From 915885e094bfcae912527f3c1f61b85621d51577 Mon Sep 17 00:00:00 2001
From: lmoss <larrymoss@gmail.com>
Date: Wed, 8 Jan 2025 19:46:39 -0500
Subject: [PATCH] changed arithmetic notebook

---
 _build/.doctrees/environment.pickle           | Bin 939194 -> 944846 bytes
 .../more_programs/arithmetic.doctree          | Bin 57870 -> 60344 bytes
 .../_sources/more_programs/arithmetic.ipynb   | 191 +++++++++---------
 _build/html/genindex.html                     |   4 +-
 _build/html/index.html                        |   4 +-
 _build/html/more_programs/arithmetic.html     |  76 ++++---
 _build/html/objects.inv                       | Bin 1187 -> 1187 bytes
 _build/html/prf-prf.html                      |   4 +-
 _build/html/search.html                       |   4 +-
 _build/html/searchindex.js                    |   2 +-
 .../more_programs/arithmetic.ipynb            |  31 +--
 _toc.yml                                      |   2 +-
 more_programs/arithmetic.ipynb                | 191 +++++++++---------
 13 files changed, 278 insertions(+), 231 deletions(-)

diff --git a/_build/.doctrees/environment.pickle b/_build/.doctrees/environment.pickle
index 8b9e7d21744787665120ed54aeb0a0414c3c1957..51ae3179666f2ee37e9afc0c6be0ab4e13eca7f8 100644
GIT binary patch
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diff --git a/_build/.doctrees/more_programs/arithmetic.doctree b/_build/.doctrees/more_programs/arithmetic.doctree
index 308110501e1c7ab1613b9a3d83a34885ad3e543f..5ef65431ea01815f839704e05976f19546677af1 100644
GIT binary patch
delta 7412
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diff --git a/_build/html/_sources/more_programs/arithmetic.ipynb b/_build/html/_sources/more_programs/arithmetic.ipynb
index e616a964..f0d6417b 100644
--- a/_build/html/_sources/more_programs/arithmetic.ipynb
+++ b/_build/html/_sources/more_programs/arithmetic.ipynb
@@ -1,47 +1,31 @@
 {
-  "nbformat": 4,
-  "nbformat_minor": 0,
-  "metadata": {
-    "colab": {
-      "provenance": [],
-      "toc_visible": true,
-      "authorship_tag": "ABX9TyPhPjEe24JYXQSUVGBFiwOH",
-      "include_colab_link": true
-    },
-    "kernelspec": {
-      "name": "python3",
-      "display_name": "Python 3"
-    },
-    "language_info": {
-      "name": "python"
-    }
-  },
   "cells": [
     {
       "cell_type": "markdown",
       "metadata": {
-        "id": "view-in-github",
-        "colab_type": "text"
+        "colab_type": "text",
+        "id": "view-in-github"
       },
       "source": [
         "<a href=\"https://colab.research.google.com/github/lmoss/onesharp/blob/main/arithmetic.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
       ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "mx5eAo_aRljE"
+      },
       "source": [
-        "## Programs for Arithmetic\n",
+        "## Programs for arithmetic\n",
         "\n",
-        "The main point of working with 1# is that it gives an explicit treatment of some of the most important results of the theory of computation. This was the reason the language was formulated and developed. Nobody ever intended 1# to be used very widely.\n",
+        "The main point of working with ```1#``` is that it gives an explicit treatment of some of the most important results of the theory of computation. This was the reason the language was formulated and developed. Nobody ever intended 1# to be used very widely.\n",
         "\n",
-        "At the same time, once we have 1# or some other programming language, it will be of some interest to check that we really can use it to do computation. This lesson is mainly a verification that you can use 1# to send a rocket to the moon. That is, it hints that one could theoretically use 1# to do any numerical computations that could be done with any computer. So this backs up the claim that our very simple language 1# is computationally complete.\n",
+        "At the same time, once we have ```1#``` or some other programming language, it will be of some interest to check that we really can use it to do computation. This lesson is mainly a verification that you can use 1# to send a rocket to the moon. That is, it hints that one could theoretically use 1# to do any numerical computations that could be done with any computer. So this backs up the claim that our very simple language 1# is computationally complete.\n",
         "\n",
-        "For the most part, this lesson is optional. The specific results on arithmetic are not going to be used later. The definitions of primitive recursive function and computable function are important in computability theory. But if you already know them from other sources and are willing to believe that 1# is a Turing-complete formalism, then you should feel free to work through this lesson quickly, or even to skip it altogether. .\n",
+        "For the most part, this lesson is optional. The specific results on arithmetic are not going to be used later. The definitions of primitive recursive function and computable function are important in computability theory. But if you already know them from other sources and are willing to believe that ```1#``` is a Turing-complete formalism, then you should feel free to work through this lesson quickly, or even to skip it altogether. .\n",
         "\n"
-      ],
-      "metadata": {
-        "id": "mx5eAo_aRljE"
-      }
+      ]
     },
     {
       "cell_type": "code",
@@ -58,6 +42,9 @@
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "eLajSKX_R2Gz"
+      },
       "source": [
         "# bb representation\n",
         "\n",
@@ -103,13 +90,13 @@
         "</td>\n",
         "</table>\n",
         "<br>\n"
-      ],
-      "metadata": {
-        "id": "eLajSKX_R2Gz"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "GJYkLkUGSmrQ"
+      },
       "source": [
         "We can't use the binary representation directly, since we don't have a symbol 0 around. So we change the 0s to #s. And after that, we turn the number around, getting the backwards binary representation. Here are the first 20 numbers again, in this form:\n",
         "\n",
@@ -148,26 +135,28 @@
         "</TABLE>        \n",
         "</td>\n",
         "</table>\n"
-      ],
-      "metadata": {
-        "id": "GJYkLkUGSmrQ"
-      }
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "1Vm2_hSaSv_x"
+      },
       "source": [
-        "### Removing superfluous #s\n",
+        "### Removing superfluous ```#```s\n",
         "\n",
-        "One touchy point about bb representations is that all bb representations should either be the word # or else end in a 1. Here is a program that strips off superfluous #s at the end of a word in R1. It uses the first three registers It also converts an empty R1 to a #.\n",
+        "One touchy point about bb representations is that all bb representations should either be the word # or else end in a ```1```. Here is a program that strips off superfluous ```#```s at the end of a word in R1. It uses the first three registers It also converts an empty R1 to a ```#```.\n",
         "\n",
         "```1##### 1111111111 11111111111 11111111111 11111111111 1111### 111### 11## 11### 11# 1##### 1111111111 1111111111### 1111111111### 11##### 1### 1111### 111## 11## 11111111#### 111# 11## 1111111111 1#### 11##### 1### 1111### 111## 11# 1111111111 1111111#### 111# 11# 1111111111 1111111111#### 11##### 1111111111### 1111111111### 111##### 111111### 111### 1## 1111#### 1# 111111#### 1111111111 11111111111 11111111111 11111#### 1111111111 1### 111# 111##### 111111### 111### 1## 1111#### 1# 111111#### 11### 1##```\n"
