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</script></head><body><div id="package-header"><ul class="links" id="page-menu"><li><a href="src/Control-Lens-Type.html">Source</a></li><li><a href="index.html">Contents</a></li><li><a href="doc-index.html">Index</a></li></ul><p class="caption">lens-4.15.4: Lenses, Folds and Traversals</p></div><div id="content"><div id="module-header"><table class="info"><tr><th valign="top">Copyright</th><td>(C) 2012-16 Edward Kmett</td></tr><tr><th>License</th><td>BSD-style (see the file LICENSE)</td></tr><tr><th>Maintainer</th><td>Edward Kmett <ekmett@gmail.com></td></tr><tr><th>Stability</th><td>provisional</td></tr><tr><th>Portability</th><td>Rank2Types</td></tr><tr><th>Safe Haskell</th><td>Safe</td></tr><tr><th>Language</th><td>Haskell98</td></tr></table><p class="caption">Control.Lens.Type</p></div><div id="table-of-contents"><p class="caption">Contents</p><ul><li><a href="#g:1">Other</a></li><li><a href="#g:2">Lenses, Folds and Traversals</a></li><li><a href="#g:3">Indexed</a></li><li><a href="#g:4">Index-Preserving</a></li><li><a href="#g:5">Common</a></li></ul></div><div id="description"><p class="caption">Description</p><div class="doc"><p>This module exports the majority of the types that need to appear in user
signatures or in documentation when talking about lenses. The remaining types
for consuming lenses are distributed across various modules in the hierarchy.</p></div></div><div id="synopsis"><p id="control.syn" class="caption expander" onclick="toggleSection('syn')">Synopsis</p><ul id="section.syn" class="hide" onclick="toggleSection('syn')"><li class="src short"><span class="keyword">type</span> <a href="#t:Equality">Equality</a> s t a b = <span class="keyword">forall</span> k3 p f. p a (f b) -> p s (f t)</li><li class="src short"><span class="keyword">type</span> <a href="#t:Equality-39-">Equality'</a> s a = <a href="Control-Lens-Type.html#t:Equality">Equality</a> s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:As">As</a> a = <a href="Control-Lens-Type.html#t:Equality-39-">Equality'</a> a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:Iso">Iso</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Iso.html#t:Profunctor">Profunctor</a> p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => p a (f b) -> p s (f t)</li><li class="src short"><span class="keyword">type</span> <a href="#t:Iso-39-">Iso'</a> s a = <a href="Control-Lens-Type.html#t:Iso">Iso</a> s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:Prism">Prism</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Prism.html#t:Choice">Choice</a> p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => p a (f b) -> p s (f t)</li><li class="src short"><span class="keyword">type</span> <a href="#t:Prism-39-">Prism'</a> s a = <a href="Control-Lens-Type.html#t:Prism">Prism</a> s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:Review">Review</a> t b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Prism.html#t:Choice">Choice</a> p, <a href="Control-Lens-Review.html#t:Bifunctor">Bifunctor</a> p, <a href="Control-Lens-Internal-Setter.html#t:Settable">Settable</a> f) => <a href="Control-Lens-Type.html#t:Optic-39-">Optic'</a> p f t b</li><li class="src short"><span class="keyword">type</span> <a href="#t:AReview">AReview</a> t b = <a href="Control-Lens-Type.html#t:Optic-39-">Optic'</a> <a href="file:///usr/share/doc/libghc-tagged-doc/html/Data-Tagged.html#t:Tagged">Tagged</a> <a href="Control-Lens-Setter.html#t:Identity">Identity</a> t b</li><li class="src short"><span class="keyword">type</span> <a href="#t:Lens">Lens</a> s t a b = <span class="keyword">forall</span> f. <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f => (a -> f b) -> s -> f t</li><li class="src short"><span class="keyword">type</span> <a href="#t:Lens-39-">Lens'</a> s a = <a href="Control-Lens-Type.html#t:Lens">Lens</a> s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:Traversal">Traversal</a> s t a b = <span class="keyword">forall</span> f. <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f => (a -> f b) -> s -> f t</li><li class="src short"><span class="keyword">type</span> <a href="#t:Traversal-39-">Traversal'</a> s a = <a href="Control-Lens-Type.html#t:Traversal">Traversal</a> s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:Traversal1">Traversal1</a> s t a b = <span class="keyword">forall</span> f. <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f => (a -> f b) -> s -> f t</li><li class="src short"><span class="keyword">type</span> <a href="#t:Traversal1-39-">Traversal1'</a> s a = <a href="Control-Lens-Type.html#t:Traversal1">Traversal1</a> s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:Setter">Setter</a> s t a b = <span class="keyword">forall</span> f. <a href="Control-Lens-Internal-Setter.html#t:Settable">Settable</a> f => (a -> f b) -> s -> f t</li><li class="src short"><span class="keyword">type</span> <a href="#t:Setter-39-">Setter'</a> s a = <a href="Control-Lens-Type.html#t:Setter">Setter</a> s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:Getter">Getter</a> s a = <span class="keyword">forall</span> f. (<a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => (a -> f a) -> s -> f s</li><li class="src short"><span class="keyword">type</span> <a href="#t:Fold">Fold</a> s a = <span class="keyword">forall</span> f. (<a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => (a -> f a) -> s -> f s</li><li class="src short"><span class="keyword">type</span> <a href="#t:Fold1">Fold1</a> s a = <span class="keyword">forall</span> f. (<a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f) => (a -> f a) -> s -> f s</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedLens">IndexedLens</a> i s t a b = <span class="keyword">forall</span> f p. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => p a (f b) -> s -> f t</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedLens-39-">IndexedLens'</a> i s a = <a href="Control-Lens-Type.html#t:IndexedLens">IndexedLens</a> i s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedTraversal">IndexedTraversal</a> i s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => p a (f b) -> s -> f t</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedTraversal-39-">IndexedTraversal'</a> i s a = <a href="Control-Lens-Type.html#t:IndexedTraversal">IndexedTraversal</a> i s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedTraversal1">IndexedTraversal1</a> i s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f) => p a (f b) -> s -> f t</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedTraversal1-39-">IndexedTraversal1'</a> i s a = <a href="Control-Lens-Type.html#t:IndexedTraversal1">IndexedTraversal1</a> i s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedSetter">IndexedSetter</a> i s t a b = <span class="keyword">forall</span> f p. