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<title>src/full/Agda/Utils/Permutation.hs</title>
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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE DeriveDataTypeable, CPP #-}</span>
<a name="line-2"></a><span class='hs-keyword'>module</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'>Permutation</span> <span class='hs-keyword'>where</span>
<a name="line-3"></a>
<a name="line-4"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Typeable</span> <span class='hs-layout'>(</span><span class='hs-conid'>Typeable</span><span class='hs-layout'>)</span>
<a name="line-5"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>List</span>
<a name="line-6"></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'>Size</span>
<a name="line-7"></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-8"></a>
<a name="line-9"></a><span class='hs-cpp'>#include "../undefined.h"</span>
<a name="line-10"></a>
<a name="line-11"></a><a name="Permutation"></a><span class='hs-comment'>-- | @permute [1,2,0] [x0,x1,x2] = [x1,x2,x0]@</span>
<a name="line-12"></a><a name="Permutation"></a><span class='hs-comment'>--</span>
<a name="line-13"></a><a name="Permutation"></a><span class='hs-comment'>-- Agda typing would be:</span>
<a name="line-14"></a><a name="Permutation"></a><span class='hs-comment'>-- @Perm : {m : Nat}(n : Nat) -> Vec (Fin n) m -> Permutation@</span>
<a name="line-15"></a><a name="Permutation"></a><span class='hs-comment'>-- @m@ is the 'size' of the permutation.</span>
<a name="line-16"></a><a name="Permutation"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Permutation</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Perm</span> <span class='hs-layout'>{</span> <span class='hs-varid'>permRange</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span><span class='hs-layout'>,</span> <span class='hs-varid'>permPicks</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-layout'>}</span>
<a name="line-17"></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'>Typeable</span><span class='hs-layout'>)</span>
<a name="line-18"></a>
<a name="line-19"></a><a name="instance%20Show%20Permutation"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-conid'>Permutation</span> <span class='hs-keyword'>where</span>
<a name="line-20"></a> <span class='hs-varid'>show</span> <span class='hs-layout'>(</span><span class='hs-conid'>Perm</span> <span class='hs-varid'>n</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>showx</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>++</span> <span class='hs-str'>" -> "</span> <span class='hs-varop'>++</span> <span class='hs-varid'>showx</span> <span class='hs-varid'>xs</span>
<a name="line-21"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>showx</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>showList</span> <span class='hs-str'>","</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-str'>"x"</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span>
<a name="line-22"></a> <span class='hs-varid'>showList</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>String</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-conid'>String</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>String</span>
<a name="line-23"></a> <span class='hs-varid'>showList</span> <span class='hs-varid'>sep</span> <span class='hs-varid'>f</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-str'>""</span>
<a name="line-24"></a> <span class='hs-varid'>showList</span> <span class='hs-varid'>sep</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>e</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-varid'>e</span>
<a name="line-25"></a> <span class='hs-varid'>showList</span> <span class='hs-varid'>sep</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>e</span><span class='hs-conop'>:</span><span class='hs-varid'>es</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-varid'>e</span> <span class='hs-varop'>++</span> <span class='hs-varid'>sep</span> <span class='hs-varop'>++</span> <span class='hs-varid'>showList</span> <span class='hs-varid'>sep</span> <span class='hs-varid'>f</span> <span class='hs-varid'>es</span>
<a name="line-26"></a>
<a name="line-27"></a><a name="instance%20Sized%20Permutation"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Sized</span> <span class='hs-conid'>Permutation</span> <span class='hs-keyword'>where</span>
<a name="line-28"></a> <span class='hs-varid'>size</span> <span class='hs-layout'>(</span><span class='hs-conid'>Perm</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>size</span> <span class='hs-varid'>xs</span>
<a name="line-29"></a>
<a name="line-30"></a><a name="permute"></a><span class='hs-comment'>-- | @permute [1,2,0] [x0,x1,x2] = [x1,x2,x0]@</span>
<a name="line-31"></a><span class='hs-comment'>-- More precisely, @permute indices list = sublist@, generates @sublist@</span>
<a name="line-32"></a><span class='hs-comment'>-- from @list@ by picking the elements of list as indicated by @indices@.</span>
<a name="line-33"></a><span class='hs-comment'>-- @permute [1,3,0] [x0,x1,x2,x3] = [x1,x3,x0]@</span>
