\nNotes

Optics

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\n 1 Obligatory Preamble\n
- \n 1.1 Why should I read this book?\n
- \n 1.2 How to read this book\n
- \n 1.3 Chosen language and optics encodings\n
- \n 1.4 Your practice environment\n
- \n 1.5 Following examples\n
- \n 1.6 About the type signatures\n
- \n 1.7 About the exercises\n
\n
\n 2 Optics\n
- \n 2.1 What are optics?\n
- \n 2.2 Strengths\n
- \n 2.3 Weaknesses\n
- \n 2.4 Practical optics at a glance\n
- \n 2.5 Impractical optics at a glance\n
\n
\n 3 Lenses\n
- \n 3.1 Introduction to Lenses\n
  - \n Anatomy\n
  - \n Exercises – Optic Anatomy\n
  \n
- \n 3.2 Lens actions\n
  - \n Viewing through lenses\n
  - \n Setting through a lens\n
  - \n Exercises - Lens Actions\n
  \n
- \n 3.3 Lenses and records\n
  - \n Lenses subsume the “accessor” pattern\n
  - \n Building a lens for a record field\n
  - \n Exercises - Records Part One\n
  - \n Getting and setting with a field lens\n
  - \n Modifying fields with a lens\n
  - \n Automatically generating field lenses\n
  - \n Exercises - Records Part Two\n
  \n
- \n 3.4 Limitations\n
  - \n Is it a Lens?\n
  - \n Is it a Lens? – Answers\n
  \n
- \n 3.5 Lens Laws\n
  - \n Why do optics have laws?\n
  - \n The Laws\n
  - \n Case Study: _1\n
  - \n Case Study: msg\n
  - \n Case Study: lensProduct\n
  - \n Exercises - Laws\n
  \n
- \n 3.6 Virtual Fields\n
  - \n What’s a virtual field\n
  - \n Writing a virtual field\n
  - \n Breakage-free refactoring\n
  - \n Exercises – Virtual Fields\n
  \n
- \n 3.7 Data correction and maintaining invariants\n
  - \n Including correction logic in lenses\n
  - \n Exercises – Self-Correcting Lenses\n
  \n
\n
\n 4 Polymorphic Optics\n
- \n 4.1 Introduction to polymorphic optics\n
  - \n Simple vs Polymorphic optics\n
  \n
- \n 4.2 When do we need polymorphic lenses\n
  - \n Type-changing focuses\n
  - \n Changing type variables with polymorphic lenses\n
  - \n Exercises – Polymorphic Lenses\n
  \n
- \n 4.3 Composing Lenses\n
  - \n How do I update fields in deeply nested records?\n
  - \n Composing update functions\n
  - \n Composing Lenses\n
  - \n How do Lens Types Compose?\n
  - \n Exercises – Lens Composition\n
  \n
\n
\n 5 Operators\n
- \n 5.1 Lens Operators\n
- \n 5.2 view a.k.a. ^.\n
- \n 5.3 set a.k.a. .~\n
- \n 5.4 Chaining many operations\n
- \n 5.5 Using %~ a.k.a. over\n
- \n 5.6 Learning Hieroglyphics\n
- \n 5.7 Modifiers\n
- \n 5.8 When to use operators vs named actions?\n
- \n 5.9 Exercises – Operators\n
\n
\n 6 Folds\n
- \n 6.1 Introduction to Folds\n
  - \n Focusing all elements of a container\n
  - \n Collapsing the Set\n
  - \n Collecting focuses as a list\n
  - \n Using lenses as folds\n
  - \n Composing folds\n
  - \n Foundational fold combinators\n
  - \n Exercises – Simple Folds\n
  \n
- \n 6.2 Custom Folds\n
  - \n Mapping over folds\n
  - \n Combining multiple folds on the same structure\n
  - \n Exercises – Custom Folds\n
  \n
- \n 6.3 Fold Actions\n
  - \n Writing queries with folds\n
  - \n Queries case study\n
  - \n Folding with effects\n
  - \n Combining fold results\n
  - \n Using ‘view’ on folds\n
  - \n Customizing monoidal folds\n
  - \n Exercises – Fold Actions\n
  \n
- \n 6.4 Higher Order Folds\n
  - \n Taking, Dropping\n
  - \n Backwards\n
  - \n TakingWhile, DroppingWhile\n
  - \n Exercises – Higher Order Folds\n
  \n
- \n 6.5 Filtering folds\n
  - \n Filtered\n
  - \n Exercises – Filtering\n
  \n
- \n 6.6 Fold Laws\n
\n
\n 7 Traversals\n
- \n 7.1 Introduction to Traversals\n
  - \n How do Traversals fit into the hierarchy?\n
  - \n A bit of Nostalgia\n
  - \n From fold to traversal\n
  \n
- \n 7.2 Traversal Combinators\n
  - \n Traversing each element of a container\n
  - \n More Combinators\n
  - \n Traversing multiple paths at once\n
  - \n Focusing a specific traversal element\n
  \n
- \n 7.3 Traversal Composition\n
  - \n Exercises – Simple Traversals\n
  \n
- \n 7.4 Traversal Actions\n
  - \n A Primer on Traversable\n
  - \n Traverse on Traversals\n
  - \n Infix traverseOf\n
  - \n Using Traversals directly\n
  - \n Exercises – Traversal Actions\n
  \n
- \n 7.5 Custom traversals\n
  - \n Optics look like traverse\n
  - \n Our first custom traversal\n
  - \n Traversals with custom logic\n
  - \n Case Study: Transaction Traversal\n
  - \n Exercises – Custom Traversals\n
  \n
- \n 7.6 Traversal Laws\n
  - \n Law One: Respect Purity\n
  - \n Law Two: Consistent Focuses\n
  - \n Good Traversal Bad Traversal\n
  - \n Exercises – Traversal Laws\n
  \n
- \n 7.7 Advanced manipulation\n
  - \n partsOf\n
  - \n Polymorphic partsOf\n
  - \n partsOf and other data structures\n
  - \n Exercises – partsOf\n
  \n
\n
\n 8 Indexable Structures\n
- \n 8.1 What’s an “indexable” structure?\n
- \n 8.2 Accessing and updating values with ‘Ixed’\n
  - \n The Ixed Class\n
  - \n Accessing and setting values with ix\n
  - \n Indexed Structures\n
  - \n Indexing monomorphic types\n
  - \n Indexing stranger structures\n
  \n
- \n 8.3 Inserting & Deleting with ‘At’\n
  - \n Map-like structures\n
  - \n Manipulating Sets\n
  - \n Exercises – Indexable Structures\n
  \n
- \n 8.4 Custom Indexed Data Structures\n
  - \n Custom Ixed: Cyclical indexing\n
  - \n Custom At: Address indexing\n
  - \n Exercises – Custom Indexed Structures\n
  \n
- \n 8.5 Handling missing values\n
  - \n Checking whether updates succeed\n
  - \n Fallbacks with ‘failing’\n
