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<title>src/full/Agda/Termination/SparseMatrix.hs</title>
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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE CPP, DeriveFunctor #-}</span>
<a name="line-2"></a>
<a name="line-3"></a><span class='hs-comment'>{- | Sparse matrices.
<a name="line-4"></a>
<a name="line-5"></a>We assume the matrices to be very sparse, so we just implement them as
<a name="line-6"></a>sorted association lists.
<a name="line-7"></a>
<a name="line-8"></a> -}</span>
<a name="line-9"></a>
<a name="line-10"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Agda</span><span class='hs-varop'>.</span><span class='hs-conid'>Termination</span><span class='hs-varop'>.</span><span class='hs-conid'>SparseMatrix</span>
<a name="line-11"></a> <span class='hs-layout'>(</span> <span class='hs-comment'>-- * Basic data types</span>
<a name="line-12"></a> <span class='hs-conid'>Matrix</span>
<a name="line-13"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>matrixInvariant</span>
<a name="line-14"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>Size</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span>
<a name="line-15"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>sizeInvariant</span>
<a name="line-16"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>MIx</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span>
<a name="line-17"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>mIxInvariant</span>
<a name="line-18"></a> <span class='hs-comment'>-- * Generating and creating matrices</span>
<a name="line-19"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>fromLists</span>
<a name="line-20"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>fromIndexList</span>
<a name="line-21"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>toLists</span>
<a name="line-22"></a><span class='hs-comment'>-- , Agda.Termination.Matrix.zipWith</span>
<a name="line-23"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>matrix</span>
<a name="line-24"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>matrixUsingRowGen</span>
<a name="line-25"></a> <span class='hs-comment'>-- * Combining and querying matrices</span>
<a name="line-26"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>size</span>
<a name="line-27"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>square</span>
<a name="line-28"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>isEmpty</span>
<a name="line-29"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>isSingleton</span>
<a name="line-30"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>add</span><span class='hs-layout'>,</span> <span class='hs-varid'>intersectWith</span>
<a name="line-31"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>mul</span>
<a name="line-32"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>transpose</span>
<a name="line-33"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>diagonal</span>
<a name="line-34"></a> <span class='hs-comment'>-- * Modifying matrices</span>
<a name="line-35"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>addRow</span>
<a name="line-36"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>addColumn</span>
<a name="line-37"></a> <span class='hs-comment'>-- * Tests</span>
<a name="line-38"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>Agda</span><span class='hs-varop'>.</span><span class='hs-conid'>Termination</span><span class='hs-varop'>.</span><span class='hs-conid'>SparseMatrix</span><span class='hs-varop'>.</span><span class='hs-varid'>tests</span>
<a name="line-39"></a> <span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-40"></a>
<a name="line-41"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Array</span>
<a name="line-42"></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'>List</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>List</span>
<a name="line-43"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Maybe</span>
<a name="line-44"></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-45"></a>
<a name="line-46"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Agda</span><span class='hs-varop'>.</span><span class='hs-conid'>Utils</span><span class='hs-varop'>.</span><span class='hs-conid'>Pretty</span> <span class='hs-varid'>hiding</span> <span class='hs-layout'>(</span><span class='hs-varid'>isEmpty</span><span class='hs-layout'>)</span>
<a name="line-47"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Agda</span><span class='hs-varop'>.</span><span class='hs-conid'>Utils</span><span class='hs-varop'>.</span><span class='hs-conid'>QuickCheck</span>
<a name="line-48"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Agda</span><span class='hs-varop'>.</span><span class='hs-conid'>Utils</span><span class='hs-varop'>.</span><span class='hs-conid'>TestHelpers</span>
<a name="line-49"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Agda</span><span class='hs-varop'>.</span><span class='hs-conid'>Termination</span><span class='hs-varop'>.</span><span class='hs-conid'>Semiring</span> <span class='hs-layout'>(</span><span class='hs-conid'>HasZero</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>SemiRing</span><span class='hs-layout'>,</span> <span class='hs-conid'>Semiring</span><span class='hs-layout'>)</span>
<a name="line-50"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Agda</span><span class='hs-varop'>.</span><span class='hs-conid'>Termination</span><span class='hs-varop'>.</span><span class='hs-conid'>Semiring</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>Semiring</span>
<a name="line-51"></a>
<a name="line-52"></a><span class='hs-cpp'>#include "../undefined.h"</span>
<a name="line-53"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Agda</span><span class='hs-varop'>.</span><span class='hs-conid'>Utils</span><span class='hs-varop'>.</span><span class='hs-conid'>Impossible</span>
<a name="line-54"></a>
<a name="line-55"></a><span class='hs-comment'>------------------------------------------------------------------------</span>
<a name="line-56"></a><span class='hs-comment'>-- Basic data types</span>
<a name="line-57"></a>
<a name="line-58"></a><span class='hs-comment'>-- | This matrix type is used for tests.</span>
<a name="line-59"></a>
<a name="line-60"></a><a name="TM"></a><span class='hs-keyword'>type</span> <span class='hs-conid'>TM</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Matrix</span> <span class='hs-conid'>Integer</span> <span class='hs-conid'>Integer</span>
<a name="line-61"></a>
<a name="line-62"></a><span class='hs-comment'>-- | Size of a matrix.</span>
<a name="line-63"></a>
<a name="line-64"></a><a name="Size"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Size</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Size</span> <span class='hs-layout'>{</span> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>cols</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>i</span> <span class='hs-layout'>}</span>
<a name="line-65"></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'>Ord</span><span class='hs-layout'>,</span> <span class='hs-conid'>Show</span><span class='hs-layout'>)</span>
<a name="line-66"></a>
<a name="line-67"></a><a name="sizeInvariant"></a><span class='hs-definition'>sizeInvariant</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Size</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-68"></a><span class='hs-definition'>sizeInvariant</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rows</span> <span class='hs-varid'>sz</span> <span class='hs-varop'>>=</span> <span class='hs-num'>0</span> <span class='hs-varop'>&&</span> <span class='hs-varid'>cols</span> <span class='hs-varid'>sz</span> <span class='hs-varop'>>=</span> <span class='hs-num'>0</span>
<a name="line-69"></a>
<a name="line-70"></a><a name="instance%20Arbitrary%20(Size%20i)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Arbitrary</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Integral</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Arbitrary</span> <span class='hs-layout'>(</span><span class='hs-conid'>Size</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-71"></a> <span class='hs-varid'>arbitrary</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-72"></a> <span class='hs-varid'>r</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>natural</span>
<a name="line-73"></a> <span class='hs-varid'>c</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>natural</span>
<a name="line-74"></a> <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Size</span> <span class='hs-layout'>{</span> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-varid'>r</span><span class='hs-layout'>,</span> <span class='hs-varid'>cols</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-varid'>c</span> <span class='hs-layout'>}</span>
<a name="line-75"></a>
<a name="line-76"></a><a name="instance%20CoArbitrary%20(Size%20i)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>CoArbitrary</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>CoArbitrary</span> <span class='hs-layout'>(</span><span class='hs-conid'>Size</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-77"></a> <span class='hs-varid'>coarbitrary</span> <span class='hs-layout'>(</span><span class='hs-conid'>Size</span> <span class='hs-varid'>rs</span> <span class='hs-varid'>cs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>coarbitrary</span> <span class='hs-varid'>rs</span> <span class='hs-varop'>.</span> <span class='hs-varid'>coarbitrary</span> <span class='hs-varid'>cs</span>
