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Page: 0 Introduction This is a programming supplement to `A Gentle Introduction to Haskell' by Hudak and Fasel. This supplement augments the tutorial by providing executable Haskell programs which you can run and experiment with. All program fragments in the tutorial are found here, as well as other examples not included in the tutorial. Using This Tutorial You should have a copy of both the `Gentle Introduction' and the report itself to make full use of this tutorial. Although the `Gentle Introduction' is meant to stand by itself, it is often easier to learn a language through actual use and experimentation than by reading alone. Once you finish this introduction, we recommend that you proceed section by section through the `Gentle Introduction' and after having read each section go back to this online tutorial. You should wait until you have finished the tutorial before attempting to read the report. This tutorial does not assume any familiarity with Haskell or other functional languages. However, knowledge of almost-functional languages such as ML or Scheme is very useful. Throughout the online component of this tutorial, we try to relate Haskell to other programming languages and clarify the written tutorial through additional examples and text. Organization of the Online Tutorial This online tutorial is divided into a series of pages. Each page covers one or more sections in the written tutorial. You can use special editor commands to move back and forth through the pages of the online tutorial. Each page is a single Haskell program. Comments in the program contain the text of the online tutorial. You can modify the program freely (this will not change the underlying tutorial file!) and ask the system to print the value of expressions defined in the program. At the beginning of each page, the sections covered by the page are listed. In addition, the start of each individual section is marked within each page. To create useful, executable examples of Haskell code, some language constructs need to be revealed well before they are explained in the tutorial. We attempt to point these out when they occur. Some small changes have been made to the examples in the written tutorial; these are usually cosmetic and should be ignored. Don't feel you have to understand everything on a page before you move on -- many times concepts become clearer as you move on and can relate them to other aspect of the language. Each page of the tutorial defines a set of variables. Some of these are named e1, e2, and so on. These `e' variables are the ones which are meant for you to evaluate as you go through the tutorial. Of course you may evaluate any other expressions or variables you wish. The Haskell Report While the report is not itself a tutorial on the Haskell language, it can be an invaluable reference to even a novice user. A very important feature of Haskell is the prelude. The prelude is a rather large chunk of Haskell code which is implicitly a part of every Haskell program. Whenever you see functions used which are not defined in the current page, these come from the Prelude. Appendix A of the report lists the entire Prelude; the index has an entry for every function in the Prelude. Looking at the definitions in the Prelude is sometimes necessary to fully understand the programs in this tutorial. Another reason to look at the report is to understand the syntax of Haskell. Appendix B contains the complete syntax for Haskell. The tutorial treats the syntax very informally; the precise details are found only in the report. The Yale Haskell System This version of the tutorial runs under version Y2.1 of Yale Haskell. The Yale Haskell system is an interactive programming environment for the Haskell language. The Macintosh version comes with a built-in Emacs-like editor. Yale Haskell is available free of change via ftp. Using the Compiler You can interact with Yale Haskell directly from editor windows. While many commands are available to the Yale Haskell user, a single command is the primary means of communicating with the compiler: C-c e. (You can also invoke this command as "Eval Expression" from the "Haskell" menu.) This command evaluates and prints an expression in the context of the program on the screen. Here is what this command does: a) A dialog box pops up to ask you for the expression to evaluate. b) The "Listener" window pops up onto your screen. c) If the program in the current page of the tutorial has not yet been compiled or the page has been modified after its most recent compilation, the entire page is compiled. This may result in a short delay. d) If there are no errors in the program, the expression entered in step a) is compiled in the context of the program. Any value defined in the current page can be referenced. e) If there are no errors in the expression, its value is printed in the Listener window. There are also a few other commands you can use. C-c i interrupts the Haskell program. Some tight loops cannot be interrupted; in this case you will have to kill the Haskell process. C-c q exits the Haskell system. Editor Commands Used by the Tutorial These commands are specific to the tutorial. The tutorial is entered by selecting "Tutorial" from the "Haskell" menu. To move among the pages of the tutorial, use C-c C-f -- go forward 1 page C-c C-b -- go back 1 page Each page of the tutorial can be modified without changing the underlying text of the tutorial. Changes are not saved as you go between pages. To revert a page to its original form use C-c C-l. You can get help regarding the editor commands with C-?. You are now ready to start the tutorial. Start by reading the `Gentle Introduction' section 1 then proceed through the online tutorial using C-c C-f to advance to the next page. You should read about each topic first before turning to the associated programming example in the online tutorial. Page: 1 Section 2 Section: 2 Values, Types, and Other Goodies This tutorial is written in `literate Haskell'. This style requires that all lines containing Haskell code start with `>'; all other lines are comments and are discarded by the compiler. Appendix C of the report defines the syntax of a literate program. This is the first line of the Haskell program on this page: > module Test(Bool) where Comments at the end of source code lines start with `--'. We use these throughout the tutorial to place explanatory text in the program. Remember to use C-c e to evaluate expressions, C-c ? for help. All Haskell programs start with a module declaration, as in the previous `module Test(Bool) where'. This can be ignored for now. We will start by defining some identifiers (variables) using equations. You can print out the value of an identifier by typing C-c e and typing the name of the identifier you wish to evaluate. This will compile the entire program, not just the line with the definition you want to see. Not all definitions are very interesting to print out - by convention, we will use variables e1, e2, ... to denote values that are interesting to print. We will start with some constants as well as their associated type. There are two ways to associate a type with a value: a type declaration and an expression type signature. Here is an equation and a type declaration: > e1 :: Int -- This is a type declaration for the identifier e1 > e1 = 5 -- This is an equation defining e1 You can evaluate the expression e1 and watch the system print `5'. Remember that C-c e is prompting for an expression. Expressions like e1 or 5 or 1+1 are all valid. However, `e1 = 5' is a definition, not an expression. Trying to evaluate it will result in a syntax error. The type declaration for e1 is not really necessary but we will try to always provide type declarations for values to help document the program and to ensure that the system infers the same type we do for an expression. If you change the value for e1 to `True', the program will no longer compile due to the type mismatch. We will briefly mention expression type signatures: these are attached to expressions instead of identifiers. Here are equivalent ways to do the previous definition: > e2 = 5 :: Int > e3 = (2 :: Int) + (3 :: Int) The :: has very low precedence in expressions and should usually be placed in parenthesis. There are two completely separate languages in Haskell: an expression language for values and a type language for type signatures. The type language is used only in the type declarations previously described and declarations of new types, described later. Haskell uses a uniform syntax so that values resemble their type signature as much as possible. However, you must always be aware of the difference between type expressions and value expressions. Here are some of the predefined types Haskell provides: type Value Syntax Type Syntax Small integers <digits> Int > e4 :: Int > e4 = 12345 Characters '<character>' Char > e5 :: Char > e5 = 'a' Strings "chars" String > e6 :: String > e6 = "abc" Boolean True, False Bool > e7 :: Bool > e7 = True Floating point <digits.digits> Float > e8 :: Float > e8 = 123.456 Homogeneous list [<exp1>,<exp2>,...] [<constituant type>] > e9 :: [Int] > e9 = [1,2,3] Tuple (<exp1>,<exp2>,...) (<exp1-type>,<exp2-type>,...) > e10 :: (Char,Int) > e10 = ('b',4) Functional described later domain type -> range type > succ :: Int -> Int -- a function which takes an Int argument and returns Int > succ x = x + 1 -- test this by evaluating `succ 4' Here's a few leftover examples from section 2: > e11 = succ (succ 3) -- you could also evaluate `succ (succ 3)' directly > -- by entering the entire expression to the C-c e If you