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="Control-Lens-Internal-Setter.html#t:Settable">Settable</a> f) => p a (f b) -> s -> f t</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedSetter-39-">IndexedSetter'</a> i s a = <a href="Control-Lens-Type.html#t:IndexedSetter">IndexedSetter</a> i s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedGetter">IndexedGetter</a> i s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => p a (f a) -> s -> f s</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedFold">IndexedFold</a> i s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => p a (f a) -> s -> f s</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedFold1">IndexedFold1</a> i s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f) => p a (f a) -> s -> f s</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexPreservingLens">IndexPreservingLens</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => p a (f b) -> p s (f t)</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexPreservingLens-39-">IndexPreservingLens'</a> s a = <a href="Control-Lens-Type.html#t:IndexPreservingLens">IndexPreservingLens</a> s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexPreservingTraversal">IndexPreservingTraversal</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => p a (f b) -> p s (f t)</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexPreservingTraversal-39-">IndexPreservingTraversal'</a> s a = <a href="Control-Lens-Type.html#t:IndexPreservingTraversal">IndexPreservingTraversal</a> s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexPreservingTraversal1">IndexPreservingTraversal1</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f) => p a (f b) -> p s (f t)</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexPreservingTraversal1-39-">IndexPreservingTraversal1'</a> s a = <a href="Control-Lens-Type.html#t:IndexPreservingTraversal1">IndexPreservingTraversal1</a> s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexPreservingSetter">IndexPreservingSetter</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="Control-Lens-Internal-Setter.html#t:Settable">Settable</a> f) => p a (f b) -> p s (f t)</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexPreservingSetter-39-">IndexPreservingSetter'</a> s a = <a href="Control-Lens-Type.html#t:IndexPreservingSetter">IndexPreservingSetter</a> s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexPreservingGetter">IndexPreservingGetter</a> s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => p a (f a) -> p s (f s)</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexPreservingFold">IndexPreservingFold</a> s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => p a (f a) -> p s (f s)</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexPreservingFold1">IndexPreservingFold1</a> s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f) => p a (f a) -> p s (f s)</li><li class="src short"><span class="keyword">type</span> <a href="#t:Simple">Simple</a> f s a = f s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:LensLike">LensLike</a> f s t a b = (a -> f b) -> s -> f t</li><li class="src short"><span class="keyword">type</span> <a href="#t:LensLike-39-">LensLike'</a> f s a = <a href="Control-Lens-Type.html#t:LensLike">LensLike</a> f s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:Over">Over</a> p f s t a b = p a (f b) -> s -> f t</li><li class="src short"><span class="keyword">type</span> <a href="#t:Over-39-">Over'</a> p f s a = <a href="Control-Lens-Type.html#t:Over">Over</a> p f s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedLensLike">IndexedLensLike</a> i f s t a b = <span class="keyword">forall</span> p. <a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p => p a (f b) -> s -> f t</li><li class="src short"><span class="keyword">type</span> <a href="#t:IndexedLensLike-39-">IndexedLensLike'</a> i f s a = <a href="Control-Lens-Type.html#t:IndexedLensLike">IndexedLensLike</a> i f s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:Optical">Optical</a> p q f s t a b = p a (f b) -> q s (f t)</li><li class="src short"><span class="keyword">type</span> <a href="#t:Optical-39-">Optical'</a> p q f s a = <a href="Control-Lens-Type.html#t:Optical">Optical</a> p q f s s a a</li><li class="src short"><span class="keyword">type</span> <a href="#t:Optic">Optic</a> p f s t a b = p a (f b) -> p s (f t)</li><li class="src short"><span class="keyword">type</span> <a href="#t:Optic-39-">Optic'</a> p f s a = <a href="Control-Lens-Type.html#t:Optic">Optic</a> p f s s a a</li></ul></div><div id="interface"><h1 id="g:1">Other</h1><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Equality" class="def">Equality</a> s t a b = <span class="keyword">forall</span> k3 p f. p a (f b) -> p s (f t) <a href="src/Control-Lens-Type.html#Equality" class="link">Source</a> <a href="#t:Equality" class="selflink">#</a></p><div class="doc"><p>A witness that <code>(a ~ s, b ~ t)</code>.</p><p>Note: Composition with an <code><a href="Control-Lens-Type.html#t:Equality">Equality</a></code> is index-preserving.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Equality-39-" class="def">Equality'</a> s a = <a href="Control-Lens-Type.html#t:Equality">Equality</a> s s a a <a href="src/Control-Lens-Type.html#Equality%27" class="link">Source</a> <a href="#t:Equality-39-" class="selflink">#</a></p><div class="doc"><p>A <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:Equality">Equality</a></code>.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:As" class="def">As</a> a = <a href="Control-Lens-Type.html#t:Equality-39-">Equality'</a> a a <a href="src/Control-Lens-Type.html#As" class="link">Source</a> <a href="#t:As" class="selflink">#</a></p><div class="doc"><p>Composable <code>asTypeOf</code>. Useful for constraining excess
polymorphism, <code>foo . (id :: As Int) . bar</code>.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Iso" class="def">Iso</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Iso.html#t:Profunctor">Profunctor</a> p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => p a (f b) -> p s (f t) <a href="src/Control-Lens-Type.html#Iso" class="link">Source</a> <a href="#t:Iso" class="selflink">#</a></p><div class="doc"><p>Isomorphism families can be composed with another <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> using (<code>.</code>) and <code>id</code>.</p><p>Since every <code><a href="Control-Lens-Type.html#t:Iso">Iso</a></code> is both a valid <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> and a valid <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code>, the laws for those types
imply the following laws for an <code><a href="Control-Lens-Type.html#t:Iso">Iso</a></code> <code>f</code>:</p><pre>f <code>.</code> <code><a href="Control-Lens-Iso.html#v:from">from</a></code> f ≡ <code>id</code>
<code><a href="Control-Lens-Iso.html#v:from">from</a></code> f <code>.</code> f ≡ <code>id</code>
</pre><p>Note: Composition with an <code><a href="Control-Lens-Type.html#t:Iso">Iso</a></code> is index- and measure- preserving.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Iso-39-" class="def">Iso'</a> s a = <a href="Control-Lens-Type.html#t:Iso">Iso</a> s s a a <a href="src/Control-Lens-Type.html#Iso%27" class="link">Source</a> <a href="#t:Iso-39-" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:Iso-39-">Iso'</a></code> = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:Iso">Iso</a></code>
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Prism" class="def">Prism</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Prism.html#t:Choice">Choice</a> p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => p a (f b) -> p s (f t) <a href="src/Control-Lens-Type.html#Prism" class="link">Source</a> <a href="#t:Prism" class="selflink">#</a></p><div class="doc"><p>A <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code> <code>l</code> is a <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> that can also be turned
around with <code><a href="Control-Lens-Review.html#v:re">re</a></code> to obtain a <code><a href="Control-Lens-Type.html#t:Getter">Getter</a></code> in the
