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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE EmptyDataDecls, MultiParamTypeClasses, FunctionalDependencies,
<a name="line-2"></a> Rank2Types, DeriveDataTypeable, FlexibleInstances,
<a name="line-3"></a> UndecidableInstances, FlexibleContexts #-}</span>
<a name="line-4"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span>
<a name="line-5"></a><span class='hs-comment'>-- |</span>
<a name="line-6"></a><span class='hs-comment'>-- Module : Data.TypeLevel.Bool</span>
<a name="line-7"></a><span class='hs-comment'>-- Copyright : (c) 2008 Alfonso Acosta, Oleg Kiselyov, Wolfgang Jeltsch</span>
<a name="line-8"></a><span class='hs-comment'>-- and KTH's SAM group </span>
<a name="line-9"></a><span class='hs-comment'>-- License : BSD-style (see the file LICENSE)</span>
<a name="line-10"></a><span class='hs-comment'>-- </span>
<a name="line-11"></a><span class='hs-comment'>-- Maintainer : alfonso.acosta@gmail.com</span>
<a name="line-12"></a><span class='hs-comment'>-- Stability : experimental (MPTC, non-standarad instances)</span>
<a name="line-13"></a><span class='hs-comment'>-- Portability : non-portable</span>
<a name="line-14"></a><span class='hs-comment'>--</span>
<a name="line-15"></a><span class='hs-comment'>-- Type-level Booleans.</span>
<a name="line-16"></a><span class='hs-comment'>-- </span>
<a name="line-17"></a><span class='hs-comment'>----------------------------------------------------------------------------</span>
<a name="line-18"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>TypeLevel</span><span class='hs-varop'>.</span><span class='hs-conid'>Bool</span> <span class='hs-layout'>(</span>
<a name="line-19"></a> <span class='hs-comment'>-- * Type-level boolean values</span>
<a name="line-20"></a> <span class='hs-conid'>Bool</span><span class='hs-layout'>,</span> <span class='hs-varid'>toBool</span><span class='hs-layout'>,</span>
<a name="line-21"></a> <span class='hs-conid'>False</span><span class='hs-layout'>,</span> <span class='hs-varid'>false</span><span class='hs-layout'>,</span>
<a name="line-22"></a> <span class='hs-conid'>True</span><span class='hs-layout'>,</span> <span class='hs-varid'>true</span><span class='hs-layout'>,</span>
<a name="line-23"></a> <span class='hs-varid'>reifyBool</span><span class='hs-layout'>,</span>
<a name="line-24"></a> <span class='hs-comment'>-- * Type-level boolean operations</span>
<a name="line-25"></a> <span class='hs-conid'>Not</span><span class='hs-layout'>,</span> <span class='hs-varid'>not</span><span class='hs-layout'>,</span>
<a name="line-26"></a> <span class='hs-conid'>And</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>
<a name="line-27"></a> <span class='hs-conid'>Or</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>
<a name="line-28"></a> <span class='hs-conid'>Xor</span><span class='hs-layout'>,</span> <span class='hs-varid'>xor</span><span class='hs-layout'>,</span>
<a name="line-29"></a> <span class='hs-conid'>Imp</span><span class='hs-layout'>,</span> <span class='hs-varid'>imp</span><span class='hs-layout'>,</span>
<a name="line-30"></a> <span class='hs-conid'>Eq</span><span class='hs-layout'>,</span> <span class='hs-varid'>eq</span>
<a name="line-31"></a>
<a name="line-32"></a><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-33"></a>
<a name="line-34"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Generics</span> <span class='hs-layout'>(</span><span class='hs-conid'>Typeable</span><span class='hs-layout'>)</span>
<a name="line-35"></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'>Bool</span><span class='hs-layout'>,</span> <span class='hs-varid'>not</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-layout'>(</span><span class='hs-varop'>||</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</span><span class='hs-layout'>)</span>
<a name="line-36"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Prelude</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>P</span>
<a name="line-37"></a>
<a name="line-38"></a><span class='hs-comment'>------------------------------------</span>
<a name="line-39"></a><span class='hs-comment'>-- Definition of type-level Booleans</span>
<a name="line-40"></a><span class='hs-comment'>------------------------------------</span>
<a name="line-41"></a>
<a name="line-42"></a><a name="True"></a><span class='hs-comment'>-- | True type-level value</span>
<a name="line-43"></a><a name="True"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>True</span> <span class='hs-keyword'>deriving</span> <span class='hs-conid'>Typeable</span>
<a name="line-44"></a>
<a name="line-45"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-conid'>True</span> <span class='hs-keyword'>where</span>
<a name="line-46"></a> <span class='hs-varid'>show</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-str'>"True"</span>
<a name="line-47"></a>
<a name="line-48"></a><a name="true"></a><span class='hs-comment'>-- | True value-level reflecting function</span>
<a name="line-49"></a><span class='hs-definition'>true</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>True</span>
<a name="line-50"></a><span class='hs-definition'>true</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-51"></a>
<a name="line-52"></a><a name="False"></a><span class='hs-comment'>-- | False type-level value</span>
<a name="line-53"></a><a name="False"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>False</span> <span class='hs-keyword'>deriving</span> <span class='hs-conid'>Typeable</span>
<a name="line-54"></a>
<a name="line-55"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-conid'>False</span> <span class='hs-keyword'>where</span>
