3 Expressions

In this chapter, we describe the syntax and informal semantics of Haskell expressions, including their translations into the Haskell kernel, where appropriate. Except in the case of let expressions, these translations preserve both the static and dynamic semantics. Free variables and constructors used in these translations always refer to entities defined by the Prelude. For example, “concatMap” used in the translation of list comprehensions (Section 3.11) means the concatMap defined by the Prelude, regardless of whether or not the identifier “concatMap” is in scope where the list comprehension is used, and (if it is in scope) what it is bound to.

Expressions involving infix operators are disambiguated by the operator’s fixity (see Section 4.4.2). Consecutive unparenthesized operators with the same precedence must both be either left or right associative to avoid a syntax error. Given an unparenthesized expression “$𝑥{\text{𝑞𝑜𝑝}}^{\left(𝑎,𝑖\right)}𝑦{\text{𝑞𝑜𝑝}}^{\left(𝑏,𝑗\right)}𝑧$” (where ${\text{𝑞𝑜𝑝}}^{\left(𝑎,𝑖\right)}$ means an operator with associativity $𝑎$ and precedence $𝑖$), parentheses must be added around either $𝑥{\text{𝑞𝑜𝑝}}^{\left(𝑎,𝑖\right)}𝑦$ or $𝑦{\text{𝑞𝑜𝑝}}^{\left(𝑏,𝑗\right)}𝑧$ when $𝑖=𝑗$ unless $𝑎=𝑏=\text{l}$ or $𝑎=𝑏=\text{r}$.

An example algorithm for resolving expressions involving infix operators is given in Section 10.6.

Negation is the only prefix operator in Haskell; it has the same precedence as the infix - operator defined in the Prelude (see Section 4.4.2, Figure 1).

The grammar is ambiguous regarding the extent of lambda abstractions, let expressions, and conditionals. The ambiguity is resolved by the meta-rule that each of these constructs extends as far to the right as possible.

Sample parses are shown below.

For the sake of clarity, the rest of this section will assume that expressions involving infix operators have been resolved according to the fixities of the operators.

3.1 Errors

Errors during expression evaluation, denoted by $\perp$ (“bottom”), are indistinguishable by a Haskell program from non-termination. Since Haskell is a non-strict language, all Haskell types include $\perp$. That is, a value of any type may be bound to a computation that, when demanded, results in an error. When evaluated, errors cause immediate program termination and cannot be caught by the user. The Prelude provides two functions to directly cause such errors:

error     :: String -> a
undefined :: a

A call to error terminates execution of the program and returns an appropriate error indication to the operating system. It should also display the string in some system-dependent manner. When undefined is used, the error message is created by the compiler.

Translations of Haskell expressions use error and undefined to explicitly indicate where execution time errors may occur. The actual program behavior when an error occurs is up to the implementation. The messages passed to the error function in these translations are only suggestions; implementations may choose to display more or less information when an error occurs.

3.2 Variables, Constructors, Operators, and Literals

Haskell provides special syntax to support infix notation. An operator is a function that can be applied using infix syntax (Section 3.4), or partially applied using a section (Section 3.5).

An operator is either an operator symbol, such as + or $$, or is an ordinary identifier enclosed in grave accents (backquotes), such as `op`. For example, instead of writing the prefix application op x y, one can write the infix application x `op` y. If no fixity declaration is given for op then it defaults to highest precedence and left associativity (see Section 4.4.2).

Dually, an operator symbol can be converted to an ordinary identifier by enclosing it in parentheses. For example, (+) x y is equivalent to x + y, and foldr (*) 1 xs is equivalent to foldr (\x y -> x*y) xs.

Special syntax is used to name some constructors for some of the built-in types, as found in the production for $\text{𝑔𝑐𝑜𝑛}$ and $\text{𝑙𝑖𝑡𝑒𝑟𝑎𝑙}$. These are described in Section 6.1.

An integer literal represents the application of the function fromInteger to the appropriate value of type Integer. Similarly, a floating point literal stands for an application of fromRational to a value of type Rational (that is, Ratio Integer).

3.3 Curried Applications and Lambda Abstractions

Function application is written ${𝑒}_1{𝑒}_2$. Application associates to the left, so the parentheses may be omitted in (f x) y. Because ${𝑒}_1$ could be a data constructor, partial applications of data constructors are allowed.

