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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE TypeOperators, FlexibleInstances, FlexibleContexts,
<a name="line-2"></a> UndecidableInstances, ScopedTypeVariables,
<a name="line-3"></a> Rank2Types #-}</span>
<a name="line-4"></a><span class='hs-comment'>{-# OPTIONS_GHC -fno-warn-incomplete-patterns -fno-warn-name-shadowing #-}</span>
<a name="line-5"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span>
<a name="line-6"></a><span class='hs-comment'>-- |</span>
<a name="line-7"></a><span class='hs-comment'>-- Module : Data.TypeLevel.Num.Sets</span>
<a name="line-8"></a><span class='hs-comment'>-- Copyright : (c) 2008 Alfonso Acosta, Oleg Kiselyov, Wolfgang Jeltsch</span>
<a name="line-9"></a><span class='hs-comment'>-- and KTH's SAM group </span>
<a name="line-10"></a><span class='hs-comment'>-- License : BSD-style (see the file LICENSE)</span>
<a name="line-11"></a><span class='hs-comment'>-- </span>
<a name="line-12"></a><span class='hs-comment'>-- Maintainer : alfonso.acosta@gmail.com</span>
<a name="line-13"></a><span class='hs-comment'>-- Stability : experimental</span>
<a name="line-14"></a><span class='hs-comment'>-- Portability : non-portable (non-standard instances)</span>
<a name="line-15"></a><span class='hs-comment'>--</span>
<a name="line-16"></a><span class='hs-comment'>-- Type-level numerical sets. Currently there is only support for Naturals and </span>
<a name="line-17"></a><span class='hs-comment'>-- Positives.</span>
<a name="line-18"></a><span class='hs-comment'>-- </span>
<a name="line-19"></a><span class='hs-comment'>----------------------------------------------------------------------------</span>
<a name="line-20"></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'>Sets</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pos</span><span class='hs-layout'>,</span> <span class='hs-conid'>Nat</span><span class='hs-layout'>,</span> <span class='hs-varid'>toNum</span><span class='hs-layout'>,</span> <span class='hs-varid'>toInt</span><span class='hs-layout'>,</span> <span class='hs-varid'>reifyIntegral</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-21"></a>
<a name="line-22"></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-23"></a>
<a name="line-24"></a><span class='hs-comment'>-----------</span>
<a name="line-25"></a><span class='hs-comment'>-- Naturals</span>
<a name="line-26"></a><span class='hs-comment'>-----------</span>
<a name="line-27"></a>
<a name="line-28"></a>
<a name="line-29"></a><span class='hs-comment'>-- The well-formedness condition, the kind predicate.</span>
<a name="line-30"></a><span class='hs-comment'>-- These classes are internal, denoted by the ending "I", which is removed in </span>
<a name="line-31"></a><span class='hs-comment'>-- the exported proxies (read below)</span>
<a name="line-32"></a>
<a name="line-33"></a><span class='hs-comment'>-- | Naturals (Positives and zero), internal version</span>
<a name="line-34"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>NatI</span> <span class='hs-varid'>n</span> <span class='hs-keyword'>where</span>
<a name="line-35"></a> <span class='hs-comment'>-- | Reflecting function</span>
<a name="line-36"></a> <span class='hs-varid'>toNum</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-37"></a>
<a name="line-38"></a>
<a name="line-39"></a><a name="toInt"></a><span class='hs-comment'>-- | Less generic reflecting function (Int)</span>
<a name="line-40"></a><span class='hs-definition'>toInt</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span>
<a name="line-41"></a><span class='hs-definition'>toInt</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toNum</span>
<a name="line-42"></a>
<a name="line-43"></a>
<a name="line-44"></a>
<a name="line-45"></a><a name="PosI"></a><span class='hs-comment'>-- | Positives (Naturals without zero), internal version</span>
<a name="line-46"></a><a name="PosI"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>NatI</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>n</span>
<a name="line-47"></a>
<a name="line-48"></a><span class='hs-comment'>-- To prevent the user from adding new instances to NatI and especially</span>
<a name="line-49"></a><span class='hs-comment'>-- to PosI (e.g., to prevent the user from adding the instance |Pos D0|)</span>
<a name="line-50"></a><span class='hs-comment'>-- we do NOT export NatI and PosI. Rather, we export the following proxies.</span>
<a name="line-51"></a><span class='hs-comment'>-- The proxies entail PosI and NatI and so can be used to add PosI and NatI</span>
<a name="line-52"></a><span class='hs-comment'>-- constraints in the signatures. However, all the constraints below</span>
<a name="line-53"></a><span class='hs-comment'>-- are expressed in terms of NatI and PosI rather than proxies. Thus,</span>
<a name="line-54"></a><span class='hs-comment'>-- even if the user adds new instances to proxies, it would not matter.</span>
<a name="line-55"></a><span class='hs-comment'>-- Besides, because the following proxy instances are most general,</span>
<a name="line-56"></a><span class='hs-comment'>-- one may not add further instances without overlapping instance extension.</span>
<a name="line-57"></a>
<a name="line-58"></a><a name="Nat"></a><span class='hs-comment'>-- | Naturals (Positives and zero)</span>
<a name="line-59"></a><a name="Nat"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>NatI</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>n</span>
<a name="line-60"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>NatI</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>n</span>
<a name="line-61"></a>
<a name="line-62"></a><a name="Pos"></a><span class='hs-comment'>-- | Positives (Naturals without zero)</span>
<a name="line-63"></a><a name="Pos"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>n</span>
