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<title>src/Data/TypeLevel/Num/Ops.hs</title>
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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies, TypeOperators,
<a name="line-2"></a> FlexibleInstances, FlexibleContexts, UndecidableInstances,
<a name="line-3"></a> EmptyDataDecls #-}</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.Num.Ops</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</span>
<a name="line-13"></a><span class='hs-comment'>-- Portability : non-portable (MPTC, non-standard instances)</span>
<a name="line-14"></a><span class='hs-comment'>--</span>
<a name="line-15"></a><span class='hs-comment'>-- Type-level numerical operations and its value-level reflection functions.</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'>Num</span><span class='hs-varop'>.</span><span class='hs-conid'>Ops</span>
<a name="line-19"></a> <span class='hs-layout'>(</span><span class='hs-comment'>-- * Successor/Predecessor</span>
<a name="line-20"></a> <span class='hs-conid'>Succ</span><span class='hs-layout'>,</span> <span class='hs-varid'>succ</span><span class='hs-layout'>,</span>
<a name="line-21"></a> <span class='hs-conid'>Pred</span><span class='hs-layout'>,</span> <span class='hs-varid'>pred</span><span class='hs-layout'>,</span>
<a name="line-22"></a> <span class='hs-comment'>-- * Addition/Subtraction</span>
<a name="line-23"></a> <span class='hs-conid'>Add</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-24"></a> <span class='hs-conid'>Sub</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-25"></a> <span class='hs-comment'>-- * Multiplication/Division</span>
<a name="line-26"></a> <span class='hs-conid'>Mul</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'>Div</span><span class='hs-layout'>,</span> <span class='hs-varid'>div</span><span class='hs-layout'>,</span>
<a name="line-28"></a> <span class='hs-conid'>Mod</span><span class='hs-layout'>,</span> <span class='hs-varid'>mod</span><span class='hs-layout'>,</span>
<a name="line-29"></a> <span class='hs-conid'>DivMod</span><span class='hs-layout'>,</span> <span class='hs-varid'>divMod</span><span class='hs-layout'>,</span>
<a name="line-30"></a> <span class='hs-conid'>IsDivBy</span><span class='hs-layout'>,</span> <span class='hs-varid'>isDivBy</span><span class='hs-layout'>,</span>
<a name="line-31"></a> <span class='hs-comment'>-- ** Special efficiency cases</span>
<a name="line-32"></a> <span class='hs-conid'>Mul10</span><span class='hs-layout'>,</span> <span class='hs-varid'>mul10</span><span class='hs-layout'>,</span>
<a name="line-33"></a> <span class='hs-conid'>Div10</span><span class='hs-layout'>,</span> <span class='hs-varid'>div10</span><span class='hs-layout'>,</span>
<a name="line-34"></a> <span class='hs-conid'>DivMod10</span><span class='hs-layout'>,</span> <span class='hs-varid'>divMod10</span><span class='hs-layout'>,</span>
<a name="line-35"></a> <span class='hs-comment'>-- * Exponientiation/Logarithm</span>
<a name="line-36"></a> <span class='hs-conid'>ExpBase</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-37"></a> <span class='hs-conid'>LogBase</span><span class='hs-layout'>,</span> <span class='hs-varid'>logBase</span><span class='hs-layout'>,</span>
<a name="line-38"></a> <span class='hs-conid'>LogBaseF</span><span class='hs-layout'>,</span> <span class='hs-varid'>logBaseF</span><span class='hs-layout'>,</span>
<a name="line-39"></a> <span class='hs-conid'>IsPowOf</span><span class='hs-layout'>,</span> <span class='hs-varid'>isPowOf</span><span class='hs-layout'>,</span>
<a name="line-40"></a> <span class='hs-comment'>-- ** Special efficiency cases</span>
<a name="line-41"></a> <span class='hs-conid'>Exp10</span><span class='hs-layout'>,</span> <span class='hs-varid'>exp10</span><span class='hs-layout'>,</span>
<a name="line-42"></a> <span class='hs-conid'>Log10</span><span class='hs-layout'>,</span> <span class='hs-varid'>log10</span><span class='hs-layout'>,</span>
<a name="line-43"></a> <span class='hs-comment'>-- * Comparison assertions</span>
<a name="line-44"></a> <span class='hs-comment'>-- ** General comparison assertion</span>
<a name="line-45"></a> <span class='hs-conid'>Trich</span><span class='hs-layout'>,</span> <span class='hs-varid'>trich</span><span class='hs-layout'>,</span>
<a name="line-46"></a> <span class='hs-comment'>-- *** Type-level values denoting comparison results</span>
<a name="line-47"></a> <span class='hs-conid'>LT</span><span class='hs-layout'>,</span> <span class='hs-conid'>EQ</span><span class='hs-layout'>,</span> <span class='hs-conid'>GT</span><span class='hs-layout'>,</span>
<a name="line-48"></a> <span class='hs-comment'>-- ** Abbreviated comparison assertions</span>
<a name="line-49"></a> <span class='hs-layout'>(</span><span class='hs-conop'>:==:</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-conop'>:>:</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-conop'>:<:</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-conop'>:>=:</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-conop'>:<=:</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-50"></a> <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-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-layout'>(</span><span class='hs-varop'><=</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-51"></a> <span class='hs-comment'>-- * Maximum/Minimum</span>
<a name="line-52"></a> <span class='hs-conid'>Max</span><span class='hs-layout'>,</span> <span class='hs-varid'>max</span><span class='hs-layout'>,</span>
<a name="line-53"></a> <span class='hs-conid'>Min</span><span class='hs-layout'>,</span> <span class='hs-varid'>min</span><span class='hs-layout'>,</span>
<a name="line-54"></a> <span class='hs-comment'>-- * Greatest Common Divisor</span>
<a name="line-55"></a> <span class='hs-conid'>GCD</span><span class='hs-layout'>,</span> <span class='hs-varid'>gcd</span>
<a name="line-56"></a> <span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-57"></a>
<a name="line-58"></a><span class='hs-keyword'>import</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'>Num</span><span class='hs-varop'>.</span><span class='hs-conid'>Reps</span>
<a name="line-59"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>TypeLevel</span><span class='hs-varop'>.</span><span class='hs-conid'>Num</span><span class='hs-varop'>.</span><span class='hs-conid'>Sets</span>
<a name="line-60"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>TypeLevel</span><span class='hs-varop'>.</span><span class='hs-conid'>Bool</span>
<a name="line-61"></a>
<a name="line-62"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Prelude</span> <span class='hs-varid'>hiding</span>
<a name="line-63"></a> <span class='hs-layout'>(</span><span class='hs-varid'>succ</span><span class='hs-layout'>,</span> <span class='hs-varid'>pred</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-comment'>-</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-varid'>div</span><span class='hs-layout'>,</span> <span class='hs-varid'>mod</span><span class='hs-layout'>,</span> <span class='hs-varid'>divMod</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'>logBase</span><span class='hs-layout'>,</span>
<a name="line-64"></a> <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-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-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-varid'>max</span><span class='hs-layout'>,</span> <span class='hs-varid'>min</span><span class='hs-layout'>,</span> <span class='hs-varid'>gcd</span><span class='hs-layout'>,</span> <span class='hs-conid'>Bool</span><span class='hs-layout'>)</span>
<a name="line-65"></a>
<a name="line-66"></a><span class='hs-comment'>-------------------------</span>
<a name="line-67"></a><span class='hs-comment'>-- Successor, Predecessor</span>
<a name="line-68"></a><span class='hs-comment'>-------------------------</span>
<a name="line-69"></a>
<a name="line-70"></a><a name="Succ"></a><span class='hs-comment'>-- | Successor type-level relation. @Succ x y@ establishes</span>
<a name="line-71"></a><a name="Succ"></a><span class='hs-comment'>-- that @succ x = y@.</span>
<a name="line-72"></a><a name="Succ"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Succ</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span>
<a name="line-73"></a>
<a name="line-74"></a>
<a name="line-75"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-conid'>IsZero</span> <span class='hs-varid'>y</span> <span class='hs-varid'>yz</span><span class='hs-layout'>,</span> <span class='hs-conid'>DivMod10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>xi</span> <span class='hs-varid'>xl</span><span class='hs-layout'>,</span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xi</span> <span class='hs-varid'>xl</span> <span class='hs-varid'>yi</span> <span class='hs-varid'>yl</span> <span class='hs-varid'>yz</span><span class='hs-layout'>,</span>
<a name="line-76"></a> <span class='hs-conid'>DivMod10</span> <span class='hs-varid'>y</span> <span class='hs-varid'>yi</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span>
<a name="line-77"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Succ</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span>
<a name="line-78"></a>
<a name="line-79"></a><a name="Succ'"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xh</span> <span class='hs-varid'>xl</span> <span class='hs-varid'>yh</span> <span class='hs-varid'>yl</span> <span class='hs-varid'>yz</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>xh</span> <span class='hs-varid'>xl</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>yh</span> <span class='hs-varid'>yl</span> <span class='hs-varid'>yz</span><span class='hs-layout'>,</span> <span class='hs-varid'>yh</span> <span class='hs-varid'>yl</span> <span class='hs-varid'>yz</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>xh</span> <span class='hs-varid'>xl</span>
<a name="line-80"></a>
<a name="line-81"></a>
<a name="line-82"></a><span class='hs-comment'>-- This intends to implement a user reporting operation when</span>
<a name="line-83"></a><span class='hs-comment'>-- trying to calculate the predecesor of 0</span>
<a name="line-84"></a><span class='hs-comment'>-- FIXME: however, the instance rule is never triggered!</span>
<a name="line-85"></a>
<a name="line-86"></a><a name="Succ'"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>Failure</span> <span class='hs-varid'>t</span>
<a name="line-87"></a><a name="Succ'"></a><span class='hs-comment'>-- No instances</span>
<a name="line-88"></a><a name="Succ'"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>PredecessorOfZeroError</span> <span class='hs-varid'>t</span>
<a name="line-89"></a>
<a name="line-90"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Failure</span> <span class='hs-layout'>(</span><span class='hs-conid'>PredecessorOfZeroError</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Succ'</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>True</span>
<a name="line-91"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D0</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>False</span>
<a name="line-92"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D1</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>False</span>
