lookup = ...
  {-# INLINABLE lookup #-}

module Client where
  import Map( lookup )

  data T = T1 | T2 deriving( Eq, Ord )
  {-# SPECIALISE lookup :: [(T,a)] -> T -> Maybe a

Here, lookup is declared INLINABLE, but it cannot be specialised for type T at its definition site, because that type does not exist yet. Instead a client module can define T and then specialise lookup at that type.

Moreover, every module that imports Client (or imports a module that imports Client, transitively) will “see”, and make use of, the specialised version of lookup. You don’t need to put a SPECIALIZE pragma in every module.

Moreover you often don’t even need the SPECIALIZE pragma in the first place. When compiling a module M, GHC’s optimiser (when given the -O flag) automatically considers each top-level overloaded function declared in M, and specialises it for the different types at which it is called in M. The optimiser also considers each imported INLINABLE overloaded function, and specialises it for the different types at which it is called in M. So in our example, it would be enough for lookup to be called at type T:

module Client where
  import Map( lookup )

  data T = T1 | T2 deriving( Eq, Ord )

  findT1 :: [(T,a)] -> Maybe a
  findT1 m = lookup m T1   -- A call of lookup at type T

However, sometimes there are no such calls, in which case the pragma can be useful.

9.23.9.3. Obsolete SPECIALIZE syntax

In earlier versions of GHC, it was possible to provide your own specialised function for a given type:

{-# SPECIALIZE hammeredLookup :: [(Int, value)] -> Int -> value = intLookup #-}

This feature has been removed, as it is now subsumed by the RULES pragma (see Specialisation).

9.23.10. SPECIALIZE instance pragma

Same idea, except for instance declarations. For example:

instance (Eq a) => Eq (Foo a) where {
   {-# SPECIALIZE instance Eq (Foo [(Int, Bar)]) #-}
   ... usual stuff ...
 }

The pragma must occur inside the where part of the instance declaration.

9.23.11. UNPACK pragma

The UNPACK indicates to the compiler that it should unpack the contents of a constructor field into the constructor itself, removing a level of indirection. For example:

data T = T {-# UNPACK #-} !Float
           {-# UNPACK #-} !Float

will create a constructor T containing two unboxed floats. This may not always be an optimisation: if the T constructor is scrutinised and the floats passed to a non-strict function for example, they will have to be reboxed (this is done automatically by the compiler).

Unpacking constructor fields should only be used in conjunction with -O [1], in order to expose unfoldings to the compiler so the reboxing can be removed as often as possible. For example:

f :: T -> Float
f (T f1 f2) = f1 + f2

The compiler will avoid reboxing f1 and f2 by inlining + on floats, but only when -O is on.

Any single-constructor data is eligible for unpacking; for example

data T = T {-# UNPACK #-} !(Int,Int)

will store the two Ints directly in the T constructor, by flattening the pair. Multi-level unpacking is also supported:

data T = T {-# UNPACK #-} !S
data S = S {-# UNPACK #-} !Int {-# UNPACK #-} !Int

will store two unboxed Int#s directly in the T constructor. The unpacker can see through newtypes, too.

See also the -funbox-strict-fields flag, which essentially has the effect of adding {-# UNPACK #-} to every strict constructor field.

[1]	in fact, UNPACK has no effect without -O , for technical reasons (see tick 5252 )

9.23.12. NOUNPACK pragma

The NOUNPACK pragma indicates to the compiler that it should not unpack the contents of a constructor field. Example:

data T = T {-# NOUNPACK #-} !(Int,Int)

Even with the flags -funbox-strict-fields and -O, the field of the constructor T is not unpacked.

9.23.13. SOURCE pragma

The {-# SOURCE #-} pragma is used only in import declarations, to break a module loop. It is described in detail in How to compile mutually recursive modules.

9.23.14. OVERLAPPING, OVERLAPPABLE, OVERLAPS, and INCOHERENT pragmas

The pragmas OVERLAPPING, OVERLAPPABLE, OVERLAPS, INCOHERENT are used to specify the overlap behavior for individual instances, as described in Section Overlapping instances. The pragmas are written immediately after the instance keyword, like this:

instance {-# OVERLAPPING #-} C t where ...

9.24. Rewrite rules

The programmer can specify rewrite rules as part of the source program (in a pragma). Here is an example:

{-# RULES
"map/map"    forall f g xs.  map f (map g xs) = map (f.g) xs
  #-}

Use the debug flag -ddump-simpl-stats to see what rules fired. If you need more information, then -ddump-rule-firings shows you each individual rule firing and -ddump-rule-rewrites also shows what the code looks like before and after the rewrite.

9.24.1. Syntax

From a syntactic point of view:

• There may be zero or more rules in a RULES pragma, separated by semicolons (which may be generated by the layout rule).

