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<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html>
<head>
<!-- Generated by HsColour, http://code.haskell.org/~malcolm/hscolour/ -->
<title>Data/Ranged/RangedSet.hs</title>
<link type='text/css' rel='stylesheet' href='hscolour.css' />
</head>
<body>
<pre><a name="line-1"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Ranged</span><span class='hs-varop'>.</span><span class='hs-conid'>RangedSet</span> <span class='hs-layout'>(</span>
<a name="line-2"></a>   <span class='hs-comment'>-- ** Ranged Set Type</span>
<a name="line-3"></a>   <span class='hs-conid'>RSet</span><span class='hs-layout'>,</span>
<a name="line-4"></a>   <span class='hs-varid'>rSetRanges</span><span class='hs-layout'>,</span>
<a name="line-5"></a>   <span class='hs-comment'>-- ** Ranged Set construction functions and their preconditions</span>
<a name="line-6"></a>   <span class='hs-varid'>makeRangedSet</span><span class='hs-layout'>,</span>
<a name="line-7"></a>   <span class='hs-varid'>unsafeRangedSet</span><span class='hs-layout'>,</span>
<a name="line-8"></a>   <span class='hs-varid'>validRangeList</span><span class='hs-layout'>,</span>
<a name="line-9"></a>   <span class='hs-varid'>normaliseRangeList</span><span class='hs-layout'>,</span>
<a name="line-10"></a>   <span class='hs-varid'>rSingleton</span><span class='hs-layout'>,</span>
<a name="line-11"></a>   <span class='hs-varid'>rSetUnfold</span><span class='hs-layout'>,</span>
<a name="line-12"></a>   <span class='hs-comment'>-- ** Predicates</span>
<a name="line-13"></a>   <span class='hs-varid'>rSetIsEmpty</span><span class='hs-layout'>,</span>
<a name="line-14"></a>   <span class='hs-varid'>rSetIsFull</span><span class='hs-layout'>,</span>
<a name="line-15"></a>   <span class='hs-layout'>(</span><span class='hs-varop'>-?-</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>  <span class='hs-varid'>rSetHas</span><span class='hs-layout'>,</span>
<a name="line-16"></a>   <span class='hs-layout'>(</span><span class='hs-varop'>-&lt;=-</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>rSetIsSubset</span><span class='hs-layout'>,</span>
<a name="line-17"></a>   <span class='hs-layout'>(</span><span class='hs-varop'>-&lt;-</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>  <span class='hs-varid'>rSetIsSubsetStrict</span><span class='hs-layout'>,</span>
<a name="line-18"></a>   <span class='hs-comment'>-- ** Set Operations</span>
<a name="line-19"></a>   <span class='hs-layout'>(</span><span class='hs-varop'>-\/-</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>rSetUnion</span><span class='hs-layout'>,</span>
<a name="line-20"></a>   <span class='hs-layout'>(</span><span class='hs-varop'>-/\-</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>rSetIntersection</span><span class='hs-layout'>,</span>
<a name="line-21"></a>   <span class='hs-layout'>(</span><span class='hs-varop'>-!-</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>  <span class='hs-varid'>rSetDifference</span><span class='hs-layout'>,</span>
<a name="line-22"></a>   <span class='hs-varid'>rSetNegation</span><span class='hs-layout'>,</span>
<a name="line-23"></a>   <span class='hs-comment'>-- ** Useful Sets</span>
<a name="line-24"></a>   <span class='hs-varid'>rSetEmpty</span><span class='hs-layout'>,</span>
<a name="line-25"></a>   <span class='hs-varid'>rSetFull</span><span class='hs-layout'>,</span>
<a name="line-26"></a>   <span class='hs-comment'>-- ** QuickCheck Properties</span>
<a name="line-27"></a>   <span class='hs-comment'>-- *** Construction</span>
<a name="line-28"></a>   <span class='hs-varid'>prop_validNormalised</span><span class='hs-layout'>,</span>
<a name="line-29"></a>   <span class='hs-varid'>prop_has</span><span class='hs-layout'>,</span>
<a name="line-30"></a>   <span class='hs-varid'>prop_unfold</span><span class='hs-layout'>,</span>
<a name="line-31"></a>   <span class='hs-comment'>-- *** Basic Operations</span>
<a name="line-32"></a>   <span class='hs-varid'>prop_union</span><span class='hs-layout'>,</span>
<a name="line-33"></a>   <span class='hs-varid'>prop_intersection</span><span class='hs-layout'>,</span>
<a name="line-34"></a>   <span class='hs-varid'>prop_difference</span><span class='hs-layout'>,</span>
<a name="line-35"></a>   <span class='hs-varid'>prop_negation</span><span class='hs-layout'>,</span>
<a name="line-36"></a>   <span class='hs-varid'>prop_not_empty</span><span class='hs-layout'>,</span>
<a name="line-37"></a>   <span class='hs-comment'>-- *** Some Identities and Inequalities</span>
<a name="line-38"></a>   <span class='hs-comment'>-- $ConstructionProperties</span>
<a name="line-39"></a>   <span class='hs-comment'>-- $BasicOperationProperties</span>
<a name="line-40"></a>   <span class='hs-comment'>-- $SomeIdentitiesAndInequalities</span>
<a name="line-41"></a>   <span class='hs-varid'>prop_empty</span><span class='hs-layout'>,</span>
<a name="line-42"></a>   <span class='hs-varid'>prop_full</span><span class='hs-layout'>,</span>
<a name="line-43"></a>   <span class='hs-varid'>prop_empty_intersection</span><span class='hs-layout'>,</span>
<a name="line-44"></a>   <span class='hs-varid'>prop_full_union</span><span class='hs-layout'>,</span>
<a name="line-45"></a>   <span class='hs-varid'>prop_union_superset</span><span class='hs-layout'>,</span>
<a name="line-46"></a>   <span class='hs-varid'>prop_intersection_subset</span><span class='hs-layout'>,</span>
<a name="line-47"></a>   <span class='hs-varid'>prop_diff_intersect</span><span class='hs-layout'>,</span>
<a name="line-48"></a>   <span class='hs-varid'>prop_subset</span><span class='hs-layout'>,</span>
<a name="line-49"></a>   <span class='hs-varid'>prop_strict_subset</span><span class='hs-layout'>,</span>
<a name="line-50"></a>   <span class='hs-varid'>prop_union_strict_superset</span><span class='hs-layout'>,</span>
<a name="line-51"></a>   <span class='hs-varid'>prop_intersection_commutes</span><span class='hs-layout'>,</span>
<a name="line-52"></a>   <span class='hs-varid'>prop_union_commutes</span><span class='hs-layout'>,</span>
<a name="line-53"></a>   <span class='hs-varid'>prop_intersection_associates</span><span class='hs-layout'>,</span>
<a name="line-54"></a>   <span class='hs-varid'>prop_union_associates</span><span class='hs-layout'>,</span>
<a name="line-55"></a>   <span class='hs-varid'>prop_de_morgan_intersection</span><span class='hs-layout'>,</span>
<a name="line-56"></a>   <span class='hs-varid'>prop_de_morgan_union</span><span class='hs-layout'>,</span>
<a name="line-57"></a><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-58"></a>
<a name="line-59"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Ranged</span><span class='hs-varop'>.</span><span class='hs-conid'>Boundaries</span>
<a name="line-60"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Ranged</span><span class='hs-varop'>.</span><span class='hs-conid'>Ranges</span>
<a name="line-61"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Monoid</span>
<a name="line-62"></a>
<a name="line-63"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>List</span>
<a name="line-64"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Test</span><span class='hs-varop'>.</span><span class='hs-conid'>QuickCheck</span>
<a name="line-65"></a>
<a name="line-66"></a><span class='hs-keyword'>infixl</span> <span class='hs-num'>7</span> <span class='hs-varop'>-/\-</span>
<a name="line-67"></a><span class='hs-keyword'>infixl</span> <span class='hs-num'>6</span> <span class='hs-varop'>-\/-</span><span class='hs-layout'>,</span> <span class='hs-varop'>-!-</span>
<a name="line-68"></a><span class='hs-keyword'>infixl</span> <span class='hs-num'>5</span> <span class='hs-varop'>-&lt;=-</span><span class='hs-layout'>,</span> <span class='hs-varop'>-&lt;-</span><span class='hs-layout'>,</span> <span class='hs-varop'>-?-</span>
<a name="line-69"></a>
<a name="line-70"></a><a name="RSet"></a><span class='hs-comment'>-- | An RSet (for Ranged Set) is a list of ranges.  The ranges must be sorted</span>
<a name="line-71"></a><a name="RSet"></a><span class='hs-comment'>-- and not overlap.</span>