want to evaluate something more complex than the `e' variables defined here, it is better to enter a complex expression, such as succ (succ 3), directly than to edit a new definition like e10 into the program. This is because any change to the program will require recompilation of the entire page. The expressions entered to C-c e are compiled separately (and very quickly!). Uncomment this next line to see a compile time type error. > -- e12 = 'a'+'b' Don't worry about the error message - it will make more sense later. Proceed to the next page using C-c C-f Page: 2 Section 2.1 Section: 2.1 Polymorphic Types > module Test(Bool) where The following line allows us to redefine functions in the standard prelude. Ignore this for now. > import Prelude hiding (length,head,tail,null) Start with some sample lists to use in test cases: > list1 :: [Int] > list1 = [1,2,3] > list2 :: [Char] -- This is the really a String > list2 = ['a','b','c'] -- This is the same as "abc"; evaluate list2 and see. > list3 :: [[a]] -- The element type of the inner list is unknown > list3 = [[],[],[],[]] -- so this list can't be printed > list4 :: [Int] > list4 = 1:2:3:4:[] -- Exactly the same as [1,2,3,4]; print it and see. This is the length function. You can test it by evaluating expressions such as `length list1'. Function application is written by simple juxtaposition: `f(x)' in other languages would be `f x' in Haskell. > length :: [a] -> Int > length [] = 0 > length (x:xs) = 1 + length xs Function application has the highest precedence, so 1 + length xs is parsed as 1 + (length xs). In general, you have to surround non-atomic arguments to a function with parens. This includes arguments which are also function applications. For example, f g x is the function f applied to arguments g and x, similar to f(g,x) in other languages. However, f (g x) is f applied to (g x), or f(g(x)), which means something quite different! Be especially careful with infix operators: f x+1 y-2 would be parsed as (f x)+(1 y)-2. This is also true on the left of the `=': the parens around (x:xs) are absolutely necessary. length x:xs would be parsed as (length x):xs. Also be careful with prefix negation, -. The application `f -1' is f-1, not f(-1). Add parens around negative numbers to avoid this problem. Here are some other list functions: > head :: [a] -> a -- returns the first element in a list (same as car in lisp) > head (x:xs) = x > tail :: [a] -> [a] -- removes the first element from a list (same as cdr) > tail (x:xs) = xs > null :: [a] -> Bool > null [] = True > null (x:xs) = False > cons :: a -> [a] -> [a] > cons x xs = x:xs > nil :: [a] > nil = [] Length could be defined using these functions. This is not good Haskell style but does illustrate these other list functions. Haskell programmers feel that the pattern matching style, as used in the previous version of length, is more natural and readable. > length' :: [a] -> Int -- Note that ' can be part of a name > length' x = if null x then 0 else 1 + length' (tail x) A test case for length', cons, and nil > e1 = length' (cons 1 (cons 2 nil)) We haven't said anything about errors yet. Each of the following examples illustrates a potential runtime or compile time error. The compile time error is commented out so that other examples will compile; you can uncomment them and see what happens. > -- e2 = cons True False -- Why is this not possible in Haskell? > e3 = tail (tail ['a']) -- What happens if you evaluate this? > e4 = [] -- This is especially mysterious! This last example, e4, is something hard to explain but is often encountered early by novices. We haven't explained yet how the system prints out the expressions you type in - this will wait until later. However, the problem here is that e4 has the type [a]. The printer for the list datatype is complaining that it needs to know a specific type for the list elements even though the list has no elements! This can be avoided by giving e4 a type such as [Int]. (To further confuse you, try giving e4 the type [Char] and see what happens.) Page: 3 Section 2.2 Section: 2.2 User-Defined Types > module Test(Bool) where The type Bool is already defined in the Prelude so there is no need to define it here. > data Color = Red | Green | Blue | Indigo | Violet deriving Text The `deriving Text' is necessary if you want to print a Color value. You can now evaluate these expressions. > e1 :: Color > e1 = Red > e2 :: [Color] > e2 = [Red,Blue] It is very important to keep the expression language and the type language in Haskell separated. The data declaration above defines the type constructor Color. This is a nullary constructor: it takes no arguments. Color is found ONLY in the type language - it can not be part of an expression. e1 = Color is meaningless. (Actually, Color could be both a data constructor and a type constructor but we'll ignore this possibility for now). On the other hand, Red, Blue, and so on are (nullary) data constructors. They can appear in expressions and in patterns (described later). The declaration e1 :: Blue would also be meaningless. Data constructors can be defined ONLY in a data declaration. In the next example, Point is a type constructor and Pt is a data constructor. Point takes one argument and Pt takes two. A data constructor like Pt is really just an ordinary function except that it can be used in a pattern. Type signatures can not be supplied directly for data constructors; their typing is completely defined by the data declaration. However, data constructors have a signature just like any variable: Pt :: a -> a -> Point a -- Not valid Haskell syntax That is, Pt is a function which takes two arguments with the same arbitrary type and returns a value containing the two argument values. > data Point a = Pt a a deriving Text > e3 :: Point Float > e3 = Pt 2.0 3.0 > e4 :: Point Char > e4 = Pt 'a' 'b' > e5 :: Point (Point Int) > e5 = Pt (Pt 1 2) (Pt 3 4) > -- e6 = Pt 'a' True -- This is a typing error The individual components of a point do not have names. Let's jump ahead a little so that we can write functions using these data types. Data constructors (Red, Blue, ..., and Pt) can be used in patterns. When more than one equation is used to define a function, pattern matching occurs top down. A function to remove red from a list of colors. > removeRed :: [Color] -> [Color] > removeRed [] = [] > removeRed (Red:cs) = removeRed cs > removeRed (c:cs) = c : removeRed cs -- c cannot be Red at this point > e7 :: [Color] > e7 = removeRed [Blue,Red,Green,Red] Pattern matching is capable of testing equality with a specific color. All equations defining a function must share a common type. A definition such as: foo Red = 1 foo (Pt x y) = x would result in a type error since the argument to foo cannot be both a Color and a Point. Similarly, the right hand sides must also share a common type; a definition such as foo Red = Blue foo Blue = Pt Red Red would also result in a type error. Here are a couple of functions defined on points. > dist :: Point Float -> Point Float -> Float > dist (Pt x1 y1) (Pt x2 y2) = sqrt ((x1-x2)^2 + (y1-y2)^2) > midpoint :: Point Float -> Point Float -> Point Float > midpoint (Pt x1 y1) (Pt x2 y2) = Pt ((x1+x2)/2) ((y1+y2)/2) > p1 :: Point Float > p1 = Pt 1.0 1.0 > p2 :: Point Float > p2 = Pt 2.0 2.0 > e8 :: Float > e8 = dist p1 p2 > e9 :: Point Float > e9 = midpoint p1 p2 The only way to take apart a point is to pattern match. That is, the two values which constitute a point must be extracted by matching a pattern containing the Pt data constructor. Much more will be said about pattern matching later. Haskell prints values in the same syntax used in expressions. Thus Pt 1 2 would print as Pt 1 2 (of course, Pt 1 (1+1) would also print as Pt 1 2). Page: 4 Section 2.3 Section: 2.3 Recursive Types > module Test where > data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving Text The following typings are implied by this declaration. As before, this is not valid Haskell syntax. Leaf :: a -> Tree a Branch :: Tree a -> Tree a -> Tree a > fringe :: Tree a -> [a] > fringe (Leaf x) = [x] > fringe (Branch left right) = fringe left ++ fringe right The following trees can be used to test functions: > tree1 :: Tree Int > tree1 = Branch (Leaf 1) (Branch (Branch (Leaf 2) (Leaf 3)) (Leaf 4)) > tree2 :: Tree Int > tree2 = Branch (Branch (Leaf 3) (Leaf 1)) (Branch (Leaf 4) (Leaf 1)) > tree3 :: Tree Int > tree3 = Branch tree1 tree2 Try evaluating `fringe tree1' and others. Here's another tree function: > twist :: Tree a -> Tree a > twist (Branch left right) = Branch right left > twist x = x -- This equation only applies to leaves Here's a function which compares two trees to see if they have the same shape. Note the signature: the two trees need not contain the same type of values. > sameShape :: Tree a -> Tree b -> Bool > sameShape (Leaf x) (Leaf y) = True > sameShape (Branch l1 r1) (Branch l2 r2) = sameShape l1 l2 && sameShape r1 r2 > sameShape x y = False -- One is a branch, the other is a leaf The && function is a boolean AND function. The entire pattern on the left hand side must match in order for the right hand side to be evaluated. The first clause requires both arguments to be a leaf' otherwise the next equation is tested. The last clause will always match: the final x and y match both leaves and branches. This compares a tree of integers to a tree of booleans. > e1 = sameShape tree1 (Branch (Leaf True) (Leaf False)) Page: 5 Sections 2.4, 2.5, 2.6 Section: 2.4 Type Synonyms > module Test(Bool) where Since type synonyms are part of the type language only, it's hard to write a program which shows what they do. Essentially, they are like macros for the type