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Joined 2026-09-21 Message
Incoming links: Pattern matching
Algebraic data type 2026-10-31
A type whose values are described by named data constructors with specified fields. For example,
data Tree a = One a | Two (Tree a) (Tree a) describes a leaf-labelled binary tree. Its two alternatives are distinguished by pattern matching; the recursive fields give a recursive type. List zip 2026-10-31
Combine corresponding entries of two lists into a list of tuples. The usual finite-list operation stops when either input ends. A lazy evaluation implementation can produce entries from infinite lists, but its exact strictness depends on the order of pattern matching in the definition.
Use pattern matching to separate the empty list from a head and tail. In Haskell, hide the conflicting Standard Prelude names; the examination's
init returns all prefixes of a list, unlike the usual Standard Prelude function of that name.import Prelude hiding (map, length, init, tails, foldr, compare)
map :: (a -> b) -> [a] -> [b]
map f [] = []
map f (x:xs) = f x : map f xs
length :: Num n => [a] -> n
length [] = 0
length (_:xs) = 1 + length xs
copy :: (Eq n, Num n) => n -> a -> [a]
copy 0 xs = []
copy n xs = xs : copy (n-1) xs
combine :: [[a]] -> [[a]] -> [[a]]
combine [] _ = []
combine _ [] = []
combine (xs:xss) (ys:yss) = (xs ++ ys) : combine xss yss
init :: [a] -> [[a]]
init [] = [[]]
init (x:xs) = [] : map (x:) (init xs)
tails :: [a] -> [[a]]
tails [] = [[]]
tails (x:xs) = (x:xs) : tails xsThe map function preserves the input list shape while transforming each entry.
length counts the cons cells. copy repeats the whole value xs, so for the specified use its second argument is a list and the result is a list of lists. Its inferred type is more general: the repeated value can have any data type. If one restricts the interface to lists, the corresponding type is (Eq n, Num n) => n -> [a] -> [[a]]. The numeric argument has the precondition that it is a nonnegative integer; the most general inferred numeric constraints do not guarantee termination on arbitrary values of their instances. An explicitly integral interface may instead use Integral n => n -> [a] -> [[a]].combine performs list concatenation head by head and stops as soon as either outer list ends. For init, every nonempty prefix of a list starts with x and continues with a prefix of a list from xs; prepending [] supplies the shortest prefix of a list. For tails, the longest suffix of a list is the whole input, followed by all suffixes of a list of its tail. For example, init [1,2] = [[],[1],[1,2]] and tails [1,2] = [[1,2],[2],[]]. Thus both include the empty list, andUse pattern matching on the two data constructors, concatenating the left subtree's inorder traversal, the root label, and the right subtree's inorder traversal:Here
data Tree = Leaf | Fork Tree Int Tree
inorder :: Tree -> [Int]
inorder Leaf = []
inorder (Fork l x r) = inorder l ++ [x] ++ inorder rLeaf represents the empty binary tree, rather than a labelled leaf. The printed declaration uses type; an executable Haskell algebraic data type requires data, as above. By structural induction, the two equations return precisely the prescribed inorder traversal:In the first expression, pattern matching discovers that the first tail is nonempty, so the ignored first label, the bottom value , is never demanded. The singleton
Cans 'A' Nil returns ZagType 'A'.In the second expression the tail is
Nal, so the result is ZigType applied to the bottom value. Under lazy evaluation this has a defined outer constructor and an undefined field: case lastZig (Cins undefined Nal) of ZigType _ -> True yields True, whereas extracting or fully printing the payload fails. Thus the result is , not a wholly undefined outer value.The original PDF really prints
Nil in the third expression. Cins requires a tail of type Zag b a, whereas Nil has type Zig c d. These distinct type constructors cannot be unified. There is no well-typed expression to evaluate, regardless of the undefined first label; replacing Nil by Nal would change the question.Use the usual left-to-right pattern matching order:
zip :: [a] -> [b] -> [(a,b)]
zip [] _ = []
zip _ [] = []
zip (x:xs) (y:ys) = (x,y) : zip xs ysThe result stops when either list ends, so its length on finite total inputs is the smaller input length. Elements remaining in the longer list are discarded.
Strictness is asymmetric. Writing the bottom value as ,
zip is strict in its first list but not uniformly strict in its second: zip undefined ys is undefined, whereas zip [] undefined = []. With a nonempty first list, an undefined second list does make the result undefined. Neither pair component is forced merely to produce the outer result constructor: zip [undefined] [undefined] already has a defined outer cons and pair. This follows from ordered pattern matching and lazy evaluation, not from the behavior on total lists alone.[ref]Haskell 2010 Language Report. §3.17.2, ordered pattern matching; Chapter 9, zip in the Standard Prelude
Use pattern matching and an optional value to distinguish absence from a present result:The representation precondition is that equal keys always have equal associated values, with a stable equality relation on keys. This permits harmless repeated identical pairs; it excludes conflicting outputs for one input.
class PartialFn[T, U](val data: List[(T, U)]) {
def get(x: T): Option[U] =
data.find { case (key, _) => key == x }.map(_._2)
def remove(x: T): PartialFn[T, U] =
new PartialFn[T, U](data.filterNot { case (key, _) => key == x })
def add(x: T, y: U): PartialFn[T, U] =
new PartialFn[T, U]((x, y) ::
data.filterNot { case (key, _) => key == x })
}get therefore returns Some(y) exactly when the represented partial function is defined at x. Removing filters out every occurrence of the key and leaves all other pairs unchanged. Adding first removes every old occurrence and prepends exactly the new pair, proving the partial function update postcondition even if the input contains duplicate identical pairs. The immutable linked list representation leaves the original mapping unchanged. Each operation has worst-case time complexity for pairs under constant-cost key equality.For every accumulator , pattern matching makes
foldl s a undefined the bottom value. Now use structural induction on a finite list prefix ending in the bottom value. The induction step is foldl s a (x:ps) = foldl s (s a x) ps; the induction hypothesis applies to every accumulator, including s a x. Consequently every such finite approximation has result .An infinite lazy list is the limit of its finite prefixes, each interpreted as a finite approximation of a lazy list. Definable Haskell operations are Scott continuous maps, so the result on the infinite lazy list is the supremum of the results on these approximations. Every result is , hence
foldl s a xs = undefined on every infinite input spine, irrespective of whether s would inspect the accumulator.For a concrete separation, choose
s = (&&), a = True, and xs = False : repeat True. The left fold is , whereas the right fold returns False after observing the first element: False && r does not demand r. It is reaching the end of the input spine, rather than strictness of the combining operation, that prevents this left fold from returning. Tagged union 2026-10-31
A data type whose values belong to one of several alternatives, with a logical tag distinguishing the chosen alternative. If
Number n and Operator op are its constructors, pattern matching can distinguish the variants even if their payloads use similar representations. The tag need not occupy a separate physical memory field in every implementation.