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Lenses Home
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Fortgeschrittene funktionale Programmierung in Haskell
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Lenses
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Webapp-Development in Haskell
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2
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Openapi-generator
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<a class='hover:underline truncate' title='Logik für Dummies' href='Logik'>
Logik für Dummies
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2
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Todo
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<h1 class='flex items-end justify-center mb-4 p-3 bg-purple-100 text-5xl font-extrabold text-black rounded'>
<a class='z-40 tracking-tighter '>
Lenses
</a>
</h1>
<article class='overflow-auto'>
<!-- What goes in this file will appear on top of note body-->
<h2 id='wofür-brauchen-wir-das-überhaupt' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Wofür brauchen wir das überhaupt?</h2>
<p class='mb-3'>
Die Idee dahinter ist, dass man Zugriffsabstraktionen über Daten verknüpfen kann. Als einfachen Datenstruktur kann man einen Record mit der entsprechenden Syntax nehmen.
</p>
<h3 id='beispiel' class='mt-6 mb-2 text-3xl font-bold text-gray-700'>Beispiel</h3><div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data Person = P { name :: String
, addr :: Address
, salary :: Int }
data Address = A { road :: String
, city :: String
, postcode :: String }
-- autogeneriert unten anderem: addr :: Person -&gt; Address
setName :: String -&gt; Person -&gt; Person
setName n p = p { name = n } --record update notation
setPostcode :: String -&gt; Person -&gt; Person
setPostcode pc p
= p { addr = addr p { postcode = pc } }
-- update of a record inside a record</code></pre></div><h3 id='probleme' class='mt-6 mb-2 text-3xl font-bold text-gray-700'>Probleme</h3>
<p class='mb-3'>
Probleme mit diesem Code:
</p>
<ul class='my-3 ml-6 space-y-1 list-disc'>
<li>
für 1-Dimensionale Felder ist die record-syntax ok.
</li>
<li>
tiefere Ebenen nur umständlich zu erreichen
</li>
<li>
eigentlich wollen wir nur pe in p setzen, müssen aber über addr etc. gehen.
</li>
<li>
wir brauchen wissen über die “Zwischenstrukturen”, an denen wir nicht interessiert sind
</li>
</ul>
<h3 id='was-wir-gern-hätten' class='mt-6 mb-2 text-3xl font-bold text-gray-700'>Was wir gern hätten</h3><div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data Person = P { name :: String
, addr :: Address
, salary :: Int }
-- a lens for each field
lname :: Lens' Person String
laddr :: Lens' Person Adress
lsalary :: Lens' Person Int
-- getter/setter for them
view :: Lens' s a -&gt; s -&gt; a
set :: Lens' s a -&gt; a -&gt; s -&gt; s
-- lens-composition
composeL :: Lens' s1 s2 -&gt; Lens s2 a -&gt; Lens' s1 a</code></pre></div><h3 id='wie-uns-das-hilft' class='mt-6 mb-2 text-3xl font-bold text-gray-700'>Wie uns das hilft</h3>
<p class='mb-3'>
Mit diesen Dingen (wenn wir sie hätten) könnte man dann
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data Person = P { name :: String
, addr :: Address
, salary :: Int }
data Address = A { road :: String
, city :: String
, postcode :: String }
setPostcode :: String -&gt; Person -&gt; Person
setPostcode pc p
= set (laddr `composeL` lpostcode) pc p</code></pre></div>
<p class='mb-3'>
machen und wäre fertig.
