D Specification of Derived Instances

If T is an algebraic datatype declared by:

(where m>=0 and the parentheses may be omitted if m=1) then a derived instance declaration is possible for a class C if these conditions hold:

1. C is one of Eq, Ord, Enum, Bounded, Show, or Read.

2. There is a context c' such that c' => C t_ij holds for each of the constituent types t_ij.

3. If C is Bounded, the type must be either an enumeration (all constructors must by nullary) or have only one constructor.

4. If C is Enum, the type must be an enumeration.

5. There must be no explicit instance declaration elsewhere in the program which makes T u1 ... u_k an instance of C.

If the deriving form is present, an instance declaration is automatically generated for T u1 ... u_k over each class C_i. If the derived instance declaration is impossible for any of the C_i then a static error results. If no derived instances are required, the deriving form may be omitted or the form deriving () may be used.

Each derived instance declaration will have the form:

instance (c, C'_1 u'_1, ..., C'_j u'_j ) => C_i (T u_1 ... u_k) where { d }

where d is derived automatically depending on C_i and the data type declaration for T (as will be described in the remainder of this section), and u'1 through u'_j form a subset of u1 through u_k. When inferring the context for the derived instances, type synonyms must be expanded out first. Free names in the declarations d are all defined in the Prelude; the qualifier `Prelude.' is implicit here. The remaining details of the derived instances for each of the derivable Prelude classes are now given.

Derived instances of Eq and Ord. The class methods automatically introduced by derived instances of Eq and Ord are (==), (/=), compare, (<), (<=), (>), (>=), max, and min. The latter seven operators are defined so as to compare their arguments lexicographically with respect to the constructor set given, with earlier constructors in the datatype declaration counting as smaller than later ones. For example, for the Bool datatype, we have that (True > False) == True.

Derived comparisons always traverse constructors from left to right. These examples illustrate this property:

(1,undefined) == (2,undefined) =>    False
(undefined,1) == (undefined,2) =>    bottom

Derived instances of Enum Derived instance declarations for the class Enum are only possible for enumerations. The nullary constructors are assumed to be numbered left-to-right with the indices 0 through n-1. Enum introduces the class methods toEnum, fromEnum, enumFrom, enumFromThen, enumFromTo, and enumFromThenTo, which are used to define arithmetic sequences as described in Section 3.10.

The toEnum and fromEnum operators map enumerated values to and from the Int type. enumFrom n returns a list corresponding to the complete enumeration of n's type starting at the value n. Similarly, enumFromThen n n' is the enumeration starting at n, but with second element n', and with subsequent elements generated at a spacing equal to the difference between n and n'. enumFromTo and enumFromThenTo are as defined by the default class methods for Enum (see Figure 5, page ). For example, given the datatype:

data  Color = Red | Orange | Yellow | Green  deriving (Enum)

[Orange..]          ==  [Orange, Yellow, Green]
fromEnum Yellow     ==  2

Derived instances of Bounded. The Bounded class introduces the class methods minBound and maxBound, which define the minimal and maximal elements of the type. For an enumeration, the first and last constructors listed in the data declaration are the bounds. For a type with a single constructor, the constructor is applied to the bounds for the constituent types. For example, the following datatype:

data  Pair a b = Pair a b deriving Bounded

instance (Bounded a,Bounded b) => Bounded (Pair a b) where
  minBound = Pair minBound minBound
  maxBound = Pair maxBound maxBound

Derived instances of Read and Show. The class methods automatically introduced by derived instances of Read and Show are showsPrec, readsPrec, showList, and readList. They are used to coerce values into strings and parse strings into values.

The function showsPrec d x r accepts a precedence level d (a number from 0 to 10), a value x, and a string r. It returns a string representing x concatenated to r. showsPrec satisfies the law:

showsPrec d x r ++ s  ==  showsPrec d x (r ++ s)

The representation will be enclosed in parentheses if the precedence of the top-level constructor operator in x is less than d. Thus, if d is 0 then the result is never surrounded in parentheses; if d is 10 it is always surrounded in parentheses, unless it is an atomic expression. The extra parameter r is essential if tree-like structures are to be printed in linear time rather than time quadratic in the size of the tree.

The function readsPrec d s accepts a precedence level d (a number from 0 to 10) and a string s, and returns a list of pairs (x,r) such that showsPrec d x r == s. readsPrec is a parse function, returning a list of (parsed value, remaining string) pairs. If there is no successful parse, the returned list is empty.

showList and readList allow lists of objects to be represented using non-standard denotations. This is especially useful for strings (lists of Char).

readsPrec will parse any valid representation of the standard types apart from lists, for which only the bracketed form [...] is accepted. See Appendix A for full details.

A precise definition of the derived Read and Show instances for general types is beyond the scope of this report. However, the derived Read and Show instances have the following properties:

• The result of show is a syntactically correct Haskell expression containing only constants given the fixity declarations in force at the point where the type is declared.
• The result of show is readable by read if all component types are readable. (This is true for all instances defined in the Prelude but may not be true for user-defined instances.)
• The instance generated by Read allows arbitrary whitespace between tokens on the input string. Extra parenthesis are also allowed.
• The result of show contains only the constructor names defined in the data type, parenthesis, and spaces. When labeled constructor fields are used, braces, commas, field names, and equal signs are also used. No leading or trailing spaces are generated. Parenthesis are only added where needed. No line breaks are added.
• If a constructor is defined using labeled field syntax then the derived show for that constructor will this same syntax; the fields will be in the order declared in the data declaration. The derived Read instance will require this same syntax: all fields must be present and the declared order must be maintained.
• If a constructor is defined in the infix style, the derived Show instance will also use infix style. The derived Read instance will require that the constructor be infix.

The derived Read and Show instances may be unsuitable for some uses. Some problems include:

• Circular structures cannot be printed or read by these instances.
• The printer loses shared substructure; the printed representation of an object may be much larger that necessary.
• The parsing techniques used by the reader are very inefficient; reading a large structure may be quite slow.
• There is no user control over the printing of types defined in the Prelude. For example, there is no way to change the formatting of floating point numbers.

D.1 An example

data Tree a = Leaf a | Tree a :^: Tree a
     deriving (Eq, Ord, Read, Show)

infix 4 :^:
data Tree a =  Leaf a  |  Tree a :^: Tree a

instance (Eq a) => Eq (Tree a) where
	Leaf m == Leaf n  =  m==n
	u:^:v  == x:^:y	  =  u==x && v==y
             _ == _       =  False

instance (Ord a) => Ord (Tree a) where
	Leaf m <= Leaf n  =  m<=n
	Leaf m <= x:^:y	  =  True
	u:^:v  <= Leaf n  =  False
	u:^:v  <= x:^:y	  =  u<x || u==x && v<=y

instance (Show a) => Show (Tree a) where

        showsPrec d (Leaf m) = showParen (d >= 10) showStr
    	  where
             showStr = showString "Leaf " . showsPrec 10 m

        showsPrec d (u :^: v) = showParen (d > 4) showStr
    	  where
             showStr = showsPrec 5 u . 
                       showString " :^: " .
                       showsPrec 5 v

instance (Read a) => Read (Tree a) where

	readsPrec d r =  readParen (d > 4)
			 (\r -> [(u:^:v,w) |
				 (u,s) <- readsPrec 5 r,
				 (":^:",t) <- lex s,
				 (v,w) <- readsPrec 5 t]) r

		      ++ readParen (d > 9)
			 (\r -> [(Leaf m,t) |
				 ("Leaf",s) <- lex r,
				 (m,t) <- readsPrec 10 s]) r

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