<a name="line-56"></a> <span class='hs-varid'>show</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-str'>"False"</span>
<a name="line-57"></a>
<a name="line-58"></a>
<a name="line-59"></a><a name="false"></a><span class='hs-comment'>-- | False value-level reflecting function</span>
<a name="line-60"></a><span class='hs-definition'>false</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>False</span>
<a name="line-61"></a><span class='hs-definition'>false</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-62"></a>
<a name="line-63"></a>
<a name="line-64"></a><a name="Bool"></a><span class='hs-comment'>-- | Booleans, internal version</span>
<a name="line-65"></a><a name="Bool"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b</span> <span class='hs-keyword'>where</span>
<a name="line-66"></a> <span class='hs-varid'>toBool</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>P</span><span class='hs-varop'>.</span><span class='hs-conid'>Bool</span>
<a name="line-67"></a>
<a name="line-68"></a>
<a name="line-69"></a><span class='hs-comment'>-- To prevent the user from adding new instances to BoolI we do NOT export </span>
<a name="line-70"></a><span class='hs-comment'>-- BoolI itself. Rather, we export the following proxy (Bool). </span>
<a name="line-71"></a><span class='hs-comment'>-- The proxy entails BoolI and so can be used to add BoolI </span>
<a name="line-72"></a><span class='hs-comment'>-- constraints in the signatures. However, all the constraints below</span>
<a name="line-73"></a><span class='hs-comment'>-- are expressed in terms of BoolI rather than the proxy. Thus, even if the </span>
<a name="line-74"></a><span class='hs-comment'>-- user adds new instances to the proxy, it would not matter. </span>
<a name="line-75"></a><span class='hs-comment'>-- Besides, because the following proxy instances are most general,</span>
<a name="line-76"></a><span class='hs-comment'>-- one may not add further instances without the overlapping instances </span>
<a name="line-77"></a><span class='hs-comment'>-- extension.</span>
<a name="line-78"></a>
<a name="line-79"></a><a name="Bool"></a><span class='hs-comment'>-- | Type-level Booleans</span>
<a name="line-80"></a><a name="Bool"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Bool</span> <span class='hs-varid'>b</span>
<a name="line-81"></a>
<a name="line-82"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Bool</span> <span class='hs-varid'>b</span>
<a name="line-83"></a>
<a name="line-84"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>BoolI</span> <span class='hs-conid'>True</span> <span class='hs-keyword'>where</span>
<a name="line-85"></a> <span class='hs-varid'>toBool</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span>
<a name="line-86"></a>
<a name="line-87"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>BoolI</span> <span class='hs-conid'>False</span> <span class='hs-keyword'>where</span>
<a name="line-88"></a> <span class='hs-varid'>toBool</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-89"></a>
<a name="line-90"></a><a name="reifyBool"></a><span class='hs-comment'>-- | Reification function. In CPS style (best possible solution)</span>
<a name="line-91"></a><span class='hs-definition'>reifyBool</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>P</span><span class='hs-varop'>.</span><span class='hs-conid'>Bool</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>b</span> <span class='hs-varop'>.</span> <span class='hs-conid'>Bool</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span>
<a name="line-92"></a><span class='hs-definition'>reifyBool</span> <span class='hs-conid'>True</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-varid'>true</span>
<a name="line-93"></a><span class='hs-definition'>reifyBool</span> <span class='hs-conid'>False</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-varid'>false</span>
<a name="line-94"></a>
<a name="line-95"></a><span class='hs-comment'>-------------</span>
<a name="line-96"></a><span class='hs-comment'>-- Operations</span>
<a name="line-97"></a><span class='hs-comment'>-------------</span>
<a name="line-98"></a>
<a name="line-99"></a><a name="Not"></a><span class='hs-comment'>-- | Boolean negation type-level relation. @Not b1 b2@ establishes that</span>
<a name="line-100"></a><a name="Not"></a><span class='hs-comment'>-- @not b1 = b2@</span>
<a name="line-101"></a><a name="Not"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>BoolI</span> <span class='hs-varid'>b1</span><span class='hs-layout'>,</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Not</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>b1</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b2</span><span class='hs-layout'>,</span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b1</span>
<a name="line-102"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Not</span> <span class='hs-conid'>False</span> <span class='hs-conid'>True</span>
<a name="line-103"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Not</span> <span class='hs-conid'>True</span> <span class='hs-conid'>False</span>
<a name="line-104"></a>
<a name="line-105"></a><a name="not"></a><span class='hs-comment'>-- | value-level reflection function for the 'Not' type-level relation</span>
<a name="line-106"></a><span class='hs-definition'>not</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Not</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b1</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b2</span>
<a name="line-107"></a><span class='hs-definition'>not</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-108"></a>
