= Pattern Synonyms =
Most language entities in Haskell can be named so that they can be abbreviated instead of written out in full.
This proposal provides the same power for patterns.
See the [wiki:PatternSynonyms/Implementation implementation] page for implementation details.
Relevant closed tickets:
* #5144 (Pattern synonyms [https://github.com/ghc/ghc/commit/4f8369bf47d27b11415db251e816ef1a2e1eb3d8 merged into master] on 20 January 2014)
* #8968 (GADTs)
* #9417 (Haddock)
* #9514 (Haddock again)
* #8584 (pattern type signatures)
Relevant open tickets:
* #8581 (explicitly bidirectional)
* #8582
* #8583
* #8761
* #8779
== Motivating example ==
Here is a simple representation of types
{{{
data Type = App String [Type]
}}}
Using this representations the arrow type looks like `App "->" [t1, t2]`.
Here are functions that collect all argument types of nested arrows and recognize the `Int` type:
{{{
collectArgs :: Type -> [Type]
collectArgs (App "->" [t1, t2]) = t1 : collectArgs t2
collectArgs _ = []
isInt (App "Int" []) = True
isInt _ = False
}}}
Matching on `App` directly is both hard to read and error prone to write.
The proposal is to introduce a way to give patterns names:
{{{
pattern Arrow t1 t2 = App "->" [t1, t2]
pattern Int = App "Int" []
}}}
And now we can write
{{{
collectArgs :: Type -> [Type]
collectArgs (Arrow t1 t2) = t1 : collectArgs t2
collectArgs _ = []
isInt Int = True
isInt _ = False
}}}
Here is a second example from [http://www.reddit.com/r/haskell/comments/1kmods/patternsynonyms_ghc_trac/ pigworker on Reddit].
Your basic sums-of-products functors can be built from this kit.
{{{
newtype K a x = K a
newtype I x = I x
newtype (:+:) f g x = Sum (Either (f x) (g x))
newtype (:*:) f g x = Prod (f x, g x)
}}}
and then you can make recursive datatypes via
{{{
newtype Fix f = In (f (Fix f))
}}}
e.g.,
{{{
type Tree = Fix (K () :+: (I :*: I))
}}}
and you can get useful generic operations cheaply because the functors in the kit are all `Traversable`, admit a partial zip operation, etc.
You can define friendly constructors for use in expressions
{{{
leaf :: Tree
leaf = In (Sum (Left (K ())))
node :: Tree -> Tree -> Tree
node l r = In (Sum (Right (Prod (I l, I r))))
}}}
but any `Tree`-specific pattern matching code you write will be wide and obscure. Turning these definitions into pattern synonyms means you can have both readable type-specific programs and handy generics without marshalling your data between views.
== Uni-directional (pattern-only) synonyms ==
The simplest form of pattern synonyms is the one from the examples above. The grammar rule is:
`pattern` ''conid'' ''varid,,1,,'' ... ''varid,,n,,'' `<-` ''pat''
`pattern` ''varid,,1,,'' ''consym'' ''varid,,2,,'' `<-` ''pat''
* Each of the variables on the left hand side must occur exactly once on the right hand side
* Pattern synonyms are not allowed to be recursive. Cf. type synonyms.
{{{#!td
There have been several proposals for the syntax of defining pattern-only synonyms:
* `pattern` ''conid'' ''varid,,1,,'' ... ''varid,,n,,'' `~` ''pat''
* `pattern` ''conid'' ''varid,,1,,'' ... ''varid,,n,,'' `:=` ''pat''
* `pattern` ''conid'' ''varid,,1,,'' ... ''varid,,n,,'' `->` ''pat''
* `pattern` ''conid'' ''varid,,1,,'' ... ''varid,,n,,'' `<-` ''pat''
}}}
Pattern synonyms can be exported and imported by prefixing the ''conid'' with the keyword `pattern`:
{{{
module Foo (pattern Arrow) where ...
}}}
This is required because pattern synonyms are in the namespace of constructors, so it's perfectly valid to have
{{{
data P = C
pattern P = 42
}}}
You may also give a type signature for a pattern, but as with most other type signatures in Haskell it is optional:
`pattern` ''conid'' `::` ''type''
E.g.
