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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE TypeFamilies, MultiParamTypeClasses, TypeOperators, FlexibleContexts, FlexibleInstances, UndecidableInstances #-}</span>
<a name="line-2"></a><span class='hs-comment'>-------------------------------------------------------------------------------------------</span>
<a name="line-3"></a><span class='hs-comment'>-- |</span>
<a name="line-4"></a><span class='hs-comment'>-- Module : Control.Category.Cartesian</span>
<a name="line-5"></a><span class='hs-comment'>-- Copyright : 2008-2010 Edward Kmett</span>
<a name="line-6"></a><span class='hs-comment'>-- License : BSD</span>
<a name="line-7"></a><span class='hs-comment'>--</span>
<a name="line-8"></a><span class='hs-comment'>-- Maintainer : Edward Kmett <ekmett@gmail.com></span>
<a name="line-9"></a><span class='hs-comment'>-- Stability : experimental</span>
<a name="line-10"></a><span class='hs-comment'>-- Portability : non-portable (class-associated types)</span>
<a name="line-11"></a><span class='hs-comment'>--</span>
<a name="line-12"></a><span class='hs-comment'>-------------------------------------------------------------------------------------------</span>
<a name="line-13"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Category</span><span class='hs-varop'>.</span><span class='hs-conid'>Cartesian</span>
<a name="line-14"></a> <span class='hs-layout'>(</span>
<a name="line-15"></a> <span class='hs-comment'>-- * Pre-(Co)Cartesian categories</span>
<a name="line-16"></a> <span class='hs-conid'>PreCartesian</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'>bimapProduct</span><span class='hs-layout'>,</span> <span class='hs-varid'>braidProduct</span><span class='hs-layout'>,</span> <span class='hs-varid'>associateProduct</span><span class='hs-layout'>,</span> <span class='hs-varid'>disassociateProduct</span>
<a name="line-18"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>PreCoCartesian</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span>
<a name="line-19"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>bimapSum</span><span class='hs-layout'>,</span> <span class='hs-varid'>braidSum</span><span class='hs-layout'>,</span> <span class='hs-varid'>associateSum</span><span class='hs-layout'>,</span> <span class='hs-varid'>disassociateSum</span>
<a name="line-20"></a> <span class='hs-comment'>-- * (Co)Cartesian categories</span>
<a name="line-21"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>Cartesian</span>
<a name="line-22"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>CoCartesian</span>
<a name="line-23"></a> <span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-24"></a>
<a name="line-25"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Category</span><span class='hs-varop'>.</span><span class='hs-conid'>Associative</span>
<a name="line-26"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Category</span><span class='hs-varop'>.</span><span class='hs-conid'>Braided</span>
<a name="line-27"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Category</span><span class='hs-varop'>.</span><span class='hs-conid'>Monoidal</span>
<a name="line-28"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Prelude</span> <span class='hs-varid'>hiding</span> <span class='hs-layout'>(</span><span class='hs-conid'>Functor</span><span class='hs-layout'>,</span> <span class='hs-varid'>map</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>.</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>id</span><span class='hs-layout'>,</span> <span class='hs-varid'>fst</span><span class='hs-layout'>,</span> <span class='hs-varid'>snd</span><span class='hs-layout'>,</span> <span class='hs-varid'>curry</span><span class='hs-layout'>,</span> <span class='hs-varid'>uncurry</span><span class='hs-layout'>)</span>
<a name="line-29"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Prelude</span> <span class='hs-layout'>(</span><span class='hs-varid'>fst</span><span class='hs-layout'>,</span><span class='hs-varid'>snd</span><span class='hs-layout'>)</span>
<a name="line-30"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Categorical</span><span class='hs-varop'>.</span><span class='hs-conid'>Bifunctor</span>
<a name="line-31"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Category</span>
<a name="line-32"></a>
<a name="line-33"></a><span class='hs-keyword'>infixr</span> <span class='hs-num'>3</span> <span class='hs-varop'>&&&</span>
<a name="line-34"></a><span class='hs-keyword'>infixr</span> <span class='hs-num'>2</span> <span class='hs-varop'>|||</span>
<a name="line-35"></a>
<a name="line-36"></a><a name="PreCartesian"></a><span class='hs-comment'>{- |
<a name="line-37"></a>NB: This is weaker than traditional category with products! That is Cartesian, below.