<a name="line-93"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D2</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>False</span>
<a name="line-94"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D3</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>False</span>
<a name="line-95"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D4</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>False</span>
<a name="line-96"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D5</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>False</span>
<a name="line-97"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D6</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>False</span>
<a name="line-98"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D7</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>False</span>
<a name="line-99"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D8</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>False</span>
<a name="line-100"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ</span> <span class='hs-varid'>xi</span> <span class='hs-varid'>yi</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Succ'</span> <span class='hs-varid'>xi</span> <span class='hs-conid'>D9</span> <span class='hs-varid'>yi</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>False</span>
<a name="line-101"></a>
<a name="line-102"></a>
<a name="line-103"></a><span class='hs-comment'>{-
<a name="line-104"></a>
<a name="line-105"></a>Nicer, but not relational implementation of Succ
<a name="line-106"></a>
<a name="line-107"></a>class (Nat x, Pos y) => Succ' x y | x -> y
<a name="line-108"></a>
<a name="line-109"></a>-- by structural induction on the first argument
<a name="line-110"></a>instance Succ' D0 D1
<a name="line-111"></a>instance Succ' D1 D2
<a name="line-112"></a>instance Succ' D2 D3
<a name="line-113"></a>instance Succ' D3 D4
<a name="line-114"></a>instance Succ' D4 D5
<a name="line-115"></a>instance Succ' D5 D6
<a name="line-116"></a>instance Succ' D6 D7
<a name="line-117"></a>instance Succ' D7 D8
<a name="line-118"></a>instance Succ' D8 D9
<a name="line-119"></a>instance Succ' D9 (D1 :* D0)
<a name="line-120"></a>instance Pos x => Succ' (x :* D0) (x :* D1)
<a name="line-121"></a>instance Pos x => Succ' (x :* D1) (x :* D2)
<a name="line-122"></a>instance Pos x => Succ' (x :* D2) (x :* D3)
<a name="line-123"></a>instance Pos x => Succ' (x :* D3) (x :* D4)
<a name="line-124"></a>instance Pos x => Succ' (x :* D4) (x :* D5)
<a name="line-125"></a>instance Pos x => Succ' (x :* D5) (x :* D6)
<a name="line-126"></a>instance Pos x => Succ' (x :* D6) (x :* D7)
<a name="line-127"></a>instance Pos x => Succ' (x :* D7) (x :* D8)
<a name="line-128"></a>instance Pos x => Succ' (x :* D8) (x :* D9)
<a name="line-129"></a>instance (Pos x, Succ' x y) => Succ' (x :* D9) (y :* D0)
<a name="line-130"></a>
<a name="line-131"></a>
<a name="line-132"></a>class (Nat x, Pos y) => Succ x y | x -> y, y -> x
<a name="line-133"></a>instance Succ' x y => Succ x y
<a name="line-134"></a>-}</span>
<a name="line-135"></a>
<a name="line-136"></a><a name="succ"></a><span class='hs-comment'>-- | value-level reflection function for the 'Succ' type-level relation</span>
<a name="line-137"></a><span class='hs-definition'>succ</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Succ</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span>
<a name="line-138"></a><span class='hs-definition'>succ</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-139"></a>
<a name="line-140"></a><a name="Pred"></a><span class='hs-comment'>-- Note: maybe redundant </span>
<a name="line-141"></a><a name="Pred"></a><span class='hs-comment'>-- | Predecessor type-level relation. @Pred x y@ establishes</span>
<a name="line-142"></a><a name="Pred"></a><span class='hs-comment'>-- that @pred x = y@.</span>
<a name="line-143"></a><a name="Pred"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Pred</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span>
<a name="line-144"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Pred</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span>
<a name="line-145"></a>
<a name="line-146"></a><a name="pred"></a><span class='hs-comment'>-- | value-level reflection function for the 'Pred' type-level relation</span>
<a name="line-147"></a><span class='hs-definition'>pred</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Pred</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span>
<a name="line-148"></a><span class='hs-definition'>pred</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-149"></a>
<a name="line-150"></a>
<a name="line-151"></a>
<a name="line-152"></a><span class='hs-comment'>--------------------</span>
<a name="line-153"></a><span class='hs-comment'>-- Add and Subtract</span>
<a name="line-154"></a><span class='hs-comment'>--------------------</span>
<a name="line-155"></a>
<a name="line-156"></a>
<a name="line-157"></a><a name="Add'"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span> <span class='hs-varid'>z</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span>
<a name="line-158"></a>
<a name="line-159"></a><span class='hs-comment'>-- by structural induction on the first argument</span>
<a name="line-160"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D0</span> <span class='hs-varid'>y</span> <span class='hs-varid'>y</span>
<a name="line-161"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Succ</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D1</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span>
<a name="line-162"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Succ</span> <span class='hs-varid'>z</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D1</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D2</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-163"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Succ</span> <span class='hs-varid'>z</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D2</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D3</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-164"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Succ</span> <span class='hs-varid'>z</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D3</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D4</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-165"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Succ</span> <span class='hs-varid'>z</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D4</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D5</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-166"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Succ</span> <span class='hs-varid'>z</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D5</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D6</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-167"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Succ</span> <span class='hs-varid'>z</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D6</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D7</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-168"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Succ</span> <span class='hs-varid'>z</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D7</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D8</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-169"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Succ</span> <span class='hs-varid'>z</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D8</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-conid'>D9</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-170"></a><span class='hs-comment'>-- multidigit addition</span>
<a name="line-171"></a><span class='hs-comment'>-- TODO: explain</span>
<a name="line-172"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>xi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>xl</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span>
<a name="line-173"></a> <span class='hs-conid'>Add'</span> <span class='hs-varid'>xi</span> <span class='hs-varid'>yi</span> <span class='hs-varid'>zi</span><span class='hs-layout'>,</span> <span class='hs-conid'>DivMod10</span> <span class='hs-varid'>y</span> <span class='hs-varid'>yi</span> <span class='hs-varid'>yl</span><span class='hs-layout'>,</span> <span class='hs-conid'>Add'</span> <span class='hs-varid'>xl</span> <span class='hs-layout'>(</span><span class='hs-varid'>zi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span>
<a name="line-174"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add'</span> <span class='hs-layout'>(</span><span class='hs-varid'>xi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>xl</span><span class='hs-layout'>)</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span>
<a name="line-175"></a>
<a name="line-176"></a><a name="Add"></a><span class='hs-comment'>-- | Addition type-level relation. @Add x y z@ establishes</span>
<a name="line-177"></a><a name="Add"></a><span class='hs-comment'>-- that @x + y = z@.</span>
<a name="line-178"></a><a name="Add"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Add'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span> <span class='hs-conid'>Add'</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span> <span class='hs-varid'>z</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-varid'>z</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span>
<a name="line-179"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Add'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span> <span class='hs-conid'>Add'</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Add</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span>
<a name="line-180"></a>
<a name="line-181"></a>
<a name="line-182"></a><a name="+"></a><span class='hs-comment'>-- | value-level reflection function for the 'Add' type-level relation </span>
<a name="line-183"></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-conid'>Add</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span>
<a name="line-184"></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-185"></a>
<a name="line-186"></a><a name="Sub"></a><span class='hs-comment'>-- | Subtraction type-level relation. @Sub x y z@ establishes</span>
<a name="line-187"></a><a name="Sub"></a><span class='hs-comment'>-- that @x - y = z@ </span>
<a name="line-188"></a><a name="Sub"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>Sub</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span> <span class='hs-varid'>z</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-varid'>z</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span>
<a name="line-189"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Add</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Sub</span> <span class='hs-varid'>z</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span>
<a name="line-190"></a>
<a name="line-191"></a><span class='hs-comment'>-- | value-level reflection function for the 'Sub' type-level relation </span>
<a name="line-192"></a><span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sub</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span>