Lambda abstractions are written $\text{}{𝑝}_1\dots {𝑝}_{𝑛}\text{->}𝑒$, where the ${𝑝}_{𝑖}$ are patterns. An expression such as \x:xs->x is syntactically incorrect; it may legally be written as \(x:xs)->x.

The set of patterns must be linear—no variable may appear more than once in the set.

Given this translation combined with the semantics of case expressions and pattern matching described in Section 3.17.3, if the pattern fails to match, then the result is $\perp$.

3.4 Operator Applications

The form ${𝑒}_1\text{𝑞𝑜𝑝}{𝑒}_2$ is the infix application of binary operator $\text{𝑞𝑜𝑝}$ to expressions ${𝑒}_1$ and ${𝑒}_2$.

The special form $-𝑒$ denotes prefix negation, the only prefix operator in Haskell, and is syntax for $\text{𝚗𝚎𝚐𝚊𝚝𝚎}\left(𝑒\right)$. The binary - operator does not necessarily refer to the definition of - in the Prelude; it may be rebound by the module system. However, unary - will always refer to the negate function defined in the Prelude. There is no link between the local meaning of the - operator and unary negation.

Prefix negation has the same precedence as the infix operator - defined in the Prelude (see Table 1). Because e1-e2 parses as an infix application of the binary operator -, one must write e1(-e2) for the alternative parsing. Similarly, (-) is syntax for \x y -> x-y, as with any infix operator, and does not denote \x -> -x—one must use negate for that.

3.5 Sections

Sections are written as $\left(\text{𝑜𝑝}𝑒\right)$ or $\left(𝑒\text{𝑜𝑝}\right)$, where $\text{𝑜𝑝}$ is a binary operator and $𝑒$ is an expression. Sections are a convenient syntax for partial application of binary operators.

Syntactic precedence rules apply to sections as follows. $\left(\text{𝑜𝑝}𝑒\right)$ is legal if and only if $\left(𝑥\text{𝑜𝑝}𝑒\right)$ parses in the same way as $\left(𝑥\text{𝑜𝑝}\left(𝑒\right)\right)$; and similarly for $(𝑒\text{𝑜𝑝}$. For example, (*a+b) is syntactically invalid, but (+a*b) and (*(a+b)) are valid. Because (+) is left associative, (a+b+) is syntactically correct, but (+a+b) is not; the latter may legally be written as (+(a+b)). As another example, the expression

  (let n = 10 in n +)

is invalid because, by the let/lambda meta-rule (Section 3), the expression

  (let n = 10 in n + x)

parses as

  (let n = 10 in (n + x))

rather than

  ((let n = 10 in n) + x)

Because - is treated specially in the grammar, $\left(-\text{𝑒𝑥𝑝}\right)$ is not a section, but an application of prefix negation, as described in the preceding section. However, there is a subtract function defined in the Prelude such that $\left(\text{𝚜𝚞𝚋𝚝𝚛𝚊𝚌𝚝}\text{𝑒𝑥𝑝}\right)$ is equivalent to the disallowed section. The expression $\left(+\left(-\text{𝑒𝑥𝑝}\right)\right)$ can serve the same purpose.

3.6 Conditionals

A conditional expression has the form $\text{𝚒𝚏}{𝑒}_1\text{𝚝𝚑𝚎𝚗}{𝑒}_2\text{𝚎𝚕𝚜𝚎}{𝑒}_3$ and returns the value of ${𝑒}_2$ if the value of ${𝑒}_1$ is True, ${𝑒}_3$ if ${𝑒}_1$ is False, and $\perp$ otherwise.

3.7 Lists

Lists are written $\left[{𝑒}_1,\dots ,{𝑒}_{𝑘}\right]$, where $𝑘\ge 1$. The list constructor is :, and the empty list is denoted []. Standard operations on lists are given in the Prelude (see Section 6.1.3, and Chapter 9 notably Section 9.1).

The constructor “:” is reserved solely for list construction; like [], it is considered part of the language syntax, and cannot be hidden or redefined. It is a right-associative operator, with precedence level 5 (Section 4.4.2).