</p>
<h2 id='trivialer-ansatz' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Trivialer Ansatz</h2><h3 id='gettersetter-als-lens-methoden' class='mt-6 mb-2 text-3xl font-bold text-gray-700'>Getter/Setter als Lens-Methoden</h3><div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data LensR s a = L { viewR :: s -&gt; a
, setR :: a -&gt; s -&gt; s }
composeL (L v1 u1) (L v2 u2)
= L (\s -&gt; v2 (v1 s))
(\a s -&gt; u1 (u2 a (v1 s)) s)</code></pre></div><h3 id='wieso-ist-das-schlecht' class='mt-6 mb-2 text-3xl font-bold text-gray-700'>Wieso ist das schlecht?</h3>
<ul class='my-3 ml-6 space-y-1 list-disc'>
<li>
extrem ineffizient
</li>
</ul>
<p class='mb-3'>
Auslesen traversiert die Datenstruktur, dann wird die Funktion angewendet und zum setzen wird die Datenstruktur erneut traversiert:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>over :: LensR s a -&gt; (a -&gt; a) -&gt; s -&gt; s
over ln f s = setR l (f (viewR l s)) s</code></pre></div>
<ul class='my-3 ml-6 space-y-1 list-disc'>
<li>
Lösung: modify-funktion hinzufügen
</li>
</ul>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data LensR s a
= L { viewR :: s -&gt; a
, setR :: a -&gt; s -&gt; s
, mod :: (a-&gt;a) -&gt; s -&gt; s
, modM :: (a-&gt;Maybe a) -&gt; s -&gt; Maybe s
, modIO :: (a-&gt;IO a) -&gt; s -&gt; IO s }</code></pre></div>
<p class='mb-3'>
Neues Problem: Für jeden Spezialfall muss die Lens erweitert werden.
</p>
<h3 id='something-in-common' class='mt-6 mb-2 text-3xl font-bold text-gray-700'>Something in common</h3>
<p class='mb-3'>
Man kann alle Monaden abstrahieren. Functor reicht schon:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data LensR s a
= L { viewR :: s -&gt; a
, setR :: a -&gt; s -&gt; s
, mod :: (a-&gt;a) -&gt; s -&gt; s
, modF :: Functor f =&gt; (a-&gt;f a) -&gt; s -&gt; f s }</code></pre></div>
<p class='mb-3'>
Idee: Die 3 darüberliegenden durch modF ausdrücken.
</p>
<h3 id='typ-einer-lens' class='mt-6 mb-2 text-3xl font-bold text-gray-700'>Typ einer Lens</h3>
<p class='mb-3'>
Wenn man das berücksichtigt, dann hat einen Lens folgenden Typ:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>type Lens' s a = forall f. Functor f
=&gt; (a -&gt; f a) -&gt; s -&gt; f s</code></pre></div>
<p class='mb-3'>
Allerdings haben wir dann noch unseren getter/setter:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data LensR s a = L { viewR :: s -&gt; a
, setR :: a -&gt; s -&gt; s }</code></pre></div>
<p class='mb-3'>
Stellt sich raus: Die sind isomorph! Auch wenn die von den Typen her komplett anders aussehen.
</p>
<h2 id='benutzen-einer-lens-als-setter' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Benutzen einer Lens als Setter</h2><div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>set :: Lens' s a -&gt; (a -&gt; s -&gt; s)
set ln a s = --...umm...
--:t ln =&gt; (a -&gt; f a) -&gt; s -&gt; f s
-- =&gt; get s out of f s to return it</code></pre></div>
<p class='mb-3'>
Wir können für f einfach die “Identity”-Monade nehmen, die wir nachher wegcasten können.
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>newtype Identity a = Identity a
-- Id :: a -&gt; Identity a
runIdentity :: Identity s -&gt; s
runIdentity (Identity x) = x
instance Functor Identity where
fmap f (Identity x) = Identity (f x)</code></pre></div>
<p class='mb-3'>
somit ist set einfach nur
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>set :: Lens' s a -&gt; (a -&gt; s -&gt; s)
set ln x s
= runIdentity (ls set_fld s)
where
set_fld :: a -&gt; Identity a
set_fld _ = Identity x
-- a was the OLD value.
-- We throw that away and set the new value</code></pre></div>
<p class='mb-3'>
oder kürzer (für nerds wie den Autor der Lens-Lib)
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>set :: Lens' s a -&gt; (a -&gt; s -&gt; s)
set ln x = runIdentity . ln (Identity . const x)</code></pre></div><h2 id='benutzen-einer-lens-als-modify' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Benutzen einer Lens als Modify</h2>
<p class='mb-3'>
Dasselbe wie Set, nur dass wir den Parameter nicht entsorgen, sondern in die mitgelieferte Funktion stopfen.
