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- <div class="section" id="using-x-to-generate-values">
+ <section id="using-x-to-generate-values">
<h1>Using <code class="docutils literal notranslate"><span class="pre">x</span></code> to Generate Values<a class="headerlink" href="#using-x-to-generate-values" title="Permalink to this headline">¶</a></h1>
<p>Cf. jp-reprod.html</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">notebook_preamble</span> <span class="k">import</span> <span class="n">J</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">define</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>from notebook_preamble import J, V, define
</pre></div>
</div>
<p>Consider the <code class="docutils literal notranslate"><span class="pre">x</span></code> combinator:</p>
</pre></div>
</div>
<p>Let’s try it:</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">'[0 swap [dup ++] dip rest cons] x'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>V('[0 swap [dup ++] dip rest cons] x')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">.</span> <span class="p">[</span><span class="mi">0</span> <span class="n">swap</span> <span class="p">[</span><span class="n">dup</span> <span class="o">++</span><span class="p">]</span> <span class="n">dip</span> <span class="n">rest</span> <span class="n">cons</span><span class="p">]</span> <span class="n">x</span>
</div>
<p>After one application of <code class="docutils literal notranslate"><span class="pre">x</span></code> the quoted program contains <code class="docutils literal notranslate"><span class="pre">1</span></code> and
<code class="docutils literal notranslate"><span class="pre">0</span></code> is below it on the stack.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'[0 swap [dup ++] dip rest cons] x x x x x pop'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('[0 swap [dup ++] dip rest cons] x x x x x pop')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">0</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">4</span>
</pre></div>
</div>
-<div class="section" id="direco">
+<section id="direco">
<h2><code class="docutils literal notranslate"><span class="pre">direco</span></code><a class="headerlink" href="#direco" title="Permalink to this headline">¶</a></h2>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'direco == dip rest cons'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('direco == dip rest cons')
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">'[0 swap [dup ++] direco] x'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>V('[0 swap [dup ++] direco] x')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">.</span> <span class="p">[</span><span class="mi">0</span> <span class="n">swap</span> <span class="p">[</span><span class="n">dup</span> <span class="o">++</span><span class="p">]</span> <span class="n">direco</span><span class="p">]</span> <span class="n">x</span>
<span class="mi">0</span> <span class="p">[</span><span class="mi">1</span> <span class="n">swap</span> <span class="p">[</span><span class="n">dup</span> <span class="o">++</span><span class="p">]</span> <span class="n">direco</span><span class="p">]</span> <span class="o">.</span>
</pre></div>
</div>
-</div>
-<div class="section" id="making-generators">
+</section>
+<section id="making-generators">
<h2>Making Generators<a class="headerlink" href="#making-generators" title="Permalink to this headline">¶</a></h2>
<p>We want to define a function that accepts <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">[C]</span></code> and builds
our quoted program:</p>
<span class="n">G</span> <span class="o">==</span> <span class="p">[</span><span class="n">direco</span><span class="p">]</span> <span class="n">cons</span> <span class="p">[</span><span class="n">swap</span><span class="p">]</span> <span class="n">swoncat</span> <span class="n">cons</span>
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'G == [direco] cons [swap] swoncat cons'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('G == [direco] cons [swap] swoncat cons')
</pre></div>
</div>
<p>Let’s try it out:</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'0 [dup ++] G'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('0 [dup ++] G')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="mi">0</span> <span class="n">swap</span> <span class="p">[</span><span class="n">dup</span> <span class="o">++</span><span class="p">]</span> <span class="n">direco</span><span class="p">]</span>
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'0 [dup ++] G x x x pop'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('0 [dup ++] G x x x pop')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">0</span> <span class="mi">1</span> <span class="mi">2</span>
</pre></div>
</div>