3.8 Tuples

Tuples are written $\left({𝑒}_1,\dots ,{𝑒}_{𝑘}\right)$, and may be of arbitrary length $𝑘\ge 2$. The constructor for an $𝑛$-tuple is denoted by $\left(,\dots ,\right)$, where there are $𝑛-1$ commas. Thus (a,b,c) and (,,) a b c denote the same value. Standard operations on tuples are given in the Prelude (see Section 6.1.4 and Chapter 9).

3.9 Unit Expressions and Parenthesized Expressions

The form $(𝑒$) is simply a parenthesized expression, and is equivalent to $𝑒$. The unit expression () has type () (see Section 4.1.2). It is the only member of that type apart from $\perp$, and can be thought of as the “nullary tuple” (see Section 6.1.5).

3.10 Arithmetic Sequences

The arithmetic sequence $\left[{𝑒}_1,{𝑒}_2..{𝑒}_3\right]$ denotes a list of values of type $𝑡$, where each of the ${𝑒}_{𝑖}$ has type $𝑡$, and $𝑡$ is an instance of class Enum.

The semantics of arithmetic sequences therefore depends entirely on the instance declaration for the type t. See Section 6.3.4 for more details of which Prelude types are in Enum and their semantics.

3.11 List Comprehensions

A list comprehension has the form $\left[𝑒|{𝑞}_1,\dots ,{𝑞}_{𝑛}\right]$, $𝑛\ge 1$, where the ${𝑞}_{𝑖}$ qualifiers are either

• generators of the form $𝑝\text{<-}𝑒$, where $𝑝$ is a pattern (see Section 3.17) of type $𝑡$ and $𝑒$ is an expression of type $\left[𝑡\right]$
• local bindings that provide new definitions for use in the generated expression $𝑒$ or subsequent boolean guards and generators
• boolean guards, which are arbitrary expressions of type Bool.

Such a list comprehension returns the list of elements produced by evaluating $𝑒$ in the successive environments created by the nested, depth-first evaluation of the generators in the qualifier list. Binding of variables occurs according to the normal pattern matching rules (see Section 3.17), and if a match fails then that element of the list is simply skipped over. Thus:

[ x |  xs   <- [ [(1,2),(3,4)], [(5,4),(3,2)] ],
      (3,x) <- xs ]

yields the list [4,2]. If a qualifier is a boolean guard, it must evaluate to True for the previous pattern match to succeed. As usual, bindings in list comprehensions can shadow those in outer scopes; for example:

As indicated by the translation of list comprehensions, variables bound by let have fully polymorphic types while those defined by <- are lambda bound and are thus monomorphic (see Section 4.5.4).

3.12 Let Expressions

Let expressions have the general form $\text{𝚕𝚎𝚝}\left\{{𝑑}_1;\dots ;{𝑑}_{𝑛}\right\}\text{𝚒𝚗}𝑒$, and introduce a nested, lexically-scoped, mutually-recursive list of declarations (let is often called letrec). The scope of the declarations is the expression $𝑒$ and the right hand side of the declarations. Declarations are described in Chapter 4. Pattern bindings are matched lazily; an implicit ~ makes these patterns irrefutable. For example,

does not cause an execution-time error until x or y is evaluated.

3.13 Case Expressions

A case expression has the general form

where each ${\text{𝑚𝑎𝑡𝑐ℎ}}_{𝑖}$ is of the general form

(Notice that in the syntax rule for $\text{𝑔𝑢𝑎𝑟𝑑𝑠}$, the “|” is a terminal symbol, not the syntactic metasymbol for alternation.) Each alternative ${𝑝}_{𝑖}{\text{𝑚𝑎𝑡𝑐ℎ}}_{𝑖}$ consists of a pattern ${𝑝}_{𝑖}$ and its matches, ${\text{𝑚𝑎𝑡𝑐ℎ}}_{𝑖}$. Each match in turn consists of a sequence of pairs of guards ${\text{𝑔𝑠}}_{\text{𝑖𝑗}}$ and bodies ${𝑒}_{\text{𝑖𝑗}}$ (expressions), followed by optional bindings (${\text{𝑑𝑒𝑐𝑙𝑠}}_{𝑖}$) that scope over all of the guards and expressions of the alternative.

