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<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html>
<head>
<!-- Generated by HsColour, http://code.haskell.org/~malcolm/hscolour/ -->
<title>src/Djinn/LJT.hs</title>
<link type='text/css' rel='stylesheet' href='hscolour.css' />
</head>
<body>
<pre><a name="line-1"></a><span class='hs-comment'>--</span>
<a name="line-2"></a><span class='hs-comment'>-- Copyright (c) 2005, 2008 Lennart Augustsson</span>
<a name="line-3"></a><span class='hs-comment'>-- See LICENSE for licensing details.</span>
<a name="line-4"></a><span class='hs-comment'>--</span>
<a name="line-5"></a><span class='hs-comment'>-- Intuitionistic theorem prover</span>
<a name="line-6"></a><span class='hs-comment'>-- Written by Roy Dyckhoff, Summer 1991</span>
<a name="line-7"></a><span class='hs-comment'>-- Modified to use the LWB syntax  Summer 1997</span>
<a name="line-8"></a><span class='hs-comment'>-- and simplified in various ways...</span>
<a name="line-9"></a><span class='hs-comment'>--</span>
<a name="line-10"></a><span class='hs-comment'>-- Translated to Haskell by Lennart Augustsson December 2005</span>
<a name="line-11"></a><span class='hs-comment'>--</span>
<a name="line-12"></a><span class='hs-comment'>-- Incorporates the Vorob'ev-Hudelmaier etc calculus (I call it LJT)</span>
<a name="line-13"></a><span class='hs-comment'>-- See RD's paper in JSL 1992:</span>
<a name="line-14"></a><span class='hs-comment'>-- "Contraction-free calculi for intuitionistic logic"</span>
<a name="line-15"></a><span class='hs-comment'>--</span>
<a name="line-16"></a><span class='hs-comment'>-- Torkel Franzen (at SICS) gave me good ideas about how to write this</span>
<a name="line-17"></a><span class='hs-comment'>-- properly, taking account of first-argument indexing,</span>
<a name="line-18"></a><span class='hs-comment'>-- and I learnt a trick or two from Neil Tennant's "Autologic" book.</span>
<a name="line-19"></a>
<a name="line-20"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Djinn</span><span class='hs-varop'>.</span><span class='hs-conid'>LJT</span> <span class='hs-layout'>(</span>
<a name="line-21"></a>  <span class='hs-keyword'>module</span> <span class='hs-conid'>Djinn</span><span class='hs-varop'>.</span><span class='hs-conid'>LJTFormula</span><span class='hs-layout'>,</span>
<a name="line-22"></a>  <span class='hs-varid'>provable</span><span class='hs-layout'>,</span>
<a name="line-23"></a>  <span class='hs-varid'>prove</span><span class='hs-layout'>,</span>
<a name="line-24"></a>  <span class='hs-conid'>Proof</span><span class='hs-layout'>,</span>
<a name="line-25"></a>  <span class='hs-conid'>MoreSolutions</span>
<a name="line-26"></a><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-27"></a>
<a name="line-28"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Applicative</span> <span class='hs-layout'>(</span><span class='hs-conid'>Applicative</span><span class='hs-layout'>,</span> <span class='hs-conid'>Alternative</span><span class='hs-layout'>,</span> <span class='hs-varid'>pure</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>&lt;*&gt;</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>empty</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>&lt;|&gt;</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-29"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span>
<a name="line-30"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>List</span> <span class='hs-layout'>(</span><span class='hs-varid'>partition</span><span class='hs-layout'>)</span>
<a name="line-31"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Debug</span><span class='hs-varop'>.</span><span class='hs-conid'>Trace</span>
<a name="line-32"></a>
<a name="line-33"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Djinn</span><span class='hs-varop'>.</span><span class='hs-conid'>LJTFormula</span>
<a name="line-34"></a>
<a name="line-35"></a><a name="mtrace"></a><span class='hs-definition'>mtrace</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>String</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>a</span>
<a name="line-36"></a><span class='hs-definition'>mtrace</span> <span class='hs-varid'>m</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>debug</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>trace</span> <span class='hs-varid'>m</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>x</span>
<a name="line-37"></a><a name="wrapM"></a><span class='hs-comment'>-- wrap :: (Show a, Show b) =&gt; String -&gt; a -&gt; b -&gt; b</span>
<a name="line-38"></a><span class='hs-comment'>-- wrap fun args ret = mtrace (fun ++ ": " ++ show args) $</span>
<a name="line-39"></a><span class='hs-comment'>--                     let o = show ret in seq o $</span>
<a name="line-40"></a><span class='hs-comment'>--                     mtrace (fun ++ " returns: " ++ o) ret</span>
<a name="line-41"></a><span class='hs-definition'>wrapM</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Show</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Show</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-conid'>Monad</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>String</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>m</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>m</span> <span class='hs-varid'>b</span>
<a name="line-42"></a><span class='hs-definition'>wrapM</span> <span class='hs-varid'>fun</span> <span class='hs-varid'>args</span> <span class='hs-varid'>mret</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-43"></a>  <span class='hs-conid'>()</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>mtrace</span> <span class='hs-layout'>(</span><span class='hs-varid'>fun</span> <span class='hs-varop'>++</span> <span class='hs-str'>": "</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>args</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>return</span> <span class='hs-conid'>()</span>
<a name="line-44"></a>  <span class='hs-varid'>ret</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>mret</span>
<a name="line-45"></a>  <span class='hs-conid'>()</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>mtrace</span> <span class='hs-layout'>(</span><span class='hs-varid'>fun</span> <span class='hs-varop'>++</span> <span class='hs-str'>" returns: "</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>ret</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>return</span> <span class='hs-conid'>()</span>
<a name="line-46"></a>  <span class='hs-varid'>return</span> <span class='hs-varid'>ret</span>
<a name="line-47"></a>
<a name="line-48"></a><a name="debug"></a><span class='hs-definition'>debug</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Bool</span>
<a name="line-49"></a><span class='hs-definition'>debug</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-50"></a>
<a name="line-51"></a><a name="MoreSolutions"></a><span class='hs-keyword'>type</span> <span class='hs-conid'>MoreSolutions</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Bool</span>
<a name="line-52"></a>
<a name="line-53"></a><a name="provable"></a><span class='hs-definition'>provable</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-54"></a><span class='hs-definition'>provable</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>not</span> <span class='hs-varop'>$</span> <span class='hs-varid'>null</span> <span class='hs-varop'>$</span> <span class='hs-varid'>prove</span> <span class='hs-conid'>False</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>a</span>
<a name="line-55"></a>
<a name="line-56"></a><a name="prove"></a><span class='hs-definition'>prove</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>MoreSolutions</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>Symbol</span><span class='hs-layout'>,</span> <span class='hs-conid'>Formula</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Proof</span><span class='hs-keyglyph'>]</span>
<a name="line-57"></a><span class='hs-definition'>prove</span> <span class='hs-varid'>more</span> <span class='hs-varid'>env</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>runP</span> <span class='hs-varop'>$</span> <span class='hs-varid'>redtop</span> <span class='hs-varid'>more</span> <span class='hs-varid'>env</span> <span class='hs-varid'>a</span>
<a name="line-58"></a>
<a name="line-59"></a><a name="redtop"></a><span class='hs-definition'>redtop</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>MoreSolutions</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>Symbol</span><span class='hs-layout'>,</span> <span class='hs-conid'>Formula</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-60"></a><span class='hs-definition'>redtop</span> <span class='hs-varid'>more</span> <span class='hs-varid'>ifs</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-61"></a>  <span class='hs-keyword'>let</span> <span class='hs-varid'>form</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>foldr</span> <span class='hs-layout'>(</span><span class='hs-conop'>:-&gt;</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span> <span class='hs-layout'>(</span><span class='hs-varid'>map</span> <span class='hs-varid'>snd</span> <span class='hs-varid'>ifs</span><span class='hs-layout'>)</span>
<a name="line-62"></a>  <span class='hs-varid'>p</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>redant</span> <span class='hs-varid'>more</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>form</span>
<a name="line-63"></a>  <span class='hs-varid'>nf</span> <span class='hs-layout'>(</span><span class='hs-varid'>foldl</span> <span class='hs-conid'>Apply</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fst</span><span class='hs-layout'>)</span> <span class='hs-varid'>ifs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-64"></a>
<a name="line-65"></a><a name="Proof"></a><span class='hs-comment'>------------------------------</span>
<a name="line-66"></a><a name="Proof"></a><span class='hs-keyword'>type</span> <span class='hs-conid'>Proof</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Term</span>
<a name="line-67"></a>
<a name="line-68"></a><a name="subst"></a><span class='hs-definition'>subst</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Symbol</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Term</span>
<a name="line-69"></a><span class='hs-definition'>subst</span> <span class='hs-varid'>b</span> <span class='hs-varid'>x</span> <span class='hs-varid'>term</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sub</span> <span class='hs-varid'>term</span>
<a name="line-70"></a>  <span class='hs-keyword'>where</span> <span class='hs-varid'>sub</span> <span class='hs-varid'>t</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>s'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>x</span> <span class='hs-varop'>==</span> <span class='hs-varid'>s'</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>copy</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>b</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>return</span> <span class='hs-varid'>t</span>
<a name="line-71"></a>        <span class='hs-varid'>sub</span> <span class='hs-layout'>(</span><span class='hs-conid'>Lam</span> <span class='hs-varid'>s</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>liftM</span> <span class='hs-layout'>(</span><span class='hs-conid'>Lam</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>sub</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-72"></a>        <span class='hs-varid'>sub</span> <span class='hs-layout'>(</span><span class='hs-conid'>Apply</span> <span class='hs-varid'>t1</span> <span class='hs-varid'>t2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>liftM2</span> <span class='hs-conid'>Apply</span> <span class='hs-layout'>(</span><span class='hs-varid'>sub</span> <span class='hs-varid'>t1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>sub</span> <span class='hs-varid'>t2</span><span class='hs-layout'>)</span>
