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Perhaps Not the Answer You Were Expecting but You Asked for It

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Perhaps Not the Answer You Were Expecting but You Asked for It Perhaps Not The Answer You Were Expecting But You Asked For It (An Accidental Blook) Conor McBride October 23, 2019 2 Contents 1 Haskell Curiosities 7 1.1 What is () in Haskell, exactly? . 7 1.2 What does () mean in Haskell? . 8 1.3 Implement the function lines in Haskell . 9 1.4 Boolean Expression evaluation (subtle type error) . 9 1.5 Purely functional data structures for text editors . 10 1.6 What are some motivating examples for Cofree CoMonad in Haskell? . 12 1.7 Find indices of things in lists . 15 1.8 How do I extend this mergeWords function to any number of strings? . 15 1.9 Example of UndecidableInstances yielding nonterminating typecheck . 16 1.10 Why do 3 and x (which was assigned 3) have different inferred types in Haskell? . 17 1.11 Use case for rank-3 (or higher) polymorphism? . 18 1.12 Why don’t Haskell compilers facilitate deterministic memory management? . 19 1.13 How does ArrowLoop work? Also, mfix? . 19 1.14 What does ) mean in a type signature? . 20 1.15 Meaning of Double and Floating point? . 21 1.16 Haskell terminology: meaning of type vs. data type, are they synonyms? . 22 1.17 Can you formulate the Bubble sort as a monoid or semigroup? . 23 1.18 Is this a correctly implemented mergesort in Haskell? . 24 1.19 Haskell type system nuances (ambiguity) . 26 1.20 Understanding a case of Haskell Type Ambiguity . 27 1.21 Why does Haskell use ! instead of =?.......................... 27 1.22 Is it possible to make a type an instance of a class if its type parameters are in the wrong order? . 28 1.23 Functor type variables for Flip data type . 29 1.24 Why does product [] return 1? . 30 1.25 Minimum of Two Maybes . 30 1.26 Is the equivalent of Haskell’s Foldable and Traversable simply a sequence in Clojure? 31 1.27 How do you keep track of multiple properties of a string without traversing it multiple times? . 32 1.28 Checking whether a binary tree is a binary search tree . 33 1.29 Finding a leaf with value x in a binary tree . 33 1.30 Taking from a list until encountering a duplicate . 34 1.31 RankNTypes and PolyKinds (quantifier alternation issues) . 35 1.32 How to write this case expression with the view pattern syntax? . 36 1.33 Recursive Type Families . 36 1.34 Determine whether a value is a function in Haskell . 38 1.35 Automatic Functor Instance (not) . 38 1.36 How do I apply the first partial function that works in Haskell? . 39 1.37 ‘Zipping’ a plain list with a nested list . 39 1.38 Bunched accumulations . 41 1.39 Functor on Phantom Type . 42 3 4 CONTENTS 1.40 Creating an Interpreter (with store) in Haskell . 43 1.41 Existential type wrappers necessity . 45 1.42 Non-linear Patterns in Type-Level Functions . 46 1.43 Initial algebra for rose trees . 46 2 Pattern Matching 49 2.1 Algorithm for typechecking ML-like pattern matching? . 49 2.2 Haskell Pattern Matching . 50 2.3 Complex pattern matching . 51 2.4 How to return an element before I entered? . 52 2.5 Buzzard Bazooka Zoom . 52 2.6 Why ++ is not allowed in pattern matching? . 54 3 Recursion 55 3.1 What are paramorphisms? . 55 3.2 Why can you reverse list with foldl, but not with foldr in Haskell . 56 3.3 Can fold be used to create infinite lists? . 59 3.4 How do I give a Functor instance to a datatype built for general recursion schemes? 59 3.5 Are there (term-transforming) morphisms in Haskell? . 61 3.6 Is this Fibonacci sequence function recursive? . 62 3.7 Can someone explain this lazy Fibonacci solution? . 63 3.8 Monoidal folds on fixed points . 64 3.9 List Created Evaluating List Elements . 65 3.10 Functions of GADTs . 66 4 Applicative Functors 69 4.1 Where to find programming exercises for applicative functors? . 69 4.2 N-ary tree traversal . 71 4.2.1 First Attempt: Hard Work . 71 4.2.2 Second Attempt: Numbering and Threading . 72 4.2.3 Third Attempt: Type-Directed Numbering . 74 4.2.4 Eventually. 75 4.3 Partial application of functions and currying, how to make a better code instead of a lot of maps? . 76 4.4 Translating monad to applicative . 77 4.5 Applicatives compose, monads don’t . 78 4.6 Examples Separating Functor, Applicative and Monad . 79 4.7 Parsec: Applicatives vs Monads . 80 4.8 Refactoring do notation into applicative style . 81 4.9 Zip with default values instead of dropping values? . 82 4.10 sum3 with zipWith3 in Haskell . 82 4.11 What is the ’Const’ applicative functor useful for? . 83 4.12 Applicative instance for free monad . 83 4.13 Examples of a monad whose Applicative part can be better optimized than the Monad part . 84 4.14 How arbitrary is the “ap” implementation for monads? . 85 4.15 Applicative without a functor (for arrays) . 85 4.16 Does this simple Haskell function already have a well-known name? (strength) . 86 4.17 The Kids are all Right . 87 4.18 Why is((,) r) a Functor that is NOT an Applicative? . 87 4.19 Applicative Rewriting (for reader) . 88 4.20 Serialised Diagonalisation . 89 4.21 Applicative style for infix operators? . 89 4.22 Where is the Monoid in Applicative? . 89 CONTENTS 5 4.23 Applicatives from Monoids including min and max . 93 5 Monads 95 5.1 Why we use monadic functions a ! m b ......................... 95 5.2 Monads with Join instead of Bind . 96 5.3 Using return versus not using return in the list monad . 97 5.4 Example showing monads don’t compose . 98 5.5 The Pause monad . 98 5.6 Haskell monad return arbitrary data type . 100 5.7 Should I avoid using Monad fail? . 100 5.8 Why isn’t Kleisli an instance of Monoid? . 101 5.9 Monads at the prompt? . 102 5.10 Is this a case to use liftM? . 102 5.11 Zappy colists do not form a monad . 104 5.12 Haskell io-streams and forever produces no output to stdout . 104 6 Differential Calculus for Types 105 6.1 Find the preceding element of an element in list . 105 6.2 Splitting a List . 106 6.3 nub as a List Comprehension . 108 6.4 How to make a binary tree zipper an instance of Comonad? . 109 6.5 What’s the absurd function in Data.Void useful for? . 115 6.6 Writing cojoin or cobind for n-dimensional grids . 117 6.6.1 Cursors in Lists . 117 6.6.2 Composing Cursors, Transposing Cursors? . 118 6.6.3 Hancock’s Tensor Product . 120 6.6.4 InContext for Tensor Products . 121 6.6.5 Naperian Functors . 121 6.7 Zipper Comonads, Generically . 122 6.8 Traversable and zippers: necessity and sufficiency . ..
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