-<div class="section" id="powers-of-2">
+<section id="powers-of-2">
<h3>Powers of 2<a class="headerlink" href="#powers-of-2" title="Permalink to this headline">¶</a></h3>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'1 [dup 1 <<] G x x x x x x x x x pop'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('1 [dup 1 <<] G x x x x x x x x x pop')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">4</span> <span class="mi">8</span> <span class="mi">16</span> <span class="mi">32</span> <span class="mi">64</span> <span class="mi">128</span> <span class="mi">256</span>
</pre></div>
</div>
-</div>
-<div class="section" id="x-times">
+</section>
+<section id="x-times">
<h3><code class="docutils literal notranslate"><span class="pre">[x]</span> <span class="pre">times</span></code><a class="headerlink" href="#x-times" title="Permalink to this headline">¶</a></h3>
<p>If we have one of these quoted programs we can drive it using <code class="docutils literal notranslate"><span class="pre">times</span></code>
with the <code class="docutils literal notranslate"><span class="pre">x</span></code> combinator.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'23 [dup ++] G 5 [x] times'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('23 [dup ++] G 5 [x] times')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">23</span> <span class="mi">24</span> <span class="mi">25</span> <span class="mi">26</span> <span class="mi">27</span> <span class="p">[</span><span class="mi">28</span> <span class="n">swap</span> <span class="p">[</span><span class="n">dup</span> <span class="o">++</span><span class="p">]</span> <span class="n">direco</span><span class="p">]</span>
</pre></div>
</div>
-</div>
-</div>
-<div class="section" id="generating-multiples-of-three-and-five">
+</section>
+</section>
+<section id="generating-multiples-of-three-and-five">
<h2>Generating Multiples of Three and Five<a class="headerlink" href="#generating-multiples-of-three-and-five" title="Permalink to this headline">¶</a></h2>
<p>Look at the treatment of the Project Euler Problem One in the
“Developing a Program” notebook and you’ll see that we might be
</div>
<p>And pick them off by masking with 3 (binary 11) and then shifting the
int right two bits.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'PE1.1 == dup [3 &] dip 2 >>'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('PE1.1 == dup [3 &] dip 2 >>')
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">'14811 PE1.1'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>V('14811 PE1.1')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">.</span> <span class="mi">14811</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span>
</pre></div>
</div>
<p>If we plug <code class="docutils literal notranslate"><span class="pre">14811</span></code> and <code class="docutils literal notranslate"><span class="pre">[PE1.1]</span></code> into our generator form…</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'14811 [PE1.1] G'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('14811 [PE1.1] G')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="mi">14811</span> <span class="n">swap</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">direco</span><span class="p">]</span>
</pre></div>
</div>
<p>…we get a generator that works for seven cycles before it reaches zero:</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'[14811 swap [PE1.1] direco] 7 [x] times'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('[14811 swap [PE1.1] direco] 7 [x] times')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="p">[</span><span class="mi">0</span> <span class="n">swap</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">direco</span><span class="p">]</span>
</pre></div>
</div>
-<div class="section" id="reset-at-zero">
+<section id="reset-at-zero">
<h3>Reset at Zero<a class="headerlink" href="#reset-at-zero" title="Permalink to this headline">¶</a></h3>
<p>We need a function that checks if the int has reached zero and resets it
if so.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'PE1.1.check == dup [pop 14811] [] branch'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('PE1.1.check == dup [pop 14811] [] branch')
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'14811 [PE1.1.check PE1.1] G'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('14811 [PE1.1.check PE1.1] G')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="mi">14811</span> <span class="n">swap</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mf">1.</span><span class="n">check</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">direco</span><span class="p">]</span>