<a name="line-38"></a>The problem is @(->)@ lacks an initial object, since every type is inhabited in Haskell.
<a name="line-39"></a>Consequently its coproduct is merely a semigroup, not a monoid (as it has no identity), and
<a name="line-40"></a>since we want to be able to describe its dual category, which has this non-traditional
<a name="line-41"></a>form being built over a category with an associative bifunctor rather than as a monoidal category
<a name="line-42"></a>for the product monoid.
<a name="line-43"></a>
<a name="line-44"></a>Minimum definition:
<a name="line-45"></a>
<a name="line-46"></a>> fst, snd, diag
<a name="line-47"></a>> fst, snd, (&&&)
<a name="line-48"></a>-}</span>
<a name="line-49"></a><a name="PreCartesian"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span> <span class='hs-conid'>Associative</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-50"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>Disassociative</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-51"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>Symmetric</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-52"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>Braided</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-53"></a> <span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PreCartesian</span> <span class='hs-varid'>k</span> <span class='hs-keyword'>where</span>
<a name="line-54"></a> <span class='hs-keyword'>type</span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>::</span> <span class='hs-varop'>*</span> <span class='hs-keyglyph'>-></span> <span class='hs-varop'>*</span> <span class='hs-keyglyph'>-></span> <span class='hs-varop'>*</span>
<a name="line-55"></a> <span class='hs-varid'>fst</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span> <span class='hs-varop'>`k`</span> <span class='hs-varid'>a</span>
<a name="line-56"></a> <span class='hs-varid'>snd</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span> <span class='hs-varop'>`k`</span> <span class='hs-varid'>b</span>
<a name="line-57"></a> <span class='hs-varid'>diag</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>a</span> <span class='hs-varop'>`k`</span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>a</span>
<a name="line-58"></a> <span class='hs-layout'>(</span><span class='hs-varop'>&&&</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-varop'>`k`</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-varop'>`k`</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-varop'>`k`</span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span>
<a name="line-59"></a>
<a name="line-60"></a> <span class='hs-varid'>diag</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>id</span> <span class='hs-varop'>&&&</span> <span class='hs-varid'>id</span>
<a name="line-61"></a> <span class='hs-varid'>f</span> <span class='hs-varop'>&&&</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>bimap</span> <span class='hs-varid'>f</span> <span class='hs-varid'>g</span> <span class='hs-varop'>.</span> <span class='hs-varid'>diag</span>
<a name="line-62"></a>
<a name="line-63"></a>
<a name="line-64"></a><span class='hs-comment'>{-- RULES
<a name="line-65"></a>"fst . diag" fst . diag = id
<a name="line-66"></a>"snd . diag" snd . diag = id
<a name="line-67"></a>"fst . f &&& g" forall f g. fst . (f &&& g) = f
<a name="line-68"></a>"snd . f &&& g" forall f g. snd . (f &&& g) = g
<a name="line-69"></a> --}</span>
<a name="line-70"></a>
<a name="line-71"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PreCartesian</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>-></span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-72"></a> <span class='hs-keyword'>type</span> <span class='hs-conid'>Product</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>-></span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>(,)</span>
<a name="line-73"></a> <span class='hs-varid'>fst</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Prelude</span><span class='hs-varop'>.</span><span class='hs-varid'>fst</span>
<a name="line-74"></a> <span class='hs-varid'>snd</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Prelude</span><span class='hs-varop'>.</span><span class='hs-varid'>snd</span>
<a name="line-75"></a> <span class='hs-varid'>diag</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-76"></a> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varop'>&&&</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>g</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-77"></a>
<a name="line-78"></a><a name="Cartesian"></a><span class='hs-comment'>-- alias</span>
<a name="line-79"></a><a name="Cartesian"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span> <span class='hs-conid'>Monoidal</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-80"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>PreCartesian</span> <span class='hs-varid'>k</span>
