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<!-- @format -->
<!doctype html>
<html lang="en">
<head>
<title>Lurch Plus!</title>
<meta charset="utf-8" />
<meta
name="description"
content="This is the home page for lurch.plus, the home of the Lurch proof assistant, plus additional content." />
<meta
name="keywords"
content="Lurch, Lurch plus, proof assistant, math academy, summer courses, math proof camp, math proofs, mathematical proof, proof class, proof course, summer math camp, summer math program, summer math class, math camp, math program, math class, MATHCOUNTS, AMC, AIME, USAMO, IMO, ARML, math contest training, math competition training, math olympiad, AP olympiad, Intel Science, ISEF, ISTS, Regeneron competition, Siemens competition, math gifted, summer school program, camps for kids, academic summer camp, summer academy, summer schools, summer school usa, summer academic programs, summer classes, academy of mathematics, university of scranton, Lehigh Valley ARML" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<link rel="shortcut icon" href="assets/media/favicon-L+.svg" />
<!-- load Font Awesome icon pack
see https://fontawesome.com/search?m=free&o=r for more icons
-->
<link
href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.0.0-beta3/css/all.min.css"
rel="stylesheet" />
<link
href="//fonts.googleapis.com/css?family=Open+Sans:400,600,400italic,700,300"
rel="stylesheet"
type="text/css" />
<link
href="https://fonts.googleapis.com/css2?family=Merriweather:wght@300;400;700&display=swap"
rel="stylesheet" />
<link rel="stylesheet" href="assets/css/monks.css" />
<link rel="stylesheet" href="assets/css/lurchplus.css" />
<!-- <link rel='stylesheet' href='assets/css/bootstrap.min.css'> -->
</head>
<body>
<div id="leftnav">
<img
src="assets/media/lurchlogo.png"
style="margin: 0px; border-radius: 5px; display: block"
width="190px" />
<ul>
<li
><span class="has-submenu"
><i class="fa-solid fa-caret-down"></i
><i class="fa-solid fa-house prefix"></i>Home</span
>
<ul class="submenu">
<li><a href="#start">Start Lurching!</a></li>
<li><a href="#examples">Example Proof Gallery</a></li>
<li><a href="#course">Example Definitions and Theorems</a></li>
<li><a href="#contexts">Other Useful Contexts</a></li>
<li><a href="#assignments">Example Assignments</a></li>
</ul></li
>
<li><a href="./getlurch">Get Lurch!</a></li>
<li><a href="./teaching">Teaching with Lurch</a></li>
<li><a href="./theory">Documentation and Theory</a></li>
<li><a href="./talks">Lurch Talk Slides</a></li>
<li><a href="./100">100 Theorem Challenge!</a></li>
<li><a href="./about">About Lurch</a></li>
</ul>
<div id="contact">
<div>
<a href="mailto:monks@scranton.edu">Ken Monks</a>
</div>
<div>
<a href="https://monks.scranton.edu" target="_blank"
>monks.scranton.edu</a
>
</div>
<div>
<a href="https://proveitmath.org" target="_blank">proveitmath.org</a>
</div>
</div>
</div>
<!-- flex column for content-block and footer-block -->
<div id="wrap">
<div id="content-block">
<div class="title-box">
<h1>Proof Verification with Lurch Plus</h1>
<h2>Building the bridge to higher mathematics</h2>
</div>
<div class="haiku">
<p
>Is my proof correct?<br />
Lurch says it is not convinced.<br />
I need to say more.</p
>
<p class="attribution">Anonymous, 2024</p>
</div>
<p class="large medkip"
>Welcome to the home of Lurch — an open source math editor that can
check your proofs!</p
>
<ul class="disc medskip large complement" id="topfeatures">
<li>Natural mathematical style and notation</li>
<li>Completely customizable</li>
<li>Designed from the ground up for ease of use</li>
</ul>
<!-- ----------------------------------------------------- -->
<div id="video-container">
<video autoplay muted loop>
<source src="assets/media/splash.mp4" type="video/mp4" />
Your browser does not support the video tag.