<a name="line-78"></a>
<a name="line-79"></a><a name="prop_Arbitrary_Size"></a><span class='hs-definition'>prop_Arbitrary_Size</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Size</span> <span class='hs-conid'>Integer</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-80"></a><span class='hs-definition'>prop_Arbitrary_Size</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sizeInvariant</span>
<a name="line-81"></a>
<a name="line-82"></a><span class='hs-comment'>-- | Converts a size to a set of bounds suitable for use with</span>
<a name="line-83"></a><span class='hs-comment'>-- the matrices in this module.</span>
<a name="line-84"></a>
<a name="line-85"></a><a name="toBounds"></a><span class='hs-definition'>toBounds</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Size</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span>
<a name="line-86"></a><span class='hs-definition'>toBounds</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-layout'>{</span> <span class='hs-varid'>row</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1</span><span class='hs-layout'>,</span> <span class='hs-varid'>col</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1</span> <span class='hs-layout'>}</span><span class='hs-layout'>,</span> <span class='hs-conid'>MIx</span> <span class='hs-layout'>{</span> <span class='hs-varid'>row</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rows</span> <span class='hs-varid'>sz</span><span class='hs-layout'>,</span> <span class='hs-varid'>col</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cols</span> <span class='hs-varid'>sz</span> <span class='hs-layout'>}</span><span class='hs-layout'>)</span>
<a name="line-87"></a>
<a name="line-88"></a><span class='hs-comment'>-- | Type of matrix indices (row, column).</span>
<a name="line-89"></a>
<a name="line-90"></a><a name="MIx"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>MIx</span> <span class='hs-layout'>{</span> <span class='hs-varid'>row</span><span class='hs-layout'>,</span> <span class='hs-varid'>col</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>i</span> <span class='hs-layout'>}</span>
<a name="line-91"></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> <span class='hs-conid'>Ix</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span><span class='hs-layout'>)</span>
<a name="line-92"></a>
<a name="line-93"></a><a name="instance%20Arbitrary%20(MIx%20i)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Arbitrary</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Integral</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Arbitrary</span> <span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-94"></a> <span class='hs-varid'>arbitrary</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-95"></a> <span class='hs-varid'>r</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>positive</span>
<a name="line-96"></a> <span class='hs-varid'>c</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>positive</span>
<a name="line-97"></a> <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-conid'>MIx</span> <span class='hs-layout'>{</span> <span class='hs-varid'>row</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>r</span><span class='hs-layout'>,</span> <span class='hs-varid'>col</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>c</span> <span class='hs-layout'>}</span>
<a name="line-98"></a>
<a name="line-99"></a><a name="instance%20CoArbitrary%20(MIx%20i)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>CoArbitrary</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>CoArbitrary</span> <span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-100"></a> <span class='hs-varid'>coarbitrary</span> <span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>r</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>coarbitrary</span> <span class='hs-varid'>r</span> <span class='hs-varop'>.</span> <span class='hs-varid'>coarbitrary</span> <span class='hs-varid'>c</span>
<a name="line-101"></a>
<a name="line-102"></a><span class='hs-comment'>-- | No nonpositive indices are allowed.</span>
<a name="line-103"></a>
<a name="line-104"></a><a name="mIxInvariant"></a><span class='hs-definition'>mIxInvariant</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-105"></a><span class='hs-definition'>mIxInvariant</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>row</span> <span class='hs-varid'>i</span> <span class='hs-varop'>>=</span> <span class='hs-num'>1</span> <span class='hs-varop'>&&</span> <span class='hs-varid'>col</span> <span class='hs-varid'>i</span> <span class='hs-varop'>>=</span> <span class='hs-num'>1</span>
<a name="line-106"></a>
<a name="line-107"></a><a name="prop_Arbitrary_MIx"></a><span class='hs-definition'>prop_Arbitrary_MIx</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>MIx</span> <span class='hs-conid'>Integer</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-108"></a><span class='hs-definition'>prop_Arbitrary_MIx</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>mIxInvariant</span>
<a name="line-109"></a>
<a name="line-110"></a><span class='hs-comment'>-- | Type of matrices, parameterised on the type of values.</span>
<a name="line-111"></a>
<a name="line-112"></a><a name="Matrix"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span> <span class='hs-layout'>{</span> <span class='hs-varid'>size</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Size</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>unM</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-layout'>}</span>
<a name="line-113"></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'>Ord</span><span class='hs-layout'>,</span> <span class='hs-conid'>Functor</span><span class='hs-layout'>)</span>
<a name="line-114"></a>
<a name="line-115"></a><a name="matrixInvariant"></a><span class='hs-definition'>matrixInvariant</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ix</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-116"></a><span class='hs-definition'>matrixInvariant</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>all</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span> <span class='hs-varid'>j</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-num'>1</span> <span class='hs-varop'><=</span> <span class='hs-varid'>i</span> <span class='hs-varop'>&&</span> <span class='hs-varid'>i</span> <span class='hs-varop'><=</span> <span class='hs-varid'>rows</span> <span class='hs-varid'>sz</span>
<a name="line-117"></a> <span class='hs-varop'>&&</span> <span class='hs-num'>1</span> <span class='hs-varop'><=</span> <span class='hs-varid'>j</span> <span class='hs-varop'>&&</span> <span class='hs-varid'>j</span> <span class='hs-varop'><=</span> <span class='hs-varid'>cols</span> <span class='hs-varid'>sz</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>unM</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-118"></a> <span class='hs-varop'>&&</span> <span class='hs-varid'>strictlySorted</span> <span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-num'>0</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>unM</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-119"></a> <span class='hs-varop'>&&</span> <span class='hs-varid'>sizeInvariant</span> <span class='hs-varid'>sz</span>
<a name="line-120"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>size</span> <span class='hs-varid'>m</span>
<a name="line-121"></a>
<a name="line-122"></a><span class='hs-comment'>-- matrix indices are lexicographically sorted with no duplicates</span>
<a name="line-123"></a><span class='hs-comment'>-- Ord MIx should be the lexicographic one already (Haskell report)</span>
<a name="line-124"></a>
<a name="line-125"></a><a name="strictlySorted"></a><span class='hs-definition'>strictlySorted</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-126"></a><span class='hs-definition'>strictlySorted</span> <span class='hs-varid'>i</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span>
<a name="line-127"></a><span class='hs-definition'>strictlySorted</span> <span class='hs-varid'>i</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>i'</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>i</span> <span class='hs-varop'><</span> <span class='hs-varid'>i'</span> <span class='hs-varop'>&&</span> <span class='hs-varid'>strictlySorted</span> <span class='hs-varid'>i'</span> <span class='hs-varid'>l</span>
<a name="line-128"></a><span class='hs-comment'>{-
<a name="line-129"></a>strictlySorted (MIx i j) [] = True
<a name="line-130"></a>strictlySorted (MIx i j) ((MIx i' j', b) : l) =
<a name="line-131"></a> (i < i' || i == i' && j < j' ) && strictlySorted (MIx i' j') b
<a name="line-132"></a>-}</span>
<a name="line-133"></a>
<a name="line-134"></a><a name="instance%20Show%20(Matrix%20i%20b)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Integral</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Enum</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ix</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Show</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Show</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Show</span> <span class='hs-layout'>(</span><span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-135"></a> <span class='hs-varid'>showsPrec</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span>