opposite direction.</p><p>There are two laws that a <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code> should satisfy:</p><p>First, if I <code><a href="Control-Lens-Review.html#v:re">re</a></code> or <code><a href="Control-Lens-Review.html#v:review">review</a></code> a value with a <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code> and then <code><a href="Control-Lens-Fold.html#v:preview">preview</a></code> or use (<code><a href="Control-Lens-Fold.html#v:-94--63-">^?</a></code>), I will get it back:</p><pre><code><a href="Control-Lens-Fold.html#v:preview">preview</a></code> l (<code><a href="Control-Lens-Review.html#v:review">review</a></code> l b) ≡ <code>Just</code> b
</pre><p>Second, if you can extract a value <code>a</code> using a <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code> <code>l</code> from a value <code>s</code>, then the value <code>s</code> is completely described by <code>l</code> and <code>a</code>:</p><p>If <code><code><a href="Control-Lens-Fold.html#v:preview">preview</a></code> l s ≡ <code>Just</code> a</code> then <code><code><a href="Control-Lens-Review.html#v:review">review</a></code> l a ≡ s</code></p><p>These two laws imply that the <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> laws hold for every <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code> and that we <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Traversable.html#v:traverse">traverse</a></code> at most 1 element:</p><pre><code><a href="Control-Lens-Fold.html#v:lengthOf">lengthOf</a></code> l x <code><=</code> 1
</pre><p>It may help to think of this as a <code><a href="Control-Lens-Type.html#t:Iso">Iso</a></code> that can be partial in one direction.</p><p>Every <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code> is a valid <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code>.</p><p>Every <code><a href="Control-Lens-Type.html#t:Iso">Iso</a></code> is a valid <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code>.</p><p>For example, you might have a <code><code><a href="Control-Lens-Type.html#t:Prism-39-">Prism'</a></code> <code>Integer</code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Numeric-Natural.html#v:Natural">Natural</a></code></code> allows you to always
go from a <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Numeric-Natural.html#v:Natural">Natural</a></code> to an <code>Integer</code>, and provide you with tools to check if an <code>Integer</code> is
a <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Numeric-Natural.html#v:Natural">Natural</a></code> and/or to edit one if it is.</p><pre><code>nat</code> :: <code><a href="Control-Lens-Type.html#t:Prism-39-">Prism'</a></code> <code>Integer</code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Numeric-Natural.html#v:Natural">Natural</a></code>
<code>nat</code> = <code><a href="Control-Lens-Prism.html#v:prism">prism</a></code> <code>toInteger</code> <code>$</code> \ i ->
if i <code><</code> 0
then <code>Left</code> i
else <code>Right</code> (<code>fromInteger</code> i)
</pre><p>Now we can ask if an <code>Integer</code> is a <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Numeric-Natural.html#v:Natural">Natural</a></code>.</p><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>5^?nat
</code></strong>Just 5
</pre><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>(-5)^?nat
</code></strong>Nothing
</pre><p>We can update the ones that are:</p><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>(-3,4) & both.nat *~ 2
</code></strong>(-3,8)
</pre><p>And we can then convert from a <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Numeric-Natural.html#v:Natural">Natural</a></code> to an <code>Integer</code>.</p><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>5 ^. re nat -- :: Natural
</code></strong>5
</pre><p>Similarly we can use a <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code> to <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Traversable.html#v:traverse">traverse</a></code> the <code>Left</code> half of an <code>Either</code>:</p><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>Left "hello" & _Left %~ length
</code></strong>Left 5
</pre><p>or to construct an <code>Either</code>:</p><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>5^.re _Left
</code></strong>Left 5
</pre><p>such that if you query it with the <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code>, you will get your original input back.</p><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>5^.re _Left ^? _Left
</code></strong>Just 5
</pre><p>Another interesting way to think of a <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code> is as the categorical dual of a <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code>
-- a co-<code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code>, so to speak. This is what permits the construction of <code><a href="Control-Lens-Prism.html#v:outside">outside</a></code>.</p><p>Note: Composition with a <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code> is index-preserving.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Prism-39-" class="def">Prism'</a> s a = <a href="Control-Lens-Type.html#t:Prism">Prism</a> s s a a <a href="src/Control-Lens-Type.html#Prism%27" class="link">Source</a> <a href="#t:Prism-39-" class="selflink">#</a></p><div class="doc"><p>A <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code>.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Review" class="def">Review</a> t b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Prism.html#t:Choice">Choice</a> p, <a href="Control-Lens-Review.html#t:Bifunctor">Bifunctor</a> p, <a href="Control-Lens-Internal-Setter.html#t:Settable">Settable</a> f) => <a href="Control-Lens-Type.html#t:Optic-39-">Optic'</a> p f t b <a href="src/Control-Lens-Type.html#Review" class="link">Source</a> <a href="#t:Review" class="selflink">#</a></p><div class="doc"><p>This is a limited form of a <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code> that can only be used for <code>re</code> operations.</p><p>Like with a <code><a href="Control-Lens-Type.html#t:Getter">Getter</a></code>, there are no laws to state for a <code><a href="Control-Lens-Type.html#t:Review">Review</a></code>.</p><p>You can generate a <code><a href="Control-Lens-Type.html#t:Review">Review</a></code> by using <code>unto</code>. You can also use any <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code> or <code><a href="Control-Lens-Type.html#t:Iso">Iso</a></code>
directly as a <code><a href="Control-Lens-Type.html#t:Review">Review</a></code>.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:AReview" class="def">AReview</a> t b = <a href="Control-Lens-Type.html#t:Optic-39-">Optic'</a> <a href="file:///usr/share/doc/libghc-tagged-doc/html/Data-Tagged.html#t:Tagged">Tagged</a> <a href="Control-Lens-Setter.html#t:Identity">Identity</a> t b <a href="src/Control-Lens-Type.html#AReview" class="link">Source</a> <a href="#t:AReview" class="selflink">#</a></p><div class="doc"><p>If you see this in a signature for a function, the function is expecting a <code><a href="Control-Lens-Type.html#t:Review">Review</a></code>
(in practice, this usually means a <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code>).</p></div></div><h1 id="g:2">Lenses, Folds and Traversals</h1><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Lens" class="def">Lens</a> s t a b = <span class="keyword">forall</span> f. <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f => (a -> f b) -> s -> f t <a href="src/Control-Lens-Type.html#Lens" class="link">Source</a> <a href="#t:Lens" class="selflink">#</a></p><div class="doc"><p>A <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> is actually a lens family as described in