• The layout rule applies in a pragma. Currently no new indentation level is set, so if you put several rules in single RULES pragma and wish to use layout to separate them, you must lay out the starting in the same column as the enclosing definitions.

  {-# RULES
  "map/map"    forall f g xs.  map f (map g xs) = map (f.g) xs
  "map/append" forall f xs ys. map f (xs ++ ys) = map f xs ++ map f ys
    #-}
  

  Furthermore, the closing #-} should start in a column to the right of the opening {-#.

• Each rule has a name, enclosed in double quotes. The name itself has no significance at all. It is only used when reporting how many times the rule fired.

• A rule may optionally have a phase-control number (see Phase control), immediately after the name of the rule. Thus:

  {-# RULES
        "map/map" [2]  forall f g xs. map f (map g xs) = map (f.g) xs
    #-}
  

  The “[2]” means that the rule is active in Phase 2 and subsequent phases. The inverse notation “[~2]” is also accepted, meaning that the rule is active up to, but not including, Phase 2.

  Rules support the special phase-control notation “[~]”, which means the rule is never active. This feature supports plugins (see Compiler Plugins), by making it possible to define a RULE that is never run by GHC, but is nevertheless parsed, typechecked etc, so that it is available to the plugin.

• Each variable mentioned in a rule must either be in scope (e.g. map), or bound by the forall (e.g. f, g, xs). The variables bound by the forall are called the pattern variables. They are separated by spaces, just like in a type forall.

• A pattern variable may optionally have a type signature. If the type of the pattern variable is polymorphic, it must have a type signature. For example, here is the foldr/build rule:

  "fold/build"  forall k z (g::forall b. (a->b->b) -> b -> b) .
                foldr k z (build g) = g k z
  

  Since g has a polymorphic type, it must have a type signature.

• The left hand side of a rule must consist of a top-level variable applied to arbitrary expressions. For example, this is not OK:

  "wrong1"   forall e1 e2.  case True of { True -> e1; False -> e2 } = e1
  "wrong2"   forall f.      f True = True
  

  In "wrong1", the LHS is not an application; in "wrong2", the LHS has a pattern variable in the head.

• A rule does not need to be in the same module as (any of) the variables it mentions, though of course they need to be in scope.

• All rules are implicitly exported from the module, and are therefore in force in any module that imports the module that defined the rule, directly or indirectly. (That is, if A imports B, which imports C, then C’s rules are in force when compiling A.) The situation is very similar to that for instance declarations.

• Inside a RULE “forall” is treated as a keyword, regardless of any other flag settings. Furthermore, inside a RULE, the language extension -XScopedTypeVariables is automatically enabled; see Lexically scoped type variables.

• Like other pragmas, RULE pragmas are always checked for scope errors, and are typechecked. Typechecking means that the LHS and RHS of a rule are typechecked, and must have the same type. However, rules are only enabled if the -fenable-rewrite-rules flag is on (see Semantics).

9.24.2. Semantics

From a semantic point of view:

• Rules are enabled (that is, used during optimisation) by the -fenable-rewrite-rules flag. This flag is implied by -O, and may be switched off (as usual) by -fno-enable-rewrite-rules. (NB: enabling -fenable-rewrite-rules without -O may not do what you expect, though, because without -O GHC ignores all optimisation information in interface files; see -fignore-interface-pragmas, -f*: platform-independent flags.) Note that -fenable-rewrite-rules is an optimisation flag, and has no effect on parsing or typechecking.

• Rules are regarded as left-to-right rewrite rules. When GHC finds an expression that is a substitution instance of the LHS of a rule, it replaces the expression by the (appropriately-substituted) RHS. By “a substitution instance” we mean that the LHS can be made equal to the expression by substituting for the pattern variables.

• GHC makes absolutely no attempt to verify that the LHS and RHS of a rule have the same meaning. That is undecidable in general, and infeasible in most interesting cases. The responsibility is entirely the programmer’s!

• GHC makes no attempt to make sure that the rules are confluent or terminating. For example:

  "loop"        forall x y.  f x y = f y x
  

  This rule will cause the compiler to go into an infinite loop.

• If more than one rule matches a call, GHC will choose one arbitrarily to apply.

• GHC currently uses a very simple, syntactic, matching algorithm for matching a rule LHS with an expression. It seeks a substitution which makes the LHS and expression syntactically equal modulo alpha conversion. The pattern (rule), but not the expression, is eta-expanded if necessary. (Eta-expanding the expression can lead to laziness bugs.) But not beta conversion (that’s called higher-order matching).

  Matching is carried out on GHC’s intermediate language, which includes type abstractions and applications. So a rule only matches if the types match too. See Specialisation below.