<a name="line-72"></a><a name="RSet"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>RSet</span> <span class='hs-layout'>{</span><span class='hs-varid'>rSetRanges</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>}</span>
<a name="line-73"></a>   <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span> <span class='hs-conid'>Show</span><span class='hs-layout'>)</span>
<a name="line-74"></a>
<a name="line-75"></a><a name="instance%20Monoid%20(RSet%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Monoid</span> <span class='hs-layout'>(</span><span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-76"></a>    <span class='hs-varid'>mappend</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetUnion</span>
<a name="line-77"></a>    <span class='hs-varid'>mempty</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetEmpty</span>
<a name="line-78"></a>
<a name="line-79"></a><a name="validRangeList"></a><span class='hs-comment'>-- | Determine if the ranges in the list are both in order and non-overlapping.</span>
<a name="line-80"></a><span class='hs-comment'>-- If so then they are suitable input for the unsafeRangedSet function.</span>
<a name="line-81"></a><span class='hs-definition'>validRangeList</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-82"></a>
<a name="line-83"></a><span class='hs-definition'>validRangeList</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span>
<a name="line-84"></a><span class='hs-definition'>validRangeList</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>lower</span> <span class='hs-varid'>upper</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>lower</span> <span class='hs-varop'>&lt;=</span> <span class='hs-varid'>upper</span>
<a name="line-85"></a><span class='hs-definition'>validRangeList</span> <span class='hs-varid'>rs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>and</span> <span class='hs-varop'>$</span> <span class='hs-varid'>zipWith</span> <span class='hs-varid'>okAdjacent</span> <span class='hs-varid'>rs</span> <span class='hs-layout'>(</span><span class='hs-varid'>tail</span> <span class='hs-varid'>rs</span><span class='hs-layout'>)</span>
<a name="line-86"></a>   <span class='hs-keyword'>where</span>
<a name="line-87"></a>      <span class='hs-varid'>okAdjacent</span> <span class='hs-layout'>(</span><span class='hs-conid'>Range</span> <span class='hs-varid'>lower1</span> <span class='hs-varid'>upper1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Range</span> <span class='hs-varid'>lower2</span> <span class='hs-varid'>upper2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-88"></a>         <span class='hs-varid'>lower1</span> <span class='hs-varop'>&lt;=</span> <span class='hs-varid'>upper1</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>upper1</span> <span class='hs-varop'>&lt;=</span> <span class='hs-varid'>lower2</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>lower2</span> <span class='hs-varop'>&lt;=</span> <span class='hs-varid'>upper2</span>
<a name="line-89"></a>
<a name="line-90"></a>
<a name="line-91"></a><a name="normaliseRangeList"></a><span class='hs-comment'>-- | Rearrange and merge the ranges in the list so that they are in order and</span>
<a name="line-92"></a><span class='hs-comment'>-- non-overlapping.</span>
<a name="line-93"></a><span class='hs-definition'>normaliseRangeList</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span>
<a name="line-94"></a><span class='hs-definition'>normaliseRangeList</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>normalise</span> <span class='hs-varop'>.</span> <span class='hs-varid'>sort</span> <span class='hs-varop'>.</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>rangeIsEmpty</span><span class='hs-layout'>)</span>
<a name="line-95"></a>
<a name="line-96"></a>
<a name="line-97"></a><a name="normalise"></a><span class='hs-comment'>-- Private routine: normalise a range list that is known to be already sorted.</span>
<a name="line-98"></a><span class='hs-comment'>-- This precondition is not checked.</span>
<a name="line-99"></a><span class='hs-definition'>normalise</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span>
<a name="line-100"></a><span class='hs-definition'>normalise</span> <span class='hs-layout'>(</span><span class='hs-varid'>r1</span><span class='hs-conop'>:</span><span class='hs-varid'>r2</span><span class='hs-conop'>:</span><span class='hs-varid'>rs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-101"></a>         <span class='hs-keyword'>if</span> <span class='hs-varid'>overlap</span> <span class='hs-varid'>r1</span> <span class='hs-varid'>r2</span>
<a name="line-102"></a>               <span class='hs-keyword'>then</span> <span class='hs-varid'>normalise</span> <span class='hs-varop'>$</span>
<a name="line-103"></a>                       <span class='hs-conid'>Range</span> <span class='hs-layout'>(</span><span class='hs-varid'>rangeLower</span> <span class='hs-varid'>r1</span><span class='hs-layout'>)</span>
<a name="line-104"></a>                             <span class='hs-layout'>(</span><span class='hs-varid'>max</span> <span class='hs-layout'>(</span><span class='hs-varid'>rangeUpper</span> <span class='hs-varid'>r1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>rangeUpper</span> <span class='hs-varid'>r2</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-105"></a>                       <span class='hs-conop'>:</span> <span class='hs-varid'>rs</span>
<a name="line-106"></a>               <span class='hs-keyword'>else</span> <span class='hs-varid'>r1</span> <span class='hs-conop'>:</span> <span class='hs-layout'>(</span><span class='hs-varid'>normalise</span> <span class='hs-varop'>$</span> <span class='hs-varid'>r2</span> <span class='hs-conop'>:</span> <span class='hs-varid'>rs</span><span class='hs-layout'>)</span>
<a name="line-107"></a>   <span class='hs-keyword'>where</span>
<a name="line-108"></a>      <span class='hs-varid'>overlap</span> <span class='hs-layout'>(</span><span class='hs-conid'>Range</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>upper1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Range</span> <span class='hs-varid'>lower2</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>upper1</span> <span class='hs-varop'>&gt;=</span> <span class='hs-varid'>lower2</span>
<a name="line-109"></a>
<a name="line-110"></a><span class='hs-definition'>normalise</span> <span class='hs-varid'>rs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rs</span>
<a name="line-111"></a>
<a name="line-112"></a>
<a name="line-113"></a><a name="makeRangedSet"></a><span class='hs-comment'>-- | Create a new Ranged Set from a list of ranges.  The list may contain</span>
<a name="line-114"></a><span class='hs-comment'>-- ranges that overlap or are not in ascending order.</span>
<a name="line-115"></a><span class='hs-definition'>makeRangedSet</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span>
<a name="line-116"></a><span class='hs-definition'>makeRangedSet</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>RSet</span> <span class='hs-varop'>.</span> <span class='hs-varid'>normaliseRangeList</span>
<a name="line-117"></a>
<a name="line-118"></a>
<a name="line-119"></a><a name="unsafeRangedSet"></a><span class='hs-comment'>-- | Create a new Ranged Set from a list of ranges. @validRangeList ranges@</span>
<a name="line-120"></a><span class='hs-comment'>-- must return @True@.  This precondition is not checked.</span>
<a name="line-121"></a><span class='hs-definition'>unsafeRangedSet</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span>
<a name="line-122"></a><span class='hs-definition'>unsafeRangedSet</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>RSet</span>
<a name="line-123"></a>
<a name="line-124"></a><a name="rSingleton"></a><span class='hs-comment'>-- | Create a Ranged Set from a single element.</span>
<a name="line-125"></a><span class='hs-definition'>rSingleton</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span>
<a name="line-126"></a><span class='hs-definition'>rSingleton</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unsafeRangedSet</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>singletonRange</span> <span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span>
<a name="line-127"></a>
