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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" />
<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><a href="#">Home</a></li>
<li><a href="#documentation">Documentation</a></li>
<li><a href="/100" target="_self">100 Theorem Challenge</a></li>
<li>
<span class="has-submenu" onclick="toggleMenu(this)">
Math 299 Content
</span>
<ul class="submenu hidden">
<li><a href="#libraries">Rule Libraries</a></li>
<li><a href="#contexts">Other Contexts</a></li>
<li><a href="#assignments">Example Assignments</a></li>
</ul>
</li>
<li>
<span class="has-submenu" onclick="toggleMenu(this)">
Blank Lurch
</span>
<ul class="submenu hidden">
<li><a href="#lurchforstudents">Lurch for Students</a></li>
<li><a href="#lurchforinstructors">Lurch for Instructors</a></li>
</ul>
</li>
<li><a href="#talkslides">Talk Slides</a></li>
<li><a href="#getlurch">Get Lurch!</a></li>
<li><a href="/about">About Lurch</a></li>
</ul>
<div id="contact">
<div>
<a href="mailto:[email protected]">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</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>
Lurch is a math editor that can check your proofs (coauthored with
Nathan Carter). It was used successfully in my four credit
undergraduate bridge course for math and math education majors.
Instructors who would like to create their own Lurch course materials
using the rule libraries and content from my course can find updated
and improved supporting materials below and on this site.
</p>
<!-- ----------------------------------------------------- -->
<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="documentation">Documentation and Examples</h2>
<ul class="disc strong-tags dash-after">
<li>
<strong>Introductory Worksheets</strong>
<ul>
<li>
<a href="./student.html?load=help/quick-start-guide.lurch">
1. Quick Start Guide
</a>
how to use the student version of Lurch
</li>
<li>
<a href="./student.html?load=help/proofs-worksheet.lurch">
2. Doing Proofs in Lurch
</a>
introduction to proofs in Lurch (
<a href="./student.html?load=help/proofs-worksheet-solns.lurch">
solutions
</a>
)
</li>
<li>
<a
href="./instructor.html?load=help/instructors-worksheet.lurch">
3. Lurch for instructors
</a>
additional features of Lurch for instructor
</li>
</ul>
</li>
<li>
<strong>Examples</strong>
– a few proofs done in Lurch using the topics defined below.
<ul class="circle">
<li>
<a href="./student.html?load=help/example-proofs.lurch">
Example Proofs
</a>
a sample of some proofs in Lurch using a variety of topics and
styles.
</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=math/examples/Topo-logical%20-%20soln.lurch">
Topo-logical
</a>
a proof that the composition of continuous functions is
continuous in Topology.
<br />
(spoiler alert: this is the solution to part D of the Final Exam
below)
</li>
<li>
<a href="/100">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>
</li>
<li>
<strong>Final Exam!</strong>
– try your hand at the Math 299 Final Exam from Spring 2024 (four
parts)
<ul>
<li class="nodash">
<a
href="./student.html?load=math/examples/A.%20The%20Logic%20of%20Love.lurch">
A. The Logic of Love
</a>
</li>
<li class="nodash">
<a
href="./student.html?load=math/examples/B.%20Nostalgia%20for%20Calculus.lurch">
B. Nostalgia for Calculus
</a>
</li>
<li class="nodash">
<a
href="./student.html?load=math/examples/C.%20Putnam%20Practice.lurch">
C. Putnam Practice
</a>
</li>
<li class="nodash">
<a
href="./student.html?load=math/examples/D.%20Topo-logical.lurch">
D. Topo-logical
</a>
</li>
</ul>
</li>
<li>
<strong>Lurch Deductive Engine</strong>
– the nitty gritty details under the hood of the validation
algorithm used by the Lurch Deductive Engine for developers (not up
to date, but mostly correct)
<ul class="circle">
<li>
<a href="./instructor.html?load=help/theory.lurch">
Validation in Lurch
</a>
introduction to the Global $n$-compact validation algorithm used
in this version of Lurch.
</li>
<li>
<a href="lde/src/experimental/docs">
Validation Algorithm API docs
</a>
the source code documentation for the Global $n$-compact
algorithm and supporting infrastructure
</li>
<li>
<a href="lde/docs">Core API docs</a>
the LC data structure, putdown notation, Matching package, and
other core utilties
</li>
<li>
<a href="https://github.com/kenmonks/lurch">
Source Code on GitHub
</a>
the source code and content of this repository on GitHub
</li>
</ul>
</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
</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>
</ul>
<div class="">
<!-- ----------------------------------------------------- -->
<h2 id="libraries">Math 299 Lurch Rule Libraries</h2>
<div class="blue-box">
<p>
<strong class="bigfont">Cumulative Topics</strong>
– each link one 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="">course lecture notes</a>
.
</p>
<dl>
<dt>
<a href="./student.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="./student.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="./student.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="./student.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="./student.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="./student.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="./student.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="./student.html?load=math/Sequences.lurch">
Sequences and Recursion
</a>
</dt>
<dd>
defines
<code>summation</code>
$\left(\sum\right)$,
<code>Fibonnaci numbers</code>
$\left(F_n\right)$,
<code>factorial</code>
$\left(!\right)$,
<code>multinomial coefficients</code>
$\binom{n}{k}$,
<code>binomial coefficients</code>
,
<code>exponentiation</code>
$\left(z^n\right)$ (
<a href="./instructor.html?load=math/Sequences-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.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="./student.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="./student.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="./student.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="./student.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="./student.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="./student.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="./student.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="./student.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="./student.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="./student.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="./student.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 one 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. We are in the process of updating them.
I also intend to revise the entire syllabus in like 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="" class="button medskip" rel="noopener noreferrer">
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 (DeMorgan):</strong>
$\neg (S\text{ and }T) \Rightarrow \neg S\text{ or } \neg T$
</li>
<li>
<strong>Theorem (DeMorgan):</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 it's 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">