<a name="line-193"></a><span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-194"></a><span class='hs-keyword'>infixl</span> <span class='hs-num'>6</span> <span class='hs-varop'>+</span><span class='hs-layout'>,</span> <span class='hs-comment'>-</span>
<a name="line-195"></a>
<a name="line-196"></a><span class='hs-comment'>------------------------------</span>
<a name="line-197"></a><span class='hs-comment'>-- Multiplication and Division</span>
<a name="line-198"></a><span class='hs-comment'>------------------------------</span>
<a name="line-199"></a>
<a name="line-200"></a><span class='hs-comment'>-----------------</span>
<a name="line-201"></a><span class='hs-comment'>-- Multiplication</span>
<a name="line-202"></a><span class='hs-comment'>-----------------</span>
<a name="line-203"></a>
<a name="line-204"></a><a name="Mul"></a><span class='hs-comment'>-- | Multiplication type-level relation. @Mul x y z@ establishes</span>
<a name="line-205"></a><a name="Mul"></a><span class='hs-comment'>-- that @x * y = z@.</span>
<a name="line-206"></a><a name="Mul"></a><span class='hs-comment'>-- Note it isn't relational (i.e. its inverse cannot be used for division,</span>
<a name="line-207"></a><a name="Mul"></a><span class='hs-comment'>-- however, even if it could, the resulting division would only</span>
<a name="line-208"></a><a name="Mul"></a><span class='hs-comment'>-- work for zero-remainder divisions)</span>
<a name="line-209"></a><a name="Mul"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span>
<a name="line-210"></a>
<a name="line-211"></a><span class='hs-comment'>-- By structural induction on the first argument</span>
<a name="line-212"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D0</span> <span class='hs-varid'>y</span> <span class='hs-conid'>D0</span>
<a name="line-213"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D1</span> <span class='hs-varid'>y</span> <span class='hs-varid'>y</span>
<a name="line-214"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Add</span> <span class='hs-varid'>y</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D2</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span>
<a name="line-215"></a><span class='hs-comment'>-- IMPORTANT: changing the line above by the commented line below</span>
<a name="line-216"></a><span class='hs-comment'>-- would make multiplication relational. However, that would</span>
<a name="line-217"></a><span class='hs-comment'>-- happen at the cost of performing a division by 2 in every </span>
<a name="line-218"></a><span class='hs-comment'>-- multiplication which doesn't pay off.</span>
<a name="line-219"></a><span class='hs-comment'>-- Besides, the Division algortihm obtained out of the </span>
<a name="line-220"></a><span class='hs-comment'>-- inverse of Mul can only work when the remainder is zero, </span>
<a name="line-221"></a><span class='hs-comment'>-- which isn't really useful.</span>
<a name="line-222"></a><span class='hs-comment'>-- instance (Add y y z, DivMod z D2 y D0) => Mul D2 y z</span>
<a name="line-223"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Add</span> <span class='hs-varid'>z</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D2</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D3</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-224"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Add</span> <span class='hs-varid'>z</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D3</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D4</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-225"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Add</span> <span class='hs-varid'>z</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D4</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D5</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-226"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Add</span> <span class='hs-varid'>z</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D5</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D6</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-227"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Add</span> <span class='hs-varid'>z</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D6</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D7</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-228"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Add</span> <span class='hs-varid'>z</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D7</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D8</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-229"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Add</span> <span class='hs-varid'>z</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D8</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-conid'>D9</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-230"></a><span class='hs-comment'>-- TODO explain.</span>
<a name="line-231"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>xi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>xl</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-conid'>Mul</span> <span class='hs-varid'>xi</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span> <span class='hs-conid'>Mul10</span> <span class='hs-varid'>z</span> <span class='hs-varid'>z10</span><span class='hs-layout'>,</span> <span class='hs-conid'>Mul</span> <span class='hs-varid'>xl</span> <span class='hs-varid'>y</span> <span class='hs-varid'>dy</span><span class='hs-layout'>,</span>
<a name="line-232"></a> <span class='hs-conid'>Add</span> <span class='hs-varid'>dy</span> <span class='hs-varid'>z10</span> <span class='hs-varid'>z'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul</span> <span class='hs-layout'>(</span><span class='hs-varid'>xi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>xl</span><span class='hs-layout'>)</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z'</span>
<a name="line-233"></a>
<a name="line-234"></a>
<a name="line-235"></a><a name="*"></a><span class='hs-comment'>-- | value-level reflection function for the multiplication type-level relation </span>
<a name="line-236"></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'>Mul</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span>
<a name="line-237"></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-238"></a><span class='hs-keyword'>infixl</span> <span class='hs-num'>7</span> <span class='hs-varop'>*</span>
<a name="line-239"></a>
<a name="line-240"></a><span class='hs-comment'>-----------</span>
<a name="line-241"></a><span class='hs-comment'>-- Division</span>
<a name="line-242"></a><span class='hs-comment'>-----------</span>
<a name="line-243"></a>
<a name="line-244"></a><a name="DivMod"></a><span class='hs-comment'>-- | Division and Remainder type-level relation. @DivMod x y q r@ establishes</span>
<a name="line-245"></a><a name="DivMod"></a><span class='hs-comment'>-- that @x/y = q + r/y@</span>
<a name="line-246"></a><a name="DivMod"></a><span class='hs-comment'>-- Note it is not relational (i.e. its inverse cannot be used </span>
<a name="line-247"></a><a name="DivMod"></a><span class='hs-comment'>-- for multiplication). </span>
<a name="line-248"></a><a name="DivMod"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span>
<a name="line-249"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-conid'>Trich</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>cmp</span><span class='hs-layout'>,</span> <span class='hs-conid'>DivMod'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span> <span class='hs-varid'>cmp</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span>
<a name="line-250"></a>
<a name="line-251"></a><a name="DivMod'"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span> <span class='hs-varid'>cmp</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>cmp</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span><span class='hs-layout'>,</span>
<a name="line-252"></a> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span> <span class='hs-varid'>cmp</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span>
<a name="line-253"></a> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span> <span class='hs-varid'>cmp</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span>
<a name="line-254"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-conid'>D0</span> <span class='hs-varid'>x</span> <span class='hs-conid'>LT</span>
<a name="line-255"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>EQ</span>
<a name="line-256"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-conid'>Sub</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pred</span> <span class='hs-varid'>q</span> <span class='hs-varid'>q'</span><span class='hs-layout'>,</span> <span class='hs-conid'>DivMod</span> <span class='hs-varid'>x'</span> <span class='hs-varid'>y</span> <span class='hs-varid'>q'</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span>
<a name="line-257"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span> <span class='hs-conid'>GT</span>
<a name="line-258"></a>
<a name="line-259"></a><a name="divMod"></a><span class='hs-comment'>-- | value-level reflection function for the 'DivMod' type-level relation</span>
<a name="line-260"></a><span class='hs-definition'>divMod</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DivMod</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>q</span><span class='hs-layout'>,</span><span class='hs-varid'>r</span><span class='hs-layout'>)</span>
<a name="line-261"></a><span class='hs-definition'>divMod</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span><span class='hs-layout'>,</span> <span class='hs-varid'>undefined</span><span class='hs-layout'>)</span>
<a name="line-262"></a>
<a name="line-263"></a><a name="Div"></a><span class='hs-comment'>-- | Division type-level relation. Remainder-discarding version of 'DivMod'. </span>
<a name="line-264"></a><a name="Div"></a><span class='hs-comment'>-- Note it is not relational (due to DivMod not being relational)</span>
<a name="line-265"></a><a name="Div"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>Div</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span>
<a name="line-266"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>DivMod</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Div</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>q</span>
<a name="line-267"></a>
<a name="line-268"></a><a name="div"></a><span class='hs-comment'>-- | value-level reflection function for the 'Div' type-level relation </span>
<a name="line-269"></a><span class='hs-definition'>div</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Div</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span>
<a name="line-270"></a><span class='hs-definition'>div</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-271"></a>
<a name="line-272"></a><a name="Mod"></a><span class='hs-comment'>-- | Remainder of division, type-level relation. @Mod x y r@ establishes that</span>
<a name="line-273"></a><a name="Mod"></a><span class='hs-comment'>-- @r@ is the reminder of dividing @x@ by @y@.</span>
<a name="line-274"></a><a name="Mod"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>Mod</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span>
