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</script></head><body><div id="package-header"><ul class="links" id="page-menu"><li><a href="src/Data-Functor-Contravariant-Divisible.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">contravariant-1.3.3: Contravariant functors</p></div><div id="content"><div id="module-header"><table class="info"><tr><th>Copyright</th><td>(C) 2014-2015 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>portable</td></tr><tr><th>Safe Haskell</th><td>Safe</td></tr><tr><th>Language</th><td>Haskell98</td></tr></table><p class="caption">Data.Functor.Contravariant.Divisible</p></div><div id="table-of-contents"><p class="caption">Contents</p><ul><li><a href="#g:1">Contravariant Applicative</a></li><li><a href="#g:2">Contravariant Alternative</a></li></ul></div><div id="description"><p class="caption">Description</p><div class="doc"><p>This module supplies contravariant analogues to the <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Applicative">Applicative</a></code> and <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Alternative">Alternative</a></code> classes.</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">class</span> <a href="Data-Functor-Contravariant.html#t:Contravariant">Contravariant</a> f => <a href="#t:Divisible">Divisible</a> f <span class="keyword">where</span><ul class="subs"><li><a href="#v:divide">divide</a> :: (a -> (b, c)) -> f b -> f c -> f a</li><li><a href="#v:conquer">conquer</a> :: f a</li></ul></li><li class="src short"><a href="#v:divided">divided</a> :: <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> f => f a -> f b -> f (a, b)</li><li class="src short"><a href="#v:conquered">conquered</a> :: <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> f => f ()</li><li class="src short"><a href="#v:liftD">liftD</a> :: <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> f => (a -> b) -> f b -> f a</li><li class="src short"><span class="keyword">class</span> <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> f => <a href="#t:Decidable">Decidable</a> f <span class="keyword">where</span><ul class="subs"><li><a href="#v:lose">lose</a> :: (a -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Void.html#t:Void">Void</a>) -> f a</li><li><a href="#v:choose">choose</a> :: (a -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#t:Either">Either</a> b c) -> f b -> f c -> f a</li></ul></li><li class="src short"><a href="#v:chosen">chosen</a> :: <a href="Data-Functor-Contravariant-Divisible.html#t:Decidable">Decidable</a> f => f b -> f c -> f (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#t:Either">Either</a> b c)</li><li class="src short"><a href="#v:lost">lost</a> :: <a href="Data-Functor-Contravariant-Divisible.html#t:Decidable">Decidable</a> f => f <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Void.html#t:Void">Void</a></li></ul></div><div id="interface"><h1 id="g:1">Contravariant Applicative</h1><div class="top"><p class="src"><span class="keyword">class</span> <a href="Data-Functor-Contravariant.html#t:Contravariant">Contravariant</a> f => <a name="t:Divisible" class="def">Divisible</a> f <span class="keyword">where</span> <a href="src/Data-Functor-Contravariant-Divisible.html#Divisible" class="link">Source</a></p><div class="doc"><p>A <code><a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a></code> contravariant functor is the contravariant analogue of <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Applicative">Applicative</a></code>.</p><p>In denser jargon, a <code><a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a></code> contravariant functor is a monoid object in the category
of presheaves from Hask to Hask, equipped with Day convolution mapping the Cartesian
product of the source to the Cartesian product of the target.</p><p>By way of contrast, an <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Applicative">Applicative</a></code> functor can be viewed as a monoid object in the
category of copresheaves from Hask to Hask, equipped with Day convolution mapping the
Cartesian product of the source to the Cartesian product of the target.</p><p>Given the canonical diagonal morphism:</p><pre>delta a = (a,a)
</pre><p><code><code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> <code>delta</code></code> should be associative with <code><a href="Data-Functor-Contravariant-Divisible.html#v:conquer">conquer</a></code> as a unit</p><pre><code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> <code>delta</code> m <code><a href="Data-Functor-Contravariant-Divisible.html#v:conquer">conquer</a></code> = m
<code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> <code>delta</code> <code><a href="Data-Functor-Contravariant-Divisible.html#v:conquer">conquer</a></code> m = m
<code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> <code>delta</code> (<code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> <code>delta</code> m n) o = <code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> <code>delta</code> m (<code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> <code>delta</code> n o)
</pre><p>With more general arguments you'll need to reassociate and project using the monoidal
structure of the source category. (Here fst and snd are used in lieu of the more restricted
lambda and rho, but this construction works with just a monoidal category.)</p><pre><code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> f m <code><a href="Data-Functor-Contravariant-Divisible.html#v:conquer">conquer</a></code> = <code><a href="Data-Functor-Contravariant.html#v:contramap">contramap</a></code> (<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Tuple.html#v:fst">fst</a></code> . f) m