<a name="line-81"></a> <span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Cartesian</span> <span class='hs-varid'>k</span>
<a name="line-82"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span> <span class='hs-conid'>Monoidal</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-83"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>PreCartesian</span> <span class='hs-varid'>k</span>
<a name="line-84"></a> <span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Cartesian</span> <span class='hs-varid'>k</span>
<a name="line-85"></a>
<a name="line-86"></a><a name="bimapProduct"></a><span class='hs-comment'>-- | free construction of 'Bifunctor' for the product 'Bifunctor' @Product k@ if @(&&&)@ is known</span>
<a name="line-87"></a><span class='hs-definition'>bimapProduct</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>PreCartesian</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'><*></span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>~</span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</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-varop'>`k`</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>b</span> <span class='hs-varop'>`k`</span> <span class='hs-varid'>d</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-varop'><*></span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-varop'>`k`</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span> <span class='hs-varop'><*></span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span>
<a name="line-88"></a><span class='hs-definition'>bimapProduct</span> <span class='hs-varid'>f</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fst</span><span class='hs-layout'>)</span> <span class='hs-varop'>&&&</span> <span class='hs-layout'>(</span><span class='hs-varid'>g</span> <span class='hs-varop'>.</span> <span class='hs-varid'>snd</span><span class='hs-layout'>)</span>
<a name="line-89"></a>
<a name="line-90"></a><a name="braidProduct"></a><span class='hs-comment'>-- | free construction of 'Braided' for the product 'Bifunctor' @Product k@</span>
<a name="line-91"></a><span class='hs-comment'>-- braidProduct :: (PreCartesian k, Product k ~ (<*>)) => a <*> b ~> b <*> a</span>
<a name="line-92"></a><span class='hs-definition'>braidProduct</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>PreCartesian</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span> <span class='hs-varop'>`k`</span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>b</span> <span class='hs-varid'>a</span>
<a name="line-93"></a><span class='hs-definition'>braidProduct</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>snd</span> <span class='hs-varop'>&&&</span> <span class='hs-varid'>fst</span>
<a name="line-94"></a>
<a name="line-95"></a><a name="associateProduct"></a><span class='hs-comment'>-- | free construction of 'Associative' for the product 'Bifunctor' @Product k@</span>
<a name="line-96"></a><span class='hs-comment'>-- associateProduct :: (PreCartesian k, (<*>) ~ Product k) => (a <*> b) <*> c ~> (a <*> (b <*> c))</span>
<a name="line-97"></a><span class='hs-definition'>associateProduct</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>PreCartesian</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-varid'>c</span> <span class='hs-varop'>`k`</span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span>
<a name="line-98"></a><span class='hs-definition'>associateProduct</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>fst</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fst</span><span class='hs-layout'>)</span> <span class='hs-varop'>&&&</span> <span class='hs-varid'>first</span> <span class='hs-varid'>snd</span>
<a name="line-99"></a>
<a name="line-100"></a><a name="disassociateProduct"></a><span class='hs-comment'>-- | free construction of 'Disassociative' for the product 'Bifunctor' @Product k@</span>
<a name="line-101"></a><span class='hs-comment'>-- disassociateProduct:: (PreCartesian k, (<*>) ~ Product k) => a <*> (b <*> c) ~> (a <*> b) <*> c</span>
<a name="line-102"></a><span class='hs-definition'>disassociateProduct</span><span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>PreCartesian</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-varop'>`k`</span> <span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-varid'>c</span>
<a name="line-103"></a><span class='hs-definition'>disassociateProduct</span><span class='hs-keyglyph'>=</span> <span class='hs-varid'>braid</span> <span class='hs-varop'>.</span> <span class='hs-varid'>second</span> <span class='hs-varid'>braid</span> <span class='hs-varop'>.</span> <span class='hs-varid'>associateProduct</span> <span class='hs-varop'>.</span> <span class='hs-varid'>first</span> <span class='hs-varid'>braid</span> <span class='hs-varop'>.</span> <span class='hs-varid'>braid</span>
<a name="line-104"></a>
