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interpreters-sol.html
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<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>
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<div id='write' class = 'is-node'><h1><a name='header-n0' class='md-header-anchor '></a>Interpreters Solutions</h1><h2><a name='header-n2' class='md-header-anchor '></a>Problems</h2><h3><a name='header-n3' class='md-header-anchor '></a>What does lexer turn these into?</h3><ol start='' ><li><pre class="md-fences md-end-block" lang="lisp"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">(</span><span class="cm-number">1</span> <span class="cm-number">2</span> <span class="cm-number">4</span><span class="cm-bracket">)</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> <code>['(', 1, 2, 4, ')']</code></p></li><li><pre class="md-fences md-end-block" lang="lisp"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">(</span><span class="cm-keyword">define</span> <span class="cm-bracket">(</span><span class="cm-variable">f</span> <span class="cm-variable">x</span><span class="cm-bracket">)</span> <span class="cm-number">5</span><span class="cm-bracket">)</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> <code>['(', 'define', '(', 'f', 'x', ')', 5, ')']</code>, </p><ul><li>Importantly, 'define' is <strong>not</strong> broken up into <code>['d','e','f','i','n','e']</code> this is because variable names are tokens in scheme so the smallest you can break it apart is into <code>'define'</code></li></ul></li></ol><h3><a name='header-n17' class='md-header-anchor '></a>What does the parser turn these into?</h3><ol start='' ><li><pre class="md-fences md-end-block" lang="python"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">[<span class="cm-string">'('</span>, <span class="cm-string">'+'</span>, <span class="cm-number">1</span>, <span class="cm-number">2</span>, <span class="cm-string">')'</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> <code>Pair('+', Pair(1, Pair(2, nil)))</code></p></li><li><pre class="md-fences md-end-block" lang="python"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">[<span class="cm-string">'('</span>, <span class="cm-number">1</span>, <span class="cm-string">'('</span>, <span class="cm-number">2</span>, <span class="cm-string">')'</span>, <span class="cm-number">3</span>, <span class="cm-string">')'</span>]</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> <code>Pair(1, Pair(Pair(2, nil), Pair(3, nil)))</code></p><ul><li>Writing this out without the quotes and commas gives: <code>(1 (2) 3)</code>, if we remember the notation for scheme <code>Pair</code>'s then we see translate this into calls to <code>cons</code> to construct this list which gives the corresponding <code>Pair</code> structure.</li><li><code>(1 (2) 3) -> (cons 1 '((2) 3) ) -> (cons 1 (cons '(2) '(3)) )</code></li><li><code>-> (cons 1 (cons (cons 2 nil) (cons 3 nil)))</code></li><li><code>-> Pair(1, Pair(Pair(2, nil), Pair(3, nil)))</code></li></ul></li></ol><h3><a name='header-n31' class='md-header-anchor '></a>How many calls to scheme_eval/scheme_apply?</h3><p><em>For problems 1-3 in this section refer to <a href='https://docs.google.com/presentation/d/1CqmCEbpM-10bfl2BaYqeesBO_ZpCx9ZSCwDU0U2gaY4/edit#slide=id.g24bf8bb8aa_0_331'>slides 92-96</a> of the review slides from Friday for further assistance</em></p><ol start='' ><li><pre class="md-fences md-end-block" lang="lisp "> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">(</span><span class="cm-keyword">define</span> <span class="cm-bracket">(</span><span class="cm-variable">f</span> <span class="cm-variable">x</span><span class="cm-bracket">)</span> <span class="cm-bracket">(</span><span class="cm-variable">*</span> <span class="cm-bracket">((</span><span class="cm-variable">lambda</span> <span class="cm-bracket">()</span> <span class="cm-variable">x</span><span class="cm-bracket">))</span> <span class="cm-number">2</span><span class="cm-bracket">))</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> 1 eval, 0 apply (function define statements don't evaluate the body so the only eval is the entire statement itself)</p></li><li><pre class="md-fences md-end-block" lang="lisp"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">((</span><span class="cm-variable">lambda</span> <span class="cm-bracket">(</span><span class="cm-variable">x</span> <span class="cm-variable">y</span><span class="cm-bracket">)</span> <span class="cm-bracket">(</span><span class="cm-variable">+</span> <span class="cm-bracket">(</span><span class="cm-variable">y</span> <span class="cm-number">2</span><span class="cm-bracket">)</span> <span class="cm-variable">x</span><span