<a name="line-128"></a><a name="rSetIsEmpty"></a><span class='hs-comment'>-- | True if the set has no members.</span>
<a name="line-129"></a><span class='hs-definition'>rSetIsEmpty</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-130"></a><span class='hs-definition'>rSetIsEmpty</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>null</span> <span class='hs-varop'>.</span> <span class='hs-varid'>rSetRanges</span>
<a name="line-131"></a>
<a name="line-132"></a>
<a name="line-133"></a><a name="rSetIsFull"></a><span class='hs-comment'>-- | True if the negation of the set has no members.</span>
<a name="line-134"></a><span class='hs-definition'>rSetIsFull</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-135"></a><span class='hs-definition'>rSetIsFull</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetIsEmpty</span> <span class='hs-varop'>.</span> <span class='hs-varid'>rSetNegation</span>
<a name="line-136"></a>
<a name="line-137"></a>
<a name="line-138"></a><a name="rSetHas"></a><span class='hs-comment'>-- | True if the value is within the ranged set.  Infix precedence is left 5.</span>
<a name="line-139"></a><span class='hs-definition'>rSetHas</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>-?-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-140"></a><span class='hs-definition'>rSetHas</span> <span class='hs-layout'>(</span><span class='hs-conid'>RSet</span> <span class='hs-varid'>ls</span><span class='hs-layout'>)</span> <span class='hs-varid'>value</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetHas1</span> <span class='hs-varid'>ls</span>
<a name="line-141"></a>   <span class='hs-keyword'>where</span>
<a name="line-142"></a>      <span class='hs-varid'>rSetHas1</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-143"></a>      <span class='hs-varid'>rSetHas1</span> <span class='hs-layout'>(</span><span class='hs-varid'>r</span><span class='hs-conop'>:</span><span class='hs-varid'>rs</span><span class='hs-layout'>)</span>
<a name="line-144"></a>         <span class='hs-keyglyph'>|</span> <span class='hs-varid'>value</span> <span class='hs-varop'>/&gt;/</span> <span class='hs-varid'>rangeLower</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rangeHas</span> <span class='hs-varid'>r</span> <span class='hs-varid'>value</span> <span class='hs-varop'>||</span> <span class='hs-varid'>rSetHas1</span> <span class='hs-varid'>rs</span>
<a name="line-145"></a>         <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span>              <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-146"></a>
<a name="line-147"></a><a name="-?-"></a><span class='hs-layout'>(</span><span class='hs-varop'>-?-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetHas</span>
<a name="line-148"></a>
<a name="line-149"></a><a name="rSetIsSubset"></a><span class='hs-comment'>-- | True if the first argument is a subset of the second argument, or is</span>
<a name="line-150"></a><span class='hs-comment'>-- equal.</span>
<a name="line-151"></a><span class='hs-comment'>--</span>
<a name="line-152"></a><span class='hs-comment'>-- Infix precedence is left 5.</span>
<a name="line-153"></a><span class='hs-definition'>rSetIsSubset</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>-&lt;=-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-154"></a><span class='hs-definition'>rSetIsSubset</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetIsEmpty</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-!-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span>
<a name="line-155"></a><a name="-%3c=-"></a><span class='hs-layout'>(</span><span class='hs-varop'>-&lt;=-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetIsSubset</span>
<a name="line-156"></a>
<a name="line-157"></a>
<a name="line-158"></a><a name="rSetIsSubsetStrict"></a><span class='hs-comment'>-- | True if the first argument is a strict subset of the second argument.</span>
<a name="line-159"></a><span class='hs-comment'>--</span>
<a name="line-160"></a><span class='hs-comment'>-- Infix precedence is left 5.</span>
<a name="line-161"></a><span class='hs-definition'>rSetIsSubsetStrict</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>-&lt;-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-162"></a><span class='hs-definition'>rSetIsSubsetStrict</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-keyglyph'>=</span>
<a name="line-163"></a>   <span class='hs-varid'>rSetIsEmpty</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-!-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span>
<a name="line-164"></a>   <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>not</span> <span class='hs-layout'>(</span><span class='hs-varid'>rSetIsEmpty</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs2</span> <span class='hs-varop'>-!-</span> <span class='hs-varid'>rs1</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-165"></a>
<a name="line-166"></a><a name="-%3c-"></a><span class='hs-layout'>(</span><span class='hs-varop'>-&lt;-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetIsSubsetStrict</span>
<a name="line-167"></a>
<a name="line-168"></a><a name="rSetUnion"></a><span class='hs-comment'>-- | Set union for ranged sets.  Infix precedence is left 6.</span>
<a name="line-169"></a><span class='hs-definition'>rSetUnion</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>-\/-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span>
<a name="line-170"></a><span class='hs-comment'>-- Implementation note: rSetUnion merges the two lists into a single</span>
<a name="line-171"></a><span class='hs-comment'>-- sorted list and then calls normalise to combine overlapping ranges.</span>
<a name="line-172"></a><span class='hs-definition'>rSetUnion</span> <span class='hs-layout'>(</span><span class='hs-conid'>RSet</span> <span class='hs-varid'>ls1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>RSet</span> <span class='hs-varid'>ls2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>RSet</span> <span class='hs-varop'>$</span> <span class='hs-varid'>normalise</span> <span class='hs-varop'>$</span> <span class='hs-varid'>merge</span> <span class='hs-varid'>ls1</span> <span class='hs-varid'>ls2</span>
<a name="line-173"></a>   <span class='hs-keyword'>where</span>
<a name="line-174"></a>      <span class='hs-varid'>merge</span> <span class='hs-varid'>ms1</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ms1</span>
<a name="line-175"></a>      <span class='hs-varid'>merge</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>ms2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ms2</span>
<a name="line-176"></a>      <span class='hs-varid'>merge</span> <span class='hs-varid'>ms1</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>h1</span><span class='hs-conop'>:</span><span class='hs-varid'>t1</span><span class='hs-layout'>)</span> <span class='hs-varid'>ms2</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>h2</span><span class='hs-conop'>:</span><span class='hs-varid'>t2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-177"></a>         <span class='hs-keyword'>if</span> <span class='hs-varid'>h1</span> <span class='hs-varop'>&lt;</span>  <span class='hs-varid'>h2</span>
<a name="line-178"></a>            <span class='hs-keyword'>then</span> <span class='hs-varid'>h1</span> <span class='hs-conop'>:</span> <span class='hs-varid'>merge</span> <span class='hs-varid'>t1</span> <span class='hs-varid'>ms2</span>
<a name="line-179"></a>            <span class='hs-keyword'>else</span> <span class='hs-varid'>h2</span> <span class='hs-conop'>:</span> <span class='hs-varid'>merge</span> <span class='hs-varid'>ms1</span> <span class='hs-varid'>t2</span>
<a name="line-180"></a>
<a name="line-181"></a><a name="-%5c/-"></a><span class='hs-layout'>(</span><span class='hs-varop'>-\/-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetUnion</span>
<a name="line-182"></a>
<a name="line-183"></a><a name="rSetIntersection"></a><span class='hs-comment'>-- | Set intersection for ranged sets.  Infix precedence is left 7.</span>
<a name="line-184"></a><span class='hs-definition'>rSetIntersection</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>-/\-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span>