<a href="http://comonad.com/reader/2012/mirrored-lenses/">http://comonad.com/reader/2012/mirrored-lenses/</a>.</p><p>With great power comes great responsibility and a <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> is subject to the
three common sense <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> laws:</p><p>1) You get back what you put in:</p><pre><code><a href="Control-Lens-Getter.html#v:view">view</a></code> l (<code><a href="Control-Lens-Setter.html#v:set">set</a></code> l v s) ≡ v
</pre><p>2) Putting back what you got doesn't change anything:</p><pre><code><a href="Control-Lens-Setter.html#v:set">set</a></code> l (<code><a href="Control-Lens-Getter.html#v:view">view</a></code> l s) s ≡ s
</pre><p>3) Setting twice is the same as setting once:</p><pre><code><a href="Control-Lens-Setter.html#v:set">set</a></code> l v' (<code><a href="Control-Lens-Setter.html#v:set">set</a></code> l v s) ≡ <code><a href="Control-Lens-Setter.html#v:set">set</a></code> l v' s
</pre><p>These laws are strong enough that the 4 type parameters of a <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> cannot
vary fully independently. For more on how they interact, read the "Why is
it a Lens Family?" section of
<a href="http://comonad.com/reader/2012/mirrored-lenses/">http://comonad.com/reader/2012/mirrored-lenses/</a>.</p><p>There are some emergent properties of these laws:</p><p>1) <code><code><a href="Control-Lens-Setter.html#v:set">set</a></code> l s</code> must be injective for every <code>s</code> This is a consequence of law #1</p><p>2) <code><code><a href="Control-Lens-Setter.html#v:set">set</a></code> l</code> must be surjective, because of law #2, which indicates that it is possible to obtain any <code>v</code> from some <code>s</code> such that <code><code><a href="Control-Lens-Setter.html#v:set">set</a></code> s v = s</code></p><p>3) Given just the first two laws you can prove a weaker form of law #3 where the values <code>v</code> that you are setting match:</p><pre><code><a href="Control-Lens-Setter.html#v:set">set</a></code> l v (<code><a href="Control-Lens-Setter.html#v:set">set</a></code> l v s) ≡ <code><a href="Control-Lens-Setter.html#v:set">set</a></code> l v s
</pre><p>Every <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> can be used directly as a <code><a href="Control-Lens-Setter.html#v:Setter">Setter</a></code> or <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code>.</p><p>You can also use a <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> for <code><a href="Control-Lens-Getter.html#v:Getting">Getting</a></code> as if it were a
<code><a href="Control-Lens-Type.html#t:Fold">Fold</a></code> or <code><a href="Control-Lens-Type.html#t:Getter">Getter</a></code>.</p><p>Since every <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> is a valid <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code>, the
<code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> laws are required of any <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> you create:</p><pre>l <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#v:pure">pure</a></code> ≡ <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#v:pure">pure</a></code>
<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#v:fmap">fmap</a></code> (l f) <code>.</code> l g ≡ <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor-Compose.html#v:getCompose">getCompose</a></code> <code>.</code> l (<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor-Compose.html#v:Compose">Compose</a></code> <code>.</code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#v:fmap">fmap</a></code> f <code>.</code> g)
</pre><pre>type <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> s t a b = forall f. <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a></code> f => <code><a href="Control-Lens-Type.html#t:LensLike">LensLike</a></code> f s t a b
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Lens-39-" class="def">Lens'</a> s a = <a href="Control-Lens-Type.html#t:Lens">Lens</a> s s a a <a href="src/Control-Lens-Type.html#Lens%27" class="link">Source</a> <a href="#t:Lens-39-" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:Lens-39-">Lens'</a></code> = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code>
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Traversal" class="def">Traversal</a> s t a b = <span class="keyword">forall</span> f. <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f => (a -> f b) -> s -> f t <a href="src/Control-Lens-Type.html#Traversal" class="link">Source</a> <a href="#t:Traversal" class="selflink">#</a></p><div class="doc"><p>A <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> can be used directly as a <code><a href="Control-Lens-Setter.html#v:Setter">Setter</a></code> or a <code><a href="Control-Lens-Type.html#t:Fold">Fold</a></code> (but not as a <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code>) and provides
the ability to both read and update multiple fields, subject to some relatively weak <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> laws.</p><p>These have also been known as multilenses, but they have the signature and spirit of</p><pre><code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Traversable.html#v:traverse">traverse</a></code> :: <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Traversable.html#v:Traversable">Traversable</a></code> f => <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> (f a) (f b) a b
</pre><p>and the more evocative name suggests their application.</p><p>Most of the time the <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> you will want to use is just <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Traversable.html#v:traverse">traverse</a></code>, but you can also pass any
<code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> or <code><a href="Control-Lens-Type.html#t:Iso">Iso</a></code> as a <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code>, and composition of a <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> (or <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> or <code><a href="Control-Lens-Type.html#t:Iso">Iso</a></code>) with a <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> (or <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> or <code><a href="Control-Lens-Type.html#t:Iso">Iso</a></code>)
using (<code>.</code>) forms a valid <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code>.</p><p>The laws for a <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> <code>t</code> follow from the laws for <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Traversable.html#v:Traversable">Traversable</a></code> as stated in "The Essence of the Iterator Pattern".</p><pre>t <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#v:pure">pure</a></code> ≡ <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#v:pure">pure</a></code>
<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#v:fmap">fmap</a></code> (t f) <code>.</code> t g ≡ <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor-Compose.html#v:getCompose">getCompose</a></code> <code>.</code> t (<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor-Compose.html#v:Compose">Compose</a></code> <code>.</code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#v:fmap">fmap</a></code> f <code>.</code> g)
</pre><p>One consequence of this requirement is that a <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> needs to leave the same number of elements as a
candidate for subsequent <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> that it started with. Another testament to the strength of these laws
is that the caveat expressed in section 5.5 of the "Essence of the Iterator Pattern" about exotic