class="cm-bracket">))</span> <span class="cm-number">3</span> <span class="cm-variable">f</span><span class="cm-bracket">)</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> 16 eval, 5 apply</p><ul><li><p>1 eval for entire line: <code>((lambda (x y) (+ (y 2) x)) 3 f)</code> </p><ul><li><p>1 eval for operator: <code>(lambda (x y) (+ (y 2) x))</code></p></li><li><p>2 evals for operands: <code>3</code> and <code>f</code></p></li><li><p>1 apply for applying <code>(lambda (x y) (+ (y 2) x))</code> to <code>3</code> and <code>f</code></p><ul><li><p>1 eval for body: <code>(+ (y 2) x)</code></p><ul><li><p>1 eval for operator: <code>+</code></p></li><li><p>1 eval for operand: <code>(y 2)</code></p><ul><li><p>1 eval for operator: <code>y</code></p></li><li><p>1 eval for operand: <code>2</code></p></li><li><p>1 apply for applying <code>y</code> to <code>2</code> <em>(note: <code>y</code> is <code>f</code> from problem 1)</em></p><ul><li><p>1 eval for body: <code>(* ((lambda () x)) 2)</code></p><ul><li><p>1 eval for operator: <code>*</code></p></li><li><p>1 eval for operand: <code>((lambda () x))</code></p><ul><li><p>1 eval for operator: <code>(lambda () x)</code></p></li><li><p>1 apply for calling <code>(lambda () x)</code></p><ul><li>1 eval for: <code>x</code> (this is 2)</li></ul></li></ul></li><li><p>1 eval for operand: <em>2</em></p></li><li><p>1 apply for applying <code>*</code> to <code>2</code> <em>(what <code>((lambda () x))</code> evaluated to)</em> and <code>2</code></p></li></ul></li></ul></li></ul></li><li><p>1 eval for operand: <code>x</code></p></li><li><p>1 apply for applying <code>+</code> to <code>(y 2)</code> and <code>x</code> </p></li></ul></li></ul></li></ul></li></ul></li><li><pre class="md-fences md-end-block" lang="lisp"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">(</span><span class="cm-keyword">let</span> <span class="cm-bracket">((</span><span class="cm-variable">f</span> <span class="cm-number">4</span><span class="cm-bracket">)</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span> <span class="cm-bracket">(</span><span class="cm-variable">g</span> <span class="cm-variable">f</span><span class="cm-bracket">))</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span> <span class="cm-bracket">(</span><span class="cm-variable">g</span> <span class="cm-bracket">(</span><span class="cm-variable">+</span> <span class="cm-variable">f</span> <span class="cm-bracket">(</span><span class="cm-variable">eval</span> <span class="cm-bracket">(</span><span class="cm-keyword">define</span> <span class="cm-variable">h</span> <span class="cm-bracket">((</span><span class="cm-variable">lambda</span> <span class="cm-bracket">(</span><span class="cm-variable">a</span> <span class="cm-variable">b</span><span class="cm-bracket">)</span> <span class="cm-bracket">(</span><span class="cm-variable">*</span> <span class="cm-variable">a</span> <span class="cm-variable">b</span><span class="cm-bracket">))</span> <span class="cm-variable">f</span> <span class="cm-number">5</span><span class="cm-bracket">)))))</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">)</span></span></pre></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 112px;"></div><div class="CodeMirror-gutters" style="display: none; height: 142px;"></div></div></div></pre><p><strong>Solution:</strong> 26 eval, 7 apply</p><ul><li><p><strong>Important note:</strong> <code>eval</code> is an ordinary operator <strong>not</strong> a special form (eval it like any normal function)</p></li><li><p>1 eval for entire statement (we notice <code>let</code> is special form so we must evaluate according to the way <code>let</code> works)</p><ul><li><p>Assignment part:</p><ul><li>1 eval for <code>4</code> (which we assign to <code>f</code>)</li><li>1 eval for <code>f</code> (which we assign to <code>g</code>)</li></ul></li><li><p>Body (1 eval for entire body):</p><ul><li><p>1 eval for operator: <code>g</code></p></li><li><p>1 eval for operand: <code>(+ f (eval (define h ((lambda (a b) (* a b)) f 5))))</code></p><ul><li><p>1 eval for operator: <code>+</code></p></li><li><p>1 eval for operand: <code>f</code></p></li><li><p>1 eval for <code>(eval (define h ((lambda (a b) (* a b)) f 5)))</code></p><ul><li><p>1 eval for operator: <code>eval</code></p></li><li><p>1 eval for operand: <code>(define h ((lambda (a b) (* a b)) f 5))</code> <em>(Define is special form!)