</video>
</div>
<!-- ----------------------------------------------------- -->
<h2 id="examples">Example Proof Gallery</h2>
<p class="smallskip"
>Here are a few example proofs done in Lurch using the topics defined
below.</p
>
<ul class="disc strong-tags dash-after">
<li
><a
href="student.html?load=math/examples/Topo-logical%20-%20soln.lurch"
>Topo-logical</a
>
a proof that the composition of continuous functions is continuous
in Topology. Imitate the video after opening the document to see
Lurch in action. (<em
>Spoiler alert: this is the solution to part D of the Final Exam
below.</em
>)</li
>
<li
><a
href="student.html?load=math/examples/Mar%2026%20-%20in%20class.lurch"
>Double induction</a
>
a proof of a binomial coefficient identity done in class using
double induction (and the recursive definition of binomial
coefficients given in the course). Colored fonts are used to help
keep track of which induction is which.</li
>
<li
><a href="student.html?load=help/example-proofs.lurch"
>Proof Style Sampler</a
>
a smorgasbord of proofs in Lurch using a variety of topics and
styles.</li
>
<li
><a href="./100" target="_self">100 Famous Theorems Challenge!</a>
Lurch's status in verifying the proofs of theorems on the 100 famous
theorems challenge list. Game on!</li
>
</ul>
<!-- ----------------------------------------------------- -->
<h2 id="start">Start Lurching!</h2>
<p class="smallskip"
>Mastering the art of proof can take a lifetime, but getting started
with Lurch is easy!</p
>
<div class="strong-tags large-tags dash-after bigskip">
<ul>
<li
><span class="list-head">How to use the Lurch App Menus</span>
<ul>
<li
><a href="student.html?load=help/quick-start-guide.lurch"
>1. Quick Start Guide for Students</a
>
how to use the student version of the Lurch app (~15 min)</li
>
<li
><a
href="instructor.html?load=help/instructors-worksheet.lurch"
>2. Quick Start Guide for Instructors</a
>
additional features of the Lurch app for instructors (~5
min)</li
>
</ul>
</li>
<li
><span class="list-head"
>How to write and check proofs with Lurch</span
>
<ul>
<li
><a
href="student.html?load=help/proofs-worksheet.lurch"
class="large"
>3. Writing and Checking Proofs with Lurch</a
>
an introduction to the proof validation tools and capabilities
of Lurch (<a
href="student.html?load=help/proofs-worksheet-solns.lurch"
>solutions</a
>) (~30 min)</li
>
</ul>
</li>
<li
><span class="list-head">How to get Lurch</span>
<ul>
<li
><a href="./getlurch">4. Get Lurch!</a> information for
instructors (or anyone who just wants to play!) who would like
to obtain and customize their own Lurch site or install it on
their computer.</li
>
</ul>
</li>
<li
><span class="list-head">How to use Lurch in your courses</span>
<ul>
<li
><a href="./teaching">5. Teaching with Lurch</a> information
for educators who want to use Lurch for instruction.</li
>
</ul>
</li>
</ul>
</div>
<!-- ----------------------------------------------------- -->
<h4 class="smallskip">Additional Information</h4>
<p class="smallskip"
>Most users can get by with just the information provided above. For
those of you who want to go deeper into the inner workings of Lurch,
here are some useful links.</p
>
<ul class="disc strong-tags dash-after bigskip">
<li
><a href="./theory">Lurch Deductive Engine</a> the nitty-gritty
details under the hood of the validation algorithm used by the Lurch
Deductive Engine for mathematicians, logicians, and computer
scientists.</li
>
<li
><a href="./lde/src/experimental/parsers/lurch-parser-docs.html"
>Lurch Syntax</a
>
a quick reference showing how to type various math expressions in
Lurch (not usually needed because you can click on any expression to
see how it is typed or guess what it should be).</li
>
<li
><a href="https://monks.scranton.edu/math299">Math 299 Home Page</a>
for my Introduction to Mathematical Proof course at the University
of Scranton, Spring 2024.</li
>
<li
><a href="./talks">Talk Slides</a> those of you who have attended my
talks on Lurch can find my 'slides' from those talks here.</li
>
</ul>
<!-- ----------------------------------------------------- -->
<h4 class="smallskip">Start a New Document</h4>
<p
>Click one of the following buttons to open a new document in either
the student version or instructor version of Lurch. The document will
have no mathematical content or context whatsoever. You can start
completely from scratch! If you prefer to start with some definitions
or theorems available, try one of the blank documents with a context
provided in the next section below.</p
>
<div class="flexrow">
<div class="buttondiv">
<p
><a
href="''"
class="button"
onclick="openLurch('./student.html'); return false;"
>Lurch for students</a
></p
>
</div>
<div class="buttondiv">
<p
><a
href="''"
class="button"
onclick="window.open('./instructor.html','target=_blank'); return false;"
>Lurch for Instructors</a
></p
>
</div>
</div>
<div id="course" class="subtitle-box">
<h2>Example Definitions, Rules, and Theorems</h2>
<h4
>Introduction to Mathematical Proof (Math 299), University of
Scranton</h4
>
</div>
<p
>Lurch allows instructors to completely customize Lurch by creating
their own definitions, theorems, notes, textbook, and other course
content.</p
>
<p
>Lurch was used successfully in my undergraduate bridge course for
math majors, Math 299 at the University of Scranton, Spring 2024.