<a name="line-73"></a>        <span class='hs-varid'>sub</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-varid'>t</span>
<a name="line-74"></a>
<a name="line-75"></a><a name="copy"></a><span class='hs-definition'>copy</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>Symbol</span><span class='hs-layout'>,</span> <span class='hs-conid'>Symbol</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Term</span>
<a name="line-76"></a><span class='hs-definition'>copy</span> <span class='hs-varid'>r</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span>       <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Var</span> <span class='hs-varop'>$</span> <span class='hs-varid'>maybe</span> <span class='hs-varid'>s</span> <span class='hs-varid'>id</span> <span class='hs-varop'>$</span> <span class='hs-varid'>lookup</span> <span class='hs-varid'>s</span> <span class='hs-varid'>r</span>
<a name="line-77"></a><span class='hs-definition'>copy</span> <span class='hs-varid'>r</span> <span class='hs-layout'>(</span><span class='hs-conid'>Lam</span> <span class='hs-varid'>s</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>     <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-78"></a>    <span class='hs-varid'>s'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>newSym</span> <span class='hs-str'>"c"</span>
<a name="line-79"></a>    <span class='hs-varid'>liftM</span> <span class='hs-layout'>(</span><span class='hs-conid'>Lam</span> <span class='hs-varid'>s'</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>copy</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>s</span><span class='hs-layout'>,</span> <span class='hs-varid'>s'</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-varid'>t</span>
<a name="line-80"></a><span class='hs-definition'>copy</span> <span class='hs-varid'>r</span> <span class='hs-layout'>(</span><span class='hs-conid'>Apply</span> <span class='hs-varid'>t1</span> <span class='hs-varid'>t2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>liftM2</span> <span class='hs-conid'>Apply</span> <span class='hs-layout'>(</span><span class='hs-varid'>copy</span> <span class='hs-varid'>r</span> <span class='hs-varid'>t1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>copy</span> <span class='hs-varid'>r</span> <span class='hs-varid'>t2</span><span class='hs-layout'>)</span>
<a name="line-81"></a><span class='hs-definition'>copy</span> <span class='hs-sel'>_r</span> <span class='hs-varid'>t</span>            <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-varid'>t</span>
<a name="line-82"></a>
<a name="line-83"></a><span class='hs-comment'>------------------------------</span>
<a name="line-84"></a>
<a name="line-85"></a><span class='hs-comment'>-- XXX The symbols used in the functions below must not clash</span>
<a name="line-86"></a><span class='hs-comment'>-- XXX with any symbols from newSym.</span>
<a name="line-87"></a>
<a name="line-88"></a><a name="applyAtom"></a><span class='hs-definition'>applyAtom</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span>
<a name="line-89"></a><span class='hs-definition'>applyAtom</span> <span class='hs-varid'>f</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Apply</span> <span class='hs-varid'>f</span> <span class='hs-varid'>a</span>
<a name="line-90"></a>
<a name="line-91"></a><a name="curryt"></a><span class='hs-definition'>curryt</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span>
<a name="line-92"></a><span class='hs-definition'>curryt</span> <span class='hs-varid'>n</span> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>foldr</span> <span class='hs-conid'>Lam</span> <span class='hs-layout'>(</span><span class='hs-conid'>Apply</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-varid'>applys</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ctuple</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>map</span> <span class='hs-conid'>Var</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span>
<a name="line-93"></a>  <span class='hs-keyword'>where</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span> <span class='hs-conid'>Symbol</span> <span class='hs-layout'>(</span><span class='hs-str'>"x_"</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span> <span class='hs-keyglyph'>..</span> <span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>]</span>
<a name="line-94"></a>
<a name="line-95"></a><a name="inj"></a><span class='hs-definition'>inj</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>ConsDesc</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span>
<a name="line-96"></a><span class='hs-definition'>inj</span> <span class='hs-varid'>cd</span> <span class='hs-varid'>i</span> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Lam</span> <span class='hs-varid'>x</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Apply</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-conid'>Apply</span> <span class='hs-layout'>(</span><span class='hs-conid'>Cinj</span> <span class='hs-varid'>cd</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-97"></a>  <span class='hs-keyword'>where</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Symbol</span> <span class='hs-str'>"x"</span>
<a name="line-98"></a>
<a name="line-99"></a><a name="applyImp"></a><span class='hs-definition'>applyImp</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span>
<a name="line-100"></a><span class='hs-definition'>applyImp</span> <span class='hs-varid'>p</span> <span class='hs-varid'>q</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Apply</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-conid'>Apply</span> <span class='hs-varid'>q</span> <span class='hs-layout'>(</span><span class='hs-conid'>Lam</span> <span class='hs-varid'>y</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Apply</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-conid'>Lam</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-101"></a>  <span class='hs-keyword'>where</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Symbol</span> <span class='hs-str'>"x"</span>
<a name="line-102"></a>        <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Symbol</span> <span class='hs-str'>"y"</span>
<a name="line-103"></a>
<a name="line-104"></a><a name="cImpDImpFalse"></a><span class='hs-comment'>-- ((c-&gt;d)-&gt;false) -&gt; ((c-&gt;false)-&gt;false, d-&gt;false)</span>
<a name="line-105"></a><span class='hs-comment'>-- p : (c-&gt;d)-&gt;false)</span>
<a name="line-106"></a><span class='hs-comment'>-- replace p1 and p2 with the components of the pair</span>
<a name="line-107"></a><span class='hs-definition'>cImpDImpFalse</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Symbol</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Symbol</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Term</span>
<a name="line-108"></a><span class='hs-definition'>cImpDImpFalse</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>p2</span> <span class='hs-varid'>cdf</span> <span class='hs-varid'>gp</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-109"></a>  <span class='hs-keyword'>let</span> <span class='hs-varid'>p1b</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Lam</span> <span class='hs-varid'>cf</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Apply</span> <span class='hs-varid'>cdf</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Lam</span> <span class='hs-varid'>x</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Apply</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ccases</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Apply</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>cf</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-110"></a>      <span class='hs-varid'>p2b</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Lam</span> <span class='hs-varid'>d</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Apply</span> <span class='hs-varid'>cdf</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Lam</span> <span class='hs-varid'>c</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Var</span> <span class='hs-varid'>d</span>
<a name="line-111"></a>      <span class='hs-varid'>cf</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Symbol</span> <span class='hs-str'>"cf"</span>
<a name="line-112"></a>      <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Symbol</span> <span class='hs-str'>"x"</span>
<a name="line-113"></a>      <span class='hs-varid'>d</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Symbol</span> <span class='hs-str'>"d"</span>
<a name="line-114"></a>      <span class='hs-varid'>c</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Symbol</span> <span class='hs-str'>"c"</span>
<a name="line-115"></a>  <span class='hs-varid'>subst</span> <span class='hs-varid'>p1b</span> <span class='hs-varid'>p1</span> <span class='hs-varid'>gp</span> <span class='hs-varop'>&gt;&gt;=</span> <span class='hs-varid'>subst</span> <span class='hs-varid'>p2b</span> <span class='hs-varid'>p2</span>
<a name="line-116"></a>
<a name="line-117"></a><span class='hs-comment'>------------------------------</span>
<a name="line-118"></a>
<a name="line-119"></a><span class='hs-comment'>-- More simplifications:</span>
<a name="line-120"></a><span class='hs-comment'>--  split where no variables used can be removed</span>
<a name="line-121"></a><span class='hs-comment'>--  either with equal RHS can me merged.</span>
<a name="line-122"></a>
<a name="line-123"></a><a name="nf"></a><span class='hs-comment'>-- Compute the normal form</span>
<a name="line-124"></a><span class='hs-definition'>nf</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Term</span>
<a name="line-125"></a><span class='hs-definition'>nf</span> <span class='hs-varid'>ee</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>spine</span> <span class='hs-varid'>ee</span> <span class='hs-conid'>[]</span>
<a name="line-126"></a>  <span class='hs-keyword'>where</span> <span class='hs-varid'>spine</span> <span class='hs-layout'>(</span><span class='hs-conid'>Apply</span> <span class='hs-varid'>f</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>as</span>     <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span> <span class='hs-varid'>a'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>nf</span> <span class='hs-varid'>a</span><span class='hs-layout'>;</span> <span class='hs-varid'>spine</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>a'</span> <span class='hs-conop'>:</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span>
<a name="line-127"></a>        <span class='hs-varid'>spine</span> <span class='hs-layout'>(</span><span class='hs-conid'>Lam</span> <span class='hs-varid'>s</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-conid'>[]</span>       <span class='hs-keyglyph'>=</span> <span class='hs-varid'>liftM</span> <span class='hs-layout'>(</span><span class='hs-conid'>Lam</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>nf</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span>