  - \n Default elements\n
  - \n Checking fold success/failure\n
  - \n Exercises – Missing Values\n
  \n
\n
\n 9 Prisms\n
- \n 9.1 Introduction to Prisms\n
  - \n How do Prisms fit into the hierarchy?\n
  - \n Simple Pattern-Matching Prisms\n
  - \n Checking pattern matches with prisms\n
  - \n Generating prisms with makePrisms\n
  - \n Embedding values with prisms\n
  - \n Other types of patterns\n
  - \n Exercises – Prisms\n
  \n
- \n 9.2 Writing Custom Prisms\n
  - \n Rebuilding _Just and _Nothing\n
  - \n Matching String Prefixes\n
  - \n Cracking the coding interview: Prisms style!\n
  - \n Exercises – Custom Prisms\n
  \n
- \n 9.3 Laws\n
  - \n Law One: Review-Preview\n
  - \n Law Two: Prism Complement\n
  - \n Law Three: Pass-through Reversion\n
  - \n Summary\n
  - \n Exercises – Prism Laws\n
  \n
- \n 9.4 Case Study: Simple Server\n
  - \n Path prefix matching\n
  - \n Altering sub-sets of functions\n
  - \n Matching on HTTP Verb\n
  \n
\n
\n 10 Isos\n
- \n 10.1 Introduction to Isos\n
  - \n How do Isos fit into the hierarchy?\n
  - \n There and back again\n
  \n
- \n 10.2 Building Isos\n
- \n 10.3 Flipping isos with from\n
- \n 10.4 Modification under isomorphism\n
- \n 10.5 Varieties of isomorphisms\n
  - \n Composing isos\n
  - \n Exercises – Intro to Isos\n
  \n
- \n 10.6 Projecting Isos\n
  - \n Exercises – Projected Isos\n
  \n
- \n 10.7 Isos and newtypes\n
  - \n Coercing with isos\n
  - \n Newtype wrapper isos\n
  \n
- \n 10.8 Laws\n
  - \n The one and only law: Reversibility\n
  - \n Exercises – Iso Laws\n
  \n
\n
\n 11 Indexed Optics\n
- \n 11.1 What are indexed optics?\n
- \n 11.2 Index Composition\n
  - \n Custom index composition\n
  - \n Exercises – Indexed Optics\n
  \n
- \n 11.3 Filtering by index\n
  - \n Exercises – Index Filters\n
  \n
- \n 11.4 Custom indexed optics\n
  - \n Custom IndexedFolds\n
  - \n Custom IndexedTraversals\n
  - \n Index helpers\n
  - \n Exercises – Custom Indexed Optics\n
  \n
- \n 11.5 Index-preserving optics\n
\n
\n 12 Dealing with Type Errors\n
- \n 12.1 Interpreting expanded optics types\n
- \n 12.2 Type Error Arena\n
  - \n First Foe: Level 1 Lenslion\n
  - \n Level 2 Tuplicant\n
  - \n Level 3 Settersiren\n
  - \n Level 4 Composicore\n
  - \n Level 5 Foldasaurus\n
  - \n Level 6 Higher Order Beast\n
  - \n Level 7 Traversacula\n
  \n
\n
\n 13 Optics and Monads\n
- \n 13.1 Reader Monad and View\n
- \n 13.2 State Monad Combinators\n
- \n 13.3 Magnify & Zoom\n
\n
\n 14 Classy Lenses\n
- \n 14.1 What are classy lenses and when do I need them?\n
  - \n No duplicate record fields\n
  - \n Separating logic and minimizing global knowledge\n
  - \n Granular dependencies with makeFields\n
  - \n Field requirements compose\n
  \n
- \n 14.2 makeFields vs makeClassy\n
\n
\n 15 JSON\n
- \n 15.1 Introspecting JSON\n
- \n 15.2 Diving deeper into JSON structures\n
- \n 15.3 Traversing into multiple JSON substructures\n
  - \n Traversing Arrays\n
  - \n Traversing Objects\n
  \n
- \n 15.4 Filtering JSON Queries\n
- \n 15.5 Serializing & Deserializing within an optics path\n
- \n 15.6 Exercises: Kubernetes API\n
  - \n BONUS Questions\n
  \n
\n
\n 16 Uniplate - Manipulating recursive data\n
- \n 16.1 A Brief History\n
- \n 16.2 Control.Lens.Plated\n
  - \n Children\n
  - \n Rewrite\n
  - \n Universe\n
  - \n Transform\n
  - \n Deep\n
  \n
- \n 16.3 Overriding plate\n
- \n 16.4 The magic of biplate\n
- \n 16.5 Exercises – Uniplate\n
\n
\n 17 generic-lens\n
- \n 17.1 Generic Lenses\n
  - \n Record Fields\n
  - \n Positional Lenses\n
  - \n Typed Lenses\n
  - \n Altogether Now\n
  - \n Subtype Lenses\n
  \n
- \n 17.2 Generic Traversals\n
  - \n Types Traversal\n
  - \n Parameter Traversals\n
  \n
- \n 17.3 Generic Prisms\n
  - \n Constructor Prisms\n
  - \n Typed Prisms\n
  \n
- \n 17.4 Exercises – Generic Lens\n
\n
\n 18 Appendices\n
- \n 18.1 Optic Composition Table\n
- \n 18.2 Optic Compatibility Chart\n
- \n 18.3 Operator Cheat Sheet\n
  - \n Legend for Getters\n
  - \n Legend for Setters/Modifiers\n
  \n
- \n 18.4 Optic Ingredients\n
\n
\n 19 Answers to Exercises\n
- \n 19.1 Optic Anatomy\n
- \n 19.2 Lens Actions\n
- \n 19.3 Records Part One\n
- \n 19.4 Records Part Two\n
  - \n Laws\n
  \n
- \n 19.5 Virtual Fields\n
- \n 19.6 Self-Correcting Lenses\n
- \n 19.7 Polymorphic Lenses\n
- \n 19.8 Lens Composition\n
- \n 19.9 Operators\n
- \n 19.10 Simple Folds\n
- \n 19.11 Writing Custom Folds\n
- \n 19.12 Querying Using Folds\n
- \n 19.13 Higher Order Folds\n
- \n 19.14 Filtering\n
- \n 19.15 Simple Traversals\n
- \n 19.16 Traversal Actions\n
- \n 19.17 Custom Traversals\n
- \n 19.18 Traversal Laws\n
- \n 19.19 partsOf\n
- \n 19.20 Indexable Structures\n
- \n 19.21 Custom Indexed Structures\n
- \n 19.22 Missing Values\n
- \n 19.23 Prisms\n
- \n 19.24 Custom Prisms\n
- \n 19.25 Prism Laws\n
- \n 19.26 Intro to Isos\n
- \n 19.27 Projected Isos\n
- \n 19.28 Iso Laws\n
- \n 19.29 Indexed Optics\n
- \n 19.30 Index Filters\n
- \n 19.31 Custom Indexed Optics\n
- \n 19.32 Type Errors\n
  - \n First Foe: Level 1 Lenslion\n
  - \n Level 2 Tuplicant\n
  - \n Level 3 Settersiren\n
  - \n Level 4 Composicore\n
  - \n Level 5 Foldasaurus\n
  - \n Level 6 Foldasaurus\n
  - \n Level 7 Traversacula\n
  \n
- \n 19.33 Kubernetes API\n
  - \n BONUS Questions\n
  \n
- \n 19.34 Uniplate\n
- \n 19.35 Generic Lens\n
\n
\n 20 Thanks\n
- \n 20.1 Patreon Supporters\n
- \n 20.2 Book Cover\n
\n
\n Notes\n

