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<pre><a name="line-1"></a><span class='hs-comment'>{- |
<a name="line-2"></a>Copyright : (c) Henning Thielemann 2007-2010
<a name="line-3"></a>
<a name="line-4"></a>Maintainer : haskell@henning-thielemann.de
<a name="line-5"></a>Stability : stable
<a name="line-6"></a>Portability : Haskell 98
<a name="line-7"></a>
<a name="line-8"></a>A type class for non-negative numbers.
<a name="line-9"></a>Prominent instances are 'Numeric.NonNegative.Wrapper.T',
<a name="line-10"></a>peano numbers and types from "Data.Word".
<a name="line-11"></a>However, there is Monoid class for Word types.
<a name="line-12"></a>Maybe Word is not a candidate, since it has wrap-around semantics.
<a name="line-13"></a>This class cannot do any checks,
<a name="line-14"></a>but it let you show to the user what arguments your function expects.
<a name="line-15"></a>Thus you must define class instances with care.
<a name="line-16"></a>In fact many standard functions ('take', '(!!)', ...)
<a name="line-17"></a>should have this type class constraint.
<a name="line-18"></a>-}</span>
<a name="line-19"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Numeric</span><span class='hs-varop'>.</span><span class='hs-conid'>NonNegative</span><span class='hs-varop'>.</span><span class='hs-conid'>Class</span> <span class='hs-layout'>(</span>
<a name="line-20"></a> <span class='hs-conid'>C</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-21"></a> <span class='hs-varid'>splitDefault</span><span class='hs-layout'>,</span>
<a name="line-22"></a>
<a name="line-23"></a> <span class='hs-layout'>(</span><span class='hs-varop'>-|</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-24"></a> <span class='hs-varid'>zero</span><span class='hs-layout'>,</span>
<a name="line-25"></a> <span class='hs-varid'>add</span><span class='hs-layout'>,</span>
<a name="line-26"></a> <span class='hs-varid'>sum</span><span class='hs-layout'>,</span>
<a name="line-27"></a> <span class='hs-varid'>maximum</span><span class='hs-layout'>,</span>
<a name="line-28"></a>
<a name="line-29"></a> <span class='hs-varid'>switchDifferenceNegative</span><span class='hs-layout'>,</span>
<a name="line-30"></a> <span class='hs-varid'>switchDifferenceOrdering</span><span class='hs-layout'>,</span>
<a name="line-31"></a> <span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-32"></a>
<a name="line-33"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Monoid</span> <span class='hs-layout'>(</span><span class='hs-conid'>Monoid</span><span class='hs-layout'>,</span> <span class='hs-layout'>)</span>
<a name="line-34"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Monoid</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>Monoid</span>
<a name="line-35"></a>
<a name="line-36"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Prelude</span> <span class='hs-varid'>hiding</span> <span class='hs-layout'>(</span><span class='hs-varid'>sum</span><span class='hs-layout'>,</span> <span class='hs-varid'>maximum</span><span class='hs-layout'>,</span> <span class='hs-layout'>)</span>
<a name="line-37"></a>
<a name="line-38"></a>
<a name="line-39"></a><a name="C"></a><span class='hs-comment'>{- |
<a name="line-40"></a>Instances of this class must ensure non-negative values.
<a name="line-41"></a>We cannot enforce this by types, but the type class constraint @NonNegative.C@
<a name="line-42"></a>avoids accidental usage of types which allow for negative numbers.
<a name="line-43"></a>
<a name="line-44"></a>The Monoid superclass contributes a zero and an addition.
<a name="line-45"></a>-}</span>
<a name="line-46"></a><a name="C"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Monoid</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>C</span> <span class='hs-varid'>a</span> <span class='hs-keyword'>where</span>
<a name="line-47"></a> <span class='hs-comment'>{- |
<a name="line-48"></a> @split x y == (m,(b,d))@ means that
<a name="line-49"></a> @b == (x<=y)@,
<a name="line-50"></a> @m == min x y@,
<a name="line-51"></a> @d == max x y - min x y@, that is @d == abs(x-y)@.
<a name="line-52"></a>
<a name="line-53"></a> We have chosen this function as base function,
<a name="line-54"></a> since it provides comparison and subtraction in one go,
<a name="line-55"></a> which is important for replacing common structures like
<a name="line-56"></a>
<a name="line-57"></a> > if x<=y
<a name="line-58"></a> > then f(x-y)
<a name="line-59"></a> > else g(y-x)
<a name="line-60"></a>
<a name="line-61"></a> that lead to a memory leak for peano numbers.