language. They can be used interchangeably with their definition: > e1 :: String > e1 = "abc" > e2 :: [Char] -- No different than String > e2 = e1 In the written tutorial the declaration of `Addr' is a data type declaration, not a synonym declaration. This shows that the data type declaration as well as a signature can reference a synonym. Section: 2.5 Built-in Types Tuples are an easy way of grouping a set of data values. Here are a few tuples. Note the consistancy in notation between the values and types. > e3 :: (Bool,Int) > e3 = (True,4) > e4 :: (Char,[Int],Char) > e4 = ('a',[1,2,3],'b') Here's a function which returns the second component of a 3 tuple. > second :: (a,b,c) -> b > second (a,b,c) = b Try out `second e3' and `second e4' - what happens? Each different size of tuple is a completely distinct type. There is no general way to append two arbitrary tuples or randomly select the i'th component of an arbitrary tuple. Here's a function built using 2-tuples to represent intervals. Use a type synonym to represent homogeneous 2 tuples > type Interval a = (a,a) > containsInterval :: Interval Int -> Interval Int -> Bool > containsInterval (xmin,xmax) (ymin,ymax) = xmin <= ymin && xmax >= ymax > p1 :: Interval Int > p1 = (2,3) > p2 :: Interval Int > p2 = (1,4) > e5 = containsInterval p1 p2 > e6 = containsInterval p2 p1 Here's a type declaration for a type isomorphic to lists: > data List a = Nil | Cons a (List a) deriving Text Except for the notation, this is completely equivalent to ordinary lists in Haskell. > length' :: List a -> Int > length' Nil = 0 > length' (Cons x y) = 1 + length' y > e7 = length' (Cons 'a' (Cons 'b' (Cons 'c' Nil))) It is hard to demonstrate much about the `non-specialness' of built-in types. However, here is a brief summary: Numbers and characters, such as 1, 2.2, or 'a', are the same as nullary type constructors. Lists have a special type constructor, [a] instead of List a, and an odd looking data constructor, []. The other data constructor, :, is not `unusual', syntactically speaking. The notation [x,y] is just syntax for x:y:[] and "abc" for 'a' : 'b' : 'c' : []. Tuples use a special syntax. In a type expression, a 2 tuple containing types a and be would be written (a,b) instead of using a prefix type constructor such as Tuple2 a b. This same notation is used to build tuple values: (1,2) would construct a 2 tuple containing the values 1 and 2. Page: 6 Sections 2.5.1, 2.5.2 > module Test(Bool) where Section: 2.5.1 List Comprehensions and Arithmetic Sequences Warning: brackets in Haskell are used in three different sorts of expressions: lists, as in [a,b,c], sequences (distinguished by the ..), as in [1..2], and list comprehensions (distinguished by the bar: |), as in [x+1 | x <- xs, x > 1]. Before list comprehensions, consider sequences: > e1 :: [Int] > e1 = [1..10] -- Step is 1 > e2 :: [Int] > e2 = [1,3..10] -- Step is 3 - 1 > e3 :: [Int] > e3 = [1,-1..-10] > e4 :: [Char] > e4 = ['a'..'z'] -- This works on chars too We'll avoid infinite sequences like [1..] for now. If you print one, use C-c i to interrupt the Haskell program. List comprehensions are very similar to nested loops. They return a list of values generated by the expression inside the loop. The filter expressions are similar to conditionals in the loop. This function does nothing at all! It just scans through a list and copies it into a new one. > doNothing :: [a] -> [a] > doNothing l = [x | x <- l] Adding a filter to the previous function allows only selected elements to be generated. This is similar to what is done in quicksort. > positives :: [Int] -> [Int] > positives l = [x | x <- l, x > 0] > e5 = positives [2,-4,5,6,-5,3] Now the full quicksort function. > quicksort :: [Char] -> [Char] -- Use Char just to be different! > quicksort [] = [] > quicksort (x:xs) = quicksort [y | y <- xs, y <= x] ++ > [x] ++ > quicksort [y | y <- xs, y > x] > e6 = quicksort "Why use Haskell?" Now for some nested loops. Each generator, <-, adds another level of nesting to the loop. The variable introduced by each generator can be used in each following generator; all variables can be used in the generated expression: > e7 :: [(Int,Int)] > e7 = [(x,y) | x <- [1..5], y <- [x..5]] Now add some guards: (the /= function is `not equal') > e8 :: [(Int,Int)] > e8 = [(x,y) | x <- [1..7], x /= 5, y <- [x..8] , x*y /= 12] This is the same as the loop: (going to a psuedo Algol notation) for x := 1 to 7 do if x <> 5 then for y := x to 8 do if x*y <> 12 generate (x,y) Section: 2.5.2 Strings > e9 = "hello" ++ " world" Page: 7 Sections 3, 3.1 > module Test(Bool) where > import Prelude hiding (map) Section: 3 Functions > add :: Int -> Int -> Int > add x y = x+y > e1 :: Int > e1 = add 1 2 This Int -> Int is the latter part of the signature of add: add :: Int -> (Int -> Int) > succ :: Int -> Int > succ = add 1 > e2 :: Int > e2 = succ 3 > map :: (a->b) -> [a] -> [b] > map f [] = [] > map f (x:xs) = f x : (map f xs) > e3 :: [Int] > e3 = map (add 1) [1,2,3] This next definition is the equivalent to e3 > e4 :: [Int] > e4 = map succ [1,2,3] Heres a more complex example. Define flist to be a list of functions: > flist :: [Int -> Int] > flist = map add [1,2,3] This returns a list of functions which add 1, 2, or 3 to their input. Haskell should print flist as something like [<<function>>,<<function>>,<<function>>] Now, define a function which takes a function and returns its value when applied to the constant 1: > applyTo1 :: (Int -> a) -> a > applyTo1 f = f 1 > e5 :: [Int] > e5 = map applyTo1 flist -- Apply each function in flist to 1 If you want to look at how the type inference works, figure out how the signatures of map, applyTo1, and flist combine to yield [Int]. Section: 3.1 Lambda Abstractions The symbol \ is like `lambda' in lisp or scheme. Anonymous functions are written as \ arg1 arg2 ... argn -> body Instead of naming every function, you can code it inline with this notation: > e6 = map (\f -> f 1) flist Be careful with the syntax here. \x->\y->x+y parses as \ x ->\ y -> x + y. The ->\ is all one token. Use spaces!! This is identical to e5 except that the applyTo1 function has no name. Function arguments on the left of an = are the same as lambda on the right: > add' = \x y -> x+y -- identical to add > succ' = \x -> x+1 -- identical to succ As with ordinary function, the parameters to anonymous functions can be patterns: > e7 :: [Int] > e7 = map (\(x,y) -> x+y) [(1,2),(3,4),(5,6)] Functions defined by more than one equation, like map, cannot be converted to anonymous lambda functions quite as easily - a case statement is also required. This is discussed later. Page: 8 Sections 3.2, 3.2.1, 3.2.2 > module Test(Bool) where > import Prelude hiding ((++),(.)) Section: 3.2 Infix operators Haskell has both identifiers, like `x', and operators, like `+'. These are just two different types of syntax for variables. However, operators are by default used in infix notation. Briefly, identifiers begin with a letter and may have numbers, _, and ' in them: x, xyz123, x'', xYz'_12a. The case of the first letter distinguishes variables from data constructors (or type variables from type constructors). An operator is a string of symbols, where :!#$%&*+./<=>?@\^| are all symbols. If the first character is : then the operator is a data constructor; otherwise it is an ordinary variable operator. The - and ~ characters may start a symbol but cannot be used after the first character. This allows a*-b to parse as a * - b instead of a *- b. Operators can be converted to identifiers by enclosing them in parens. This is required in signature declarations. Operators can be defined as well as used in the infix style: > (++) :: [a] -> [a] -> [a] > [] ++ y = y > (x:xs) ++ y = x : (xs ++ y) Table 2 (Page 54) of the report is invaluable for sorting out the precedences of the many predefined infix operators. > e1 = "Foo" ++ "Bar" This is the same function without operator syntax > appendList :: [a] -> [a] -> [a] > appendList [] y = y > appendList (x:xs) y = x : appendList xs y > (.) :: (b -> c) -> (a -> b) -> (a -> c) > f . g = \x -> f (g x) > add1 :: Int -> Int > add1 x = x+1 > e2 = (add1 . add1) 3 Section: 3.2.1 Sections Sections are a way of creating unary functions from infix binary functions. When a parenthesized expression starts or ends in an operator, it is a section. Another definition of add1: > add1' :: Int -> Int > add1' = (+ 1) > e3 = add1' 4 Here are a few section examples: > e4 = map (++ "abc") ["x","y","z"] > e5 = map ("abc" ++) ["x","y","z"] Section: 3.2.2 Fixity Declarations We'll avoid any demonstration of fixity declarations. The Prelude contains numerous examples. Page: 9 Sections 3.3, 3.4, 3.5 > module Test(Bool) where > import Prelude hiding (take,zip) Section: 3.3 Functions are Non-strict Observing lazy evaluation can present difficulties. The essential question is `does an expression get evaluated?'