<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Traversable.html#v:Traversable">Traversable</a></code> instances that <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Traversable.html#v:traverse">traverse</a></code> the same entry multiple times was actually already ruled out by the
second law in that same paper!</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Traversal-39-" class="def">Traversal'</a> s a = <a href="Control-Lens-Type.html#t:Traversal">Traversal</a> s s a a <a href="src/Control-Lens-Type.html#Traversal%27" class="link">Source</a> <a href="#t:Traversal-39-" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:Traversal-39-">Traversal'</a></code> = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code>
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Traversal1" class="def">Traversal1</a> s t a b = <span class="keyword">forall</span> f. <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f => (a -> f b) -> s -> f t <a href="src/Control-Lens-Type.html#Traversal1" class="link">Source</a> <a href="#t:Traversal1" class="selflink">#</a></p></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Traversal1-39-" class="def">Traversal1'</a> s a = <a href="Control-Lens-Type.html#t:Traversal1">Traversal1</a> s s a a <a href="src/Control-Lens-Type.html#Traversal1%27" class="link">Source</a> <a href="#t:Traversal1-39-" class="selflink">#</a></p></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Setter" class="def">Setter</a> s t a b = <span class="keyword">forall</span> f. <a href="Control-Lens-Internal-Setter.html#t:Settable">Settable</a> f => (a -> f b) -> s -> f t <a href="src/Control-Lens-Type.html#Setter" class="link">Source</a> <a href="#t:Setter" class="selflink">#</a></p><div class="doc"><p>The only <code><a href="Control-Lens-Type.html#t:LensLike">LensLike</a></code> law that can apply to a <code><a href="Control-Lens-Type.html#t:Setter">Setter</a></code> <code>l</code> is that</p><pre><code><a href="Control-Lens-Setter.html#v:set">set</a></code> l y (<code><a href="Control-Lens-Setter.html#v:set">set</a></code> l x a) ≡ <code><a href="Control-Lens-Setter.html#v:set">set</a></code> l y a
</pre><p>You can't <code><a href="Control-Lens-Getter.html#v:view">view</a></code> a <code><a href="Control-Lens-Type.html#t:Setter">Setter</a></code> in general, so the other two laws are irrelevant.</p><p>However, two <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a></code> laws apply to a <code><a href="Control-Lens-Type.html#t:Setter">Setter</a></code>:</p><pre><code><a href="Control-Lens-Setter.html#v:over">over</a></code> l <code>id</code> ≡ <code>id</code>
<code><a href="Control-Lens-Setter.html#v:over">over</a></code> l f <code>.</code> <code><a href="Control-Lens-Setter.html#v:over">over</a></code> l g ≡ <code><a href="Control-Lens-Setter.html#v:over">over</a></code> l (f <code>.</code> g)
</pre><p>These can be stated more directly:</p><pre>l <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#v:pure">pure</a></code> ≡ <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#v:pure">pure</a></code>
l f <code>.</code> <code><a href="Control-Lens-Internal-Setter.html#v:untainted">untainted</a></code> <code>.</code> l g ≡ l (f <code>.</code> <code><a href="Control-Lens-Internal-Setter.html#v:untainted">untainted</a></code> <code>.</code> g)
</pre><p>You can compose a <code><a href="Control-Lens-Type.html#t:Setter">Setter</a></code> with a <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> or a <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> using (<code>.</code>) from the <code>Prelude</code>
and the result is always only a <code><a href="Control-Lens-Type.html#t:Setter">Setter</a></code> and nothing more.</p><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>over traverse f [a,b,c,d]
</code></strong>[f a,f b,f c,f d]
</pre><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>over _1 f (a,b)
</code></strong>(f a,b)
</pre><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>over (traverse._1) f [(a,b),(c,d)]
</code></strong>[(f a,b),(f c,d)]
</pre><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>over both f (a,b)
</code></strong>(f a,f b)
</pre><pre class="screen"><code class="prompt">>>> </code><strong class="userinput"><code>over (traverse.both) f [(a,b),(c,d)]
</code></strong>[(f a,f b),(f c,f d)]
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Setter-39-" class="def">Setter'</a> s a = <a href="Control-Lens-Type.html#t:Setter">Setter</a> s s a a <a href="src/Control-Lens-Type.html#Setter%27" class="link">Source</a> <a href="#t:Setter-39-" class="selflink">#</a></p><div class="doc"><p>A <code><a href="Control-Lens-Type.html#t:Setter-39-">Setter'</a></code> is just a <code><a href="Control-Lens-Type.html#t:Setter">Setter</a></code> that doesn't change the types.</p><p>These are particularly common when talking about monomorphic containers. <em>e.g.</em></p><pre><code>sets</code> Data.Text.map :: <code><a href="Control-Lens-Type.html#t:Setter-39-">Setter'</a></code> <code><a href="file:///usr/share/doc/libghc-text-doc/html/Data-Text-Internal.html#v:Text">Text</a></code> <code>Char</code>
</pre><pre>type <code><a href="Control-Lens-Type.html#t:Setter-39-">Setter'</a></code> = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:Setter">Setter</a></code>
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Getter" class="def">Getter</a> s a = <span class="keyword">forall</span> f. (<a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => (a -> f a) -> s -> f s <a href="src/Control-Lens-Type.html#Getter" class="link">Source</a> <a href="#t:Getter" class="selflink">#</a></p><div class="doc"><p>A <code><a href="Control-Lens-Type.html#t:Getter">Getter</a></code> describes how to retrieve a single value in a way that can be
composed with other <code><a href="Control-Lens-Type.html#t:LensLike">LensLike</a></code> constructions.</p><p>Unlike a <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> a <code><a href="Control-Lens-Type.html#t:Getter">Getter</a></code> is read-only. Since a <code><a href="Control-Lens-Type.html#t:Getter">Getter</a></code>
cannot be used to write back there are no <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> laws that can be applied to
it. In fact, it is isomorphic to an arbitrary function from <code>(s -> a)</code>.</p><p>Moreover, a <code><a href="Control-Lens-Type.html#t:Getter">Getter</a></code> can be used directly as a <code><a href="Control-Lens-Fold.html#v:Fold">Fold</a></code>,
since it just ignores the <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a></code>.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Fold" class="def">Fold</a> s a = <span class="keyword">forall</span> f. (<a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => (a -> f a) -> s -> f s <a href="src/Control-Lens-Type.html#Fold" class="link">Source</a> <a href="#t:Fold" class="selflink">#</a></p><div class="doc"><p>A <code><a href="Control-Lens-Type.html#t:Fold">Fold</a></code> describes how to retrieve multiple values in a way that can be composed
with other <code><a href="Control-Lens-Type.html#t:LensLike">LensLike</a></code> constructions.</p><p>A <code><code><a href="Control-Lens-Type.html#t:Fold">Fold</a></code> s a</code> provides a structure with operations very similar to those of the <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Foldable.html#v:Foldable">Foldable</a></code>
