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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /><title>Numeric.Additive.Class</title><link href="ocean.css" rel="stylesheet" type="text/css" title="Ocean" /><script src="haddock-util.js" type="text/javascript"></script><script type="text/javascript">//<![CDATA[
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</script></head><body><div id="package-header"><ul class="links" id="page-menu"><li><a href="src/Numeric-Additive-Class.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">algebra-2.0.7: Constructive abstract algebra</p></div><div id="content"><div id="module-header"><table class="info"><tr><th>Safe Haskell</th><td>Safe-Infered</td></tr></table><p class="caption">Numeric.Additive.Class</p></div><div id="table-of-contents"><p class="caption">Contents</p><ul><li><a href="#g:1">Additive Semigroups
</a></li><li><a href="#g:2">Additive Abelian semigroups
</a></li><li><a href="#g:3">Additive Monoids
</a></li><li><a href="#g:4">Partitionable semigroups
</a></li></ul></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="#t:Additive">Additive</a> r <span class="keyword">where</span><ul class="subs"><li><a href="#v:-43-">(+)</a> :: r -> r -> r</li><li><a href="#v:sinnum1p">sinnum1p</a> :: <a href="Numeric-Algebra.html#t:Whole">Whole</a> n => n -> r -> r</li><li><a href="#v:sumWith1">sumWith1</a> :: <a href="/usr/share/doc/libghc-semigroupoids-doc/html/Data-Semigroup-Foldable.html#t:Foldable1">Foldable1</a> f => (a -> r) -> f a -> r</li></ul></li><li class="src short"><a href="#v:sum1">sum1</a> :: (<a href="/usr/share/doc/libghc-semigroupoids-doc/html/Data-Semigroup-Foldable.html#t:Foldable1">Foldable1</a> f, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r) => f r -> r</li><li class="src short"><span class="keyword">class</span> <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="#t:Abelian">Abelian</a> r </li><li class="src short"><span class="keyword">class</span> <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="#t:Idempotent">Idempotent</a> r </li><li class="src short"><a href="#v:sinnum1pIdempotent">sinnum1pIdempotent</a> :: <a href="Numeric-Algebra.html#t:Natural">Natural</a> -> r -> r</li><li class="src short"><span class="keyword">class</span> <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> m => <a href="#t:Partitionable">Partitionable</a> m <span class="keyword">where</span><ul class="subs"><li><a href="#v:partitionWith">partitionWith</a> :: (m -> m -> r) -> m -> <a href="/usr/share/doc/libghc-semigroups-doc/html/Data-List-NonEmpty.html#t:NonEmpty">NonEmpty</a> r</li></ul></li></ul></div><div id="interface"><h1 id="g:1">Additive Semigroups
</h1><div class="top"><p class="src"><span class="keyword">class</span> <a name="t:Additive" class="def">Additive</a> r <span class="keyword">where</span><a href="src/Numeric-Additive-Class.html#Additive" class="link">Source</a></p><div class="doc"><pre> (a + b) + c = a + (b + c)
sinnum 1 a = a
sinnum (2 * n) a = sinnum n a + sinnum n a
sinnum (2 * n + 1) a = sinnum n a + sinnum n a + a
</pre></div><div class="subs methods"><p class="caption">Methods</p><p class="src"><a name="v:-43-" class="def">(+)</a> :: r -> r -> r<a href="src/Numeric-Additive-Class.html#%2B" class="link">Source</a></p><p class="src"><a name="v:sinnum1p" class="def">sinnum1p</a> :: <a href="Numeric-Algebra.html#t:Whole">Whole</a> n => n -> r -> r<a href="src/Numeric-Additive-Class.html#sinnum1p" class="link">Source</a></p><div class="doc"><p>sinnum1p n r = sinnum (1 + n) r
</p></div><p class="src"><a name="v:sumWith1" class="def">sumWith1</a> :: <a href="/usr/share/doc/libghc-semigroupoids-doc/html/Data-Semigroup-Foldable.html#t:Foldable1">Foldable1</a> f => (a -> r) -> f a -> r<a href="src/Numeric-Additive-Class.html#sumWith1" class="link">Source</a></p></div><div class="subs instances"><p id="control.i:Additive" class="caption collapser" onclick="toggleSection('i:Additive')">Instances</p><div id="section.i:Additive" class="show"><table><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Bool.html#t:Bool">Bool</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Int.html#t:Int">Int</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Int.html#t:Int8">Int8</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Int.html#t:Int16">Int16</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Int.html#t:Int32">Int32</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Int.html#t:Int64">Int64</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Prelude.html#t:Integer">Integer</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Word.html#t:Word">Word</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Word.html#t:Word8">Word8</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Word.html#t:Word16">Word16</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Word.html#t:Word32">Word32</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Word.html#t:Word64">Word64</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> ()</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="Numeric-Algebra.html#t:Natural">Natural</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> <a