A guard has one of the following forms:

• pattern guards are of the form $𝑝\text{<-}𝑒$, where $𝑝$ is a pattern (see Section 3.17) of type $𝑡$ and $𝑒$ is an expression type $𝑡$. They succeed if the expression $𝑒$ matches the pattern $𝑝$, and introduce the bindings of the pattern to the environment.
• local bindings are of the form $\text{𝚕𝚎𝚝}\text{𝑑𝑒𝑐𝑙𝑠}$. They always succeed, and they introduce the names defined in $\text{𝑑𝑒𝑐𝑙𝑠}$ to the environment.
• boolean guards are arbitrary expressions of type Bool. They succeed if the expression evaluates to True, and they do not introduce new names to the environment. A boolean guard, $𝑔$, is semantically equivalent to the pattern guard $\text{𝚃𝚛𝚞𝚎 <-}𝑔$.

An alternative of the form

is treated as shorthand for:

A case expression must have at least one alternative and each alternative must have at least one body. Each body must have the same type, and the type of the whole expression is that type.

A case expression is evaluated by pattern matching the expression $𝑒$ against the individual alternatives. The alternatives are tried sequentially, from top to bottom. If $𝑒$ matches the pattern of an alternative, then the guarded expressions for that alternative are tried sequentially from top to bottom in the environment of the case expression extended first by the bindings created during the matching of the pattern, and then by the ${\text{𝑑𝑒𝑐𝑙𝑠}}_{𝑖}$ in the where clause associated with that alternative.

For each guarded expression, the comma-separated guards are tried sequentially from left to right. If all of them succeed, then the corresponding expression is evaluated in the environment extended with the bindings introduced by the guards. That is, the bindings that are introduced by a guard (either by using a let clause or a pattern guard) are in scope in the following guards and the corresponding expression. If any of the guards fail, then this guarded expression fails and the next guarded expression is tried.

If none of the guarded expressions for a given alternative succeed, then matching continues with the next alternative. If no alternative succeeds, then the result is $\perp$. Pattern matching is described in Section 3.17, with the formal semantics of case expressions in Section 3.17.3.

A note about parsing. The expression

  case x of { (a,_) | let b = not a in b :: Bool -> a }

is tricky to parse correctly. It has a single unambiguous parse, namely

  case x of { (a,_) | (let b = not a in b :: Bool) -> a }

However, the phrase Bool -> a is syntactically valid as a type, and parsers with limited lookahead may incorrectly commit to this choice, and hence reject the program. Programmers are advised, therefore, to avoid guards that end with a type signature — indeed that is why a $\text{𝑔𝑢𝑎𝑟𝑑}$ contains an $\text{𝑖𝑛𝑓𝑖𝑥𝑒𝑥𝑝}$ not an $\text{𝑒𝑥𝑝}$.

3.14 Do Expressions

A do expression provides a more conventional syntax for monadic programming. It allows an expression such as

  putStr "x: "    >>
  getLine         >>= \l ->
  return (words l)

to be written in a more traditional way as:

  do putStr "x: "
     l <- getLine
     return (words l)

As indicated by the translation of do, variables bound by let have fully polymorphic types while those defined by <- are lambda bound and are thus monomorphic.

3.15 Datatypes with Field Labels

A datatype declaration may optionally define field labels (see Section 4.2.1). These field labels can be used to construct, select from, and update fields in a manner that is independent of the overall structure of the datatype.

Different datatypes cannot share common field labels in the same scope. A field label can be used at most once in a constructor. Within a datatype, however, a field label can be used in more than one constructor provided the field has the same typing in all constructors. To illustrate the last point, consider:

  data S = S1 { x :: Int } | S2 { x :: Int }   -- OK
  data T = T1 { y :: Int } | T2 { y :: Bool }  -- BAD

Here S is legal but T is not, because y is given inconsistent typings in the latter.

3.15.1 Field Selection

Field labels are used as selector functions. When used as a variable, a field label serves as a function that extracts the field from an object. Selectors are top level bindings and so they may be shadowed by local variables but cannot conflict with other top level bindings of the same name. This shadowing only affects selector functions; in record construction (Section 3.15.2) and update (Section 3.15.3), field labels cannot be confused with ordinary variables.