• GHC keeps trying to apply the rules as it optimises the program. For example, consider:

  let s = map f
      t = map g
  in
  s (t xs)
  

  The expression s (t xs) does not match the rule "map/map", but GHC will substitute for s and t, giving an expression which does match. If s or t was (a) used more than once, and (b) large or a redex, then it would not be substituted, and the rule would not fire.

9.24.3. How rules interact with INLINE/NOINLINE pragmas

Ordinary inlining happens at the same time as rule rewriting, which may lead to unexpected results. Consider this (artificial) example

f x = x
g y = f y
h z = g True

{-# RULES "f" f True = False #-}

Since f‘s right-hand side is small, it is inlined into g, to give

g y = y

Now g is inlined into h, but f‘s RULE has no chance to fire. If instead GHC had first inlined g into h then there would have been a better chance that f‘s RULE might fire.

The way to get predictable behaviour is to use a NOINLINE pragma, or an INLINE[⟨phase⟩] pragma, on f, to ensure that it is not inlined until its RULEs have had a chance to fire. The warning flag -fwarn-inline-rule-shadowing (see Warnings and sanity-checking) warns about this situation.

9.24.4. How rules interact with CONLIKE pragmas

GHC is very cautious about duplicating work. For example, consider

f k z xs = let xs = build g
           in ...(foldr k z xs)...sum xs...
{-# RULES "foldr/build" forall k z g. foldr k z (build g) = g k z #-}

Since xs is used twice, GHC does not fire the foldr/build rule. Rightly so, because it might take a lot of work to compute xs, which would be duplicated if the rule fired.

Sometimes, however, this approach is over-cautious, and we do want the rule to fire, even though doing so would duplicate redex. There is no way that GHC can work out when this is a good idea, so we provide the CONLIKE pragma to declare it, thus:

{-# INLINE CONLIKE [1] f #-}
f x = blah

CONLIKE is a modifier to an INLINE or NOINLINE pragma. It specifies that an application of f to one argument (in general, the number of arguments to the left of the = sign) should be considered cheap enough to duplicate, if such a duplication would make rule fire. (The name “CONLIKE” is short for “constructor-like”, because constructors certainly have such a property.) The CONLIKE pragma is a modifier to INLINE/NOINLINE because it really only makes sense to match f on the LHS of a rule if you are sure that f is not going to be inlined before the rule has a chance to fire.

9.24.5. How rules interact with class methods

Giving a RULE for a class method is a bad idea:

class C a where
  op :: a -> a -> a

instance C Bool where
  op x y = ...rhs for op at Bool...

{-# RULES "f" op True y = False #-}

In this example, op is not an ordinary top-level function; it is a class method. GHC rapidly rewrites any occurrences of op-used-at-type-Bool to a specialised function, say opBool, where

opBool :: Bool -> Bool -> Bool
opBool x y = ..rhs for op at Bool...

So the RULE never has a chance to fire, for just the same reasons as in How rules interact with INLINE/NOINLINE pragmas.

The solution is to define the instance-specific function yourself, with a pragma to prevent it being inlined too early, and give a RULE for it:

instance C Bool where
  op x y = opBool

opBool :: Bool -> Bool -> Bool
{-# NOINLINE [1] opBool #-}
opBool x y = ..rhs for op at Bool...

{-# RULES "f" opBool True y = False #-}

If you want a RULE that truly applies to the overloaded class method, the only way to do it is like this:

class C a where
  op_c :: a -> a -> a

op :: C a => a -> a -> a
{-# NOINLINE [1] op #-}
op = op_c

{-# RULES "reassociate" op (op x y) z = op x (op y z) #-}

Now the inlining of op is delayed until the rule has a chance to fire. The down-side is that instance declarations must define op_c, but all other uses should go via op.

9.24.6. List fusion

The RULES mechanism is used to implement fusion (deforestation) of common list functions. If a “good consumer” consumes an intermediate list constructed by a “good producer”, the intermediate list should be eliminated entirely.

The following are good producers:

• List comprehensions
• Enumerations of Int, Integer and Char (e.g. ['a'..'z']).
• Explicit lists (e.g. [True, False])
• The cons constructor (e.g 3:4:[])
• ++
• map
• take, filter
• iterate, repeat
• zip, zipWith

The following are good consumers:

• List comprehensions
• array (on its second argument)
• ++ (on its first argument)
• foldr
• map
• take, filter
• concat
• unzip, unzip2, unzip3, unzip4
• zip, zipWith (but on one argument only; if both are good producers, zip will fuse with one but not the other)
• partition
• head
• and, or, any, all
• sequence_
• msum

So, for example, the following should generate no intermediate lists:

array (1,10) [(i,i*i) | i <- map (+ 1) [0..9]]

This list could readily be extended; if there are Prelude functions that you use a lot which are not included, please tell us.

If you want to write your own good consumers or producers, look at the Prelude definitions of the above functions to see how to do so.

