.. _rewrite-rules:

Rewrite rules
=============

.. index::
   single: rewrite rules

.. pragma:: RULES "⟨name⟩" forall ⟨binder⟩ ... . ⟨expr⟩ = ⟨expr⟩ ...

    :where: top-level

    Define a rewrite rule to be used to optimize a source program.

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 :ghc-flag:`-ddump-simpl-stats` to see what rules fired. If
you need more information, then :ghc-flag:`-ddump-rule-firings` shows you each
individual rule firing and :ghc-flag:`-ddump-rule-rewrites` also shows what the
code looks like before and after the rewrite.

.. ghc-flag:: -fenable-rewrite-rules
    :shortdesc: Switch on all rewrite rules (including rules generated by
        automatic specialisation of overloaded functions). Implied by
        :ghc-flag:`-O`.
    :type: dynamic
    :reverse: -fno-enable-rewrite-rules
    :category: optimization

    Allow the compiler to apply rewrite rules to the source program.

Syntax
------

From a syntactic point of view:

-  There may be zero or more rules in a :pragma:`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
   :ref:`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
   :ref:`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 (term) 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.

-  If :extension:`ExplicitForAll` is enabled, type/kind variables can also be
   explicitly bound. For example: ::

       {-# RULES "id" forall a. forall (x :: a). id @a x = x #-}

   When a type-level explicit ``forall`` is present, each type/kind variable
   mentioned must now also be either in scope or bound by the ``forall``. In
   particular, unlike some other places in Haskell, this means free kind
   variables will not be implicitly bound. For example: ::

       "this_is_bad" forall (c :: k). forall (x :: Proxy c) ...
       "this_is_ok"  forall k (c :: k). forall (x :: Proxy c) ...

   When bound type/kind variables are needed, both foralls must always be
   included, though if no pattern variables are needed, the second can be left
   empty. For example: ::

       {-# RULES "map/id" forall a. forall. map (id @a) = id @[a] #-}

-  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
       "wrong3"   forall x.      Just x = Nothing

   In ``"wrong1"``, the LHS is not an application; in ``"wrong2"``, the
   LHS has a pattern variable in the head. In ``"wrong3"``, the LHS consists
   of a *constructor*, rather than a *variable*, applied to an argument.

-  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 :pragma:`RULES` "``forall``" is treated as a keyword, regardless of any
   other flag settings. Furthermore, inside a :pragma:`RULES`, the language
   extension :extension:`ScopedTypeVariables` is automatically enabled; see
   :ref:`scoped-type-variables`.

-  Like other pragmas, :pragma:`RULES` 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 :ghc-flag:`-fenable-rewrite-rules` flag is on (see
   :ref:`rule-semantics`).

.. _rule-semantics:

Semantics
---------

From a semantic point of view:

-  Rules are enabled (that is, used during optimisation) by the
   :ghc-flag:`-fenable-rewrite-rules` flag. This flag is implied by
   :ghc-flag:`-O`, and may be switched off (as usual) by
   :ghc-flag:`-fno-enable-rewrite-rules <-fenable-rewrite-rules>`. (NB: enabling
   :ghc-flag:`-fenable-rewrite-rules` without :ghc-flag:`-O` may not do what you
   expect, though, because without :ghc-flag:`-O` GHC ignores all optimisation
   information in interface files; see :ghc-flag:`-fignore-interface-pragmas`).
   Note that :ghc-flag:`-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 :ref:`rule-spec` 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.

-  GHC will never match a forall'd variable in a template with an expression
   which contains locally bound variables. For example, it is permitted to write
   a rule which contains a case expression::

        {-# RULES
          "test/case-tup" forall (x :: (Int, Int)) (y :: Int) (z :: Int).
            test (case x of (l, r) -> y) z = case x of (l, r) -> test y z
          #-}

   But the rule will not match when ``y`` contains either of ``l`` or ``r`` because
   they are locally bound. Therefore the following application will fail to trigger
   the rule::

        prog :: (Int, Int) -> (Int, Int)
        prog x = test (case x of (p, q) -> p) 0

   because ``y`` would have to match against ``p`` (which is locally bound)
   but it will fire for::

        prog :: (Int, Int) -> (Int, Int)
        prog x = test (case x of (p, q) -> 0) 0

   because ``y`` can match against ``0``.