\n1 Obligatory Preamble

\n\n\n

This content is not available in the sample book. The book can be purchased on Leanpub at http://leanpub.com/optics-by-example.

\n\n

I’ll take just a moment to expand on a few strengths:

\n\n

Composition: Each optic focuses on some subset of data, when composing with other optics they can pick up from where the previous optic left off and dive down even further. This means that each optic you learn becomes a member of your growing vocabulary of optics. Just as words in natural language can be strung together to form a sentence which communicates any intent you might want, a sufficiently complete vocabulary of optics can be arranged to effectively manipulate data to achieve your goals.
Separation of concerns: Optics are the abstraction most programmers didn’t know they needed. They allow us to cleanly separate concerns in stronger ways than either of Object-Oriented or Functional-Programming styles allow on their own. Optics allow us to specify which portions of data we wish to work with separately from the operations we wish to perform on them. We could, for example, encode a pre-order traversal of a tree structure, and combine it with a behaviour which prints the elements. We can swap out either of the data-selector or the action without affecting the other. Say goodbye to the Visitor Pattern!
Concision: Although there are many other good reasons to love optics, they have the nice property of being very succinct. Most tasks can be expressed in a single line of code, and in many cases the resulting code even reads like a simple sentence. For instance: sumOf (key \"transactions\" . values . key \"cost\" . _Number) will accept a JSON blob, will look into the “transactions” key, will then dive into each of the elements of the array, and in each of those objects will collect the “cost” of the transaction as a number, summing them all up into a total. In a typical imperative language this operation would likely take a few lines of code, or would at the very least require some ugly nested brackets. If we wish to instead take the average of these numbers we need only swap the sumOf action without fussing about with any variables and loops.
Enforcing interface boundaries: Optics can serve as an external interface which remains consistent despite changes to your data layer. They provide an abstraction layer similar to getters & setters which one might write in Java or Python. This allows you to alter your underlying data structures without breaking external consumers. You can even enforce data-consistency invariants when both getting and setting values! This replaces the idea of class-based “getters” and “setters” while also covering significant ground which used to require interfaces.
A principled and mature ecosystem: Optics have been around for long enough now that the ecosystem has ironed out most bugs and performance issues. There are a wide variety of libraries available, and many popular libraries provide optics-based interfaces (e.g. there are optics wrappers around JSON and XML libraries). Optics have a simple universal construction with the surprising benefit that writing an optics combinator does not require a dependency on any optics libraries! This allows us to use optics as a primitive building block or interface across libraries without worrying about large transitive or cyclic dependencies.

\n\n

Hopefully all of this is sounding “too good to be true”! I assure you that optics can deliver on these promises. I can also guarantee that it’ll take a little work and more than a few terrible, horrible, no good, very bad type errors to get there, but we’re all in this together!

\n\n

\n2.3 Weaknesses

\n\n

Can’t always have your cake and eat it too; here are a few areas where optics aren’t perfect yet:

\n\n

Type Errors: Most optics libraries (especially lens) can spew out some pretty ugly type errors when something goes wrong. This is one of the most common complaints, however it’s a problem which is not easily solved. A great deal of type-level complexity is required to keep these libraries polymorphic and performant! We’ll approach new types carefully and will address some common mistakes as well as talking about how to read these terrible beasts, so hopefully we can mitigate this one slightly.
Complex Implementation: Most (all?) optics implementations have their fair share of magic (see: complex category theory and/or dirty hacks) going on behind the scenes. To make matters worse, most libraries are implemented in completely different ways! The Scala implementation is different from the Haskell implementation which is different from the Purescript implementation! They all follow a lot of the same theories and encode the same concepts, but a perfect backing implementation hasn’t been discovered yet, though the profunctor encoding is looking pretty good so far. Luckily most libraries provide helpers which abstract over the underlying implementation so you won’t typically need to worry about it.
Vast collection of combinators: This is one of those “weaknesses” you put on your CV that’s actually a strength in disguise. There are a LOT of helpers and combinators provided in most optics libraries, it’s overwhelming at first, but you’ll learn how to search through them and better find the ones you need; and when you can do that effectively it means you’ll usually be able to find a helper for performing almost any optics task! Just have a little patience, and finish reading this book of course!