<a name="line-64"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>n</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'>-- Natural Instances</span>
<a name="line-68"></a><span class='hs-comment'>--------------------</span>
<a name="line-69"></a>
<a name="line-70"></a><span class='hs-comment'>-- Note: TH would be helpful to sistematically define instances </span>
<a name="line-71"></a><span class='hs-comment'>-- (our type level operations)</span>
<a name="line-72"></a><span class='hs-comment'>-- However, type-splicing is not yet implemented in GHC :S</span>
<a name="line-73"></a>
<a name="line-74"></a><span class='hs-comment'>-- monodigit naturals</span>
<a name="line-75"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>NatI</span> <span class='hs-conid'>D0</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>0</span>
<a name="line-76"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>NatI</span> <span class='hs-conid'>D1</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>1</span>
<a name="line-77"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>NatI</span> <span class='hs-conid'>D2</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>2</span>
<a name="line-78"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>NatI</span> <span class='hs-conid'>D3</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>3</span>
<a name="line-79"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>NatI</span> <span class='hs-conid'>D4</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>4</span>
<a name="line-80"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>NatI</span> <span class='hs-conid'>D5</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>5</span>
<a name="line-81"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>NatI</span> <span class='hs-conid'>D6</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>6</span>
<a name="line-82"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>NatI</span> <span class='hs-conid'>D7</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>7</span>
<a name="line-83"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>NatI</span> <span class='hs-conid'>D8</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>8</span>
<a name="line-84"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>NatI</span> <span class='hs-conid'>D9</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>9</span>
<a name="line-85"></a>
<a name="line-86"></a><span class='hs-comment'>-- multidigit naturals</span>
<a name="line-87"></a><span class='hs-comment'>-- Note: The PosI constraint guarantees that all valid representations are </span>
<a name="line-88"></a><span class='hs-comment'>-- normalized (i.e. D0 :* D1 will lead to a compiler error)</span>
<a name="line-89"></a><span class='hs-comment'>-- Note as well that ill-formed representations such as</span>
<a name="line-90"></a><span class='hs-comment'>-- (D1 :* D2) :* (D3 :* D4) are not recognized as instances of</span>
<a name="line-91"></a><span class='hs-comment'>-- naturals nor positives.</span>
<a name="line-92"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>NatI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>subLastDec</span> <span class='hs-varid'>n</span>
<a name="line-93"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>NatI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D1</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>subLastDec</span> <span class='hs-varid'>n</span> <span class='hs-varop'>+</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>1</span>
<a name="line-94"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>NatI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D2</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>subLastDec</span> <span class='hs-varid'>n</span> <span class='hs-varop'>+</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>2</span>
<a name="line-95"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>NatI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D3</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>subLastDec</span> <span class='hs-varid'>n</span> <span class='hs-varop'>+</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>3</span>
<a name="line-96"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>NatI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D4</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>subLastDec</span> <span class='hs-varid'>n</span> <span class='hs-varop'>+</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>4</span>
<a name="line-97"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>NatI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D5</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>subLastDec</span> <span class='hs-varid'>n</span> <span class='hs-varop'>+</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>5</span>
<a name="line-98"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>NatI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D6</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>subLastDec</span> <span class='hs-varid'>n</span> <span class='hs-varop'>+</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>6</span>
<a name="line-99"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>NatI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D7</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>subLastDec</span> <span class='hs-varid'>n</span> <span class='hs-varop'>+</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>7</span>
<a name="line-100"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>NatI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D8</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>subLastDec</span> <span class='hs-varid'>n</span> <span class='hs-varop'>+</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>8</span>