<a name="line-136"></a> <span class='hs-varid'>showString</span> <span class='hs-str'>"Agda.Termination.Matrix.fromLists "</span> <span class='hs-varop'>.</span> <span class='hs-varid'>shows</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span>
<a name="line-137"></a> <span class='hs-varid'>showString</span> <span class='hs-str'>" "</span> <span class='hs-varop'>.</span> <span class='hs-varid'>shows</span> <span class='hs-layout'>(</span><span class='hs-varid'>toLists</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-138"></a>
<a name="line-139"></a><a name="instance%20Pretty%20(Matrix%20i%20b)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Show</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Integral</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ix</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pretty</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-140"></a> <span class='hs-conid'>Pretty</span> <span class='hs-layout'>(</span><span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-141"></a> <span class='hs-varid'>pretty</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>vcat</span> <span class='hs-varop'>.</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>hsep</span> <span class='hs-varop'>.</span> <span class='hs-varid'>map</span> <span class='hs-varid'>pretty</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>toLists</span>
<a name="line-142"></a>
<a name="line-143"></a><a name="instance%20Arbitrary%20(Matrix%20i%20b)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Arbitrary</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Integral</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Arbitrary</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span>
<a name="line-144"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Arbitrary</span> <span class='hs-layout'>(</span><span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-145"></a> <span class='hs-varid'>arbitrary</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>matrix</span> <span class='hs-varop'>=<<</span> <span class='hs-varid'>arbitrary</span>
<a name="line-146"></a>
<a name="line-147"></a><a name="instance%20CoArbitrary%20(Matrix%20i%20b)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Show</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Integral</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Enum</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ix</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>CoArbitrary</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>CoArbitrary</span> <span class='hs-layout'>(</span><span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-148"></a> <span class='hs-varid'>coarbitrary</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>coarbitrary</span> <span class='hs-layout'>(</span><span class='hs-varid'>toLists</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-149"></a>
<a name="line-150"></a>
<a name="line-151"></a><a name="prop_Arbitrary_Matrix"></a><span class='hs-definition'>prop_Arbitrary_Matrix</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>TM</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-152"></a><span class='hs-definition'>prop_Arbitrary_Matrix</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>matrixInvariant</span>
<a name="line-153"></a>
<a name="line-154"></a>
<a name="line-155"></a><span class='hs-comment'>------------------------------------------------------------------------</span>
<a name="line-156"></a><span class='hs-comment'>-- Generating and creating matrices</span>
<a name="line-157"></a>
<a name="line-158"></a><span class='hs-comment'>-- | Generates a matrix of the given size, using the given generator</span>
<a name="line-159"></a><span class='hs-comment'>-- to generate the rows.</span>
<a name="line-160"></a>
<a name="line-161"></a><a name="matrixUsingRowGen"></a><span class='hs-definition'>matrixUsingRowGen</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Arbitrary</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Integral</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Arbitrary</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span>
<a name="line-162"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Size</span> <span class='hs-varid'>i</span>
<a name="line-163"></a> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Gen</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>b</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span>
<a name="line-164"></a> <span class='hs-comment'>-- ^ The generator is parameterised on the size of the row.</span>
<a name="line-165"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Gen</span> <span class='hs-layout'>(</span><span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span>
<a name="line-166"></a><span class='hs-definition'>matrixUsingRowGen</span> <span class='hs-varid'>sz</span> <span class='hs-varid'>rowGen</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-167"></a> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>vectorOf</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromIntegral</span> <span class='hs-varop'>$</span> <span class='hs-varid'>rows</span> <span class='hs-varid'>sz</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>rowGen</span> <span class='hs-varop'>$</span> <span class='hs-varid'>cols</span> <span class='hs-varid'>sz</span><span class='hs-layout'>)</span>
<a name="line-168"></a> <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-varid'>fromLists</span> <span class='hs-varid'>sz</span> <span class='hs-varid'>rows</span>
<a name="line-169"></a>
<a name="line-170"></a><span class='hs-comment'>-- | Generates a matrix of the given size.</span>
<a name="line-171"></a>
<a name="line-172"></a><a name="matrix"></a><span class='hs-definition'>matrix</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Arbitrary</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Integral</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Arbitrary</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span>
<a name="line-173"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Size</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Gen</span> <span class='hs-layout'>(</span><span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span>
<a name="line-174"></a><span class='hs-definition'>matrix</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>matrixUsingRowGen</span> <span class='hs-varid'>sz</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>n</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>vectorOf</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromIntegral</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-varid'>arbitrary</span><span class='hs-layout'>)</span>
<a name="line-175"></a>
<a name="line-176"></a><a name="prop_matrix"></a><span class='hs-definition'>prop_matrix</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>forAll</span> <span class='hs-layout'>(</span><span class='hs-varid'>matrix</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Gen</span> <span class='hs-conid'>TM</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span><span class='hs-varid'>m</span> <span class='hs-keyglyph'>-></span>
<a name="line-177"></a><span class='hs-comment'>-- matrixInvariant m &&</span>
<a name="line-178"></a> <span class='hs-varid'>size</span> <span class='hs-varid'>m</span> <span class='hs-varop'>==</span> <span class='hs-varid'>sz</span>
<a name="line-179"></a>
<a name="line-180"></a><span class='hs-comment'>-- | Constructs a matrix from a list of (index, value)-pairs.</span>
<a name="line-181"></a>
<a name="line-182"></a><span class='hs-comment'>-- compareElt = (\ (i,_) (j,_) -> compare i j)</span>
<a name="line-183"></a><span class='hs-comment'>-- normalize = filter (\ (i,b) -> b /= zeroElement)</span>
<a name="line-184"></a>
<a name="line-185"></a><a name="fromIndexList"></a><span class='hs-definition'>fromIndexList</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Size</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span>
<a name="line-186"></a><span class='hs-definition'>fromIndexList</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span> <span class='hs-varid'>sz</span> <span class='hs-varop'>.</span> <span class='hs-conid'>List</span><span class='hs-varop'>.</span><span class='hs-varid'>sortBy</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>j</span><span class='hs-layout'>,</span><span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>compare</span> <span class='hs-varid'>i</span> <span class='hs-varid'>j</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>zeroElement</span><span class='hs-layout'>)</span>
<a name="line-187"></a>
<a name="line-188"></a><a name="prop_fromIndexList"></a><span class='hs-definition'>prop_fromIndexList</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>TM</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-189"></a><span class='hs-definition'>prop_fromIndexList</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>matrixInvariant</span> <span class='hs-varid'>m'</span> <span class='hs-varop'>&&</span> <span class='hs-varid'>m'</span> <span class='hs-varop'>==</span> <span class='hs-varid'>m</span>
<a name="line-190"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>vs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unM</span> <span class='hs-varid'>m</span>
<a name="line-191"></a> <span class='hs-varid'>m'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromIndexList</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-varid'>vs</span>