<a name="line-185"></a><span class='hs-definition'>rSetIntersection</span> <span class='hs-layout'>(</span><span class='hs-conid'>RSet</span> <span class='hs-varid'>ls1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>RSet</span> <span class='hs-varid'>ls2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-186"></a>   <span class='hs-conid'>RSet</span> <span class='hs-varop'>$</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>.</span> <span class='hs-varid'>rangeIsEmpty</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>merge</span> <span class='hs-varid'>ls1</span> <span class='hs-varid'>ls2</span>
<a name="line-187"></a>   <span class='hs-keyword'>where</span>
<a name="line-188"></a>      <span class='hs-varid'>merge</span> <span class='hs-varid'>ms1</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>h1</span><span class='hs-conop'>:</span><span class='hs-varid'>t1</span><span class='hs-layout'>)</span> <span class='hs-varid'>ms2</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>h2</span><span class='hs-conop'>:</span><span class='hs-varid'>t2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-189"></a>         <span class='hs-varid'>rangeIntersection</span> <span class='hs-varid'>h1</span> <span class='hs-varid'>h2</span>
<a name="line-190"></a>         <span class='hs-conop'>:</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>rangeUpper</span> <span class='hs-varid'>h1</span> <span class='hs-varop'>&lt;</span> <span class='hs-varid'>rangeUpper</span> <span class='hs-varid'>h2</span>
<a name="line-191"></a>               <span class='hs-keyword'>then</span> <span class='hs-varid'>merge</span> <span class='hs-varid'>t1</span> <span class='hs-varid'>ms2</span>
<a name="line-192"></a>               <span class='hs-keyword'>else</span> <span class='hs-varid'>merge</span> <span class='hs-varid'>ms1</span> <span class='hs-varid'>t2</span>
<a name="line-193"></a>      <span class='hs-varid'>merge</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-194"></a>
<a name="line-195"></a><a name="-/%5c-"></a><span class='hs-layout'>(</span><span class='hs-varop'>-/\-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetIntersection</span>
<a name="line-196"></a>
<a name="line-197"></a>
<a name="line-198"></a><a name="rSetDifference"></a><span class='hs-comment'>-- | Set difference.  Infix precedence is left 6.</span>
<a name="line-199"></a><span class='hs-definition'>rSetDifference</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>-!-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span>
<a name="line-200"></a><span class='hs-definition'>rSetDifference</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rs1</span> <span class='hs-varop'>-/\-</span> <span class='hs-layout'>(</span><span class='hs-varid'>rSetNegation</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span>
<a name="line-201"></a><a name="-!-"></a><span class='hs-layout'>(</span><span class='hs-varop'>-!-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetDifference</span>
<a name="line-202"></a>
<a name="line-203"></a>
<a name="line-204"></a><a name="rSetNegation"></a><span class='hs-comment'>-- | Set negation.</span>
<a name="line-205"></a><span class='hs-definition'>rSetNegation</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span>
<a name="line-206"></a><span class='hs-definition'>rSetNegation</span> <span class='hs-varid'>set</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>RSet</span> <span class='hs-varop'>$</span> <span class='hs-varid'>ranges1</span> <span class='hs-varop'>$</span> <span class='hs-varid'>setBounds1</span>
<a name="line-207"></a>   <span class='hs-keyword'>where</span>
<a name="line-208"></a>      <span class='hs-varid'>ranges1</span> <span class='hs-layout'>(</span><span class='hs-varid'>b1</span><span class='hs-conop'>:</span><span class='hs-varid'>b2</span><span class='hs-conop'>:</span><span class='hs-varid'>bs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Range</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-conop'>:</span> <span class='hs-varid'>ranges1</span> <span class='hs-varid'>bs</span>
<a name="line-209"></a>      <span class='hs-varid'>ranges1</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>BoundaryAboveAll</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-210"></a>      <span class='hs-varid'>ranges1</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>b</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>b</span> <span class='hs-conid'>BoundaryAboveAll</span><span class='hs-keyglyph'>]</span>
<a name="line-211"></a>      <span class='hs-varid'>ranges1</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-212"></a>      <span class='hs-varid'>setBounds1</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>setBounds</span> <span class='hs-keyword'>of</span>
<a name="line-213"></a>         <span class='hs-layout'>(</span><span class='hs-conid'>BoundaryBelowAll</span> <span class='hs-conop'>:</span> <span class='hs-varid'>bs</span><span class='hs-layout'>)</span>  <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>bs</span>
<a name="line-214"></a>         <span class='hs-keyword'>_</span>                        <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>BoundaryBelowAll</span> <span class='hs-conop'>:</span> <span class='hs-varid'>setBounds</span>
<a name="line-215"></a>      <span class='hs-varid'>setBounds</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>bounds</span> <span class='hs-varop'>$</span> <span class='hs-varid'>rSetRanges</span> <span class='hs-varid'>set</span>
<a name="line-216"></a>      <span class='hs-varid'>bounds</span> <span class='hs-layout'>(</span><span class='hs-varid'>r</span><span class='hs-conop'>:</span><span class='hs-varid'>rs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rangeLower</span> <span class='hs-varid'>r</span> <span class='hs-conop'>:</span> <span class='hs-varid'>rangeUpper</span> <span class='hs-varid'>r</span> <span class='hs-conop'>:</span> <span class='hs-varid'>bounds</span> <span class='hs-varid'>rs</span>
<a name="line-217"></a>      <span class='hs-varid'>bounds</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-218"></a>
<a name="line-219"></a><a name="rSetEmpty"></a><span class='hs-comment'>-- | The empty set.</span>
<a name="line-220"></a><span class='hs-definition'>rSetEmpty</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span>
<a name="line-221"></a><span class='hs-definition'>rSetEmpty</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>RSet</span> <span class='hs-conid'>[]</span>
<a name="line-222"></a>
<a name="line-223"></a><a name="rSetFull"></a><span class='hs-comment'>-- | The set that contains everything.</span>
<a name="line-224"></a><span class='hs-definition'>rSetFull</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span>
<a name="line-225"></a><span class='hs-definition'>rSetFull</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>RSet</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-conid'>BoundaryBelowAll</span> <span class='hs-conid'>BoundaryAboveAll</span><span class='hs-keyglyph'>]</span>
<a name="line-226"></a>
<a name="line-227"></a><a name="rSetUnfold"></a><span class='hs-comment'>-- | Construct a range set.</span>
<a name="line-228"></a><span class='hs-definition'>rSetUnfold</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=&gt;</span>
<a name="line-229"></a>   <span class='hs-conid'>Boundary</span> <span class='hs-varid'>a</span>
<a name="line-230"></a>      <span class='hs-comment'>-- ^ A first lower boundary.</span>
<a name="line-231"></a>   <span class='hs-keyglyph'>-&gt;</span> <span class='hs-layout'>(</span><span class='hs-conid'>Boundary</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Boundary</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-232"></a>      <span class='hs-comment'>-- ^ A function from a lower boundary to an upper boundary, which must</span>
<a name="line-233"></a>      <span class='hs-comment'>-- return a result greater than the argument (not checked).</span>
<a name="line-234"></a>   <span class='hs-keyglyph'>-&gt;</span> <span class='hs-layout'>(</span><span class='hs-conid'>Boundary</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Maybe</span> <span class='hs-layout'>(</span><span class='hs-conid'>Boundary</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-235"></a>      <span class='hs-comment'>-- ^ A function from a lower boundary to @Maybe@ the successor lower</span>