<a name="line-101"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>NatI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D9</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>toNum</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>subLastDec</span> <span class='hs-varid'>n</span> <span class='hs-varop'>+</span> <span class='hs-varid'>fromInteger</span> <span class='hs-num'>9</span>
<a name="line-102"></a>
<a name="line-103"></a><span class='hs-comment'>-- monodigit positives</span>
<a name="line-104"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-conid'>D1</span>
<a name="line-105"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-conid'>D2</span>
<a name="line-106"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-conid'>D3</span>
<a name="line-107"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-conid'>D4</span>
<a name="line-108"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-conid'>D5</span>
<a name="line-109"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-conid'>D6</span>
<a name="line-110"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-conid'>D7</span>
<a name="line-111"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-conid'>D8</span>
<a name="line-112"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-conid'>D9</span>
<a name="line-113"></a>
<a name="line-114"></a><span class='hs-comment'>-- multidigit positives</span>
<a name="line-115"></a><span class='hs-comment'>-- Note: The PosI constraint guarantees that all valid representations are </span>
<a name="line-116"></a><span class='hs-comment'>-- normalized (i.e. D0 :* D1 will lead to a compiler error)</span>
<a name="line-117"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PosI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-118"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PosI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D1</span><span class='hs-layout'>)</span>
<a name="line-119"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PosI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D2</span><span class='hs-layout'>)</span>
<a name="line-120"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PosI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D3</span><span class='hs-layout'>)</span>
<a name="line-121"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PosI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D4</span><span class='hs-layout'>)</span>
<a name="line-122"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PosI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D5</span><span class='hs-layout'>)</span>
<a name="line-123"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PosI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D6</span><span class='hs-layout'>)</span>
<a name="line-124"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PosI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D7</span><span class='hs-layout'>)</span>
<a name="line-125"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PosI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D8</span><span class='hs-layout'>)</span>
<a name="line-126"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>PosI</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>PosI</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D9</span><span class='hs-layout'>)</span>
<a name="line-127"></a>
<a name="line-128"></a>
<a name="line-129"></a><a name="reifyIntegral"></a><span class='hs-comment'>-- | Reification function. In CPS style (best possible solution)</span>
<a name="line-130"></a><span class='hs-definition'>reifyIntegral</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Integral</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>n</span> <span class='hs-varop'>.</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span>
<a name="line-131"></a><span class='hs-definition'>reifyIntegral</span> <span class='hs-varid'>i</span> <span class='hs-varid'>f</span>
<a name="line-132"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-varop'><</span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"reifyIntegral: integral < 0"</span>
<a name="line-133"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-varop'>==</span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>D0</span><span class='hs-layout'>)</span>
<a name="line-134"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>i</span> <span class='hs-varid'>f</span>
<a name="line-135"></a> <span class='hs-comment'>-- reifyIntegral for positives</span>
<a name="line-136"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Integral</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>n</span> <span class='hs-varop'>.</span> <span class='hs-conid'>Pos</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span>
<a name="line-137"></a> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>i</span> <span class='hs-varid'>f</span>
<a name="line-138"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-varop'><</span> <span class='hs-num'>10</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>i</span> <span class='hs-keyword'>of</span>
<a name="line-139"></a> <span class='hs-num'>1</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>D1</span><span class='hs-layout'>)</span>
<a name="line-140"></a> <span class='hs-num'>2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>D2</span><span class='hs-layout'>)</span><span class='hs-layout'>;</span> <span class='hs-num'>3</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>D3</span><span class='hs-layout'>)</span>
<a name="line-141"></a> <span class='hs-num'>4</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>D4</span><span class='hs-layout'>)</span><span class='hs-layout'>;</span> <span class='hs-num'>5</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>D5</span><span class='hs-layout'>)</span>