-      ],
-      "metadata": {
-        "id": "1Vm2_hSaSv_x"
-      }
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "ytRP-YfJS4g6"
+      },
       "source": [
         "\n",
         "\n",
@@ -206,46 +195,48 @@
         "</TABLE>        \n",
         "\n",
         "\n",
-        "Here is how the program works: as long as a 1 is seen in R1, a # is put in R2; the program then loops to the top. On encountering the first #, the program goes to the blue part: it adds a 1 in R2 and then copies the rest of R1 to the end of R2. If the original input is empty, a 1 is put in R2. And after all of this, R2 is moved back to R1. The full code is shown below.\n",
+        "Here is how the program works: as long as a ```1``` is seen in R1, a ```#``` is put in R2; the program then loops to the top. On encountering the first ```#```, the program goes to the blue part: it adds a ```1``` in R2 and then copies the rest of R1 to the end of R2. If the original input is empty, a ```1``` is put in R2. And after all of this, R2 is moved back to R1. The full code is shown below.\n",
         "\n",
         "1##### 1111111111 111### 1111111111### 11# 1##### 111111### 111### 11## 1111#### 11# 111111#### 1111### 11## 1111111111 111#### 11# 11##### 111111### 111### 1## 1111#### 1# 111111####\n",
         "\n",
         "\n",
         "\n",
         "\n"
-      ],
-      "metadata": {
-        "id": "ytRP-YfJS4g6"
-      }
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "rQGxHtJEVEdy"
+      },
       "source": [
-        "###Comparison\n",
+        "### Comparison\n",
         "\n",
-        "Here is a program compare which takes words in R1 and R2 and decides if they are symbol-for-symbol the same. If so, it outputs a 1 in R1; if not, it leaves R1 empty.\n",
+        "Here is a program compare which takes words in R1 and R2 and decides if they are symbol-for-symbol the same. If so, it outputs a ```1``` in R1; if not, it leaves R1 empty.\n",
         "\n",
         "1##### 111111### 111111111### 11##### 1111111111 1### 1111111111### 111111#### 11##### 1111111111 11111### 111111### 11111### 11##### 111### 1111111111 111#### 1### 1##### 111### 11#### 111#### 11##### 1111### 11#### 111#### 1#\n",
         "\n",
         "The original contents of R1 and R2 are destroyed in the process.\n"
-      ],
-      "metadata": {
-        "id": "rQGxHtJEVEdy"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "<img src=\"https://github.com/lmoss/onesharp/blob/main/pianotrumpet.jpg?raw=1\" width=\"200\" height=\"160\">"
-      ],
       "metadata": {
         "id": "BnRthH3zU7rR"
-      }
+      },
+      "source": [
+        "<img src=\"https://github.com/lmoss/onesharp/blob/main/pianotrumpet.jpg?raw=1\" width=\"200\" height=\"160\">"
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "87sh9zNERS52"
+      },
       "source": [
-        "#Addition\n",
+        "# Addition\n",
         "\n",
         "Here is a program add which adds numbers. Given m in R1 and n in R2 (represented in bb), the program computes the sum m + n (again, represented in bb) and puts it in R1. At the end, all other registers are empty (and in particular the original inputs m and n are lost).\n",
         "\n",
@@ -256,24 +247,25 @@
         "\n",
         "\n",
         "\n"
-      ],
-      "metadata": {
-        "id": "87sh9zNERS52"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "<img src=\"https://github.com/lmoss/onesharp/blob/main/harp.jpg?raw=1\" width=\"200\" height=\"160\">"
-      ],
       "metadata": {
         "id": "yijcynmaU3Ko"
-      }
+      },
+      "source": [
+        "<img src=\"https://github.com/lmoss/onesharp/blob/main/harp.jpg?raw=1\" width=\"200\" height=\"160\">"
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "ZjTIbR-QTLFN"
+      },
       "source": [
-        "#Multiplication\n",
+        "# Multiplication\n",
         "\n",
         "Having seen a program for addition, we next look at multiplication.\n",
         "\n",
@@ -386,63 +378,78 @@
         "1##### 111### 11#### 111#### 11##### 111### 11#### 111#### 111##### 111### 11#### 111#### 1111##### 111111### 111### 1## 1111#### 1# 111111####\n",
         "\n",
         "The parts of the program that are shaded red are special to the multiplication algorithm. The point is that if we wish to write a program for something similar, we could keep the outside \"shell\" and just modify the red lines above. Your first exercise below asks you to do just that.\n"
-      ],
-      "metadata": {
-        "id": "ZjTIbR-QTLFN"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "<img src=\"https://github.com/lmoss/onesharp/blob/main/drum.jpg?raw=1\" width=\"200\" height=\"160\">"
-      ],
       "metadata": {
         "id": "Luszku-QUrPm"
-      }
+      },
+      "source": [
+        "<img src=\"https://github.com/lmoss/onesharp/blob/main/drum.jpg?raw=1\" width=\"200\" height=\"160\">"
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "ghMwz69oUbcL"
+      },
       "source": [
         "```{exercise}\n",
-        "Modify the multiplication program to compute exponentiation. This would be a program exp with the following property: when started with the bb representation of m in R1 and n in R2, the program eventually halts with the bb representation of mn in R1.\n",
+        "Modify the multiplication program to compute exponentiation. This would be a program ```exp``` with the following property: when started with the bb representation of m in R1 and n in R2, the program eventually halts with the bb representation of mn in R1.\n",
         "```"
-      ],
-      "metadata": {
-        "id": "ghMwz69oUbcL"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "S1Ikp4jAUayy"
+      },
       "source": [
         "```{exercise}\n",
         "In our work on multiplication, we assumed addition and worked with it to get multiplication. Now forget about our program for addition, and using only the program ```succ``` for the successor function, construct a program for addition.\n",
         "```"
-      ],
-      "metadata": {
-        "id": "S1Ikp4jAUayy"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "wJj8FgM6Ub9e"
+      },
       "source": [
         "```{exercise}\n",
         "Write a program which, when started with n in bb in R1, gives n! in R1. [Here n! is the factorial function. Remember that 0! = 1 and (n+1)! = (n+1) x n!.]\n",
         "```\n"
-      ],
-      "metadata": {
-        "id": "wJj8FgM6Ub9e"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "_V4TPS-_Un-v"
+      },
       "source": [
         "```{exercise}\n",
         "Write two programs which convert back and forth from unary to binary representation.\n",
         "```"
-      ],
-      "metadata": {
-        "id": "_V4TPS-_Un-v"
-      }
+      ]
+    }
+  ],
+  "metadata": {
+    "colab": {
+      "authorship_tag": "ABX9TyPhPjEe24JYXQSUVGBFiwOH",
+      "include_colab_link": true,
+      "provenance": [],
+      "toc_visible": true
+    },
+    "kernelspec": {
+      "display_name": "Python 3",
+      "name": "python3"
+    },
+    "language_info": {
+      "name": "python"
     }
-  ]
+  },
+  "nbformat": 4,
+  "nbformat_minor": 0
 }
diff --git a/_build/html/genindex.html b/_build/html/genindex.html
index 1d12db94..54fcddd1 100644
--- a/_build/html/genindex.html
+++ b/_build/html/genindex.html
@@ -203,7 +203,9 @@
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="more_programs/sanity.html">Tools to help write programs</a></li>
 