These definitions and notes are very specific to my course, but are
included here for instructors who want to use or modify them. Thus,
these are not any sort of 'official' contexts or math content built
into Lurch itself, but rather are just one example of specific content
made by a specific instructor for a specific course. We hope to add
additional mathematical content here from other instructors and
courses in the future.</p
>
<div class="blue-box">
<p
><strong class="bigfont">Cumulative Topics</strong> – each link
opens a blank Lurch document whose context consists of the rules for
that topic, and those from the topics above it. The Theorem numbers
refer to exercises in the draft of the
<a
href="https://monks.scranton.edu/files/courses/Math299/math-299-lecture.pdf"
>course lecture notes</a
>.</p
>
<dl>
<dt
><a href="instructor.html?load=math/Prop.lurch"
>Propositional Logic</a
></dt
>
<dd
>defines <code>and</code>, <code>or</code>, <code>not</code>,
<code>implies</code>, <code>iff</code>, and
<code>contradiction</code> (<a
href="instructor.html?load=math/Prop-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Pred.lurch"
>Predicate Logic with Equality</a
></dt
>
<dd
>defines <code>forall</code> $\left(\forall\right)$,
<code>exists</code> $\left(\exists\right)$,
<code>equality</code> $\left(=\right)$,
<code>unique existence</code> $\left(\exists!\right)$ (<a
href="instructor.html?load=math/Pred-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Logic-theorems.lurch"
>Logic Theorems</a
></dt
>
<dd
>provides some common theorems from Logic (<a
href="instructor.html?load=math/Logic-theorems-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Peano.lurch"
>Natural Numbers</a
></dt
>
<dd
>the Peano Axioms for the Natural Numbers (<a
href="instructor.html?load=math/Peano-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Number-theory.lurch"
>Number Theory Definitions</a
></dt
>
<dd
>defines <code>less than</code> $\left(\lt\right)$,
<code>divides</code> $\left(\mid\right)$, <code>prime</code>,
<code>even</code>, <code>odd</code> (<a
href="instructor.html?load=math/Number-theory-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Equations.lurch"
>Equations</a
></dt
>
<dd
>defines a single ‘rule’ that enables Lurch to validate transitive
chains of equalities without needing substitution, reflexivity,
symmetry, or transitivity (<a
href="instructor.html?load=math/Equations-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Number-theory-theorems.lurch"
>Number Theory Theorems</a
></dt
>
<dd
>provides some useful theorems from Number Theory that are
provable from the Peano Axioms (<a
href="instructor.html?load=math/Number-theory-theorems-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Sequences.lurch"
>Sequences and Recursion</a
></dt
>
<dd
>defines <code>summation</code> $\left(\sum\right)$,
<code>Fibonacci numbers</code> $\left(F_n\right)$,
<code>factorial</code> $\left(!\right)$,
<code>multinomial coefficients</code> $\left(m,n\right)$,
<code>binomial coefficients</code> $\binom{n}{k}$,
<code>exponentiation</code> $\left(z^n\right)$ (<a
href="instructor.html?load=math/Sequences-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Reals.lurch"
>Real Numbers</a
></dt
>
<dd
>defines the field axioms for the Real Numbers (<a
href="instructor.html?load=math/Reals-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Sets.lurch">Set Theory</a></dt
>
<dd
>defines <code>in</code> $\left(\in\right)$,
<code>subset</code> $\left(\subseteq\right)$,
<code>intersection</code> $\left(\cap\right)$,
<code>union</code> $\left(\cup\right)$,
<code>complement</code> $\left('\right)$,
<code>set difference</code> $\left(\setminus\right)$,
<code>powerset</code> $\left(\mathscr{P}\right)$,
<code>Cartesian product</code> $\left(\times\right)$,
<code>finite set</code> $\left\{ \ldots \right\}$,
<code>tuple</code> $\langle\ldots\rangle$,
<code>Indexed Union</code> $\left(\bigcup\right)$,