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>over :: Lens' s a -&gt; (a -&gt; a) -&gt; s -&gt; s
over ln f = runIdentity . ln (Identity . f)</code></pre></div><h2 id='benutzen-einer-lens-als-getter' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Benutzen einer Lens als Getter</h2><div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>view :: Lens' s a -&gt; (s -&gt; a)
view ln s = --...umm...
--:t ln =&gt; (a -&gt; f a) -&gt; s -&gt; f s
-- =&gt; get a out of the (f s) return-value
-- Wait, WHAT?</code></pre></div>
<p class='mb-3'>
Auch hier gibt es einen netten Funktor. Wir packen das “a” einfach in das “f” und werfen das “s” am Ende weg.
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>newtype Const v a = Const v
getConst :: Const v a -&gt; v
getConst (Const x) = x
instance Functor (Const v) where
fmap f (Const x) = Const x
-- throw f away. Nothing changes our const!</code></pre></div>
<p class='mb-3'>
somit ergibt sich
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>view :: Lens' s a -&gt; (s -&gt; a)
view ln s
= getConst (ln Const s)
-- Const :: s -&gt; Const a s</code></pre></div>
<p class='mb-3'>
oder nerdig
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>view :: Lens' s a -&gt; (s -&gt; a)
view ln = getConst . ln Const</code></pre></div><h2 id='lenses-bauen' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Lenses bauen</h2>
<p class='mb-3'>
Nochmal kurz der Typ:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>type Lens' s a = forall f. Functor f
=&gt; (a -&gt; f a) -&gt; s -&gt; f s</code></pre></div>
<p class='mb-3'>
Für unser Personen-Beispiel vom Anfang:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data Person = P { _name :: String, _salary :: Int }
name :: Lens' Person String
-- name :: Functor f =&gt; (String -&gt; f String)
-- -&gt; Person -&gt; f Person
name elt_fn (P n s)
= fmap (\n' -&gt; P n' s) (elt_fn n)
-- fmap :: Functor f =&gt; (a-&gt;b) -&gt; f a -&gt; f b - der Funktor, der alles verknüpft
-- \n' -&gt; .. :: String -&gt; Person - Funktion um das Element zu lokalisieren (WO wird ersetzt/gelesen/...)
-- elt_fn n :: f String - Funktion um das Element zu verändern (setzen, ändern, ...)</code></pre></div>
<p class='mb-3'>
Die Lambda-Funktion ersetzt einfach den Namen. Häufig sieht man auch
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>name elt_fn (P n s)
= (\n' -&gt; P n' s) &lt;$&gt; (elt_fn n)
-- | Focus | |Function|</code></pre></div><h2 id='wie-funktioniert-das-intern' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Wie funktioniert das intern?</h2><div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>view name (P {_name="Fred", _salary=100})
-- inline view-function
= getConst (name Const (P {_name="Fred", _salary=100})
-- inline name
= getConst (fmap (\n' -&gt; P n' 100) (Const "Fred"))
-- fmap f (Const x) = Const x - Definition von Const
= getConst (Const "Fred")
-- getConst (Const x) = x
= "Fred"</code></pre></div>
<p class='mb-3'>
Dieser Aufruf hat KEINE Runtime-Kosten, weil der Compiler direkt die Adresse des Feldes einsetzen kann. Der gesamte Boilerplate-Code wird vom Compiler wegoptimiert.
</p>
<p class='mb-3'>
Dies gilt für jeden Funktor mit newtype, da das nur ein Typalias ist.
</p>
<h2 id='composing-lenses-und-deren-benutzung' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Composing Lenses und deren Benutzung</h2>
<p class='mb-3'>
Wie sehen denn die Typen aus?