-<li class="toctree-l1"><a class="reference internal" href="more_programs/arithmetic.html">Programs for Arithmetic</a></li>
+<li class="toctree-l1"><a class="reference internal" href="more_programs/arithmetic.html">Programs for arithmetic</a></li>
+
+
 
 
 <li class="toctree-l1"><a class="reference internal" href="more_programs/primitive_recursion.html">Primitive Recursion</a></li>
diff --git a/_build/html/index.html b/_build/html/index.html
index 28bd6457..f6e1d612 100644
--- a/_build/html/index.html
+++ b/_build/html/index.html
@@ -206,7 +206,9 @@
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="more_programs/sanity.html">Tools to help write programs</a></li>
 
-<li class="toctree-l1"><a class="reference internal" href="more_programs/arithmetic.html">Programs for Arithmetic</a></li>
+<li class="toctree-l1"><a class="reference internal" href="more_programs/arithmetic.html">Programs for arithmetic</a></li>
+
+
 
 
 <li class="toctree-l1"><a class="reference internal" href="more_programs/primitive_recursion.html">Primitive Recursion</a></li>
diff --git a/_build/html/more_programs/arithmetic.html b/_build/html/more_programs/arithmetic.html
index 5cc63639..fefe916d 100644
--- a/_build/html/more_programs/arithmetic.html
+++ b/_build/html/more_programs/arithmetic.html
@@ -8,7 +8,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>Programs for Arithmetic &#8212; Invitation to Computability</title>
+    <title>Programs for arithmetic &#8212; Invitation to Computability</title>
   
   
   
@@ -205,12 +205,22 @@
 <ul class="current nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="sanity.html">Tools to help write programs</a></li>
 
-<li class="toctree-l1 current active"><a class="current reference internal" href="#">Programs for Arithmetic</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Programs for arithmetic</a></li>
+
+
 
 
 <li class="toctree-l1"><a class="reference internal" href="primitive_recursion.html">Primitive Recursion</a></li>
 
 
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Self-Writing Programs and the Recursion Theorem</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../self_writing.html">Self-Replicating Programs</a></li>
+
+
+
+
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Universal Programs</span></p>
 <ul class="nav bd-sidenav">
@@ -231,14 +241,6 @@
 
 <li class="toctree-l1"><a class="reference internal" href="../undecidability/post.html">Post’s Correspondence Problem</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../undecidability/church.html">Logic</a></li>
-</ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Self-Writing Programs and the Recursion Theorem</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../self_writing.html">Self-Replicating Programs</a></li>
-
-
-
-
 </ul>
 
     </div>
@@ -424,7 +426,7 @@
               
 
 <div id="jb-print-docs-body" class="onlyprint">
-    <h1>Programs for Arithmetic</h1>
+    <h1>Programs for arithmetic</h1>
     <!-- Table of contents -->
     <div id="print-main-content">
         <div id="jb-print-toc">
@@ -434,12 +436,17 @@ <h2> Contents </h2>
             </div>
             <nav aria-label="Page">
                 <ul class="visible nav section-nav flex-column">
-<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">Programs for Arithmetic</a></li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">Programs for arithmetic</a></li>
 <li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#bb-representation">bb representation</a><ul class="visible nav section-nav flex-column">
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#removing-superfluous-s">Removing superfluous #s</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#removing-superfluous-s">Removing superfluous <code class="docutils literal notranslate"><span class="pre">#</span></code>s</a></li>
 </ul>
 </li>
-<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#the-successor-function">The successor function</a></li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#the-successor-function">The successor function</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#comparison">Comparison</a></li>
+</ul>
+</li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#addition">Addition</a></li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#multiplication">Multiplication</a></li>
 </ul>
 