<a name="line-128"></a>        <span class='hs-varid'>spine</span> <span class='hs-layout'>(</span><span class='hs-conid'>Lam</span> <span class='hs-varid'>s</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-conop'>:</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span> <span class='hs-varid'>e'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>subst</span> <span class='hs-varid'>a</span> <span class='hs-varid'>s</span> <span class='hs-varid'>e</span><span class='hs-layout'>;</span> <span class='hs-varid'>spine</span> <span class='hs-varid'>e'</span> <span class='hs-keyword'>as</span>
<a name="line-129"></a>        <span class='hs-varid'>spine</span> <span class='hs-layout'>(</span><span class='hs-conid'>Csplit</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>b</span> <span class='hs-conop'>:</span> <span class='hs-varid'>tup</span> <span class='hs-conop'>:</span> <span class='hs-varid'>args</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>istup</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>n</span> <span class='hs-varop'>&lt;=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>spine</span> <span class='hs-layout'>(</span><span class='hs-varid'>applys</span> <span class='hs-varid'>b</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-varid'>args</span>
<a name="line-130"></a>          <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-varid'>istup</span><span class='hs-layout'>,</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>getTup</span> <span class='hs-varid'>tup</span>
<a name="line-131"></a>                <span class='hs-varid'>getTup</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ctuple</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span>  <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-conid'>True</span><span class='hs-layout'>,</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-132"></a>                <span class='hs-varid'>getTup</span> <span class='hs-layout'>(</span><span class='hs-conid'>Apply</span> <span class='hs-varid'>f</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>tf</span><span class='hs-layout'>,</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>getTup</span> <span class='hs-varid'>f</span> <span class='hs-keyword'>in</span> <span class='hs-layout'>(</span><span class='hs-varid'>tf</span><span class='hs-layout'>,</span> <span class='hs-varid'>a</span><span class='hs-conop'>:</span><span class='hs-keyword'>as</span><span class='hs-layout'>)</span>
<a name="line-133"></a>                <span class='hs-varid'>getTup</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-conid'>False</span><span class='hs-layout'>,</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-134"></a>        <span class='hs-varid'>spine</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ccases</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>e</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>Apply</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ccases</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>spine</span> <span class='hs-varid'>e</span> <span class='hs-keyword'>as</span>
<a name="line-135"></a>        <span class='hs-varid'>spine</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ccases</span> <span class='hs-varid'>cds</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Apply</span> <span class='hs-layout'>(</span><span class='hs-conid'>Cinj</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>length</span> <span class='hs-keyword'>as</span> <span class='hs-varop'>&gt;=</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>spine</span> <span class='hs-layout'>(</span><span class='hs-conid'>Apply</span> <span class='hs-layout'>(</span><span class='hs-keyword'>as</span><span class='hs-varop'>!!</span><span class='hs-varid'>i</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'>drop</span> <span class='hs-varid'>n</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span>
<a name="line-136"></a>          <span class='hs-keyword'>where</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>cds</span>
<a name="line-137"></a>        <span class='hs-varid'>spine</span> <span class='hs-varid'>f</span> <span class='hs-keyword'>as</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-varid'>applys</span> <span class='hs-varid'>f</span> <span class='hs-keyword'>as</span>
<a name="line-138"></a>
<a name="line-139"></a>
<a name="line-140"></a><span class='hs-comment'>------------------------------</span>
<a name="line-141"></a><span class='hs-comment'>----- Our Proof monad, P, a monad with state and multiple results</span>
<a name="line-142"></a>
<a name="line-143"></a><a name="P"></a><span class='hs-comment'>-- Note, this is the non-standard way to combine state with multiple</span>
<a name="line-144"></a><a name="P"></a><span class='hs-comment'>-- results.  But this is much better for backtracking.</span>
<a name="line-145"></a><a name="P"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>P</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>P</span> <span class='hs-layout'>{</span> <span class='hs-varid'>unP</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>PS</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>PS</span><span class='hs-layout'>,</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-layout'>}</span>
<a name="line-146"></a>
<a name="line-147"></a><a name="instance%20Applicative%20P"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Applicative</span> <span class='hs-conid'>P</span> <span class='hs-keyword'>where</span>
<a name="line-148"></a>  <span class='hs-varid'>pure</span>  <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span>
<a name="line-149"></a>  <span class='hs-layout'>(</span><span class='hs-varop'>&lt;*&gt;</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ap</span>
<a name="line-150"></a>
<a name="line-151"></a><a name="instance%20Monad%20P"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Monad</span> <span class='hs-conid'>P</span> <span class='hs-keyword'>where</span>
<a name="line-152"></a>  <span class='hs-varid'>return</span> <span class='hs-varid'>x</span>  <span class='hs-keyglyph'>=</span> <span class='hs-conid'>P</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>s</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-153"></a>  <span class='hs-conid'>P</span> <span class='hs-varid'>m</span> <span class='hs-varop'>&gt;&gt;=</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>P</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>-&gt;</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'>s'</span><span class='hs-layout'>,</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>m</span> <span class='hs-varid'>s</span><span class='hs-layout'>,</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>unP</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varid'>s'</span> <span class='hs-keyglyph'>]</span>
<a name="line-154"></a>
<a name="line-155"></a><a name="instance%20Functor%20P"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Functor</span> <span class='hs-conid'>P</span> <span class='hs-keyword'>where</span>
<a name="line-156"></a>  <span class='hs-varid'>fmap</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>P</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>P</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span> <span class='hs-layout'>(</span><span class='hs-varid'>s'</span><span class='hs-layout'>,</span> <span class='hs-varid'>f</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>s'</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>m</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>]</span>
<a name="line-157"></a>
<a name="line-158"></a><a name="instance%20Alternative%20P"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Alternative</span> <span class='hs-conid'>P</span> <span class='hs-keyword'>where</span>
<a name="line-159"></a>  <span class='hs-varid'>empty</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>mzero</span>
<a name="line-160"></a>  <span class='hs-layout'>(</span><span class='hs-varop'>&lt;|&gt;</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>mplus</span>
<a name="line-161"></a>
<a name="line-162"></a><a name="instance%20MonadPlus%20P"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MonadPlus</span> <span class='hs-conid'>P</span> <span class='hs-keyword'>where</span>
<a name="line-163"></a>  <span class='hs-varid'>mzero</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>P</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span> <span class='hs-sel'>_s</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>[]</span>
<a name="line-164"></a>  <span class='hs-conid'>P</span> <span class='hs-varid'>fxs</span> <span class='hs-varop'>`mplus`</span> <span class='hs-conid'>P</span> <span class='hs-varid'>fys</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>P</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>fxs</span> <span class='hs-varid'>s</span> <span class='hs-varop'>++</span> <span class='hs-varid'>fys</span> <span class='hs-varid'>s</span>
<a name="line-165"></a>
<a name="line-166"></a><a name="PS"></a><span class='hs-comment'>-- The state, just an integer for generating new variables</span>
<a name="line-167"></a><a name="PS"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>PS</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>PS</span> <span class='hs-varop'>!</span><span class='hs-conid'>Integer</span>
<a name="line-168"></a>
<a name="line-169"></a><a name="startPS"></a><span class='hs-definition'>startPS</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>PS</span>
<a name="line-170"></a><span class='hs-definition'>startPS</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>PS</span> <span class='hs-num'>1</span>
<a name="line-171"></a>
<a name="line-172"></a><a name="nextInt"></a><span class='hs-definition'>nextInt</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Integer</span>
<a name="line-173"></a><span class='hs-definition'>nextInt</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>P</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-conid'>PS</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>PS</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-174"></a>
<a name="line-175"></a><a name="none"></a><span class='hs-definition'>none</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>P</span> <span class='hs-varid'>a</span>
<a name="line-176"></a><span class='hs-definition'>none</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>mzero</span>
<a name="line-177"></a>
<a name="line-178"></a><a name="many"></a><span class='hs-definition'>many</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-varid'>a</span>
<a name="line-179"></a><span class='hs-definition'>many</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>P</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>zip</span> <span class='hs-layout'>(</span><span class='hs-varid'>repeat</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span>
<a name="line-180"></a>
<a name="line-181"></a><a name="atMostOne"></a><span class='hs-definition'>atMostOne</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>P</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-varid'>a</span>
<a name="line-182"></a><span class='hs-definition'>atMostOne</span> <span class='hs-layout'>(</span><span class='hs-conid'>P</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>P</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>take</span> <span class='hs-num'>1</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span>
<a name="line-183"></a>