<a name="line-192"></a>
<a name="line-193"></a><span class='hs-comment'>-- | @'fromLists' sz rs@ constructs a matrix from a list of lists of</span>
<a name="line-194"></a><span class='hs-comment'>-- values (a list of rows).</span>
<a name="line-195"></a><span class='hs-comment'>--</span>
<a name="line-196"></a><span class='hs-comment'>-- Precondition: @'length' rs '==' 'rows' sz '&&' 'all' (('==' 'cols' sz) . 'length') rs@.</span>
<a name="line-197"></a>
<a name="line-198"></a><a name="fromLists"></a><span class='hs-definition'>fromLists</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Enum</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Size</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</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'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span>
<a name="line-199"></a><span class='hs-definition'>fromLists</span> <span class='hs-varid'>sz</span> <span class='hs-varid'>bs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromIndexList</span> <span class='hs-varid'>sz</span> <span class='hs-varop'>$</span> <span class='hs-varid'>zip</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span> <span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span> <span class='hs-varid'>j</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'><-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>rows</span> <span class='hs-varid'>sz</span><span class='hs-keyglyph'>]</span>
<a name="line-200"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>j</span> <span class='hs-keyglyph'><-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>cols</span> <span class='hs-varid'>sz</span><span class='hs-keyglyph'>]</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>bs</span><span class='hs-layout'>)</span>
<a name="line-201"></a>
<a name="line-202"></a><span class='hs-comment'>-- | Converts a sparse matrix to a sparse list of rows</span>
<a name="line-203"></a>
<a name="line-204"></a><a name="toSparseRows"></a><span class='hs-definition'>toSparseRows</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Enum</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</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-keyglyph'>]</span>
<a name="line-205"></a><span class='hs-definition'>toSparseRows</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>aux</span> <span class='hs-num'>1</span> <span class='hs-conid'>[]</span> <span class='hs-layout'>(</span><span class='hs-varid'>unM</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-206"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>aux</span> <span class='hs-varid'>i'</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-207"></a> <span class='hs-varid'>aux</span> <span class='hs-varid'>i'</span> <span class='hs-varid'>row</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i'</span><span class='hs-layout'>,</span> <span class='hs-varid'>reverse</span> <span class='hs-varid'>row</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-208"></a> <span class='hs-varid'>aux</span> <span class='hs-varid'>i'</span> <span class='hs-varid'>row</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span> <span class='hs-varid'>j</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-209"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i'</span> <span class='hs-varop'>==</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>aux</span> <span class='hs-varid'>i'</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>j</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>row</span><span class='hs-layout'>)</span> <span class='hs-varid'>m</span>
<a name="line-210"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>i'</span><span class='hs-layout'>,</span> <span class='hs-varid'>reverse</span> <span class='hs-varid'>row</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>aux</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>j</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-varid'>m</span>
<a name="line-211"></a>
<a name="line-212"></a><a name="blowUpSparseVec"></a><span class='hs-comment'>-- sparse vectors cannot have two entries in one column</span>
<a name="line-213"></a><span class='hs-definition'>blowUpSparseVec</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Show</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Enum</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>b</span><span class='hs-keyglyph'>]</span>
<a name="line-214"></a><span class='hs-definition'>blowUpSparseVec</span> <span class='hs-varid'>zero</span> <span class='hs-varid'>n</span> <span class='hs-varid'>l</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>aux</span> <span class='hs-num'>1</span> <span class='hs-varid'>l</span>
<a name="line-215"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>aux</span> <span class='hs-varid'>i</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-varop'>></span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-216"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zero</span> <span class='hs-conop'>:</span> <span class='hs-varid'>aux</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-conid'>[]</span>
<a name="line-217"></a> <span class='hs-varid'>aux</span> <span class='hs-varid'>i</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>j</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-varop'><=</span> <span class='hs-varid'>n</span> <span class='hs-varop'>&&</span> <span class='hs-varid'>j</span> <span class='hs-varop'>==</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:</span> <span class='hs-varid'>aux</span> <span class='hs-layout'>(</span><span class='hs-varid'>succ</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-varid'>l</span>
<a name="line-218"></a> <span class='hs-varid'>aux</span> <span class='hs-varid'>i</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>j</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-varop'><=</span> <span class='hs-varid'>n</span> <span class='hs-varop'>&&</span> <span class='hs-varid'>j</span> <span class='hs-varop'>>=</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zero</span> <span class='hs-conop'>:</span> <span class='hs-varid'>aux</span> <span class='hs-layout'>(</span><span class='hs-varid'>succ</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>j</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>l</span><span class='hs-layout'>)</span>
<a name="line-219"></a> <span class='hs-varid'>aux</span> <span class='hs-varid'>i</span> <span class='hs-varid'>l</span> <span class='hs-keyglyph'>=</span> <span class='hs-sel'>__IMPOSSIBLE__</span>
<a name="line-220"></a> <span class='hs-comment'>-- error $ "blowUpSparseVec (n = " ++ show n ++ ") aux i=" ++ show i ++ " j=" ++ show (fst (head l)) ++ " length l = " ++ show (length l)</span>
<a name="line-221"></a>
<a name="line-222"></a><span class='hs-comment'>-- | Converts a matrix to a list of row lists.</span>
<a name="line-223"></a>
<a name="line-224"></a><a name="toLists"></a><span class='hs-definition'>toLists</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Show</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Integral</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Enum</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ix</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</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-varid'>b</span><span class='hs-keyglyph'>]</span><span class='hs-keyglyph'>]</span>
<a name="line-225"></a><span class='hs-definition'>toLists</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-comment'>-- if not $ matrixInvariant m then __IMPOSSIBLE__ else</span>
<a name="line-226"></a> <span class='hs-varid'>blowUpSparseVec</span> <span class='hs-varid'>emptyRow</span> <span class='hs-varid'>nr</span> <span class='hs-varop'>$</span>
<a name="line-227"></a> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>blowUpSparseVec</span> <span class='hs-varid'>zeroElement</span> <span class='hs-varid'>nc</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>toSparseRows</span> <span class='hs-varid'>m</span>
<a name="line-228"></a> <span class='hs-keyword'>where</span>
<a name="line-229"></a> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>size</span> <span class='hs-varid'>m</span>
<a name="line-230"></a> <span class='hs-varid'>nr</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rows</span> <span class='hs-varid'>sz</span>
<a name="line-231"></a> <span class='hs-varid'>nc</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cols</span> <span class='hs-varid'>sz</span>
<a name="line-232"></a> <span class='hs-varid'>emptyRow</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>take</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromIntegral</span> <span class='hs-varid'>nc</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>repeat</span> <span class='hs-varid'>zeroElement</span>
<a name="line-233"></a>
<a name="line-234"></a><a name="prop_fromLists_toLists"></a><span class='hs-definition'>prop_fromLists_toLists</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>TM</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-235"></a><span class='hs-definition'>prop_fromLists_toLists</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromLists</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>toLists</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-varid'>m</span>
<a name="line-236"></a>