\n\n

\n2.4 Practical optics at a glance

\n\n

I’ve talked at a high level about how optics help you perform actions over portions of data. Simple actions you can perform involve variants of viewing, modifying or traversing the selected data.

\n\n

Here are a few examples of varying difficulty and usefulness which represent a few things we’ll see:

\n\n\n

-- View nested fields of some record type\n>>> view (address . country) person\n\"Canada\"\n\n-- Update portions of immutable data structures\n>>> set _3 False ('a', 'b', 'c')\n('a', 'b', False)\n\n-- These selectors compose! \n-- We can perform a task over deeply nested subsets of data.\n-- Let's sum all numbers wrapped in a 'Left' within the right half of each tuple\n>>> sumOf (folded . _2 . _Left) \n      [(True, Left 10), (False, Right \"pepperoni\"), (True, Left 20)]\n30\n\n-- Truncate any stories longer than 10 characters, leaving shorter ones alone.\n>>> let stories = \n      [\"This one time at band camp\", \"Nuff said.\", \"This is a short story\"]\n>>> over\n      (traversed . filtered ((>10) . length))\n      (\\story -> take 10 story ++ \"...\")\n      stories \n[\"This one t...\",\"Nuff said.\",\"This is a ...\"]\n

\n\n\n\n\n

\n2.5 Impractical optics at a glance

\n\n

Here are a few of the more arcane and interesting things optics can do. It’s not important that you understand how these work or what they’re doing, they’re just here to help demonstrate the sheer adaptability of optics. Note how each operation is only one line of code! Hopefully they spark a bit of curiosity!

\n\n\n

-- Summarize a list of numbers, subtracting the 'Left's, adding the 'Right's!\n>>> import Numeric.Lens (negated)\n>>> sumOf (folded . beside negated id) [Left 1, Right 10, Left 2, Right 20]\n27\n\n-- Capitalize each word in a sentence\n>>> \"why is a raven like a writing desk\" & worded . _head %~ toUpper\n\"Why Is A Raven Like A Writing Desk\"\n\n-- Multiply every Integer by 100 no matter where they are in the structure:\n>>> import Data.Data.Lens (biplate)\n>>> (Just 3, Left (\"hello\", [13, 15, 17])) & biplate *~ 100\n(Just 300,Left (\"hello\",[1300,1500,1700]))\n\n-- Reverse the ordering of all even numbers in a sequence.\n-- We leave the odd numbers alone!\n>>> [1, 2, 3, 4, 5, 6, 7, 8] & partsOf (traversed . filtered even) %~ reverse\n[1,8,3,6,5,4,7,2]\n\n-- Sort all the characters in all strings, across word boundaries!\n>>> import Data.List (sort)\n>>> (\"one\", \"two\", \"three\") & partsOf (each . traversed) %~ sort\n(\"eee\",\"hno\",\"orttw\")\n\n-- Flip the 2nd bit of each number\n>>> import Data.Bits.Lens (bitAt)\n>>> [1, 2, 3, 4] & traversed . bitAt 1 %~ not\n[3,0,1,6]\n\n-- Prompt the user with each question in a tuple,\n-- then return the tuple with each prompt replaced with the user's input,\n>>> let prompts = ( \"What is your name?\"\n                  , \"What is your quest?\"\n                  , \"What is your favourite color?\"\n                  ) \n>>> prompts & each  %%~ (\\prompt -> putStrLn prompt >> getLine)\nWhat is your name?\n> Sir Galahad\nWhat is your quest?\n> To seek the holy grail\nWhat is your favourite color?\n> Blue I think?\n(\"Sir Galahad\",\"To seek the holy grail\",\"Blue I think?\")\n

\n\n\n\n\n

I hope that was a sufficiently strange list of examples to spark some wonder and creativity. These were meant to show the versatility, expressivity, and concision of optics! These examples are contrived and complex of course, but we’ll see some more practical examples as we go on.

\n\n\n

\n3 Lenses

\n\n\n

This content is not available in the sample book. The book can be purchased on Leanpub at http://leanpub.com/optics-by-example.

\n\n\n

\n3.1 Introduction to Lenses

\n\n

We’ll start our journey with lenses!

\n\n

I mentioned in the optics section that optics allow us to separate concerns; i.e. split up the action we perform on data from the selection of data we want to perform it on. To be clear I’ll refer to operations which can be performed on data as actions, whereas the data selectors are the actual optics. Each type of optic comes with a set of compatible actions.

\n\n

Each type of optic has a different balance of constraint vs flexibility, moving to and fro on this spectrum results in several different but useful behaviours. Lenses lean closer to the constrained side of things, which means you have a lot of guarantees about their behaviour, but also means that you need to prove those guarantees to make a lens, so there are fewer lenses in the world than there are of the more flexible optics.

\n\n

Lenses have the following concrete guarantees:

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A Lens focuses (i.e. selects) a single piece of data within a larger structure.
A Lens must never fail to get or modify that focus.

\n\n

These constraints unlock a few actions we can perform on lenses:

\n\n

We can use a lens to view the focus within a structure.
We can use a lens to set the focus within a structure.
We can use a lens to modify the focus within a structure.

\n\n

Before we talk too much at a high level, let’s take a look at a concrete usage of a lens and understand the different parts.

\n\n

\nAnatomy

\n\n

Here’s a simple snippet which gets the String “hello” out from a couple nested tuples:

\n\n\n

>>> view (_2 . _1) (42, (\"hello\", False))\n\"hello\"\n

\n\n\n\n\n

We won’t worry about exactly what it’s doing or how it works yet; for now we’ll pick out and name individual pieces of this construction so that I can save myself some typing for the rest of the book.

\n\n

Let’s break it down into its anatomy:

\n\n\n\n

      +-> The Action         +-> The Structure\n      |                        |\n    /-+-\\           /----------+----------\\\n>>> view  (_2 . _1) (42, (\"hello\", False)))\n          \\---+---/       \\--+--/\n              |              |\n              |              +-> The Focus\n              +-> The Path  \n

\n\n\n\n\n

Note that the names for these things aren’t really standardized yet, so you may have poor luck searching for them on Google; but if you start using them confidently around the water cooler I’m sure they’ll catch on eventually.