<a name="line-34"></a><span class='hs-comment'>--</span>
<a name="line-35"></a><span class='hs-comment'>-- Agda typing:</span>
<a name="line-36"></a><span class='hs-comment'>-- @permute (Perm {m} n is) : Vec A m -> Vec A n@</span>
<a name="line-37"></a><span class='hs-definition'>permute</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Permutation</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span>
<a name="line-38"></a><span class='hs-definition'>permute</span> <span class='hs-layout'>(</span><span class='hs-conid'>Perm</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>is</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>xs</span> <span class='hs-varop'>!!!</span><span class='hs-layout'>)</span> <span class='hs-varid'>is</span>
<a name="line-39"></a> <span class='hs-keyword'>where</span>
<a name="line-40"></a> <span class='hs-conid'>[]</span> <span class='hs-varop'>!!!</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-sel'>__IMPOSSIBLE__</span>
<a name="line-41"></a> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-varop'>!!!</span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span>
<a name="line-42"></a> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-varop'>!!!</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>xs</span> <span class='hs-varop'>!!!</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span>
<a name="line-43"></a>
<a name="line-44"></a><a name="idP"></a><span class='hs-definition'>idP</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span>
<a name="line-45"></a><span class='hs-definition'>idP</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Perm</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span><span class='hs-keyglyph'>]</span>
<a name="line-46"></a>
<a name="line-47"></a><a name="takeP"></a><span class='hs-definition'>takeP</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span>
<a name="line-48"></a><span class='hs-definition'>takeP</span> <span class='hs-varid'>n</span> <span class='hs-layout'>(</span><span class='hs-conid'>Perm</span> <span class='hs-varid'>m</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Perm</span> <span class='hs-varid'>n</span> <span class='hs-varop'>$</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varop'><</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span>
<a name="line-49"></a>
<a name="line-50"></a><a name="liftP"></a><span class='hs-comment'>-- | @liftP k@ takes a @Perm {m} n@ to a @Perm {m+k} (n+k)@.</span>
<a name="line-51"></a><span class='hs-comment'>-- Analogous to 'Agda.TypeChecking.Substitution.liftS',</span>
<a name="line-52"></a><span class='hs-comment'>-- but Permutations operate on de Bruijn LEVELS, not indices.</span>
<a name="line-53"></a><span class='hs-definition'>liftP</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span>
<a name="line-54"></a><span class='hs-definition'>liftP</span> <span class='hs-varid'>n</span> <span class='hs-layout'>(</span><span class='hs-conid'>Perm</span> <span class='hs-varid'>m</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Perm</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span> <span class='hs-varop'>+</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>xs</span> <span class='hs-varop'>++</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>m</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>m</span><span class='hs-varop'>+</span><span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-keyglyph'>]</span>
<a name="line-55"></a><span class='hs-comment'>-- liftP n (Perm m xs) = Perm (n + m) $ [0..n-1] ++ map (n+) xs -- WRONG, works for indices, but not for levels</span>
<a name="line-56"></a>
<a name="line-57"></a><a name="composeP"></a><span class='hs-comment'>-- | @permute (compose p1 p2) == permute p1 . permute p2@</span>
<a name="line-58"></a><span class='hs-definition'>composeP</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Permutation</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span>
<a name="line-59"></a><span class='hs-definition'>composeP</span> <span class='hs-varid'>p1</span> <span class='hs-layout'>(</span><span class='hs-conid'>Perm</span> <span class='hs-varid'>n</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Perm</span> <span class='hs-varid'>n</span> <span class='hs-varop'>$</span> <span class='hs-varid'>permute</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>xs</span>
<a name="line-60"></a> <span class='hs-comment'>{- proof:
<a name="line-61"></a> permute (compose (Perm xs) (Perm ys)) zs
<a name="line-62"></a> == permute (Perm (permute (Perm xs) ys)) zs
<a name="line-63"></a> == map (zs !!) (permute (Perm xs) ys)
<a name="line-64"></a> == map (zs !!) (map (ys !!) xs)
<a name="line-65"></a> == map (zs !! . ys !!) xs
<a name="line-66"></a> == map (\x -> zs !! (ys !! x)) xs
<a name="line-67"></a> == map (\x -> map (zs !!) ys !! x) xs {- map f xs !! n == f (xs !! n) -}
<a name="line-68"></a> == map (map (zs !!) ys !!) xs
<a name="line-69"></a> == permute (Perm xs) (permute (Perm ys) zs)
<a name="line-70"></a> -}</span>