<a name="line-184"></a><a name="runP"></a><span class='hs-definition'>runP</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>P</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span>
<a name="line-185"></a><span class='hs-definition'>runP</span> <span class='hs-layout'>(</span><span class='hs-conid'>P</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-varid'>snd</span> <span class='hs-layout'>(</span><span class='hs-varid'>m</span> <span class='hs-varid'>startPS</span><span class='hs-layout'>)</span>
<a name="line-186"></a>
<a name="line-187"></a>
<a name="line-188"></a><span class='hs-comment'>------------------------------</span>
<a name="line-189"></a><span class='hs-comment'>----- Atomic formulae</span>
<a name="line-190"></a>
<a name="line-191"></a><a name="AtomF"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>AtomF</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>AtomF</span> <span class='hs-conid'>Term</span> <span class='hs-conid'>Symbol</span>
<a name="line-192"></a>           <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>)</span>
<a name="line-193"></a>
<a name="line-194"></a><a name="instance%20Show%20AtomF"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-conid'>AtomF</span> <span class='hs-keyword'>where</span>
<a name="line-195"></a>  <span class='hs-varid'>show</span> <span class='hs-layout'>(</span><span class='hs-conid'>AtomF</span> <span class='hs-varid'>p</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>show</span> <span class='hs-varid'>p</span> <span class='hs-varop'>++</span> <span class='hs-str'>":"</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>s</span>
<a name="line-196"></a>
<a name="line-197"></a><a name="AtomFs"></a><span class='hs-keyword'>type</span> <span class='hs-conid'>AtomFs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>AtomF</span><span class='hs-keyglyph'>]</span>
<a name="line-198"></a>
<a name="line-199"></a><a name="findAtoms"></a><span class='hs-definition'>findAtoms</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Symbol</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>AtomFs</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Term</span><span class='hs-keyglyph'>]</span>
<a name="line-200"></a><span class='hs-definition'>findAtoms</span> <span class='hs-varid'>s</span> <span class='hs-varid'>atoms</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>AtomF</span> <span class='hs-varid'>p</span> <span class='hs-varid'>s'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>atoms</span><span class='hs-layout'>,</span> <span class='hs-varid'>s</span> <span class='hs-varop'>==</span> <span class='hs-varid'>s'</span> <span class='hs-keyglyph'>]</span>
<a name="line-201"></a>
<a name="line-202"></a><span class='hs-comment'>--removeAtom :: Symbol -&gt; AtomFs -&gt; AtomFs</span>
<a name="line-203"></a><span class='hs-comment'>--removeAtom s atoms = [ a | a@(AtomF _ s') &lt;- atoms, s /= s' ]</span>
<a name="line-204"></a>
<a name="line-205"></a><a name="addAtom"></a><span class='hs-definition'>addAtom</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>AtomF</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>AtomFs</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>AtomFs</span>
<a name="line-206"></a><span class='hs-definition'>addAtom</span> <span class='hs-varid'>a</span> <span class='hs-keyword'>as</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>a</span> <span class='hs-varop'>`elem`</span> <span class='hs-keyword'>as</span> <span class='hs-keyword'>then</span> <span class='hs-keyword'>as</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>a</span> <span class='hs-conop'>:</span> <span class='hs-keyword'>as</span>
<a name="line-207"></a>
<a name="line-208"></a><span class='hs-comment'>------------------------------</span>
<a name="line-209"></a><span class='hs-comment'>----- Implications of one atom</span>
<a name="line-210"></a>
<a name="line-211"></a><a name="AtomImp"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>AtomImp</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>AtomImp</span> <span class='hs-conid'>Symbol</span> <span class='hs-conid'>Antecedents</span>
<a name="line-212"></a>     <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Show</span><span class='hs-layout'>)</span>
<a name="line-213"></a><a name="AtomImps"></a><span class='hs-keyword'>type</span> <span class='hs-conid'>AtomImps</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>AtomImp</span><span class='hs-keyglyph'>]</span>
<a name="line-214"></a>
<a name="line-215"></a><a name="extract"></a><span class='hs-definition'>extract</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>AtomImps</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Symbol</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-conid'>Antecedent</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span> <span class='hs-conid'>AtomImps</span><span class='hs-layout'>)</span>
<a name="line-216"></a><span class='hs-definition'>extract</span> <span class='hs-varid'>aatomImps</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>atomImp</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>AtomImp</span> <span class='hs-varid'>a'</span> <span class='hs-varid'>bs</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>atomImps</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span>
<a name="line-217"></a>  <span class='hs-keyword'>case</span> <span class='hs-varid'>compare</span> <span class='hs-varid'>a</span> <span class='hs-varid'>a'</span> <span class='hs-keyword'>of</span>
<a name="line-218"></a>    <span class='hs-conid'>GT</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>rbs</span><span class='hs-layout'>,</span> <span class='hs-varid'>restImps</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>extract</span> <span class='hs-varid'>atomImps</span> <span class='hs-varid'>a</span> <span class='hs-keyword'>in</span> <span class='hs-layout'>(</span><span class='hs-varid'>rbs</span><span class='hs-layout'>,</span> <span class='hs-varid'>atomImp</span> <span class='hs-conop'>:</span> <span class='hs-varid'>restImps</span><span class='hs-layout'>)</span>
<a name="line-219"></a>    <span class='hs-conid'>EQ</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-layout'>(</span><span class='hs-varid'>bs</span><span class='hs-layout'>,</span> <span class='hs-varid'>atomImps</span><span class='hs-layout'>)</span>
<a name="line-220"></a>    <span class='hs-conid'>LT</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-layout'>(</span><span class='hs-conid'>[]</span><span class='hs-layout'>,</span> <span class='hs-varid'>aatomImps</span><span class='hs-layout'>)</span>
<a name="line-221"></a><span class='hs-definition'>extract</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-conid'>[]</span><span class='hs-layout'>,</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-222"></a>
<a name="line-223"></a><a name="insert"></a><span class='hs-definition'>insert</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>AtomImps</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>AtomImp</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>AtomImps</span>
<a name="line-224"></a><span class='hs-definition'>insert</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>ai</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span> <span class='hs-varid'>ai</span> <span class='hs-keyglyph'>]</span>
<a name="line-225"></a><span class='hs-definition'>insert</span> <span class='hs-varid'>aatomImps</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-varid'>atomImp</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>AtomImp</span> <span class='hs-varid'>a'</span> <span class='hs-varid'>bs'</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>atomImps</span><span class='hs-layout'>)</span> <span class='hs-varid'>ai</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>AtomImp</span> <span class='hs-varid'>a</span> <span class='hs-varid'>bs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-226"></a>  <span class='hs-keyword'>case</span> <span class='hs-varid'>compare</span> <span class='hs-varid'>a</span> <span class='hs-varid'>a'</span> <span class='hs-keyword'>of</span>
<a name="line-227"></a>    <span class='hs-conid'>GT</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>atomImp</span> <span class='hs-conop'>:</span> <span class='hs-varid'>insert</span> <span class='hs-varid'>atomImps</span> <span class='hs-varid'>ai</span>
<a name="line-228"></a>    <span class='hs-conid'>EQ</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>AtomImp</span> <span class='hs-varid'>a</span> <span class='hs-layout'>(</span><span class='hs-varid'>bs</span> <span class='hs-varop'>++</span> <span class='hs-varid'>bs'</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>atomImps</span>
<a name="line-229"></a>    <span class='hs-conid'>LT</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>ai</span> <span class='hs-conop'>:</span> <span class='hs-varid'>aatomImps</span>
<a name="line-230"></a>
<a name="line-231"></a><span class='hs-comment'>------------------------------</span>
<a name="line-232"></a><span class='hs-comment'>----- Nested implications, (a -&gt; b) -&gt; c</span>
<a name="line-233"></a>
<a name="line-234"></a><a name="NestImp"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>NestImp</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>NestImp</span> <span class='hs-conid'>Term</span> <span class='hs-conid'>Formula</span> <span class='hs-conid'>Formula</span> <span class='hs-conid'>Formula</span> <span class='hs-comment'>-- NestImp a b c represents (a :-&gt; b) :-&gt; c</span>
<a name="line-235"></a>             <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>)</span>
<a name="line-236"></a>
<a name="line-237"></a><a name="instance%20Show%20NestImp"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-conid'>NestImp</span> <span class='hs-keyword'>where</span>
<a name="line-238"></a>  <span class='hs-varid'>show</span> <span class='hs-layout'>(</span><span class='hs-conid'>NestImp</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>show</span> <span class='hs-varop'>$</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>c</span>
<a name="line-239"></a>
<a name="line-240"></a><a name="NestImps"></a><span class='hs-keyword'>type</span> <span class='hs-conid'>NestImps</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>NestImp</span><span class='hs-keyglyph'>]</span>
<a name="line-241"></a>
<a name="line-242"></a><a name="addNestImp"></a><span class='hs-definition'>addNestImp</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>NestImp</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>NestImps</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>NestImps</span>
<a name="line-243"></a><span class='hs-definition'>addNestImp</span> <span class='hs-varid'>n</span> <span class='hs-varid'>ns</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>n</span> <span class='hs-varop'>`elem`</span> <span class='hs-varid'>ns</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>ns</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>n</span> <span class='hs-conop'>:</span> <span class='hs-varid'>ns</span>
<a name="line-244"></a>
<a name="line-245"></a><a name="heuristics"></a><span class='hs-comment'>------------------------------</span>
<a name="line-246"></a><span class='hs-comment'>----- Ordering of nested implications</span>
<a name="line-247"></a><span class='hs-definition'>heuristics</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Bool</span>