<a name="line-237"></a><span class='hs-comment'>------------------------------------------------------------------------</span>
<a name="line-238"></a><span class='hs-comment'>-- Combining and querying matrices</span>
<a name="line-239"></a>
<a name="line-240"></a><span class='hs-comment'>-- | The size of a matrix.</span>
<a name="line-241"></a>
<a name="line-242"></a><span class='hs-comment'>{-
<a name="line-243"></a>size :: Ix i => Matrix i b -> Size i
<a name="line-244"></a>size m = Size { rows = row b, cols = col b }
<a name="line-245"></a> where (_, b) = bounds $ unM m
<a name="line-246"></a>-}</span>
<a name="line-247"></a>
<a name="line-248"></a><a name="prop_size"></a><span class='hs-definition'>prop_size</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>TM</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-249"></a><span class='hs-definition'>prop_size</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sizeInvariant</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-250"></a>
<a name="line-251"></a>
<a name="line-252"></a><a name="prop_size_fromIndexList"></a><span class='hs-definition'>prop_size_fromIndexList</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Size</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-253"></a><span class='hs-definition'>prop_size_fromIndexList</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span>
<a name="line-254"></a> <span class='hs-varid'>size</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromIndexList</span> <span class='hs-varid'>sz</span> <span class='hs-layout'>(</span><span class='hs-conid'>[]</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-conid'>Int</span><span class='hs-layout'>,</span> <span class='hs-conid'>Integer</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-varid'>sz</span>
<a name="line-255"></a>
<a name="line-256"></a><span class='hs-comment'>-- | 'True' iff the matrix is square.</span>
<a name="line-257"></a>
<a name="line-258"></a><a name="square"></a><span class='hs-definition'>square</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Ix</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-259"></a><span class='hs-definition'>square</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rows</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-varid'>cols</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-260"></a>
<a name="line-261"></a><span class='hs-comment'>-- | Returns 'True' iff the matrix is empty.</span>
<a name="line-262"></a>
<a name="line-263"></a><a name="isEmpty"></a><span class='hs-definition'>isEmpty</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ix</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-264"></a><span class='hs-definition'>isEmpty</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rows</span> <span class='hs-varid'>sz</span> <span class='hs-varop'><=</span> <span class='hs-num'>0</span> <span class='hs-varop'>||</span> <span class='hs-varid'>cols</span> <span class='hs-varid'>sz</span> <span class='hs-varop'><=</span> <span class='hs-num'>0</span>
<a name="line-265"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>size</span> <span class='hs-varid'>m</span>
<a name="line-266"></a>
<a name="line-267"></a><span class='hs-comment'>-- | Returns 'Just b' iff it is a 1x1 matrix with just one entry 'b'.</span>
<a name="line-268"></a>
<a name="line-269"></a><a name="isSingleton"></a><span class='hs-definition'>isSingleton</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ix</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Maybe</span> <span class='hs-varid'>b</span>
<a name="line-270"></a><span class='hs-definition'>isSingleton</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-layout'>(</span><span class='hs-varid'>rows</span> <span class='hs-varid'>sz</span> <span class='hs-varop'>==</span> <span class='hs-num'>1</span> <span class='hs-varop'>||</span> <span class='hs-varid'>cols</span> <span class='hs-varid'>sz</span> <span class='hs-varop'>==</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-keyword'>then</span>
<a name="line-271"></a> <span class='hs-keyword'>case</span> <span class='hs-varid'>unM</span> <span class='hs-varid'>m</span> <span class='hs-keyword'>of</span>
<a name="line-272"></a> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Just</span> <span class='hs-varid'>b</span>
<a name="line-273"></a> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-></span> <span class='hs-sel'>__IMPOSSIBLE__</span>
<a name="line-274"></a> <span class='hs-keyword'>else</span> <span class='hs-conid'>Nothing</span>
<a name="line-275"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>size</span> <span class='hs-varid'>m</span>
<a name="line-276"></a>
<a name="line-277"></a><a name="transposeSize"></a><span class='hs-comment'>-- | Transposition</span>
<a name="line-278"></a><span class='hs-definition'>transposeSize</span> <span class='hs-layout'>(</span><span class='hs-conid'>Size</span> <span class='hs-layout'>{</span> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>n</span><span class='hs-layout'>,</span> <span class='hs-varid'>cols</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>m</span> <span class='hs-layout'>}</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Size</span> <span class='hs-layout'>{</span> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>m</span><span class='hs-layout'>,</span> <span class='hs-varid'>cols</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>n</span> <span class='hs-layout'>}</span>
<a name="line-279"></a><a name="transpose"></a><span class='hs-definition'>transpose</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span> <span class='hs-layout'>{</span> <span class='hs-varid'>size</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>transposeSize</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-280"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>unM</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>List</span><span class='hs-varop'>.</span><span class='hs-varid'>sortBy</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</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-varid'>j</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>compare</span> <span class='hs-varid'>i</span> <span class='hs-varid'>j</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-281"></a> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span> <span class='hs-varid'>j</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>j</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>unM</span> <span class='hs-varid'>m</span> <span class='hs-layout'>}</span>
<a name="line-282"></a>
<a name="line-283"></a><a name="prop_transpose"></a><span class='hs-definition'>prop_transpose</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>TM</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-284"></a><span class='hs-definition'>prop_transpose</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>matrixInvariant</span> <span class='hs-varid'>m'</span> <span class='hs-varop'>&&</span> <span class='hs-varid'>m</span> <span class='hs-varop'>==</span> <span class='hs-varid'>transpose</span> <span class='hs-varid'>m'</span>
<a name="line-285"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>m'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>transpose</span> <span class='hs-varid'>m</span>
<a name="line-286"></a>
<a name="line-287"></a><span class='hs-comment'>-- | @'add' (+) m1 m2@ adds @m1@ and @m2@. Uses @(+)@ to add values.</span>
<a name="line-288"></a><span class='hs-comment'>--</span>
<a name="line-289"></a><span class='hs-comment'>-- No longer precondition: @'size' m1 == 'size' m2@.</span>
<a name="line-290"></a>
<a name="line-291"></a><a name="add"></a><span class='hs-definition'>add</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>i</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'>a</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-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>a</span>
<a name="line-292"></a><span class='hs-definition'>add</span> <span class='hs-varid'>plus</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>m2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span> <span class='hs-layout'>(</span><span class='hs-varid'>supSize</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>m2</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>mergeAssocWith</span> <span class='hs-varid'>plus</span> <span class='hs-layout'>(</span><span class='hs-varid'>unM</span> <span class='hs-varid'>m1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>unM</span> <span class='hs-varid'>m2</span><span class='hs-layout'>)</span>
<a name="line-293"></a>
<a name="line-294"></a><a name="supSize"></a><span class='hs-comment'>-- | Compute the matrix size of the union of two matrices.</span>
<a name="line-295"></a><span class='hs-definition'>supSize</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>m2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Size</span> <span class='hs-layout'>{</span> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>r</span><span class='hs-layout'>,</span> <span class='hs-varid'>cols</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>c</span> <span class='hs-layout'>}</span>
<a name="line-296"></a> <span class='hs-keyword'>where</span>
<a name="line-297"></a> <span class='hs-varid'>sz1</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>size</span> <span class='hs-varid'>m1</span>
