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+<!DOCTYPE html>
+<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
+<head>
+  <meta charset="utf-8" />
+  <meta name="generator" content="pandoc" />
+  <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
+  <meta name="author" content="Keith A. Lewis" />
+  <meta name="dcterms.date" content="2024-11-29" />
+  <title>APL</title>
+  <style>
+    code{white-space: pre-wrap;}
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+  <link rel="stylesheet" href="math.css" />
+  <script defer=""
+  src="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.js"></script>
+  <script>document.addEventListener("DOMContentLoaded", function () {
+ var mathElements = document.getElementsByClassName("math");
+ var macros = [];
+ for (var i = 0; i < mathElements.length; i++) {
+  var texText = mathElements[i].firstChild;
+  if (mathElements[i].tagName == "SPAN") {
+   katex.render(texText.data, mathElements[i], {
+    displayMode: mathElements[i].classList.contains('display'),
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+   });
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+</head>
+<body>
+<header id="title-block-header">
+<h1 class="title">APL</h1>
+<p class="author">Keith A. Lewis</p>
+<p class="date">November 29, 2024</p>
+<div class="abstract">
+<div class="abstract-title">Abstract</div>
+A Programming Language
+</div>
+</header>
+<p>Let’s give names to things.</p>
+<section id="primitive-type" class="level2">
+<h2>Primitive Type</h2>
+<p>The <em>primitive types</em> are booleans <span
+class="math inline">\mathbf{B}</span>, integers <span
+class="math inline">\mathbf{Z}</span>, non-negative integers <span
+class="math inline">\mathbf{N}</span>, real numbers <span
+class="math inline">\mathbf{R}</span>, and characters <span
+class="math inline">\mathbf{C}</span>. We have <span
+class="math inline">\mathbf{B}\subseteq\mathbf{N}\subseteq\mathbf{Z}\subseteq\mathbf{R}</span>
+as rings. We assume <span
+class="math inline">u\colon\mathbf{C}\to\mathbf{N}</span> is an encoding
+of the set of characters. (E.g., UTF-8 codepoints)</p>
+<p>Primitive types are classified by mathematical objects: set, monoid,
+group, ring, field: boolean (field), integer (commutative ring), natural
+number (monoid), real number (field), character (set).</p>
+<p>We write <span class="math inline">\mathbf{B}=
+\{\square,\blacksquare\}</span> where <span
+class="math inline">\square</span> represents false and <span
+class="math inline">\blacksquare</span> represents true.</p>
+<p><span class="math inline">(A\times B)\to C\cong A\to (B\to C)</span>
+by <span class="math inline">f(a,b) = c \leftrightarrow f,(a)b =
+c</span>.</p>
+<p><span class="math inline">A\to (B\to C) \cong (A\times B)\to C</span>
+by <span class="math inline">g(a)b = c \leftrightarrow g`(a,b) =
+c</span>.</p>
+</section>
+<section id="type" class="level2">
+<h2>Type</h2>
+<p>If <span class="math inline">X</span>, <span
+class="math inline">X_i</span> are types</p>
+<p><em>array</em> <span class="math inline">x = [x_1,\ldots,x_n]\in
+X^n\cong n\to X</span>, <span class="math inline">x[i] = x_i</span>.</p>
+<p><span class="math inline">(I\times n)^A x (-n\times I)^A \to
+I^A</span></p>
+<p><em>tuple</em> <span class="math inline">x = \langle
+x_1,\ldots,x_n\rangle\in\prod X_{i\in\mathbf{N}}</span>, <span
+class="math inline">n\to(\prod X_i\to X_i)</span>, <span
+class="math inline">n\mapsto π_i</span>, <span
+class="math inline">x\langle i\rangle = π_i(x) = x_i</span>.</p>