<a name="line-248"></a><span class='hs-definition'>heuristics</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span>
<a name="line-249"></a>
<a name="line-250"></a><a name="order"></a><span class='hs-definition'>order</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>NestImps</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>AtomImps</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>NestImps</span>
<a name="line-251"></a><span class='hs-definition'>order</span> <span class='hs-varid'>nestImps</span> <span class='hs-varid'>g</span> <span class='hs-varid'>atomImps</span> <span class='hs-keyglyph'>=</span>
<a name="line-252"></a>  <span class='hs-keyword'>if</span> <span class='hs-varid'>heuristics</span>
<a name="line-253"></a>     <span class='hs-keyword'>then</span> <span class='hs-varid'>nestImps</span>
<a name="line-254"></a>     <span class='hs-keyword'>else</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>good_for</span> <span class='hs-layout'>(</span><span class='hs-conid'>NestImp</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-layout'>(</span><span class='hs-conid'>Disj</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span>
<a name="line-255"></a>              <span class='hs-varid'>good_for</span> <span class='hs-layout'>(</span><span class='hs-conid'>NestImp</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>g'</span><span class='hs-layout'>)</span>        <span class='hs-keyglyph'>=</span> <span class='hs-varid'>g</span> <span class='hs-varop'>==</span> <span class='hs-varid'>g'</span>
<a name="line-256"></a>              <span class='hs-varid'>nice_for</span> <span class='hs-layout'>(</span><span class='hs-conid'>NestImp</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-layout'>(</span><span class='hs-conid'>PVar</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-257"></a>                <span class='hs-keyword'>case</span> <span class='hs-varid'>extract</span> <span class='hs-varid'>atomImps</span> <span class='hs-varid'>s</span> <span class='hs-keyword'>of</span>
<a name="line-258"></a>                  <span class='hs-layout'>(</span><span class='hs-varid'>bs'</span><span class='hs-layout'>,</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>bs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>A</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>bs'</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>in</span> <span class='hs-varid'>g</span> <span class='hs-varop'>`elem`</span> <span class='hs-varid'>bs</span> <span class='hs-varop'>||</span> <span class='hs-varid'>false</span> <span class='hs-varop'>`elem`</span> <span class='hs-varid'>bs</span>
<a name="line-259"></a>              <span class='hs-varid'>nice_for</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-260"></a>              <span class='hs-layout'>(</span><span class='hs-varid'>good</span><span class='hs-layout'>,</span> <span class='hs-varid'>ok</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>partition</span> <span class='hs-varid'>good_for</span> <span class='hs-varid'>nestImps</span>
<a name="line-261"></a>              <span class='hs-layout'>(</span><span class='hs-varid'>nice</span><span class='hs-layout'>,</span> <span class='hs-varid'>bad</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>partition</span> <span class='hs-varid'>nice_for</span> <span class='hs-varid'>ok</span>
<a name="line-262"></a>           <span class='hs-keyword'>in</span> <span class='hs-varid'>good</span> <span class='hs-varop'>++</span> <span class='hs-varid'>nice</span> <span class='hs-varop'>++</span> <span class='hs-varid'>bad</span>
<a name="line-263"></a>
<a name="line-264"></a><a name="newSym"></a><span class='hs-comment'>------------------------------</span>
<a name="line-265"></a><span class='hs-comment'>----- Generate a new unique variable</span>
<a name="line-266"></a><span class='hs-definition'>newSym</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>String</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Symbol</span>
<a name="line-267"></a><span class='hs-definition'>newSym</span> <span class='hs-varid'>pre</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-268"></a>  <span class='hs-varid'>i</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>nextInt</span>
<a name="line-269"></a>  <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Symbol</span> <span class='hs-varop'>$</span> <span class='hs-varid'>pre</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>i</span>
<a name="line-270"></a>
<a name="line-271"></a><a name="select"></a><span class='hs-comment'>------------------------------</span>
<a name="line-272"></a><span class='hs-comment'>----- Generate all ways to select one element of a list</span>
<a name="line-273"></a><span class='hs-definition'>select</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span>
<a name="line-274"></a><span class='hs-definition'>select</span> <span class='hs-varid'>zs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>many</span> <span class='hs-keyglyph'>[</span> <span class='hs-varid'>del</span> <span class='hs-varid'>n</span> <span class='hs-varid'>zs</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span> <span class='hs-keyglyph'>..</span> <span class='hs-varid'>length</span> <span class='hs-varid'>zs</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>]</span>
<a name="line-275"></a>  <span class='hs-keyword'>where</span> <span class='hs-varid'>del</span> <span class='hs-num'>0</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-276"></a>        <span class='hs-varid'>del</span> <span class='hs-varid'>n</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>del</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-keyword'>in</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span>
<a name="line-277"></a>        <span class='hs-varid'>del</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"select"</span>
<a name="line-278"></a>
<a name="line-279"></a><span class='hs-comment'>------------------------------</span>
<a name="line-280"></a><span class='hs-comment'>-----</span>
<a name="line-281"></a>
<a name="line-282"></a><a name="Antecedent"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Antecedent</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>A</span> <span class='hs-conid'>Term</span> <span class='hs-conid'>Formula</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Show</span><span class='hs-layout'>)</span>
<a name="line-283"></a><a name="Antecedents"></a><span class='hs-keyword'>type</span> <span class='hs-conid'>Antecedents</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Antecedent</span><span class='hs-keyglyph'>]</span>
<a name="line-284"></a>
<a name="line-285"></a><a name="Goal"></a><span class='hs-keyword'>type</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Formula</span>
<a name="line-286"></a>
<a name="line-287"></a><a name="redant"></a><span class='hs-comment'>--</span>
<a name="line-288"></a><span class='hs-comment'>-- This is the main loop of the proof search.</span>
<a name="line-289"></a><span class='hs-comment'>--</span>
<a name="line-290"></a><span class='hs-comment'>-- The redant functions reduce antecedents and the redsucc</span>
<a name="line-291"></a><span class='hs-comment'>-- function reduces the goal (succedent).</span>
<a name="line-292"></a><span class='hs-comment'>--</span>
<a name="line-293"></a><span class='hs-comment'>-- The antecedents are kept in four groups: Antecedents, AtomImps, NestImps, AtomFs</span>
<a name="line-294"></a><span class='hs-comment'>--   Antecedents contains as yet unclassified antecedents; the redant functions</span>
<a name="line-295"></a><span class='hs-comment'>--     go through them one by one and reduces and classifies them.</span>
<a name="line-296"></a><span class='hs-comment'>--   AtomImps contains implications of the form (a -&gt; b), where `a' is an atom.</span>
<a name="line-297"></a><span class='hs-comment'>--     To speed up the processing it is stored as a map from the `a' to all the</span>
<a name="line-298"></a><span class='hs-comment'>--     formulae it implies.</span>
<a name="line-299"></a><span class='hs-comment'>--   NestImps contains implications of the form ((b -&gt; c) -&gt; d)</span>
<a name="line-300"></a><span class='hs-comment'>--   AtomFs contains atomic formulae.</span>
<a name="line-301"></a><span class='hs-comment'>--</span>
<a name="line-302"></a><span class='hs-comment'>-- There is also a proof object associated with each antecedent.</span>
<a name="line-303"></a><span class='hs-comment'>--</span>
<a name="line-304"></a><span class='hs-definition'>redant</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>MoreSolutions</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Antecedents</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>AtomImps</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>NestImps</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>AtomFs</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-305"></a><span class='hs-definition'>redant</span> <span class='hs-varid'>more</span> <span class='hs-varid'>antes</span> <span class='hs-varid'>atomImps</span> <span class='hs-varid'>nestImps</span> <span class='hs-varid'>atoms</span> <span class='hs-varid'>goal</span> <span class='hs-keyglyph'>=</span>
<a name="line-306"></a>  <span class='hs-varid'>wrapM</span> <span class='hs-str'>"redant"</span> <span class='hs-layout'>(</span><span class='hs-varid'>antes</span><span class='hs-layout'>,</span> <span class='hs-varid'>atomImps</span><span class='hs-layout'>,</span> <span class='hs-varid'>nestImps</span><span class='hs-layout'>,</span> <span class='hs-varid'>atoms</span><span class='hs-layout'>,</span> <span class='hs-varid'>goal</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-307"></a>    <span class='hs-keyword'>case</span> <span class='hs-varid'>antes</span> <span class='hs-keyword'>of</span>
<a name="line-308"></a>      <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>redsucc</span> <span class='hs-varid'>goal</span>
<a name="line-309"></a>      <span class='hs-varid'>a</span><span class='hs-conop'>:</span><span class='hs-varid'>l</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>redant1</span> <span class='hs-varid'>a</span> <span class='hs-varid'>l</span> <span class='hs-varid'>goal</span>
<a name="line-310"></a>  <span class='hs-keyword'>where</span> <span class='hs-varid'>redant0</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>redant</span> <span class='hs-varid'>more</span> <span class='hs-varid'>l</span> <span class='hs-varid'>atomImps</span> <span class='hs-varid'>nestImps</span> <span class='hs-varid'>atoms</span> <span class='hs-varid'>g</span>
<a name="line-311"></a>        <span class='hs-varid'>redant1</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Antecedent</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Antecedents</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-312"></a>        <span class='hs-varid'>redant1</span> <span class='hs-varid'>a</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>A</span> <span class='hs-varid'>p</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span>
<a name="line-313"></a>          <span class='hs-varid'>wrapM</span> <span class='hs-str'>"redant1"</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>atomImps</span><span class='hs-layout'>,</span> <span class='hs-varid'>nestImps</span><span class='hs-layout'>,</span> <span class='hs-varid'>atoms</span><span class='hs-layout'>,</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-314"></a>          <span class='hs-keyword'>if</span> <span class='hs-varid'>f</span> <span class='hs-varop'>==</span> <span class='hs-varid'>g</span>