3.15.2 Construction Using Field Labels

A constructor with labeled fields may be used to construct a value in which the components are specified by name rather than by position. Unlike the braces used in declaration lists, these are not subject to layout; the { and } characters must be explicit. (This is also true of field updates and field patterns.) Construction using field labels is subject to the following constraints:

• Only field labels declared with the specified constructor may be mentioned.
• A field label may not be mentioned more than once.
• Fields not mentioned are initialized to $\perp$.
• A compile-time error occurs when any strict fields (fields whose declared types are prefixed by !) are omitted during construction. Strict fields are discused in Section 4.2.1.

The expression F {}, where F is a data constructor, is legal whether or not F was declared with record syntax (provided F has no strict fields — see the fourth bullet above); it denotes $𝐹{\perp }_1\dots {\perp }_{𝑛}$, where $𝑛$ is the arity of F.

3.15.3 Updates Using Field Labels

Values belonging to a datatype with field labels may be non-destructively updated. This creates a new value in which the specified field values replace those in the existing value. Updates are restricted in the following ways:

• All labels must be taken from the same datatype.
• At least one constructor must define all of the labels mentioned in the update.
• No label may be mentioned more than once.
• An execution error occurs when the value being updated does not contain all of the specified labels.

Here are some examples using labeled fields:

data T    = C1 {f1,f2 :: Int}
          | C2 {f1 :: Int,
                f3,f4 :: Char}

The field f1 is common to both constructors in T. This example translates expressions using constructors in field-label notation into equivalent expressions using the same constructors without field labels. A compile-time error will result if no single constructor defines the set of field labels used in an update, such as x {f2 = 1, f3 = 'x'}.

3.16 Expression Type-Signatures

Expression type-signatures have the form $𝑒\text{::}𝑡$, where $𝑒$ is an expression and $𝑡$ is a type (Section 4.1.2); they are used to type an expression explicitly and may be used to resolve ambiguous typings due to overloading (see Section 4.3.4). The value of the expression is just that of $\text{𝑒𝑥𝑝}$. As with normal type signatures (see Section 4.4.1), the declared type may be more specific than the principal type derivable from $\text{𝑒𝑥𝑝}$, but it is an error to give a type that is more general than, or not comparable to, the principal type.

3.17 Pattern Matching

Patterns appear in lambda abstractions, function definitions, pattern bindings, list comprehensions, do expressions, and case expressions. However, the first five of these ultimately translate into case expressions, so defining the semantics of pattern matching for case expressions is sufficient.

3.17.1 Patterns

Patterns have this syntax:

All patterns must be linear—no variable may appear more than once. For example, this definition is illegal:

f (x,x) = x     -- ILLEGAL; x used twice in pattern

Patterns of the form $\text{𝑣𝑎𝑟}\text{@}\text{𝑝𝑎𝑡}$ are called as-patterns, and allow one to use $\text{𝑣𝑎𝑟}$ as a name for the value being matched by $\text{𝑝𝑎𝑡}$. For example,

case e of { xs@(x:rest) -> if x==0 then rest else xs }

is equivalent to:

let { xs = e } in
  case xs of { (x:rest) -> if x==0 then rest else xs }

Patterns of the form _ are wildcards and are useful when some part of a pattern is not referenced on the right-hand-side. It is as if an identifier not used elsewhere were put in its place. For example,

case e of { [x,_,_]  ->  if x==0 then True else False }

is equivalent to:

case e of { [x,y,z]  ->  if x==0 then True else False }

3.17.2 Informal Semantics of Pattern Matching

Patterns are matched against values. Attempting to match a pattern can have one of three results: it may fail; it may succeed, returning a binding for each variable in the pattern; or it may diverge (i.e. return $\perp$). Pattern matching proceeds from left to right, and outside to inside, according to the following rules:

1. Matching the pattern $\text{𝑣𝑎𝑟}$ against a value $𝑣$ always succeeds and binds $\text{𝑣𝑎𝑟}$ to $𝑣$.

2. Matching the pattern $\sim \text{𝑎𝑝𝑎𝑡}$ against a value $𝑣$ always succeeds. The free variables in $\text{𝑎𝑝𝑎𝑡}$ are bound to the appropriate values if matching $\text{𝑎𝑝𝑎𝑡}$ against $𝑣$ would otherwise succeed, and to $\perp$ if matching $\text{𝑎𝑝𝑎𝑡}$ against $𝑣$ fails or diverges. (Binding does not imply evaluation.)