typeclass, see <code><a href="Control-Lens-Fold.html#v:foldMapOf">foldMapOf</a></code> and the other <code><a href="Control-Lens-Type.html#t:Fold">Fold</a></code> combinators.</p><p>By convention, if there exists a <code>foo</code> method that expects a <code><code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Foldable.html#v:Foldable">Foldable</a></code> (f a)</code>, then there should be a
<code>fooOf</code> method that takes a <code><code><a href="Control-Lens-Type.html#t:Fold">Fold</a></code> s a</code> and a value of type <code>s</code>.</p><p>A <code><a href="Control-Lens-Type.html#t:Getter">Getter</a></code> is a legal <code><a href="Control-Lens-Type.html#t:Fold">Fold</a></code> that just ignores the supplied <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Monoid.html#v:Monoid">Monoid</a></code>.</p><p>Unlike a <code><a href="Control-Lens-Traversal.html#v:Traversal">Traversal</a></code> a <code><a href="Control-Lens-Type.html#t:Fold">Fold</a></code> is read-only. Since a <code><a href="Control-Lens-Type.html#t:Fold">Fold</a></code> cannot be used to write back
there are no <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> laws that apply.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Fold1" class="def">Fold1</a> s a = <span class="keyword">forall</span> f. (<a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f) => (a -> f a) -> s -> f s <a href="src/Control-Lens-Type.html#Fold1" class="link">Source</a> <a href="#t:Fold1" class="selflink">#</a></p><div class="doc"><p>A relevant Fold (aka <code><a href="Control-Lens-Type.html#t:Fold1">Fold1</a></code>) has one or more targets.</p></div></div><h1 id="g:3">Indexed</h1><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedLens" class="def">IndexedLens</a> i s t a b = <span class="keyword">forall</span> f p. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => p a (f b) -> s -> f t <a href="src/Control-Lens-Type.html#IndexedLens" class="link">Source</a> <a href="#t:IndexedLens" class="selflink">#</a></p><div class="doc"><p>Every <code><a href="Control-Lens-Type.html#t:IndexedLens">IndexedLens</a></code> is a valid <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> and a valid <code><a href="Control-Lens-Traversal.html#v:IndexedTraversal">IndexedTraversal</a></code>.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedLens-39-" class="def">IndexedLens'</a> i s a = <a href="Control-Lens-Type.html#t:IndexedLens">IndexedLens</a> i s s a a <a href="src/Control-Lens-Type.html#IndexedLens%27" class="link">Source</a> <a href="#t:IndexedLens-39-" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:IndexedLens-39-">IndexedLens'</a></code> i = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> (<code><a href="Control-Lens-Type.html#t:IndexedLens">IndexedLens</a></code> i)
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedTraversal" class="def">IndexedTraversal</a> i s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => p a (f b) -> s -> f t <a href="src/Control-Lens-Type.html#IndexedTraversal" class="link">Source</a> <a href="#t:IndexedTraversal" class="selflink">#</a></p><div class="doc"><p>Every <code><a href="Control-Lens-Type.html#t:IndexedTraversal">IndexedTraversal</a></code> is a valid <code><a href="Control-Lens-Traversal.html#v:Traversal">Traversal</a></code> or
<code><a href="Control-Lens-Fold.html#v:IndexedFold">IndexedFold</a></code>.</p><p>The <code><a href="Control-Lens-Internal-Indexed.html#t:Indexed">Indexed</a></code> constraint is used to allow an <code><a href="Control-Lens-Type.html#t:IndexedTraversal">IndexedTraversal</a></code> to be used
directly as a <code><a href="Control-Lens-Traversal.html#v:Traversal">Traversal</a></code>.</p><p>The <code><a href="Control-Lens-Traversal.html#v:Traversal">Traversal</a></code> laws are still required to hold.</p><p>In addition, the index <code>i</code> should satisfy the requirement that it stays
unchanged even when modifying the value <code>a</code>, otherwise traversals like
<code>indices</code> break the <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> laws.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedTraversal-39-" class="def">IndexedTraversal'</a> i s a = <a href="Control-Lens-Type.html#t:IndexedTraversal">IndexedTraversal</a> i s s a a <a href="src/Control-Lens-Type.html#IndexedTraversal%27" class="link">Source</a> <a href="#t:IndexedTraversal-39-" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:IndexedTraversal-39-">IndexedTraversal'</a></code> i = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> (<code><a href="Control-Lens-Type.html#t:IndexedTraversal">IndexedTraversal</a></code> i)
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedTraversal1" class="def">IndexedTraversal1</a> i s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f) => p a (f b) -> s -> f t <a href="src/Control-Lens-Type.html#IndexedTraversal1" class="link">Source</a> <a href="#t:IndexedTraversal1" class="selflink">#</a></p></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedTraversal1-39-" class="def">IndexedTraversal1'</a> i s a = <a href="Control-Lens-Type.html#t:IndexedTraversal1">IndexedTraversal1</a> i s s a a <a href="src/Control-Lens-Type.html#IndexedTraversal1%27" class="link">Source</a> <a href="#t:IndexedTraversal1-39-" class="selflink">#</a></p></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedSetter" class="def">IndexedSetter</a> i s t a b = <span class="keyword">forall</span> f p. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="Control-Lens-Internal-Setter.html#t:Settable">Settable</a> f) => p a (f b) -> s -> f t <a href="src/Control-Lens-Type.html#IndexedSetter" class="link">Source</a> <a href="#t:IndexedSetter" class="selflink">#</a></p><div class="doc"><p>Every <code><a href="Control-Lens-Type.html#t:IndexedSetter">IndexedSetter</a></code> is a valid <code><a href="Control-Lens-Type.html#t:Setter">Setter</a></code>.</p><p>The <code><a href="Control-Lens-Type.html#t:Setter">Setter</a></code> laws are still required to hold.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedSetter-39-" class="def">IndexedSetter'</a> i s a = <a href="Control-Lens-Type.html#t:IndexedSetter">IndexedSetter</a> i s s a a <a href="src/Control-Lens-Type.html#IndexedSetter%27" class="link">Source</a> <a href="#t:IndexedSetter-39-" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:IndexedSetter-39-">IndexedSetter'</a></code> i = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> (<code><a href="Control-Lens-Type.html#t:IndexedSetter">IndexedSetter</a></code> i)
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedGetter" class="def">IndexedGetter</a> i s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => p a (f a) -> s -> f s <a href="src/Control-Lens-Type.html#IndexedGetter" class="link">Source</a> <a href="#t:IndexedGetter" class="selflink">#</a></p><div class="doc"><p>Every <code><a href="Control-Lens-Type.html#t:IndexedGetter">IndexedGetter</a></code> is a valid <code><a href="Control-Lens-Fold.html#v:IndexedFold">IndexedFold</a></code> and can be used for <code><a href="Control-Lens-Getter.html#v:Getting">Getting</a></code> like a <code><a href="Control-Lens-Type.html#t:Getter">Getter</a></code>.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedFold" class="def">IndexedFold</a> i s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => p a (f a) -> s -> f s <a href="src/Control-Lens-Type.html#IndexedFold" class="link">Source</a> <a href="#t:IndexedFold" class="selflink">#</a></p><div class="doc"><p>Every <code><a href="Control-Lens-Type.html#t:IndexedFold">IndexedFold</a></code> is a valid <code><a href="Control-Lens-Fold.html#v:Fold">Fold</a></code> and can be used for <code><a href="Control-Lens-Getter.html#v:Getting">Getting</a></code>.