<code>Indexed Intersection</code> $\left(\bigcap\right)$,
<code>Set Builder notation</code> $\left\{ z :\ldots \right\}$ (<a
href="instructor.html?load=math/Sets-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Functions.lurch"
>Functions</a
></dt
>
<dd
>defines <code>maps</code> $\left(f\colon A \to B\right)$,
<code>function application</code> $\left(f(x)\right)$,
<code>maps to</code> $\left(\mapsto\right)$,
<code>image</code> $\left(f(U)\right)$,
<code>identity map</code> $\left(\text{id}_A\right)$
<code>inverse image</code> $\left(f^\text{inv}(U)\right)$,
<code>composition</code> $\left(\circ\right)$,
<code>inverse function</code> $\left(f^{-1}\right)$,
<code>injective</code>, <code>surjective</code>,
<code>bijective</code> (<a
href="instructor.html?load=math/Functions-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Relations.lurch"
>Relations</a
></dt
>
<dd
>defines <code>reflexive</code>, <code>symmetric</code>,
<code>transitive</code>, <code>irreflexive</code>,
<code>antisymmetric</code>, <code>total</code>,
<code>partial order</code>, <code>strict partial order</code>,
<code>total order</code>, <code>equivalence relation</code>,
<code>partition</code>, <code>equivalence class</code> $[a]$ (<a
href="instructor.html?load=math/Relations-Rules.lurch"
>view rules</a
>)</dd
>
</dl>
</div>
<!-- blue-box -->
<!-- ----------------------------------------------------- -->
<h2 id="contexts">Other Useful Contexts</h2>
<div class="blue-box">
<p
><strong>Additional Rules.</strong> The following rules can be
optionally added by the instructor to any context above in a
particular document. Currently, only one of the three Arithmetic
rules can be used in a single document.</p
>
<dl>
<dt
><a href="instructor.html?load=math/Arithmetic-naturals.lurch"
>Arithmetic in the Natural Numbers</a
></dt
>
<dd
>Allows the user to say <code>'by</code>
<code>arithmetic'</code> after an expression to try to justify it
using a CAS. The expression can only contain natural number
constants (e.g. $0$, $1$, $2$, etc) the operators $+$, $\cdot$,
and $\text{^}$, and the relations $=$, $\leq$, and $\lt$. (<a
href="instructor.html?load=math/Arithmetic-naturals-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Arithmetic-integers.lurch"
>Arithmetic in the Integers</a
></dt
>
<dd
>Allows the user to say <code>'by</code>
<code>arithmetic'</code> after an expression to try to justify it
using a CAS. The expression can only contain natural number
constants (e.g. $0$, $1$, $2$, etc) the operators $+$, $\cdot$,
$\text{^}$, and $-$ and the relations $=$, $\leq$, and $\lt$.
Exponents cannot be negative. (<a
href="instructor.html?load=math/Arithmetic-integers-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Arithmetic-rationals.lurch"
>Arithmetic in the Rationals</a
></dt
>
<dd
>Allows the user to say <code>'by</code>
<code>arithmetic'</code> after an expression to try to justify it
using a CAS. The expression can only contain natural number
constants (e.g. $0$, $1$, $2$, etc) the operators $+$, $\cdot$,
$\text{^}$, $-$, and $/$, and the relations $=$, $\leq$, and
$\lt$. Exponents must be integers and you cannot divide by zero.
(<a
href="instructor.html?load=math/Arithmetic-rationals-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Algebra.lurch">Algebra</a></dt
>
<dd
>Allows the user to say <code>'by</code>
<code>algebra'</code> after an expression to try to justify it
using a CAS. There are no restrictions on what the expression can
contain, so use at your own discretion. The identity is evaluated
using <a href="http://algebrite.org">Algebrite</a>. (<a
href="instructor.html?load=math/Algebra-Rules.lurch"
>view rules</a
>)</dd
>
</dl>
<p
><strong>Useful Cumulative Contexts.</strong> These contexts combine
some proper subsets of the cumulative Math 299 contexts listed
above.</p
>
<dl>
<dt
><a href="instructor.html?load=math/Equations-Logic.lurch"
>Equations and Logic Only</a
></dt
>
<dd
>Equations, Logic Theorems, Predicate Logic, Propositional Logic.