\n\n

The Action™: An action executes some operation over the focus of a path. E.g. view is an action which gets the focus of a path from a structure. Actions are often written as an infix operator; e.g. %~, ^. or even <<%@=!
The Path™: The path indicates which data to focus and where to find it within the structure. A path can be a single optic, or several optics chained together through composition. If you consider dot-notation from most Object-Oriented languages you’ll see similarities.
The Structure™: The structure is the hunk of data that we want to work with. The path selects data from within the structure, and that data will be passed to the action.
The Focus™: The smaller piece of the structure indicated by the path. The focus will be passed to the action. E.g. we may want to get, set, or modify the focus.

\n\n

\nExercises – Optic Anatomy

\n\n

Jump to answers

\n\n

For each of the following, identify the action, path and structure, don’t worry about understanding how they actually work just yet. If you want a real challenge, try to identify the focus too! Note that certain optics allow multiple focuses, and some actions accept parameters other than the focus.

\n\n\n

>>> view (_1 . _2)  ((1, 2), 3)\n2\n\n>>> set (_2 . _Left) \"new\" (False, Left \"old\")\n(False, Left \"new\")\n\n>>> over (taking 2 worded . traversed) toUpper \"testing one two three\"\n\"TESTING ONE two three\"\n\n>>> foldOf (both . each) ([\"super\", \"cali\"],[\"fragilistic\", \"expialidocious\"])\n\"supercalifragilisticexpialidocious\"\n

\n\n\n

\nAutomatically generating field lenses

\n\n

Writing our field accessors manually taught us about the relationship between lenses, getters, and setters, but writing them by hand is mechanical, boring, and error-prone! Did I hear someone yell boilerplate from the back!? There’s only one correct way to get or set a record field, let’s let the computer figure it out for us.

\n\n

We can use Template Haskell to write our lenses for us! It’s basically a macro system for generating Haskell code. To use it we’ll need to enable the GHC extension by adding the following pragma to the top of our Haskell module:

\n\n\n

{-# LANGUAGE TemplateHaskell #-}\n

\n\n\n\n\n

With that enabled, we can add the appropriate Template Haskell expression right after our data declaration:

\n\n\n

data Ship =\n    Ship { _name    :: String\n         , _numCrew :: Int\n         }\n    deriving (Show)\n\nmakeLenses ''Ship\n

\n\n\n\n\n\n\n

This will generate the appropriate lens for each field in our Ship record type. By default makeLenses chooses names for the lenses by stripping the leading underscore _ from the field name. It’ll generate the exact same lens we wrote by hand and will even have the same name! You’ll need to delete, move, or rename your lens for the numCrew field if you still have that sitting around.

\n\n

Note that it won’t generate lenses for fields that aren’t named with underscores, so don’t forget to add it!

\n\n

\n\n

I mentioned earlier how lenses subsume the Accessor Pattern from Object-Oriented programming. We’ve already seen how lenses take care of the getters and setters for record fields, this chapter covers how to represent “virtual fields” with lenses.

\n\n

\nWhat’s a virtual field

\n\n

I’m using the term virtual field to mean any conceptual piece of data that doesn’t exist as an actual field in your record definition. These are sometimes called “computed properties” or “managed attributes” in lanuages like Java, Python, etc. They’re often used to present the data from concrete fields in a more convenient or enriched way. Sometimes they combine several concrete fields together, other times they’re just used to avoid breaking changes when refactoring the structure of the record.

\n\n

At the end of the day; they’re really just normal lenses! Let’s look at a few examples.

\n\n

\nWriting a virtual field

\n\n

For a simple example let’s look at the following type:

\n\n\n

data Temperature =\n    Temperature { _location :: String\n                , _celsius  :: Float\n                }\n    deriving (Show)\n

\n\n\n\n\n

This generates the field lens:

\n\n\n

celsius :: Lens' Temperature Float\n

\n\n\n\n\n

Which we can use to get or set the temperature in celsius.

\n\n\n

>>> let temp = Temperature \"Berlin\" 7.0\n>>> view celsius temp\n7.0\n\n>>> set celsius 13.5 temp\nTemperature {_location = \"Berlin\", _celsius = 13.5}\n\n-- Bump the temperature up by 10 degrees Celsius\n>>> over celsius (+10) temp\nTemperature {_location = \"Berlin\", _celsius = 17.0}\n

\n\n\n\n\n

But what about our American colleagues who’d prefer Fahrenheit? It’d be easy enough to write a function which convert Celsius to Fahrenheit and call that on the result, but you’d still need to set new temperatures using Celsius!

\n\n

First we’ll define our conversion functions back and forth, nothing too interesting there:

\n\n\n

celsiusToFahrenheit :: Float -> Float\ncelsiusToFahrenheit c = (c * (9/5)) + 32\n\nfahrenheitToCelsius :: Float -> Float\nfahrenheitToCelsius f = (f - 32) * (5/9)\n

\n\n\n\n\n

Here’s how we could get and set using Fahrenheit:

\n\n\n

>>> let temp = Temperature \"Berlin\" 7.0\n>>> celsiusToFahrenheit . view celsius temp\n44.6\n\n>>> set celsius (fahrenheitToCelsius 56.3) temp\nTemperature {_location = \"Berlin\", _celsius = 13.5}\n\n-- Bump the temp by 18 degrees Fahrenheit\n>>> over celsius (fahrenheitToCelsius . (+18) . celsiusToFahrenheit) temp\nTemperature {_location = \"Berlin\", _celsius = 17.0}\n

\n\n\n\n\n

The first two aren’t too bad, but the over example is getting a bit clunky and error prone!

\n\n

If we instead encode the Fahrenheit version of the temperature as a virtual field we gain better usability, cleaner code, and avoid a lot of possible mistakes.

\n\n

Now for the fun part! We can write a fahrenheit lens using the existing celsius lens! We simply convert back and forth when getting and setting.