<code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> f <code><a href="Data-Functor-Contravariant-Divisible.html#v:conquer">conquer</a></code> m = <code><a href="Data-Functor-Contravariant.html#v:contramap">contramap</a></code> (<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Tuple.html#v:snd">snd</a></code> . f) m
<code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> f (<code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> g m n) o = <code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> f' m (<code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:id">id</a></code> n o) where
f' a = case f a of (bc,d) -> case g bc of (b,c) -> (a,(b,c))
</pre></div><div class="subs methods"><p class="caption">Methods</p><p class="src"><a name="v:divide" class="def">divide</a> :: (a -> (b, c)) -> f b -> f c -> f a <a href="src/Data-Functor-Contravariant-Divisible.html#divide" class="link">Source</a></p><p class="src"><a name="v:conquer" class="def">conquer</a> :: f a <a href="src/Data-Functor-Contravariant-Divisible.html#conquer" class="link">Source</a></p><div class="doc"><p>The underlying theory would suggest that this should be:</p><pre>conquer :: (a -> ()) -> f a
</pre><p>However, as we are working over a Cartesian category (Hask) and the Cartesian product, such an input
morphism is uniquely determined to be <code><code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:const">const</a></code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Monoid.html#v:mempty">mempty</a></code></code>, so we elide it.</p></div></div><div class="subs instances"><p id="control.i:Divisible" class="caption collapser" onclick="toggleSection('i:Divisible')">Instances</p><div id="section.i:Divisible" class="show"><table><tr><td class="src clearfix"><span class="inst-left"><a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> <a href="Data-Functor-Contravariant.html#t:Equivalence">Equivalence</a></span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-123" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> <a href="Data-Functor-Contravariant.html#t:Comparison">Comparison</a></span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-117" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> <a href="Data-Functor-Contravariant.html#t:Predicate">Predicate</a></span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-129" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Monoid.html#t:Monoid">Monoid</a> m => <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Const">Const</a> m)</span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-134" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Monoid.html#t:Monoid">Monoid</a> r => <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> (<a href="Data-Functor-Contravariant.html#t:Op">Op</a> r)</span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-112" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left">(<a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> f, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Applicative">Applicative</a> g) => <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> (<a href="Data-Functor-Contravariant-Compose.html#t:ComposeCF">ComposeCF</a> f g)</span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-60" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left">(<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Applicative">Applicative</a> f, <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> g) => <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> (<a href="Data-Functor-Contravariant-Compose.html#t:ComposeFC">ComposeFC</a> f g)</span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-43" class="link">Source</a></td><td class="doc empty"> </td></tr></table></div></div></div><div class="top"><p class="src"><a name="v:divided" class="def">divided</a> :: <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> f => f a -> f b -> f (a, b) <a href="src/Data-Functor-Contravariant-Divisible.html#divided" class="link">Source</a></p><div class="doc"><pre><code><a href="Data-Functor-Contravariant-Divisible.html#v:divided">divided</a></code> = <code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:id">id</a></code>
</pre></div></div><div class="top"><p class="src"><a name="v:conquered" class="def">conquered</a> :: <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> f => f () <a href="src/Data-Functor-Contravariant-Divisible.html#conquered" class="link">Source</a></p><div class="doc"><p>Redundant, but provided for symmetry.</p><pre><code><a href="Data-Functor-Contravariant-Divisible.html#v:conquered">conquered</a></code> = 'conquer
</pre></div></div><div class="top"><p class="src"><a name="v:liftD" class="def">liftD</a> :: <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> f => (a -> b) -> f b -> f a <a href="src/Data-Functor-Contravariant-Divisible.html#liftD" class="link">Source</a></p><div class="doc"><p>This is the divisible analogue of <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#v:liftA">liftA</a></code>. It gives a viable default definition for <code><a href="Data-Functor-Contravariant.html#v:contramap">contramap</a></code> in terms
of the members of <code><a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a></code>.</p><pre><code><a href="Data-Functor-Contravariant-Divisible.html#v:liftD">liftD</a></code> f = <code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> ((,) () . f) <code><a href="Data-Functor-Contravariant-Divisible.html#v:conquer">conquer</a></code>
</pre></div></div><h1 id="g:2">Contravariant Alternative</h1><div class="top"><p class="src"><span class="keyword">class</span> <a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a> f => <a name="t:Decidable" class="def">Decidable</a> f <span class="keyword">where</span> <a href="src/Data-Functor-Contravariant-Divisible.html#Decidable" class="link">Source</a></p><div class="doc"><p>A <code><a href="Data-Functor-Contravariant-Divisible.html#t:Divisible">Divisible</a></code> contravariant functor is a monoid object in the category of presheaves