<a name="line-236"></a>      <span class='hs-comment'>-- boundary, which must return a result greater than the argument</span>
<a name="line-237"></a>      <span class='hs-comment'>-- (not checked).  If ranges overlap then they will be merged.</span>
<a name="line-238"></a>   <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span>
<a name="line-239"></a><span class='hs-definition'>rSetUnfold</span> <span class='hs-varid'>bound</span> <span class='hs-varid'>upperFunc</span> <span class='hs-varid'>succFunc</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>RSet</span> <span class='hs-varop'>$</span> <span class='hs-varid'>normalise</span> <span class='hs-varop'>$</span> <span class='hs-varid'>ranges1</span> <span class='hs-varid'>bound</span>
<a name="line-240"></a>   <span class='hs-keyword'>where</span>
<a name="line-241"></a>      <span class='hs-varid'>ranges1</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>=</span>
<a name="line-242"></a>         <span class='hs-conid'>Range</span> <span class='hs-varid'>b</span> <span class='hs-layout'>(</span><span class='hs-varid'>upperFunc</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span>
<a name="line-243"></a>         <span class='hs-conop'>:</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>succFunc</span> <span class='hs-varid'>b</span> <span class='hs-keyword'>of</span>
<a name="line-244"></a>            <span class='hs-conid'>Just</span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>ranges1</span> <span class='hs-varid'>b2</span>
<a name="line-245"></a>            <span class='hs-conid'>Nothing</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>[]</span>
<a name="line-246"></a>
<a name="line-247"></a>
<a name="line-248"></a><span class='hs-comment'>-- QuickCheck Generators</span>
<a name="line-249"></a>
<a name="line-250"></a><a name="instance%20Arbitrary%20(RSet%20v)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Arbitrary</span> <span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-conid'>Show</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span>
<a name="line-251"></a>      <span class='hs-conid'>Arbitrary</span> <span class='hs-layout'>(</span><span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span>
<a name="line-252"></a>   <span class='hs-keyword'>where</span>
<a name="line-253"></a>   <span class='hs-varid'>arbitrary</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>frequency</span> <span class='hs-keyglyph'>[</span>
<a name="line-254"></a>      <span class='hs-layout'>(</span><span class='hs-num'>1</span><span class='hs-layout'>,</span> <span class='hs-varid'>return</span> <span class='hs-varid'>rSetEmpty</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-255"></a>      <span class='hs-layout'>(</span><span class='hs-num'>1</span><span class='hs-layout'>,</span> <span class='hs-varid'>return</span> <span class='hs-varid'>rSetFull</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-256"></a>      <span class='hs-layout'>(</span><span class='hs-num'>18</span><span class='hs-layout'>,</span> <span class='hs-keyword'>do</span>
<a name="line-257"></a>         <span class='hs-varid'>ls</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>arbitrary</span>
<a name="line-258"></a>         <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-varid'>makeRangedSet</span> <span class='hs-varop'>$</span> <span class='hs-varid'>rangeList</span> <span class='hs-varop'>$</span> <span class='hs-varid'>sort</span> <span class='hs-varid'>ls</span>
<a name="line-259"></a>      <span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-260"></a>      <span class='hs-keyword'>where</span>
<a name="line-261"></a>         <span class='hs-comment'>-- Arbitrary lists of ranges don't give many interesting sets after</span>
<a name="line-262"></a>         <span class='hs-comment'>-- normalisation.  So instead generate a sorted list of boundaries</span>
<a name="line-263"></a>         <span class='hs-comment'>-- and pair them off.  Odd boundaries are dropped.</span>
<a name="line-264"></a>         <span class='hs-varid'>rangeList</span> <span class='hs-layout'>(</span><span class='hs-varid'>b1</span><span class='hs-conop'>:</span><span class='hs-varid'>b2</span><span class='hs-conop'>:</span><span class='hs-varid'>bs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Range</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-conop'>:</span> <span class='hs-varid'>rangeList</span> <span class='hs-varid'>bs</span>
<a name="line-265"></a>         <span class='hs-varid'>rangeList</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-266"></a>
<a name="line-267"></a><a name="instance%20CoArbitrary%20(RSet%20v)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>CoArbitrary</span> <span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-conid'>Show</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span>
<a name="line-268"></a>      <span class='hs-conid'>CoArbitrary</span> <span class='hs-layout'>(</span><span class='hs-conid'>RSet</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span>
<a name="line-269"></a>   <span class='hs-keyword'>where</span>
<a name="line-270"></a>   <span class='hs-varid'>coarbitrary</span> <span class='hs-layout'>(</span><span class='hs-conid'>RSet</span> <span class='hs-varid'>ls</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>variant</span> <span class='hs-layout'>(</span><span class='hs-num'>0</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>coarbitrary</span> <span class='hs-varid'>ls</span>
<a name="line-271"></a>
<a name="line-272"></a><span class='hs-comment'>-- ==================================================================</span>
<a name="line-273"></a><span class='hs-comment'>-- QuickCheck Properties</span>
<a name="line-274"></a><span class='hs-comment'>-- ==================================================================</span>
<a name="line-275"></a>
<a name="line-276"></a><span class='hs-comment'>---------------------------------------------------------------------</span>
<a name="line-277"></a><span class='hs-comment'>-- Construction properties</span>
<a name="line-278"></a><span class='hs-comment'>---------------------------------------------------------------------</span>
<a name="line-279"></a>
<a name="line-280"></a><a name="prop_validNormalised"></a><span class='hs-comment'>-- | A normalised range list is valid for unsafeRangedSet</span>
<a name="line-281"></a><span class='hs-comment'>--</span>
<a name="line-282"></a><span class='hs-comment'>-- &gt; prop_validNormalised ls = validRangeList $ normaliseRangeList ls</span>
<a name="line-283"></a><span class='hs-definition'>prop_validNormalised</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-284"></a><span class='hs-definition'>prop_validNormalised</span> <span class='hs-varid'>ls</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>validRangeList</span> <span class='hs-varop'>$</span> <span class='hs-varid'>normaliseRangeList</span> <span class='hs-varid'>ls</span>
<a name="line-285"></a>
<a name="line-286"></a>
<a name="line-287"></a><a name="prop_has"></a><span class='hs-comment'>-- | Iff a value is in a range list then it is in a ranged set</span>
<a name="line-288"></a><span class='hs-comment'>-- constructed from that list.</span>
<a name="line-289"></a><span class='hs-comment'>--</span>
<a name="line-290"></a><span class='hs-comment'>-- &gt; prop_has ls v = (ls `rangeListHas` v) == makeRangedSet ls -?- v</span>
<a name="line-291"></a><span class='hs-definition'>prop_has</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Range</span> <span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-292"></a><span class='hs-definition'>prop_has</span> <span class='hs-varid'>ls</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>ls</span> <span class='hs-varop'>`rangeListHas`</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-varid'>makeRangedSet</span> <span class='hs-varid'>ls</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span>
<a name="line-293"></a>
<a name="line-294"></a>