\n\n\n

fahrenheit :: Lens' Temperature Float\nfahrenheit = lens getter setter\n  where\n    getter = celsiusToFahrenheit . view celsius\n    setter temp f = set celsius (fahrenheitToCelsius f) temp\n

\n\n\n\n\n

Look how it cleans up the call site:

\n\n\n

>>> let temp = Temperature \"Berlin\" 7.0\n>>> view fahrenheit temp\n44.6\n\n>>> set fahrenheit 56.3 temp\nTemperature {_location = \"Berlin\", _celsius = 13.5}\n\n>>> over fahrenheit (+18) temp\nTemperature {_location = \"Berlin\", _celsius = 17.0}\n

\n\n\n\n\n

Much cleaner! Even though our Temperature record doesn’t have a field for Fahrenheit we faked it using lenses to create a virtual field!

\n\n

\nBreakage-free refactoring

\n\n

Another great benefit of using lenses instead of field accessors for interacting with our data is that we gain more freedom when refactoring. To continue with the Temperature example, let’s say as we’ve developed our wonderful weather app further we’ve discovered that Kelvin is a much better canonical representation for temperature data. We’d love to swap our _celsius field for a _kelvin field instead.

\n\n

We’ll consider two possible universes, in one this book was never written, so we didn’t use lenses to access our fields. In the other (the one you’re living in) we finished the book and decided to use lenses as our external interface instead.

\n\n

\nThe universe without lenses

\n\n

In the sad universe without lenses we had the following code scattered throughout our app:

\n\n\n

updateTempReading :: Temperature -> IO Temperature\nupdateTempReading temp = do\n  newTempInCelsius <- readTemp\n  return temp{_celsius=newTempInCelsius}\n

\n\n\n\n\n

Then we refactored our Temperature object to the following:

\n\n\n

data Temperature =\n    Temperature { _location :: String\n                , _kelvin  :: Float\n                }\n    deriving (Show)\n

\n\n\n\n\n

And unfortunately every file that used record update syntax now fails to compile, because the _celsius field no longer exists. If we had instead used pattern matching, the situation would be even worse:

\n\n\n

updateTempReading :: Temperature -> IO Temperature\nupdateTempReading (Temperature location _) = do\n  newTempInCelsius <- readTemp\n  return (Temperature location newTempInCelsius)\n

\n\n\n\n\n

In this case the code will still compile, but we’ve completely switched units, this will behave completely incorrectly!

\n\n

\nThe glorius utopian lenses universe

\n\n

Come with me now to the happy universe. In this universe we decided to use lenses as our interface for interacting with Temperatures, meaning we didn’t expose the field accessors and thus disallowed fragile record-update syntax. We used the celsius lens to perform the update instead:

\n\n\n

updateTempReading :: Temperature -> IO Temperature\nupdateTempReading temp = do\n  newTempInCelsius <- readTemp\n  return $ set celsius newTempInCelsius temp\n

\n\n\n\n\n

Now when we refactor, we can simply export a replacement celsius lens in place of the old one:

\n\n\n

data Temperature =\n    Temperature { _location :: String\n                , _kelvin  :: Float\n                }\n    deriving (Show)\nmakeLenses ''Temperature\n\ncelsius :: Lens' Temperature Float\ncelsius = lens getter setter\n  where\n    getter = (subtract 273.15) . view kelvin\n    setter temp c = set kelvin (c + 273.15) temp\n

\n\n\n\n\n

By adding the replacement lens we avoid breaking any external users of the type! Even our fahrenheit lens was defined in terms of celsius, so it will continue to work perfectly.

\n\n

This is a simple example, but this idea works for more complex refactorings as well. When adopting this style it’s important to avoid exporting the data type constructor or field accessors. Export a “smart constructor” function and the lenses for each field instead.

\n\n

\nExercises – Virtual Fields

\n\n

Jump to answers

\n\n

Consider this data type for the following exercises:

\n\n\n

data User =\n  User { _firstName :: String\n       , _lastName :: String\n       , _username :: String\n       , _email :: String\n       } deriving (Show)\nmakeLenses ''User\n

\n\n\n\n\n

We’ve decided we’re no longer going to have separate usernames and emails; now the email will be used in place of a username. Your task is to delete the _username field and write a replacement username lens which reads and writes from/to the _email field instead. The change should be unnoticed by those importing the module.
Write a lens for the user’s fullName. It should append the first and last names when “getting”. When “setting” treat everything till the first space as the first name, and everything following it as the last name.

\n\n

It should behave something like this:

\n\n\n

>>> let user = User \"John\" \"Cena\" \"invisible@example.com\"\n>>> view fullName user\n\"John Cena\"\n>>> set fullName \"Doctor of Thuganomics\" user\nUser\n    { _firstName = \"Doctor\"\n    , _lastName = \"of Thuganomics\"\n    , _email = \"invisible@example.com\"\n    }\n

\n\n\n\n\n\n

\n3.7 Data correction and maintaining invariants

\n\n

We just learned about using lenses for computed and virtual fields; there’s an extension to this idea where we can use lenses to perform certain types of data correction to ensure our data remains in a valid state. This is easiest explained with an example so we’ll jump right in.

\n\n

\nIncluding correction logic in lenses

\n\n

Imagine we’ve got a rudimentary data type for storing clock time:

\n\n\n

data Time =\n    Time { _hours :: Int\n         , _mins  :: Int\n         }\n    deriving (Show)\n

\n\n\n\n\n

We want to allow users to edit the time of the clock, so we’ll expose some lenses! However, we want to make sure that no matter what, our hours value remains between 0-23, and the minutes remain between 0-59. If the user tries to set the values outside of that range we’ll simply clamp the value to fit the range instead. We can do this pretty easily by adding some simple logic to our setters

\n\n\n

clamp :: Int -> Int -> Int -> Int\nclamp minVal maxVal a = min maxVal . max minVal $ a\n\nhours :: Lens' Time Int\nhours = lens getter setter\n  where\n    getter (Time h _) = h\n    setter (Time _ m) newHours = Time (clamp 0 23 newHours) m\n\nmins :: Lens' Time Int\nmins = lens getter setter\n  where\n    getter (Time _ m) = m\n    setter (Time h _) newMinutes = Time h (clamp 0 59 newMinutes)\n

\n\n\n\n\n

These custom lenses clamp any new values we’re setting to be within the expected range.