<a name="line-315"></a>             <span class='hs-keyword'>then</span> <span class='hs-comment'>-- The goal is the antecedent, we're done.</span>
<a name="line-316"></a>               <span class='hs-keyword'>if</span> <span class='hs-varid'>more</span>
<a name="line-317"></a>                  <span class='hs-keyword'>then</span> <span class='hs-varid'>return</span> <span class='hs-varid'>p</span> <span class='hs-varop'>`mplus`</span> <span class='hs-varid'>redant1'</span> <span class='hs-varid'>a</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span>
<a name="line-318"></a>                  <span class='hs-keyword'>else</span> <span class='hs-varid'>return</span> <span class='hs-varid'>p</span>
<a name="line-319"></a>             <span class='hs-keyword'>else</span> <span class='hs-varid'>redant1'</span> <span class='hs-varid'>a</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span>
<a name="line-320"></a>
<a name="line-321"></a>        <span class='hs-comment'>-- Reduce the first antecedent</span>
<a name="line-322"></a>        <span class='hs-varid'>redant1'</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Antecedent</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Antecedents</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-323"></a>        <span class='hs-varid'>redant1'</span> <span class='hs-layout'>(</span><span class='hs-conid'>A</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-conid'>PVar</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span>
<a name="line-324"></a>          <span class='hs-keyword'>let</span> <span class='hs-varid'>af</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>AtomF</span> <span class='hs-varid'>p</span> <span class='hs-varid'>s</span>
<a name="line-325"></a>              <span class='hs-layout'>(</span><span class='hs-varid'>bs</span><span class='hs-layout'>,</span> <span class='hs-varid'>restAtomImps</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>extract</span> <span class='hs-varid'>atomImps</span> <span class='hs-varid'>s</span>
<a name="line-326"></a>           <span class='hs-keyword'>in</span> <span class='hs-varid'>redant</span> <span class='hs-varid'>more</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-conid'>A</span> <span class='hs-layout'>(</span><span class='hs-conid'>Apply</span> <span class='hs-varid'>f</span> <span class='hs-varid'>p</span><span class='hs-layout'>)</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>A</span> <span class='hs-varid'>f</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>bs</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>++</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-varid'>restAtomImps</span> <span class='hs-varid'>nestImps</span> <span class='hs-layout'>(</span><span class='hs-varid'>addAtom</span> <span class='hs-varid'>af</span> <span class='hs-varid'>atoms</span><span class='hs-layout'>)</span> <span class='hs-varid'>g</span>
<a name="line-327"></a>        <span class='hs-varid'>redant1'</span> <span class='hs-layout'>(</span><span class='hs-conid'>A</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-conid'>Conj</span> <span class='hs-varid'>bs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-328"></a>          <span class='hs-varid'>vs</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>mapM</span> <span class='hs-layout'>(</span><span class='hs-varid'>const</span> <span class='hs-layout'>(</span><span class='hs-varid'>newSym</span> <span class='hs-str'>"v"</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>bs</span>
<a name="line-329"></a>          <span class='hs-varid'>gp</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>redant0</span> <span class='hs-layout'>(</span><span class='hs-varid'>zipWith</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>A</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>bs</span> <span class='hs-varop'>++</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-varid'>g</span>
<a name="line-330"></a>          <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-varid'>applys</span> <span class='hs-layout'>(</span><span class='hs-conid'>Csplit</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>bs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>foldr</span> <span class='hs-conid'>Lam</span> <span class='hs-varid'>gp</span> <span class='hs-varid'>vs</span><span class='hs-layout'>,</span> <span class='hs-varid'>p</span><span class='hs-keyglyph'>]</span>
<a name="line-331"></a>        <span class='hs-varid'>redant1'</span> <span class='hs-layout'>(</span><span class='hs-conid'>A</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-conid'>Disj</span> <span class='hs-varid'>ds</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-332"></a>          <span class='hs-varid'>vs</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>mapM</span> <span class='hs-layout'>(</span><span class='hs-varid'>const</span> <span class='hs-layout'>(</span><span class='hs-varid'>newSym</span> <span class='hs-str'>"d"</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>ds</span>
<a name="line-333"></a>          <span class='hs-varid'>ps</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>mapM</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-layout'>,</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>redant1</span> <span class='hs-layout'>(</span><span class='hs-conid'>A</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>zip</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>ds</span><span class='hs-layout'>)</span>
<a name="line-334"></a>          <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>ds</span> <span class='hs-varop'>&amp;&amp;</span> <span class='hs-varid'>g</span> <span class='hs-varop'>==</span> <span class='hs-conid'>Disj</span> <span class='hs-conid'>[]</span>
<a name="line-335"></a>             <span class='hs-keyword'>then</span> <span class='hs-varid'>return</span> <span class='hs-varid'>p</span>
<a name="line-336"></a>             <span class='hs-keyword'>else</span> <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-varid'>applys</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ccases</span> <span class='hs-layout'>(</span><span class='hs-varid'>map</span> <span class='hs-varid'>fst</span> <span class='hs-varid'>ds</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-conop'>:</span> <span class='hs-varid'>zipWith</span> <span class='hs-conid'>Lam</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>ps</span><span class='hs-layout'>)</span>
<a name="line-337"></a>        <span class='hs-varid'>redant1'</span> <span class='hs-layout'>(</span><span class='hs-conid'>A</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>redantimp</span> <span class='hs-varid'>p</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span>
<a name="line-338"></a>
<a name="line-339"></a>        <span class='hs-varid'>redantimp</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Antecedents</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-340"></a>        <span class='hs-varid'>redantimp</span> <span class='hs-varid'>t</span> <span class='hs-varid'>c</span> <span class='hs-varid'>d</span> <span class='hs-varid'>a</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span>
<a name="line-341"></a>          <span class='hs-varid'>wrapM</span> <span class='hs-str'>"redantimp"</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span><span class='hs-layout'>,</span><span class='hs-varid'>d</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span><span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-342"></a>          <span class='hs-varid'>redantimp'</span> <span class='hs-varid'>t</span> <span class='hs-varid'>c</span> <span class='hs-varid'>d</span> <span class='hs-varid'>a</span> <span class='hs-varid'>g</span>
<a name="line-343"></a>
<a name="line-344"></a>        <span class='hs-comment'>-- Reduce an implication antecedent</span>
<a name="line-345"></a>        <span class='hs-varid'>redantimp'</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Antecedents</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-346"></a>        <span class='hs-comment'>-- p : PVar s -&gt; b</span>
<a name="line-347"></a>        <span class='hs-varid'>redantimp'</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-conid'>PVar</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-varid'>b</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>redantimpatom</span> <span class='hs-varid'>p</span> <span class='hs-varid'>s</span> <span class='hs-varid'>b</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span>
<a name="line-348"></a>        <span class='hs-comment'>-- p : (c &amp; d) -&gt; b</span>
<a name="line-349"></a>        <span class='hs-varid'>redantimp'</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-conid'>Conj</span> <span class='hs-varid'>cs</span><span class='hs-layout'>)</span> <span class='hs-varid'>b</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-350"></a>          <span class='hs-varid'>x</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>newSym</span> <span class='hs-str'>"x"</span>
<a name="line-351"></a>          <span class='hs-keyword'>let</span> <span class='hs-varid'>imp</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>foldr</span> <span class='hs-layout'>(</span><span class='hs-conop'>:-&gt;</span><span class='hs-layout'>)</span> <span class='hs-varid'>b</span> <span class='hs-varid'>cs</span>
<a name="line-352"></a>          <span class='hs-varid'>gp</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>redant1</span> <span class='hs-layout'>(</span><span class='hs-conid'>A</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varid'>imp</span><span class='hs-layout'>)</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span>
<a name="line-353"></a>          <span class='hs-varid'>subst</span> <span class='hs-layout'>(</span><span class='hs-varid'>curryt</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>cs</span><span class='hs-layout'>)</span> <span class='hs-varid'>p</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-varid'>gp</span>
<a name="line-354"></a>        <span class='hs-comment'>-- p : (c | d) -&gt; b</span>
<a name="line-355"></a>        <span class='hs-varid'>redantimp'</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-conid'>Disj</span> <span class='hs-varid'>ds</span><span class='hs-layout'>)</span> <span class='hs-varid'>b</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-356"></a>          <span class='hs-varid'>vs</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>mapM</span> <span class='hs-layout'>(</span><span class='hs-varid'>const</span> <span class='hs-layout'>(</span><span class='hs-varid'>newSym</span> <span class='hs-str'>"d"</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>ds</span>
<a name="line-357"></a>          <span class='hs-varid'>gp</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>redant0</span> <span class='hs-layout'>(</span><span class='hs-varid'>zipWith</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-varid'>v</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-layout'>,</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>A</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>d</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>ds</span> <span class='hs-varop'>++</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-varid'>g</span>