(<a href="instructor.html?load=math/Equations-Logic-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Sets-Equations-Logic.lurch"
>Sets, Equations, and Logic</a
></dt
>
<dd
>Set Theory, Equations, Logic Theorems, Predicate Logic,
Propositional Logic. This does not include Set Theory Theorems.
(<a
href="instructor.html?load=math/Sets-Equations-Logic-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a
href="instructor.html?load=math/Functions-Sets-Equations-Logic.lurch"
>Functions, Sets, Equations, and Logic</a
></dt
>
<dd
>Functions, Set Theory Theorems, Set Theory, Equations, Logic
Theorems, Predicate Logic, Propositional Logic. (<a
href="instructor.html?load=math/Functions-Sets-Equations-Logic-Rules.lurch"
>view rules</a
>)</dd
>
<dt
><a href="instructor.html?load=math/Topology.lurch"
>Topology</a
></dt
>
<dd
>Defines <code>topological space</code>, <code>open</code>,
<code>closed</code>, and <code>continuous</code> (in a toplogical
space). Includes the Functions library above. (<a
href="instructor.html?load=math/Topology-Rules.lurch"
>view rules</a
>)</dd
>
</dl>
</div>
<!-- blue-box -->
<!-- ----------------------------------------------------- -->
<h2 id="assignments">Example Assignments</h2>
<p class="smallskip"
><strong class="bigfont">Math 299 Spring 2024 Assignments</strong> –
each link opens a Lurch document containing a homework assignment from
the course. Note that Lurch was under development during this semester
so that some assignments are not good examples of what can be done
today if these were rewritten to take advantage of some of the new
features. I am in the process of updating them. I also intend to
revise the entire syllabus as a result of having Lurch available to
spend less time on formal logic and more time on the later topics in
the course (and perhaps add a few new ones).</p
>
<p class="medskip"
>The theorem numbers for the assigned problems below refer to the
following course lecture notes. These are not fully updated to be
consistent with the libraries used in the assignments below and may be
revised frequently.</p
>
<p
><a
href="https://monks.scranton.edu/files/courses/Math299/math-299-lecture.pdf"
class="button medskip"
rel="noopener noreferrer"
target="_blank"
>Lecture Notes</a
></p
>
<div class="blue-box thmlist">
<dl>
<dt
><a href="student.html?load=assignments/assignment-06.lurch"
>Assignment #06</a
>
Propositional logic</dt
>
<dd>
<ol>
<li
><strong>Theorem (Iff is stronger):</strong>
$(P\Leftrightarrow Q)\Rightarrow(Q\Rightarrow P)$</li
>
<li
><strong>Theorem (the other way around):</strong> $\neg(\neg
A)\Rightarrow A$</li
>
<li
><strong>Theorem (Modus Tollens):</strong> $(S \Rightarrow
T)\text{ and } \neg T \Rightarrow \neg S$</li
>
<li
><strong
>Theorem (atomic statements don't have to be
variables):</strong
>
$0 \lt x\text{ or }x\lt 10 \Rightarrow x\lt 10\text{ or } 0\lt
x$</li
>
<li
><strong>Theorem (excluded middle):</strong> $K\text{ or }
\neg K$</li
>
</ol>
</dd>
<dt
><a href="student.html?load=assignments/assignment-07.lurch"
>Assignment #07</a
>
Propositional logic</dt
>
<dd>
<ol>
<li
><strong>Theorem (implies is transitive):</strong>