<a name="line-71"></a>
<a name="line-72"></a><a name="invertP"></a><span class='hs-definition'>invertP</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Permutation</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span>
<a name="line-73"></a><span class='hs-definition'>invertP</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>Perm</span> <span class='hs-varid'>n</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Perm</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map</span> <span class='hs-varid'>inv</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span><span class='hs-keyglyph'>]</span>
<a name="line-74"></a> <span class='hs-keyword'>where</span>
<a name="line-75"></a> <span class='hs-varid'>inv</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>findIndex</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-varop'>==</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-keyword'>of</span>
<a name="line-76"></a> <span class='hs-conid'>Just</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>fromIntegral</span> <span class='hs-varid'>y</span>
<a name="line-77"></a> <span class='hs-conid'>Nothing</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>error</span> <span class='hs-varop'>$</span> <span class='hs-str'>"invertP: non-surjective permutation "</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>p</span>
<a name="line-78"></a>
<a name="line-79"></a><a name="compactP"></a><span class='hs-comment'>-- | Turn a possible non-surjective permutation into a surjective permutation.</span>
<a name="line-80"></a><span class='hs-definition'>compactP</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Permutation</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span>
<a name="line-81"></a><span class='hs-definition'>compactP</span> <span class='hs-layout'>(</span><span class='hs-conid'>Perm</span> <span class='hs-varid'>n</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Perm</span> <span class='hs-varid'>m</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map</span> <span class='hs-varid'>adjust</span> <span class='hs-varid'>xs</span>
<a name="line-82"></a> <span class='hs-keyword'>where</span>
<a name="line-83"></a> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>genericLength</span> <span class='hs-varid'>xs</span>
<a name="line-84"></a> <span class='hs-varid'>missing</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>\\</span> <span class='hs-varid'>xs</span>
<a name="line-85"></a> <span class='hs-varid'>holesBelow</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>genericLength</span> <span class='hs-varop'>$</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varop'><</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-varid'>missing</span>
<a name="line-86"></a> <span class='hs-varid'>adjust</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>k</span> <span class='hs-comment'>-</span> <span class='hs-varid'>holesBelow</span> <span class='hs-varid'>k</span>
<a name="line-87"></a>
<a name="line-88"></a><a name="reverseP"></a><span class='hs-comment'>-- | @permute (reverseP p) xs ==</span>
<a name="line-89"></a><span class='hs-comment'>-- reverse $ permute p $ reverse xs@</span>
<a name="line-90"></a><span class='hs-comment'>--</span>
<a name="line-91"></a><span class='hs-comment'>-- Example:</span>
<a name="line-92"></a><span class='hs-comment'>-- @</span>
<a name="line-93"></a><span class='hs-comment'>-- permute (reverseP (Perm 4 [1,3,0])) [x0,x1,x2,x3]</span>
<a name="line-94"></a><span class='hs-comment'>-- == permute (Perm 4 $ map (3-) [0,3,1]) [x0,x1,x2,x3]</span>
<a name="line-95"></a><span class='hs-comment'>-- == permute (Perm 4 [3,0,2]) [x0,x1,x2,x3]</span>
<a name="line-96"></a><span class='hs-comment'>-- == [x3,x0,x2]</span>
<a name="line-97"></a><span class='hs-comment'>-- == reverse [x2,x0,x3]</span>
<a name="line-98"></a><span class='hs-comment'>-- == reverse $ permute (Perm 4 [1,3,0]) [x3,x2,x1,x0]</span>
<a name="line-99"></a><span class='hs-comment'>-- == reverse $ permute (Perm 4 [1,3,0]) $ reverse [x0,x1,x2,x3]</span>
<a name="line-100"></a><span class='hs-comment'>-- @</span>
<a name="line-101"></a><span class='hs-definition'>reverseP</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Permutation</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span>
<a name="line-102"></a><span class='hs-definition'>reverseP</span> <span class='hs-layout'>(</span><span class='hs-conid'>Perm</span> <span class='hs-varid'>n</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Perm</span> <span class='hs-varid'>n</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>n</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-comment'>-</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>reverse</span> <span class='hs-varid'>xs</span>
<a name="line-103"></a>
<a name="line-104"></a><a name="expandP"></a><span class='hs-comment'>-- | @expandP i n π@ in the domain of @π@ replace the /i/th element by /n/ elements.</span>
<a name="line-105"></a><span class='hs-definition'>expandP</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Permutation</span>