<a name="line-295"></a><span class='hs-comment'>-- | Verifies the correct membership of a set containing all integers</span>
<a name="line-296"></a><span class='hs-comment'>-- starting with the digit \"1\" up to 19999.</span>
<a name="line-297"></a><span class='hs-comment'>--</span>
<a name="line-298"></a><span class='hs-comment'>-- &gt; prop_unfold = (v &lt;= 99999 &amp;&amp; head (show v) == '1') == (initial1 -?- v)</span>
<a name="line-299"></a><span class='hs-comment'>-- &gt;    where</span>
<a name="line-300"></a><span class='hs-comment'>-- &gt;       initial1 = rSetUnfold (BoundaryBelow 1) addNines times10</span>
<a name="line-301"></a><span class='hs-comment'>-- &gt;       addNines (BoundaryBelow n) = BoundaryAbove $ n * 2 - 1</span>
<a name="line-302"></a><span class='hs-comment'>-- &gt;       times10 (BoundaryBelow n) =</span>
<a name="line-303"></a><span class='hs-comment'>-- &gt;          if n &lt;= 1000 then Just $ BoundaryBelow $ n * 10 else Nothing</span>
<a name="line-304"></a>
<a name="line-305"></a><a name="prop_unfold"></a><span class='hs-definition'>prop_unfold</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Integer</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-306"></a><span class='hs-definition'>prop_unfold</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span> <span class='hs-varop'>&lt;=</span> <span class='hs-num'>99999</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>head</span> <span class='hs-layout'>(</span><span class='hs-varid'>show</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-chr'>'1'</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-varid'>initial1</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span>
<a name="line-307"></a>   <span class='hs-keyword'>where</span>
<a name="line-308"></a>      <span class='hs-varid'>initial1</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetUnfold</span> <span class='hs-layout'>(</span><span class='hs-conid'>BoundaryBelow</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-varid'>addNines</span> <span class='hs-varid'>times10</span>
<a name="line-309"></a>      <span class='hs-varid'>addNines</span> <span class='hs-layout'>(</span><span class='hs-conid'>BoundaryBelow</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>BoundaryAbove</span> <span class='hs-varop'>$</span> <span class='hs-varid'>n</span> <span class='hs-varop'>*</span> <span class='hs-num'>2</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span>
<a name="line-310"></a>      <span class='hs-varid'>addNines</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"Can't happen"</span>
<a name="line-311"></a>      <span class='hs-varid'>times10</span> <span class='hs-layout'>(</span><span class='hs-conid'>BoundaryBelow</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-312"></a>         <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>&lt;=</span> <span class='hs-num'>10000</span> <span class='hs-keyword'>then</span> <span class='hs-conid'>Just</span> <span class='hs-varop'>$</span> <span class='hs-conid'>BoundaryBelow</span> <span class='hs-varop'>$</span> <span class='hs-varid'>n</span> <span class='hs-varop'>*</span> <span class='hs-num'>10</span> <span class='hs-keyword'>else</span> <span class='hs-conid'>Nothing</span>
<a name="line-313"></a>      <span class='hs-varid'>times10</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"Can't happen"</span>
<a name="line-314"></a>
<a name="line-315"></a><span class='hs-comment'>---------------------------------------------------------------------</span>
<a name="line-316"></a><span class='hs-comment'>-- Basic operation properties</span>
<a name="line-317"></a><span class='hs-comment'>---------------------------------------------------------------------</span>
<a name="line-318"></a>
<a name="line-319"></a><a name="prop_union"></a><span class='hs-comment'>-- | Iff a value is in either of two ranged sets then it is in the union of</span>
<a name="line-320"></a><span class='hs-comment'>-- those two sets.</span>
<a name="line-321"></a><span class='hs-comment'>--</span>
<a name="line-322"></a><span class='hs-comment'>-- &gt; prop_union rs1 rs2 v =</span>
<a name="line-323"></a><span class='hs-comment'>-- &gt;    (rs1 -?- v || rs2 -?- v) == ((rs1 -\/- rs2) -?- v)</span>
<a name="line-324"></a><span class='hs-definition'>prop_union</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span> <span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-325"></a><span class='hs-definition'>prop_union</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span> <span class='hs-varop'>||</span> <span class='hs-varid'>rs2</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-\/-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span>
<a name="line-326"></a>
<a name="line-327"></a><a name="prop_intersection"></a><span class='hs-comment'>-- | Iff a value is in both of two ranged sets then it is n the intersection</span>
<a name="line-328"></a><span class='hs-comment'>-- of those two sets.</span>
<a name="line-329"></a><span class='hs-comment'>--</span>
<a name="line-330"></a><span class='hs-comment'>-- &gt; prop_intersection rs1 rs2 v =</span>
<a name="line-331"></a><span class='hs-comment'>-- &gt;    (rs1 -?- v &amp;&amp; rs2 -?- v) == ((rs1 -/\- rs2) -?- v)</span>
<a name="line-332"></a><span class='hs-definition'>prop_intersection</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-333"></a><span class='hs-definition'>prop_intersection</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span>
<a name="line-334"></a>   <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>rs2</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-/\-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span>
<a name="line-335"></a>
<a name="line-336"></a><a name="prop_difference"></a><span class='hs-comment'>-- | Iff a value is in ranged set 1 and not in ranged set 2 then it is in the</span>
<a name="line-337"></a><span class='hs-comment'>-- difference of the two.</span>
<a name="line-338"></a><span class='hs-comment'>--</span>
<a name="line-339"></a><span class='hs-comment'>-- &gt; prop_difference rs1 rs2 v =</span>
<a name="line-340"></a><span class='hs-comment'>-- &gt;    (rs1 -?- v &amp;&amp; not (rs2 -?- v)) == ((rs1 -!- rs2) -?- v)</span>
<a name="line-341"></a><span class='hs-definition'>prop_difference</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-342"></a><span class='hs-definition'>prop_difference</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span>
<a name="line-343"></a>   <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>not</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs2</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-!-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span>
<a name="line-344"></a>
<a name="line-345"></a><a name="prop_negation"></a><span class='hs-comment'>-- | Iff a value is not in a ranged set then it is in its negation.</span>
<a name="line-346"></a><span class='hs-comment'>--</span>
<a name="line-347"></a><span class='hs-comment'>-- &gt; prop_negation rs v = rs -?- v == not (rSetNegation rs -?- v)</span>
<a name="line-348"></a><span class='hs-definition'>prop_negation</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-349"></a><span class='hs-definition'>prop_negation</span> <span class='hs-varid'>rs</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rs</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span> <span class='hs-varop'>==</span> <span class='hs-varid'>not</span> <span class='hs-layout'>(</span><span class='hs-varid'>rSetNegation</span> <span class='hs-varid'>rs</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span>
<a name="line-350"></a>
<a name="line-351"></a><a name="prop_not_empty"></a><span class='hs-comment'>-- | A set that contains a value is not empty</span>
<a name="line-352"></a><span class='hs-comment'>--</span>
<a name="line-353"></a><span class='hs-comment'>-- &gt; prop_not_empty rs v = (rs -?- v) ==&gt; not (rSetIsEmpty rs)</span>
<a name="line-354"></a><span class='hs-definition'>prop_not_empty</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Property</span>