</em></p><ul><li><p>1 eval for: <code>((lambda (a b) (* a b)) f 5)</code></p><ul><li><p>1 eval for operator: <code>(lambda (a b) (* a b))</code></p></li><li><p>2 eval for operands: <code>f</code> and <code>5</code></p></li><li><p>1 apply for applying <code>(lambda (a b) (* a b))</code> to <code>f</code> and <code>5</code></p><ul><li>1 eval for operator: <code>*</code></li><li>2 evals for operands: <code>a</code> and <code>b</code></li><li>1 apply for applying <code>*</code> to <code>a</code> and <code>b</code></li></ul></li></ul></li><li><p>*Note: here the expression resembles: `h = (lambda a, b: a * b)(f, 5)<code>in python. In order to set</code>h<code>to the value of the right side we have to evaluate it to figure out what to assign</code>h` to*</p></li><li><p><em>In contrast, with function definitions we more closely resemble <code>def f(x): return x</code> importantly we <strong>don't</strong> evaluate the interior until it is called</em></p></li></ul></li><li><p>1 apply for applying <code>eval</code> to <code>(define h ((lambda (a b) (* a b)) f 5))</code> which evaluated to just <code>h</code> since define returns th variable just assigned. <code>eval</code> calls <code>scheme_eval</code> on the operand so we also have 1 eval in this part</p></li></ul></li><li><p>1 apply for applying <code>+</code> to <code>f</code> and <code>(define h ((lambda (a b) (* a b)) f 5))</code> which evaluates to 20</p></li></ul></li><li><p>1 apply for applying <code>g</code>, which evaluates to <code>f</code> from prob 1, onto <code>(+ f (eval (define h ((lambda (a b) (* a b)) f 5))))</code>, which evaluates to 24</p><ul><li><p>1 eval for function body: <code>(* ((lambda () x)) 2)</code></p><ul><li><p>1 eval for operator: <code>*</code></p></li><li><p>1 eval for operand: <code>((lambda () x))</code></p><ul><li><p>1 eval for operator: <code>(lambda () x)</code></p></li><li><p>1 apply for calling <code>(lambda () x)</code></p><ul><li>1 eval for: <code>x</code> which is 24</li></ul></li></ul></li><li><p>1 eval for operand: <code>2</code></p></li><li><p>1 apply for applying <code>*</code> to <code>2</code> and <code>x</code> or 24</p></li></ul></li></ul></li></ul></li></ul></li></ul></li><li><pre class="md-fences md-end-block" lang="lisp"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">(</span><span class="cm-variable">+</span> <span class="cm-bracket">(</span><span class="cm-variable">*</span> <span class="cm-bracket">(</span><span class="cm-keyword">if</span> <span class="cm-bracket">(</span><span class="cm-variable">and</span> <span class="cm-error">#t</span><span class="cm-bracket">)</span> <span class="cm-number">1</span> <span class="cm-number">2</span><span class="cm-bracket">)</span> <span class="cm-number">4</span><span class="cm-bracket">)</span> <span class="cm-bracket">(</span><span class="cm-variable">begin</span> <span class="cm-bracket">(</span><span class="cm-variable">+</span> <span class="cm-number">1</span> <span class="cm-number">2</span><span class="cm-bracket">)</span> <span class="cm-bracket">(</span><span class="cm-variable">+</span> <span class="cm-number">3</span> <span class="cm-number">4</span><span class="cm-bracket">)))</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> 17 eval, 4 apply</p><ul><li><p>1 eval for entire line</p><ul><li><p>1 eval for operator: <code>+</code></p></li><li><p>1 eval for operand: <code>(* (if (and #t) 1 2) 4)</code></p><ul><li><p>1 eval for operator: <code>*</code></p></li><li><p>1 eval for operand: <code>(if (and #t) 1 2)</code></p><ul><li><p><code>if</code> is special form so we follows the <code>if</code> eval process</p></li><li><p>1 eval for <code>(and #t)</code></p><ul><li><code>and</code> is special form so it will short circuit after the first true</li><li>1 eval for: <code>#t</code> and <code>and</code> ends due to short circuiting </li></ul></li><li><p>because the condition eval'd to true we only eval <code>1</code> in the then case</p></li></ul></li><li><p>1 eval for operand: 4</p></li><li><p>1 apply for applying <code>*</code> to <code>1</code> <em>(from if statement)</em> and <code>4</code></p></li></ul></li><li><p>1 eval for operand: <code>(begin (+ 1 2) (+ 3 4))</code></p><ul><li><p><code>begin</code> is special form so we follow <code>begin</code>'s evaling process (evaluate each then return last)</p></li><li><p>1 eval for <code>(+ 1 2)</code></p><ul><li><p>1 eval for operator: <code>+</code></p></li><li><p>2 evals for operand: <code>1</code> and <code>2</code></p><ul><li>1 apply for applying <code>+</code> to <code>1</code> and <code>2</code></li></ul></li></ul></li><li><p>1 eval for <code>(+ 3 4)</code></p><ul><li><p>1 eval for operator: <code>+</code></p></li><li><p>2 evals for operand: <code>3</code> and <code>4</code></p><ul><li>1 apply for applying <code>+</code> to <code>3</code> and <code>4</code></li></ul></li></ul></li></ul></li><li><p>1 apply for applying <code>+</code> to <code>(* (if (and #t) 1 2) 4)</code> and <code>(begin (+ 1 2) (+ 3 4))</code></p></li></ul></li></ul></li></ol><h3><a