\n\n\n

>>> let time = Time 3 10\n>>> time\nTime {_hours = 3, _mins = 10}\n\n>>> set hours 40 time\nTime {_hours = 23, _mins = 10}\n\n>>> set mins (-10) time\nTime {_hours = 3, _mins = 0}\n

\n\n\n\n\n

This ensures that the values are within the expected ranges when setting! If you’re you’re a bit paranoid you could also clamp the getters.

\n\n

Hopefully at some point during this section you thought “wait a minute, is this lawful”? The answer is no, these are not lawful lenses. If we set a bad value, we’ll get the corrected value instead. This is usually fine, but it’s good to think carefully about whether this behaviour is acceptable to you or not.

\n\n

This isn’t the only type of correction we could make in this scenario. If we wanted we could actually have the “minutes” and “hours” fields roll over when out of bounds. This makes it possible to do operations like adding 90 minutes to a time and still getting a sensible answer. Let’s see how that would look:

\n\n\n

hours :: Lens' Time Int\nhours = lens getter setter\n  where\n    getter (Time h _) = h\n    -- Take the hours 'mod' 24 so we always end up in the right range\n    setter (Time _ m) newHours = Time (newHours `mod` 24) m\n\nmins :: Lens' Time Int\nmins = lens getter setter\n  where\n    getter (Time _ m) = m\n    -- Minutes overflow into hours\n    setter (Time h _) newMinutes\n      = Time ((h + (newMinutes `div` 60)) `mod` 24) (newMinutes `mod` 60)\n

\n\n\n\n\n

In this new configuration we can add or subtract minutes and hours from the clock time and the lens will automatically normalize the minutes and hours!

\n\n\n

>>> let time = Time 3 10\n>>> time\nTime {_hours = 3, _mins = 10}\n\n>>> over mins (+ 55) time\nTime {_hours = 4, _mins = 5}\n\n>>> over mins (subtract 20) time\nTime {_hours = 2, _mins = 50}\n\n>>> over mins (+1) (Time 23 59)\nTime {_hours = 0, _mins = 0}\n

\n\n\n\n\n

Nifty! Again; these lenses are unlawful, but still useful!

\n\n

You’re probably wondering whether there are ways to provide an error message on invalid input rather than silently correcting it; and indeed there are! We’ll just need to learn a few more things before we’re ready to take that on.

\n\n

\nExercises – Self-Correcting Lenses

\n\n

Jump to answers

\n\n

Consider the following:

\n\n\n

data ProducePrices =\n    ProducePrices { _limePrice  :: Float\n                 , _lemonPrice :: Float\n                 }\n    deriving Show\n

\n\n\n\n\n

We’re handling a system for pricing our local grocery store’s citrus produce! Our first job is to write lenses for setting the prices of limes and lemons. Write lenses for limePrice and lemonPrice which prevent negative prices by rounding up to 0 (we’re okay with given produce out for free, but certainly aren’t going to pay others to take it).
The owner has informed us that it’s VERY important that the prices of limes and lemons must NEVER be further than 50 cents apart or the produce world would descend into total chaos. Update your lenses so that when setting lime-cost the lemon-cost is rounded to within 50 cents; (and vice versa).

\n\n

It should behave something like this; don’t worry if you can’t get it exactly right, this one is tricky!

\n\n\n

>>> let prices = ProducePrices 1.50 1.48\n>>> set limePrice 2 prices\nProducePrices\n    { _limePrice = 2.0\n    , _lemonPrice = 1.5\n    }\n>>> set limePrice 1.8 prices\nProducePrices\n    { _limePrice = 1.8\n    , _lemonPrice = 1.48\n    }\n>>> set limePrice 1.63 prices\nProducePrices\n    { _limePrice = 1.63\n    , _lemonPrice = 1.48\n    }\n>>> set limePrice  (-1.00) prices\nProducePrices\n    { _limePrice = 0.0\n    , _lemonPrice = 0.5\n    }\n

\n\n

”–” signifies that the optics are incompatible and do not compose.

\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n

	Fold	Getter	Setter	Traversal	Prism	Lens	Iso
Fold	Fold	Fold	–	Fold	Fold	Fold	Fold
Getter	Fold	Getter	–	Fold	Fold	Getter	Getter
Setter	–	–	Setter	Setter	Setter	Setter	Setter
Traversal	Fold	Fold	Setter	Traversal	Traversal	Traversal	Traversal
Prism	Fold	Fold	Setter	Traversal	Prism	Traversal	Prism
Lens	Fold	Getter	Setter	Traversal	Traversal	Lens	Lens
Iso	Fold	Getter	Setter	Traversal	Prism	Lens	Iso

\n\n

For example, to determine which type we get by composing traverse with _Just we first check each of their types to discover that traverse is a Traversal and _Just is a Prism. We then look up the column (or row) with the header Traversal, then find the cell with the corresponding header Prism on the other axis. Performing this look up we see that the composition traversed . _Just results in a Traversal.

\n\n\n

\n18.2 Optic Compatibility Chart

\n\n

The following chart details which optics are valid substitutions for one another.

\n\n

As an example, let’s say we were curious if all Prisms are a valid Traversal; we first find the row with Prism in the first column; then find the corresponding Traversal column and find a Yes; meaning that a Prism is a valid substitution for a Traversal.

	Fold	Getter	Setter	Traversal	Lens	Review	Prism	Iso
Fold	Yes	No	No	No	No	No	No	No
Getter	Yes	Yes	No	No	No	No	No	No
Setter	No	No	Yes	No	No	No	No	No
Traversal	Yes	Yes	Yes	Yes	No	No	No	No
Lens	Yes	Yes	Yes	Yes	Yes	No	No	No
Review	No	No	No	No	No	Yes	No	No
Prism	Yes	No	Yes	Yes	No	Yes	Yes	No
Iso	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes

\n\n\n

\n18.3 Operator Cheat Sheet

\n\n

Operators may look like an earthquake hit the mechanical keyboard factory, but there’s actually a bit of a language to the whole thing which starts to make sense after a bit of practice.

\n\n

Once you get used to the ideas you can usually guess the name of a symbol which does what you need, and it’ll usually exist!