             </nav>
@@ -454,10 +461,10 @@ <h2> Contents </h2>
                   
   <p><a href="https://colab.research.google.com/github/lmoss/onesharp/blob/main/arithmetic.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a></p>
 <div class="section" id="programs-for-arithmetic">
-<h1>Programs for Arithmetic<a class="headerlink" href="#programs-for-arithmetic" title="Permalink to this heading">#</a></h1>
-<p>The main point of working with 1# is that it gives an explicit treatment of some of the most important results of the theory of computation. This was the reason the language was formulated and developed. Nobody ever intended 1# to be used very widely.</p>
-<p>At the same time, once we have 1# or some other programming language, it will be of some interest to check that we really can use it to do computation. This lesson is mainly a verification that you can use 1# to send a rocket to the moon. That is, it hints that one could theoretically use 1# to do any numerical computations that could be done with any computer. So this backs up the claim that our very simple language 1# is computationally complete.</p>
-<p>For the most part, this lesson is optional. The specific results on arithmetic are not going to be used later. The definitions of primitive recursive function and computable function are important in computability theory. But if you already know them from other sources and are willing to believe that 1# is a Turing-complete formalism, then you should feel free to work through this lesson quickly, or even to skip it altogether. .</p>
+<h1>Programs for arithmetic<a class="headerlink" href="#programs-for-arithmetic" title="Permalink to this heading">#</a></h1>
+<p>The main point of working with <code class="docutils literal notranslate"><span class="pre">1#</span></code> is that it gives an explicit treatment of some of the most important results of the theory of computation. This was the reason the language was formulated and developed. Nobody ever intended 1# to be used very widely.</p>
+<p>At the same time, once we have <code class="docutils literal notranslate"><span class="pre">1#</span></code> or some other programming language, it will be of some interest to check that we really can use it to do computation. This lesson is mainly a verification that you can use 1# to send a rocket to the moon. That is, it hints that one could theoretically use 1# to do any numerical computations that could be done with any computer. So this backs up the claim that our very simple language 1# is computationally complete.</p>
+<p>For the most part, this lesson is optional. The specific results on arithmetic are not going to be used later. The definitions of primitive recursive function and computable function are important in computability theory. But if you already know them from other sources and are willing to believe that <code class="docutils literal notranslate"><span class="pre">1#</span></code> is a Turing-complete formalism, then you should feel free to work through this lesson quickly, or even to skip it altogether. .</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-python notranslate"><div class="highlight"><pre><span></span>!python -m pip install -U setuptools
@@ -546,8 +553,8 @@ <h1>bb representation<a class="headerlink" href="#bb-representation" title="Perm
 </td>
 </table>
 <div class="section" id="removing-superfluous-s">
-<h2>Removing superfluous #s<a class="headerlink" href="#removing-superfluous-s" title="Permalink to this heading">#</a></h2>
-<p>One touchy point about bb representations is that all bb representations should either be the word # or else end in a 1. Here is a program that strips off superfluous #s at the end of a word in R1. It uses the first three registers It also converts an empty R1 to a #.</p>
+<h2>Removing superfluous <code class="docutils literal notranslate"><span class="pre">#</span></code>s<a class="headerlink" href="#removing-superfluous-s" title="Permalink to this heading">#</a></h2>
+<p>One touchy point about bb representations is that all bb representations should either be the word # or else end in a <code class="docutils literal notranslate"><span class="pre">1</span></code>. Here is a program that strips off superfluous <code class="docutils literal notranslate"><span class="pre">#</span></code>s at the end of a word in R1. It uses the first three registers It also converts an empty R1 to a <code class="docutils literal notranslate"><span class="pre">#</span></code>.</p>
 <p><code class="docutils literal notranslate"><span class="pre">1#####</span> <span class="pre">1111111111</span> <span class="pre">11111111111</span> <span class="pre">11111111111</span> <span class="pre">11111111111</span> <span class="pre">1111###</span> <span class="pre">111###</span> <span class="pre">11##</span> <span class="pre">11###</span> <span class="pre">11#</span> <span class="pre">1#####</span> <span class="pre">1111111111</span> <span class="pre">1111111111###</span> <span class="pre">1111111111###</span> <span class="pre">11#####</span> <span class="pre">1###</span> <span class="pre">1111###</span> <span class="pre">111##</span> <span class="pre">11##</span> <span class="pre">11111111####</span> <span class="pre">111#</span> <span class="pre">11##</span> <span class="pre">1111111111</span> <span class="pre">1####</span> <span class="pre">11#####</span> <span class="pre">1###</span> <span class="pre">1111###</span> <span class="pre">111##</span> <span class="pre">11#</span> <span class="pre">1111111111</span> <span class="pre">1111111####</span> <span class="pre">111#</span> <span class="pre">11#</span> <span class="pre">1111111111</span> <span class="pre">1111111111####</span> <span class="pre">11#####</span> <span class="pre">1111111111###</span> <span class="pre">1111111111###</span> <span class="pre">111#####</span> <span class="pre">111111###</span> <span class="pre">111###</span> <span class="pre">1##</span> <span class="pre">1111####</span> <span class="pre">1#</span> <span class="pre">111111####</span> <span class="pre">1111111111</span> <span class="pre">11111111111</span> <span class="pre">11111111111</span> <span class="pre">11111####</span> <span class="pre">1111111111</span> <span class="pre">1###</span> <span class="pre">111#</span> <span class="pre">111#####</span> <span class="pre">111111###</span> <span class="pre">111###</span> <span class="pre">1##</span> <span class="pre">1111####</span> <span class="pre">1#</span> <span class="pre">111111####</span> <span class="pre">11###</span> <span class="pre">1##</span></code></p>
 </div>
 </div>
@@ -583,17 +590,23 @@ <h1>The successor function<a class="headerlink" href="#the-successor-function" t
                 <TD><i>move</i><sub>2,1</sub> </TD> <TD>from Lesson 1</TD>                          
         </TR>
 </TABLE>        
-<p>Here is how the program works: as long as a 1 is seen in R1, a # is put in R2; the program then loops to the top. On encountering the first #, the program goes to the blue part: it adds a 1 in R2 and then copies the rest of R1 to the end of R2. If the original input is empty, a 1 is put in R2. And after all of this, R2 is moved back to R1. The full code is shown below.</p>
+<p>Here is how the program works: as long as a <code class="docutils literal notranslate"><span class="pre">1</span></code> is seen in R1, a <code class="docutils literal notranslate"><span class="pre">#</span></code> is put in R2; the program then loops to the top. On encountering the first <code class="docutils literal notranslate"><span class="pre">#</span></code>, the program goes to the blue part: it adds a <code class="docutils literal notranslate"><span class="pre">1</span></code> in R2 and then copies the rest of R1 to the end of R2. If the original input is empty, a <code class="docutils literal notranslate"><span class="pre">1</span></code> is put in R2. And after all of this, R2 is moved back to R1. The full code is shown below.</p>
 <p>1##### 1111111111 111### 1111111111### 11# 1##### 111111### 111### 11## 1111#### 11# 111111#### 1111### 11## 1111111111 111#### 11# 11##### 111111### 111### 1## 1111#### 1# 111111####</p>
-<p>###Comparison</p>
-<p>Here is a program compare which takes words in R1 and R2 and decides if they are symbol-for-symbol the same. If so, it outputs a 1 in R1; if not, it leaves R1 empty.</p>
+<div class="section" id="comparison">
+<h2>Comparison<a class="headerlink" href="#comparison" title="Permalink to this heading">#</a></h2>
+<p>Here is a program compare which takes words in R1 and R2 and decides if they are symbol-for-symbol the same. If so, it outputs a <code class="docutils literal notranslate"><span class="pre">1</span></code> in R1; if not, it leaves R1 empty.</p>
 <p>1##### 111111### 111111111### 11##### 1111111111 1### 1111111111### 111111#### 11##### 1111111111 11111### 111111### 11111### 11##### 111### 1111111111 111#### 1### 1##### 111### 11#### 111#### 11##### 1111### 11#### 111#### 1#</p>
 <p>The original contents of R1 and R2 are destroyed in the process.</p>
-<img src="https://github.com/lmoss/onesharp/blob/main/pianotrumpet.jpg?raw=1" width="200" height="160"><p>#Addition</p>
+<img src="https://github.com/lmoss/onesharp/blob/main/pianotrumpet.jpg?raw=1" width="200" height="160"></div>
+</div>
+<div class="tex2jax_ignore mathjax_ignore section" id="addition">
+<h1>Addition<a class="headerlink" href="#addition" title="Permalink to this heading">#</a></h1>
 <p>Here is a program add which adds numbers. Given m in R1 and n in R2 (represented in bb), the program computes the sum m + n (again, represented in bb) and puts it in R1. At the end, all other registers are empty (and in particular the original inputs m and n are lost).</p>
 <p>1##### 111### 111111### 111111111### 11##### 1111111111 11111111111 11111111111 111### 1111111111 11111111111 11111### 1111111111 11111111111 111111### 11##### 1111111111 11111111111 11### 1111111111 11111111111 1111111### 1111111111 11111111111### 11##### 1111111111 11111111111### 1111111111 11111111### 1111111111 111111111### 1##### 111### 111111### 111111111### 11##### 1111111111 1### 1111111111 111111### 111111111### 11##### 1111111111 111### 1111111111### 1111111111 1### 11##### 111### 11111111### 1### 111# 1111111111 11111111111 11111111111 1#### 111## 1111111111 11111111111 11111111111 111#### 111# 1111111111 11111111111#### 111## 1111111111 11111111111 11#### 1### 111##### 111111### 111### 1## 1111#### 1# 111111####</p>
 <p>This program implements the usual algorithm for addition using carrying. It is also possible to get a program for addition using some of the work that we’ll see soon on primitive recursion.</p>
-<img src="https://github.com/lmoss/onesharp/blob/main/harp.jpg?raw=1" width="200" height="160"><p>#Multiplication</p>
+<img src="https://github.com/lmoss/onesharp/blob/main/harp.jpg?raw=1" width="200" height="160"></div>
+<div class="tex2jax_ignore mathjax_ignore section" id="multiplication">
+<h1>Multiplication<a class="headerlink" href="#multiplication" title="Permalink to this heading">#</a></h1>
 <p>Having seen a program for addition, we next look at multiplication.</p>
 <p>We seek a program times with the following features: Given m in R1 and n in R2 the program computes the sum m x n and puts it in R1. At the end, all other registers are empty (and in particular the original inputs m and n are lost). As with addition, we want to work with bb representations.</p>
 <p>Recall that with addition, our program was designed to implement the usual carrying algorithm. We could do the same for multiplication, but this is not the best choice for us. Although it is more familiar, it is less general; so the ideas used in the traditional algorithm are not going to be useful in other contexts. The basic idea in our treatment is that multiplication satisfies two recursion equations:</p>
@@ -689,7 +702,7 @@ <h1>The successor function<a class="headerlink" href="#the-successor-function" t
 