<a name="line-109"></a><a name="And"></a><span class='hs-comment'>-- | 'And' type-level relation. @And b1 b2 b3@ establishes that</span>
<a name="line-110"></a><a name="And"></a><span class='hs-comment'>-- @b1 && b2 = b3@</span>
<a name="line-111"></a><a name="And"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>BoolI</span> <span class='hs-varid'>b1</span><span class='hs-layout'>,</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b2</span><span class='hs-layout'>,</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b3</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>And</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-varid'>b3</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b3</span>
<a name="line-112"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>And</span> <span class='hs-conid'>False</span> <span class='hs-conid'>False</span> <span class='hs-conid'>False</span>
<a name="line-113"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>And</span> <span class='hs-conid'>False</span> <span class='hs-conid'>True</span> <span class='hs-conid'>False</span>
<a name="line-114"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>And</span> <span class='hs-conid'>True</span> <span class='hs-conid'>False</span> <span class='hs-conid'>False</span>
<a name="line-115"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>And</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span>
<a name="line-116"></a>
<a name="line-117"></a>
<a name="line-118"></a><a name="&&"></a><span class='hs-comment'>-- | value-level reflection function for the 'And' type-level relation</span>
<a name="line-119"></a><span class='hs-layout'>(</span><span class='hs-varop'>&&</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>And</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-varid'>b3</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b1</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b3</span>
<a name="line-120"></a><span class='hs-layout'>(</span><span class='hs-varop'>&&</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-121"></a><span class='hs-keyword'>infixr</span> <span class='hs-num'>3</span> <span class='hs-varop'>&&</span>
<a name="line-122"></a>
<a name="line-123"></a><a name="Or"></a><span class='hs-comment'>-- | Or type-level relation. @Or b1 b2 b3@ establishes that</span>
<a name="line-124"></a><a name="Or"></a><span class='hs-comment'>-- @b1 || b2 = b3@</span>
<a name="line-125"></a><a name="Or"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>BoolI</span> <span class='hs-varid'>b1</span><span class='hs-layout'>,</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b2</span><span class='hs-layout'>,</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b3</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Or</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-varid'>b3</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b3</span>
<a name="line-126"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Or</span> <span class='hs-conid'>False</span> <span class='hs-conid'>False</span> <span class='hs-conid'>False</span>
<a name="line-127"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Or</span> <span class='hs-conid'>False</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span>
<a name="line-128"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Or</span> <span class='hs-conid'>True</span> <span class='hs-conid'>False</span> <span class='hs-conid'>True</span>
<a name="line-129"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Or</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span>
<a name="line-130"></a>
<a name="line-131"></a>
<a name="line-132"></a><a name="||"></a><span class='hs-comment'>-- | value-level reflection function for the 'Or' type-level relation</span>
<a name="line-133"></a><span class='hs-layout'>(</span><span class='hs-varop'>||</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Or</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-varid'>b3</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b1</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b3</span>
<a name="line-134"></a><span class='hs-layout'>(</span><span class='hs-varop'>||</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-135"></a><span class='hs-keyword'>infixr</span> <span class='hs-num'>2</span> <span class='hs-varop'>||</span>
<a name="line-136"></a>
<a name="line-137"></a><a name="Xor"></a><span class='hs-comment'>-- | Exclusive or type-level relation. @Xor b1 b2 b3@ establishes that</span>
<a name="line-138"></a><a name="Xor"></a><span class='hs-comment'>-- @xor b1 b2 = b3@</span>
<a name="line-139"></a><a name="Xor"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>BoolI</span> <span class='hs-varid'>b1</span><span class='hs-layout'>,</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b2</span><span class='hs-layout'>,</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b3</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Xor</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-varid'>b3</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b3</span>
<a name="line-140"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Xor</span> <span class='hs-conid'>False</span> <span class='hs-conid'>False</span> <span class='hs-conid'>False</span>
<a name="line-141"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Xor</span> <span class='hs-conid'>False</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span>
<a name="line-142"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Xor</span> <span class='hs-conid'>True</span> <span class='hs-conid'>False</span> <span class='hs-conid'>True</span>