<a name="line-298"></a> <span class='hs-varid'>sz2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>size</span> <span class='hs-varid'>m2</span>
<a name="line-299"></a> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>max</span> <span class='hs-layout'>(</span><span class='hs-varid'>rows</span> <span class='hs-varid'>sz1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>rows</span> <span class='hs-varid'>sz2</span><span class='hs-layout'>)</span>
<a name="line-300"></a> <span class='hs-varid'>c</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>max</span> <span class='hs-layout'>(</span><span class='hs-varid'>cols</span> <span class='hs-varid'>sz1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>cols</span> <span class='hs-varid'>sz2</span><span class='hs-layout'>)</span>
<a name="line-301"></a>
<a name="line-302"></a><a name="mergeAssocWith"></a><span class='hs-comment'>-- | assoc list union</span>
<a name="line-303"></a><span class='hs-definition'>mergeAssocWith</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>i</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'>a</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-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-304"></a><span class='hs-definition'>mergeAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>m</span>
<a name="line-305"></a><span class='hs-definition'>mergeAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-varid'>l</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>l</span>
<a name="line-306"></a><span class='hs-definition'>mergeAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-varid'>l</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>l'</span><span class='hs-layout'>)</span> <span class='hs-varid'>m</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>j</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>m'</span><span class='hs-layout'>)</span>
<a name="line-307"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-varop'><</span> <span class='hs-varid'>j</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>mergeAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-varid'>l'</span> <span class='hs-varid'>m</span>
<a name="line-308"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-varop'>></span> <span class='hs-varid'>j</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>j</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>mergeAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-varid'>l</span> <span class='hs-varid'>m'</span>
<a name="line-309"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>f</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>mergeAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-varid'>l'</span> <span class='hs-varid'>m'</span>
<a name="line-310"></a>
<a name="line-311"></a><span class='hs-comment'>-- | @'intersectWith' f m1 m2@ build the pointwise conjunction @m1@ and @m2@.</span>
<a name="line-312"></a><span class='hs-comment'>-- Uses @f@ to combine non-zero values.</span>
<a name="line-313"></a><span class='hs-comment'>--</span>
<a name="line-314"></a><span class='hs-comment'>-- No longer precondition: @'size' m1 == 'size' m2@.</span>
<a name="line-315"></a>
<a name="line-316"></a><a name="intersectWith"></a><span class='hs-definition'>intersectWith</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>i</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'>a</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-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>a</span>
<a name="line-317"></a><span class='hs-definition'>intersectWith</span> <span class='hs-varid'>f</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>m2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span> <span class='hs-layout'>(</span><span class='hs-varid'>infSize</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>m2</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>interAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>unM</span> <span class='hs-varid'>m1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>unM</span> <span class='hs-varid'>m2</span><span class='hs-layout'>)</span>
<a name="line-318"></a>
<a name="line-319"></a><a name="infSize"></a><span class='hs-comment'>-- | Compute the matrix size of the intersection of two matrices.</span>
<a name="line-320"></a><span class='hs-definition'>infSize</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>m2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Size</span> <span class='hs-layout'>{</span> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>r</span><span class='hs-layout'>,</span> <span class='hs-varid'>cols</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>c</span> <span class='hs-layout'>}</span>
<a name="line-321"></a> <span class='hs-keyword'>where</span>
<a name="line-322"></a> <span class='hs-varid'>sz1</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>size</span> <span class='hs-varid'>m1</span>
<a name="line-323"></a> <span class='hs-varid'>sz2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>size</span> <span class='hs-varid'>m2</span>
<a name="line-324"></a> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>min</span> <span class='hs-layout'>(</span><span class='hs-varid'>rows</span> <span class='hs-varid'>sz1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>rows</span> <span class='hs-varid'>sz2</span><span class='hs-layout'>)</span>
<a name="line-325"></a> <span class='hs-varid'>c</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>min</span> <span class='hs-layout'>(</span><span class='hs-varid'>cols</span> <span class='hs-varid'>sz1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>cols</span> <span class='hs-varid'>sz2</span><span class='hs-layout'>)</span>
<a name="line-326"></a>
<a name="line-327"></a><a name="interAssocWith"></a><span class='hs-comment'>-- | assoc list intersection</span>
<a name="line-328"></a><span class='hs-definition'>interAssocWith</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>i</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'>a</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-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-329"></a><span class='hs-definition'>interAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-330"></a><span class='hs-definition'>interAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-varid'>l</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-331"></a><span class='hs-definition'>interAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-varid'>l</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>l'</span><span class='hs-layout'>)</span> <span class='hs-varid'>m</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>j</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>m'</span><span class='hs-layout'>)</span>
<a name="line-332"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-varop'><</span> <span class='hs-varid'>j</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>interAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-varid'>l'</span> <span class='hs-varid'>m</span>
<a name="line-333"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-varop'>></span> <span class='hs-varid'>j</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>interAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-varid'>l</span> <span class='hs-varid'>m'</span>
<a name="line-334"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>f</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>interAssocWith</span> <span class='hs-varid'>f</span> <span class='hs-varid'>l'</span> <span class='hs-varid'>m'</span>
<a name="line-335"></a>
<a name="line-336"></a><a name="prop_add"></a><span class='hs-definition'>prop_add</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span>
<a name="line-337"></a> <span class='hs-varid'>forAll</span> <span class='hs-layout'>(</span><span class='hs-varid'>three</span> <span class='hs-layout'>(</span><span class='hs-varid'>matrix</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Gen</span> <span class='hs-conid'>TM</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span><span class='hs-layout'>(</span><span class='hs-varid'>m1</span><span class='hs-layout'>,</span> <span class='hs-varid'>m2</span><span class='hs-layout'>,</span> <span class='hs-varid'>m3</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span>
<a name="line-338"></a> <span class='hs-keyword'>let</span> <span class='hs-varid'>m'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>add</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>)</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>m2</span> <span class='hs-keyword'>in</span>
<a name="line-339"></a> <span class='hs-varid'>associative</span> <span class='hs-layout'>(</span><span class='hs-varid'>add</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'>m1</span> <span class='hs-varid'>m2</span> <span class='hs-varid'>m3</span> <span class='hs-varop'>&&</span>
<a name="line-340"></a> <span class='hs-varid'>commutative</span> <span class='hs-layout'>(</span><span class='hs-varid'>add</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'>m1</span> <span class='hs-varid'>m2</span> <span class='hs-varop'>&&</span>
<a name="line-341"></a> <span class='hs-varid'>matrixInvariant</span> <span class='hs-varid'>m'</span> <span class='hs-varop'>&&</span>
<a name="line-342"></a> <span class='hs-varid'>size</span> <span class='hs-varid'>m'</span> <span class='hs-varop'>==</span> <span class='hs-varid'>size</span> <span class='hs-varid'>m1</span>
<a name="line-343"></a>