href="Numeric-Coalgebra-Geometric.html#t:Euclidean">Euclidean</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Algebra-Complex.html#t:Complex">Complex</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Algebra-Quaternion.html#t:Quaternion">Quaternion</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Algebra-Dual.html#t:Dual">Dual</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Algebra-Hyperbolic.html#t:Hyper-39-">Hyper'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Coalgebra-Hyperbolic.html#t:Hyper">Hyper</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Coalgebra-Dual.html#t:Dual-39-">Dual'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Coalgebra-Geometric.html#t:BasisCoblade">BasisCoblade</a> m)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Coalgebra-Quaternion.html#t:Quaternion-39-">Quaternion'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Coalgebra-Trigonometric.html#t:Trig">Trig</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Algebra-Class.html#t:Multiplicative">Multiplicative</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Log.html#t:Log">Log</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Ring-Endomorphism.html#t:End">End</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Ring-Opposite.html#t:Opposite">Opposite</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Ring-Rng.html#t:RngRing">RngRing</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Rng-Zero.html#t:ZeroRng">ZeroRng</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (b -> r)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Additive">Additive</a> a, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> b) => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (a, b)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="/usr/share/doc/libghc-representable-tries-doc/html/Data-Functor-Representable-Trie.html#t:HasTrie">HasTrie</a> b, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r) => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="/usr/share/doc/libghc-representable-tries-doc/html/Data-Functor-Representable-Trie.html#t::-45--62-:">:->:</a> b r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Covector.html#t:Covector">Covector</a> r a)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Additive">Additive</a> a, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> b, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> c) => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (a, b, c)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (<a href="Numeric-Map.html#t:Map">Map</a> r b a)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Additive">Additive</a> a, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> b, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> c, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> d) => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (a, b, c, d)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Additive">Additive</a> a, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> b, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> c, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> d, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> e) => <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> (a, b, c, d, e)</td><td class="doc empty"> </td></tr></table></div></div></div><div class="top"><p class="src"><a name="v:sum1" class="def">sum1</a> :: (<a href="/usr/share/doc/libghc-semigroupoids-doc/html/Data-Semigroup-Foldable.html#t:Foldable1">Foldable1</a> f, <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r) => f r -> r<a href="src/Numeric-Additive-Class.html#sum1" class="link">Source</a></p></div><h1 id="g:2">Additive Abelian semigroups
</h1><div class="top"><p class="src"><span class="keyword">class</span> <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a name="t:Abelian" class="def">Abelian</a> r <a href="src/Numeric-Additive-Class.html#Abelian" class="link">Source</a></p><div class="doc"><p>an additive abelian semigroup
</p><p>a + b = b + a