 <p class="admonition-title"><span class="caption-number">Exercise 33 </span></p>
 <div class="section" id="exercise-content">
-<p>Modify the multiplication program to compute exponentiation. This would be a program exp with the following property: when started with the bb representation of m in R1 and n in R2, the program eventually halts with the bb representation of mn in R1.</p>
+<p>Modify the multiplication program to compute exponentiation. This would be a program <code class="docutils literal notranslate"><span class="pre">exp</span></code> with the following property: when started with the bb representation of m in R1 and n in R2, the program eventually halts with the bb representation of mn in R1.</p>
 </div>
 </div>
 <div class="exercise admonition" id="more_programs/arithmetic-exercise-1">
@@ -778,12 +791,17 @@ <h1>The successor function<a class="headerlink" href="#the-successor-function" t
   </div>
   <nav class="bd-toc-nav page-toc">
     <ul class="visible nav section-nav flex-column">
-<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">Programs for Arithmetic</a></li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">Programs for arithmetic</a></li>
 <li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#bb-representation">bb representation</a><ul class="visible nav section-nav flex-column">
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#removing-superfluous-s">Removing superfluous #s</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#removing-superfluous-s">Removing superfluous <code class="docutils literal notranslate"><span class="pre">#</span></code>s</a></li>
+</ul>
+</li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#the-successor-function">The successor function</a><ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#comparison">Comparison</a></li>
 </ul>
 </li>
-<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#the-successor-function">The successor function</a></li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#addition">Addition</a></li>
+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#multiplication">Multiplication</a></li>
 </ul>
 
   </nav></div>
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diff --git a/_build/html/prf-prf.html b/_build/html/prf-prf.html
index 7a667331..1479b96c 100644
--- a/_build/html/prf-prf.html
+++ b/_build/html/prf-prf.html
@@ -210,7 +210,9 @@
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="more_programs/sanity.html">Tools to help write programs</a></li>
 
-<li class="toctree-l1"><a class="reference internal" href="more_programs/arithmetic.html">Programs for Arithmetic</a></li>
+<li class="toctree-l1"><a class="reference internal" href="more_programs/arithmetic.html">Programs for arithmetic</a></li>
+
+
 
 
 <li class="toctree-l1"><a class="reference internal" href="more_programs/primitive_recursion.html">Primitive Recursion</a></li>
diff --git a/_build/html/search.html b/_build/html/search.html
index 08668135..378a42e7 100644
--- a/_build/html/search.html
+++ b/_build/html/search.html
@@ -205,7 +205,9 @@
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="more_programs/sanity.html">Tools to help write programs</a></li>
 
-<li class="toctree-l1"><a class="reference internal" href="more_programs/arithmetic.html">Programs for Arithmetic</a></li>
+<li class="toctree-l1"><a class="reference internal" href="more_programs/arithmetic.html">Programs for arithmetic</a></li>
+
+
 