<a name="line-105"></a><span class='hs-comment'>-- * Co-PreCartesian categories</span>
<a name="line-106"></a>
<a name="line-107"></a><a name="PreCoCartesian"></a><span class='hs-comment'>-- a category that has finite coproducts, weakened the same way as PreCartesian above was weakened</span>
<a name="line-108"></a><a name="PreCoCartesian"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span> <span class='hs-conid'>Associative</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-109"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>Disassociative</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-110"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>Symmetric</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-111"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>Braided</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-112"></a> <span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PreCoCartesian</span> <span class='hs-varid'>k</span> <span class='hs-keyword'>where</span>
<a name="line-113"></a> <span class='hs-keyword'>type</span> <span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>::</span> <span class='hs-varop'>*</span> <span class='hs-keyglyph'>-></span> <span class='hs-varop'>*</span> <span class='hs-keyglyph'>-></span> <span class='hs-varop'>*</span>
<a name="line-114"></a> <span class='hs-varid'>inl</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>a</span> <span class='hs-varop'>`k`</span> <span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span>
<a name="line-115"></a> <span class='hs-varid'>inr</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>b</span> <span class='hs-varop'>`k`</span> <span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span>
<a name="line-116"></a> <span class='hs-varid'>codiag</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>a</span> <span class='hs-varop'>`k`</span> <span class='hs-varid'>a</span>
<a name="line-117"></a> <span class='hs-layout'>(</span><span class='hs-varop'>|||</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-varop'>`k`</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>b</span> <span class='hs-varop'>`k`</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span> <span class='hs-varop'>`k`</span> <span class='hs-varid'>c</span>
<a name="line-118"></a>
<a name="line-119"></a> <span class='hs-varid'>codiag</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>id</span> <span class='hs-varop'>|||</span> <span class='hs-varid'>id</span>
<a name="line-120"></a> <span class='hs-varid'>f</span> <span class='hs-varop'>|||</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>codiag</span> <span class='hs-varop'>.</span> <span class='hs-varid'>bimap</span> <span class='hs-varid'>f</span> <span class='hs-varid'>g</span>
<a name="line-121"></a>
<a name="line-122"></a><span class='hs-comment'>{-- RULES
<a name="line-123"></a>"codiag . inl" codiag . inl = id
<a name="line-124"></a>"codiag . inr" codiag . inr = id
<a name="line-125"></a>"(f ||| g) . inl" forall f g. (f ||| g) . inl = f
<a name="line-126"></a>"(f ||| g) . inr" forall f g. (f ||| g) . inr = g
<a name="line-127"></a> --}</span>
<a name="line-128"></a>
<a name="line-129"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PreCoCartesian</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>-></span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-130"></a> <span class='hs-keyword'>type</span> <span class='hs-conid'>Sum</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>-></span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Either</span>
<a name="line-131"></a> <span class='hs-varid'>inl</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Left</span>
<a name="line-132"></a> <span class='hs-varid'>inr</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Right</span>
<a name="line-133"></a> <span class='hs-varid'>codiag</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>a</span>
<a name="line-134"></a> <span class='hs-varid'>codiag</span> <span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>a</span>
<a name="line-135"></a> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varop'>|||</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-varid'>a</span>
<a name="line-136"></a> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span> <span class='hs-varop'>|||</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>g</span> <span class='hs-varid'>a</span>
<a name="line-137"></a>
<a name="line-138"></a><a name="bimapSum"></a><span class='hs-comment'>-- | free construction of 'Bifunctor' for the coproduct 'Bifunctor' @Sum k@ if @(|||)@ is known</span>