</p></div><div class="subs instances"><p id="control.i:Abelian" class="caption collapser" onclick="toggleSection('i:Abelian')">Instances</p><div id="section.i:Abelian" class="show"><table><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Bool.html#t:Bool">Bool</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Int.html#t:Int">Int</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Int.html#t:Int8">Int8</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Int.html#t:Int16">Int16</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Int.html#t:Int32">Int32</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Int.html#t:Int64">Int64</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Prelude.html#t:Integer">Integer</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Word.html#t:Word">Word</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Word.html#t:Word8">Word8</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Word.html#t:Word16">Word16</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Word.html#t:Word32">Word32</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Word.html#t:Word64">Word64</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> ()</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="Numeric-Algebra.html#t:Natural">Natural</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> <a href="Numeric-Coalgebra-Geometric.html#t:Euclidean">Euclidean</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Algebra-Complex.html#t:Complex">Complex</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Algebra-Quaternion.html#t:Quaternion">Quaternion</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Algebra-Dual.html#t:Dual">Dual</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Algebra-Hyperbolic.html#t:Hyper-39-">Hyper'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Coalgebra-Hyperbolic.html#t:Hyper">Hyper</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Coalgebra-Dual.html#t:Dual-39-">Dual'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Coalgebra-Geometric.html#t:BasisCoblade">BasisCoblade</a> m)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Coalgebra-Quaternion.html#t:Quaternion-39-">Quaternion'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Coalgebra-Trigonometric.html#t:Trig">Trig</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Algebra-Commutative.html#t:Commutative">Commutative</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Log.html#t:Log">Log</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Ring-Endomorphism.html#t:End">End</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Ring-Opposite.html#t:Opposite">Opposite</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Ring-Rng.html#t:RngRing">RngRing</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Rng-Zero.html#t:ZeroRng">ZeroRng</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (e -> r)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> a, <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> b) => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (a, b)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="/usr/share/doc/libghc-representable-tries-doc/html/Data-Functor-Representable-Trie.html#t:HasTrie">HasTrie</a> e, <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> r) => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="/usr/share/doc/libghc-representable-tries-doc/html/Data-Functor-Representable-Trie.html#t::-45--62-:">:->:</a> e r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> s => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Covector.html#t:Covector">Covector</a> s a)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> a, <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> b, <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> c) => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (a, b, c)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> s => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (<a href="Numeric-Map.html#t:Map">Map</a> s b a)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> a, <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> b, <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> c, <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> d) => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (a, b, c, d)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> a, <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> b, <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> c, <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> d, <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> e) => <a href="Numeric-Additive-Class.html#t:Abelian">Abelian</a> (a, b, c, d, e)</td><td class="doc empty"> </td></tr></table></div></div></div><h1 id="g:3">Additive Monoids
</h1><div class="top"><p class="src"><span class="keyword">class</span> <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> r => <a name="t:Idempotent" class="def">Idempotent</a> r <a href="src/Numeric-Additive-Class.html#Idempotent" class="link">Source</a></p><div class="doc"><p>An additive semigroup with idempotent addition.