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'[14811 swap [PE1.1.check PE1.1] direco] 21 [x] times'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('[14811 swap [PE1.1.check PE1.1] direco] 21 [x] times')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="p">[</span><span class="mi">0</span> <span class="n">swap</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mf">1.</span><span class="n">check</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">direco</span><span class="p">]</span>
that’s a little beyond the scope of this article. This solution does
extra work, but not much, and we’re not using it “in production” as they
say.)</p>
-</div>
-<div class="section" id="run-466-times">
+</section>
+<section id="run-466-times">
<h3>Run 466 times<a class="headerlink" href="#run-466-times" title="Permalink to this headline">¶</a></h3>
<p>In the PE1 problem we are asked to sum all the multiples of three and
five less than 1000. It’s worked out that we need to use all seven
numbers sixty-six times and then four more.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'7 66 * 4 +'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('7 66 * 4 +')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">466</span>
</pre></div>
</div>
<p>If we drive our generator 466 times and sum the stack we get 999.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'[14811 swap [PE1.1.check PE1.1] direco] 466 [x] times'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('[14811 swap [PE1.1.check PE1.1] direco] 466 [x] times')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">3</span> <span class="mi">2</span> <span class="mi">1</span> <span class="mi">3</span> <span class="p">[</span><span class="mi">57</span> <span class="n">swap</span> <span class="p">[</span><span class="n">PE1</span><span class="o">.</span><span class="mf">1.</span><span class="n">check</span> <span class="n">PE1</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">direco</span><span class="p">]</span>
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'[14811 swap [PE1.1.check PE1.1] direco] 466 [x] times pop enstacken sum'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('[14811 swap [PE1.1.check PE1.1] direco] 466 [x] times pop enstacken sum')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">999</span>
</pre></div>
</div>
-</div>
-</div>
-<div class="section" id="project-euler-problem-one">
+</section>
+</section>
+<section id="project-euler-problem-one">
<h2>Project Euler Problem One<a class="headerlink" href="#project-euler-problem-one" title="Permalink to this headline">¶</a></h2>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'PE1.2 == + dup [+] dip'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('PE1.2 == + dup [+] dip')
</pre></div>
</div>
<p>Now we can add <code class="docutils literal notranslate"><span class="pre">PE1.2</span></code> to the quoted program given to <code class="docutils literal notranslate"><span class="pre">G</span></code>.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'0 0 0 [PE1.1.check PE1.1] G 466 [x [PE1.2] dip] times popop'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('0 0 0 [PE1.1.check PE1.1] G 466 [x [PE1.2] dip] times popop')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">233168</span>
</pre></div>
</div>
-</div>
-<div class="section" id="a-generator-for-the-fibonacci-sequence">
+</section>
+<section id="a-generator-for-the-fibonacci-sequence">
<h2>A generator for the Fibonacci Sequence.<a class="headerlink" href="#a-generator-for-the-fibonacci-sequence" title="Permalink to this headline">¶</a></h2>
<p>Consider:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="n">b</span> <span class="n">a</span> <span class="n">F</span><span class="p">]</span> <span class="n">x</span>
<span class="n">fib_gen</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">1</span> <span class="n">F</span><span class="p">]</span>
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'fib == + [popdd over] cons infra uncons'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('fib == + [popdd over] cons infra uncons')
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'fib_gen == [1 1 fib]'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('fib_gen == [1 1 fib]')
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'fib_gen 10 [x] times'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('fib_gen 10 [x] times')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span> <span class="mi">5</span> <span class="mi">8</span> <span class="mi">13</span> <span class="mi">21</span> <span class="mi">34</span> <span class="mi">55</span> <span class="mi">89</span> <span class="p">[</span><span class="mi">144</span> <span class="mi">89</span> <span class="n">fib</span><span class="p">]</span>
</pre></div>
</div>
-</div>
-<div class="section" id="project-euler-problem-two">
+</section>
+<section id="project-euler-problem-two">