-  GHC implements **higher order matching** as described by
   `GHC Proposal #555 <https://github.com/ghc-proposals/ghc-proposals/blob/master/proposals/0555-template-patterns.rst>`_.
   When a pattern variable is applied to distinct locally bound variables it forms
   what we call a **higher order pattern**.
   When matching, higher order patterns are treated like pattern variables, but they are
   allowed to match expressions that contain the locally bound variables that are part of
   the higher order patterns.

   For example, we can use this to fix the broken rule from the example from the
   previous bullet point::

        {-# RULES
          "test/case-tup" forall (x :: (Int, Int)) (f :: Int -> Int -> Int) (z :: Int).
            test (case x of (l, r) -> f l r) z = case x of (m, n) -> test (f m n) z
          #-}

   This modified rule does fire for::

        prog :: (Int, Int) -> (Int, Int)
        prog x = test (case x of (p, q) -> p) 0

   Under higher order matching, ``f p q`` matches ``p`` by assigning ``f = \p q -> p``.
   The resulting code after the rewrite is::

        prog x = case x of (m, n) -> test ((\p q -> p) m n) 0

-  A rule that has a forall binder with a polymorphic type, is likely to fail to fire. E. g., ::

        {-# RULES forall (x :: forall a. Num a => a -> a).  f x = blah #-}

   Here ``x`` has a polymorphic type.  This applies to a forall'd binder with a type class constraint, such as::

        {-# RULES forall @m (x :: KnownNat m => Proxy m).  g x = blah #-}

   See :ghc-ticket:`21093` for discussion.

.. _rules-inline:

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 :pragma:`RULES` might fire.

The way to get predictable behaviour is to use a :pragma:`NOINLINE` pragma, or an
``INLINE[⟨phase⟩]`` pragma, on ``f``, to ensure that it is not inlined until
its :pragma:`RULES` have had a chance to fire. The warning flag
:ghc-flag:`-Winline-rule-shadowing` (see :ref:`options-sanity`) warns about
this situation.

All of this applies similarly to :pragma:`INLINABLE` (also accepted as
``INLINABLE``; see :ref:`inline-noinline-pragma`). As with
the other pragmas, you can use explicit :ref:`phase-control` to delay
inlining so that a :pragma:`RULES` rewrite has an opportunity to match first.

The key difference from :pragma:`INLINE` (see :ref:`inline-pragma`) is that
:pragma:`INLINABLE` does not guarantee inlining, leaving the inlining decision
to GHC's usual heuristics. Use :pragma:`INLINE` when your RULE setup relies
on inlining happening reliably, and :pragma:`INLINABLE` otherwise.

.. _conlike:

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 :pragma:`CONLIKE` pragma is a modifier to
:pragma:`INLINE`/:pragma:`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.

.. _rules-class-methods:

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
:ref:`rules-inline`.

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 = 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``.

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.

.. _rule-spec:

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 a
:pragma:`SPECIALIZE` pragma with a right-hand-side: ::

    {-# 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.

.. _controlling-rules:

Controlling what's going on in rewrite rules
--------------------------------------------

-  Use :ghc-flag:`-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 :ghc-flag:`-ddump-simpl-stats` to see what rules are being fired. If you
   add :ghc-flag:`-dppr-debug` you get a more detailed listing.

-  Use :ghc-flag:`-ddump-rule-firings` or :ghc-flag:`-ddump-rule-rewrites` to see in
   great detail what rules are being fired. If you add :ghc-flag:`-dppr-debug`
   you get a still more detailed listing.

-  The definition of (say) ``build`` in ``GHC/Base.hs`` looks like
   this: ::

               build   :: forall a. (forall b. (a -> b -> b) -> b -> b) -> [a]
               {-# INLINE build #-}
               build g = g (:) []

   Notice the :pragma:`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.hs`` 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.hs``.