</p><pre> a + a = a
</pre></div><div class="subs instances"><p id="control.i:Idempotent" class="caption collapser" onclick="toggleSection('i:Idempotent')">Instances</p><div id="section.i:Idempotent" class="show"><table><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Bool.html#t:Bool">Bool</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> ()</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Algebra-Complex.html#t:Complex">Complex</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Algebra-Quaternion.html#t:Quaternion">Quaternion</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Algebra-Dual.html#t:Dual">Dual</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Algebra-Hyperbolic.html#t:Hyper-39-">Hyper'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Coalgebra-Hyperbolic.html#t:Hyper">Hyper</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Coalgebra-Dual.html#t:Dual-39-">Dual'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Coalgebra-Quaternion.html#t:Quaternion-39-">Quaternion'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Coalgebra-Trigonometric.html#t:Trig">Trig</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Algebra-Idempotent.html#t:Band">Band</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Log.html#t:Log">Log</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Ring-Opposite.html#t:Opposite">Opposite</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Rng-Zero.html#t:ZeroRng">ZeroRng</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (e -> r)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> a, <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> b) => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (a, b)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="/usr/share/doc/libghc-representable-tries-doc/html/Data-Functor-Representable-Trie.html#t:HasTrie">HasTrie</a> e, <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r) => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="/usr/share/doc/libghc-representable-tries-doc/html/Data-Functor-Representable-Trie.html#t::-45--62-:">:->:</a> e r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> r => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (<a href="Numeric-Covector.html#t:Covector">Covector</a> r a)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> a, <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> b, <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> c) => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (a, b, c)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> a, <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> b, <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> c, <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> d) => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (a, b, c, d)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> a, <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> b, <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> c, <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> d, <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> e) => <a href="Numeric-Additive-Class.html#t:Idempotent">Idempotent</a> (a, b, c, d, e)</td><td class="doc empty"> </td></tr></table></div></div></div><div class="top"><p class="src"><a name="v:sinnum1pIdempotent" class="def">sinnum1pIdempotent</a> :: <a href="Numeric-Algebra.html#t:Natural">Natural</a> -> r -> r<a href="src/Numeric-Additive-Class.html#sinnum1pIdempotent" class="link">Source</a></p></div><h1 id="g:4">Partitionable semigroups
</h1><div class="top"><p class="src"><span class="keyword">class</span> <a href="Numeric-Additive-Class.html#t:Additive">Additive</a> m => <a name="t:Partitionable" class="def">Partitionable</a> m <span class="keyword">where</span><a href="src/Numeric-Additive-Class.html#Partitionable" class="link">Source</a></p><div class="subs methods"><p class="caption">Methods</p><p class="src"><a name="v:partitionWith" class="def">partitionWith</a> :: (m -> m -> r) -> m -> <a href="/usr/share/doc/libghc-semigroups-doc/html/Data-List-NonEmpty.html#t:NonEmpty">NonEmpty</a> r<a href="src/Numeric-Additive-Class.html#partitionWith" class="link">Source</a></p><div class="doc"><p>partitionWith f c returns a list containing f a b for each a b such that a + b = c,
</p></div></div><div class="subs instances"><p id="control.i:Partitionable" class="caption collapser" onclick="toggleSection('i:Partitionable')">Instances</p><div id="section.i:Partitionable" class="show"><table><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.5.0.0/Data-Bool.html#t:Bool">Bool</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> ()</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> <a href="Numeric-Algebra.html#t:Natural">Natural</a></td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> r => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (<a href="Numeric-Algebra-Complex.html#t:Complex">Complex</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> r => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (<a href="Numeric-Algebra-Quaternion.html#t:Quaternion">Quaternion</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> r => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (<a href="Numeric-Algebra-Dual.html#t:Dual">Dual</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> r => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (<a href="Numeric-Algebra-Hyperbolic.html#t:Hyper-39-">Hyper'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> r => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (<a href="Numeric-Coalgebra-Hyperbolic.html#t:Hyper">Hyper</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> r => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (<a href="Numeric-Coalgebra-Dual.html#t:Dual-39-">Dual'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> r => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (<a href="Numeric-Coalgebra-Quaternion.html#t:Quaternion-39-">Quaternion'</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> r => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (<a href="Numeric-Coalgebra-Trigonometric.html#t:Trig">Trig</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="Numeric-Algebra-Factorable.html#t:Factorable">Factorable</a> r => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (<a href="Numeric-Log.html#t:Log">Log</a> r)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> a, <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> b) => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (a, b)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> a, <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> b, <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> c) => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (a, b, c)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> a, <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> b, <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> c, <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> d) => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (a, b, c, d)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> a, <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> b, <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> c, <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> d, <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> e) => <a href="Numeric-Additive-Class.html#t:Partitionable">Partitionable</a> (a, b, c, d, e)</td><td class="doc empty"> </td></tr></table></div></div></div></div></div><div id="footer"><p>Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.10.0</p></div></body></html>
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