9.24.7. Specialisation

Rewrite rules can be used to get the same effect as a feature present in earlier versions of GHC. For example, suppose that:

genericLookup :: Ord a => Table a b   -> a   -> b
intLookup     ::          Table Int b -> Int -> b

where intLookup is an implementation of genericLookup that works very fast for keys of type Int. You might wish to tell GHC to use intLookup instead of genericLookup whenever the latter was called with type Table Int b -> Int -> b. It used to be possible to write

{-# SPECIALIZE genericLookup :: Table Int b -> Int -> b = intLookup #-}

This feature is no longer in GHC, but rewrite rules let you do the same thing:

{-# RULES "genericLookup/Int" genericLookup = intLookup #-}

This slightly odd-looking rule instructs GHC to replace genericLookup by intLookup whenever the types match. What is more, this rule does not need to be in the same file as genericLookup, unlike the SPECIALIZE pragmas which currently do (so that they have an original definition available to specialise).

It is Your Responsibility to make sure that intLookup really behaves as a specialised version of genericLookup!!!

An example in which using RULES for specialisation will Win Big:

toDouble :: Real a => a -> Double
toDouble = fromRational . toRational

{-# RULES "toDouble/Int" toDouble = i2d #-}
i2d (I# i) = D# (int2Double# i) -- uses Glasgow prim-op directly

The i2d function is virtually one machine instruction; the default conversion—via an intermediate Rational-is obscenely expensive by comparison.

9.24.8. Controlling what’s going on in rewrite rules

• Use -ddump-rules to see the rules that are defined in this module. This includes rules generated by the specialisation pass, but excludes rules imported from other modules.

• Use -ddump-simpl-stats to see what rules are being fired. If you add -dppr-debug you get a more detailed listing.

• Use -ddump-rule-firings or -ddump-rule-rewrites to see in great detail what rules are being fired. If you add -dppr-debug you get a still more detailed listing.

• The definition of (say) build in GHC/Base.lhs looks like this:

  build   :: forall a. (forall b. (a -> b -> b) -> b -> b) -> [a]
  {-# INLINE build #-}
  build g = g (:) []
  

  Notice the INLINE! That prevents (:) from being inlined when compiling PrelBase, so that an importing module will “see” the (:), and can match it on the LHS of a rule. INLINE prevents any inlining happening in the RHS of the INLINE thing. I regret the delicacy of this.

• In libraries/base/GHC/Base.lhs look at the rules for map to see how to write rules that will do fusion and yet give an efficient program even if fusion doesn’t happen. More rules in GHC/List.lhs.

9.25. Special built-in functions

GHC has a few built-in functions with special behaviour. In particular:

• inline allows control over inlining on a per-call-site basis.
• lazy restrains the strictness analyser.
• oneShot gives a hint to the compiler about how often a function is being called.

9.26. Generic classes

GHC used to have an implementation of generic classes as defined in the paper “Derivable type classes”, Ralf Hinze and Simon Peyton Jones, Haskell Workshop, Montreal Sept 2000, pp. 94-105. These have been removed and replaced by the more general support for generic programming.

9.27. Generic programming

Using a combination of -XDeriveGeneric (Deriving Typeable instances), -XDefaultSignatures (Default method signatures), and -XDeriveAnyClass (Deriving any other class), you can easily do datatype-generic programming using the GHC.Generics framework. This section gives a very brief overview of how to do it.

Generic programming support in GHC allows defining classes with methods that do not need a user specification when instantiating: the method body is automatically derived by GHC. This is similar to what happens for standard classes such as Read and Show, for instance, but now for user-defined classes.

9.27.1. Deriving representations

The first thing we need is generic representations. The GHC.Generics module defines a couple of primitive types that are used to represent Haskell datatypes:

-- | Unit: used for constructors without arguments
data U1 p = U1

-- | Constants, additional parameters and recursion of kind *
newtype K1 i c p = K1 { unK1 :: c }

-- | Meta-information (constructor names, etc.)
newtype M1 i c f p = M1 { unM1 :: f p }

-- | Sums: encode choice between constructors
infixr 5 :+:
data (:+:) f g p = L1 (f p) | R1 (g p)

-- | Products: encode multiple arguments to constructors
infixr 6 :*:
data (:*:) f g p = f p :*: g p

The Generic and Generic1 classes mediate between user-defined datatypes and their internal representation as a sum-of-products:

class Generic a where
  -- Encode the representation of a user datatype
  type Rep a :: * -> *
  -- Convert from the datatype to its representation
  from  :: a -> (Rep a) x
  -- Convert from the representation to the datatype
  to    :: (Rep a) x -> a

class Generic1 f where
  type Rep1 f :: * -> *

  from1  :: f a -> Rep1 f a
  to1    :: Rep1 f a -> f a

Generic1 is used for functions that can only be defined over type containers, such as map. Instances of these classes can be derived by GHC with the -XDeriveGeneric (Deriving Typeable instances), and are necessary to be able to define generic instances automatically.