{{{
pattern Arrow :: Type -> Type -> Type
pattern Arrow t1 t2 <- App "->" [t1, t2]
}}}
Together with ViewPatterns we can now create patterns that look like regular patterns to match on existing (perhaps abstract) types in new ways:
{{{
import qualified Data.Sequence as Seq
pattern Empty <- (Seq.viewl -> Seq.EmptyL)
pattern x :< xs <- (Seq.viewl -> x Seq.:< xs)
pattern xs :> x <- (Seq.viewr -> xs Seq.:> x)
}}}
== Simply-bidirectional pattern synonyms ==
In cases where ''pat'' is in the intersection of the grammars for patterns and expressions (i.e. is valid both as an expression and a pattern), the pattern synonym can be made bidirectional, and can be used in expression contexts as well. Bidirectional pattern synonyms have the following syntax:
`pattern` ''conid'' ''varid,,1,,'' ... ''varid,,n,,'' `=` ''pat''
`pattern` ''varid,,1,,'' ''consym'' ''varid,,2,,'' `=` ''pat''
For example, the following two pattern synonym definitions are rejected, because they are not bidirectional (but they would be valid as pattern-only synonyms)
{{{
pattern ThirdElem x = _:_:x:_
pattern Snd y = (x, y)
}}}
since the right-hand side is not a closed expression of {''x''} and {''y''} respectively.
In contrast, the pattern synonyms for ''Arrow'' and ''Int'' above are bidirectional, so you can e.g. write:
{{{
arrows :: [Type] -> Type -> Type
arrows = flip $ foldr Arrow
}}}
== Explicitly-bidirectional pattern synonyms ==
What if you want to use `Succ` in an expression:
{{{
pattern Succ n <- n1 | let n = n1 -1, n >= 0
}}}
It's clearly impossible since its expansion is a pattern that has no meaning as an expression.
Nevertheless, if we want to make what looks like a constructor for a type we will often want to use it in both patterns and expressions.
This is the rationale for the most complicated synonyms, the bidirectional ones. They provide two expansions, one for patterns and one for expressions.
`pattern` ''conid'' ''varid,,1,,'' ... ''varid,,n,,'' `<-` ''pat'' `where` ''cfunlhs'' ''rhs''
where ''cfunlhs'' is like ''funlhs'', except that the functions symbol is a ''conid'' instead of a ''varid''.
Example:
{{{
pattern Succ n <- n1 | let n = n1-1, n >= 0 where
Succ n = n + 1
}}}
'''TODO''': Rewrite this example to not use ViewPatternsAlternative
The first part as is before and describes the expansion of the synonym in patterns. The second part describes the expansion in expressions.
{{{
fac 0 = 0
fac (Succ n) = Succ n * fac n
}}}
== Associated pattern synonyms ==
Just like data types and type synonyms can be part of a class declaration, it would be possible to have pattern synonyms as well.
Example:
{{{
class ListLike l where
pattern Nil :: l a
pattern Cons :: a -> l a -> a
isNil :: l a -> Bool
isNil Nil = True
isNil (Cons _ _) = False
append :: l a -> l a -> l a
instance ListLike [] where
pattern Nil = []
pattern Cons x xs = x:xs
append = (++)
headOf :: (ListLike l) => l a -> Maybe a
headOf Nil = Nothing
headOf (Cons x _) = Just x
}}}
One could go one step further and leave out the `pattern` keyword to obtain ''associated constructors'', which are required to be bidirectional. The capitalized identifier would indicate that a pattern synonym is being defined. For complicated cases one could resort to the `where` syntax (shown above).
'''TODO''': Syntax for associated pattern synonym declarations to discern between pattern-only and bidirectional pattern synonyms
== Static semantics ==
A unidirectional pattern synonym declaration has the form
{{{
pattern P var1 var2 ... varN <- pat
}}}
The formal pattern synonym arguments `var1`, `var2`, ..., `varN` are brought
into scope by the pattern pat on the right-hand side. The declaration
brings the name `P` as a pattern synonym into the module-level scope.
The pattern synonym `P` is assigned a //pattern type// of the form
{{{
pattern P :: CProv => CReq => t1 -> t2 -> ... -> tN -> t
}}}
where `t1`, ..., `tN` are the types of the parameters `var1`, ..., `varN`, `t` is the simple type (with no context) of the thing getting matched, and `CReq` and `CProv` are type contexts.