name='header-n311' class='md-header-anchor '></a>Generalized counting Eval/Apply</h3><p>Given these general forms, write the number 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E3-MJTT-79" x="2100" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E3-MJTT-5F" x="2625" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E3-MJTT-63" x="3150" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E3-MJTT-6F" x="3675" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E3-MJTT-75" x="4200" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E3-MJTT-6E" x="4725" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E3-MJTT-74" x="5250" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-3">\texttt{apply_count}</script>):</p><ol start='' ><li><pre class="md-fences md-end-block" lang="lisp"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">(</span><span class="cm-keyword">define</span> <span class="cm-variable">a</span> <span class="cm-bracket">(</span><span class="cm-variable">+</span> <span class="cm-variable">b</span> <span class="cm-variable">c</span><span class="cm-bracket">))</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> </p><ul><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-4-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="36.843ex" height="2.601ex" viewBox="0 -817.3 15862.9 1119.7" role="img" focusable="false" style="vertical-align: -0.702ex;"><defs><path stroke-width="0" id="E4-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E4-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E4-MJTT-65" d="M48 217Q48 295 100 361T248 439L258 440Q268 440 274 440Q329 438 369 416T428 359T456 292T464 228Q464 215 461 208T454 198T442 190L288 189H135L138 179Q153 132 199 102T303 71Q336 71 353 86T380 120T398 143Q404 146 422 146Q453 146 462 126Q464 120 464 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140Q132 115 157 93T224 70Q269 70 302 87T344 133Q346 139 347 175V211H339Q256 209 194 190T132 140Z"></path><path stroke-width="0" id="E4-MJTT-6C" d="M51 573Q51 602 73 610H76Q79 610 84 610T97 610T113 610T133 611T155 611T179 611H282Q301 598 304 586V76H452Q466 67 470 60T474 38Q474 10 452 1H73Q51 9 51 32V38Q51 54 54 60T73 76H220V535H146L73 536Q51 545 51 567V573Z"></path><path stroke-width="0" id="E4-MJTT-5F" d="M57 -60Q57 -33 86 -25H438Q468 -34 468 -60T438 -95H86Q57 -86 57 -60Z"></path><path stroke-width="0" id="E4-MJTT-63" d="M291 -6Q196 -6 131 60T66 216Q66 296 119 361Q154 403 200 421T273 439Q275 440 293 440H313Q400 440 433 409Q454 388 454 359Q454 335 439 321T402 306Q380 306 365 321T350 357V362L340 363Q339 363 326 363T303 364Q280 364 266 362Q217 352 184 313T151 215Q151 153 199 112T313 70Q341 70 357 85T381 118T394 140Q402 146 424 146Q443 146 447 144Q466 137 466 117Q466 106 457 88T429 47T374 10T291 -6Z"></path><path stroke-width="0" id="E4-MJTT-6F" d="M52 216Q52 318 118 379T261 440Q343 440 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d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path><path stroke-width="0" id="E4-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E4-MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-32" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-2B" x="722" y="0"></use><g transform="translate(1722,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJTT-65"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJTT-76" x="525" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJTT-61" x="1050" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJTT-6C" x="1575" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJTT-5F" x="2100" y="0"></use><use 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xlink:href="#E5-MJMATHI-63" x="16090" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E5-MJMAIN-29" x="16523" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-5">1+\texttt{apply_count}(b)+\texttt{apply_count}(c)</script> applies</li></ul></li><li><pre class="md-fences md-end-block" lang="lisp"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">(</span><span class="cm-keyword">if</span> <span class="cm-variable">a</span> <span class="cm-variable">b</span> <span class="cm-variable">c</span><span class="cm-bracket">)</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> </p><ul><li><p>if <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.229ex" height="1.405ex" viewBox="0 -516.9 529 604.8" role="img" focusable="false" style="vertical-align: -0.204ex;"><defs><path stroke-width="0" id="E8-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMATHI-61" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-8">a</script> evals to <code>True</code> then:</p><ul><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-6-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="33.065ex" height="2.601ex" viewBox="0 -817.3 14236.4 1119.7" role="img" focusable="false" style="vertical-align: -0.702ex;"><defs><path stroke-width="0" id="E6-MJTT-65" d="M48 217Q48 295 100 361T248 439L258 440Q268 440 274 440Q329 438 369 416T428 359T456 292T464 228Q464 215 461 208T454 198T442 190L288 189H135L138 179Q153 132 199 102T303 71Q336 71 353 86T380 120T398 143Q404 146 422 146Q453 146 462 126Q464 120 464 116Q464 84 416 39T285 -6Q187 -6 118 59T48 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transform="translate(7779,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJTT-65"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJTT-76" x="525" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJTT-61" x="1050" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJTT-6C" x="1575" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJTT-5F" x="2100" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJTT-63" x="2625" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJTT-6F" x="3150" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJTT-75" x="3675" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJTT-6E" x="4200" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJTT-74" x="4725" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-28" x="13029" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMATHI-62" x="13418" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-29" x="13847" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-6">\texttt{eval_count}(a) +\texttt{eval_count}(b)</script> evals</li><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-7-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="35.504ex" height="2.601ex" viewBox="0 -817.3 15286.4 1119.7" role="img" focusable="false" style="vertical-align: -0.702ex;"><defs><path stroke-width="0" id="E7-MJTT-61" d="M126 306Q105 306 90 321T74 359Q74 439 211 439Q268 439 276 438Q343 426 383 390T430 306Q431 301 431 190V81Q446 79 465 78T492 76T509 72T521 60T524 38Q524 11 506 3Q502 1 466 1Q426 1 406 5T379 14T355 36L345 30Q284 -6 205 -6Q135 -6 92 39T48 141Q48 182 79 212T158 256T252 278T342 285H347V290Q347 315 325 335T267 362Q258 363 224 363Q189 363 185 362H179L178 358Q178 353 178 352T176 345T174 337T170 330T165 322T158 316T150 311T139 308T126 306ZM132 140Q132 115 157 93T224 70Q269 70 302 87T344 133Q346 139 347 175V211H339Q256 209 194 190T132 140Z"></path><path stroke-width="0" id="E7-MJTT-70" d="M89 431Q93 431 104 431T121 432Q173 432 173 401V396L182 402Q237 437 305 437Q376 437 434 378T492 217Q492 146 459 93T382 17T291 -6Q261 -6 232 5T188 26L174 37Q173 37 173 -54V-146H236Q250 -155 254 -162T258 -184Q258 -213 236 -221H27Q4 -214 4 -184Q4 -169 8 -162T27 -146H89V355H58L27 356Q4 363 4 393Q4 408 8 415T27 431H89ZM409 215Q409 269 377 315T283 361Q255 361 224 344T177 297L173 290V167Q189 124 213 97T278 70Q330 70 369 111T409 215Z"></path><path stroke-width="0" id="E7-MJTT-6C" d="M51 573Q51 602 73 610H76Q79 610 84 610T97 610T113 610T133 611T155 611T179 611H282Q301 598 304 586V76H452Q466 67 470 60T474 38Q474 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E10-MJTT-74" x="5250" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E10-MJMAIN-28" x="14079" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E10-MJMATHI-63" x="14468" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E10-MJMAIN-29" x="14901" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-10">\texttt{apply_count}(a)+\texttt{apply_count}(c)</script> applies</li></ul></li></ul></li><li><pre class="md-fences md-end-block" lang="lisp"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">(</span><span class="cm-keyword">define</span> <span class="cm-bracket">(</span><span class="cm-variable">fn</span> <span class="cm-variable">a</span> <span class="cm-variable">b</span><span class="cm-bracket">)</span> <span class="cm-variable">c</span><span class="cm-bracket">)</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> 1 eval, 0 apply <em>(we don't evaluate the interior of a function <strong>ever</strong>)</em></p></li><li><pre class="md-fences md-end-block" lang="lisp"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">(</span><span class="cm-keyword">let</span> <span class="cm-bracket">((</span><span class="cm-variable">a</span> <span class="cm-variable">b</span><span class="cm-bracket">)</span> <span class="cm-bracket">(</span><span class="cm-variable">c</span> <span class="cm-variable">d</span><span class="cm-bracket">))</span> <span class="cm-variable">e</span><span class="cm-bracket">)</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> </p><ul><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="50.974ex" height="2.601ex" viewBox="0 -817.3 21946.9 1119.7" role="img" focusable="false" style="vertical-align: -0.702ex;"><defs><path