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedFold1" class="def">IndexedFold1</a> i s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f) => p a (f a) -> s -> f s <a href="src/Control-Lens-Type.html#IndexedFold1" class="link">Source</a> <a href="#t:IndexedFold1" class="selflink">#</a></p></div><h1 id="g:4">Index-Preserving</h1><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexPreservingLens" class="def">IndexPreservingLens</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => p a (f b) -> p s (f t) <a href="src/Control-Lens-Type.html#IndexPreservingLens" class="link">Source</a> <a href="#t:IndexPreservingLens" class="selflink">#</a></p><div class="doc"><p>An <code><a href="Control-Lens-Type.html#t:IndexPreservingLens">IndexPreservingLens</a></code> leaves any index it is composed with alone.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexPreservingLens-39-" class="def">IndexPreservingLens'</a> s a = <a href="Control-Lens-Type.html#t:IndexPreservingLens">IndexPreservingLens</a> s s a a <a href="src/Control-Lens-Type.html#IndexPreservingLens%27" class="link">Source</a> <a href="#t:IndexPreservingLens-39-" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:IndexPreservingLens-39-">IndexPreservingLens'</a></code> = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:IndexPreservingLens">IndexPreservingLens</a></code>
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexPreservingTraversal" class="def">IndexPreservingTraversal</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => p a (f b) -> p s (f t) <a href="src/Control-Lens-Type.html#IndexPreservingTraversal" class="link">Source</a> <a href="#t:IndexPreservingTraversal" class="selflink">#</a></p><div class="doc"><p>An <code><a href="Control-Lens-Type.html#t:IndexPreservingLens">IndexPreservingLens</a></code> leaves any index it is composed with alone.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexPreservingTraversal-39-" class="def">IndexPreservingTraversal'</a> s a = <a href="Control-Lens-Type.html#t:IndexPreservingTraversal">IndexPreservingTraversal</a> s s a a <a href="src/Control-Lens-Type.html#IndexPreservingTraversal%27" class="link">Source</a> <a href="#t:IndexPreservingTraversal-39-" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:IndexPreservingTraversal-39-">IndexPreservingTraversal'</a></code> = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:IndexPreservingTraversal">IndexPreservingTraversal</a></code>
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexPreservingTraversal1" class="def">IndexPreservingTraversal1</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f) => p a (f b) -> p s (f t) <a href="src/Control-Lens-Type.html#IndexPreservingTraversal1" class="link">Source</a> <a href="#t:IndexPreservingTraversal1" class="selflink">#</a></p></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexPreservingTraversal1-39-" class="def">IndexPreservingTraversal1'</a> s a = <a href="Control-Lens-Type.html#t:IndexPreservingTraversal1">IndexPreservingTraversal1</a> s s a a <a href="src/Control-Lens-Type.html#IndexPreservingTraversal1%27" class="link">Source</a> <a href="#t:IndexPreservingTraversal1-39-" class="selflink">#</a></p></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexPreservingSetter" class="def">IndexPreservingSetter</a> s t a b = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="Control-Lens-Internal-Setter.html#t:Settable">Settable</a> f) => p a (f b) -> p s (f t) <a href="src/Control-Lens-Type.html#IndexPreservingSetter" class="link">Source</a> <a href="#t:IndexPreservingSetter" class="selflink">#</a></p><div class="doc"><p>An <code><a href="Control-Lens-Type.html#t:IndexPreservingSetter">IndexPreservingSetter</a></code> can be composed with a <code><a href="Control-Lens-Type.html#t:IndexedSetter">IndexedSetter</a></code>, <code><a href="Control-Lens-Type.html#t:IndexedTraversal">IndexedTraversal</a></code> or <code><a href="Control-Lens-Type.html#t:IndexedLens">IndexedLens</a></code>
and leaves the index intact, yielding an <code><a href="Control-Lens-Type.html#t:IndexedSetter">IndexedSetter</a></code>.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexPreservingSetter-39-" class="def">IndexPreservingSetter'</a> s a = <a href="Control-Lens-Type.html#t:IndexPreservingSetter">IndexPreservingSetter</a> s s a a <a href="src/Control-Lens-Type.html#IndexPreservingSetter%27" class="link">Source</a> <a href="#t:IndexPreservingSetter-39-" class="selflink">#</a></p><div class="doc"><pre>type <code>IndexedPreservingSetter'</code> i = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code>IndexedPreservingSetter</code>
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexPreservingGetter" class="def">IndexPreservingGetter</a> s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a> f) => p a (f a) -> p s (f s) <a href="src/Control-Lens-Type.html#IndexPreservingGetter" class="link">Source</a> <a href="#t:IndexPreservingGetter" class="selflink">#</a></p><div class="doc"><p>An <code><a href="Control-Lens-Type.html#t:IndexPreservingGetter">IndexPreservingGetter</a></code> can be used as a <code><a href="Control-Lens-Type.html#t:Getter">Getter</a></code>, but when composed with an <code><a href="Control-Lens-Type.html#t:IndexedTraversal">IndexedTraversal</a></code>,
<code><a href="Control-Lens-Type.html#t:IndexedFold">IndexedFold</a></code>, or <code><a href="Control-Lens-Type.html#t:IndexedLens">IndexedLens</a></code> yields an <code><a href="Control-Lens-Type.html#t:IndexedFold">IndexedFold</a></code>, <code><a href="Control-Lens-Type.html#t:IndexedFold">IndexedFold</a></code> or <code><a href="Control-Lens-Type.html#t:IndexedGetter">IndexedGetter</a></code> respectively.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexPreservingFold" class="def">IndexPreservingFold</a> s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a> f) => p a (f a) -> p s (f s) <a href="src/Control-Lens-Type.html#IndexPreservingFold" class="link">Source</a> <a href="#t:IndexPreservingFold" class="selflink">#</a></p><div class="doc"><p>An <code><a href="Control-Lens-Type.html#t:IndexPreservingFold">IndexPreservingFold</a></code> can be used as a <code><a href="Control-Lens-Type.html#t:Fold">Fold</a></code>, but when composed with an <code><a href="Control-Lens-Type.html#t:IndexedTraversal">IndexedTraversal</a></code>,
<code><a href="Control-Lens-Type.html#t:IndexedFold">IndexedFold</a></code>, or <code><a href="Control-Lens-Type.html#t:IndexedLens">IndexedLens</a></code> yields an <code><a href="Control-Lens-Type.html#t:IndexedFold">IndexedFold</a></code> respectively.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexPreservingFold1" class="def">IndexPreservingFold1</a> s a = <span class="keyword">forall</span> p f. (<a href="Control-Lens-Internal-Indexed.html#t:Conjoined">Conjoined</a> p, <a href="Control-Lens-Getter.html#t:Contravariant">Contravariant</a> f, <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f) => p a (f a) -> p s (f s) <a href="src/Control-Lens-Type.html#IndexPreservingFold1" class="link">Source</a> <a href="#t:IndexPreservingFold1" class="selflink">#</a></p></div><h1 id="g:5">Common</h1><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Simple" class="def">Simple</a> f s a = f s s a a <a href="src/Control-Lens-Type.html#Simple" class="link">Source</a> <a href="#t:Simple" class="selflink">#</a></p><div class="doc"><p>A <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code>, <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code>, ... can
be used instead of a <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code>,<code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code>, ...