For example, a user-defined datatype of trees

::

data UserTree a = Node a (UserTree a) (UserTree a) | Leaf

in a Main module in a package named foo will get the following representation:

instance Generic (UserTree a) where
  -- Representation type
  type Rep (UserTree a) =
    M1 D ('MetaData "UserTree" "Main" "package-name" "foo" 'False) (
          M1 C ('MetaCons "Node" 'PrefixI 'False) (
                M1 S 'MetaNoSel (K1 R a)
            :*: M1 S 'MetaNoSel (K1 R (UserTree a))
            :*: M1 S 'MetaNoSel (K1 R (UserTree a)))
      :+: M1 C ('MetaCons "Leaf" 'PrefixI 'False) U1)

  -- Conversion functions
  from (Node x l r) = M1 (L1 (M1 (M1 (K1 x) :*: M1 (K1 l) :*: M1 (K1 r))))
  from Leaf         = M1 (R1 (M1 U1))
  to (M1 (L1 (M1 (M1 (K1 x) :*: M1 (K1 l) :*: M1 (K1 r))))) = Node x l r
  to (M1 (R1 (M1 U1)))                                      = Leaf

This representation is generated automatically if a deriving Generic clause is attached to the datatype. Standalone deriving can also be used.

9.27.2. Writing generic functions

A generic function is defined by creating a class and giving instances for each of the representation types of GHC.Generics. As an example we show generic serialization:

data Bin = O | I

class GSerialize f where
  gput :: f a -> [Bin]

instance GSerialize U1 where
  gput U1 = []

instance (GSerialize a, GSerialize b) => GSerialize (a :*: b) where
  gput (x :*: y) = gput x ++ gput y

instance (GSerialize a, GSerialize b) => GSerialize (a :+: b) where
  gput (L1 x) = O : gput x
  gput (R1 x) = I : gput x

instance (GSerialize a) => GSerialize (M1 i c a) where
  gput (M1 x) = gput x

instance (Serialize a) => GSerialize (K1 i a) where
  gput (K1 x) = put x

Typically this class will not be exported, as it only makes sense to have instances for the representation types.

9.27.3. Unlifted representation types

The data family URec is provided to enable generic programming over datatypes with certain unlifted arguments. There are six instances corresponding to common unlifted types:

data family URec a p

data instance URec (Ptr ()) p = UAddr   { uAddr#   :: Addr#   }
data instance URec Char     p = UChar   { uChar#   :: Char#   }
data instance URec Double   p = UDouble { uDouble# :: Double# }
data instance URec Int      p = UInt    { uInt#    :: Int#    }
data instance URec Float    p = UFloat  { uFloat#  :: Float#  }
data instance URec Word     p = UWord   { uWord#   :: Word#   }

Six type synonyms are provided for convenience:

type UAddr   = URec (Ptr ())
type UChar   = URec Char
type UDouble = URec Double
type UFloat  = URec Float
type UInt    = URec Int
type UWord   = URec Word

As an example, this data declaration:

data IntHash = IntHash Int#
  deriving Generic

results in the following Generic instance:

instance Generic IntHash where
  type Rep IntHash =
    D1 D1IntHash
      (C1 C1_0IntHash
        (S1 NoSelector UInt))

A user could provide, for example, a GSerialize UInt instance so that a Serialize IntHash instance could be easily defined in terms of GSerialize.

9.27.4. Generic defaults

The only thing left to do now is to define a “front-end” class, which is exposed to the user:

class Serialize a where
  put :: a -> [Bin]

  default put :: (Generic a, GSerialize (Rep a)) => a -> [Bit]
  put = gput . from

Here we use a default signature to specify that the user does not have to provide an implementation for put, as long as there is a Generic instance for the type to instantiate. For the UserTree type, for instance, the user can just write:

instance (Serialize a) => Serialize (UserTree a)

The default method for put is then used, corresponding to the generic implementation of serialization. If you are using -XDeriveAnyClass, the same instance is generated by simply attaching a deriving Serialize clause to the UserTree datatype declaration. For more examples of generic functions please refer to the generic-deriving package on Hackage.

9.27.5. More information

For more details please refer to the HaskellWiki page or the original paper [Generics2010].

[Generics2010]

Jose Pedro Magalhaes, Atze Dijkstra, Johan Jeuring, and Andres Loeh. A generic deriving mechanism for Haskell. Proceedings of the third ACM Haskell symposium on Haskell (Haskell‘2010), pp. 37-48, ACM, 2010.