<a name="line-62"></a> We have choosen the simple check @x<=y@
<a name="line-63"></a> instead of a full-blown @compare@,
<a name="line-64"></a> since we want @Zero <= undefined@ for peano numbers.
<a name="line-65"></a> Because of undefined values 'split' is in general
<a name="line-66"></a> not commutative in the sense
<a name="line-67"></a>
<a name="line-68"></a> > let (m0,(b0,d0)) = split x y
<a name="line-69"></a> > (m1,(b1,d1)) = split y x
<a name="line-70"></a> > in m0==m1 && d0==d1
<a name="line-71"></a>
<a name="line-72"></a> The result values are in the order
<a name="line-73"></a> in which they are generated for Peano numbers.
<a name="line-74"></a> We have chosen the nested pair instead of a triple
<a name="line-75"></a> in order to prevent a memory leak
<a name="line-76"></a> that occurs if you only use @b@ and @d@ and ignore @m@.
<a name="line-77"></a> This is demonstrated by test cases
<a name="line-78"></a> Chunky.splitSpaceLeak3 and Chunky.splitSpaceLeak4.
<a name="line-79"></a> -}</span>
<a name="line-80"></a> <span class='hs-varid'>split</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-conid'>Bool</span><span class='hs-layout'>,</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-81"></a>
<a name="line-82"></a>
<a name="line-83"></a><a name="splitDefault"></a><span class='hs-comment'>{- |
<a name="line-84"></a>Default implementation for wrapped types of 'Ord' and 'Num' class.
<a name="line-85"></a>-}</span>
<a name="line-86"></a><span class='hs-comment'>{-# INLINE splitDefault #-}</span>
<a name="line-87"></a><span class='hs-definition'>splitDefault</span> <span class='hs-keyglyph'>::</span>
<a name="line-88"></a> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-89"></a> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-conid'>Bool</span><span class='hs-layout'>,</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-90"></a><span class='hs-definition'>splitDefault</span> <span class='hs-varid'>unpack</span> <span class='hs-varid'>pack</span> <span class='hs-varid'>px</span> <span class='hs-varid'>py</span> <span class='hs-keyglyph'>=</span>
<a name="line-91"></a> <span class='hs-keyword'>let</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unpack</span> <span class='hs-varid'>px</span>
<a name="line-92"></a> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unpack</span> <span class='hs-varid'>py</span>
<a name="line-93"></a> <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>x</span><span class='hs-varop'><=</span><span class='hs-varid'>y</span>
<a name="line-94"></a> <span class='hs-keyword'>then</span> <span class='hs-layout'>(</span><span class='hs-varid'>pack</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-conid'>True</span><span class='hs-layout'>,</span> <span class='hs-varid'>pack</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-comment'>-</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-95"></a> <span class='hs-keyword'>else</span> <span class='hs-layout'>(</span><span class='hs-varid'>pack</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-conid'>False</span><span class='hs-layout'>,</span> <span class='hs-varid'>pack</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-comment'>-</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-96"></a>
<a name="line-97"></a>
<a name="line-98"></a><a name="zero"></a><span class='hs-definition'>zero</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>C</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>a</span>
<a name="line-99"></a><span class='hs-definition'>zero</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Monoid</span><span class='hs-varop'>.</span><span class='hs-varid'>mempty</span>
<a name="line-100"></a>
<a name="line-101"></a><span class='hs-comment'>-- like (+)</span>
<a name="line-102"></a><span class='hs-keyword'>infixl</span> <span class='hs-num'>6</span> <span class='hs-varop'>`add`</span>
<a name="line-103"></a>
<a name="line-104"></a><a name="add"></a><span class='hs-definition'>add</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>C</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-105"></a><span class='hs-definition'>add</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Monoid</span><span class='hs-varop'>.</span><span class='hs-varid'>mappend</span>
<a name="line-106"></a>
<a name="line-107"></a><a name="sum"></a><span class='hs-definition'>sum</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>C</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-108"></a><span class='hs-definition'>sum</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Monoid</span><span class='hs-varop'>.</span><span class='hs-varid'>mconcat</span>
<a name="line-109"></a>
<a name="line-110"></a><a name="maximum"></a><span class='hs-comment'>{- |
<a name="line-111"></a>Left biased maximum of a list of numbers that can also be empty.