<a name="line-275"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mod</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>r</span>
<a name="line-276"></a>
<a name="line-277"></a><a name="mod"></a><span class='hs-comment'>-- | value-level reflection function for the 'Mod' type-level relation </span>
<a name="line-278"></a><span class='hs-definition'>mod</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Mod</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span>
<a name="line-279"></a><span class='hs-definition'>mod</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-280"></a><span class='hs-keyword'>infixl</span> <span class='hs-num'>7</span> <span class='hs-varop'>`div`</span><span class='hs-layout'>,</span> <span class='hs-varop'>`mod`</span>
<a name="line-281"></a>
<a name="line-282"></a>
<a name="line-283"></a><span class='hs-comment'>----------------------------------------</span>
<a name="line-284"></a><span class='hs-comment'>-- Multiplication/Division special cases</span>
<a name="line-285"></a><span class='hs-comment'>----------------------------------------</span>
<a name="line-286"></a>
<a name="line-287"></a><a name="Mul10"></a><span class='hs-comment'>-- | Multiplication by 10 type-level relation (based on 'DivMod10').</span>
<a name="line-288"></a><a name="Mul10"></a><span class='hs-comment'>-- @Mul10 x y@ establishes that @10 * x = y@.</span>
<a name="line-289"></a><a name="Mul10"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>q</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>q</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>q</span><span class='hs-layout'>,</span> <span class='hs-varid'>q</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span>
<a name="line-290"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>q</span> <span class='hs-conid'>D0</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Mul10</span> <span class='hs-varid'>q</span> <span class='hs-varid'>x</span>
<a name="line-291"></a>
<a name="line-292"></a><a name="mul10"></a><span class='hs-comment'>-- | value-level reflection function for 'Mul10' </span>
<a name="line-293"></a><span class='hs-definition'>mul10</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Mul10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>q</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>q</span>
<a name="line-294"></a><span class='hs-definition'>mul10</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-295"></a>
<a name="line-296"></a><a name="DivMod10"></a><span class='hs-comment'>-- | Division by 10 and Remainer type-level relation (similar to 'DivMod'). </span>
<a name="line-297"></a><a name="DivMod10"></a><span class='hs-comment'>--</span>
<a name="line-298"></a><a name="DivMod10"></a><span class='hs-comment'>-- This operation is much faster than DivMod. Furthermore, it is </span>
<a name="line-299"></a><a name="DivMod10"></a><span class='hs-comment'>-- the general, non-structural, constructor/deconstructor since it</span>
<a name="line-300"></a><a name="DivMod10"></a><span class='hs-comment'>-- splits a decimal numeral into its initial digits and last digit.</span>
<a name="line-301"></a><a name="DivMod10"></a><span class='hs-comment'>-- Thus, it allows to inspect the structure of a number and is normally</span>
<a name="line-302"></a><a name="DivMod10"></a><span class='hs-comment'>-- used to create type-level operations.</span>
<a name="line-303"></a><a name="DivMod10"></a><span class='hs-comment'>--</span>
<a name="line-304"></a><a name="DivMod10"></a><span class='hs-comment'>-- Note that contrary to 'DivMod', 'DivMod10' is relational (it can be used to</span>
<a name="line-305"></a><a name="DivMod10"></a><span class='hs-comment'>-- multiply by 10)</span>
<a name="line-306"></a><a name="DivMod10"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>i</span> <span class='hs-varid'>l</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-varid'>l</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>i</span> <span class='hs-varid'>l</span>
<a name="line-307"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D0</span>
<a name="line-308"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D1</span>
<a name="line-309"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D2</span>
<a name="line-310"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D3</span>
<a name="line-311"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D4</span>
<a name="line-312"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D5</span>
<a name="line-313"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D6</span>
<a name="line-314"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D7</span>
<a name="line-315"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D8</span>
<a name="line-316"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D9</span>
<a name="line-317"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>D1</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod10</span> <span class='hs-layout'>(</span><span class='hs-conid'>D1</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-conid'>D1</span> <span class='hs-varid'>l</span>
<a name="line-318"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>D2</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod10</span> <span class='hs-layout'>(</span><span class='hs-conid'>D2</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-conid'>D2</span> <span class='hs-varid'>l</span>
<a name="line-319"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>D3</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod10</span> <span class='hs-layout'>(</span><span class='hs-conid'>D3</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-conid'>D3</span> <span class='hs-varid'>l</span>
<a name="line-320"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>D4</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod10</span> <span class='hs-layout'>(</span><span class='hs-conid'>D4</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-conid'>D4</span> <span class='hs-varid'>l</span>
<a name="line-321"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>D5</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod10</span> <span class='hs-layout'>(</span><span class='hs-conid'>D5</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-conid'>D5</span> <span class='hs-varid'>l</span>
<a name="line-322"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>D6</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod10</span> <span class='hs-layout'>(</span><span class='hs-conid'>D6</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-conid'>D6</span> <span class='hs-varid'>l</span>
<a name="line-323"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>D7</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod10</span> <span class='hs-layout'>(</span><span class='hs-conid'>D7</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-conid'>D7</span> <span class='hs-varid'>l</span>
<a name="line-324"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>D8</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod10</span> <span class='hs-layout'>(</span><span class='hs-conid'>D8</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-conid'>D8</span> <span class='hs-varid'>l</span>
<a name="line-325"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>D9</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>DivMod10</span> <span class='hs-layout'>(</span><span class='hs-conid'>D9</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-conid'>D9</span> <span class='hs-varid'>l</span>
<a name="line-326"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l'</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-327"></a> <span class='hs-conid'>DivMod10</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l'</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-varid'>l'</span>
<a name="line-328"></a>
<a name="line-329"></a><a name="divMod10"></a><span class='hs-comment'>-- | value-level reflection function for DivMod10 </span>
<a name="line-330"></a><span class='hs-definition'>divMod10</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>DivMod10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</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'>q</span><span class='hs-layout'>,</span><span class='hs-varid'>r</span><span class='hs-layout'>)</span>
<a name="line-331"></a><span class='hs-definition'>divMod10</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span><span class='hs-layout'>,</span> <span class='hs-varid'>undefined</span><span class='hs-layout'>)</span>
<a name="line-332"></a>
<a name="line-333"></a>
<a name="line-334"></a><a name="Div10"></a><span class='hs-comment'>-- | Division by 10 type-level relation (based on DivMod10)</span>
<a name="line-335"></a><a name="Div10"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>q</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Div10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>q</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>q</span><span class='hs-layout'>,</span> <span class='hs-varid'>q</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span>
<a name="line-336"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>DivMod10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Div10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>q</span>
<a name="line-337"></a>
<a name="line-338"></a><a name="div10"></a><span class='hs-comment'>-- | value-level reflection function for Mul10 </span>
<a name="line-339"></a><span class='hs-definition'>div10</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Div10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>q</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>q</span>
<a name="line-340"></a><span class='hs-definition'>div10</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-341"></a>
<a name="line-342"></a><span class='hs-comment'>----------------------------</span>
<a name="line-343"></a><span class='hs-comment'>-- Is-Divisible-By assertion</span>
<a name="line-344"></a><span class='hs-comment'>----------------------------</span>
<a name="line-345"></a>
<a name="line-346"></a><a name="IsDivBy"></a><span class='hs-comment'>-- | Is-divisible-by type-level assertion. e.g @IsDivBy d x@ establishes that</span>
<a name="line-347"></a><a name="IsDivBy"></a><span class='hs-comment'>-- @x@ is divisible by @d@.</span>
<a name="line-348"></a><a name="IsDivBy"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>d</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>IsDivBy</span> <span class='hs-varid'>d</span> <span class='hs-varid'>x</span>
<a name="line-349"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>DivMod</span> <span class='hs-varid'>x</span> <span class='hs-varid'>d</span> <span class='hs-varid'>q</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>IsDivBy</span> <span class='hs-varid'>d</span> <span class='hs-varid'>x</span>
<a name="line-350"></a>
<a name="line-351"></a><a name="isDivBy"></a><span class='hs-comment'>-- | value-level reflection function for IsDivBy</span>
<a name="line-352"></a><span class='hs-definition'>isDivBy</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>IsDivBy</span> <span class='hs-varid'>d</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>d</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span>