9.28. Roles

Using -XGeneralizedNewtypeDeriving (Generalising the deriving clause), a programmer can take existing instances of classes and “lift” these into instances of that class for a newtype. However, this is not always safe. For example, consider the following:

newtype Age = MkAge { unAge :: Int }

type family Inspect x
type instance Inspect Age = Int
type instance Inspect Int = Bool

class BadIdea a where
  bad :: a -> Inspect a

instance BadIdea Int where
  bad = (> 0)

deriving instance BadIdea Age    -- not allowed!

If the derived instance were allowed, what would the type of its method bad be? It would seem to be Age -> Inspect Age, which is equivalent to Age -> Int, according to the type family Inspect. Yet, if we simply adapt the implementation from the instance for Int, the implementation for bad produces a Bool, and we have trouble.

The way to identify such situations is to have roles assigned to type variables of datatypes, classes, and type synonyms.

Roles as implemented in GHC are a from a simplified version of the work described in Generative type abstraction and type-level computation, published at POPL 2011.

9.28.1. Nominal, Representational, and Phantom

The goal of the roles system is to track when two types have the same underlying representation. In the example above, Age and Int have the same representation. But, the corresponding instances of BadIdea would not have the same representation, because the types of the implementations of bad would be different.

Suppose we have two uses of a type constructor, each applied to the same parameters except for one difference. (For example, T Age Bool c and T Int Bool c for some type T.) The role of a type parameter says what we need to know about the two differing type arguments in order to know that the two outer types have the same representation (in the example, what must be true about Age and Int in order to show that T Age Bool c has the same representation as T Int Bool c).

GHC supports three different roles for type parameters: nominal, representational, and phantom. If a type parameter has a nominal role, then the two types that differ must not actually differ at all: they must be identical (after type family reduction). If a type parameter has a representational role, then the two types must have the same representation. (If T‘s first parameter’s role is representational, then T Age Bool c and T Int Bool c would have the same representation, because Age and Int have the same representation.) If a type parameter has a phantom role, then we need no further information.

Here are some examples:

data Simple a = MkSimple a          -- a has role representational

type family F
type instance F Int = Bool
type instance F Age = Char

data Complex a = MkComplex (F a)    -- a has role nominal

data Phant a = MkPhant Bool         -- a has role phantom

The type Simple has its parameter at role representational, which is generally the most common case. Simple Age would have the same representation as Simple Int. The type Complex, on the other hand, has its parameter at role nominal, because Simple Age and Simple Int are not the same. Lastly, Phant Age and Phant Bool have the same representation, even though Age and Bool are unrelated.

9.28.2. Role inference

What role should a given type parameter should have? GHC performs role inference to determine the correct role for every parameter. It starts with a few base facts: (->) has two representational parameters; (~) has two nominal parameters; all type families’ parameters are nominal; and all GADT-like parameters are nominal. Then, these facts are propagated to all places where these types are used. The default role for datatypes and synonyms is phantom; the default role for classes is nominal. Thus, for datatypes and synonyms, any parameters unused in the right-hand side (or used only in other types in phantom positions) will be phantom. Whenever a parameter is used in a representational position (that is, used as a type argument to a constructor whose corresponding variable is at role representational), we raise its role from phantom to representational. Similarly, when a parameter is used in a nominal position, its role is upgraded to nominal. We never downgrade a role from nominal to phantom or representational, or from representational to phantom. In this way, we infer the most-general role for each parameter.

Classes have their roles default to nominal to promote coherence of class instances. If a C Int were stored in a datatype, it would be quite bad if that were somehow changed into a C Age somewhere, especially if another C Age had been declared!

There is one particularly tricky case that should be explained:

data Tricky a b = MkTricky (a b)

What should Tricky‘s roles be? At first blush, it would seem that both a and b should be at role representational, since both are used in the right-hand side and neither is involved in a type family. However, this would be wrong, as the following example shows:

data Nom a = MkNom (F a)   -- type family F from example above

Is Tricky Nom Age representationally equal to Tricky Nom Int? No! The former stores a Char and the latter stores a Bool. The solution to this is to require all parameters to type variables to have role nominal. Thus, GHC would infer role representational for a but role nominal for b.

9.28.3. Role annotations -XRoleAnnotations

Sometimes the programmer wants to constrain the inference process. For example, the base library contains the following definition:

data Ptr a = Ptr Addr#

The idea is that a should really be a representational parameter, but role inference assigns it to phantom. This makes some level of sense: a pointer to an Int really is representationally the same as a pointer to a Bool. But, that’s not at all how we want to use Ptrs! So, we want to be able to say

type role Ptr representational
data Ptr a = Ptr Addr#

The type role (enabled with -XRoleAnnotations) declaration forces the parameter a to be at role representational, not role phantom. GHC then checks the user-supplied roles to make sure they don’t break any promises. It would be bad, for example, if the user could make BadIdea‘s role be representational.