. While in theory using a non-terminating computation is the way evaluation issues are examined, we need a more practical approach. In expressions, Haskell uses `_' to denote bottom. Evaluation of `_' will halt execution and print an informative error message giving the location of the _ which was evaluated. While it would seem that `_' is of little use to the ordinary programmer, this construct is actually quite handy. Bottom can be used to create stub functions for program components which have not been written yet or as a value to insert into data structures where a data value is required but should never be used. > e1 :: Bool -- This can be any type at all! > e1 = _ -- evaluate this and see what happens. > const1 :: a -> Int > const1 x = 1 > e2 :: Int > e2 = const1 _ -- The bottom is not needed and will thus not be evaluated. Section: 3.4 "Infinite" Data Structures Data structures are constructed lazily. A constructor like : will not evaluate its arguments until they are demanded. All demands arise from the need to print the result of the computation -- components not needed to compute the printed result will not be evaluated. > list1 :: [Int] > list1 = (1:_) > e3 = head list1 -- does not evaluate _ > e4 = tail list1 -- does evaluate _ Some infinite data structures. Don't print these! If you do, you will need to interrupt the system (C-c i) or kill the Haskell process. > ones :: [Int] > ones = 1 : ones > numsFrom :: Int -> [Int] > numsFrom n = n : numsFrom (n+1) An alternate numsFrom using series notation: > numsFrom' :: Int -> [Int] > numsFrom' n = [n..] > squares :: [Int] > squares = map (^2) (numsFrom 0) Before we start printing anything, we need a function to truncate these infinite lists down to a more manageable size. The `take' function extracts the first k elements of a list: > take :: Int -> [a] -> [a] > take 0 x = [] -- two base cases: k = 0 > take k [] = [] -- or the list is empty > take k (x:xs) = x : take (k-1) xs now some printable lists: > e5 :: [Int] > e5 = take 5 ones > e6 :: [Int] > e6 = take 5 (numsFrom 10) > e7 :: [Int] > e7 = take 5 (numsFrom' 0) > e8 :: [Int] > e8 = take 5 squares zip is a function which turns two lists into a list of 2 tuples. If the lists are of differing sizes, the result is as long as the shortest list. > zip (x:xs) (y:ys) = (x,y) : zip xs ys > zip xs ys = [] -- one of the lists is [] > e9 :: [(Int,Int)] > e9 = zip [1,2,3] [4,5,6] > e10 :: [(Int,Int)] > e10 = zip [1,2,3] ones > fib :: [Int] > fib = 1 : 1 : [x+y | (x,y) <- zip fib (tail fib)] > e11 = take 5 fib This can be done without the list comprehension: > fib' :: [Int] > fib' = 1 : 1 : map (\(x,y) -> x+y) (zip fib (tail fib)) This could be written even more cleanly using a map function which maps a binary function over two lists at once. This is in the Prelude and is called zipWith (the name map2 would possibly be clearer!). > fib'' :: [Int] > fib'' = 1 : 1 : zipWith (+) fib (tail fib) For more examples using infinite structures look in the demo files that come with Yale Haskell. Both the pascal program and the primes program use infinite lists. Section: 3.5 The Error Function Too late - we already used it! One thing to note is that `_' is not part of Haskell 1.2 but will be a part of Haskell 1.3 when it is ready. Meanwhile, we have added _ to the Yale system in anticipation of the new report. Page: 10 Sections 4, 4.1, 4.2 > module Test(Bool) where > import Prelude hiding (take,(^)) Section: 4 Case Expressions and Pattern Matching Now for details of pattern matching. We use [Int] instead of [a] since the only value of type [a] is []. > contrived :: ([Int], Char, (Int, Float), String, Bool) -> Bool > contrived ([], 'b', (1, 2.0), "hi", True) = False > contrived x = True -- add a second equation to avoid runtime errors > e1 :: Bool > e1 = contrived ([], 'b', (1, 2.0), "hi", True) > e2 :: Bool > e2 = contrived ([1], 'b', (1, 2.0), "hi", True) Contrived just tests its input against a big constant. Linearity in pattern matching implies that patterns can only compare values with constants. The following is not valid Haskell: member x [] = False member x (x:ys) = True -- Invalid since x appears twice member x (y:ys) = member x ys The use of `_' in patterns differs from the use of `_' in an expression. In either case `_' can be thought of as a "don't care" object; in an expression it means that you don't care what the value of the expression is since you never intend to use it. In a pattern, it means that you don't care what value the `_' is being matched against. > f :: [a] -> [a] > f s@(x:xs) = x:s > f _ = [] > e3 = f "abc" Another use of _: > middle :: (a,b,c) -> b > middle (_,x,_) = x > e4 :: Char > e4 = middle (True, 'a', "123") > (^) :: Int -> Int -> Int > x ^ 0 = 1 > x ^ (n+1) = x*(x^n) > e5 :: Int > e5 = 3^3 > e6 :: Int > e6 = 4^(-2) -- Notice the behavior of the + pattern on this one Section: 4.1 Pattern Matching Semantics Here's an extended example to illustrate the left -> right, top -> bottom semantics of pattern matching. > foo :: (Int,[Int],Int) -> Int > foo (1,[2],3) = 1 > foo (2,(3:_),3) = 2 > foo (1,_,3) = 3 > foo _ = 4 > e7 = foo (1,[],3) > e8 = foo (1,_,3) > e9 = foo (1,1:_,3) > e10 = foo (2,_,2) > e11 = foo (3,_,_) Now add some guards: > sign :: Int -> Int > sign x | x > 0 = 1 > | x == 0 = 0 > | x < 0 = -1 > e12 = sign 3 The last guard is often `True' to catch all other cases. The identifier `otherwise' is defined as True for use in guards: > max' :: Int -> Int -> Int > max' x y | x > y = x > | otherwise = y Guards can refer to any variables bound by pattern matching. When no guard is true, pattern matching resumes at the next equation. Guards may also refer to values bound in an associated where declaration. > inOrder :: [Int] -> Bool > inOrder (x1:x2:xs) | x1 <= x2 = True > inOrder _ = False > e13 = inOrder [1,2,3] > e14 = inOrder [2,1] Section: 4.2 An Example > take :: Int -> [a] -> [a] > take 0 _ = [] > take _ [] = [] > take (n+1) (x:xs) = x:take n xs > take' :: Int -> [a] -> [a] > take' _ [] = [] > take' 0 _ = [] > take' (n+1) (x:xs) = x:take' n xs > e15, e16, e17, e18 :: [Int] > e15 = take 0 _ > e16 = take' 0 _ > e17 = take _ [] > e18 = take' _ [] Page: 11 Sections 4.3, 4.4, 4.5, 4.6 > module Test(Bool) where > import Prelude hiding (take) Section: 4.3 Case Expressions The function `take' using a case statement instead of multiple equations: > take :: Int -> [a] -> [a] > take m ys = case (m,ys) of > (0 ,_) -> [] > (_ ,[]) -> [] > (n+1,x:xs) -> x : take n xs The function take using if then else. We can also eliminate the n+k pattern just for fun. The original version of take is much easier to read! > take' :: Int -> [a] -> [a] > take' m ys = if m == 0 then [] else > if null ys then [] else > if m > 0 then head ys : take (m-1) (tail ys) > else error "m < 0" Section: 4.4 Lazy Patterns Before the client-server example, here is a contrived example of lazy patterns. The first version will fail to pattern match whenever the the first argument is []. The second version will always pattern match initially but x will fail if used when the list is []. > nonlazy :: [Int] -> Bool -> [Int] > nonlazy (x:xs) isNull = if isNull then [] else [x] > e1 = nonlazy [1,2] False > e2 = nonlazy [] True > e3 = nonlazy [] False This version will never fail the initial pattern match > lazy :: [Int] -> Bool -> [Int] > lazy ~(x:xs) isNull = if isNull then [] else [x] > e4 = lazy [1,2] False > e5 = lazy [] True > e6 = lazy [] False The server - client example is a little hard to demonstrate. We'll avoid the initial version which loops. Here is the version with irrefutable patterns. > type Response = Int > type Request = Int > client :: Request -> [Response] -> [Request] > client init ~(resp:resps) = init : client (next resp) resps > server :: [Request] -> [Response] > server (req : reqs) = process req : server reqs Next maps the response from the previous request onto the next request > next :: Response -> Request > next resp = resp Process maps a request to a response > process :: Request -> Response > process req = req+1 > requests :: [Request] > requests = client 0 responses > responses :: [Response] > responses = server requests > e7 = take 5 responses The lists of requests and responses are infinite - there is no need to check for [] in this program. These lists correspond to streams in other languages. Here is fib again: > fib :: [Int] > fib@(_:tfib) = 1 : 1 : [ a+b | (a,b) <- zip fib tfib] > e8 = take 10 fib Section: 4.5 Lexical Scoping and Nested Forms One thing that is important to note is that the order of the definitions in a program, let expression, or where clauses is completely arbitrary. Definitions can be arranged 'top down' or `bottom up' without changing the program. > e9 = let y = 2 :: Float > f x = (x+y)/y > in f 1 + f 2 > f :: Int -> Int -> String > f x y | y > z = "y > x^2" > | y == z = "y = x^2" > | y < z = "y < x^2" > where > z = x*x > e10 = f 2 5 > e11 = f 2 4 Section: 4.6 Layout There's nothing much to demonstrate here. We have been using layout all through the tutorial. The main thing is to be careful line up the first character of each definition. For example, if you change the indentation of the definition of f in e9 you will get a parse error. Page: 12 Section 5 > module Test(Bool) where > import Prelude hiding (elem) Section: 5 Type Classes Names in the basic class structure of Haskell cannot be hidden (they are in PreludeCore) so we have to modify the names used in the tutorial. Here is a new Eq class: > class Eq' a where > eq :: a -> a -> Bool Now we can define elem using eq from above: > elem :: (Eq' a) => a -> [a] -> Bool > x `elem` [] = False > x `elem` (y:ys) = x `eq` y || x `elem` ys Before this is of any use, we need to admit some types to Eq' > instance Eq' Int where > x `eq` y = abs (x-y) < 3 -- Let's make this `nearly equal' just for fun > instance Eq' Float where > x `eq` y = abs (x-y) < 0.1 > list1 :: [Int] > list1 = [1,5,9,23] > list2 :: [Float] > list2 = [0.2,5.6,33,12.34] > e1 = 2 `elem` list1 > e2 = 100 `elem` list1 > e3 = 0.22 `elem` list2 Watch out! Integers in Haskell are overloaded - without a type signature to designate an integer as an Int, expressions like 3 `eq` 3 will be ambiguous. See 5.5.4 about this problem. Now to add the tree type: > data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving Text > instance (Eq' a) => Eq' (Tree a) where > (Leaf a) `eq` (Leaf b) = a `eq` b > (Branch l1 r1) `eq` (Branch l2 r2) = (l1 `eq` l2) && (r1 `eq` r2) > _ `eq` _ = False > tree1,tree2 :: Tree Int > tree1 = Branch (Leaf 1) (Leaf 2) > tree2 = Branch (Leaf 2) (Leaf 1) > e4 = tree1 `eq` tree2 Now make a new class with Eq' as a super class: > class (Eq' a) => Ord' a where > lt,le :: a -> a -> Bool -- lt and le are operators in Ord' > x `le` y = x `eq` y || x `lt` y -- This is a default for le The typing of lt & le is le,lt :: (Ord' a) => a -> a -> Bool This is identical to le,lt :: (Eq' a,Ord' a) => a -> a -> Bool Make Int an instance of Ord: > instance Ord' Int where > x `lt` y = x < y+1 > i :: Int -- Avoid ambiguity > i = 3 > e5 :: Bool > e5 = i `lt` i Some constraints on instance declarations: A program can never have more than one instance declaration for a given combination of data type and class. If a type is declared to be a member of a class, it must also be declared in all superclasses of that class. An instance declaration does not need to supply a method for every operator in the class. When a method is not supplied in an instance declaration and no default is present in the class declaration, a runtime error occurs if the method is invoked. You must supply the correct context for an instance declaration -- this context is not inferred automatically. Section: 5.1 Equality and Ordered Classes Section: 5.2 Enumeration and Index Classes No examples are provided for 5.1 or 5.2. The standard Prelude contains many instance declarations which illustrate the Eq, Ord, and Enum classes. Page: 13 Section 5.3 > module Test(Bool) where Section: 5.3 Text and Binary Classes This is the slow showTree. The `show' function is part of the Text class and works with all the built-in types. The context `Text a' arises from the call to show for leaf values. > data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving Text > showTree :: (Text a) => Tree a -> String > showTree (Leaf x) = show x > showTree (Branch l r) = "<" ++ showTree l ++ "|" ++ showTree r ++ ">" > tree1 :: Tree Int > tree1 = Branch (Leaf 1) (Branch (Leaf 3) (Leaf 6)) > e1 = showTree tree1 Now the improved showTree; shows is already defined for all built in types. > showsTree :: Text a => Tree a -> String -> String > showsTree (Leaf x) s = shows x s > showsTree (Branch l r) s = '<' : showsTree l ('|' : showsTree r ('>' : s)) > e2 = showsTree tree1 "" The final polished version. ShowS is predefined in the Prelude so we don't need it here. > showsTree' :: Text a => Tree a -> ShowS > showsTree' (Leaf x) = shows x > showsTree' (Branch l r) = ('<' :) . showsTree' l . ('|' :) . > showsTree' r . ('>' :) > e3 = showsTree' tree1 "" Page: 14 This page break is just to keep recompilation from getting too long. The compiler takes a little longer to compile this page than other pages. > module Test(Bool) where > data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving Text Now for the reading function. Again, ReadS is predefined and reads works for all built-in types. The generators in the list comprehensions are patterns: p <- l binds pattern p to successive elements of l which match p. Elements not matching p are skipped. > readsTree :: (Text a) => ReadS (Tree a) > readsTree ('<':s) = [(Branch l r, u) | (l, '|':t) <- readsTree s, > (r, '>':u) <- readsTree t ] > readsTree s = [(Leaf x,t) | (x,t) <- reads s] > e4 :: [(Int,String)] > e4 = reads "5 golden rings" > e5 :: [(Tree Int,String)] > e5 = readsTree "<1|<2|3>>" > e6 :: [(Tree Int,String)] > e6 = readsTree "<1|2" > e7 :: [(Tree Int,String)] > e7 = readsTree "<1|<<2|3>|<4|5>>> junk at end" Before we do the next readTree, let's play with the lex function. > e8 :: [(String,String)] > e8 = lex "foo bar bletch" Here's a function to completely lex a string. This does not handle lexical ambiguity - lex would return more than one possible lexeme when an ambiguity is encountered and the patterns used here would not match. > lexAll :: String -> [String] > lexAll s = case lex s of > [("",_)] -> [] -- lex returns an empty token if none is found > [(token,rest)] -> token : lexAll rest > e9 = lexAll "lexAll :: String -> [String]" > e10 = lexAll "<1|<a|3>>" Finally, the complete reader. This is not sensitive to white space as were the previous versions. When you derive the Text class for a data type the reader generated automatically is similar to this in style. > readsTree' :: (Text a) => ReadS (Tree a) > readsTree' s = [(Branch l r, x) | ("<", t) <- lex s, > (l, u) <- readsTree' t, > ("|", v) <- lex u, > (r, w) <- readsTree' v, > (">", x) <- lex w ] > ++ > [(Leaf x, t) | (x, t) <- reads s] When testing this program, you must make sure the input conforms to Haskell lexical syntax. If you remove spaces between | and < or > they will lex as a single token as you should have noticed with e10. > e11 :: [(Tree Int,String)] > e11 = readsTree' "<1 | <2 | 3> >" > e12 :: [(Tree Bool,String)] > e12 = readsTree' "<True|False>" Finally, here is a simple version of read for trees only: > read' :: (Text a) => String -> (Tree a) > read' s = case (readsTree' s) of > [(tree,"")] -> tree -- Only one parse, no junk at end > [] -> error "Couldn't parse tree" > [_] -> error "Crud after the tree" -- unread chars at end > _ -> error "Ambiguous parse of tree" > e13 :: Tree Int > e13 = read' "foo" > e14 :: Tree Int > e14 = read' "< 1 | < 2 | 3 > >" > e15 :: Tree Int > e15 = read' "3 xxx" Page: 15 Section 5.4 > module Test(Bool) where Section: 5.4 Derived Instances We have actually been using the derived Text instances all along for printing out trees and other structures we have defined. The code in the tutorial for the Eq and Ord instance of Tree is created implicitly by the deriving clause so there is no need to write it here. > data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving (Eq,Ord,Text) Now we can fire up both Eq and Ord functions for trees: > tree1, tree2, tree3, tree4 :: Tree Int > tree1 = Branch (Leaf 1) (Leaf 3) > tree2 = Branch (Leaf 1) (Leaf 5) > tree3 = Leaf 4 > tree4 = Branch (Branch (Leaf 4) (Leaf 3)) (Leaf 5) > e1 = tree1 == tree1 > e2 = tree1 == tree2 > e3 = tree1 < tree2 > quicksort :: Ord a => [a] -> [a] > quicksort [] = [] > quicksort (x:xs) = quicksort [y | y <- xs, y <= x] ++ > [x] ++ > quicksort [y | y <- xs, y > x] > e4 = quicksort [tree1,tree2,tree3,tree4] Now for Enum: > data Day = Sunday | Monday | Tuesday | Wednesday | Thursday | > Friday | Saturday deriving (Text,Eq,Ord,Enum) > e5 = quicksort [Monday,Saturday,Friday,Sunday] > e6 = [Wednesday .. Friday] > e7 = [Monday, Wednesday ..] > e8 = [Saturday, Friday ..] Page: 16 Sections 5.5, 5.5.1, 5.5.2, 5.5.3 > module Test(Bool) where Section: 5.5 Numbers Section: 5.5.1 Numeric Class Structure Section: 5.5.2 Constructed Numbers Here's a brief summary of Haskell numeric classes. Class Num Most general numeric class. Has addition, subtraction, multiplication. Integers can be coerced to any instance of Num with fromInteger. All integer constants are in this class. Instances: Int, Integer, Float, Double, Ratio a, Complex a Class Real This class contains ordered numbers which can be converted to rationals. Instances: Int, Integer, Float, Double, Ratio a Class Integral This class deals with integer division. All values in Integral can be mapped onto Integer. Instances: Int, Integer Class Fractional These are numbers which can be divided. Any rational number can be converted to a fractional. Floating point constants are in this class: 1.2 would be 12/10. Instances: Float, Double, Ratio a Class Floating This class contains all the standard floating point functions such as sqrt and sin. Instances: Float, Double, Complex a Class RealFrac These values can be rounded to integers and approximated by rationals. Instances: Float, Double, Ratio a Class RealFloat These are floating point numbers constructed from a fixed precision mantissa and exponent. Instances: Float, Double There are only a few sensible combinations of the constructed numerics with built-in types: Ratio Integer (same as Rational): arbitrary precision rationals Ratio Int: limited precision rationals Complex Float: complex numbers with standard precision components Complex Double: complex numbers with double precision components The following function works for arbitrary numerics: > fact :: (Num a) => a -> a > fact 0 = 1 > fact n = n*(fact (n-1)) Note the behavior when applied to different types of numbers: > e1 :: Int > e1 = fact 6 > e2 :: Int > e2 = fact 20 -- Yale Haskell may not handle overflow gracefully! > e3 :: Integer > e3 = fact 20 > e4 :: Rational > e4 = fact 6 > e5 :: Float > e5 = fact 6 > e6 :: Complex Float > e6 = fact 6 Be careful: values like `fact 1.5' will loop. As a practical matter, Int operations are much faster than Integer operations. Also, overloaded functions can be much slower than non- overloaded functions. Giving a function like fact a precise typing: fact :: Int -> Int will yield much faster code. In general, numeric expressions work as expected. Literals are a little tricky - they are coerced to the appropriate value. A constant like 1 can be used as ANY numeric type. > e7 :: Float > e7 = sqrt 2 > e8 :: Rational > e8 = ((4%5) * (1%2)) / (3%4) > e9 :: Rational > e9 = 2.2 * (3%11) - 1 > e10 :: Complex Float > e10 = (2 * (3:+3)) / (1.1:+2.0 - 1) > e11 :: Complex Float > e11 = sqrt (-1) > e12 :: Integer > e12 = numerator (4%2) > e13 :: Complex Float > e13 = conjugate (4:+5.2) A function using pattern matching on complex numbers: > mag :: (RealFloat a) => Complex a -> a > mag (a:+b) = sqrt (a^2 + b^2) > e14 :: Float > e14 = mag (1:+1) Section: 5.5.3 Numeric Coercions and Overloaded Literals The Haskell type system does NOT implicitly coerce values between the different numeric types! Although overloaded constants are coerced when the overloading is resolved, no implicit coercion goes on when