<a name="line-353"></a><span class='hs-definition'>isDivBy</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-354"></a>
<a name="line-355"></a><span class='hs-comment'>---------------------------</span>
<a name="line-356"></a><span class='hs-comment'>-- Exponentiation/Logarithm</span>
<a name="line-357"></a><span class='hs-comment'>---------------------------</span>
<a name="line-358"></a>
<a name="line-359"></a><a name="ExpBase"></a><span class='hs-comment'>-- | Exponentation type-level relation. @ExpBase b e r@ establishes</span>
<a name="line-360"></a><a name="ExpBase"></a><span class='hs-comment'>-- that @b^e = r@</span>
<a name="line-361"></a><a name="ExpBase"></a><span class='hs-comment'>-- Note it is not relational (i.e. it cannot be used to express logarithms)</span>
<a name="line-362"></a><a name="ExpBase"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>e</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-varid'>e</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>b</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span>
<a name="line-363"></a>
<a name="line-364"></a><span class='hs-comment'>-- structural induction over the exponent</span>
<a name="line-365"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D1</span>
<a name="line-366"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D1</span> <span class='hs-varid'>b</span>
<a name="line-367"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Mul</span> <span class='hs-varid'>b</span> <span class='hs-varid'>b</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D2</span> <span class='hs-varid'>r</span>
<a name="line-368"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Mul</span> <span class='hs-varid'>r</span> <span class='hs-varid'>b</span> <span class='hs-varid'>r'</span><span class='hs-layout'>,</span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D2</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D3</span> <span class='hs-varid'>r'</span>
<a name="line-369"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Mul</span> <span class='hs-varid'>r</span> <span class='hs-varid'>b</span> <span class='hs-varid'>r'</span><span class='hs-layout'>,</span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D3</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D4</span> <span class='hs-varid'>r'</span>
<a name="line-370"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Mul</span> <span class='hs-varid'>r</span> <span class='hs-varid'>b</span> <span class='hs-varid'>r'</span><span class='hs-layout'>,</span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D4</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D5</span> <span class='hs-varid'>r'</span>
<a name="line-371"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Mul</span> <span class='hs-varid'>r</span> <span class='hs-varid'>b</span> <span class='hs-varid'>r'</span><span class='hs-layout'>,</span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D5</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D6</span> <span class='hs-varid'>r'</span>
<a name="line-372"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Mul</span> <span class='hs-varid'>r</span> <span class='hs-varid'>b</span> <span class='hs-varid'>r'</span><span class='hs-layout'>,</span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D6</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D7</span> <span class='hs-varid'>r'</span>
<a name="line-373"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Mul</span> <span class='hs-varid'>r</span> <span class='hs-varid'>b</span> <span class='hs-varid'>r'</span><span class='hs-layout'>,</span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D7</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D8</span> <span class='hs-varid'>r'</span>
<a name="line-374"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Mul</span> <span class='hs-varid'>r</span> <span class='hs-varid'>b</span> <span class='hs-varid'>r'</span><span class='hs-layout'>,</span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D8</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D9</span> <span class='hs-varid'>r'</span>
<a name="line-375"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>ei</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>el</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>r</span><span class='hs-layout'>,</span>
<a name="line-376"></a> <span class='hs-conid'>Mul</span> <span class='hs-varid'>b</span> <span class='hs-varid'>r</span> <span class='hs-varid'>r'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pred</span> <span class='hs-layout'>(</span><span class='hs-varid'>ei</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>el</span><span class='hs-layout'>)</span> <span class='hs-varid'>e'</span><span class='hs-layout'>,</span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-varid'>e'</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span>
<a name="line-377"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-layout'>(</span><span class='hs-varid'>ei</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>el</span><span class='hs-layout'>)</span> <span class='hs-varid'>r'</span>
<a name="line-378"></a><a name="^"></a><span class='hs-comment'>-- | value-level reflection function for the ExpBase type-level relation</span>
<a name="line-379"></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'>ExpBase</span> <span class='hs-varid'>b</span> <span class='hs-varid'>e</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span>
<a name="line-380"></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-381"></a><span class='hs-keyword'>infixr</span> <span class='hs-num'>8</span> <span class='hs-varop'>^</span>
<a name="line-382"></a>
<a name="line-383"></a><a name="LogBase"></a><span class='hs-comment'>-- Logarithm type-level relation. @LogBase b x e@ establishes that </span>
<a name="line-384"></a><a name="LogBase"></a><span class='hs-comment'>-- @log_base_b x = e@</span>
<a name="line-385"></a><a name="LogBase"></a><span class='hs-comment'>-- Note it is not relational (i.e. cannot be used to express exponentiation)</span>
<a name="line-386"></a><a name="LogBase"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:>=:</span> <span class='hs-conid'>D2</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>LogBase</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>e</span>
<a name="line-387"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>LogBaseF</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>e</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>LogBase</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>e</span>
<a name="line-388"></a>
<a name="line-389"></a>
<a name="line-390"></a><a name="logBase"></a><span class='hs-comment'>-- | value-level reflection function for LogBase</span>
<a name="line-391"></a><span class='hs-definition'>logBase</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>LogBaseF</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>e</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>e</span>
<a name="line-392"></a><span class='hs-definition'>logBase</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-393"></a>
<a name="line-394"></a>
<a name="line-395"></a><a name="LogBaseF"></a><span class='hs-comment'>-- | Version of LogBase which also outputs if the logarithm</span>
<a name="line-396"></a><a name="LogBaseF"></a><span class='hs-comment'>-- calculated was exact.</span>
<a name="line-397"></a><a name="LogBaseF"></a><span class='hs-comment'>-- f indicates if the resulting logarithm has no fractional part (i.e.</span>
<a name="line-398"></a><a name="LogBaseF"></a><span class='hs-comment'>-- tells if the result provided is exact)</span>
<a name="line-399"></a><a name="LogBaseF"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:>=:</span> <span class='hs-conid'>D2</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>e</span><span class='hs-layout'>,</span> <span class='hs-conid'>Bool</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span>
<a name="line-400"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>LogBaseF</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>e</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>e</span> <span class='hs-varid'>f</span>
<a name="line-401"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Trich</span> <span class='hs-varid'>x</span> <span class='hs-varid'>b</span> <span class='hs-varid'>cmp</span><span class='hs-layout'>,</span> <span class='hs-conid'>LogBaseF'</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>e</span> <span class='hs-varid'>f</span> <span class='hs-varid'>cmp</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>LogBaseF</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>e</span> <span class='hs-varid'>f</span>
<a name="line-402"></a>
<a name="line-403"></a><a name="LogBaseF'"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:>=:</span> <span class='hs-conid'>D2</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>e</span><span class='hs-layout'>,</span> <span class='hs-conid'>Bool</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span>
<a name="line-404"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>LogBaseF'</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>e</span> <span class='hs-varid'>f</span> <span class='hs-varid'>cmp</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>cmp</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>e</span> <span class='hs-varid'>f</span>
<a name="line-405"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:>=:</span> <span class='hs-conid'>D2</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>LogBaseF'</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>False</span> <span class='hs-conid'>LT</span>
<a name="line-406"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:>=:</span> <span class='hs-conid'>D2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>LogBaseF'</span> <span class='hs-varid'>b</span> <span class='hs-varid'>b</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>True</span> <span class='hs-conid'>EQ</span>
<a name="line-407"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:>=:</span> <span class='hs-conid'>D2</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>DivMod</span> <span class='hs-varid'>x</span> <span class='hs-varid'>b</span> <span class='hs-varid'>q</span> <span class='hs-varid'>r</span><span class='hs-layout'>,</span> <span class='hs-conid'>IsZero</span> <span class='hs-varid'>r</span> <span class='hs-varid'>rz</span><span class='hs-layout'>,</span> <span class='hs-conid'>And</span> <span class='hs-varid'>rz</span> <span class='hs-varid'>f'</span> <span class='hs-varid'>f</span><span class='hs-layout'>,</span>
<a name="line-408"></a> <span class='hs-conid'>Pred</span> <span class='hs-varid'>e</span> <span class='hs-varid'>e'</span><span class='hs-layout'>,</span> <span class='hs-conid'>LogBaseF</span> <span class='hs-varid'>b</span> <span class='hs-varid'>q</span> <span class='hs-varid'>e'</span> <span class='hs-varid'>f'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>LogBaseF'</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>e</span> <span class='hs-varid'>f</span> <span class='hs-conid'>GT</span>
<a name="line-409"></a>