</p>
<p class='mb-3'>
Wir wollen ein
</p>
<blockquote class='py-0.5 px-4 mb-3 italic border-l-4 bg-gray-50 text-gray-600 border-gray-400 quote'>
<p class='mb-3'>
Lens s1 s2 -&gt; Lens s2 a -&gt; Lens s1 a
</p>
</blockquote>
<p class='mb-3'>
Wir haben 2 Lenses
</p>
<blockquote class='py-0.5 px-4 mb-3 italic border-l-4 bg-gray-50 text-gray-600 border-gray-400 quote'>
<p class='mb-3'>
ln1 :: (s2 -&gt; f s2) -&gt; (s1 -&gt; f s1) ln2 :: (a -&gt; f a) -&gt; (s2 -&gt; f s2)
</p>
</blockquote>
<p class='mb-3'>
wenn man scharf hinsieht, kann man die verbinden
</p>
<blockquote class='py-0.5 px-4 mb-3 italic border-l-4 bg-gray-50 text-gray-600 border-gray-400 quote'>
<p class='mb-3'>
ln1 . ln2 :: (a -&gt; f s) -&gt; (s1 -&gt; f s1)
</p>
</blockquote>
<p class='mb-3'>
und erhält eine Lens. Sogar die Gewünschte!<br />Somit ist Lens-Composition einfach nur Function-Composition (.).
</p>
<h2 id='automatisieren-mit-template-haskell' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Automatisieren mit Template-Haskell</h2>
<p class='mb-3'>
Der Code um die Lenses zu bauen ist für records immer Identisch:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data Person = P { _name :: String, _salary :: Int }
name :: Lens' Person String
name elt_fn (P n s) = (\n' -&gt; P n' s) &lt;$&gt; (elt_fn n)</code></pre></div>
<p class='mb-3'>
Daher kann man einfach
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>import Control.Lens.TH
data Person = P { _name :: String, _salary :: Int }
$(makeLenses ''Person)</code></pre></div>
<p class='mb-3'>
nehmen, was einem eine Lens für “name” und eine Lens für “salary” generiert.<br />Mit anderen Templates kann man auch weitere Dinge steuern (etwa wofür Lenses generiert werden, welches Prefix (statt _) man haben will etc. pp.).
</p>
<p class='mb-3'>
Will man das aber haben, muss man selbst in den Control.Lens.TH-Code schauen.
</p>
<h2 id='lenses-für-den-beispielcode' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Lenses für den Beispielcode</h2><div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>import Control.Lens.TH
data Person = P { _name :: String
, _addr :: Address
, _salary :: Int }
data Address = A { _road :: String
, _city :: String
, _postcode :: String }
$(makeLenses ''Person)
$(makeLenses ''Address)
setPostcode :: String -&gt; Person -&gt; Person
setPostcode pc p = set (addr . postcode) pc p</code></pre></div><h2 id='shortcuts-mit-line-noise' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Shortcuts mit “Line-Noise”</h2><div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>-- ...
setPostcode :: String -&gt; Person -&gt; Person
setPostcode pc p = addr . postcode .~ pc $ p
-- | Focus |set|to what|in where
getPostcode :: Person -&gt; String
getPostcode p = p ^. $ addr . postcode
-- |from|get| Focus |</code></pre></div>
<p class='mb-3'>
Es gibt drölf-zillionen weitere Infix-Operatoren (für Folds, Listenkonvertierungen, -traversierungen, …)
</p>
<h2 id='virtuelle-felder' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Virtuelle Felder</h2>
<p class='mb-3'>
Man kann mit Lenses sogar Felder emulieren, die gar nicht da sind. Angenommen folgender Code:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data Temp = T { _fahrenheit :: Float }
$(makeLenses ''Temp)
-- liefert Lens: fahrenheit :: Lens Temp Float
centigrade :: Lens Temp Float
centigrade centi_fn (T faren)
= (\centi' -&gt; T (cToF centi'))
&lt;$&gt; (centi_fn (fToC faren))
-- cToF & fToC as Converter-Functions defined someplace else</code></pre></div>
<p class='mb-3'>
Hiermit kann man dann auch Funktionen, die auf Grad-Celsius rechnen auf Daten anwenden, die eigenlich nur Fahrenheit speichern, aber eine Umrechnung bereitstellen. Analog kann man auch einen Zeit-Datentypen definieren, der intern mit Sekunden rechnet (und somit garantiert frei von Fehlern wie -3 Minuten oder 37 Stunden ist)
