There may be zero or more rules in a RULES pragma.

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 Section 7.10.3.3, “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.

Layout applies in a RULES pragma. Currently no new indentation level is set, so you must lay out your rules starting in the same column as the enclosing definitions.

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.

Rules are automatically exported from a module, just as instance declarations are.

Rules are only applied if you use the -O flag.

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.

The LHS and RHS of a rule are typechecked, and must have the same type.

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 Section 7.11.4, “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.

In the earlier phases of compilation, GHC inlines nothing that appears on the LHS of a rule, because once you have substituted for something you can't match against it (given the simple minded matching). So if you write the rule

        "map/map"       forall f,g.  map f . map g = map (f.g)

this won't match the expression map f (map g xs). It will only match something written with explicit use of ".". Well, not quite. It will match the expression

wibble f g xs

where wibble is defined:

wibble f g = map f . map g

because wibble will be inlined (it's small). Later on in compilation, GHC starts inlining even things on the LHS of rules, but still leaves the rules enabled. This inlining policy is controlled by the per-simplification-pass flag -finline-phasen.

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.

List comprehensions

Enumerations of Int 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

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

sortBy

Use -ddump-rules to see what transformation rules GHC is using.

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

The definition of (say) build in GHC/Base.lhs looks llike 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.