6.7.4. Pattern synonyms

PatternSynonyms
Since

7.8.1

Allow the definition of pattern synonyms.

Pattern synonyms are enabled by the language extension PatternSynonyms, which is required for defining them, but not for using them. More information and examples of pattern synonyms can be found on the Wiki page.

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.

Explicitly bidirectional pattern synonyms offer greater flexibility than implicitly bidirectional ones in terms of the syntax that is permitted. For instance, the following is not a legal implicitly bidirectional pattern synonym:

pattern StrictJust a = Just !a

This is illegal because the use of BangPatterns on the right-hand sides prevents it from being a well formed expression. However, constructing a strict pattern synonym is quite possible with an explicitly bidirectional pattern synonym:

pattern StrictJust a <- Just !a where
  StrictJust !a = Just a

Constructing an explicitly bidirectional pattern synonym also:

  • can create different data constructors from the underlying data type, not just the one appearing in the pattern match;

  • can call any functions or conditional logic, especially validation, of course providing it constructs a result of the right type;

  • can use guards on the lhs of the =;

  • can have multiple equations.

For example:

data PosNeg = Pos Int | Neg Int
pattern Smarter{ nonneg } <- Pos nonneg  where
  Smarter x = if x >= 0 then (Pos x) else (Neg x)

Or using guards:

pattern Smarter{ nonneg } <- Pos nonneg  where
  Smarter x | x >= 0    = (Pos x)
            | otherwise = (Neg x)

There is an extensive Haskell folk art of smart constructors, essentially functions that wrap validation around a constructor, and avoid exposing its representation. The downside is that the underlying constructor can’t be used as a matcher. Pattern synonyms can be used as genuinely smart constructors, for both validation and matching.

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)

6.7.4.1. 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 -> (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. There are also lots more details in the paper.

See the Wiki page for more details.

6.7.4.2. 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                      -- lhs restricted, see below

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}

The pat_lhs for explicitly bidirectional construction cannot use Record syntax. (Because the rhs expr might be constructing different data constructors.) It can use guards with multiple equations.

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.

COMPLETE pragmas can be specified in order to tell the pattern match exhaustiveness checker that a set of pattern synonyms is complete.

6.7.4.3. 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.

As of GHC 8.0.1 you may also “bundle” pattern synonyms with an exported type constructor, making that pattern appear as a data constructor of that type. To bundle a pattern synonym, 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 pattern synonyms 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.

6.7.4.4. 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 => CProv => t1 -> t2 -> ... -> tN -> t

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

  • ⟨CReq⟩ are the constraints required to match the pattern.

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

For example, consider

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

f1 :: (Num a, Eq 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 CProv is empty, (i.e., ()), it can be omitted altogether in the above pattern type signature for P.

  • However, if CProv is non-empty, while CReq is, the above pattern type signature for P must be specified as

    P :: () => CProv => t1 -> t2 -> .. -> tN -> t
    
  • The GHCi :info command shows pattern types in this format.

  • 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]
    

    Just like signatures on value-level bindings, pattern synonym signatures can apply to more than one pattern. For instance,

    pattern Left', Right' :: a -> Either a a
    pattern Left' x  = Left x
    pattern Right' x = Right x
    
  • The rules for lexically-scoped type variables (see Lexically scoped type variables) apply to pattern-synonym signatures. As those rules specify, only the type variables from an explicit, syntactically-visible outer forall (the universals) scope over the definition of the pattern synonym; the existentials, bound by the inner forall, do not. For example

    data T a where
       MkT :: Bool -> b -> (b->Int) -> a -> T a
    
    pattern P :: forall a. forall b. b -> (b->Int) -> a -> T a
    pattern P x y v <- MkT True x y (v::a)
    

    Here the universal type variable a scopes over the definition of P, but the existential b does not. (c.f. discussion on #14998.)

  • 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 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 #9953 for discussion of this choice.)

6.7.4.5. 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.

More precisely, the semantics of pattern matching is given in Section 3.17 of the Haskell 2010 report. To the informal semantics in Section 3.17.2 we add this extra rule:

  • If the pattern is a constructor pattern (P p1 ... pn), where P is a pattern synonym defined by P x1 ... xn = p or P x1 ... xn <- p, then:

    1. Match the value v against p. If this match fails or diverges, so does the whole (pattern synonym) match. Otherwise the match against p must bind the variables x1 ... xn; let them be bound to values v1 ... vn.

    2. Match v1 against p1, v2 against p2 and so on. If any of these matches fail or diverge, so does the whole match.

    3. If all the matches against the pi succeed, the match succeeds, binding the variables bound by the pi . (The xi are not bound; they remain local to the pattern synonym declaration.)

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