 
 <li class="toctree-l1"><a class="reference internal" href="more_programs/primitive_recursion.html">Primitive Recursion</a></li>
diff --git a/_build/html/searchindex.js b/_build/html/searchindex.js
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19, "mth": 19, "1111111111111111": 19, "11111111111111111": 19, "handl": 19, "376": 19, "608": 19, "wherea": 19, "322": 19, "liter": [3, 19], "self1": 19, "ipywidget": 3, "widget": 3, "date_input": 3, "month": 3, "dai": 3, "on_submit": 3, "handle_submit": 3, "press": 3, "field": 3, "stackoverflow": 3, "43744999": 3, "button_styl": 3, "on_button_click": 3, "on_click": 3, "be182edc98d2": 3, "syntaxerror": 3, "eol": 3, "scan": 3, "largest": 14, "p_": 14, "p_n": 14, "quadeq": [14, 16], "linear": 14, "lclclcl": 14, "contradict": [14, 16], "colon": 14, "estim": 14, "atom": 14, "82": 14, "50": 14, "satisfy": 15, "incomplet": 15, "though": 15, "deserv": 15, "philosoph": 15, "famou": 16, "2_p": 16, "toward": 16, "1_q": 16, "phifn_a": 16, "phifn_q": 16, "quadiff": [16, 18], "geoffrei": 16, "pullum": 16, "spirit": 16, "dr": 16, "seuss": 16, "tidi": [16, 18], "emtpi": 16, "h_f": 16, "mathcal": 18, "plane": 18, "anyth": 18, "mathbb": 18, "subset": 18, "easili": 18, "leftrightarrow": 18, "ey": 18, "reliabl": 18, "issu": 18, "avoid": 18, "font": 18, "card": 18, "heartsuit": 18, "diamondsuit": 18, "page": 18, "oblong": 18, "rectangl": 18, "copyright": 18, "circl": 18, "propert": 18, "devot": 18, "forev": 18, "induct": 18, "131": 18, "view": 18, "139": 18, "68": 18, "remark": 18, "4743": 18, "24": 18, "4744": 18, "uppos": 18, "25": 18, "heart": 18, "whole": 18, "uniqu": 18, "_": 18, "crucial": 18, "koenig": 18, "infin": 18, "lemma": 18, "map": 18, "noteworthi": 18, "ne": 18, "nw": 18, "se": 18, "sw": 18, "northwest": 18, "northeast": 18, "hao": 18, "wang": 18, "1961": 18, "entitl": 18, "pattern": 18, "recognit": 18, "ii": 18, "bell": 18, "journal": 18, "40": 18, "41": 18, "area": 18, "known": 18, "raphael": 18, "robinson": 18, "nonperiod": 18, "invention": 18, "mathematica": 18, "1971": 18, "pp": 18, "177": 18, "209": 18, "newer": 18, "simpler": 18}, "objects": {}, "objtypes": {}, "objnames": {}, "titleterms": {"invit": 0, "comput": [0, 1, 7], "statu": [], "march": [], "15": [], 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16, 19], "anoth": 9, "first": 3, "step": 3, "mathtt": [], "1": [3, 11], "instruct": [3, 19], "ar": [3, 11], "run": [3, 11], "notebook": [3, 11], "cell": 3, "full": 3, "command": 3, "line": [3, 11], "tool": [3, 12], "oper": 3, "insid": 3, "jupyt": 3, "arithmet": 10, "bb": 10, "represent": 10, "remov": 10, "superflu": 10, "successor": 10, "primit": 11, "recurs": 11, "term": 11, "which": 11, "thei": 11, "denot": 11, "basic": 11, "thi": [11, 19], "special": 11, "note": 11, "in_place_program": 11, "do": 11, "sens": 11, "cours": 11, "instead": 11, "preserv": 11, "For": 11, "associ": 11, "add": [11, 19], "below": 11, "shown": 11, "That": 11, "take": 11, "awai": 11, "from": 11, "begin": 11, "after": 11, "2": 11, "result": 11, "goe": 11, "regist": [11, 16, 19], "3": 11, "syntax": 11, "check": 11, "you": 11, "can": 11, "enter": 11, "like": 11, "ones": 11, "an": 11, "express": 11, "realli": 11, "find": 11, "error": 11, "output": 11, "long": 11, "becaus": 11, "syntax_check": 11, "thu": 11, "report": 11, "inform": 11, "about": 11, "all": 11, "subterm": 11, "whatev": 11, "how": [11, 12], "insepct": 11, "i": 11, "am": 11, "go": [11, 12], "chang": 11, "5": 11, "c": 11, "littl": 11, "test": 11, "prime": 11, "help": 12, "workflow": 12, "saniti": 12, "mai": 12, "segment": 12, "label": 12, "work": 12, "further": 12, "univers": 19, "plan": 19, "lesson": 19, "simplif": 19, "unboundedli": 19, "mani": 19, "project": 19, "A": 19, "remark": 19, "beig": 19, "green": 19, "sub": 19, "implement": 19, "than": 19, "busi": 14, "beaver": 14, "logic": 15, "corollari": 16, "improv": 16, "our": 18, "goal": 18, "tt": 18, "p": 18, "some": 18, "observ": 18, "restat": 18, "point": 18, "statement": 18, "convers": 18, "comparison": 10, "addit": 10, "multipl": 10}, "envversion": {"sphinx.domains.c": 2, "sphinx.domains.changeset": 1, "sphinx.domains.citation": 1, "sphinx.domains.cpp": 6, "sphinx.domains.index": 1, "sphinx.domains.javascript": 2, "sphinx.domains.math": 2, "sphinx.domains.python": 3, 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\ No newline at end of file
diff --git a/_build/jupyter_execute/more_programs/arithmetic.ipynb b/_build/jupyter_execute/more_programs/arithmetic.ipynb
index faab90a9..37dc8b75 100644
--- a/_build/jupyter_execute/more_programs/arithmetic.ipynb
+++ b/_build/jupyter_execute/more_programs/arithmetic.ipynb
@@ -11,18 +11,19 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {
     "id": "mx5eAo_aRljE"
    },
    "source": [
-    "## Programs for Arithmetic\n",
+    "## Programs for arithmetic\n",
     "\n",
-    "The main point of working with 1# is that it gives an explicit treatment of some of the most important results of the theory of computation. This was the reason the language was formulated and developed. Nobody ever intended 1# to be used very widely.\n",
+    "The main point of working with ```1#``` is that it gives an explicit treatment of some of the most important results of the theory of computation. This was the reason the language was formulated and developed. Nobody ever intended 1# to be used very widely.\n",
     "\n",
-    "At the same time, once we have 1# or some other programming language, it will be of some interest to check that we really can use it to do computation. This lesson is mainly a verification that you can use 1# to send a rocket to the moon. That is, it hints that one could theoretically use 1# to do any numerical computations that could be done with any computer. So this backs up the claim that our very simple language 1# is computationally complete.\n",