+<p><em>union</em> <span class="math inline">x = \{\ldots,x_i,\ldots\} =
+\langle i,x_i\rangle\in\coprod_{i\in n} X_i</span>, <span
+class="math inline">n\to(X_i\to\coprod X_i)</span>, <span
+class="math inline">n\mapsto ν_i</span>, <span
+class="math inline">x\{i\} = ν_i(x) = \langle i,x_i\rangle</span>.</p>
+<p><em>sequence</em> <span class="math inline">x = (x_0,\ldots,x_{n-1})
+\in X^* = \coprod_{n\in\mathbf{N}} X^n</span> and define <span
+class="math inline">*x = x_0</span>, <span class="math inline">+x =
+(x_1,\ldots,x_{n-1})</span>, <span class="math inline">?x</span> if and
+only if <span class="math inline">n\ge 0</span>, i.e., <span
+class="math inline">x</span> is not empty. Note <span
+class="math inline">x\in\operatorname{dom}*</span> iff <span
+class="math inline">?x</span>. Define <span class="math inline">x =
+y</span> if <span class="math inline">*x = *y</span> and <span
+class="math inline">+x = +y</span> or if <span
+class="math inline">x</span> and <span class="math inline">y</span> are
+both empty.</p>
+<p>The function <em>concatenate</em> <span class="math inline">,\colon
+X^*\times X^*\to X^*</span> written <span class="math inline">,(x,y) =
+x,y</span> is defined by <span class="math inline">*(x,y) = *x</span> if
+<span class="math inline">?x</span> and <span
+class="math inline">*y</span> if not <span
+class="math inline">?x</span>, <span class="math inline">+(x,y) =
+(+x,y)</span> if <span class="math inline">?x</span> and <span
+class="math inline">(x,+y)</span> if not <span
+class="math inline">?x</span>, <span class="math inline">?(x,y) = ?x\vee
+?y</span> where <span class="math inline">\vee</span> is logical or.
+Note <span class="math inline">x,() = x</span> and <span
+class="math inline">(),y = y</span> for <span class="math inline">x,y\in
+X^*</span> so sequences are a monoid under concatenation with unit the
+empty sequence.</p>
+<p>If <span class="math inline">x = (x_0,\ldots,x_{n-1})</span> is a
+sequence let <span class="math inline">[x] = [x_0,\ldots,x_{n-1}]\in
+X^n</span> be the array it determines and <span
+class="math inline">\langle x\rangle = \langle
+x_0,\ldots,x_{n-1}\rangle</span> be the (homogeneous) tuple it
+determines.</p>
+<p><em>exponential</em> <span class="math inline">f\in Y^X = \{f\colon
+X\to Y\}</span> is the set of all functions from <span
+class="math inline">X</span> to <span class="math inline">Y</span>. We
+also write <span class="math inline">X\to Y</span> for <span
+class="math inline">Y^X</span>.</p>
+<p><em>relation</em> <span class="math inline">R\subseteq X\times
+Y</span>. Write <span class="math inline">xRy</span> for <span
+class="math inline">\langle x,y\rangle\in R</span> and <span
+class="math inline">R&#39; = \{\langle y, x\rangle \mid xRy\}\subseteq
+Y\times X</span> for its <em>transpose</em>. The <em>domain</em> of
+<span class="math inline">R</span> is <span
+class="math inline">\operatorname{dom}R = π_X(R)\subseteq X</span> and
+the <em>range</em> is <span class="math inline">\operatorname{ran}R =
+π_Y(R)\subseteq Y</span>. The <em>right coset</em> <span
+class="math inline">xR = \{y\in Y \mid xRy\}\subseteq Y</span> for <span
+class="math inline">x\in X</span>, the <em>left coset</em> <span
+class="math inline">Ry = \{x\in X \mid xRy\}\subseteq X</span> for <span
+class="math inline">y\in Y</span>. The right cosets <span
+class="math inline">\{xR \mid x\in X\}</span> parition <span
+class="math inline">\operatorname{ran}R</span> and the left cosets <span