As another example, we can consider a type Set a that represents a set of data, ordered according to a‘s Ord instance. While it would generally be type-safe to consider a to be at role representational, it is possible that a newtype and its base type have different orderings encoded in their respective Ord instances. This would lead to misbehavior at runtime. So, the author of the Set datatype would like its parameter to be at role nominal. This would be done with a declaration

type role Set nominal

Role annotations can also be used should a programmer wish to write a class with a representational (or phantom) role. However, as a class with non-nominal roles can quickly lead to class instance incoherence, it is necessary to also specify -XIncoherentInstances to allow non-nominal roles for classes.

The other place where role annotations may be necessary are in hs-boot files (How to compile mutually recursive modules), where the right-hand sides of definitions can be omitted. As usual, the types/classes declared in an hs-boot file must match up with the definitions in the hs file, including down to the roles. The default role for datatypes is representational in hs-boot files, corresponding to the common use case.

Role annotations are allowed on data, newtype, and class declarations. A role annotation declaration starts with type role and is followed by one role listing for each parameter of the type. (This parameter count includes parameters implicitly specified by a kind signature in a GADT-style data or newtype declaration.) Each role listing is a role (nominal, representational, or phantom) or a _. Using a _ says that GHC should infer that role. The role annotation may go anywhere in the same module as the datatype or class definition (much like a value-level type signature). Here are some examples:

type role T1 _ phantom
data T1 a b = MkT1 a     -- b is not used; annotation is fine but unnecessary

type role T2 _ phantom
data T2 a b = MkT2 b     -- ERROR: b is used and cannot be phantom

type role T3 _ nominal
data T3 a b = MkT3 a     -- OK: nominal is higher than necessary, but safe

type role T4 nominal
data T4 a = MkT4 (a Int) -- OK, but nominal is higher than necessary

type role C representational _   -- OK, with -XIncoherentInstances
class C a b where ...    -- OK, b will get a nominal role

type role X nominal
type X a = ...           -- ERROR: role annotations not allowed for type synonyms

9.29. Strict Haskell

High-performance Haskell code (e.g. numeric code) can sometimes be littered with bang patterns, making it harder to read. The reason is that lazy evaluation isn’t the right default in this particular code but the programmer has no way to say that except by repeatedly adding bang patterns. Below -XStrictData and -XStrict are detailed that allows the programmer to switch the default behavior on a per-module basis.

9.29.1. Strict-by-default data types

Informally the StrictData language extension switches data type declarations to be strict by default allowing fields to be lazy by adding a ~ in front of the field.

When the user writes

data T = C a
data T' = C' ~a

we interpret it as if they had written

data T = C !a
data T' = C' a

The extension only affects definitions in this module.

9.29.2. Strict-by-default pattern bindings

Informally the Strict language extension switches functions, data types, and bindings to be strict by default, allowing optional laziness by adding ~ in front of a variable. This essentially reverses the present situation where laziness is default and strictness can be optionally had by adding ! in front of a variable.

Strict implies StrictData.

• Function definitions.

  When the user writes

  f x = ...
  

  we interpret it as if they had written

  f !x = ...
  

  Adding ~ in front of x gives the regular lazy behavior.

• Let/where bindings.

  When the user writes

  let x = ...
  let pat = ...
  

  we interpret it as if they had written

  let !x = ...
  let !pat = ...
  

  Adding ~ in front of x gives the regular lazy behavior. Notice that we do not put bangs on nested patterns. For example

  let (p,q) = if flob then (undefined, undefined) else (True, False)
  in ...
  

  will behave like

  let !(p,q) = if flob then (undefined, undefined) else (True,False)
  in ...
  

  which will strictly evaluate the right hand side, and bind p and q to the components of the pair. But the pair itself is lazy (unless we also compile the Prelude with Strict; see Modularity below). So p and q may end up bound to undefined. See also Recursive and polymorphic let bindings below.

• Case expressions.

  The patterns of a case expression get an implicit bang, unless disabled with ~. For example

  case x of (a,b) -> rhs
  

  is interpreted as

  case x of !(a,b) -> rhs
  

  Since the semantics of pattern matching in case expressions is strict, this usually has no effect whatsoever. But it does make a difference in the degenerate case of variables and newtypes. So

  case x of y -> rhs
  

  is lazy in Haskell, but with Strict is interpreted as

  case x of !y -> rhs
  

  which evalutes x. Similarly, if newtype Age = MkAge Int, then

  case x of MkAge i -> rhs
  

  is lazy in Haskell; but with Strict the added bang makes it strict.

• Top level bindings.

  are unaffected by Strict. For example:

  x = factorial 20
  (y,z) = if x > 10 then True else False
  

  Here x and the pattern binding (y,z) remain lazy. Reason: there is no good moment to force them, until first use.

• Newtypes.