<a name="line-142"></a> <span class='hs-num'>6</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>D6</span><span class='hs-layout'>)</span><span class='hs-layout'>;</span> <span class='hs-num'>7</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>D7</span><span class='hs-layout'>)</span>
<a name="line-143"></a> <span class='hs-num'>8</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>D8</span><span class='hs-layout'>)</span><span class='hs-layout'>;</span> <span class='hs-num'>9</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>D9</span><span class='hs-layout'>)</span>
<a name="line-144"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span>
<a name="line-145"></a> <span class='hs-keyword'>case</span> <span class='hs-varid'>m</span> <span class='hs-keyword'>of</span>
<a name="line-146"></a> <span class='hs-num'>0</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>::</span><span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>e</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-147"></a> <span class='hs-num'>1</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>::</span><span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>e</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D1</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-148"></a> <span class='hs-num'>2</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>::</span><span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>e</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D2</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-149"></a> <span class='hs-num'>3</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>::</span><span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>e</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D3</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-150"></a> <span class='hs-num'>4</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>::</span><span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>e</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D4</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-151"></a> <span class='hs-num'>5</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>::</span><span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>e</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D5</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-152"></a> <span class='hs-num'>6</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>::</span><span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>e</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D6</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-153"></a> <span class='hs-num'>7</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>::</span><span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>e</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D7</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-154"></a> <span class='hs-num'>8</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>::</span><span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>e</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D8</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-155"></a> <span class='hs-num'>9</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>reifyIntegralp</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>::</span><span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>undefined</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>e</span> <span class='hs-conop'>:*</span> <span class='hs-conid'>D9</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-156"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-varid'>d</span><span class='hs-layout'>,</span><span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>divMod</span> <span class='hs-varid'>i</span> <span class='hs-num'>10</span>
<a name="line-157"></a>
<a name="line-158"></a>
<a name="line-159"></a><span class='hs-comment'>---------------------</span>
<a name="line-160"></a><span class='hs-comment'>-- Internal functions</span>
<a name="line-161"></a><span class='hs-comment'>---------------------</span>
<a name="line-162"></a>
<a name="line-163"></a><a name="subLastDec"></a><span class='hs-comment'>-- substract the last digit of a decimal type-level numeral and obtain </span>
<a name="line-164"></a><span class='hs-comment'>-- the result's reflected value </span>
<a name="line-165"></a><span class='hs-comment'>{-# INLINE subLastDec #-}</span>
<a name="line-166"></a><span class='hs-definition'>subLastDec</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>NatI</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-layout'>,</span> <span class='hs-conid'>NatI</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>d</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-167"></a><span class='hs-definition'>subLastDec</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-num'>10</span><span class='hs-varop'>*</span><span class='hs-layout'>)</span><span class='hs-varop'>.</span><span class='hs-varid'>toNum</span><span class='hs-varop'>.</span><span class='hs-varid'>div10Dec</span>
<a name="line-168"></a>
<a name="line-169"></a><span class='hs-comment'>-- Divide a decimal type-level numeral by 10 </span>
<a name="line-170"></a><span class='hs-comment'>{-# INLINE div10Dec #-}</span>
<a name="line-171"></a><a name="div10Dec"></a><span class='hs-definition'>div10Dec</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>NatI</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-keyglyph'>=></span> <span class='hs-varid'>x</span> <span class='hs-conop'>:*</span> <span class='hs-varid'>d</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span>
<a name="line-172"></a><span class='hs-definition'>div10Dec</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>undefined</span>
</pre></body>
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