<a name="line-143"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Xor</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span> <span class='hs-conid'>False</span>
<a name="line-144"></a>
<a name="line-145"></a><a name="xor"></a><span class='hs-comment'>-- | value-level reflection function for the 'Xor' type-level relation</span>
<a name="line-146"></a><span class='hs-definition'>xor</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Xor</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-varid'>b3</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b1</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b3</span>
<a name="line-147"></a><span class='hs-definition'>xor</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-148"></a><span class='hs-keyword'>infixl</span> <span class='hs-num'>6</span> <span class='hs-varop'>`xor`</span> <span class='hs-comment'>-- infix declaration from Data.Bits</span>
<a name="line-149"></a>
<a name="line-150"></a><a name="Imp"></a><span class='hs-comment'>-- | Implication type-level relation. @Imp b1 b2 b3@ establishes that</span>
<a name="line-151"></a><a name="Imp"></a><span class='hs-comment'>-- @b1 =>b2 = b3@</span>
<a name="line-152"></a><a name="Imp"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>BoolI</span> <span class='hs-varid'>b1</span><span class='hs-layout'>,</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b2</span><span class='hs-layout'>,</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b3</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Imp</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-varid'>b3</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b3</span>
<a name="line-153"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Imp</span> <span class='hs-conid'>False</span> <span class='hs-conid'>False</span> <span class='hs-conid'>True</span>
<a name="line-154"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Imp</span> <span class='hs-conid'>False</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span>
<a name="line-155"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Imp</span> <span class='hs-conid'>True</span> <span class='hs-conid'>False</span> <span class='hs-conid'>False</span>
<a name="line-156"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Imp</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span>
<a name="line-157"></a>
<a name="line-158"></a>
<a name="line-159"></a><a name="imp"></a><span class='hs-comment'>-- | value-level reflection function for the Imp type-level relation</span>
<a name="line-160"></a><span class='hs-definition'>imp</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Imp</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-varid'>b3</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b1</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b3</span>
<a name="line-161"></a><span class='hs-definition'>imp</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-162"></a>
<a name="line-163"></a>
<a name="line-164"></a><span class='hs-comment'>-- Although equality can be defined as the composition of Xor and Not</span>
<a name="line-165"></a><span class='hs-comment'>-- we define it specifically</span>
<a name="line-166"></a>
<a name="line-167"></a><a name="Eq"></a><span class='hs-comment'>-- | Boolean equality type-level relation</span>
<a name="line-168"></a><a name="Eq"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>BoolI</span> <span class='hs-varid'>b1</span><span class='hs-layout'>,</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b2</span><span class='hs-layout'>,</span> <span class='hs-conid'>BoolI</span> <span class='hs-varid'>b3</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Eq</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-varid'>b3</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b3</span>
<a name="line-169"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Eq</span> <span class='hs-conid'>False</span> <span class='hs-conid'>False</span> <span class='hs-conid'>True</span>
<a name="line-170"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Eq</span> <span class='hs-conid'>False</span> <span class='hs-conid'>True</span> <span class='hs-conid'>False</span>
<a name="line-171"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Eq</span> <span class='hs-conid'>True</span> <span class='hs-conid'>False</span> <span class='hs-conid'>False</span>
<a name="line-172"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Eq</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span> <span class='hs-conid'>True</span>
<a name="line-173"></a>
<a name="line-174"></a>
<a name="line-175"></a><span class='hs-comment'>-- FIXME: eq should be named (==) but it clashes with the (==) defined</span>
<a name="line-176"></a><span class='hs-comment'>-- in Data.TypeLevel.Num . The chosen (and ugly) workaround was </span>
<a name="line-177"></a><span class='hs-comment'>-- to rename it to eq.</span>
<a name="line-178"></a>
<a name="line-179"></a><a name="eq"></a><span class='hs-comment'>-- | value-level reflection function for the 'Eq' type-level relation</span>
<a name="line-180"></a><span class='hs-definition'>eq</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Eq</span> <span class='hs-varid'>b1</span> <span class='hs-varid'>b2</span> <span class='hs-varid'>b3</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b1</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b3</span>
<a name="line-181"></a><span class='hs-definition'>eq</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-182"></a>
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