+class="math inline">\{Ry \mid y\in Y\}</span> partition <span
+class="math inline">\operatorname{dom}R</span>.</p>
+<p>The <em>graph</em> of a function <span class="math inline">f\in
+Y^X</span> is <span class="math inline">\{\langle x,y\rangle \mid f(x) =
+y\}\subseteq X\times Y</span>. A relation <span
+class="math inline">R</span> is a function if the <em>right coset</em>
+<span class="math inline">xR</span> is a singleton for all <span
+class="math inline">x\in X</span> and we write <span
+class="math inline">R(x)</span> for that unique element.</p>
+<p><em>itoa</em> is <span class="math inline">ι =
+(0,1,2,\ldots)\in\mathbf{N}^*</span></p>
+</section>
+<section id="function" class="level2">
+<h2>Function</h2>
+<p>The function <em>take</em> is <span
+class="math inline">\uparrow\colon\mathbf{N}\times X^*\to X^*</span>
+defined by <span class="math inline">\uparrow(n,x) = (*x), \uparrow(n -
+1, +x))</span>, <span class="math inline">\uparrow(0,x) = ()</span>,
+<span class="math inline">\uparrow(n,()) = ()</span>, where <span
+class="math inline">(x_0, ()) = x_0</span>.</p>
+<p>The function <em>drop</em> is <span
+class="math inline">\downarrow\colon\mathbf{N}\times X^*\to X^*</span>
+defined by <span class="math inline">\downarrow(n,x) = \downarrow(n - 1,
++x)</span>, <span class="math inline">\downarrow(0,x) = x</span>, <span
+class="math inline">\downarrow(n,()) = ()</span>.</p>
+<p>The function <em>itoa</em> is <span
+class="math inline">ι\colon\mathbf{N}\to \mathbf{N}^*</span> <span
+class="math inline">ι\mapsto (0,\ldots,n-1)</span>.</p>
+</section>
+<section id="operation" class="level2">
+<h2>Operation</h2>
+<p>The <em>evaluation function</em> <span class="math inline">ε_Y^X =
+ε\colon (X\to Y)\times X\to Y</span> is <span class="math inline">ε(f,x)
+= f(x)\in Y</span>, <span class="math inline">f\in Y^X</span>, <span
+class="math inline">x\in X</span>.</p>
+<p>monoid folds <span class="math inline">\star\colon X\times X\to
+X</span></p>
+<p>left fold <span class="math inline">(\star\colon X^n\to X</span>,
+<span class="math inline">(\star(x_0,\ldots,x_n) = x_0\star(x_1\dots
+x_n)</span></p>
+<p>right fold <span class="math inline">)\star\colon X^n\to X</span>,
+<span class="math inline">)\star(x_0,\ldots,x_n) = (x_0\dots
+x_{n-1})\star x_n</span></p>
+<section id="curry" class="level3">
+<h3>Curry</h3>
+<p>The <em>curry function</em> <span class="math inline">γ\colon
+((X\times Y)\to Z)\to (X\to(Y\to Z))</span> is <span
+class="math inline">(γ(f)(x))(y) = f(x,y)\in Z</span>, <span
+class="math inline">f\in X\times Y\to Z</span>, <span
+class="math inline">x\in X</span>, <span class="math inline">y\in
+Y</span>.</p>
+<p>The <em>uncurry function</em> <span class="math inline">γ^*\colon
+((X\to (Y\to Z))\to (X\times Y\to Z)</span> is <span
+class="math inline">(γ^*(f))(x,y) = f(x,y)\in Z</span>, <span
+class="math inline">f\in X\to(Y\to Z)</span>, <span
+class="math inline">x\in X</span>, <span class="math inline">y\in
+Y</span>.</p>
+<p>Use <span class="math inline">.</span> for composition. If <span
+class="math inline">f\colon X\to Y</span> identify <span
+class="math inline">x</span> with the constant function <span
+class="math inline">K_x</span>.</p>
+<p>Interpret <span class="math inline">f.x</span> as <span
+class="math inline">f.K_x</span> so <span class="math inline">f.x(y) =
+f.K_x(y) = f(K_x(y)) = f(x)</span> for all <span
+class="math inline">y</span>.</p>
+<p>If <span class="math inline">x\in X</span>, <span
+class="math inline">y\in Y</span> write <span