$((P\Rightarrow Q)\text{ and }(Q \Rightarrow R)) \Rightarrow
(P \Rightarrow R)$</li
>
<li
><strong>Theorem (alternate or-):</strong> $(S\text{ or }
T)\text{ and } \neg S \Rightarrow T$</li
>
</ol>
</dd>
<dt
><a href="student.html?load=assignments/assignment-08.lurch"
>Assignment #08</a
>
Predicate logic with Equality</dt
>
<dd>
<ol>
<li
><strong>Theorem (alpha equivalence warm up):</strong>
$(\forall x.P(x)) \Rightarrow (\forall y.P(y))$</li
>
<li
><strong>Theorem (same thing but for exists):</strong>
$(\exists x.P(x)) \Rightarrow (\exists y.P(y))$</li
>
<li
><strong>Theorem (commuting quantifiers):</strong> $(\forall
x.\forall y.x\text{ loves }y) \Rightarrow (\forall y.\forall
x.x\text{ loves }y)$</li
>
<li
><strong>Theorem (same thing for exists):</strong> $(\exists
x.\exists y.x\text{ loves }y) \Rightarrow (\exists y.\exists
x.x\text{ loves }y)$</li
>
<li
><strong>Theorem (De Morgan):</strong> $\neg (S\text{ and }T)
\Rightarrow \neg S\text{ or } \neg T$</li
>
<li
><strong>Theorem (De Morgan):</strong> $\neg (\forall x.R(x))
\Rightarrow (\exists x.\neg R(x))$</li
>
<li
><strong>Theorem ($=$ is transitive):</strong> $x=y\text{ and
}y=z \Rightarrow x=z$</li
>
</ol>
</dd>
<dt
><a href="student.html?load=assignments/assignment-09.lurch"
>Assignment #09</a
>
Logic theorems</dt
>
<dd>
<ol>
<li
><strong>Theorem 5.10 (some are the same):</strong> $(\forall
x.\forall y.P(x,y)) \Rightarrow (\forall z.P(z,z))$</li
>
<li
><strong>Theorem 5.16 (excluded middle):</strong> $(\forall
x.Q(x))\text{ or } (\exists x.\neg Q(x))$</li
>
<li
><strong>Theorem 4.22 (alternate $\Rightarrow$):</strong> $(R
\Rightarrow S) \Rightarrow \neg R\text{ or }S$</li
>
</ol>
</dd>
<dt
><a href="student.html?load=assignments/assignment-10.lurch"
>Assignment #10</a
>
Logic review</dt
>
<dd>
<ol>
<li
><strong>Theorem (ex falso quodlibet):</strong> If Dracula
fears Alice and Dracula does not fear Alice then Bob loves
Alice.</li
>
<li
><strong>Theorem 4.29 (the most beautiful?):</strong> $S
\Rightarrow (S \Leftrightarrow S\text{ or }(\neg S \text{ and
} S))$</li
>
<li
><strong>Theorem 4.25 (shunting):</strong> $((P \text{ and }
Q) \Rightarrow R) \Leftrightarrow (P \Rightarrow (Q
\Rightarrow R))$</li
>
<li
><strong>Theorem 5.21 (avoiding vacuous domains):</strong>
$(\exists x. x = x)\text{ and }(\forall y. T(y)) \Rightarrow
(\exists z. T(z))$</li
>
<li
><strong>Theorem 5.5 (distributivity):</strong> $(\exists x.
A(x) \Rightarrow B(x)) \Leftrightarrow (\forall y. A(y))
\Rightarrow (\exists z. B(z))$</li
>
</ol>
</dd>
<dt
><a href="student.html?load=assignments/assignment-11.lurch"
>Assignment #11</a
>
Logic review</dt
>
<dd>
<ol>
<li
><strong>Theorem (Pierce's Law):</strong> $((S \Rightarrow T)
\Rightarrow S) \Rightarrow S$</li
>
<li
><strong>Theorem (quantifier fun):</strong> $(\forall
x.\exists y.\forall z.A(x,y,z)) \Rightarrow (\forall z.\forall
x.\exists y.A(x,y,z))$</li
>
</ol>
</dd>
<dt
><a href="student.html?load=assignments/assignment-12.lurch"
>Assignment #12</a
>
Peano arithmetic and Induction</dt
>
<dd>
<ol>
<li
><strong
>Theorem 7.3 (no number is its own successor):</strong
>
$n\neq\sigma(n)$</li
>
<li
><strong