<h2>Project Euler Problem Two<a class="headerlink" href="#project-euler-problem-two" title="Permalink to this headline">¶</a></h2>
<blockquote>
-<div>By considering the terms in the Fibonacci sequence whose values do
-not exceed four million, find the sum of the even-valued terms.</div></blockquote>
+<div><p>By considering the terms in the Fibonacci sequence whose values do
+not exceed four million, find the sum of the even-valued terms.</p>
+</div></blockquote>
<p>Now that we have a generator for the Fibonacci sequence, we need a
function that adds a term in the sequence to a sum if it is even, and
<code class="docutils literal notranslate"><span class="pre">pop</span></code>s it otherwise.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'PE2.1 == dup 2 % [+] [pop] branch'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('PE2.1 == dup 2 % [+] [pop] branch')
</pre></div>
</div>
<p>And a predicate function that detects when the terms in the series
“exceed four million”.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'>4M == 4000000 >'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('>4M == 4000000 >')
</pre></div>
</div>
<p>Now it’s straightforward to define <code class="docutils literal notranslate"><span class="pre">PE2</span></code> as a recursive function that
generates terms in the Fibonacci sequence until they exceed four million
and sums the even ones.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'PE2 == 0 fib_gen x [pop >4M] [popop] [[PE2.1] dip x] primrec'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('PE2 == 0 fib_gen x [pop >4M] [popop] [[PE2.1] dip x] primrec')
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'PE2'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('PE2')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">4613732</span>
<span class="n">PE2</span> <span class="o">==</span> <span class="mi">0</span> <span class="n">fib_gen</span> <span class="n">x</span> <span class="p">[</span><span class="n">pop</span> <span class="o">></span><span class="mi">4</span><span class="n">M</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span><span class="p">]</span> <span class="p">[[</span><span class="n">PE2</span><span class="o">.</span><span class="mi">1</span><span class="p">]</span> <span class="n">dip</span> <span class="n">x</span><span class="p">]</span> <span class="n">primrec</span>
</pre></div>
</div>
-<div class="section" id="even-valued-fibonacci-terms">
+<section id="even-valued-fibonacci-terms">
<h3>Even-valued Fibonacci Terms<a class="headerlink" href="#even-valued-fibonacci-terms" title="Permalink to this headline">¶</a></h3>
<p>Using <code class="docutils literal notranslate"><span class="pre">o</span></code> for odd and <code class="docutils literal notranslate"><span class="pre">e</span></code> for even:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">o</span> <span class="o">+</span> <span class="n">o</span> <span class="o">=</span> <span class="n">e</span>
</pre></div>
</div>
<p>Every third term is even.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'[1 0 fib] x x x'</span><span class="p">)</span> <span class="c1"># To start the sequence with 1 1 2 3 instead of 1 2 3.</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('[1 0 fib] x x x') # To start the sequence with 1 1 2 3 instead of 1 2 3.
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="mi">1</span> <span class="mi">2</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">2</span> <span class="n">fib</span><span class="p">]</span>
</pre></div>
</div>
<p>Drive the generator three times and <code class="docutils literal notranslate"><span class="pre">popop</span></code> the two odd terms.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'[1 0 fib] x x x [popop] dipd'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('[1 0 fib] x x x [popop] dipd')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">2</span> <span class="p">[</span><span class="mi">3</span> <span class="mi">2</span> <span class="n">fib</span><span class="p">]</span>
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'PE2.2 == x x x [popop] dipd'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('PE2.2 == x x x [popop] dipd')
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'[1 0 fib] 10 [PE2.2] times'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('[1 0 fib] 10 [PE2.2] times')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">2</span> <span class="mi">8</span> <span class="mi">34</span> <span class="mi">144</span> <span class="mi">610</span> <span class="mi">2584</span> <span class="mi">10946</span> <span class="mi">46368</span> <span class="mi">196418</span> <span class="mi">832040</span> <span class="p">[</span><span class="mi">1346269</span> <span class="mi">832040</span> <span class="n">fib</span><span class="p">]</span>