+class="math inline">x,y</span> for <span class="math inline">(x,y)\in
+X\times Y</span>.</p>
+</section>
+</section>
+<section id="shape" class="level2">
+<h2>Shape</h2>
+<p>The <em>shape</em> function is <span
+class="math inline">ρ\colon\mathbf{N}^k\to
+(\mathbf{N}^{k-1}\to\mathbf{N})</span>, <span
+class="math inline">ρ\mapsto(n \mapsto n)</span>, <span
+class="math inline">ρ(n,\ldots)\mapsto ((i,\ldots) \mapsto i + n
+ρ(\ldots)(\ldots))</span></p>
+<section id="partition" class="level3">
+<h3>Partition</h3>
+<p>A function <span class="math inline">p\colon X\to\iota n</span>
+determines a <em>partition</em> on <span class="math inline">X</span>
+where <span class="math inline">i\sim i&#39;</span> iff <span
+class="math inline">p(i) = p(i&#39;)</span>.</p>
+<p>Example: s = “a,bc,def” is a string in <span
+class="math inline">\mathbf{C}^{\iota 8}</span>. Using cyclic arrays
+<span class="math inline">= &quot;,&quot; s</span> is <span
+class="math inline">\square\blacksquare\square\square\blacksquare\square\square\square</span>
+so <span class="math inline">+\backslash= &quot;,&quot; s</span> is
+<span class="math inline">0111222</span>.</p>
+<p>Define <span class="math inline">ν\colon X^n\to X^m</span> by <span
+class="math inline">νx_0 x_1 \ldots = x_0 \nu x_2 \ldots</span> if <span
+class="math inline">x1 = x0</span> and <span class="math inline">νx_0
+x_1 \ldots = x_0 \nu x_1 \ldots</span> if <span class="math inline">x_1
+\not= x_0</span>.</p>
+<!--
+## Remarks
+
+Natural numbers $\NN = \{0, 1, \ldots\}$.
+
+$\iota n = \{0, \ldots, n - 1\}$, $n\in\NN$.
+
+$\iota\colon\NN\to\NN^*$ where $\NN^* = \coprod{n\ge0}\NN^n$.  
+$\iota n\in\NN^n$.
+
+Every expression is a list of data or functions in Polish notation.
+
+Expressions are parsed right to left.
+
+Data is pushed on the call stack.
+
+Functions know what is on the call stack.
+
+_Each_ is $f␣\scan␣x␣y␣\ldots$ is $f␣x␣f␣y␣\ldots$, $f␣g\scan ␣x$ is $f␣x␣g␣x$.
+
+_Fold_ is $f \fold a␣b \ldots$ is $f␣a␣f␣b␣\ldots))$.
+
+$K$ is the constant function $K\colon X\to\{Y\to X\}$ where $Kxy = x$.
+
+Use $!$ for evaluation. 
+
+_Array_ $X^{\iota n}$, $[␣x_0␣x_1␣\ldots\colon \iota n\to X$ by $i\mapsto x_i$.
+
+_Tuple_ $\sqcap_{i\in I} X_i$, $\langle␣x_0␣x_1␣\ldots\colon I\to\coprod{i\in I}X_i$ by $i\mapsto (i, x_i)$.
+
+## Examples
+
+_Average_ $\div␣\stop␣\add\fold␣\add\fold K 1\scan␣x_0␣x_1␣\ldots$
+
+## Types
+
+_boolean_ $\BB$.
+
+_integer_ $\ZZ$.
+
+_unsigned_ $\NN$.
+
+_number_ $\RR$.
+
+_character_ $\CC$ utf-8 _code points_.
+
+## Structures
+
+
+
+Storage: stack (life of function), heap (life of process), persistent (across processes), lazy generators.
+-->
+</section>
+</section>
+</body>
+</html>
diff --git a/docs/a_z.html b/docs/a_z.html
@@ -117,9 +117,9 @@ <h2>Derivation</h2>
 <p>Let’s derive the less unremarkable fact <span class="math inline">(x
 + y)^2 = x^2 + 2xy + y^2</span> using this technique. You may have
 learned the FOIL method for this – First, Outer, Inner, Last so <span
-class="math inline">(x + y)^2 = (x + y)(x + y) = xx + xy + yx + yy = x^2
-+ 2xy + y^2</span>. Mathematics uses axioms and substituion for rigorous
-derivations instead of cute mnemonics.</p>
+class="math inline">(x + y)^2 = (x + y)(x + y)
+= xx + xy + yx + yy = x^2 + 2xy + y^2</span>. Mathematics uses axioms
+and substituion for rigorous derivations instead of cute mnemonics.</p>
 <p><em>Derivation</em>. Start by writing <span
 class="math inline">A</span></p>
 <p><span class="math inline">\underline{(x + y)}^2</span></p>