whenever the type variables don't change upon setting a value.</p><pre><code><a href="Data-Complex-Lens.html#v:_imagPart">_imagPart</a></code> :: <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> (<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Complex.html#v:Complex">Complex</a></code> a) a
<code><a href="Control-Lens-Traversal.html#v:traversed">traversed</a></code> :: <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> (<code><a href="Control-Lens-Type.html#t:IndexedTraversal">IndexedTraversal</a></code> <code>Int</code>) [a] a
</pre><p>Note: To use this alias in your own code with <code><code><a href="Control-Lens-Type.html#t:LensLike">LensLike</a></code> f</code> or
<code><a href="Control-Lens-Type.html#t:Setter">Setter</a></code>, you may have to turn on <code>LiberalTypeSynonyms</code>.</p><p>This is commonly abbreviated as a "prime" marker, <em>e.g.</em> <code><a href="Control-Lens-Type.html#t:Lens-39-">Lens'</a></code> = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code>.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:LensLike" class="def">LensLike</a> f s t a b = (a -> f b) -> s -> f t <a href="src/Control-Lens-Type.html#LensLike" class="link">Source</a> <a href="#t:LensLike" class="selflink">#</a></p><div class="doc"><p>Many combinators that accept a <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code> can also accept a
<code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code> in limited situations.</p><p>They do so by specializing the type of <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a></code> that they require of the
caller.</p><p>If a function accepts a <code><code><a href="Control-Lens-Type.html#t:LensLike">LensLike</a></code> f s t a b</code> for some <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Data-Functor.html#t:Functor">Functor</a></code> <code>f</code>,
then they may be passed a <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code>.</p><p>Further, if <code>f</code> is an <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#t:Applicative">Applicative</a></code>, they may also be passed a
<code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code>.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:LensLike-39-" class="def">LensLike'</a> f s a = <a href="Control-Lens-Type.html#t:LensLike">LensLike</a> f s s a a <a href="src/Control-Lens-Type.html#LensLike%27" class="link">Source</a> <a href="#t:LensLike-39-" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:LensLike-39-">LensLike'</a></code> f = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> (<code><a href="Control-Lens-Type.html#t:LensLike">LensLike</a></code> f)
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Over" class="def">Over</a> p f s t a b = p a (f b) -> s -> f t <a href="src/Control-Lens-Type.html#Over" class="link">Source</a> <a href="#t:Over" class="selflink">#</a></p><div class="doc"><p>This is a convenient alias for use when you need to consume either indexed or non-indexed lens-likes based on context.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Over-39-" class="def">Over'</a> p f s a = <a href="Control-Lens-Type.html#t:Over">Over</a> p f s s a a <a href="src/Control-Lens-Type.html#Over%27" class="link">Source</a> <a href="#t:Over-39-" class="selflink">#</a></p><div class="doc"><p>This is a convenient alias for use when you need to consume either indexed or non-indexed lens-likes based on context.</p><pre>type <code><a href="Control-Lens-Type.html#t:Over-39-">Over'</a></code> p f = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> (<code><a href="Control-Lens-Type.html#t:Over">Over</a></code> p f)
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedLensLike" class="def">IndexedLensLike</a> i f s t a b = <span class="keyword">forall</span> p. <a href="Control-Lens-Internal-Indexed.html#t:Indexable">Indexable</a> i p => p a (f b) -> s -> f t <a href="src/Control-Lens-Type.html#IndexedLensLike" class="link">Source</a> <a href="#t:IndexedLensLike" class="selflink">#</a></p><div class="doc"><p>Convenient alias for constructing indexed lenses and their ilk.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:IndexedLensLike-39-" class="def">IndexedLensLike'</a> i f s a = <a href="Control-Lens-Type.html#t:IndexedLensLike">IndexedLensLike</a> i f s s a a <a href="src/Control-Lens-Type.html#IndexedLensLike%27" class="link">Source</a> <a href="#t:IndexedLensLike-39-" class="selflink">#</a></p><div class="doc"><p>Convenient alias for constructing simple indexed lenses and their ilk.</p></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Optical" class="def">Optical</a> p q f s t a b = p a (f b) -> q s (f t) <a href="src/Control-Lens-Type.html#Optical" class="link">Source</a> <a href="#t:Optical" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:LensLike">LensLike</a></code> f s t a b = <code><a href="Control-Lens-Type.html#t:Optical">Optical</a></code> (->) (->) f s t a b
</pre><pre>type <code><a href="Control-Lens-Type.html#t:Over">Over</a></code> p f s t a b = <code><a href="Control-Lens-Type.html#t:Optical">Optical</a></code> p (->) f s t a b
</pre><pre>type <code><a href="Control-Lens-Type.html#t:Optic">Optic</a></code> p f s t a b = <code><a href="Control-Lens-Type.html#t:Optical">Optical</a></code> p p f s t a b
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Optical-39-" class="def">Optical'</a> p q f s a = <a href="Control-Lens-Type.html#t:Optical">Optical</a> p q f s s a a <a href="src/Control-Lens-Type.html#Optical%27" class="link">Source</a> <a href="#t:Optical-39-" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:Optical-39-">Optical'</a></code> p q f s a = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> (<code><a href="Control-Lens-Type.html#t:Optical">Optical</a></code> p q f) s a
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Optic" class="def">Optic</a> p f s t a b = p a (f b) -> p s (f t) <a href="src/Control-Lens-Type.html#Optic" class="link">Source</a> <a href="#t:Optic" class="selflink">#</a></p><div class="doc"><p>A valid <code><a href="Control-Lens-Type.html#t:Optic">Optic</a></code> <code>l</code> should satisfy the laws:</p><pre>l <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#v:pure">pure</a></code> ≡ <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.1.0/Control-Applicative.html#v:pure">pure</a></code>
l (<code>Procompose</code> f g) = <code>Procompose</code> (l f) (l g)
</pre><p>This gives rise to the laws for <code><a href="Control-Lens-Type.html#t:Equality">Equality</a></code>, <code><a href="Control-Lens-Type.html#t:Iso">Iso</a></code>, <code><a href="Control-Lens-Type.html#t:Prism">Prism</a></code>, <code><a href="Control-Lens-Type.html#t:Lens">Lens</a></code>,
<code><a href="Control-Lens-Type.html#t:Traversal">Traversal</a></code>, <code><a href="Control-Lens-Type.html#t:Traversal1">Traversal1</a></code>, <code><a href="Control-Lens-Type.html#t:Setter">Setter</a></code>, <code><a href="Control-Lens-Type.html#t:Fold">Fold</a></code>, <code><a href="Control-Lens-Type.html#t:Fold1">Fold1</a></code>, and <code><a href="Control-Lens-Type.html#t:Getter">Getter</a></code> as well
along with their index-preserving variants.</p><pre>type <code><a href="Control-Lens-Type.html#t:LensLike">LensLike</a></code> f s t a b = <code><a href="Control-Lens-Type.html#t:Optic">Optic</a></code> (->) f s t a b
</pre></div></div><div class="top"><p class="src"><span class="keyword">type</span> <a id="t:Optic-39-" class="def">Optic'</a> p f s a = <a href="Control-Lens-Type.html#t:Optic">Optic</a> p f s s a a <a href="src/Control-Lens-Type.html#Optic%27" class="link">Source</a> <a href="#t:Optic-39-" class="selflink">#</a></p><div class="doc"><pre>type <code><a href="Control-Lens-Type.html#t:Optic-39-">Optic'</a></code> p f s a = <code><a href="Control-Lens-Type.html#t:Simple">Simple</a></code> (<code><a href="Control-Lens-Type.html#t:Optic">Optic</a></code> p f) s a
</pre></div></div></div></div><div id="footer"><p>Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.17.3</p></div></body></html>
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