from Hask to Hask, equipped with Day convolution mapping the cartesian product of the
source to the Cartesian product of the target.</p><pre><code><a href="Data-Functor-Contravariant-Divisible.html#v:choose">choose</a></code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#v:Left">Left</a></code> m (<code><a href="Data-Functor-Contravariant-Divisible.html#v:lose">lose</a></code> f) = m
<code><a href="Data-Functor-Contravariant-Divisible.html#v:choose">choose</a></code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#v:Right">Right</a></code> (<code><a href="Data-Functor-Contravariant-Divisible.html#v:lose">lose</a></code> f) m = m
<code><a href="Data-Functor-Contravariant-Divisible.html#v:choose">choose</a></code> f (<code><a href="Data-Functor-Contravariant-Divisible.html#v:choose">choose</a></code> g m n) o = <code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> f' m (<code><a href="Data-Functor-Contravariant-Divisible.html#v:divide">divide</a></code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:id">id</a></code> n o) where
f' bcd = <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#v:either">either</a></code> (<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#v:either">either</a></code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:id">id</a></code> (<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#v:Right">Right</a></code> . <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#v:Left">Left</a></code>) . g) (<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#v:Right">Right</a></code> . <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#v:Right">Right</a></code>) . f
</pre><p>In addition, we expect the same kind of distributive law as is satisfied by the usual
covariant <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Alternative">Alternative</a></code>, w.r.t <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Applicative">Applicative</a></code>, which should be fully formulated and
added here at some point!</p></div><div class="subs methods"><p class="caption">Methods</p><p class="src"><a name="v:lose" class="def">lose</a> :: (a -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Void.html#t:Void">Void</a>) -> f a <a href="src/Data-Functor-Contravariant-Divisible.html#lose" class="link">Source</a></p><div class="doc"><p>The only way to win is not to play.</p></div><p class="src"><a name="v:choose" class="def">choose</a> :: (a -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#t:Either">Either</a> b c) -> f b -> f c -> f a <a href="src/Data-Functor-Contravariant-Divisible.html#choose" class="link">Source</a></p></div><div class="subs instances"><p id="control.i:Decidable" class="caption collapser" onclick="toggleSection('i:Decidable')">Instances</p><div id="section.i:Decidable" class="show"><table><tr><td class="src clearfix"><span class="inst-left"><a href="Data-Functor-Contravariant-Divisible.html#t:Decidable">Decidable</a> <a href="Data-Functor-Contravariant.html#t:Equivalence">Equivalence</a></span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-195" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="Data-Functor-Contravariant-Divisible.html#t:Decidable">Decidable</a> <a href="Data-Functor-Contravariant.html#t:Comparison">Comparison</a></span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-185" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="Data-Functor-Contravariant-Divisible.html#t:Decidable">Decidable</a> <a href="Data-Functor-Contravariant.html#t:Predicate">Predicate</a></span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-205" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Monoid.html#t:Monoid">Monoid</a> r => <a href="Data-Functor-Contravariant-Divisible.html#t:Decidable">Decidable</a> (<a href="Data-Functor-Contravariant.html#t:Op">Op</a> r)</span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-209" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left">(<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Applicative">Applicative</a> f, <a href="Data-Functor-Contravariant-Divisible.html#t:Decidable">Decidable</a> g) => <a href="Data-Functor-Contravariant-Divisible.html#t:Decidable">Decidable</a> (<a href="Data-Functor-Contravariant-Compose.html#t:ComposeFC">ComposeFC</a> f g)</span> <a href="src/Data-Functor-Contravariant-Divisible.html#line-47" class="link">Source</a></td><td class="doc empty"> </td></tr></table></div></div></div><div class="top"><p class="src"><a name="v:chosen" class="def">chosen</a> :: <a href="Data-Functor-Contravariant-Divisible.html#t:Decidable">Decidable</a> f => f b -> f c -> f (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Either.html#t:Either">Either</a> b c) <a href="src/Data-Functor-Contravariant-Divisible.html#chosen" class="link">Source</a></p><div class="doc"><pre><code><a href="Data-Functor-Contravariant-Divisible.html#v:chosen">chosen</a></code> = <code><a href="Data-Functor-Contravariant-Divisible.html#v:choose">choose</a></code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:id">id</a></code>
</pre></div></div><div class="top"><p class="src"><a name="v:lost" class="def">lost</a> :: <a href="Data-Functor-Contravariant-Divisible.html#t:Decidable">Decidable</a> f => f <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Void.html#t:Void">Void</a> <a href="src/Data-Functor-Contravariant-Divisible.html#lost" class="link">Source</a></p><div class="doc"><pre><code><a href="Data-Functor-Contravariant-Divisible.html#v:lost">lost</a></code> = <code><a href="Data-Functor-Contravariant-Divisible.html#v:lose">lose</a></code> <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:id">id</a></code>
</pre></div></div></div></div><div id="footer"><p>Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.16.1</p></div></body></html>
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