</p>
<h2 id='non-record-strukturen' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Non-Record Strukturen</h2>
<p class='mb-3'>
Das ganze kann man auch parametrisieren und auf Non-Record-Strukturen anwenden. Beispielhaft an einer Map verdeutlicht:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>-- from Data.Lens.At
at :: Ord k =&gt; k -&gt; Lens' (Map k v) (Maybe v)
-- oder identisch, wenn man die Lens' auflöst:
at :: Ord k, forall f. Functor f =&gt; k -&gt; (Maybe v -&gt; f Maybe v) -&gt; Map k v -&gt; f Map k v
at k mb_fn m
= wrap &lt;$&gt; (mb_fn mv)
where
mv = Map.lookup k m
wrap :: Maybe v -&gt; Map k v
wrap (Just v') = Map.insert k v' m
wrap Nothing = case mv of
Nothing -&gt; m
Just _ -&gt; Map.delete k m
-- mb_fn :: Maybe v -&gt; f Maybe v</code></pre></div><h2 id='weitere-beispiele' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Weitere Beispiele</h2>
<ul class='my-3 ml-6 space-y-1 list-disc'>
<li>
<p class='mb-3'>
Bitfields auf Strukturen die Bits haben (Ints, …) in Data.Bits.Lens
</p>
</li>
<li>
<p class='mb-3'>
Web-scraper in Package hexpat-lens
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>p ^.. _HTML' . to allNodes
. traverse . named "a"
. traverse . ix "href"
. filtered isLocal
. to trimSpaces</code></pre></div>
<p class='mb-3'>
Zieht alle externen Links aus dem gegebenen HTML-Code in p um weitere ziele fürs crawlen zu finden.
</p>
</li>
</ul>
<h2 id='erweiterungen' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Erweiterungen</h2>
<p class='mb-3'>
Bisher hatten wir Lenses nur auf Funktoren F. Die nächstmächtigere Klasse ist Applicative.
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>type Traversal' s a = forall f. Applicative f
=&gt; (a -&gt; f a) -&gt; (s -&gt; f s)</code></pre></div>
<p class='mb-3'>
Da wir den Container identisch lassen (weder s noch a wurde angefasst) muss sich etwas anderes ändern. Statt eines einzelnen Focus erhalten wir viele Foci.
</p>
<p class='mb-3'>
Was ist ein Applicative überhaupt? Eine schwächere Monade (nur 1x Anwendung und kein Bind - dafür kann man die beliebig oft hintereinanderhängen).
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>class Functor f =&gt; Applicative f where
pure :: a -&gt; f a
(&lt;*&gt;) :: f (a -&gt; b) -&gt; f a -&gt; f b
-- Monade als Applicative:
pure = return
mf &lt;*&gt; mx = do { f &lt;- mf; x &lt;- mx; return (f x) }</code></pre></div>
<p class='mb-3'>
Recap: Was macht eine Lens:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>data Adress = A { _road :: String
, _city :: String
, _postcode :: String }
road :: Lens' Adress String
road elt_fn (A r c p) = (\r' -&gt; A r' c p) &lt;$&gt; (elt_fn r)
-- | "Hole" | | Thing to put in|</code></pre></div>
<p class='mb-3'>
Wenn man nun road & city gleichzeitig bearbeiten will:
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>addr_strs :: Traversal' Address String
addr_strs elt_fn (A r c p)
= ... (\r' c' -&gt; A r' c' p) .. (elt_fn r) .. (elt_fn c) ..
-- | function with 2 "Holes"| first Thing | second Thing</code></pre></div>
<p class='mb-3'>
fmap kann nur 1 Loch stopfen, aber nicht mit n Löchern umgehen. Applicative mit &lt;*&gt; kann das.<br />Somit gibt sich
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>addr_strs :: Traversal' Address String
addr_strs elt_fn (A r c p)
= pure (\r' c' -&gt; A r' c' p) &lt;*&gt; (elt_fn r) &lt;*&gt; (elt_fn c)
-- lift in Appl. | function with 2 "Holes"| first Thing | second Thing
-- oder kürzer
addr_strs :: Traversal' Address String
addr_strs elt_fn (A r c p)
= (\r' c' -&gt; A r' c' p) &lt;$&gt; (elt_fn r) &lt;*&gt; (elt_fn c)
-- pure x &lt;*&gt; y == x &lt;$&gt; y</code></pre></div>
<p class='mb-3'>
Wie würd eine modify-funktion aussehen?
