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!

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 OK

g :: Eq [a] => ... g :: Ord (T a ()) => ...

The flag `-XFlexibleContexts`

also lifts the corresponding
restriction on class declarations (Section 7.6.1.2, “The superclasses of a class declaration”) and instance declarations
(Section 7.6.3.3, “Relaxed rules for instance contexts”).

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 ::f = ...blah... g ::`type`

g = f`type`

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

is
ambiguious if and only if `ty`

`((undefined :: `

would fail to typecheck. We use a
very similar test for * ty*)
::

`ty`

*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

*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.
*End of historical note.*

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 parameterized 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 parameterized sort function in
terms of an explicitly parameterized `sortBy`

function:

sortBy :: (a -> a -> Bool) -> [a] -> [a] sort :: (?cmp :: a -> a -> Bool) => [a] -> [a] sort = sortBy ?cmp

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`

.

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`let`

s 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`

isf :: (?x::Int) => Int -> Int

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.

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`

.

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.

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`

, Section 7.13.1, “Explicit universal quantification (forall)”)
enables higher-rank types.
That is, you can nest `forall`

s
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 arrowAs 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 Section 7.13.8, “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.

The obselete 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 specifciations.)

In a `data`

or `newtype`

declaration one can quantify
the types of the constructor arguments. Here are several examples:

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.

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 (Section 7.13.8, “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.

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 innermost 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 for-alls explicitly. Indeed, doing so is strongly advised for rank-2 types.

GHC supports *impredicative polymorphism*,
enabled with `-XImpredicativeTypes`

.
This means
that you can 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])
```

.

The technical details of this extension are described in the paper Boxy types: type inference for higher-rank types and impredicativity, which appeared at ICFP 2006.

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`

(Section 7.13.8.2, “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`

.

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 (Section 7.13.8.2, “Declaration type signatures”)

An expression type signature (Section 7.13.8.3, “Expression type signatures”)

A pattern type signature (Section 7.13.8.4, “Pattern type signatures”)

Class and instance declarations (Section 7.13.8.5, “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)

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.

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)`

.

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.

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 Section 2.8.3, “Setting options for interactive evaluation only”).

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

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 ifthe 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 ifeach 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!)