<a name="line-358"></a>          <span class='hs-varid'>foldM</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-varid'>r</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varid'>cd</span><span class='hs-layout'>,</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>subst</span> <span class='hs-layout'>(</span><span class='hs-varid'>inj</span> <span class='hs-varid'>cd</span> <span class='hs-varid'>i</span> <span class='hs-varid'>p</span><span class='hs-layout'>)</span> <span class='hs-varid'>v</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-varid'>gp</span> <span class='hs-layout'>(</span><span class='hs-varid'>zip3</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-keyglyph'>]</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>ds</span><span class='hs-layout'>)</span>
<a name="line-359"></a>        <span class='hs-comment'>-- p : (c -&gt; d) -&gt; b</span>
<a name="line-360"></a>        <span class='hs-varid'>redantimp'</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-varid'>b</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>redantimpimp</span> <span class='hs-varid'>p</span> <span class='hs-varid'>c</span> <span class='hs-varid'>d</span> <span class='hs-varid'>b</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span>
<a name="line-361"></a>
<a name="line-362"></a>        <span class='hs-varid'>redantimpimp</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Antecedents</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-363"></a>        <span class='hs-varid'>redantimpimp</span> <span class='hs-varid'>f</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span> <span class='hs-varid'>d</span> <span class='hs-varid'>a</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span>
<a name="line-364"></a>          <span class='hs-varid'>wrapM</span> <span class='hs-str'>"redantimpimp"</span> <span class='hs-layout'>(</span><span class='hs-varid'>b</span><span class='hs-layout'>,</span><span class='hs-varid'>c</span><span class='hs-layout'>,</span><span class='hs-varid'>d</span><span class='hs-layout'>,</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span><span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-365"></a>          <span class='hs-varid'>redantimpimp'</span> <span class='hs-varid'>f</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span> <span class='hs-varid'>d</span> <span class='hs-varid'>a</span> <span class='hs-varid'>g</span>
<a name="line-366"></a>
<a name="line-367"></a>        <span class='hs-comment'>-- Reduce a double implication antecedent</span>
<a name="line-368"></a>        <span class='hs-varid'>redantimpimp'</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Antecedents</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-369"></a>        <span class='hs-comment'>-- next clause exploits ~(C-&gt;D) &lt;=&gt; (~~C &amp; ~D)</span>
<a name="line-370"></a>        <span class='hs-comment'>-- which isn't helpful when D = false</span>
<a name="line-371"></a>        <span class='hs-varid'>redantimpimp'</span> <span class='hs-varid'>p</span> <span class='hs-varid'>c</span> <span class='hs-varid'>d</span> <span class='hs-layout'>(</span><span class='hs-conid'>Disj</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>d</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>false</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-372"></a>          <span class='hs-varid'>x</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>newSym</span> <span class='hs-str'>"x"</span>
<a name="line-373"></a>          <span class='hs-varid'>y</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>newSym</span> <span class='hs-str'>"y"</span>
<a name="line-374"></a>          <span class='hs-varid'>gp</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>redantimpimp</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varid'>c</span> <span class='hs-varid'>false</span> <span class='hs-varid'>false</span> <span class='hs-layout'>(</span><span class='hs-conid'>A</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>d</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>false</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-varid'>g</span>
<a name="line-375"></a>          <span class='hs-varid'>cImpDImpFalse</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-varid'>p</span> <span class='hs-varid'>gp</span>
<a name="line-376"></a>        <span class='hs-comment'>-- p : (c -&gt; d) -&gt; b</span>
<a name="line-377"></a>        <span class='hs-varid'>redantimpimp'</span> <span class='hs-varid'>p</span> <span class='hs-varid'>c</span> <span class='hs-varid'>d</span> <span class='hs-varid'>b</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>redant</span> <span class='hs-varid'>more</span> <span class='hs-varid'>l</span> <span class='hs-varid'>atomImps</span> <span class='hs-layout'>(</span><span class='hs-varid'>addNestImp</span> <span class='hs-layout'>(</span><span class='hs-conid'>NestImp</span> <span class='hs-varid'>p</span> <span class='hs-varid'>c</span> <span class='hs-varid'>d</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-varid'>nestImps</span><span class='hs-layout'>)</span> <span class='hs-varid'>atoms</span> <span class='hs-varid'>g</span>
<a name="line-378"></a>
<a name="line-379"></a>        <span class='hs-comment'>-- Reduce an atomic implication</span>
<a name="line-380"></a>        <span class='hs-varid'>redantimpatom</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Symbol</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Antecedents</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-381"></a>        <span class='hs-varid'>redantimpatom</span> <span class='hs-varid'>p</span> <span class='hs-varid'>s</span> <span class='hs-varid'>b</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span>
<a name="line-382"></a>          <span class='hs-varid'>wrapM</span> <span class='hs-str'>"redantimpatom"</span> <span class='hs-layout'>(</span><span class='hs-varid'>s</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>,</span><span class='hs-varid'>l</span><span class='hs-layout'>,</span><span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-383"></a>          <span class='hs-varid'>redantimpatom'</span> <span class='hs-varid'>p</span> <span class='hs-varid'>s</span> <span class='hs-varid'>b</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span>
<a name="line-384"></a>
<a name="line-385"></a>        <span class='hs-varid'>redantimpatom'</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Term</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Symbol</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Antecedents</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-386"></a>        <span class='hs-varid'>redantimpatom'</span> <span class='hs-varid'>p</span> <span class='hs-varid'>s</span> <span class='hs-varid'>b</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span>
<a name="line-387"></a>          <span class='hs-keyword'>do</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>cutSearch</span> <span class='hs-varid'>more</span> <span class='hs-varop'>$</span> <span class='hs-varid'>many</span> <span class='hs-layout'>(</span><span class='hs-varid'>findAtoms</span> <span class='hs-varid'>s</span> <span class='hs-varid'>atoms</span><span class='hs-layout'>)</span>
<a name="line-388"></a>             <span class='hs-varid'>x</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>newSym</span> <span class='hs-str'>"x"</span>
<a name="line-389"></a>             <span class='hs-varid'>gp</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>redant1</span> <span class='hs-layout'>(</span><span class='hs-conid'>A</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-varid'>l</span> <span class='hs-varid'>g</span>
<a name="line-390"></a>             <span class='hs-varid'>mtrace</span> <span class='hs-str'>"redantimpatom: LLL"</span> <span class='hs-varop'>$</span> <span class='hs-varid'>subst</span> <span class='hs-layout'>(</span><span class='hs-varid'>applyAtom</span> <span class='hs-varid'>p</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-varid'>gp</span>
<a name="line-391"></a>          <span class='hs-varop'>`mplus`</span>
<a name="line-392"></a>             <span class='hs-layout'>(</span><span class='hs-varid'>mtrace</span> <span class='hs-str'>"redantimpatom: RRR"</span> <span class='hs-varop'>$</span>
<a name="line-393"></a>               <span class='hs-varid'>redant</span> <span class='hs-varid'>more</span> <span class='hs-varid'>l</span> <span class='hs-layout'>(</span><span class='hs-varid'>insert</span> <span class='hs-varid'>atomImps</span> <span class='hs-layout'>(</span><span class='hs-conid'>AtomImp</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>A</span> <span class='hs-varid'>p</span> <span class='hs-varid'>b</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>nestImps</span> <span class='hs-varid'>atoms</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span>
<a name="line-394"></a>
<a name="line-395"></a>        <span class='hs-comment'>-- Reduce the goal, with all antecedents already being classified</span>
<a name="line-396"></a>        <span class='hs-varid'>redsucc</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-397"></a>        <span class='hs-varid'>redsucc</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span>
<a name="line-398"></a>          <span class='hs-varid'>wrapM</span> <span class='hs-str'>"redsucc"</span> <span class='hs-layout'>(</span><span class='hs-varid'>g</span><span class='hs-layout'>,</span> <span class='hs-varid'>atomImps</span><span class='hs-layout'>,</span> <span class='hs-varid'>nestImps</span><span class='hs-layout'>,</span> <span class='hs-varid'>atoms</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-399"></a>          <span class='hs-varid'>redsucc'</span> <span class='hs-varid'>g</span>
<a name="line-400"></a>
<a name="line-401"></a>        <span class='hs-varid'>redsucc'</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-402"></a>        <span class='hs-varid'>redsucc'</span> <span class='hs-varid'>a</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>PVar</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-403"></a>            <span class='hs-layout'>(</span><span class='hs-varid'>cutSearch</span> <span class='hs-varid'>more</span> <span class='hs-varop'>$</span> <span class='hs-varid'>many</span> <span class='hs-layout'>(</span><span class='hs-varid'>findAtoms</span> <span class='hs-varid'>s</span> <span class='hs-varid'>atoms</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-404"></a>          <span class='hs-varop'>`mplus`</span>
<a name="line-405"></a>            <span class='hs-comment'>-- The posin check is an optimization.  It gets a little slower without the test.</span>
<a name="line-406"></a>            <span class='hs-layout'>(</span><span class='hs-keyword'>if</span> <span class='hs-varid'>posin</span> <span class='hs-varid'>s</span> <span class='hs-varid'>atomImps</span> <span class='hs-varid'>nestImps</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>redsucc_choice</span> <span class='hs-varid'>a</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>none</span><span class='hs-layout'>)</span>
<a name="line-407"></a>        <span class='hs-varid'>redsucc'</span> <span class='hs-layout'>(</span><span class='hs-conid'>Conj</span> <span class='hs-varid'>cs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-408"></a>          <span class='hs-varid'>ps</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>mapM</span> <span class='hs-varid'>redsucc</span> <span class='hs-varid'>cs</span>