<a name="line-106"></a><span class='hs-definition'>expandP</span> <span class='hs-varid'>i</span> <span class='hs-varid'>n</span> <span class='hs-layout'>(</span><span class='hs-conid'>Perm</span> <span class='hs-varid'>m</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Perm</span> <span class='hs-layout'>(</span><span class='hs-varid'>m</span> <span class='hs-varop'>+</span> <span class='hs-varid'>n</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>concatMap</span> <span class='hs-varid'>expand</span> <span class='hs-varid'>xs</span>
<a name="line-107"></a> <span class='hs-keyword'>where</span>
<a name="line-108"></a> <span class='hs-varid'>expand</span> <span class='hs-varid'>j</span>
<a name="line-109"></a> <span class='hs-keyglyph'>|</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-keyglyph'>[</span><span class='hs-varid'>i</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-comment'>-</span> <span class='hs-num'>1</span><span class='hs-keyglyph'>]</span>
<a name="line-110"></a> <span class='hs-keyglyph'>|</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-keyglyph'>[</span><span class='hs-varid'>j</span><span class='hs-keyglyph'>]</span>
<a name="line-111"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>j</span> <span class='hs-varop'>+</span> <span class='hs-varid'>n</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span><span class='hs-keyglyph'>]</span>
<a name="line-112"></a>
<a name="line-113"></a><a name="topoSort"></a><span class='hs-comment'>-- | Stable topologic sort. The first argument decides whether its first</span>
<a name="line-114"></a><span class='hs-comment'>-- argument is an immediate parent to its second argument.</span>
<a name="line-115"></a><span class='hs-definition'>topoSort</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-conid'>Bool</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Maybe</span> <span class='hs-conid'>Permutation</span>
<a name="line-116"></a><span class='hs-definition'>topoSort</span> <span class='hs-varid'>parent</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-conid'>Perm</span> <span class='hs-layout'>(</span><span class='hs-varid'>size</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>topo</span> <span class='hs-varid'>g</span>
<a name="line-117"></a> <span class='hs-keyword'>where</span>
<a name="line-118"></a> <span class='hs-varid'>nodes</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zip</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-keyglyph'>]</span> <span class='hs-varid'>xs</span>
<a name="line-119"></a> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-layout'>,</span> <span class='hs-varid'>parents</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>nodes</span> <span class='hs-keyglyph'>]</span>
<a name="line-120"></a> <span class='hs-varid'>parents</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-layout'>,</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>nodes</span><span class='hs-layout'>,</span> <span class='hs-varid'>parent</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>]</span>
<a name="line-121"></a>
<a name="line-122"></a> <span class='hs-varid'>topo</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Eq</span> <span class='hs-varid'>node</span> <span class='hs-keyglyph'>=></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>node</span><span class='hs-layout'>,</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>node</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Maybe</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>node</span><span class='hs-keyglyph'>]</span>
<a name="line-123"></a> <span class='hs-varid'>topo</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-conid'>[]</span>
<a name="line-124"></a> <span class='hs-varid'>topo</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>xs</span> <span class='hs-keyword'>of</span>
<a name="line-125"></a> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>fail</span> <span class='hs-str'>"cycle detected"</span>
<a name="line-126"></a> <span class='hs-varid'>x</span> <span class='hs-conop'>:</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyword'>do</span>
<a name="line-127"></a> <span class='hs-varid'>ys</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>topo</span> <span class='hs-varop'>$</span> <span class='hs-varid'>remove</span> <span class='hs-varid'>x</span> <span class='hs-varid'>g</span>
<a name="line-128"></a> <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:</span> <span class='hs-varid'>ys</span>
<a name="line-129"></a> <span class='hs-keyword'>where</span>
<a name="line-130"></a> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>]</span>
<a name="line-131"></a> <span class='hs-varid'>remove</span> <span class='hs-varid'>x</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-varid'>filter</span> <span class='hs-layout'>(</span><span class='hs-varop'>/=</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>g</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>]</span>
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