<a name="line-410"></a><a name="logBaseF"></a><span class='hs-comment'>-- | value-level reflection function for LogBaseF</span>
<a name="line-411"></a><span class='hs-definition'>logBaseF</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>LogBaseF</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>e</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b</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'>e</span><span class='hs-layout'>,</span><span class='hs-varid'>f</span><span class='hs-layout'>)</span>
<a name="line-412"></a><span class='hs-definition'>logBaseF</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span><span class='hs-layout'>,</span> <span class='hs-varid'>undefined</span><span class='hs-layout'>)</span>
<a name="line-413"></a>
<a name="line-414"></a>
<a name="line-415"></a><span class='hs-comment'>-- We could reuse LogBaseF for IsPowOf but it would be inneficient.</span>
<a name="line-416"></a><span class='hs-comment'>-- LogBaseF continues calculating the logarithm even if after knowing its</span>
<a name="line-417"></a><span class='hs-comment'>-- not exact. Thus, it is desirable to include a custom definition of</span>
<a name="line-418"></a><span class='hs-comment'>-- IsPowOf which can "abort" the calculation forcing the Divisions to be</span>
<a name="line-419"></a><span class='hs-comment'>-- exact</span>
<a name="line-420"></a>
<a name="line-421"></a>
<a name="line-422"></a><a name="IsPowOf"></a><span class='hs-comment'>-- | Assert that a number (@x@) can be expressed as the power of another one</span>
<a name="line-423"></a><a name="IsPowOf"></a><span class='hs-comment'>-- (@b@) (i.e. the fractional part of @log_base_b x = 0@, or, </span>
<a name="line-424"></a><a name="IsPowOf"></a><span class='hs-comment'>-- in a different way, @exists y . b\^y = x@). </span>
<a name="line-425"></a><a name="IsPowOf"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:>=:</span> <span class='hs-conid'>D2</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>IsPowOf</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span>
<a name="line-426"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Trich</span> <span class='hs-varid'>x</span> <span class='hs-varid'>b</span> <span class='hs-varid'>cmp</span><span class='hs-layout'>,</span> <span class='hs-conid'>IsPowOf'</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>cmp</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>IsPowOf</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span>
<a name="line-427"></a>
<a name="line-428"></a><a name="IsPowOf'"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:>=:</span> <span class='hs-conid'>D2</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>IsPowOf'</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>cmp</span>
<a name="line-429"></a><span class='hs-comment'>-- If lower (x < b), then the logarithm is not exact </span>
<a name="line-430"></a><span class='hs-comment'>-- instance (Pos b, b :>=: D2, Pos x) => IsPowOf' b x LT</span>
<a name="line-431"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:>=:</span> <span class='hs-conid'>D2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>IsPowOf'</span> <span class='hs-varid'>b</span> <span class='hs-varid'>b</span> <span class='hs-conid'>EQ</span>
<a name="line-432"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span> <span class='hs-conop'>:>=:</span> <span class='hs-conid'>D2</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>DivMod</span> <span class='hs-varid'>x</span> <span class='hs-varid'>b</span> <span class='hs-varid'>q</span> <span class='hs-conid'>D0</span><span class='hs-layout'>,</span> <span class='hs-conid'>IsPowOf</span> <span class='hs-varid'>b</span> <span class='hs-varid'>q</span><span class='hs-layout'>)</span>
<a name="line-433"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>IsPowOf'</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-conid'>GT</span>
<a name="line-434"></a><a name="isPowOf"></a><span class='hs-comment'>-- | </span>
<a name="line-435"></a><span class='hs-definition'>isPowOf</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>IsPowOf</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>()</span>
<a name="line-436"></a><span class='hs-definition'>isPowOf</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-437"></a>
<a name="line-438"></a><span class='hs-comment'>-----------------------------------</span>
<a name="line-439"></a><span class='hs-comment'>-- Base-10 Exponentiation/Logarithm</span>
<a name="line-440"></a><span class='hs-comment'>-----------------------------------</span>
<a name="line-441"></a>
<a name="line-442"></a><a name="Exp10"></a><span class='hs-comment'>-- | Base-10 Exponentiation type-level relation</span>
<a name="line-443"></a><a name="Exp10"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Exp10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span>
<a name="line-444"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Exp10</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D1</span>
<a name="line-445"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Exp10</span> <span class='hs-conid'>D1</span> <span class='hs-layout'>(</span><span class='hs-conid'>D1</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-446"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Exp10</span> <span class='hs-conid'>D2</span> <span class='hs-layout'>(</span><span class='hs-conid'>D1</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-447"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Exp10</span> <span class='hs-conid'>D3</span> <span class='hs-layout'>(</span><span class='hs-conid'>D1</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-448"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Exp10</span> <span class='hs-conid'>D4</span> <span class='hs-layout'>(</span><span class='hs-conid'>D1</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-449"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Exp10</span> <span class='hs-conid'>D5</span> <span class='hs-layout'>(</span><span class='hs-conid'>D1</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-450"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Exp10</span> <span class='hs-conid'>D6</span> <span class='hs-layout'>(</span><span class='hs-conid'>D1</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-451"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Exp10</span> <span class='hs-conid'>D7</span> <span class='hs-layout'>(</span><span class='hs-conid'>D1</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-452"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Exp10</span> <span class='hs-conid'>D8</span> <span class='hs-layout'>(</span><span class='hs-conid'>D1</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-453"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Exp10</span> <span class='hs-conid'>D9</span> <span class='hs-layout'>(</span><span class='hs-conid'>D1</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-454"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pred</span> <span class='hs-layout'>(</span><span class='hs-varid'>xi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>xl</span><span class='hs-layout'>)</span> <span class='hs-varid'>x'</span><span class='hs-layout'>,</span>
<a name="line-455"></a> <span class='hs-conid'>Exp10</span> <span class='hs-varid'>x'</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-456"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Exp10</span> <span class='hs-layout'>(</span><span class='hs-varid'>xi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>xl</span><span class='hs-layout'>)</span>
<a name="line-457"></a> <span class='hs-layout'>(</span><span class='hs-varid'>y</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-458"></a>
<a name="line-459"></a><a name="exp10"></a><span class='hs-comment'>-- | value-level reflection function for Exp10</span>
<a name="line-460"></a><span class='hs-definition'>exp10</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Exp10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span>
<a name="line-461"></a><span class='hs-definition'>exp10</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-462"></a>
<a name="line-463"></a><a name="Log10"></a><span class='hs-comment'>-- | Base-10 logarithm type-level relation</span>
<a name="line-464"></a><a name="Log10"></a><span class='hs-comment'>-- Note it is not relational (cannot be used to express Exponentation to 10)</span>
<a name="line-465"></a><a name="Log10"></a><span class='hs-comment'>-- However, it works with any positive numeral (not just powers of 10)</span>
<a name="line-466"></a><a name="Log10"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Log10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span>
<a name="line-467"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Log10</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D0</span>
<a name="line-468"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Log10</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D0</span>
<a name="line-469"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Log10</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D0</span>
<a name="line-470"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Log10</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D0</span>
<a name="line-471"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Log10</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D0</span>
<a name="line-472"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Log10</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D0</span>
<a name="line-473"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Log10</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D0</span>
<a name="line-474"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Log10</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D0</span>
<a name="line-475"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Log10</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D0</span>
<a name="line-476"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>xi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>xl</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pred</span> <span class='hs-varid'>y</span> <span class='hs-varid'>y'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Log10</span> <span class='hs-varid'>xi</span> <span class='hs-varid'>y'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Log10</span> <span class='hs-layout'>(</span><span class='hs-varid'>xi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>xl</span><span class='hs-layout'>)</span> <span class='hs-varid'>y</span>
<a name="line-477"></a>
<a name="line-478"></a><a name="log10"></a><span class='hs-comment'>-- | value-level reflection function for 'Log10'</span>
<a name="line-479"></a><span class='hs-definition'>log10</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Log10</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span>
<a name="line-480"></a><span class='hs-definition'>log10</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-481"></a>
<a name="line-482"></a><span class='hs-comment'>{- Log10': Alternative implementation of Log10
<a name="line-483"></a>
<a name="line-484"></a>Relational, but it only works for results of Exp10 (i.e. powers of 10).