  There is no effect on newtypes, which simply rename existing types. For example:

  newtype T = C a
  f (C x)  = rhs1
  g !(C x) = rhs2
  

  In ordinary Haskell, f is lazy in its argument and hence in x; and g is strict in its argument and hence also strict in x. With Strict, both become strict because f‘s argument gets an implict bang.

9.29.3. Modularity

Strict and StrictData only affects definitions in the module they are used in. Functions and data types imported from other modules are unaffected. For example, we won’t evaluate the argument to Just before applying the constructor. Similarly we won’t evaluate the first argument to Data.Map.findWithDefault before applying the function.

This is crucial to preserve correctness. Entities defined in other modules might rely on laziness for correctness (whether functional or performance).

Tuples, lists, Maybe, and all the other types from Prelude continue to have their existing, lazy, semantics.

9.29.4. Recursive and polymorphic let bindings

Static semantics

Exactly as in Haskell, unaffected by Strict. This is more permissive than past rules for bang patterns in let bindings, because it supports bang-patterns for polymorphic and recursive bindings.

Dynamic semantics

Consider the rules in the box of Section 3.12 of the Haskell report. Replace these rules with the following ones, where v stands for a variable:

FORCE

Replace any binding !p = e with v = e; p = v and replace e0 with v seq e0, where v is fresh. This translation works fine if p is already a variable x, but can obviously be optimised by not introducing a fresh variable v.

SPLIT

Replace any binding p = e, where p is not a variable, with v = e; x1 = case v of p -> x1; ...; xn = case v of p -> xn, where v is fresh and x1.. xn are the bound variables of p. Again if e is a variable, you can optimised his by not introducing a fresh variable.

The result will be a (possibly) recursive set of bindings, binding only simple variables on the left hand side. (One could go one step further, as in the Haskell Report and make the recursive bindings non-recursive using fix, but we do not do so in Core, and it only obfuscates matters, so we do not do so here.)

Here are some examples of how this translation works. The first expression of each sequence is Haskell source; the subsequent ones are Core.

Here is a simple non-recursive case:

let x :: Int     -- Non-recursive
    !x = factorial y
in body

===> (FORCE)
    let x = factorial y in x `seq` body

===> (inline seq)
    let x = factorial y in case x of x -> body

===> (inline x)
    case factorial y of x -> body

Same again, only with a pattern binding:

let !(x,y) = if blob then (factorial p, factorial q) else (0,0)
in body

===> (FORCE)
    let v = if blob then (factorial p, factorial q) else (0,0)
        (x,y) = v
    in v `seq` body

===> (SPLIT)
    let v = if blob then (factorial p, factorial q) else (0,0)
        x = case v of (x,y) -> x
        y = case v of (x,y) -> y
    in v `seq` body

===> (inline seq, float x,y bindings inwards)
    let v = if blob then (factorial p, factorial q) else (0,0)
    in case v of v -> let x = case v of (x,y) -> x
                            y = case v of (x,y) -> y
                        in body

===> (fluff up v's pattern; this is a standard Core optimisation)
    let v = if blob then (factorial p, factorial q) else (0,0)
    in case v of v@(p,q) -> let x = case v of (x,y) -> x
                                y = case v of (x,y) -> y
                            in body

===> (case of known constructor)
    let v = if blob then (factorial p, factorial q) else (0,0)
    in case v of v@(p,q) -> let x = p
                                y = q
                            in body

===> (inline x,y)
    let v = if blob then (factorial p, factorial q) else (0,0)
    in case v of (p,q) -> body[p/x, q/y]

The final form is just what we want: a simple case expression.

Here is a recursive case

letrec xs :: [Int]  -- Recursive
        !xs = factorial y : xs
in body

===> (FORCE)
    letrec xs = factorial y : xs in xs `seq` body

===> (inline seq)
    letrec xs = factorial y : xs in case xs of xs -> body

===> (eliminate case of value)
    letrec xs = factorial y : xs in body

and a polymorphic one:

let f :: forall a. [a] -> [a]    -- Polymorphic
    !f = fst (reverse, True)
in body

===> (FORCE)
    let f = /\a. fst (reverse a, True) in f `seq` body
===> (inline seq, inline f)
    case (/\a. fst (reverse a, True)) of f -> body

Notice that the seq is added only in the translation to Core If we did it in Haskell source, thus

let f = ... in f `seq` body

then f‘s polymorphic type would get intantiated, so the Core translation would be

let f = ... in f Any `seq` body

When overloading is involved, the results might be slightly counter intuitive:

let f :: forall a. Eq a => a -> [a] -> Bool    -- Overloaded
    !f = fst (member, True)
in body

===> (FORCE)
    let f = /\a \(d::Eq a). fst (member, True) in f `seq` body

===> (inline seq, case of value)
    let f = /\a \(d::Eq a). fst (member, True) in body

Note that the bang has no effect at all in this case