<a name="line-344"></a><span class='hs-comment'>-- | @'mul' semiring m1 m2@ multiplies @m1@ and @m2@. Uses the</span>
<a name="line-345"></a><span class='hs-comment'>-- operations of the semiring @semiring@ to perform the</span>
<a name="line-346"></a><span class='hs-comment'>-- multiplication.</span>
<a name="line-347"></a><span class='hs-comment'>--</span>
<a name="line-348"></a><span class='hs-comment'>-- Precondition: @'cols' ('size' m1) == rows ('size' m2)@.</span>
<a name="line-349"></a>
<a name="line-350"></a><span class='hs-comment'>{- mul A B works as follows:
<a name="line-351"></a>* turn A into a list of sparse rows and the transposed B as well
<a name="line-352"></a>* form the crossproduct using the inner vector product to compute els
<a name="line-353"></a>* the inner vector product is summing up
<a name="line-354"></a> after intersecting with the muliplication op of the semiring
<a name="line-355"></a>-}</span>
<a name="line-356"></a>
<a name="line-357"></a><a name="mul"></a><span class='hs-definition'>mul</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Enum</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ix</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-358"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Semiring</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>a</span>
<a name="line-359"></a><span class='hs-definition'>mul</span> <span class='hs-varid'>semiring</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>m2</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span> <span class='hs-layout'>(</span><span class='hs-conid'>Size</span> <span class='hs-layout'>{</span> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rows</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m1</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>cols</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cols</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m2</span><span class='hs-layout'>)</span> <span class='hs-layout'>}</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-360"></a> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span> <span class='hs-varop'>/=</span> <span class='hs-conid'>Semiring</span><span class='hs-varop'>.</span><span class='hs-varid'>zero</span> <span class='hs-varid'>semiring</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-361"></a> <span class='hs-keyglyph'>[</span> <span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span> <span class='hs-varid'>j</span><span class='hs-layout'>,</span> <span class='hs-conid'>List</span><span class='hs-varop'>.</span><span class='hs-varid'>foldl'</span> <span class='hs-layout'>(</span><span class='hs-conid'>Semiring</span><span class='hs-varop'>.</span><span class='hs-varid'>add</span> <span class='hs-varid'>semiring</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Semiring</span><span class='hs-varop'>.</span><span class='hs-varid'>zero</span> <span class='hs-varid'>semiring</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-362"></a> <span class='hs-varid'>map</span> <span class='hs-varid'>snd</span> <span class='hs-varop'>$</span> <span class='hs-varid'>interAssocWith</span> <span class='hs-layout'>(</span><span class='hs-conid'>Semiring</span><span class='hs-varop'>.</span><span class='hs-varid'>mul</span> <span class='hs-varid'>semiring</span><span class='hs-layout'>)</span> <span class='hs-varid'>v</span> <span class='hs-varid'>w</span><span class='hs-layout'>)</span>
<a name="line-363"></a> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>toSparseRows</span> <span class='hs-varid'>m1</span>
<a name="line-364"></a> <span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varid'>j</span><span class='hs-layout'>,</span><span class='hs-varid'>w</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>toSparseRows</span> <span class='hs-varop'>$</span> <span class='hs-varid'>transpose</span> <span class='hs-varid'>m2</span> <span class='hs-keyglyph'>]</span>
<a name="line-365"></a>
<a name="line-366"></a><a name="prop_mul"></a><span class='hs-definition'>prop_mul</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span>
<a name="line-367"></a> <span class='hs-varid'>sized</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span><span class='hs-varid'>n</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>resize</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span> <span class='hs-varop'>`div`</span> <span class='hs-num'>2</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-368"></a> <span class='hs-varid'>forAll</span> <span class='hs-layout'>(</span><span class='hs-varid'>two</span> <span class='hs-varid'>natural</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span><span class='hs-layout'>(</span><span class='hs-varid'>c2</span><span class='hs-layout'>,</span> <span class='hs-varid'>c3</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span>
<a name="line-369"></a> <span class='hs-varid'>forAll</span> <span class='hs-layout'>(</span><span class='hs-varid'>matrix</span> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Gen</span> <span class='hs-conid'>TM</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span><span class='hs-varid'>m1</span> <span class='hs-keyglyph'>-></span>
<a name="line-370"></a> <span class='hs-varid'>forAll</span> <span class='hs-layout'>(</span><span class='hs-varid'>matrix</span> <span class='hs-layout'>(</span><span class='hs-conid'>Size</span> <span class='hs-layout'>{</span> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cols</span> <span class='hs-varid'>sz</span><span class='hs-layout'>,</span> <span class='hs-varid'>cols</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>c2</span> <span class='hs-layout'>}</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span><span class='hs-varid'>m2</span> <span class='hs-keyglyph'>-></span>
<a name="line-371"></a> <span class='hs-varid'>forAll</span> <span class='hs-layout'>(</span><span class='hs-varid'>matrix</span> <span class='hs-layout'>(</span><span class='hs-conid'>Size</span> <span class='hs-layout'>{</span> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>c2</span><span class='hs-layout'>,</span> <span class='hs-varid'>cols</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>c3</span> <span class='hs-layout'>}</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span><span class='hs-varid'>m3</span> <span class='hs-keyglyph'>-></span>
<a name="line-372"></a> <span class='hs-keyword'>let</span> <span class='hs-varid'>m'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>mult</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>m2</span> <span class='hs-keyword'>in</span>
<a name="line-373"></a> <span class='hs-varid'>associative</span> <span class='hs-varid'>mult</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>m2</span> <span class='hs-varid'>m3</span> <span class='hs-varop'>&&</span>
<a name="line-374"></a> <span class='hs-varid'>matrixInvariant</span> <span class='hs-varid'>m'</span> <span class='hs-varop'>&&</span>
<a name="line-375"></a> <span class='hs-varid'>size</span> <span class='hs-varid'>m'</span> <span class='hs-varop'>==</span> <span class='hs-conid'>Size</span> <span class='hs-layout'>{</span> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rows</span> <span class='hs-varid'>sz</span><span class='hs-layout'>,</span> <span class='hs-varid'>cols</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>c2</span> <span class='hs-layout'>}</span>
<a name="line-376"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>mul</span> <span class='hs-conid'>Semiring</span><span class='hs-varop'>.</span><span class='hs-varid'>integerSemiring</span>
<a name="line-377"></a>
<a name="line-378"></a><span class='hs-comment'>-- | @'diagonal' m@ extracts the diagonal of @m@.</span>
<a name="line-379"></a><span class='hs-comment'>--</span>
<a name="line-380"></a><span class='hs-comment'>-- No longer precondition: @'square' m@.</span>
<a name="line-381"></a>
<a name="line-382"></a><a name="diagonal"></a><span class='hs-definition'>diagonal</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Show</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Enum</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ix</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Array</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span>
<a name="line-383"></a><span class='hs-definition'>diagonal</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-comment'>-- if r /= c then __IMPOSSIBLE__ else -- works also for non-square</span>
<a name="line-384"></a> <span class='hs-varid'>listArray</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span><span class='hs-layout'>,</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>blowUpSparseVec</span> <span class='hs-varid'>zeroElement</span> <span class='hs-varid'>n</span> <span class='hs-varop'>$</span>
<a name="line-385"></a> <span class='hs-varid'>mapMaybe</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-conid'>MIx</span> <span class='hs-varid'>i</span> <span class='hs-varid'>j</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyword'>if</span> <span class='hs-varid'>i</span><span class='hs-varop'>==</span><span class='hs-varid'>j</span> <span class='hs-keyword'>then</span> <span class='hs-conid'>Just</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span> <span class='hs-conid'>Nothing</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>unM</span> <span class='hs-varid'>m</span>