   Operationally, this means that no matching is done on a $\sim \text{𝑎𝑝𝑎𝑡}$ pattern until one of the variables in $\text{𝑎𝑝𝑎𝑡}$ is used. At that point the entire pattern is matched against the value, and if the match fails or diverges, so does the overall computation.

3. Matching the wildcard pattern _ against any value always succeeds, and no binding is done.

4. Matching the pattern $\text{𝑐𝑜𝑛}\text{𝑝𝑎𝑡}$ against a value, where $\text{𝑐𝑜𝑛}$ is a constructor defined by newtype, depends on the value:

   • If the value is of the form $\text{𝑐𝑜𝑛}𝑣$, then $\text{𝑝𝑎𝑡}$ is matched against $𝑣$.
   • If the value is $\perp$, then $\text{𝑝𝑎𝑡}$ is matched against $\perp$.

   That is, constructors associated with newtype serve only to change the type of a value.

5. Matching the pattern $\text{𝑐𝑜𝑛}{\text{𝑝𝑎𝑡}}_1\dots {\text{𝑝𝑎𝑡}}_{𝑛}$ against a value, where $\text{𝑐𝑜𝑛}$ is a constructor defined by data, depends on the value:

   • If the value is of the form $\text{𝑐𝑜𝑛}{𝑣}_1\dots {𝑣}_{𝑛}$, sub-patterns are matched left-to-right against the components of the data value; if all matches succeed, the overall match succeeds; the first to fail or diverge causes the overall match to fail or diverge, respectively.
   • If the value is of the form ${\text{𝑐𝑜𝑛}}^{\prime }{𝑣}_1\dots {𝑣}_{𝑚}$, where $\text{𝑐𝑜𝑛}$ is a different constructor to ${\text{𝑐𝑜𝑛}}^{\prime }$, the match fails.
   • If the value is $\perp$, the match diverges.
6. Matching against a constructor using labeled fields is the same as matching ordinary constructor patterns except that the fields are matched in the order they are named in the field list. All fields listed must be declared by the constructor; fields may not be named more than once. Fields not named by the pattern are ignored (matched against _).

7. Matching a numeric, character, or string literal pattern $𝑘$ against a value $𝑣$ succeeds if $𝑣\text{==}𝑘$, where == is overloaded based on the type of the pattern. The match diverges if this test diverges.

   The interpretation of numeric literals is exactly as described in Section 3.2; that is, the overloaded function fromInteger or fromRational is applied to an Integer or Rational literal (resp) to convert it to the appropriate type.

8. Matching an as-pattern $\text{𝑣𝑎𝑟}\text{@}\text{𝑎𝑝𝑎𝑡}$ against a value $𝑣$ is the result of matching $\text{𝑎𝑝𝑎𝑡}$ against $𝑣$, augmented with the binding of $\text{𝑣𝑎𝑟}$ to $𝑣$. If the match of $\text{𝑎𝑝𝑎𝑡}$ against $𝑣$ fails or diverges, then so does the overall match.

Aside from the obvious static type constraints (for example, it is a static error to match a character against a boolean), the following static class constraints hold:

• An integer literal pattern can only be matched against a value in the class Num.
• A floating literal pattern can only be matched against a value in the class Fractional.

It is sometimes helpful to distinguish two kinds of patterns. Matching an irrefutable pattern is non-strict: the pattern matches even if the value to be matched is $\perp$. Matching a refutable pattern is strict: if the value to be matched is $\perp$ the match diverges. The irrefutable patterns are as follows: a variable, a wildcard, $𝑁\text{𝑎𝑝𝑎𝑡}$ where $𝑁$ is a constructor defined by newtype and $\text{𝑎𝑝𝑎𝑡}$ is irrefutable (see Section 4.2.3), $\text{𝑣𝑎𝑟}\text{@}\text{𝑎𝑝𝑎𝑡}$ where $\text{𝑎𝑝𝑎𝑡}$ is irrefutable, or of the form $\sim \text{𝑎𝑝𝑎𝑡}$ (whether or not $\text{𝑎𝑝𝑎𝑡}$ is irrefutable). All other patterns are refutable.