<a name="line-485"></a>
<a name="line-486"></a>class (Pos x, Nat y) => Log10' x y | x -> y, y -> x
<a name="line-487"></a>instance Exp10 x y => Log10' y x
<a name="line-488"></a>-}</span>
<a name="line-489"></a>
<a name="line-490"></a>
<a name="line-491"></a><span class='hs-comment'>-------------</span>
<a name="line-492"></a><span class='hs-comment'>-- Comparison</span>
<a name="line-493"></a><span class='hs-comment'>-------------</span>
<a name="line-494"></a>
<a name="line-495"></a><span class='hs-comment'>-- type-level values denoting comparison results</span>
<a name="line-496"></a>
<a name="line-497"></a><a name="Trich"></a><span class='hs-comment'>-- | Lower than </span>
<a name="line-498"></a><a name="Trich"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>LT</span>
<a name="line-499"></a><a name="Trich"></a><span class='hs-comment'>-- | Equal</span>
<a name="line-500"></a><a name="Trich"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>EQ</span>
<a name="line-501"></a><a name="Trich"></a><span class='hs-comment'>-- | Greater than</span>
<a name="line-502"></a><a name="Trich"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>GT</span>
<a name="line-503"></a>
<a name="line-504"></a><a name="Trich"></a><span class='hs-comment'>-- | Trichotomy type-level relation. 'Trich x y r' establishes</span>
<a name="line-505"></a><a name="Trich"></a><span class='hs-comment'>-- the relation (@r@) between @x@ and @y@. The obtained relation (@r@)</span>
<a name="line-506"></a><a name="Trich"></a><span class='hs-comment'>-- Can be 'LT' (if @x@ is lower than @y@), 'EQ' (if @x@ equals @y@) or</span>
<a name="line-507"></a><a name="Trich"></a><span class='hs-comment'>-- 'GT' (if @x@ is greater than @y@)</span>
<a name="line-508"></a><a name="Trich"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span>
<a name="line-509"></a>
<a name="line-510"></a><a name="trich"></a><span class='hs-comment'>-- | value-level reflection function for the comparison type-level assertion </span>
<a name="line-511"></a><span class='hs-definition'>trich</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Trich</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span>
<a name="line-512"></a><span class='hs-definition'>trich</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-513"></a>
<a name="line-514"></a><span class='hs-comment'>-- by structural induction on the first, and then the second argument</span>
<a name="line-515"></a><span class='hs-comment'>-- D0</span>
<a name="line-516"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>EQ</span>
<a name="line-517"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>LT</span>
<a name="line-518"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>LT</span>
<a name="line-519"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>LT</span>
<a name="line-520"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>LT</span>
<a name="line-521"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>LT</span>
<a name="line-522"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>LT</span>
<a name="line-523"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>LT</span>
<a name="line-524"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>LT</span>
<a name="line-525"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>LT</span>
<a name="line-526"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D0</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>LT</span>
<a name="line-527"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>GT</span>
<a name="line-528"></a><span class='hs-comment'>-- D1</span>
<a name="line-529"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>GT</span>
<a name="line-530"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>EQ</span>
<a name="line-531"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>LT</span>
<a name="line-532"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>LT</span>
<a name="line-533"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>LT</span>
<a name="line-534"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>LT</span>
<a name="line-535"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>LT</span>
<a name="line-536"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>LT</span>
<a name="line-537"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>LT</span>
<a name="line-538"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>LT</span>
<a name="line-539"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D1</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>LT</span>
<a name="line-540"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>GT</span>
<a name="line-541"></a><span class='hs-comment'>-- D2</span>
<a name="line-542"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>GT</span>
<a name="line-543"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>GT</span>
<a name="line-544"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>EQ</span>
<a name="line-545"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>LT</span>
<a name="line-546"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>LT</span>
<a name="line-547"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>LT</span>
<a name="line-548"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>LT</span>
<a name="line-549"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>LT</span>
<a name="line-550"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>LT</span>
<a name="line-551"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>LT</span>
<a name="line-552"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D2</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>LT</span>
<a name="line-553"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>GT</span>
<a name="line-554"></a><span class='hs-comment'>-- D3</span>
<a name="line-555"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>GT</span>
<a name="line-556"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>GT</span>
<a name="line-557"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>GT</span>
<a name="line-558"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>EQ</span>
<a name="line-559"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>LT</span>
<a name="line-560"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>LT</span>
<a name="line-561"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>LT</span>
<a name="line-562"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>LT</span>
<a name="line-563"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>LT</span>
<a name="line-564"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>LT</span>
<a name="line-565"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D3</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>LT</span>
<a name="line-566"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>GT</span>
<a name="line-567"></a><span class='hs-comment'>-- D4</span>
<a name="line-568"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>GT</span>
<a name="line-569"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>GT</span>
<a name="line-570"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>GT</span>
<a name="line-571"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>GT</span>
<a name="line-572"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>EQ</span>
<a name="line-573"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>LT</span>
<a name="line-574"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>LT</span>
<a name="line-575"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>LT</span>
<a name="line-576"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>LT</span>
<a name="line-577"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>LT</span>
<a name="line-578"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D4</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>LT</span>
<a name="line-579"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>GT</span>
<a name="line-580"></a><span class='hs-comment'>-- D5</span>
<a name="line-581"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>GT</span>
<a name="line-582"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>GT</span>
<a name="line-583"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>GT</span>
<a name="line-584"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>GT</span>
<a name="line-585"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>GT</span>
<a name="line-586"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>EQ</span>
<a name="line-587"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>LT</span>
<a name="line-588"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>LT</span>
<a name="line-589"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>LT</span>
<a name="line-590"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>LT</span>
<a name="line-591"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D5</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>LT</span>
<a name="line-592"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>GT</span>
<a name="line-593"></a><span class='hs-comment'>-- D6</span>
<a name="line-594"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>GT</span>
<a name="line-595"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>GT</span>
<a name="line-596"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>GT</span>
<a name="line-597"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>GT</span>
<a name="line-598"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>GT</span>
<a name="line-599"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>GT</span>
<a name="line-600"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>EQ</span>
<a name="line-601"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>LT</span>
<a name="line-602"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>LT</span>
<a name="line-603"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>LT</span>
<a name="line-604"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D6</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>LT</span>
<a name="line-605"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>GT</span>
<a name="line-606"></a><span class='hs-comment'>-- D7</span>
<a name="line-607"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>GT</span>
<a name="line-608"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>GT</span>
<a name="line-609"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>GT</span>
<a name="line-610"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>GT</span>
<a name="line-611"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>GT</span>
<a name="line-612"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>GT</span>
<a name="line-613"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>GT</span>
<a name="line-614"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>EQ</span>
<a name="line-615"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>LT</span>
<a name="line-616"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>LT</span>
<a name="line-617"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D7</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>LT</span>
<a name="line-618"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>GT</span>
<a name="line-619"></a><span class='hs-comment'>-- D8</span>
<a name="line-620"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>GT</span>
<a name="line-621"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>GT</span>
<a name="line-622"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>GT</span>
<a name="line-623"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>GT</span>
<a name="line-624"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>GT</span>
<a name="line-625"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>GT</span>
<a name="line-626"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>GT</span>
<a name="line-627"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>GT</span>
<a name="line-628"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>EQ</span>
<a name="line-629"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>LT</span>
<a name="line-630"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D8</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>LT</span>
<a name="line-631"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>GT</span>
<a name="line-632"></a><span class='hs-comment'>-- D9</span>
<a name="line-633"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>GT</span>
<a name="line-634"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>GT</span>
<a name="line-635"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>GT</span>
<a name="line-636"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>GT</span>
<a name="line-637"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>GT</span>
<a name="line-638"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>GT</span>
<a name="line-639"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>GT</span>
<a name="line-640"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>GT</span>
<a name="line-641"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>GT</span>
<a name="line-642"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>EQ</span>
<a name="line-643"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-conid'>D9</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>LT</span>
<a name="line-644"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>GT</span>
<a name="line-645"></a>
<a name="line-646"></a>
<a name="line-647"></a><span class='hs-comment'>-- multidigit comparison</span>
<a name="line-648"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>xi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>xl</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>Pos</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>Trich</span> <span class='hs-varid'>xl</span> <span class='hs-varid'>yl</span> <span class='hs-varid'>rl</span><span class='hs-layout'>,</span> <span class='hs-conid'>Trich</span> <span class='hs-varid'>xi</span> <span class='hs-varid'>yi</span> <span class='hs-varid'>ri</span><span class='hs-layout'>,</span>
<a name="line-649"></a> <span class='hs-conid'>CS</span> <span class='hs-varid'>ri</span> <span class='hs-varid'>rl</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trich</span> <span class='hs-layout'>(</span><span class='hs-varid'>xi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>xl</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>yi</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>yl</span><span class='hs-layout'>)</span> <span class='hs-varid'>r</span>
<a name="line-650"></a>
<a name="line-651"></a><span class='hs-comment'>-- strengthen the comparison relation</span>
<a name="line-652"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>CS</span> <span class='hs-varid'>r1</span> <span class='hs-varid'>r2</span> <span class='hs-varid'>r3</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>r1</span> <span class='hs-varid'>r2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r3</span>
<a name="line-653"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>CS</span> <span class='hs-conid'>EQ</span> <span class='hs-varid'>r</span> <span class='hs-varid'>r</span>
<a name="line-654"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>CS</span> <span class='hs-conid'>GT</span> <span class='hs-varid'>r</span> <span class='hs-conid'>GT</span>