values of different types are mixed. For example: > f :: Float > f = 1.1 > i1 :: Int > i1 = 1 > i2 :: Integer > i2 = 2 All of these expressions would result in a type error (try them!): > -- g = i1 + f > -- h = i1 + i2 > -- i3 :: Int > -- i3 = i2 Appropriate coercions must be introduced by the user to allow the mixing of types in arithmetic expressions. > e15 :: Float > e15 = f + fromIntegral i1 > e16 :: Integer > e16 = fromIntegral i1 + i2 > e17 :: Int > e17 = i1 + fromInteger i2 -- fromIntegral would have worked too. Page: 17 Section 5.5.4 > module Test(Bool) where Section: 5.5.4 Default Numeric Types Ambiguous contexts arise frequently in numeric expressions. When an expression which produces a value with a general type, such as `1' (same as `fromInteger 1'; the type is (Num a) => a), with another expression which `consumes' the type, such as `show' or `toInteger', ambiguity arises. This ambiguity can be resolved using expression type signatures, but this gets tedious fast! Assigning a type to the top level of an ambiguous expression does not help: the ambiguity does not propagate to the top level. > e1 :: String -- This type does not influence the type of the argument to show > e1 = show 1 -- Does this mean to show an Int or a Float or ... > e2 :: String > e2 = show (1 :: Float) > e3 :: String > e3 = show (1 :: Complex Float) The reason the first example works is that ambiguous numeric types are resolved using defaults. The defaults in effect here are Int and Double. Since Int `fits' in the expression for e1, Int is used. When Int is not valid (due to other context constraints), Double will be tried. This function defaults the type of the 2's to be Int > rms :: (Floating a) => a -> a -> a > rms x y = sqrt ((x^2 + y^2) * 0.5) The C-c e evaluation used to the Haskell system also makes use of defaulting. When you type an expression, the system creates a simple program to print the value of the expression using a function like show. If no type signature for the printed expression is given, defaulting may occur. One of the reasons for adding type signatures throughout these examples is to avoid unexpected defaulting. Many of the top level signatures are required to avoid ambiguity. Defaulting can lead to overflow problems when values exceed Int limits. Evaluate a very large integer without a type signature to observe this (unfortunately this may cause a core dump or other unpleasantness). Notice that defaulting applies only to numeric classes. The > -- show (read "xyz") -- Try this if you want! example uses only class Text so no defaulting occurs. Ambiguity also arises with polymorphic types. As discussed previously, expressions like [] have a similar problem. > e4 = [] -- Won't print since [] has type [a] and `a' is not known. Note the difference: even though the lists have no components, the type of component makes a difference in printing. > e5 = ([] :: [Int]) > e6 = ([] :: [Char]) Page: 18 Sections 6, 6.1, 6.2 Section: 6 Modules > module Tree ( Tree(Leaf,Branch), fringe ) where Tree(..) would work also > data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving Text > fringe :: Tree a -> [a] > fringe (Leaf x) = [x] > fringe (Branch left right) = fringe left ++ fringe right The editor interface to Haskell performs evaluation within the module containing the cursor. To evaluate e1 you must place the cursor in module Main. > module Main (Tree) where > import Tree ( Tree(Leaf, Branch), fringe) > e1 :: [Int] > e1 = fringe (Branch (Leaf 1) (Leaf 2)) You could also just `import Tree' and get everything from Tree without explicitly naming the entities you need. This interactive Haskell environment can evaluate expressions in any module. The use of module Main is optional. Section: 6.1 Original Names and Renaming > module Renamed where > import Tree ( Tree(Leaf,Branch), fringe) > renaming (Leaf to Root, Branch to Twig) > e2 :: Tree Int > e2 = Twig (Root 1) (Root 2) -- Printing always uses the original names Section: 6.2 Interfaces and Implementations Yale Haskell allows separate compilation of modules using interfaces and unit files. These are described in the user's guide. Page: 19 Sections 6.3, 6.4 Section: 6.3 Abstract Data Types Since TreeADT does not import Tree it can use the name Tree without any conflict. Each module has its own separate namespace. > module TreeADT (Tree, leaf, branch, cell, left, > right, isLeaf) where > data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving Text > leaf = Leaf > branch = Branch > cell (Leaf a) = a > left (Branch l r) = l > right (Branch l r) = r > isLeaf (Leaf _) = True > isLeaf _ = False > module Test where > import TreeADT Since the constructors for type Tree are hidden, pattern matching cannot be used. > fringe :: Tree a -> [a] > fringe x = if isLeaf x then [cell x] > else fringe (left x) ++ fringe (right x) > e1 :: [Int] > e1 = fringe (branch (branch (leaf 3) (leaf 2)) (leaf 1)) Section: 6.4 No examples are provided for 6.4. Page: 20 Sections 7, 7.1, 7.2, 7.3 Section: 7 Typing Pitfalls Section: 7.1 Let-Bound Polymorphism > module Test(e2) where > -- f g = (g 'a',g []) -- This won't typecheck. Section: 7.2 Overloaded Numerals Overloaded numerics were covered previously - here is one more example. sum is a prelude function which sums the elements of a list. > average :: (Fractional a) => [a] -> a > average xs = sum xs / fromIntegral (length xs) > e1 :: Float -- Note that e1 would default to Double instead of Int - > -- this is due to the Fractional context. > e1 = average [1,2,3] Section: 7.3 The Monomorphism Restriction The monomorphism restriction is usually encountered when functions are defined without parameters. If you remove the signature for sum' the monomorphism restriction will apply. This will generate an error if either: sum' is added to the module export list at the start of this section both sumInt and sumFloat remain in the program. If sum' is not exported and all uses of sum' have the same overloading, there is no type error. > sum' :: (Num a) => [a] -> a > sum' = foldl (+) 0 -- foldl reduces a list with a binary function > -- 0 is the initial value. > sumInt :: Int > sumInt = sum' [1,2,3] > sumFloat :: Float > sumFloat = sum' [1,2,3] If you use overloaded constants you also may encounter monomorphism: > x :: Num a => a > x = 1 -- The type of x is Num a => a > y :: Int > y = x -- Uses x as an Int > z :: Integer > z = x -- Uses x as an Integer. A monomorphism will occur of the > -- signature for x is removed. The error message that this generates is somewhat arbitrary. When the monomorphism restriction applies, the first use of x will determine the type and the second will cause the type error. There is no way to be sure which use of x the type checker will encounter first; either the `y = x' or the `z = x' may be flagged as the place where the error occurs depending on the order in which type checking scans the program. Finally, if a value is exported it must not be overloaded unless bound by a function binding. e2 is the only value exported. > e2 :: Int -- Remove this to get an error. Without this line e1 will > -- be overloaded. > e2 = 1 To prevent annoying error messages about exported monomorphic variables, most modules in this tutorial do not implicitly export everything - they only export a single value, Bool, which was chosen to keep the export list non-empty (a syntactic restriction!). In Haskell systems without the evaluator used here, a module which does not export any names would be useless. module Test where -- this would export everything in the module module Test(Bool) -- exports only Bool module Test() -- this is what we really want to do but is not valid. Page: 21 Sections 8, 8.1 > module Test(Bool) where Section: 8 Input/Output Section: 8.1 Introduction to Continuations Simplify f here to be 1/x. > data Possibly a = Ok a | Oops String deriving Text > f :: Float -> Possibly Float > f x = if x == 0 then Oops "Divide by 0" else Ok (1/x) g is a `safe' call to x. The call to error could be replaced by some explicit value like Oops msg -> 0. > g x = case f x of > Ok y -> y > Oops msg -> error msg > e1 = f 0 > e2 = g 0 > e3 = g 1 Here is the same example using continuations: > f' :: Float -> (String -> Float) -> Float > f' x c = if x == 0 then c "Divide by 0" > else 1/x > g' x = f' x error -- calls error on divide by 0 > g'' x = f' x (\s -> 0) -- returns 0 on divide by 0 > e4 = g' 0 > e5 = g'' 0 Page: 22 Section 8 > module Test where > import Prelude hiding (putStr, getLine) Section: 8.1 The I/O Monad The I/O monad divides the Haskell world into values and actions. So far, we have only needed to look at values. To deal with actions, we need to introduce a new editor command: C-c r. We use the term `dialogue' to denote an action returning no value, as indicated by the type `IO ()'. Instead of printing a value, C-c r runs a dialogue. (Actually C-c e creates an action on the fly and runs it in the same manner as C-c r). As with C-c e you are prompted for an expression; here the type of the expression must be IO (). We use d1,d2,... for dialogues to be executed by C-c r. In an interactive environment such as this, there is no real need to use `main' in module `Main' to designate the dialogue associated with a program since C-c r queries for the dialogue to be executed. However, many commands such as C-c m make use of this convention. The first example is putStr: > putStr :: String -> IO () > putStr s = foldr (>>) (return ()) (map putChar s) > d1 = putStr "Hello World" Both putStr and getLine are actually in the prelude. > getLine :: IO String > getLine = getChar >>= (\c -> > if c == '\n' then return "" > else getLine >>= (\l -> return (c:l))) > d2 = putStr "Type Something: " >> getLine >>= (\str -> > putStr "You typed: " >> putStr str >> putStr "\n") To experiment with I/O errors, we need to get a bit creative. Generating an error is generally OS specific so instead we use a new version of getLine that raises an error when a blank line is entered: > getLineErr = getLine >>= (\l -> > if l == "" then failwith EOF > else return l) First, enter a blank line with no active error handler: > d3 = getLineErr >>= putStr Now with an error handler: > d4 = try getLineErr (\ e -> return (show e)) >>= putStr This is the file copier: > d5 = getAndOpenFile "Copy from: " ReadMode >>= (\fromHandle -> > getAndOpenFile "Copy to: " WriteMode >>= (\toHandle -> > getContents fromHandle >>= (\contents -> > hPutStr toHandle contents >> close toHandle >> putStr "Done.