>Theorem 7.4 (alternate definition of $\sigma$):</strong
>
$\sigma(n)=n+1$</li
>
<li
><strong>Theorem 7.5 (associativity of addition):</strong>
$(m+n)+p=m+(n+p)$</li
>
<li
><strong>Theorem 7.6 (additive identity, part 2):</strong>
$0+n=n$</li
>
<li
><strong>Theorem 7.7 (commutativity of adding $1$):</strong>
$1+n=n+1$</li
>
<li
><strong>Theorem 7.8 (commutativity of addition):</strong>
$m+n=n+m$</li
>
</ol>
</dd>
<dt
><a href="student.html?load=assignments/assignment-13.lurch"
>Assignment #13</a
>
Peano arithmetic and Induction</dt
>
<dd>
<ol>
<li
><strong>Theorem 7.11 (left multiplication by zero):</strong>
$0\cdot m=0$</li
>
<li
><strong
>Theorem 7.19 (nonzero naturals are positive):</strong
>
<ul>
<li>(a) $0\leq n$</li>
<li>(b) $0\lt n \Leftrightarrow n\neq 0$</li>
</ul>
</li>
</ol>
</dd>
<dt
><a href="student.html?load=assignments/assignment-14.lurch"
>Assignment #14</a
>
Sequences, Recursion, and Induction</dt
>
<dd>
<ol>
<li
><strong>Theorem 8.3(a) (a useful identity):</strong>
$n^2=n\cdot n$</li
>
<li
><strong>Theorem 8.3(b) (useful identity):</strong> $2\cdot
n=n+n$</li
>
<li
><strong>Theorem 8.2(b) (power law):</strong> $z^m\cdot
z^n=z^{m+n}$</li
>
<li
><strong>Theorem 8.2(c) (power law):</strong>
$(z^m)^n=z^{m\cdot n}$</li
>
<li
><strong>Theorem 8.4 (quadratic beats linear):</strong> If
$a+b\leq n$ then $a\cdot n+b\leq n^2$</li
>
</ol>
</dd>
<dt
><a href="student.html?load=assignments/assignment-15.lurch"
>Assignment #15</a
>
Sequences, Recursion, and Induction</dt
>
<dd>
<ol>
<li
><strong>Theorem 8.7 (Fib fun):</strong> $F_{n+3}+F_n=2\cdot
F_{n+2}$</li
>
<li
><strong>Theorem 8.5(b) (basic sum property):</strong>
$\displaystyle \sum_{i=0}^n s\cdot a_i = s\cdot \sum_{i=0}^n
a_i$</li
>
<li
><strong
>Theorem 8.11 (closed formula for binomial
coefficients):</strong
>
$\displaystyle(m!\cdot n!)\cdot \binom{n+m}{m}=(n+m)!$</li
>
</ol>
</dd>
<dt
><a href="student.html?load=assignments/assignment-16.lurch"
>Assignment #16</a
>
Real numbers and Field axioms</dt
>
<dd>
<ol>
<li
><strong>Theorem 9.3 (cancellation for addition):</strong> If
$x+z=y+z$ then $x=y$</li
>
<li
><strong>Theorem 9.2(b) (signed product):</strong>
$(-1)\cdot(-1)=1$</li
>
<li
><strong>Theorem 9.2(c) (harder than it looks):</strong> $0\lt
1$</li
>
<li><strong>Theorem 9.2(d) (not mod 2):</strong> $-1\neq 1$</li>
</ol>
</dd>
<dt
><a href="student.html?load=assignments/assignment-17.lurch"
>Assignment #17</a
>
Real numbers and Field axioms</dt
>
<dd>
<ol>
<li
><strong
>Theorem 9.4 (cancellation for multiplication):</strong
>
If $z\neq 0$ and $z\cdot x=z\cdot y$ then $x=y$.</li
>
<li
><strong
>Theorem 9.6 (multiplicative inverses are unique):</strong
>
If $z\cdot x=1$ and $z\cdot y=1$ then $x=y$.</li
>
<li
><strong>Theorem 9.10 (alternate additive inverse):</strong>
$-x=-1\cdot x$</li
>
<li
><strong>Theorem 9.18 (adding inequalities):</strong> If $w\lt
y$ and $x\lt z$ then $w+x\lt y+z$.</li
>
<li
><strong>Theorem 9.23 (scaling by a negative):</strong> If
$z\lt 0$ and $x\lt y$ then $z\cdot y\lt z\cdot x$.</li
>
<li
><strong>Theorem (between):</strong> There exists a real
number strictly between 0 and 1, i.e., $\exists r.0\lt r
\text{ and }r\lt 1$.</li