<a name="line-655"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>CS</span> <span class='hs-conid'>LT</span> <span class='hs-varid'>r</span> <span class='hs-conid'>LT</span>
<a name="line-656"></a>
<a name="line-657"></a><span class='hs-comment'>-- Abbreviated comparison assertions</span>
<a name="line-658"></a>
<a name="line-659"></a><span class='hs-comment'>-- | Equality abbreviated type-level assertion</span>
<a name="line-660"></a><span class='hs-keyword'>class</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:==:</span> <span class='hs-varid'>y</span>
<a name="line-661"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Trich</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-conid'>EQ</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-conop'>:==:</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-comment'>-- ??? x :==: y fires an error </span>
<a name="line-662"></a> <span class='hs-comment'>-- with ghc 6.8.2 </span>
<a name="line-663"></a>
<a name="line-664"></a><a name="=="></a><span class='hs-comment'>-- | value-level reflection function for the equality abbreviated </span>
<a name="line-665"></a><span class='hs-comment'>-- type-level assertion </span>
<a name="line-666"></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'>x</span> <span class='hs-conop'>:==:</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>()</span>
<a name="line-667"></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-668"></a>
<a name="line-669"></a><span class='hs-comment'>-- | Greater-than abbreviated type-level assertion</span>
<a name="line-670"></a><span class='hs-keyword'>class</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:>:</span> <span class='hs-varid'>y</span>
<a name="line-671"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Trich</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-conid'>GT</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-conop'>:>:</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span>
<a name="line-672"></a>
<a name="line-673"></a><a name=">"></a><span class='hs-comment'>-- | value-level reflection function for the equality abbreviated </span>
<a name="line-674"></a><span class='hs-comment'>-- type-level assertion </span>
<a name="line-675"></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'>x</span> <span class='hs-conop'>:>:</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>()</span>
<a name="line-676"></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-677"></a>
<a name="line-678"></a><span class='hs-comment'>-- | Lower-than abbreviated type-level assertion</span>
<a name="line-679"></a><span class='hs-keyword'>class</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:<:</span> <span class='hs-varid'>y</span>
<a name="line-680"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Trich</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-conid'>LT</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-conop'>:<:</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span>
<a name="line-681"></a>
<a name="line-682"></a><a name="<"></a><span class='hs-comment'>-- | value-level reflection function for the lower-than abbreviated </span>
<a name="line-683"></a><span class='hs-comment'>-- type-level assertion </span>
<a name="line-684"></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'>x</span> <span class='hs-conop'>:<:</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>()</span>
<a name="line-685"></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-686"></a>
<a name="line-687"></a><span class='hs-comment'>-- | Greater-than or equal abbreviated type-level assertion</span>
<a name="line-688"></a><span class='hs-keyword'>class</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:>=:</span> <span class='hs-varid'>y</span>
<a name="line-689"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Succ</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x'</span><span class='hs-layout'>,</span> <span class='hs-conid'>Trich</span> <span class='hs-varid'>x'</span> <span class='hs-varid'>y</span> <span class='hs-conid'>GT</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-conop'>:>=:</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span>
<a name="line-690"></a>
<a name="line-691"></a><a name=">="></a><span class='hs-comment'>-- | value-level reflection function for the greater-than or equal abbreviated </span>
<a name="line-692"></a><span class='hs-comment'>-- type-level assertion </span>
<a name="line-693"></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'>x</span> <span class='hs-conop'>:>=:</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>()</span>
<a name="line-694"></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-695"></a>
<a name="line-696"></a><span class='hs-comment'>-- | Lower-than or equal abbreviated type-level assertion</span>
<a name="line-697"></a><span class='hs-keyword'>class</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:<=:</span> <span class='hs-varid'>y</span>
<a name="line-698"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Succ</span> <span class='hs-varid'>x'</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Trich</span> <span class='hs-varid'>x'</span> <span class='hs-varid'>y</span> <span class='hs-conid'>LT</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-conop'>:<=:</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span>
<a name="line-699"></a>
<a name="line-700"></a><a name="<="></a><span class='hs-comment'>-- | value-level reflection function for the lower-than or equal abbreviated </span>
<a name="line-701"></a><span class='hs-comment'>-- type-level assertion </span>
<a name="line-702"></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'>x</span> <span class='hs-conop'>:<=:</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>()</span>
<a name="line-703"></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-704"></a><span class='hs-keyword'>infix</span> <span class='hs-num'>4</span> <span class='hs-varop'><</span><span class='hs-layout'>,</span><span class='hs-varop'><=</span><span class='hs-layout'>,</span><span class='hs-varop'>>=</span><span class='hs-layout'>,</span><span class='hs-varop'>></span><span class='hs-layout'>,</span><span class='hs-varop'>==</span>
<a name="line-705"></a><span class='hs-comment'>------------------</span>
<a name="line-706"></a><span class='hs-comment'>-- Maximum/Minimum</span>
<a name="line-707"></a><span class='hs-comment'>------------------</span>
<a name="line-708"></a>
<a name="line-709"></a><a name="Max"></a><span class='hs-comment'>-- Choose the largest of x and y in the order b</span>
<a name="line-710"></a><a name="Max"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>Max'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>b</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span>
<a name="line-711"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Max'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-conid'>LT</span> <span class='hs-varid'>y</span>
<a name="line-712"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Max'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-conid'>EQ</span> <span class='hs-varid'>y</span>
<a name="line-713"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Max'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-conid'>GT</span> <span class='hs-varid'>x</span>
<a name="line-714"></a>
<a name="line-715"></a><a name="Max"></a><span class='hs-comment'>-- | Maximum type-level relation</span>
<a name="line-716"></a><a name="Max"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>Max</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span>
<a name="line-717"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Max'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>b</span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span> <span class='hs-conid'>Trich</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Max</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span>
<a name="line-718"></a>
<a name="line-719"></a><a name="max"></a><span class='hs-comment'>-- | value-level reflection function for the maximum type-level relation</span>
<a name="line-720"></a><span class='hs-definition'>max</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Max</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span>
<a name="line-721"></a><span class='hs-definition'>max</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-722"></a>
<a name="line-723"></a><a name="Min"></a><span class='hs-comment'>-- | Minimum type-level relation</span>
<a name="line-724"></a><a name="Min"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>Min</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span>
<a name="line-725"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Max'</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span> <span class='hs-varid'>b</span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span> <span class='hs-conid'>Trich</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Min</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span>
<a name="line-726"></a>
<a name="line-727"></a>
<a name="line-728"></a><a name="min"></a><span class='hs-comment'>-- | value-level reflection function for the minimum type-level relation</span>
<a name="line-729"></a><span class='hs-definition'>min</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Min</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span>
<a name="line-730"></a><span class='hs-definition'>min</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-731"></a>
<a name="line-732"></a><span class='hs-comment'>-------</span>
<a name="line-733"></a><span class='hs-comment'>-- GCD</span>
<a name="line-734"></a><span class='hs-comment'>-------</span>
<a name="line-735"></a>
<a name="line-736"></a><a name="GCD"></a><span class='hs-comment'>-- | Greatest Common Divisor type-level relation</span>
<a name="line-737"></a><a name="GCD"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>gcd</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>GCD</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>gcd</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>gcd</span>
<a name="line-738"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-conid'>Trich</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>cmp</span><span class='hs-layout'>,</span> <span class='hs-conid'>IsZero</span> <span class='hs-varid'>y</span> <span class='hs-varid'>yz</span><span class='hs-layout'>,</span> <span class='hs-conid'>GCD'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>yz</span> <span class='hs-varid'>cmp</span> <span class='hs-varid'>gcd</span><span class='hs-layout'>)</span>
<a name="line-739"></a> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>GCD</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>gcd</span>
<a name="line-740"></a>
<a name="line-741"></a><a name="GCD'"></a><span class='hs-comment'>-- Euclidean algorithm </span>
<a name="line-742"></a><a name="GCD'"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>gcd</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>GCD'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>yz</span> <span class='hs-varid'>cmp</span> <span class='hs-varid'>gcd</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>yz</span> <span class='hs-varid'>cmp</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>gcd</span>
<a name="line-743"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>GCD'</span> <span class='hs-varid'>x</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>True</span> <span class='hs-varid'>cmp</span> <span class='hs-conid'>D0</span>
<a name="line-744"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-conid'>GCD</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x</span> <span class='hs-varid'>gcd</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>GCD'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-conid'>False</span> <span class='hs-conid'>LT</span> <span class='hs-varid'>gcd</span>
<a name="line-745"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>GCD'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>x</span> <span class='hs-conid'>False</span> <span class='hs-conid'>EQ</span> <span class='hs-varid'>x</span>
<a name="line-746"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-conid'>Sub</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>x'</span><span class='hs-layout'>,</span> <span class='hs-conid'>GCD</span> <span class='hs-varid'>x'</span> <span class='hs-varid'>y</span> <span class='hs-varid'>gcd</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>GCD'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-conid'>False</span> <span class='hs-conid'>GT</span> <span class='hs-varid'>gcd</span>
<a name="line-747"></a>
<a name="line-748"></a><a name="gcd"></a><span class='hs-comment'>-- | value-level reflection function for the GCD type-level relation</span>
<a name="line-749"></a><span class='hs-definition'>gcd</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>GCD</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>z</span>
<a name="line-750"></a><span class='hs-definition'>gcd</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
<a name="line-751"></a>
<a name="line-752"></a><span class='hs-comment'>---------------------</span>
<a name="line-753"></a><span class='hs-comment'>-- Internal functions</span>
<a name="line-754"></a><span class='hs-comment'>---------------------</span>
<a name="line-755"></a>
<a name="line-756"></a><a name="IsZero"></a><span class='hs-comment'>-- classify a natural as positive or zero</span>
<a name="line-757"></a><a name="IsZero"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>IsZero</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span>
<a name="line-758"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>IsZero</span> <span class='hs-conid'>D0</span> <span class='hs-conid'>True</span>
<a name="line-759"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>IsZero</span> <span class='hs-conid'>D1</span> <span class='hs-conid'>False</span>
<a name="line-760"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>IsZero</span> <span class='hs-conid'>D2</span> <span class='hs-conid'>False</span>
<a name="line-761"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>IsZero</span> <span class='hs-conid'>D3</span> <span class='hs-conid'>False</span>
<a name="line-762"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>IsZero</span> <span class='hs-conid'>D4</span> <span class='hs-conid'>False</span>
<a name="line-763"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>IsZero</span> <span class='hs-conid'>D5</span> <span class='hs-conid'>False</span>
<a name="line-764"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>IsZero</span> <span class='hs-conid'>D6</span> <span class='hs-conid'>False</span>
<a name="line-765"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>IsZero</span> <span class='hs-conid'>D7</span> <span class='hs-conid'>False</span>
<a name="line-766"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>IsZero</span> <span class='hs-conid'>D8</span> <span class='hs-conid'>False</span>
<a name="line-767"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>IsZero</span> <span class='hs-conid'>D9</span> <span class='hs-conid'>False</span>
<a name="line-768"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>IsZero</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-conid'>False</span>
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