\n"))) > getAndOpenFile :: String -> IOMode -> IO Handle > getAndOpenFile prompt mode = > putStr prompt >> > getLine >>= (\name -> > try (openFile mode name) > (\err -> putStr ("Cannot open "++ name ++ "\n") >> > getAndOpenFile prompt mode)) Finally, an example not in the tutorial. Reading stdin using getContents makes the demands for evaluation of the input stream visible to the user since the program will stop for input whenever demand reaches a new line in the input stream. This reads the entire input stream, breaks it into lines, and hands a list of lines to munchInput: > d6 = getContents stdin >>= \i -> munchInput (lines i) The munchInput function looks at the next line of input. It prints a prompt and then looks at a line of input. The command `stop' halts execution, the command `skip' skips over the next input line, and anything else is echoed. > munchInput (l:ls) = > putStr "* " >> > case l of > "stop" -> return () > "skip" -> munchInput (tail ls) > _ -> putStr (l ++ " not understood.\n") >> munchInput ls Run this program. There is a problem with the prompting: it reads the input line before the prompt is printed out. While the case statement would be expected to demand a line of input, this demand actually occurs earlier. The pattern (l:ls) can only be matched by stripping a line of the input stream. Since this pattern is matched upon entry to munchInput, the input demand occurs before the prompt can be printed. Change the first line of munchInput to: munchInput ~(l:ls) = This will allow the prompt to print out before the case statement looks at the value of l. Run the program again. Note the behavior when you type "skip". Again, you need to understand when the input will be demanded to figure out why the prompt prints before the ignored input line. Now change this by putting a something in the recursive call to munchInput which will look at the line being discarded before the recursive call: "skip" -> case ls of (_ : t) -> munchInput t This reads the skipped line before the prompt on the recursive call. If you find all of this confusing, then the lesson here is that it is probably best to avoid reading stdin with getContents. Using getLine to read a line at a time is much easier to understand. Since getLine is strict with respect to the input, there is no way that later demands will suddenly stop execution to wait for input. For more examples using the I/O system look in the demo programs that come with haskell (in $HASKELL/progs/demo) and the report. Page: 23 Sections 9, 9.1, 9.2 > module Test(Bool) where Section: 9 Arrays Section: 9.1 Index Types Arrays are built on the class Ix. Here are some quick examples of Ix: > e1 :: [Int] > e1 = range (0,4) > e2 :: Int > e2 = index (0,4) 2 > low,high :: (Int,Int) > low = (1,1) > high = (3,4) > e3 = range (low,high) > e4 = index (low,high) (3,2) > e5 = inRange (low,high) (4,3) Section: 9.2 Array Creation > squares :: Array Int Int > squares = array (1,100) [i := i*i | i <- [1..100]] We can also parameterize this a little: > squares' :: Int -> Array Int Int > squares' n = array (1,n) [i := i*i | i <- [1..n]] > e6 :: Int > e6 = squares!6 > e7 :: (Int,Int) > e7 = bounds squares > e8 :: Array Int Int > e8 = squares' 10 Here is a function which corresponds to `take' for lists. It takes an arbitrary slice out of an array. > atake :: (Ix a) => Array a b -> (a,a) -> Array a b > atake a (l,u) | inRange (bounds a) l && inRange (bounds a) u = > array (l,u) [i := a!i | i <- range (l,u)] > | otherwise = error "Subarray out of range" > e9 = atake squares (4,8) > mkArray :: Ix a => (a -> b) -> (a,a) -> Array a b > mkArray f bnds = array bnds [i := f i | i <- range bnds] > e10 :: Array Int Int > e10 = mkArray (\i -> i*i) (1,10) > fibs :: Int -> Array Int Int > fibs n = a where > a = array (0,n) ([0 := 1, 1 := 1] ++ > [i := a!(i-1) + a!(i-2) | i <- [2..n]]) > e11 = atake (fibs 50) (3,10) > wavefront :: Int -> Array (Int,Int) Int > wavefront n = a where > a = array ((1,1),(n,n)) > ([(1,j) := 1 | j <- [1..n]] ++ > [(i,1) := 1 | i <- [2..n]] ++ > [(i,j) := a!(i,j-1) + a!(i-1,j-1) + a!(i-1,j) > | i <- [2..n], j <- [2..n]]) > wave = wavefront 20 > e12 = atake wave ((1,1),(3,3)) > e13 = atake wave ((3,3),(5,5)) Here are some errors in array operations: > e14 :: Int > e14 = wave ! (0,0) -- Out of bounds > arr1 :: Array Int Int > arr1 = array (1,10) [1 := 1] -- No value provided for 2..10 > e15,e16 :: Int > e15 = arr1 ! 1 -- works OK > e16 = arr1 ! 2 -- undefined by array Page: 24 Sections 9.3, 9.4 > module Test(Bool) where Section: 9.3 Accumulation > hist :: (Ix a, Integral b) => (a,a) -> [a] -> Array a b > hist bnds is = accumArray (+) 0 bnds [i := 1 | i <- is, inRange bnds i] > e1 :: Array Char Int > e1 = hist ('a','z') "This counts the frequencies of each lowercase letter" > decades :: (RealFrac a) => a -> a -> [a] -> Array Int Int > decades a b = hist (0,9) . map decade > where > decade x = floor ((x-a) * s) > s = 10 / (b - a) > test1 :: [Float] > test1 = map sin [0..100] -- take the sine of the 0 - 100 > e2 = decades 0 1 test1 Section: 9.4 Incremental Updates > swapRows :: (Ix a, Ix b, Enum b) => a -> a -> Array (a,b) c -> Array (a,b) c > swapRows i i' a = a // ([(i,j) := a!(i',j) | j <- [jLo..jHi]] ++ > [(i',j) := a!(i,j) | j <- [jLo..jHi]]) > where ((iLo,jLo),(iHi,jHi)) = bounds a > arr1 :: Array (Int,Int) (Int,Int) > arr1 = array ((1,1),(5,5)) [(i,j) := (i,j) | (i,j) <- range ((1,1),(5,5))] > e3 = swapRows 2 3 arr1 Printing the arrays in more readable form makes the results easier to view. This is a printer for 2d arrays > aprint a width = shows (bounds a) . showChar '\n' . showRows lx ly where > showRows r c | r > ux = showChar '\n' > showRows r c | c > uy = showChar '\n' . showRows (r+1) ly > showRows r c = showElt (a!(r,c)) . showRows r (c+1) > showElt e = showString (take width (show e ++ repeat ' ')) . showChar ' ' > ((lx,ly),(ux,uy)) = bounds a > showArray a w = appendChan stdout (aprint a w "") abort done > d1 = showArray e3 6 > swapRows' :: (Ix a, Ix b, Enum b) => a -> a -> Array (a,b) c -> Array (a,b) c > swapRows' i i' a = a // [assoc | j <- [jLo..jHi], > assoc <- [(i,j) := a!(i',j), > (i',j) := a!(i,j)]] > where ((iLo,jLo),(iHi,jHi)) = bounds a > d2 = showArray (swapRows' 1 5 arr1) 6 Page: 25 Section 9.5 > module Test(Bool) where Section: 9.5 An example: Matrix Multiplication > aprint a width = shows (bounds a) . showChar '\n' . showRows lx ly where > showRows r c | r > ux = showChar '\n' > showRows r c | c > uy = showChar '\n' . showRows (r+1) ly > showRows r c = showElt (a!(r,c)) . showRows r (c+1) > showElt e = showString (take width (show e ++ repeat ' ')) . showChar ' ' > ((lx,ly),(ux,uy)) = bounds a > showArray a w = appendChan stdout (aprint a w "") abort done > matMult :: (Ix a, Ix b, Ix c, Num d) => > Array (a,b) d -> Array (b,c) d -> Array (a,c) d > matMult x y = > array resultBounds > [(i,j) := sum [x!(i,k) * y!(k,j) | k <- range (lj,uj)] > | i <- range (li,ui), > j <- range (lj',uj')] > where > ((li,lj),(ui,uj)) = bounds x > ((li',lj'),(ui',uj')) = bounds y > resultBounds > | (lj,uj)==(li',ui') = ((li,lj'),(ui,uj')) > | otherwise = error "matMult: incompatible bounds" > mat1,mat2,mat3,mat4 :: Array (Int,Int) Int > mat1 = array ((0,0),(1,1)) [(0,0) := 1,(0,1) := 0,(1,0) := 0,(1,1) := 1] > mat2 = array ((0,0),(1,1)) [(0,0) := 1,(0,1) := 1,(1,0) := 1,(1,1) := 1] > mat3 = array ((0,0),(1,1)) [(0,0) := 1,(0,1) := 2,(1,0) := 3,(1,1) := 4] > mat4 = array ((0,0),(1,2)) [(0,0) := 1,(0,1) := 2,(0,2) := 3, > (1,0) := 4,(1,1) := 5,(1,2) := 6] > d1 = showArray (matMult mat1 mat2) 4 > d2 = showArray (matMult mat2 mat3) 4 > d3 = showArray (matMult mat1 mat4) 4 > d4 = showArray (matMult mat4 mat1) 4 > matMult' :: (Ix a, Ix b, Ix c, Num d) => > Array (a,b) d -> Array (b,c) d -> Array (a,c) d > matMult' x y = > accumArray (+) 0 ((li,lj'),(ui,uj')) > [(i,j) := x!(i,k) * y!(k,j) > | i <- range (li,ui), > j <- range (lj',uj'), > k <- range (lj,uj)] > where > ((li,lj),(ui,uj)) = bounds x > ((li',lj'),(ui',uj')) = bounds y > resultBounds > | (lj,uj)==(li',ui') = ((li,lj'),(ui,uj')) > | otherwise = error "matMult: incompatible bounds" > d5 = showArray (matMult mat1 mat2) 4 > d6 = showArray (matMult mat2 mat3) 4 > genMatMul :: (Ix a, Ix b, Ix c) => > ([f] -> g) -> (d -> e -> f) -> > Array (a,b) d -> Array (b,c) e -> Array (a,c) g > genMatMul f g x y = > array ((li,lj'),(ui,uj')) > [(i,j) := f [(x!(i,k)) `g` (y!