</div>
<p>Replace <code class="docutils literal notranslate"><span class="pre">x</span></code> with our new driver function <code class="docutils literal notranslate"><span class="pre">PE2.2</span></code> and start our
<code class="docutils literal notranslate"><span class="pre">fib</span></code> generator at <code class="docutils literal notranslate"><span class="pre">1</span> <span class="pre">0</span></code>.</p>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'0 [1 0 fib] PE2.2 [pop >4M] [popop] [[PE2.1] dip PE2.2] primrec'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('0 [1 0 fib] PE2.2 [pop >4M] [popop] [[PE2.1] dip PE2.2] primrec')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">4613732</span>
</pre></div>
</div>
-</div>
-</div>
-<div class="section" id="how-to-compile-these">
+</section>
+</section>
+<section id="how-to-compile-these">
<h2>How to compile these?<a class="headerlink" href="#how-to-compile-these" title="Permalink to this headline">¶</a></h2>
<p>You would probably start with a special version of <code class="docutils literal notranslate"><span class="pre">G</span></code>, and perhaps
modifications to the default <code class="docutils literal notranslate"><span class="pre">x</span></code>?</p>
-</div>
-<div class="section" id="an-interesting-variation">
+</section>
+<section id="an-interesting-variation">
<h2>An Interesting Variation<a class="headerlink" href="#an-interesting-variation" title="Permalink to this headline">¶</a></h2>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'codireco == cons dip rest cons'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('codireco == cons dip rest cons')
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">'[0 [dup ++] codireco] x'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>V('[0 [dup ++] codireco] x')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">.</span> <span class="p">[</span><span class="mi">0</span> <span class="p">[</span><span class="n">dup</span> <span class="o">++</span><span class="p">]</span> <span class="n">codireco</span><span class="p">]</span> <span class="n">x</span>
<span class="mi">0</span> <span class="p">[</span><span class="mi">1</span> <span class="p">[</span><span class="n">dup</span> <span class="o">++</span><span class="p">]</span> <span class="n">codireco</span><span class="p">]</span> <span class="o">.</span>
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">'G == [codireco] cons cons'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>define('G == [codireco] cons cons')
</pre></div>
</div>
-<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">'230 [dup ++] G 5 [x] times pop'</span><span class="p">)</span>
+<div class="highlight-ipython2 notranslate"><div class="highlight"><pre><span></span>J('230 [dup ++] G 5 [x] times pop')
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">230</span> <span class="mi">231</span> <span class="mi">232</span> <span class="mi">233</span> <span class="mi">234</span>
</pre></div>
</div>
-</div>
-</div>
+</section>
+</section>
</div>
+
</div>
</div>
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- <ul>
-<li><a class="reference internal" href="#">Using <code class="docutils literal notranslate"><span class="pre">x</span></code> to Generate Values</a><ul>
-<li><a class="reference internal" href="#direco"><code class="docutils literal notranslate"><span class="pre">direco</span></code></a></li>
-<li><a class="reference internal" href="#making-generators">Making Generators</a><ul>
-<li><a class="reference internal" href="#powers-of-2">Powers of 2</a></li>
-<li><a class="reference internal" href="#x-times"><code class="docutils literal notranslate"><span class="pre">[x]</span> <span class="pre">times</span></code></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#generating-multiples-of-three-and-five">Generating Multiples of Three and Five</a><ul>
-<li><a class="reference internal" href="#reset-at-zero">Reset at Zero</a></li>
-<li><a class="reference internal" href="#run-466-times">Run 466 times</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#project-euler-problem-one">Project Euler Problem One</a></li>
-<li><a class="reference internal" href="#a-generator-for-the-fibonacci-sequence">A generator for the Fibonacci Sequence.</a></li>
-<li><a class="reference internal" href="#project-euler-problem-two">Project Euler Problem Two</a><ul>
-<li><a class="reference internal" href="#even-valued-fibonacci-terms">Even-valued Fibonacci Terms</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#how-to-compile-these">How to compile these?</a></li>
-<li><a class="reference internal" href="#an-interesting-variation">An Interesting Variation</a></li>
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