+    "At the same time, once we have ```1#``` or some other programming language, it will be of some interest to check that we really can use it to do computation. This lesson is mainly a verification that you can use 1# to send a rocket to the moon. That is, it hints that one could theoretically use 1# to do any numerical computations that could be done with any computer. So this backs up the claim that our very simple language 1# is computationally complete.\n",
     "\n",
-    "For the most part, this lesson is optional. The specific results on arithmetic are not going to be used later. The definitions of primitive recursive function and computable function are important in computability theory. But if you already know them from other sources and are willing to believe that 1# is a Turing-complete formalism, then you should feel free to work through this lesson quickly, or even to skip it altogether. .\n",
+    "For the most part, this lesson is optional. The specific results on arithmetic are not going to be used later. The definitions of primitive recursive function and computable function are important in computability theory. But if you already know them from other sources and are willing to believe that ```1#``` is a Turing-complete formalism, then you should feel free to work through this lesson quickly, or even to skip it altogether. .\n",
     "\n"
    ]
   },
@@ -137,19 +138,21 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {
     "id": "1Vm2_hSaSv_x"
    },
    "source": [
-    "### Removing superfluous #s\n",
+    "### Removing superfluous ```#```s\n",
     "\n",
-    "One touchy point about bb representations is that all bb representations should either be the word # or else end in a 1. Here is a program that strips off superfluous #s at the end of a word in R1. It uses the first three registers It also converts an empty R1 to a #.\n",
+    "One touchy point about bb representations is that all bb representations should either be the word # or else end in a ```1```. Here is a program that strips off superfluous ```#```s at the end of a word in R1. It uses the first three registers It also converts an empty R1 to a ```#```.\n",
     "\n",
     "```1##### 1111111111 11111111111 11111111111 11111111111 1111### 111### 11## 11### 11# 1##### 1111111111 1111111111### 1111111111### 11##### 1### 1111### 111## 11## 11111111#### 111# 11## 1111111111 1#### 11##### 1### 1111### 111## 11# 1111111111 1111111#### 111# 11# 1111111111 1111111111#### 11##### 1111111111### 1111111111### 111##### 111111### 111### 1## 1111#### 1# 111111#### 1111111111 11111111111 11111111111 11111#### 1111111111 1### 111# 111##### 111111### 111### 1## 1111#### 1# 111111#### 11### 1##```\n"
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {
     "id": "ytRP-YfJS4g6"
@@ -192,7 +195,7 @@
     "</TABLE>        \n",
     "\n",
     "\n",
-    "Here is how the program works: as long as a 1 is seen in R1, a # is put in R2; the program then loops to the top. On encountering the first #, the program goes to the blue part: it adds a 1 in R2 and then copies the rest of R1 to the end of R2. If the original input is empty, a 1 is put in R2. And after all of this, R2 is moved back to R1. The full code is shown below.\n",
+    "Here is how the program works: as long as a ```1``` is seen in R1, a ```#``` is put in R2; the program then loops to the top. On encountering the first ```#```, the program goes to the blue part: it adds a ```1``` in R2 and then copies the rest of R1 to the end of R2. If the original input is empty, a ```1``` is put in R2. And after all of this, R2 is moved back to R1. The full code is shown below.\n",
     "\n",
     "1##### 1111111111 111### 1111111111### 11# 1##### 111111### 111### 11## 1111#### 11# 111111#### 1111### 11## 1111111111 111#### 11# 11##### 111111### 111### 1## 1111#### 1# 111111####\n",
     "\n",
@@ -202,14 +205,15 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {
     "id": "rQGxHtJEVEdy"
    },
    "source": [
-    "###Comparison\n",
+    "### Comparison\n",
     "\n",
-    "Here is a program compare which takes words in R1 and R2 and decides if they are symbol-for-symbol the same. If so, it outputs a 1 in R1; if not, it leaves R1 empty.\n",
+    "Here is a program compare which takes words in R1 and R2 and decides if they are symbol-for-symbol the same. If so, it outputs a ```1``` in R1; if not, it leaves R1 empty.\n",
     "\n",
     "1##### 111111### 111111111### 11##### 1111111111 1### 1111111111### 111111#### 11##### 1111111111 11111### 111111### 11111### 11##### 111### 1111111111 111#### 1### 1##### 111### 11#### 111#### 11##### 1111### 11#### 111#### 1#\n",
     "\n",
@@ -226,12 +230,13 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {
     "id": "87sh9zNERS52"
    },
    "source": [
-    "#Addition\n",
+    "# Addition\n",
     "\n",
     "Here is a program add which adds numbers. Given m in R1 and n in R2 (represented in bb), the program computes the sum m + n (again, represented in bb) and puts it in R1. At the end, all other registers are empty (and in particular the original inputs m and n are lost).\n",
     "\n",
@@ -254,12 +259,13 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {
     "id": "ZjTIbR-QTLFN"
    },
    "source": [
-    "#Multiplication\n",
+    "# Multiplication\n",
     "\n",
     "Having seen a program for addition, we next look at multiplication.\n",
     "\n",
@@ -384,13 +390,14 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
    "metadata": {
     "id": "ghMwz69oUbcL"
    },
    "source": [
     "```{exercise}\n",
-    "Modify the multiplication program to compute exponentiation. This would be a program exp with the following property: when started with the bb representation of m in R1 and n in R2, the program eventually halts with the bb representation of mn in R1.\n",
+    "Modify the multiplication program to compute exponentiation. This would be a program ```exp``` with the following property: when started with the bb representation of m in R1 and n in R2, the program eventually halts with the bb representation of mn in R1.\n",
     "```"
    ]
   },
diff --git a/_toc.yml b/_toc.yml
index d725fce0..5ba07b1e 100644
--- a/_toc.yml
+++ b/_toc.yml
@@ -29,7 +29,7 @@ parts:
   