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>over :: Lens' s a -&gt; (a -&gt; a) -&gt; s -&gt; s
over ln f = runIdentity . ln (Identity . f)
over :: Traversal' s a -&gt; (a -&gt; a) -&gt; s -&gt; s
over ln f = runIdentity . ln (Identity . f)</code></pre></div>
<p class='mb-3'>
Der Code ist derselbe - nur der Typ ist generischer. Auch die anderen Dinge funktioniert diese Erweiterung (für Identity und Const muss man noch ein paar dummy-Instanzen schreiben um sie von Functor auf Applicative oder Monad zu heben
</p>
<ul class='my-3 ml-6 space-y-1 list-disc'>
<li>
konkret reicht hier die Instanzierung von Monoid). In der Lens-Library ist daher meist Monad m statt Functor f gefordert.
</li>
</ul>
<h2 id='wozu-dienen-die-erweiterungen' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Wozu dienen die Erweiterungen?</h2>
<p class='mb-3'>
Man kann mit Foci sehr selektiv vorgehen. Auch kann man diese durch Funktionen steuern. Beispisweise eine Funktion anwenden auf
</p>
<ul class='my-3 ml-6 space-y-1 list-disc'>
<li>
Jedes 2. Listenelement
</li>
<li>
Alle graden Elemente in einem Baum
</li>
<li>
Alle Namen in einer Tabelle, deren Gehalt &gt; 10.000€ ist
</li>
</ul>
<p class='mb-3'>
Traversals und Lenses kann man trivial kombinieren (<code class='py-0.5 px-0.5 bg-gray-100'>lens . lens</code> =&gt; <code class='py-0.5 px-0.5 bg-gray-100'>lens</code>, <code class='py-0.5 px-0.5 bg-gray-100'>lens . traversal</code> =&gt; <code class='py-0.5 px-0.5 bg-gray-100'>traversal</code> etc.)
</p>
<h2 id='wie-es-in-lens-wirklich-aussieht' class='inline-block mt-6 mb-4 text-4xl font-bold text-gray-700 border-b-2'>Wie es in Lens wirklich aussieht</h2>
<p class='mb-3'>
In diesem Artikel wurde nur auf Monomorphic Lenses eingegangen. In der richtigen Library ist eine Lens
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>type Lens' s a = Lens s s a a
type Lens s t a b = forall f. Functor f =&gt; (a -&gt; f b) -&gt; (s -&gt; f t)</code></pre></div>
<p class='mb-3'>
sodass sich auch die Typen ändern können um z.B. automatisch einen Konvertierten (sicheren) Typen aus einer unsicheren Datenstruktur zu geben.
</p>
<p class='mb-3'>
Die modify-Funktion over ist auch
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>&gt; over :: Profunctor p =&gt; Setting p s t a b -&gt; p a b -&gt; s -&gt; t</code></pre></div>
<blockquote class='py-0.5 px-4 mb-3 italic border-l-4 bg-gray-50 text-gray-600 border-gray-400 quote'>
<p class='mb-3'>
<em>Edward is deeply in thrall to abstractionitis</em> - Simon Peyton Jones
</p>
</blockquote>
<p class='mb-3'>
Lens alleine definiert 39 newtypes, 34 data-types und 194 Typsynonyme…<br />Ausschnitt
</p>
<div class='py-0.5 mb-3 text-sm'><pre><code class='haskell language-haskell'>-- traverseOf :: Functor f =&gt; Iso s t a b -&gt; (a -&gt; f b) -&gt; s -&gt; f t
-- traverseOf :: Functor f =&gt; Lens s t a b -&gt; (a -&gt; f b) -&gt; s -&gt; f t
-- traverseOf :: Applicative f =&gt; Traversal s t a b -&gt; (a -&gt; f b) -&gt; s -&gt; f t
traverseOf :: Over p f s t a b -&gt; p a (f b) -&gt; s -&gt; f t</code></pre></div>
<p class='mb-3'>
dafuq?
</p>
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