\n\n

\nLegend for Getters

\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n

Symbol	Description
`^`	Denotes a Getter\n
`@`	Include the index with the result
`.`	Get a single value
`..`	Get a List of values
`?`	\nMaybe get the first value
`!`	\nForce a result or throw an exception if missing

\n\n

\nExamples

\n\n

(^@..) :: s -> IndexedFold i s a -> [(i, a)]: A getter (^) which includes the index (@) in a list of all focuses (..).

\n\n\n

\"Yarrr\" ^@.. folded\n[(0,'Y'),(1,'a'),(2,'r'),(3,'r'),(4,'r')]\n

\n\n\n\n\n

(^?!) :: s -> Traversal' s a -> a: A getter (^) which forcibly gets (!) a possibly missing (?) value.

\n\n\n

>>> Just \"Nemo\" ^?! _Just\n\"Nemo\"\n>>> Nothing ^?! _Just\n*** Exception: (^?!): empty Fold\nCallStack (from HasCallStack):\n  error, called at src/Control/Lens/Fold.hs:1285:28 in lens\nE:Control.Lens.Fold\n  ^?!, called at <interactive>:1:1 in interactive:Ghci4\n

\n\n\n\n\n

\nLegend for Setters/Modifiers

Symbol	Description
`.`	\nSet the focus
`%`	\nModify the focus
`~`	Denotes a Setter/Modifier\n
`=`	Denotes a Setter/Modifier over a `MonadState` context
`<`	Include the altered focus with the result
`<<`	Include the unaltered focus with the result
`%%`	Perform a traversal over the focus
`<>`	\n`mappend` over the focus
`?`	Wrap in `Just` before setting
`+`	\nAdd to the focus
`-`	\nSubtract from the focus
`*`	\nMultiply the focus
`//`	\nDivide the focus
`\|\|`	Logically `or` the focus
`&&`	Logically `and` the focus
`@`	Pass the index to the modification function

\n\n

\nExamples

\n\n

(<>~) :: Monoid a => Traversal' s a -> a -> s -> s: A setter (~) which mappends (<>) a new value to the focus.

\n\n\n

>>> [\"Polly want a\", \"Salty as a soup\"] & traverse <>~ \" cracker!\"\n[\"Polly want a cracker!\",\"Salty as a soup cracker!\"]\n

\n\n\n\n\n

(<<%@=) :: MonadState s m => Traversal s s a b -> (i -> a -> b) -> m a: \nModify (%) the focus from within a MonadState (=), passing the index (@) to the function as well. Also return unaltered (<<) original value.

\n\n

This one’s a bit tricky:

\n\n\n

>>> runState (itraversed <<%@= \\k v -> \"item: \" <> k <> \", quantity: \" <> v) (M.sing\\\nleton \"bananas\" \"32\")\n(\"32\",fromList [(\"bananas\",\"item: bananas, quantity: 32\")])\n

\n\n\n\n\n

(%%~) :: Traversal s t a b -> (a -> f b) -> s -> f t: A setter (~) which traverses (%%) a function over the focus.

\n\n\n

>>> (1, 2) & both %%~ (\\n -> if even n then Just n else Nothing)\nNothing\n\n>>> (2, 4) & both %%~ (\\n -> if even n then Just n else Nothing)\nJust (2,4)\n

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\nNotes

Optics

Table of Contents

\n1 Obligatory Preamble

\n1.1 Why should I read this book?

\n1.2 How to read this book

\n1.3 Chosen language and optics encodings

\n1.4 Your practice environment

\n1.5 Following examples

\n1.6 About the type signatures

\n1.7 About the exercises

\n2 Optics

\n2.1 What are optics?

\n2.2 Strengths

\n2.3 Weaknesses

\n2.4 Practical optics at a glance

\n2.5 Impractical optics at a glance

\n3 Lenses

\n3.1 Introduction to Lenses

\nAnatomy

\nExercises – Optic Anatomy

\n3.2 Lens actions

\nViewing through lenses

\nSetting through a lens

\nExercises - Lens Actions

\n3.3 Lenses and records

\nLenses subsume the “accessor” pattern

\nBuilding a lens for a record field

\nExercises - Records Part One

\nGetting and setting with a field lens

\nModifying fields with a lens

\nAutomatically generating field lenses

\nExercises - Records Part Two

\n3.4 Limitations

\nIs it a Lens?

\nIs it a Lens? – Answers

\n3.5 Lens Laws

\nWhy do optics have laws?

\nThe Laws

\nYou get back what you set (set-get)

\nSetting back what you got doesn’t do anything (get-set)

\nSetting twice is the same as setting once (set-set)

\nCase Study: _1\n

\nYou get back what you set (set-get)

\nSetting back what you got doesn’t do anything (get-set)

\nSetting twice is the same as setting once (set-set)

\nCase Study: msg\n

\nYou get back what you set (set-get)

\nCase Study: lensProduct\n

\nYou get back what you set (set-get)

\nExercises - Laws

\n3.6 Virtual Fields

\nWhat’s a virtual field

\nWriting a virtual field

\nBreakage-free refactoring

\nThe universe without lenses

\nThe glorius utopian lenses universe

\nExercises – Virtual Fields

\n3.7 Data correction and maintaining invariants

\nIncluding correction logic in lenses

\nExercises – Self-Correcting Lenses

\n4 Polymorphic Optics

\n4.1 Introduction to polymorphic optics

\nSimple vs Polymorphic optics

\n4.2 When do we need polymorphic lenses

\nType-changing focuses

\nChanging type variables with polymorphic lenses

\nExercises – Polymorphic Lenses

\n4.3 Composing Lenses

\nHow do I update fields in deeply nested records?

\nComposing update functions

\nComposing Lenses

\nHow do Lens Types Compose?

\nExercises – Lens Composition

\n5 Operators

\n5.1 Lens Operators

\n5.2 view a.k.a. ^.\n

\n5.3 set a.k.a. .~\n

\n5.4 Chaining many operations

\n5.5 Using %~ a.k.a. over\n

\nCase Study: `_1`\n

\nCase Study: `msg`\n

\nCase Study: `lensProduct`\n

\n5.2 `view` a.k.a. `^.`\n

\n5.3 `set` a.k.a. `.~`\n

\n5.5 Using `%~` a.k.a. `over`\n

\nOptics look like `traverse`\n