<a name="line-355"></a><span class='hs-definition'>prop_not_empty</span> <span class='hs-varid'>rs</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varop'>==&gt;</span> <span class='hs-varid'>not</span> <span class='hs-layout'>(</span><span class='hs-varid'>rSetIsEmpty</span> <span class='hs-varid'>rs</span><span class='hs-layout'>)</span>
<a name="line-356"></a>
<a name="line-357"></a><span class='hs-comment'>---------------------------------------------------------------------</span>
<a name="line-358"></a><span class='hs-comment'>-- Some identities and inequalities of sets</span>
<a name="line-359"></a><span class='hs-comment'>---------------------------------------------------------------------</span>
<a name="line-360"></a>
<a name="line-361"></a><a name="prop_empty"></a><span class='hs-comment'>-- | The empty set has no members.</span>
<a name="line-362"></a><span class='hs-comment'>--</span>
<a name="line-363"></a><span class='hs-comment'>-- &gt; prop_empty v = not (rSetEmpty -?- v)</span>
<a name="line-364"></a><span class='hs-definition'>prop_empty</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-365"></a><span class='hs-definition'>prop_empty</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>not</span> <span class='hs-layout'>(</span><span class='hs-varid'>rSetEmpty</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span>
<a name="line-366"></a>
<a name="line-367"></a><a name="prop_full"></a><span class='hs-comment'>-- | The full set has every member.</span>
<a name="line-368"></a><span class='hs-comment'>--</span>
<a name="line-369"></a><span class='hs-comment'>-- &gt; prop_full v = rSetFull -?- v</span>
<a name="line-370"></a><span class='hs-definition'>prop_full</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-371"></a><span class='hs-definition'>prop_full</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetFull</span> <span class='hs-varop'>-?-</span> <span class='hs-varid'>v</span>
<a name="line-372"></a>
<a name="line-373"></a><a name="prop_empty_intersection"></a><span class='hs-comment'>-- | The intersection of a set with its negation is empty.</span>
<a name="line-374"></a><span class='hs-comment'>--</span>
<a name="line-375"></a><span class='hs-comment'>-- &gt; prop_empty_intersection rs =</span>
<a name="line-376"></a><span class='hs-comment'>-- &gt;    rSetIsEmpty (rs -/\- rSetNegation rs)</span>
<a name="line-377"></a><span class='hs-definition'>prop_empty_intersection</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-378"></a><span class='hs-definition'>prop_empty_intersection</span> <span class='hs-varid'>rs</span> <span class='hs-keyglyph'>=</span>
<a name="line-379"></a>   <span class='hs-varid'>rSetIsEmpty</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs</span> <span class='hs-varop'>-/\-</span> <span class='hs-varid'>rSetNegation</span> <span class='hs-varid'>rs</span><span class='hs-layout'>)</span>
<a name="line-380"></a>
<a name="line-381"></a><a name="prop_full_union"></a><span class='hs-comment'>-- | The union of a set with its negation is full.</span>
<a name="line-382"></a><span class='hs-comment'>--</span>
<a name="line-383"></a><span class='hs-comment'>-- &gt; prop_full_union rs v =</span>
<a name="line-384"></a><span class='hs-comment'>-- &gt;    rSetIsFull (rs -\/- rSetNegation rs)</span>
<a name="line-385"></a><span class='hs-definition'>prop_full_union</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-386"></a><span class='hs-definition'>prop_full_union</span> <span class='hs-varid'>rs</span> <span class='hs-keyglyph'>=</span>
<a name="line-387"></a>   <span class='hs-varid'>rSetIsFull</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs</span> <span class='hs-varop'>-\/-</span> <span class='hs-varid'>rSetNegation</span> <span class='hs-varid'>rs</span><span class='hs-layout'>)</span>
<a name="line-388"></a>
<a name="line-389"></a><a name="prop_union_superset"></a><span class='hs-comment'>-- | The union of two sets is the non-strict superset of both.</span>
<a name="line-390"></a><span class='hs-comment'>--</span>
<a name="line-391"></a><span class='hs-comment'>-- &gt; prop_union_superset rs1 rs2 =</span>
<a name="line-392"></a><span class='hs-comment'>-- &gt;    rs1 -&lt;=- u &amp;&amp; rs2 -&lt;=- u</span>
<a name="line-393"></a><span class='hs-comment'>-- &gt;    where</span>
<a name="line-394"></a><span class='hs-comment'>-- &gt;       u = rs1 -\/- rs2</span>
<a name="line-395"></a><span class='hs-definition'>prop_union_superset</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-396"></a><span class='hs-definition'>prop_union_superset</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-keyglyph'>=</span>
<a name="line-397"></a>   <span class='hs-varid'>rs1</span> <span class='hs-varop'>-&lt;=-</span> <span class='hs-varid'>u</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>rs2</span> <span class='hs-varop'>-&lt;=-</span> <span class='hs-varid'>u</span>
<a name="line-398"></a>   <span class='hs-keyword'>where</span>
<a name="line-399"></a>      <span class='hs-varid'>u</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rs1</span> <span class='hs-varop'>-\/-</span> <span class='hs-varid'>rs2</span>
<a name="line-400"></a>
<a name="line-401"></a><a name="prop_intersection_subset"></a><span class='hs-comment'>-- | The intersection of two sets is the non-strict subset of both.</span>
<a name="line-402"></a><span class='hs-comment'>--</span>
<a name="line-403"></a><span class='hs-comment'>-- &gt; prop_intersection_subset rs1 rs2 =</span>
<a name="line-404"></a><span class='hs-comment'>-- &gt;    i -&lt;=- rs1 &amp;&amp; i -&lt;=- rs2</span>
<a name="line-405"></a><span class='hs-comment'>-- &gt;    where</span>
<a name="line-406"></a><span class='hs-comment'>-- &gt;       i = rs1 -/\- rs2</span>
<a name="line-407"></a><span class='hs-definition'>prop_intersection_subset</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-408"></a><span class='hs-definition'>prop_intersection_subset</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>i</span> <span class='hs-varop'>-&lt;=-</span> <span class='hs-varid'>rs1</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>i</span> <span class='hs-varop'>-&lt;=-</span> <span class='hs-varid'>rs2</span>
<a name="line-409"></a>   <span class='hs-keyword'>where</span>
<a name="line-410"></a>      <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rs1</span> <span class='hs-varop'>-/\-</span> <span class='hs-varid'>rs2</span>
<a name="line-411"></a>
<a name="line-412"></a><a name="prop_diff_intersect"></a><span class='hs-comment'>-- | The difference of two sets intersected with the subtractand is empty.</span>
<a name="line-413"></a><span class='hs-comment'>--</span>
<a name="line-414"></a><span class='hs-comment'>-- &gt; prop_diff_intersect rs1 rs2 =</span>
<a name="line-415"></a><span class='hs-comment'>-- &gt;    rSetIsEmpty ((rs1 -!- rs2) -/\- rs2)</span>
<a name="line-416"></a><span class='hs-definition'>prop_diff_intersect</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-417"></a><span class='hs-definition'>prop_diff_intersect</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rSetIsEmpty</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-!-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span> <span class='hs-varop'>-/\-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span>
<a name="line-418"></a>
<a name="line-419"></a><a name="prop_subset"></a><span class='hs-comment'>-- | A set is the non-strict subset of itself.</span>
<a name="line-420"></a><span class='hs-comment'>--</span>
<a name="line-421"></a><span class='hs-comment'>-- &gt; prop_subset rs = rs -&lt;=- rs</span>
<a name="line-422"></a><span class='hs-definition'>prop_subset</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-423"></a><span class='hs-definition'>prop_subset</span> <span class='hs-varid'>rs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rs</span> <span class='hs-varop'>-&lt;=-</span> <span class='hs-varid'>rs</span>
<a name="line-424"></a>
<a name="line-425"></a><a name="prop_strict_subset"></a><span class='hs-comment'>-- | A set is not the strict subset of itself.</span>
<a name="line-426"></a><span class='hs-comment'>--</span>
<a name="line-427"></a><span class='hs-comment'>-- &gt; prop_strict_subset rs = not (rs -&lt;- rs)</span>