 - caption: Universal Programs
   chapters:
-#   - flle: universal/twoByTwo ### skip this one
+#   - flle: universal/twoByTwo 
    - file: universal/universal
 
 #- caption: The Enumeration Theorem
diff --git a/more_programs/arithmetic.ipynb b/more_programs/arithmetic.ipynb
index e616a964..f0d6417b 100644
--- a/more_programs/arithmetic.ipynb
+++ b/more_programs/arithmetic.ipynb
@@ -1,47 +1,31 @@
 {
-  "nbformat": 4,
-  "nbformat_minor": 0,
-  "metadata": {
-    "colab": {
-      "provenance": [],
-      "toc_visible": true,
-      "authorship_tag": "ABX9TyPhPjEe24JYXQSUVGBFiwOH",
-      "include_colab_link": true
-    },
-    "kernelspec": {
-      "name": "python3",
-      "display_name": "Python 3"
-    },
-    "language_info": {
-      "name": "python"
-    }
-  },
   "cells": [
     {
       "cell_type": "markdown",
       "metadata": {
-        "id": "view-in-github",
-        "colab_type": "text"
+        "colab_type": "text",
+        "id": "view-in-github"
       },
       "source": [
         "<a href=\"https://colab.research.google.com/github/lmoss/onesharp/blob/main/arithmetic.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
       ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "mx5eAo_aRljE"
+      },
       "source": [
-        "## Programs for Arithmetic\n",
+        "## Programs for arithmetic\n",
         "\n",
-        "The main point of working with 1# is that it gives an explicit treatment of some of the most important results of the theory of computation. This was the reason the language was formulated and developed. Nobody ever intended 1# to be used very widely.\n",
+        "The main point of working with ```1#``` is that it gives an explicit treatment of some of the most important results of the theory of computation. This was the reason the language was formulated and developed. Nobody ever intended 1# to be used very widely.\n",
         "\n",
-        "At the same time, once we have 1# or some other programming language, it will be of some interest to check that we really can use it to do computation. This lesson is mainly a verification that you can use 1# to send a rocket to the moon. That is, it hints that one could theoretically use 1# to do any numerical computations that could be done with any computer. So this backs up the claim that our very simple language 1# is computationally complete.\n",
+        "At the same time, once we have ```1#``` or some other programming language, it will be of some interest to check that we really can use it to do computation. This lesson is mainly a verification that you can use 1# to send a rocket to the moon. That is, it hints that one could theoretically use 1# to do any numerical computations that could be done with any computer. So this backs up the claim that our very simple language 1# is computationally complete.\n",
         "\n",
-        "For the most part, this lesson is optional. The specific results on arithmetic are not going to be used later. The definitions of primitive recursive function and computable function are important in computability theory. But if you already know them from other sources and are willing to believe that 1# is a Turing-complete formalism, then you should feel free to work through this lesson quickly, or even to skip it altogether. .\n",
+        "For the most part, this lesson is optional. The specific results on arithmetic are not going to be used later. The definitions of primitive recursive function and computable function are important in computability theory. But if you already know them from other sources and are willing to believe that ```1#``` is a Turing-complete formalism, then you should feel free to work through this lesson quickly, or even to skip it altogether. .\n",
         "\n"
-      ],
-      "metadata": {
-        "id": "mx5eAo_aRljE"
-      }
+      ]
     },
     {
       "cell_type": "code",
@@ -58,6 +42,9 @@
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "eLajSKX_R2Gz"
+      },
       "source": [
         "# bb representation\n",
         "\n",
@@ -103,13 +90,13 @@
         "</td>\n",
         "</table>\n",
         "<br>\n"
-      ],
-      "metadata": {
-        "id": "eLajSKX_R2Gz"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "GJYkLkUGSmrQ"
+      },
       "source": [
         "We can't use the binary representation directly, since we don't have a symbol 0 around. So we change the 0s to #s. And after that, we turn the number around, getting the backwards binary representation. Here are the first 20 numbers again, in this form:\n",
         "\n",
@@ -148,26 +135,28 @@
         "</TABLE>        \n",
         "</td>\n",
         "</table>\n"
-      ],
-      "metadata": {
-        "id": "GJYkLkUGSmrQ"
-      }
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "1Vm2_hSaSv_x"
+      },
       "source": [
-        "### Removing superfluous #s\n",
+        "### Removing superfluous ```#```s\n",
         "\n",
-        "One touchy point about bb representations is that all bb representations should either be the word # or else end in a 1. Here is a program that strips off superfluous #s at the end of a word in R1. It uses the first three registers It also converts an empty R1 to a #.\n",
+        "One touchy point about bb representations is that all bb representations should either be the word # or else end in a ```1```. Here is a program that strips off superfluous ```#```s at the end of a word in R1. It uses the first three registers It also converts an empty R1 to a ```#```.\n",
         "\n",
         "```1##### 1111111111 11111111111 11111111111 11111111111 1111### 111### 11## 11### 11# 1##### 1111111111 1111111111### 1111111111### 11##### 1### 1111### 111## 11## 11111111#### 111# 11## 1111111111 1#### 11##### 1### 1111### 111## 11# 1111111111 1111111#### 111# 11# 1111111111 1111111111#### 11##### 1111111111### 1111111111### 111##### 111111### 111### 1## 1111#### 1# 111111#### 1111111111 11111111111 11111111111 11111#### 1111111111 1### 111# 111##### 111111### 111### 1## 1111#### 1# 111111#### 11### 1##```\n"
-      ],
-      "metadata": {
-        "id": "1Vm2_hSaSv_x"
-      }
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "ytRP-YfJS4g6"
+      },
       "source": [
         "\n",
         "\n",
@@ -206,46 +195,48 @@
         "</TABLE>        \n",
         "\n",
         "\n",
-        "Here is how the program works: as long as a 1 is seen in R1, a # is put in R2; the program then loops to the top. On encountering the first #, the program goes to the blue part: it adds a 1 in R2 and then copies the rest of R1 to the end of R2. If the original input is empty, a 1 is put in R2. And after all of this, R2 is moved back to R1. The full code is shown below.\n",
+        "Here is how the program works: as long as a ```1``` is seen in R1, a ```#``` is put in R2; the program then loops to the top. On encountering the first ```#```, the program goes to the blue part: it adds a ```1``` in R2 and then copies the rest of R1 to the end of R2. If the original input is empty, a ```1``` is put in R2. And after all of this, R2 is moved back to R1. The full code is shown below.\n",
         "\n",
         "1##### 1111111111 111### 1111111111### 11# 1##### 111111### 111### 11## 1111#### 11# 111111#### 1111### 11## 1111111111 111#### 11# 11##### 111111### 111### 1## 1111#### 1# 111111####\n",
         "\n",
         "\n",
         "\n",
         "\n"
-      ],
-      "metadata": {
-        "id": "ytRP-YfJS4g6"
-      }
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "rQGxHtJEVEdy"
+      },
       "source": [
-        "###Comparison\n",
+        "### Comparison\n",
         "\n",
-        "Here is a program compare which takes words in R1 and R2 and decides if they are symbol-for-symbol the same. If so, it outputs a 1 in R1; if not, it leaves R1 empty.\n",
+        "Here is a program compare which takes words in R1 and R2 and decides if they are symbol-for-symbol the same. If so, it outputs a ```1``` in R1; if not, it leaves R1 empty.\n",
         "\n",
         "1##### 111111### 111111111### 11##### 1111111111 1### 1111111111### 111111#### 11##### 1111111111 11111### 111111### 11111### 11##### 111### 1111111111 111#### 1### 1##### 111### 11#### 111#### 11##### 1111### 11#### 111#### 1#\n",
         "\n",
         "The original contents of R1 and R2 are destroyed in the process.\n"
-      ],
-      "metadata": {
-        "id": "rQGxHtJEVEdy"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "<img src=\"https://github.com/lmoss/onesharp/blob/main/pianotrumpet.jpg?raw=1\" width=\"200\" height=\"160\">"
-      ],
       "metadata": {
         "id": "BnRthH3zU7rR"
-      }
+      },
+      "source": [
+        "<img src=\"https://github.com/lmoss/onesharp/blob/main/pianotrumpet.jpg?raw=1\" width=\"200\" height=\"160\">"
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "87sh9zNERS52"
+      },
       "source": [
-        "#Addition\n",
+        "# Addition\n",
         "\n",
         "Here is a program add which adds numbers. Given m in R1 and n in R2 (represented in bb), the program computes the sum m + n (again, represented in bb) and puts it in R1. At the end, all other registers are empty (and in particular the original inputs m and n are lost).\n",
         "\n",
@@ -256,24 +247,25 @@
         "\n",
         "\n",
         "\n"
-      ],
-      "metadata": {
-        "id": "87sh9zNERS52"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "<img src=\"https://github.com/lmoss/onesharp/blob/main/harp.jpg?raw=1\" width=\"200\" height=\"160\">"
-      ],
       "metadata": {
         "id": "yijcynmaU3Ko"
-      }
+      },
+      "source": [
+        "<img src=\"https://github.com/lmoss/onesharp/blob/main/harp.jpg?raw=1\" width=\"200\" height=\"160\">"
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "ZjTIbR-QTLFN"
+      },
       "source": [
-        "#Multiplication\n",
+        "# Multiplication\n",
         "\n",
         "Having seen a program for addition, we next look at multiplication.\n",
         "\n",
@@ -386,63 +378,78 @@
         "1##### 111### 11#### 111#### 11##### 111### 11#### 111#### 111##### 111### 11#### 111#### 1111##### 111111### 111### 1## 1111#### 1# 111111####\n",
         "\n",
         "The parts of the program that are shaded red are special to the multiplication algorithm. The point is that if we wish to write a program for something similar, we could keep the outside \"shell\" and just modify the red lines above. Your first exercise below asks you to do just that.\n"
-      ],
-      "metadata": {
-        "id": "ZjTIbR-QTLFN"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
-      "source": [
-        "<img src=\"https://github.com/lmoss/onesharp/blob/main/drum.jpg?raw=1\" width=\"200\" height=\"160\">"
-      ],
       "metadata": {
         "id": "Luszku-QUrPm"
-      }
+      },
+      "source": [
+        "<img src=\"https://github.com/lmoss/onesharp/blob/main/drum.jpg?raw=1\" width=\"200\" height=\"160\">"
+      ]
     },
     {
+      "attachments": {},
       "cell_type": "markdown",
+      "metadata": {
+        "id": "ghMwz69oUbcL"
+      },
       "source": [
         "```{exercise}\n",
-        "Modify the multiplication program to compute exponentiation. This would be a program exp with the following property: when started with the bb representation of m in R1 and n in R2, the program eventually halts with the bb representation of mn in R1.\n",
+        "Modify the multiplication program to compute exponentiation. This would be a program ```exp``` with the following property: when started with the bb representation of m in R1 and n in R2, the program eventually halts with the bb representation of mn in R1.\n",
         "```"
-      ],
-      "metadata": {
-        "id": "ghMwz69oUbcL"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "S1Ikp4jAUayy"
+      },
       "source": [
         "```{exercise}\n",
         "In our work on multiplication, we assumed addition and worked with it to get multiplication. Now forget about our program for addition, and using only the program ```succ``` for the successor function, construct a program for addition.\n",
         "```"
-      ],
-      "metadata": {
-        "id": "S1Ikp4jAUayy"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "wJj8FgM6Ub9e"
+      },
       "source": [
         "```{exercise}\n",
         "Write a program which, when started with n in bb in R1, gives n! in R1. [Here n! is the factorial function. Remember that 0! = 1 and (n+1)! = (n+1) x n!.]\n",
         "```\n"
-      ],
-      "metadata": {
-        "id": "wJj8FgM6Ub9e"
-      }
+      ]
     },
     {
       "cell_type": "markdown",
+      "metadata": {
+        "id": "_V4TPS-_Un-v"
+      },
       "source": [
         "```{exercise}\n",
         "Write two programs which convert back and forth from unary to binary representation.\n",
         "```"
-      ],
-      "metadata": {
-        "id": "_V4TPS-_Un-v"
-      }
+      ]
+    }
+  ],
+  "metadata": {
+    "colab": {
+      "authorship_tag": "ABX9TyPhPjEe24JYXQSUVGBFiwOH",
+      "include_colab_link": true,
+      "provenance": [],
+      "toc_visible": true
+    },
+    "kernelspec": {
+      "display_name": "Python 3",
+      "name": "python3"
+    },
+    "language_info": {
+      "name": "python"
     }
-  ]
+  },
+  "nbformat": 4,
+  "nbformat_minor": 0
 }