<a name="line-386"></a><span class='hs-comment'>-- map (\ ((MIx i j),b) -> (i,b)) $ filter (\ ((MIx i j),b) -> i==j) (unM m)</span>
<a name="line-387"></a> <span class='hs-keyword'>where</span>
<a name="line-388"></a> <span class='hs-varid'>sz</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>size</span> <span class='hs-varid'>m</span>
<a name="line-389"></a> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rows</span> <span class='hs-varid'>sz</span>
<a name="line-390"></a> <span class='hs-varid'>c</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cols</span> <span class='hs-varid'>sz</span>
<a name="line-391"></a> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>max</span> <span class='hs-varid'>r</span> <span class='hs-varid'>c</span>
<a name="line-392"></a>
<a name="line-393"></a><a name="prop_diagonal"></a><span class='hs-definition'>prop_diagonal</span> <span class='hs-keyglyph'>=</span>
<a name="line-394"></a> <span class='hs-varid'>forAll</span> <span class='hs-varid'>natural</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span><span class='hs-varid'>n</span> <span class='hs-keyglyph'>-></span>
<a name="line-395"></a> <span class='hs-varid'>forAll</span> <span class='hs-layout'>(</span><span class='hs-varid'>matrix</span> <span class='hs-layout'>(</span><span class='hs-conid'>Size</span> <span class='hs-varid'>n</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Gen</span> <span class='hs-conid'>TM</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span><span class='hs-varid'>m</span> <span class='hs-keyglyph'>-></span>
<a name="line-396"></a> <span class='hs-varid'>bounds</span> <span class='hs-layout'>(</span><span class='hs-varid'>diagonal</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-layout'>(</span><span class='hs-num'>1</span><span class='hs-layout'>,</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span>
<a name="line-397"></a>
<a name="line-398"></a><span class='hs-comment'>------------------------------------------------------------------------</span>
<a name="line-399"></a><span class='hs-comment'>-- Modifying matrices</span>
<a name="line-400"></a>
<a name="line-401"></a><span class='hs-comment'>-- | @'addColumn' x m@ adds a new column to @m@, after the columns</span>
<a name="line-402"></a><span class='hs-comment'>-- already existing in the matrix. All elements in the new column get</span>
<a name="line-403"></a><span class='hs-comment'>-- set to @x@.</span>
<a name="line-404"></a>
<a name="line-405"></a><a name="addColumn"></a><span class='hs-definition'>addColumn</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span>
<a name="line-406"></a><span class='hs-definition'>addColumn</span> <span class='hs-varid'>x</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varop'>==</span> <span class='hs-varid'>zeroElement</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>m</span> <span class='hs-layout'>{</span> <span class='hs-varid'>size</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-layout'>{</span> <span class='hs-varid'>cols</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cols</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-num'>1</span> <span class='hs-layout'>}</span><span class='hs-layout'>}</span>
<a name="line-407"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-sel'>__IMPOSSIBLE__</span>
<a name="line-408"></a>
<a name="line-409"></a><a name="prop_addColumn"></a><span class='hs-definition'>prop_addColumn</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>TM</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-410"></a><span class='hs-definition'>prop_addColumn</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span>
<a name="line-411"></a> <span class='hs-varid'>matrixInvariant</span> <span class='hs-varid'>m'</span>
<a name="line-412"></a> <span class='hs-varop'>&&</span>
<a name="line-413"></a> <span class='hs-varid'>map</span> <span class='hs-varid'>init</span> <span class='hs-layout'>(</span><span class='hs-varid'>toLists</span> <span class='hs-varid'>m'</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-varid'>toLists</span> <span class='hs-varid'>m</span>
<a name="line-414"></a> <span class='hs-keyword'>where</span>
<a name="line-415"></a> <span class='hs-varid'>m'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>addColumn</span> <span class='hs-varid'>zeroElement</span> <span class='hs-varid'>m</span>
<a name="line-416"></a>
<a name="line-417"></a><span class='hs-comment'>-- | @'addRow' x m@ adds a new row to @m@, after the rows already</span>
<a name="line-418"></a><span class='hs-comment'>-- existing in the matrix. All elements in the new row get set to @x@.</span>
<a name="line-419"></a>
<a name="line-420"></a><a name="addRow"></a><span class='hs-definition'>addRow</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>HasZero</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Matrix</span> <span class='hs-varid'>i</span> <span class='hs-varid'>b</span>
<a name="line-421"></a><span class='hs-definition'>addRow</span> <span class='hs-varid'>x</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varop'>==</span> <span class='hs-varid'>zeroElement</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>m</span> <span class='hs-layout'>{</span> <span class='hs-varid'>size</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-layout'>{</span> <span class='hs-varid'>rows</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rows</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-num'>1</span> <span class='hs-layout'>}</span><span class='hs-layout'>}</span>
<a name="line-422"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-sel'>__IMPOSSIBLE__</span>
<a name="line-423"></a>
<a name="line-424"></a><a name="prop_addRow"></a><span class='hs-definition'>prop_addRow</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>TM</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-425"></a><span class='hs-definition'>prop_addRow</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span>
<a name="line-426"></a> <span class='hs-varid'>matrixInvariant</span> <span class='hs-varid'>m'</span>
<a name="line-427"></a> <span class='hs-varop'>&&</span>
<a name="line-428"></a> <span class='hs-varid'>init</span> <span class='hs-layout'>(</span><span class='hs-varid'>toLists</span> <span class='hs-varid'>m'</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-varid'>toLists</span> <span class='hs-varid'>m</span>
<a name="line-429"></a> <span class='hs-keyword'>where</span>
<a name="line-430"></a> <span class='hs-varid'>m'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>addRow</span> <span class='hs-varid'>zeroElement</span> <span class='hs-varid'>m</span>
<a name="line-431"></a>
<a name="line-432"></a><span class='hs-comment'>------------------------------------------------------------------------</span>
<a name="line-433"></a><span class='hs-comment'>-- Zipping (assumes non-empty matrices)</span>
<a name="line-434"></a>
<a name="line-435"></a><span class='hs-comment'>{- use mergeAssocList or interAssocList instead
<a name="line-436"></a>zipWith :: (a -> b -> c) ->
<a name="line-437"></a> Matrix Integer a -> Matrix Integer b -> Matrix Integer c
<a name="line-438"></a>zipWith f m1 m2
<a name="line-439"></a> = fromLists (Size { rows = toInteger $ length ll,
<a name="line-440"></a> cols = toInteger $ length (head ll) }) ll
<a name="line-441"></a> where ll = List.zipWith (List.zipWith f) (toLists m1) (toLists m2)
<a name="line-442"></a>-}</span>
<a name="line-443"></a>
<a name="line-444"></a><span class='hs-comment'>------------------------------------------------------------------------</span>
<a name="line-445"></a><span class='hs-comment'>-- All tests</span>
<a name="line-446"></a>
<a name="line-447"></a><a name="tests"></a><span class='hs-definition'>tests</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>IO</span> <span class='hs-conid'>Bool</span>
<a name="line-448"></a><span class='hs-definition'>tests</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>runTests</span> <span class='hs-str'>"Agda.Termination.Matrix"</span>
<a name="line-449"></a> <span class='hs-keyglyph'>[</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_transpose</span>
<a name="line-450"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_Arbitrary_Size</span>
<a name="line-451"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_Arbitrary_Matrix</span>
<a name="line-452"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_Arbitrary_MIx</span>
<a name="line-453"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_fromIndexList</span>
<a name="line-454"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_matrix</span>
<a name="line-455"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_size</span>
<a name="line-456"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_size_fromIndexList</span>
<a name="line-457"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_fromLists_toLists</span>
<a name="line-458"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_add</span>
<a name="line-459"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_mul</span>
<a name="line-460"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_diagonal</span>
<a name="line-461"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_addColumn</span>
<a name="line-462"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>quickCheck'</span> <span class='hs-varid'>prop_addRow</span>
<a name="line-463"></a> <span class='hs-keyglyph'>]</span>
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