<a name="line-428"></a><span class='hs-definition'>prop_strict_subset</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-429"></a><span class='hs-definition'>prop_strict_subset</span> <span class='hs-varid'>rs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>not</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs</span> <span class='hs-varop'>-&lt;-</span> <span class='hs-varid'>rs</span><span class='hs-layout'>)</span>
<a name="line-430"></a>
<a name="line-431"></a><a name="prop_union_strict_superset"></a><span class='hs-comment'>-- | If rs1 - rs2 is not empty then the union of rs1 and rs2 will be a strict</span>
<a name="line-432"></a><span class='hs-comment'>-- superset of rs2.</span>
<a name="line-433"></a><span class='hs-comment'>--</span>
<a name="line-434"></a><span class='hs-comment'>-- &gt; prop_union_strict_superset rs1 rs2 =</span>
<a name="line-435"></a><span class='hs-comment'>-- &gt;    (not $ rSetIsEmpty (rs1 -!- rs2))</span>
<a name="line-436"></a><span class='hs-comment'>-- &gt;    ==&gt; (rs2 -&lt;- (rs1 -\/- rs2))</span>
<a name="line-437"></a><span class='hs-definition'>prop_union_strict_superset</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Property</span>
<a name="line-438"></a><span class='hs-definition'>prop_union_strict_superset</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-keyglyph'>=</span>
<a name="line-439"></a>   <span class='hs-layout'>(</span><span class='hs-varid'>not</span> <span class='hs-varop'>$</span> <span class='hs-varid'>rSetIsEmpty</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-!-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>==&gt;</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs2</span> <span class='hs-varop'>-&lt;-</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-\/-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-440"></a>
<a name="line-441"></a><a name="prop_intersection_commutes"></a><span class='hs-comment'>-- | Intersection commutes.</span>
<a name="line-442"></a><span class='hs-comment'>--</span>
<a name="line-443"></a><span class='hs-comment'>-- &gt; prop_intersection_commutes rs1 rs2 = (rs1 -/\- rs2) == (rs2 -/\- rs1)</span>
<a name="line-444"></a><span class='hs-definition'>prop_intersection_commutes</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-445"></a><span class='hs-definition'>prop_intersection_commutes</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-/\-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs2</span> <span class='hs-varop'>-/\-</span> <span class='hs-varid'>rs1</span><span class='hs-layout'>)</span>
<a name="line-446"></a>
<a name="line-447"></a><a name="prop_union_commutes"></a><span class='hs-comment'>-- | Union commutes.</span>
<a name="line-448"></a><span class='hs-comment'>--</span>
<a name="line-449"></a><span class='hs-comment'>-- &gt; prop_union_commutes rs1 rs2 = (rs1 -\/- rs2) == (rs2 -\/- rs1)</span>
<a name="line-450"></a><span class='hs-definition'>prop_union_commutes</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-451"></a><span class='hs-definition'>prop_union_commutes</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-\/-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs2</span> <span class='hs-varop'>-\/-</span> <span class='hs-varid'>rs1</span><span class='hs-layout'>)</span>
<a name="line-452"></a>
<a name="line-453"></a><a name="prop_intersection_associates"></a><span class='hs-comment'>-- | Intersection associates.</span>
<a name="line-454"></a><span class='hs-comment'>--</span>
<a name="line-455"></a><span class='hs-comment'>-- &gt; prop_intersection_associates rs1 rs2 rs3 =</span>
<a name="line-456"></a><span class='hs-comment'>-- &gt;    ((rs1 -/\- rs2) -/\- rs3) == (rs1 -/\- (rs2 -/\- rs3))</span>
<a name="line-457"></a><span class='hs-definition'>prop_intersection_associates</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span>
<a name="line-458"></a>   <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span>  <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-459"></a><span class='hs-definition'>prop_intersection_associates</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-varid'>rs3</span> <span class='hs-keyglyph'>=</span>
<a name="line-460"></a>   <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-/\-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span> <span class='hs-varop'>-/\-</span> <span class='hs-varid'>rs3</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-/\-</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs2</span> <span class='hs-varop'>-/\-</span> <span class='hs-varid'>rs3</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-461"></a>
<a name="line-462"></a><a name="prop_union_associates"></a><span class='hs-comment'>-- | Union associates.</span>
<a name="line-463"></a><span class='hs-comment'>--</span>
<a name="line-464"></a><span class='hs-comment'>-- &gt; prop_union_associates rs1 rs2 rs3 =</span>
<a name="line-465"></a><span class='hs-comment'>-- &gt;    ((rs1 -\/- rs2) -\/- rs3) == (rs1 -\/- (rs2 -\/- rs3))</span>
<a name="line-466"></a><span class='hs-definition'>prop_union_associates</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span>
<a name="line-467"></a>   <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span>  <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-468"></a><span class='hs-definition'>prop_union_associates</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-varid'>rs3</span> <span class='hs-keyglyph'>=</span>
<a name="line-469"></a>   <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-\/-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span> <span class='hs-varop'>-\/-</span> <span class='hs-varid'>rs3</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-\/-</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs2</span> <span class='hs-varop'>-\/-</span> <span class='hs-varid'>rs3</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-470"></a>
<a name="line-471"></a><a name="prop_de_morgan_intersection"></a><span class='hs-comment'>-- | De Morgan's Law for Intersection.</span>
<a name="line-472"></a><span class='hs-comment'>--</span>
<a name="line-473"></a><span class='hs-comment'>-- &gt; prop_de_morgan_intersection rs1 rs2 =</span>
<a name="line-474"></a><span class='hs-comment'>-- &gt;    rSetNegation (rs1 -/\- rs2) == (rSetNegation rs1 -\/- rSetNegation rs2)</span>
<a name="line-475"></a><span class='hs-definition'>prop_de_morgan_intersection</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-476"></a><span class='hs-definition'>prop_de_morgan_intersection</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-keyglyph'>=</span>
<a name="line-477"></a>   <span class='hs-varid'>rSetNegation</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-/\-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-varid'>rSetNegation</span> <span class='hs-varid'>rs1</span> <span class='hs-varop'>-\/-</span> <span class='hs-varid'>rSetNegation</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span>
<a name="line-478"></a>
<a name="line-479"></a><span class='hs-comment'>-- | De Morgan's Law for Union.</span>
<a name="line-480"></a><span class='hs-comment'>--</span>
<a name="line-481"></a><span class='hs-comment'>-- &gt; prop_de_morgan_union rs1 rs2 =</span>
<a name="line-482"></a><span class='hs-comment'>-- &gt;    rSetNegation (rs1 -\/- rs2) == (rSetNegation rs1 -/\- rSetNegation rs2)</span>
<a name="line-483"></a>
<a name="line-484"></a><a name="prop_de_morgan_union"></a><span class='hs-definition'>prop_de_morgan_union</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>DiscreteOrdered</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>RSet</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-485"></a><span class='hs-definition'>prop_de_morgan_union</span> <span class='hs-varid'>rs1</span> <span class='hs-varid'>rs2</span> <span class='hs-keyglyph'>=</span>
<a name="line-486"></a>   <span class='hs-varid'>rSetNegation</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs1</span> <span class='hs-varop'>-\/-</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-varid'>rSetNegation</span> <span class='hs-varid'>rs1</span> <span class='hs-varop'>-/\-</span> <span class='hs-varid'>rSetNegation</span> <span class='hs-varid'>rs2</span><span class='hs-layout'>)</span>
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