`CReq` can be omitted if it is empty. If `CProv` is empty, but `CReq` is not, `()` is used. The following example shows cases:
{{{
data Showable where
MkShowable :: (Show a) => a -> Showable
-- Required context is empty
pattern Sh :: (Show a) => a -> Showable
pattern Sh x <- MkShowable x
-- Provided context is empty, but required context is not
pattern One :: () => (Num a, Eq a) => a
pattern One <- 1
}}}
A pattern synonym can be used in a pattern if the
instatiated (monomorphic) type satisfies the constraints of
`CReq`. In this case, it extends the context available in the
right-hand side of the match with `CProv`, just like how an
existentially-typed data constructor can extend the context.
As with function and variable types, the pattern type signature can be inferred, or it can be explicitly written out on the program.
Here's a more complex example. Let's look at the following definition:
{{{
{-# LANGUAGE PatternSynonyms, GADTs, ViewPatterns #-}
module ShouldCompile where
data T a where
MkT :: (Eq b) => a -> b -> T a
f :: (Show a) => a -> Bool
pattern P x <- MkT (f -> True) x
}}}
Here, the inferred type of `P` is
{{{
pattern P :: (Eq b) => (Show a) => b -> T a
}}}
A bidirectional pattern synonym declaration has the form
{{{
pattern P var1 var2 ... varN = pat
}}}
where both of the following are well-typed declarations:
{{{
pattern P1 var1 var2 ... varN <- pat
P2 = \var1 var2 ... varN -> pat
}}}
In this case, the //pattern type// of `P` is simply the pattern type
of `P1`, and its //expression type// is the type of `P2`. The name `P`
is brought into the module-level scope both as a pattern synonym and
as an expression.
== Dynamic semantics ==
A pattern synonym occurance in a pattern is evaluated by first
matching against the pattern synonym itself, and then on the argument
patterns. For example, given the following definitions:
{{{
pattern P x y <- [x, y]
f (P True True) = True
f _ = False
g [True, True] = True
g _ = False
}}}
the behaviour of `f` is the same as
{{{
f [x, y] | True <- x, True <- y = True
f _ = False
}}}
Because of this, the eagerness of `f` and `g` differ:
{{{
*Main> f (False:undefined)
*** Exception: Prelude.undefined
*Main> g (False:undefined)
False
}}}
== Typed pattern synonyms ==
So far patterns only had ''syntactic'' meaning. In comparison [http://code.google.com/p/omega Ωmega] has ''typed'' pattern synonyms, so they become first class values. For bidirectional pattern synonyms this seems to be the case
{{{
data Nat = Z | S Nat deriving Show
pattern Ess p = S p
}}}
And it works:
{{{
*Main> map S [Z, Z, S Z]
[S Z,S Z,S (S Z)]
*Main> map Ess [Z, Z, S Z]
[S Z,S Z,S (S Z)]
}}}
== Branching pattern-only synonyms ==
''N.B. this is a speculative suggestion!
Sometimes you want to match against several summands of an ADT simultaneously. E.g. in a data type of potentially unbounded natural numbers:
{{{
data Nat = Zero | Succ Nat
type UNat = Maybe Nat -- Nothing meaning unbounded
}}}
Conceptually `Nothing` means ''infinite'', so it makes sense to interpret it as a ''successor'' of something. We wish it to have a predecessor just like `Just (Succ Zero)`!
I suggest ''branching pattern synonyms'' for this purpose:
{{{
pattern S pred <- pred@Nothing | pred@(Just a <- Just (Succ a))
pattern Z = Just Zero
}}}
Here `pred@(Just a <- Just (Succ a))` means that the pattern invocation `S pred` matches against `Just (Succ a)` and - if successful - binds `Just a` to `pred`.
This means we can syntactically address unbound naturals just like bounded ones:
{{{
greetTimes :: UNat -> String -> IO ()
greetTimes Z _ = return ()
greetTimes (S rest) message = putStrLn message >> greetTimes rest message
}}}
As a nice collateral win this proposal handles `pattern Name name <- Person name workplace | Dog name vet` too.