stroke-width="0" id="E11-MJTT-65" d="M48 217Q48 295 100 361T248 439L258 440Q268 440 274 440Q329 438 369 416T428 359T456 292T464 228Q464 215 461 208T454 198T442 190L288 189H135L138 179Q153 132 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E12-MJMAIN-29" x="23132" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-12">\texttt{apply_count}(b)+\texttt{apply_count}(d)+\texttt{apply_count}(e)</script> applies</li></ul></li><li><pre class="md-fences md-end-block" lang="lisp"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">((</span><span class="cm-variable">lambda</span> <span class="cm-bracket">(</span><span class="cm-variable">a</span><span class="cm-bracket">)</span> <span class="cm-variable">b</span><span class="cm-bracket">)</span> <span class="cm-variable">c</span><span class="cm-bracket">)</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> </p><ul><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-13-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="36.843ex" height="2.601ex" viewBox="0 -817.3 15862.9 1119.7" role="img" focusable="false" style="vertical-align: -0.702ex;"><defs><path stroke-width="0" id="E13-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 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class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">(</span><span class="cm-variable">and</span> <span class="cm-variable">a</span> <span class="cm-variable">b</span> <span class="cm-variable">c</span><span class="cm-bracket">)</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> The number of evals and applies is dependent on <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-15-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.296ex" height="2.401ex" viewBox="0 -774.4 2280.3 1033.9" role="img" focusable="false" style="vertical-align: -0.603ex;"><defs><path stroke-width="0" id="E15-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 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79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path><path stroke-width="0" id="E15-MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E15-MJMATHI-61" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E15-MJMAIN-2C" x="529" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E15-MJMATHI-62" x="973" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E15-MJMAIN-2C" x="1402" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E15-MJMATHI-63" x="1847" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-15">a,b,c</script> (because of short circuiting)</p><ul><li><p>If <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.229ex" height="1.405ex" viewBox="0 -516.9 529 604.8" role="img" focusable="false" style="vertical-align: -0.204ex;"><defs><path stroke-width="0" id="E8-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E8-MJMATHI-61" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-8">a</script> evals to <code>False</code> then we have the minimum number of evals and applies (<code>and</code> short circuits on the first <code>False</code> value)</p><ul><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-16-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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xlink:href="#E21-MJTT-5F" x="2625" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E21-MJTT-63" x="3150" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E21-MJTT-6F" x="3675" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E21-MJTT-75" x="4200" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E21-MJTT-6E" x="4725" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E21-MJTT-74" x="5250" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E21-MJMAIN-28" x="22283" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E21-MJMATHI-63" x="22672" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E21-MJMAIN-29" x="23105" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-21">\texttt{apply_count}(a)+\texttt{apply_count}(b)+\texttt{apply_count}(c)</script> applies</li></ul></li></ul></li><li><pre class="md-fences md-end-block" lang="lisp"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 4px; left: 4px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">(</span><span class="cm-variable">begin</span> <span class="cm-variable">a</span> <span class="cm-variable">b</span> <span class="cm-variable">c</span><span class="cm-bracket">)</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 34px;"></div><div class="CodeMirror-gutters" style="display: none; height: 64px;"></div></div></div></pre><p><strong>Solution:</strong> </p><ul><li><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-20-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" 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