<a name="line-139"></a><span class='hs-definition'>bimapSum</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>PreCoCartesian</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>~</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>)</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-varop'>`k`</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>b</span> <span class='hs-varop'>`k`</span> <span class='hs-varid'>d</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-varop'>+</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-varop'>`k`</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span> <span class='hs-varop'>+</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span>
<a name="line-140"></a><span class='hs-definition'>bimapSum</span> <span class='hs-varid'>f</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>inl</span> <span class='hs-varop'>.</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-varop'>|||</span> <span class='hs-layout'>(</span><span class='hs-varid'>inr</span> <span class='hs-varop'>.</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span>
<a name="line-141"></a>
<a name="line-142"></a><a name="braidSum"></a><span class='hs-comment'>-- | free construction of 'Braided' for the coproduct 'Bifunctor' @Sum k@</span>
<a name="line-143"></a><span class='hs-definition'>braidSum</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>PreCoCartesian</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>~</span> <span class='hs-conid'>Sum</span> <span class='hs-varid'>k</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-varop'>+</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-varop'>`k`</span> <span class='hs-layout'>(</span><span class='hs-varid'>b</span> <span class='hs-varop'>+</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-144"></a><span class='hs-definition'>braidSum</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>inr</span> <span class='hs-varop'>|||</span> <span class='hs-varid'>inl</span>
<a name="line-145"></a>
<a name="line-146"></a><a name="associateSum"></a><span class='hs-comment'>-- | free construction of 'Associative' for the coproduct 'Bifunctor' @Sum k@</span>
<a name="line-147"></a><span class='hs-comment'>-- associateSum :: (PreCoCartesian k, (+) ~ Sum k) => ((a + b) + c) ~> (a + (b + c))</span>
<a name="line-148"></a><span class='hs-definition'>associateSum</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>PreCoCartesian</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-varid'>c</span> <span class='hs-varop'>`k`</span> <span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span>
<a name="line-149"></a><span class='hs-definition'>associateSum</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>braid</span> <span class='hs-varop'>.</span> <span class='hs-varid'>first</span> <span class='hs-varid'>braid</span> <span class='hs-varop'>.</span> <span class='hs-varid'>disassociateSum</span> <span class='hs-varop'>.</span> <span class='hs-varid'>second</span> <span class='hs-varid'>braid</span> <span class='hs-varop'>.</span> <span class='hs-varid'>braid</span>
<a name="line-150"></a>
<a name="line-151"></a><a name="disassociateSum"></a><span class='hs-comment'>-- | free construction of 'Disassociative' for the coproduct 'Bifunctor' @Sum k@</span>
<a name="line-152"></a><span class='hs-comment'>-- disassociateSum :: (PreCoCartesian k, (+) ~ Sum k) => (a + (b + c)) ~> ((a + b) + c)</span>
<a name="line-153"></a><span class='hs-definition'>disassociateSum</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>PreCoCartesian</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-varop'>`k`</span> <span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-varid'>c</span>
<a name="line-154"></a><span class='hs-definition'>disassociateSum</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>inl</span> <span class='hs-varop'>.</span> <span class='hs-varid'>inl</span><span class='hs-layout'>)</span> <span class='hs-varop'>|||</span> <span class='hs-varid'>first</span> <span class='hs-varid'>inr</span>
<a name="line-155"></a>
<a name="line-156"></a><a name="CoCartesian"></a><span class='hs-keyword'>class</span>
<a name="line-157"></a> <span class='hs-layout'>(</span> <span class='hs-conid'>Comonoidal</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-158"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>PreCoCartesian</span> <span class='hs-varid'>k</span>
<a name="line-159"></a> <span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>CoCartesian</span> <span class='hs-varid'>k</span>
<a name="line-160"></a><span class='hs-keyword'>instance</span>
<a name="line-161"></a> <span class='hs-layout'>(</span> <span class='hs-conid'>Comonoidal</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sum</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span>
<a name="line-162"></a> <span class='hs-layout'>,</span> <span class='hs-conid'>PreCoCartesian</span> <span class='hs-varid'>k</span>
<a name="line-163"></a> <span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>CoCartesian</span> <span class='hs-varid'>k</span>
</pre></body>
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