(k,j)) | k <- range (lj,uj)] > | i <- range (li,ui), > j <- range (lj',uj')] > where > ((li,lj),(ui,uj)) = bounds x > ((li',lj'),(ui',uj')) = bounds y > resultBounds > | (lj,uj)==(li',ui') = ((li,lj'),(ui,uj')) > | otherwise = error "matMult: incompatible bounds" > d7 = showArray (genMatMul maximum (-) mat2 mat1) 4 > d8 = showArray (genMatMul and (==) mat1 mat2) 6 > d9 = showArray (genMatMul and (==) mat1 mat1) 6 Page: 26 More about Haskell This is the end of the tutorial on official Haskell. If you wish to see more examples of Haskell programming, Yale Haskell comes with a set of demo programs. These can be found in $HASKELL/progs/demo. Once you have mastered the tutorial, both the report and the user manual for Yale Haskell should be understandable. Many examples of Haskell programming can be found in the Prelude. The directory $HASKELL/progs/prelude contains the sources for the Prelude. The following pages describe extensions which have been added to the Yale system and are not officially a part of the Haskell language. Proceed at your own risk! We appreciate any comments you have on this tutorial. Send any comments to haskell-requests@cs.yale.edu. The Yale Haskell Group Page: 27 Basic Dynamic Typing > module Foo(Bool) where > import Dynamic The next few pages present an overview of the Yale dynamic typing system. A tech report describing this approach to dynamic typing is distributed in the $HASKELL/doc/dynamic directory. The prelude files $PRELUDE/PreludeDynamic.hs $PRELUDE/PreludeDeriving.hs have quite a bit of documentation in them also. Any module using dynamic typing must import Dynamic A value of any type can be converted to a single type, Dynamic, using the 'toDynamic' construct: > e1 = toDynamic True > e2 = toDynamic ("abc",False,'z') While it uses function calling syntax, toDynamic is not a function but a special construct. Thus `map toDynamic l' would be meaningless. When printed, only the signature of the dynamic value is shown. The value inside the dynamic is not printed. Dynamic values may be overloaded: > e3 = toDynamic 1 -- remember that Haskell adds an implicit 'fromInteger' here > e4 = toDynamic (+) The inverse of toDynamic is fromDynamic. This integrates a dynamic into the existing type context. This function requires that the result of fromDynamic is an Int: > f :: Dynamic -> Int > f x = fromDynamic x + 1 > e5 = f (toDynamic 2) fromDynamic will fail at runtime if the type of the dynamic is not appropriate. The dynamic type may be more general than the desired type (as in e5, since the type of 2 is Num a => a) > e6 = f (toDynamic False) -- This will fail at run time. To interrogate the type of a dynamic value, pattern matching can be used. The pattern (p :: Signature) matches a dynamic whose type is type is the same as (or can be coereced to) the signature. > t (x :: Bool) = if x then "True" else "False" > t (x :: Int) = "Int" > t (x :: Integer) = "Integer" > t _ = "Something else" > e7 = t (toDynamic True) > e8 = t (toDynamic (1 :: Integer)) > e9 = t (toDynamic 1) -- This could match Int or Integer More general patterns are also useful: > sh (x :: Text a => a) = show x > sh _ = "<<Error>>" > e10 = sh e1 > e11 = sh e2 > data NotInText = NotInText > e12 = sh (toDynamic NotInText) To round off the special dynamic type operators available, 'typeOf' is provided to compute the type of an object. This returns the 'Signature' type which is defined in the module Dynamic. > e13 = typeOf foldr > e14 = typeOf ((+),"abc",1) Page: 28 Operating in the Dynamic domain > module Foo(Bool) where > import DynamicInternal Many operations are supplied for the values and types used by the dynamic typing system. > dyn1 = toDynamic (1 :: Int) > dyn2 = toDynamic "abc" > dyn3 = toDynamic (+) > dyn4 = toDynamic (error "Evaluating dyn4" :: String) > dyn5 = toDynamic (error "Evaluating dyn5") The type of a dynamic can be obtained by dType: dType :: Dynamic -> Signature > e1 = dType dyn1 > e2 = dType dyn3 The top level datatype is returned by dDataType: dDataType :: Dynamic -> DataType > e3 = dDataType dyn1 > e4 = dDataType dyn2 > e5 = dDataType dyn4 > e6 = dDataType dyn5 -- This value has no data type Note: the Haskell committee has decided on some official names for unnamed types, such as (->) for the function type, [] for the list type, and (,) for tuple types. The dynamic types for these constructs do not yet have these names. dConstructor returns the data constructor which created the value. This evaluates the data value. > e7 = dConstructor dyn1 -- Numerics have a bogus internal constructor > e8 = dConstructor dyn2 > e9 = dConstructor dyn3 -- Internal constructor for functions > e10 = dConstructor dyn4 The dSlots function take apart a dynamic value: > e11 = dSlots dyn2 -- This would be meaningless for the other dynamics Finally, dShow converts a dynamic to a string. This is possible even when Text instances are not available for the type. > e12 = dShow dyn1 > e13 = dShow dyn2 > e14 = map dShow e11 Function application in the dynamic domain is performed by dApply: dApply :: Dynamic -> [Dynamic] -> Dynamic > e15 = dShow (dApply dyn3 [dyn1,dyn1]) > e16 = dShow (dApply dyn3 [dyn2]) -- A special dynamic type is used for errors. Many more dynamic operations are available - see accompanying documentation. Page: 29 Existential Types and Dynamic Typing > module Foo(Bool) where > import Dynamic When polymorphism is present, dynamic typing cannot completely capture the type of a object. > f :: a -> Dynamic > f x = toDynamic x This function does not have any information about the type of its argument. What happens is that a new type is created at runtime every time f is called. These types are called skolem types. > e1 = f True > e2 = f "abc" The type captured by f is basicly useless since this type has no 'interesting' properties. A much more useful function would be > f1 :: Num a => a -> Dynamic > f1 x = toDynamic x > e3 = f1 (1 :: Int) The skolem type introduced by f1 will be an instance of Num. The signature on f1 is necessary - without it this would have been the same as f. The following function can be applied to any numeric type, including the skolem type found in e3. > f2 (x :: Num a => a) = toDynamic (x+2) > e4 = f2 e3 Note that the type in e4 is the same as the type in e3. Unfortunately there is no way to recover the original type, Int. However, the value can be obtained through 'show': > e5 = case e4 of (x :: Text a => a) -> show x This works because Text is a superclass of Num. Warning: skolemization is impure! Any program that 'looks at' a skolem type the wrong way will lose referential transparency. While it is very convenient to have skolem types, they should be used with great care. More examples of type skolemization can be found in the documentation of the dynamic typing system but no further examples will be given here. Page: 30 Generalizing Derived Instances with Dynamics > module Foo(Bool) where > import Dynamic One important application of dynamic typing is a generalization of derived instances. In the Haskell report, the various derived instances are specified using code templates which are based on datatype declarations. No formal mechanism is provided to add new code templates to Haskell - there is no way to add a new derivable instance to the language. The Yale compiler provides a mechanism which allows the user to introduce new derived instances. The declaration: deriving DI t where instance C t where decl1 decl2 defines a new derived instance, DI, which is expanded by creating an instance declaration containing the given decls. For example, you could define a class: > class NamedType a where > typeName :: a -> String > deriving NamedType a where > instance NamedType a where > typeName x = dDataTypeName (dDataType (toDynamic x)) > data T1 = T1 deriving NamedType > data T2 = T2 deriving NamedType > e1 = typeName T1 > e2 = typeName T2 All of the standard derived instances propagate their context down to the components of a datatype. For example, for a type to be an instance of Eq, all components must also be an instance of Eq. This is a new version of Eq: > class MyEq a where > eq :: a -> a -> Bool This deriving declaration creates a new derived instance, MyEq, and uses dynamic typing to perform the computation of eq. The context in the class declaration, MyEq t, is not applied to the type t but to all component types in t. It may have been cleaner to use something like instance MyEq (subTypes t) => MyEq t but this would add extra syntax to the language. > deriving MyEq t where > instance MyEq t => MyEq t where > eq x y = dynamicEq (toDynamic x) (toDynamic y) This dynamicEq function has different strictness properties from the standard Eq class. It compares right to left instead of left to right. > dynamicEq x y = tag x == tag y && foldr (flip (&&)) True slotEqs where > tag x = dConstructorTag (dConstructor x) > slotEqs = zipWith dynEq (dSlots x) (dSlots y) > dynEq x (y :: MyEq a => a) = eq (fromDynamic x) y > data Foo = C1 Foo Foo | C2 deriving (Eq,MyEq,Text) > e3 = (C1 C2 (C1 C2 C2)) `eq` (C1 C2 (C1 C2 C2)) > e4 = (C1 C2 (C1 C2 C2)) == (C1 C2 (C1 C2 C2)) > e5 = (C1 C2 (error "2nd slot")) `eq` (C1 (C1 C2 C2) C2) > e6 = (C1 C2 (error "2nd slot")) == (C1 (C1 C2 C2) C2) > e7 = (C1 (error "1st slot") C2) `eq` (C1 C2 (C1 C2 C2)) > e8 = (C1 (error "1st slot") C2) == (C1 C2 (C1 C2 C2)) The class does not need to match the name of the derived instance. For example, an alternative Text instance could be supplied: > deriving Text' t where > instance Text t where > showsPrec p x = showDyn (toDynamic x) > showDyn x = showString "<<" . > showString (dDataTypeName (dDataType x)) . > showString ">>" > data A = B | C deriving Text' > e9 = (B,True,[C,B]) Some other features: More than one instance declaration can be supplied in a deriving declaration. A context in the deriving declaration can be used to require the presence of other instances for the type: deriving Eq t => Bar t where instance Ord t where ... This deriving will fail if no Eq instance is supplied for the type t. The data type can be restricted to enumerated or tuple data types. Look into PreludeDeriving for more examples of the deriving declaration. This is the end of the tutorial.