<a name="line-112"></a>It holds
<a name="line-113"></a>
<a name="line-114"></a>> maximum [] == zero
<a name="line-115"></a>-}</span>
<a name="line-116"></a><span class='hs-definition'>maximum</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>C</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-117"></a><span class='hs-definition'>maximum</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>foldl</span> <span class='hs-varid'>max</span> <span class='hs-varid'>zero</span>
<a name="line-118"></a>
<a name="line-119"></a><a name="switchDifferenceNegative"></a><span class='hs-comment'>{- |
<a name="line-120"></a>In @switchDifferenceNegative x y branchXminusY branchYminusX@
<a name="line-121"></a>the function @branchXminusY@ is applied to @x-y@
<a name="line-122"></a>if this difference is non-negative,
<a name="line-123"></a>otherwise @branchYminusX@ is applied to @y-x@.
<a name="line-124"></a>-}</span>
<a name="line-125"></a><span class='hs-definition'>switchDifferenceNegative</span> <span class='hs-keyglyph'>::</span>
<a name="line-126"></a> <span class='hs-conid'>C</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span>
<a name="line-127"></a> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span>
<a name="line-128"></a><span class='hs-definition'>switchDifferenceNegative</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>branchXminusY</span> <span class='hs-varid'>branchYminusX</span> <span class='hs-keyglyph'>=</span>
<a name="line-129"></a> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>b</span><span class='hs-layout'>,</span><span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>snd</span> <span class='hs-varop'>$</span> <span class='hs-varid'>split</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span>
<a name="line-130"></a> <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>b</span>
<a name="line-131"></a> <span class='hs-keyword'>then</span> <span class='hs-varid'>branchXminusY</span> <span class='hs-varid'>d</span>
<a name="line-132"></a> <span class='hs-keyword'>else</span> <span class='hs-varid'>branchYminusX</span> <span class='hs-varid'>d</span>
<a name="line-133"></a>
<a name="line-134"></a>
<a name="line-135"></a><a name="switchDifferenceOrdering"></a><span class='hs-comment'>{- |
<a name="line-136"></a>In @switchDifferenceOrdering x y branchZero branchXminusY branchYminusX@
<a name="line-137"></a>-}</span>
<a name="line-138"></a><span class='hs-definition'>switchDifferenceOrdering</span> <span class='hs-keyglyph'>::</span>
<a name="line-139"></a> <span class='hs-conid'>C</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span>
<a name="line-140"></a> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span>
<a name="line-141"></a><span class='hs-definition'>switchDifferenceOrdering</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>branchZero</span> <span class='hs-varid'>branchXminusY</span> <span class='hs-varid'>branchYminusX</span> <span class='hs-keyglyph'>=</span>
<a name="line-142"></a> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>b</span><span class='hs-layout'>,</span><span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>snd</span> <span class='hs-varop'>$</span> <span class='hs-varid'>split</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span>
<a name="line-143"></a> <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>b</span>
<a name="line-144"></a> <span class='hs-keyword'>then</span>
<a name="line-145"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>d</span><span class='hs-varop'>==</span><span class='hs-varid'>zero</span>
<a name="line-146"></a> <span class='hs-keyword'>then</span> <span class='hs-varid'>branchZero</span>
<a name="line-147"></a> <span class='hs-keyword'>else</span> <span class='hs-varid'>branchXminusY</span> <span class='hs-varid'>d</span>
<a name="line-148"></a> <span class='hs-keyword'>else</span> <span class='hs-varid'>branchYminusX</span> <span class='hs-varid'>d</span>
<a name="line-149"></a>
<a name="line-150"></a>
<a name="line-151"></a><a name="-%7c"></a><span class='hs-comment'>{- |
<a name="line-152"></a>@x -| y == max 0 (x-y)@
<a name="line-153"></a>
<a name="line-154"></a>The default implementation is not efficient,
<a name="line-155"></a>because it compares the values and then subtracts, again, if safe.
<a name="line-156"></a>@max 0 (x-y)@ is more elegant and efficient
<a name="line-157"></a>but not possible in the general case,
<a name="line-158"></a>since @x-y@ may already yield a negative number.
<a name="line-159"></a>-}</span>
<a name="line-160"></a><span class='hs-layout'>(</span><span class='hs-varop'>-|</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>C</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-161"></a><a name="x"></a><span class='hs-definition'>x</span> <span class='hs-varop'>-|</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span>
<a name="line-162"></a> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>b</span><span class='hs-layout'>,</span><span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>snd</span> <span class='hs-varop'>$</span> <span class='hs-varid'>split</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span>
<a name="line-163"></a> <span class='hs-keyword'>in</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>b</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>d</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>zero</span>
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