Here are some examples:

1. If the pattern ['a','b'] is matched against $\left[\prime {𝑥}^{\prime },\perp \right]$, then 'a' fails to match against 'x', and the result is a failed match. But if ['a','b'] is matched against $\left[\perp {,}^{\prime }{𝑥}^{\prime }\right]$, then attempting to match 'a' against $\perp$ causes the match to diverge.

2. These examples demonstrate refutable vs. irrefutable matching:

   (\ ~(x,y) -> 0) $\perp$	$\Rightarrow$	$0$
   (\ (x,y) -> 0) $\perp$	$\Rightarrow$	$\perp$
   (\ ~[x] -> 0) []	$\Rightarrow$	$0$
   (\ ~[x] -> x) []	$\Rightarrow$	$\perp$
   (\ ~[x, ~(a,b)] -> x $\left[\left(0,1\right),\perp \right]$	$\Rightarrow$	$\left(0,1\right)$
   (\ ~[x, (a,b)] -> x $\left[\left(0,1\right),\perp \right]$	$\Rightarrow$	$\perp$
   (\ (x:xs) -> x:x:xs) $\perp$	$\Rightarrow$	$\perp$
   (\ ~(x:xs) -> x:x:xs) $\perp$	$\Rightarrow$	$\perp :\perp :\perp$

3. Consider the following declarations:

   newtype N = N Bool
   data    D = D !Bool

   These examples illustrate the difference in pattern matching between types defined by data and newtype:

   (\ (N True) -> True) $\perp$	$\Rightarrow$	$\perp$
   (\ (D True) -> True) $\perp$	$\Rightarrow$	$\perp$
   (\ ~(D True) -> True) $\perp$	$\Rightarrow$	True

   Additional examples may be found in Section 4.2.3.

Top level patterns in case expressions and the set of top level patterns in function or pattern bindings may have zero or more associated guards. See Section 3.13 for the syntax and semantics of guards.

The guard semantics have an influence on the strictness characteristics of a function or case expression. In particular, an otherwise irrefutable pattern may be evaluated because of a guard. For example, in

f :: (Int,Int,Int) -> [Int] -> Int
f ~(x,y,z) [a] | (a == y) = 1

both a and y will be evaluated by == in the guard.

3.17.3 Formal Semantics of Pattern Matching

The semantics of all pattern matching constructs other than case expressions are defined by giving identities that relate those constructs to case expressions. The semantics of case expressions themselves are in turn given as a series of identities, in Figure 1 – Figure 2. Any implementation should behave so that these identities hold; it is not expected that it will use them directly, since that would generate rather inefficient code.

In Figure 1 – Figure 2: $𝑒$, ${𝑒}^{\prime }$ and ${𝑒}_{𝑖}$ are expressions; ${𝑔}_{𝑖}$ and ${\text{𝑔𝑠}}_{𝑖}$ are guards and sequences of guards respectively; $𝑝$ and ${𝑝}_{𝑖}$ are patterns; $𝑣$, $𝑥$, and ${𝑥}_{𝑖}$ are variables; $𝐾$ and ${𝐾}^{\prime }$ are algebraic datatype (data) constructors (including tuple constructors); and $𝑁$ is a newtype constructor.

Rule (b) matches a general source-language case expression, regardless of whether it actually includes guards—if no guards are written, then True is substituted for the guards ${\text{𝑔𝑠}}_{𝑖,𝑗}$ in the ${\text{𝑚𝑎𝑡𝑐ℎ}}_{𝑖}$ forms. Subsequent identities manipulate the resulting case expression into simpler and simpler forms.

Rule (h) in Listing 3 involves the overloaded operator ==; it is this rule that defines the meaning of pattern matching against overloaded constants.

These identities all preserve the static semantics. Rules (d), (e), (j), and (q) use a lambda rather than a let; this indicates that variables bound by case are monomorphically typed (Section 4.1.4).