<a name="line-409"></a>          <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-varid'>applys</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ctuple</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>cs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>ps</span>
<a name="line-410"></a>        <span class='hs-comment'>-- next clause deals with succedent (A v B) by pushing the</span>
<a name="line-411"></a>        <span class='hs-comment'>-- non-determinism into the treatment of implication on the left</span>
<a name="line-412"></a>        <span class='hs-varid'>redsucc'</span> <span class='hs-layout'>(</span><span class='hs-conid'>Disj</span> <span class='hs-varid'>ds</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-413"></a>          <span class='hs-varid'>s1</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>newSym</span> <span class='hs-str'>"_"</span>
<a name="line-414"></a>          <span class='hs-keyword'>let</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>PVar</span> <span class='hs-varid'>s1</span>
<a name="line-415"></a>          <span class='hs-varid'>redant0</span> <span class='hs-keyglyph'>[</span> <span class='hs-conid'>A</span> <span class='hs-layout'>(</span><span class='hs-conid'>Cinj</span> <span class='hs-varid'>cd</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>d</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varid'>cd</span><span class='hs-layout'>,</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>zip</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-keyglyph'>]</span> <span class='hs-varid'>ds</span> <span class='hs-keyglyph'>]</span> <span class='hs-varid'>v</span>
<a name="line-416"></a>        <span class='hs-varid'>redsucc'</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-417"></a>          <span class='hs-varid'>s</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>newSym</span> <span class='hs-str'>"x"</span>
<a name="line-418"></a>          <span class='hs-varid'>p</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>redant1</span> <span class='hs-layout'>(</span><span class='hs-conid'>A</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>b</span>
<a name="line-419"></a>          <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-conid'>Lam</span> <span class='hs-varid'>s</span> <span class='hs-varid'>p</span>
<a name="line-420"></a>
<a name="line-421"></a>        <span class='hs-comment'>-- Now we have the hard part; maybe lots of formulae</span>
<a name="line-422"></a>        <span class='hs-comment'>-- of form (C-&gt;D)-&gt;B  in nestImps to choose from!</span>
<a name="line-423"></a>        <span class='hs-comment'>-- Which one to take first? We use the order heuristic.</span>
<a name="line-424"></a>        <span class='hs-varid'>redsucc_choice</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-425"></a>        <span class='hs-varid'>redsucc_choice</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span>
<a name="line-426"></a>          <span class='hs-varid'>wrapM</span> <span class='hs-str'>"redsucc_choice"</span> <span class='hs-varid'>g</span> <span class='hs-varop'>$</span>
<a name="line-427"></a>          <span class='hs-varid'>redsucc_choice'</span> <span class='hs-varid'>g</span>
<a name="line-428"></a>
<a name="line-429"></a>        <span class='hs-varid'>redsucc_choice'</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Goal</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-conid'>Proof</span>
<a name="line-430"></a>        <span class='hs-varid'>redsucc_choice'</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>do</span>
<a name="line-431"></a>          <span class='hs-keyword'>let</span> <span class='hs-varid'>ordImps</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>order</span> <span class='hs-varid'>nestImps</span> <span class='hs-varid'>g</span> <span class='hs-varid'>atomImps</span>
<a name="line-432"></a>          <span class='hs-layout'>(</span><span class='hs-conid'>NestImp</span> <span class='hs-varid'>p</span> <span class='hs-varid'>c</span> <span class='hs-varid'>d</span> <span class='hs-varid'>b</span><span class='hs-layout'>,</span> <span class='hs-varid'>restImps</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span>
<a name="line-433"></a>              <span class='hs-varid'>mtrace</span> <span class='hs-layout'>(</span><span class='hs-str'>"redsucc_choice: order="</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>ordImps</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span>
<a name="line-434"></a>              <span class='hs-varid'>select</span> <span class='hs-varid'>ordImps</span>
<a name="line-435"></a>          <span class='hs-varid'>x</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>newSym</span> <span class='hs-str'>"x"</span>
<a name="line-436"></a>          <span class='hs-varid'>z</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>newSym</span> <span class='hs-str'>"z"</span>
<a name="line-437"></a>          <span class='hs-varid'>qz</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>redant</span> <span class='hs-varid'>more</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>A</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>d</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>b</span><span class='hs-keyglyph'>]</span> <span class='hs-varid'>atomImps</span> <span class='hs-varid'>restImps</span> <span class='hs-varid'>atoms</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span>
<a name="line-438"></a>          <span class='hs-varid'>gp</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>redant</span> <span class='hs-varid'>more</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>A</span> <span class='hs-layout'>(</span><span class='hs-conid'>Var</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varid'>b</span><span class='hs-keyglyph'>]</span> <span class='hs-varid'>atomImps</span> <span class='hs-varid'>restImps</span> <span class='hs-varid'>atoms</span> <span class='hs-varid'>g</span>
<a name="line-439"></a>          <span class='hs-varid'>subst</span> <span class='hs-layout'>(</span><span class='hs-varid'>applyImp</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-conid'>Lam</span> <span class='hs-varid'>z</span> <span class='hs-varid'>qz</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-varid'>gp</span>
<a name="line-440"></a>
<a name="line-441"></a><a name="posin"></a><span class='hs-definition'>posin</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Symbol</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>AtomImps</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>NestImps</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-442"></a><span class='hs-definition'>posin</span> <span class='hs-varid'>g</span> <span class='hs-varid'>atomImps</span> <span class='hs-varid'>nestImps</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>posin1</span> <span class='hs-varid'>g</span> <span class='hs-varid'>atomImps</span> <span class='hs-varop'>||</span> <span class='hs-varid'>posin2</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>[</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>c</span> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>NestImp</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>nestImps</span> <span class='hs-keyglyph'>]</span>
<a name="line-443"></a>
<a name="line-444"></a><a name="posin1"></a><span class='hs-definition'>posin1</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Symbol</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>AtomImps</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-445"></a><span class='hs-definition'>posin1</span> <span class='hs-varid'>g</span> <span class='hs-varid'>atomImps</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>any</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span> <span class='hs-layout'>(</span><span class='hs-conid'>AtomImp</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>bs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>posin2</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>[</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>A</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>bs</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-varid'>atomImps</span>
<a name="line-446"></a>
<a name="line-447"></a><a name="posin2"></a><span class='hs-definition'>posin2</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Symbol</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Formula</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-448"></a><span class='hs-definition'>posin2</span> <span class='hs-varid'>g</span> <span class='hs-varid'>bs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>any</span> <span class='hs-layout'>(</span><span class='hs-varid'>posin3</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-varid'>bs</span>
<a name="line-449"></a>
<a name="line-450"></a><a name="posin3"></a><span class='hs-definition'>posin3</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Symbol</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Formula</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-451"></a><span class='hs-definition'>posin3</span> <span class='hs-varid'>g</span> <span class='hs-layout'>(</span><span class='hs-conid'>Disj</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>all</span> <span class='hs-layout'>(</span><span class='hs-varid'>posin3</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>map</span> <span class='hs-varid'>snd</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span>
<a name="line-452"></a><span class='hs-definition'>posin3</span> <span class='hs-varid'>g</span> <span class='hs-layout'>(</span><span class='hs-conid'>Conj</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>any</span> <span class='hs-layout'>(</span><span class='hs-varid'>posin3</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-keyword'>as</span>
<a name="line-453"></a><span class='hs-definition'>posin3</span> <span class='hs-varid'>g</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span> <span class='hs-conop'>:-&gt;</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>posin3</span> <span class='hs-varid'>g</span> <span class='hs-varid'>b</span>
<a name="line-454"></a><span class='hs-definition'>posin3</span> <span class='hs-varid'>s</span> <span class='hs-layout'>(</span><span class='hs-conid'>PVar</span> <span class='hs-varid'>s'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>s</span> <span class='hs-varop'>==</span> <span class='hs-varid'>s'</span>
<a name="line-455"></a>
<a name="line-456"></a><a name="cutSearch"></a><span class='hs-definition'>cutSearch</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>MoreSolutions</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>P</span> <span class='hs-varid'>a</span>
<a name="line-457"></a><span class='hs-definition'>cutSearch</span> <span class='hs-conid'>False</span> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>atMostOne</span> <span class='hs-varid'>p</span>
<a name="line-458"></a><span class='hs-definition'>cutSearch</span> <span class='hs-conid'>True</span> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p</span>
<a name="line-459"></a>
<a name="line-460"></a><span class='hs-comment'>---------------------------</span>
</pre></body>
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