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Normalize formatting further, macros for lean commands.

finite-set-exercises
Joshua Potter 2023-05-17 12:28:02 -06:00
parent ca3dc196c7
commit 9ac70c15c9
4 changed files with 121 additions and 115 deletions
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128
Bookshelf/Apostol.tex
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@ -1,14 +1,10 @@
\documentclass{report}
\usepackage{graphicx}
\input{../preamble}
\graphicspath{{./Apostol/images/}}
\newcommand{\lean}[2]{\leanref{../#1.html\##2}{#2}}
\newcommand{\leanPretty}[3]{\leanref{../#1.html\##2}{#3}}
\newcommand{\ubar}[1]{\text{\b{$#1$}}}
\input{../preamble}
\makeleancommands{..}
\begin{document}
@ -228,7 +224,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\section{\verified{Lemma 1}}%
\label{sec:lemma-1}
\begin{lemma}{1}
\begin{lemma}[1]
Nonempty set $S$ has supremum $L$ if and only if set $-S$ has infimum $-L$.
@ -239,8 +235,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\lean{Bookshelf/Apostol/Chapter\_I\_03}
{Apostol.Chapter\_I\_03.is\_lub\_neg\_set\_iff\_is\_glb\_set\_neg}
\divider
Suppose $L = \sup{S}$ and fix $x \in S$.
By definition of the \nameref{sec:def-supremum}, $x \leq L$ and $L$ is the
smallest value satisfying this inequality.
@ -253,7 +247,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\section{\verified{Theorem I.27}}%
\label{sec:theorem-i.27}
\begin{theorem}{I.27}
\begin{theorem}[I.27]
Every nonempty set $S$ that is bounded below has a greatest lower bound; that
is, there is a real number $L$ such that $L = \inf{S}$.
@ -265,8 +259,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\lean{Bookshelf/Apostol/Chapter\_I\_03}
{Apostol.Chapter\_I\_03.exists\_isGLB}
\divider
Let $S$ be a nonempty set bounded below by $x$.
Then $-S$ is nonempty and bounded above by $x$.
By the \nameref{sec:completeness-axiom}, there exists a
@ -279,7 +271,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\section{\verified{Theorem I.29}}%
\label{sec:theorem-i.29}
\begin{theorem}{I.29}
\begin{theorem}[I.29]
For every real $x$ there exists a positive integer $n$ such that $n > x$.
@ -290,8 +282,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\lean{Bookshelf/Apostol/Chapter\_I\_03}
{Apostol.Chapter\_I\_03.exists\_pnat\_geq\_self}
\divider
Let $n = \abs{\ceil{x}} + 1$.
It is trivial to see $n$ is a positive integer satisfying $n \geq 1$.
Thus all that remains to be shown is that $n > x$.
@ -305,7 +295,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\label{sec:archimedean-property-reals}
\label{sec:theorem-i.30}
\begin{theorem}{I.30}
\begin{theorem}[I.30]
If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
integer $n$ such that $nx > y$.
@ -317,8 +307,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\lean{Bookshelf/Apostol/Chapter\_I\_03}
{Apostol.Chapter\_I\_03.exists\_pnat\_mul\_self\_geq\_of\_pos}
\divider
Let $x > 0$ and $y$ be an arbitrary real number.
By \nameref{sec:theorem-i.29}, there exists a positive integer $n$ such that
$n > y / x$.
@ -329,7 +317,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\section{\verified{Theorem I.31}}%
\label{sec:theorem-i.31}
\begin{theorem}{I.31}
\begin{theorem}[I.31]
If three real numbers $a$, $x$, and $y$ satisfy the inequalities
$$a \leq x \leq a + \frac{y}{n}$$ for every integer $n \geq 1$, then
@ -342,8 +330,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\lean{Bookshelf/Apostol/Chapter\_I\_03}
{Apostol.Chapter\_I\_03.forall\_pnat\_leq\_self\_leq\_frac\_imp\_eq}
\divider
By the trichotomy of the reals, there are three cases to consider:
\paragraph{Case 1}%
@ -378,7 +364,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\section{\verified{Lemma 2}}%
\label{sec:lemma-2}
\begin{lemma}{2}
\begin{lemma}[2]
If three real numbers $a$, $x$, and $y$ satisfy the inequalities
$$a - y / n \leq x \leq a$$ for every integer $n \geq 1$, then $x = a$.
@ -390,8 +376,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
\lean{Bookshelf/Apostol/Chapter\_I\_03}
{Apostol.Chapter\_I\_03.forall\_pnat\_frac\_leq\_self\_leq\_imp\_eq}
\divider
By the trichotomy of the reals, there are three cases to consider:
\paragraph{Case 1}%
@ -431,7 +415,7 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
\subsection{\verified{Theorem I.32a}}%
\label{sub:theorem-i.32a}
\begin{theorem}{I.32a}
\begin{theorem}[I.32a]
If $S$ has a supremum, then for some $x$ in $S$ we have $x > \sup{S} - h$.
@ -442,8 +426,6 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
\lean{Bookshelf/Apostol/Chapter\_I\_03}
{Apostol.Chapter\_I\_03.sup\_imp\_exists\_gt\_sup\_sub\_delta}
\divider
By definition of a \nameref{sec:def-supremum}, $\sup{S}$ is the least upper
bound of $S$.
For the sake of contradiction, suppose for all $x \in S$,
@ -459,7 +441,7 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
\subsection{\verified{Theorem I.32b}}%
\label{sub:theorem-i.32b}
\begin{theorem}{I.32b}
\begin{theorem}[I.32b]
If $S$ has an infimum, then for some $x$ in $S$ we have $x < \inf{S} + h$.
@ -470,8 +452,6 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
\lean{Bookshelf/Apostol/Chapter\_I\_03}
{Apostol.Chapter\_I\_03.inf\_imp\_exists\_lt\_inf\_add\_delta}
\divider
By definition of an \nameref{sec:def-infimum}, $\inf{S}$ is the greatest lower
bound of $S$.
For the sake of contradiction, suppose for all $x \in S$,
@ -495,7 +475,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
\subsection{\verified{Theorem I.33a}}%
\label{sub:theorem-i.33a}
\begin{theorem}{I.33a}
\begin{theorem}[I.33a]
If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
$$\sup{C} = \sup{A} + \sup{B}.$$
@ -507,8 +487,6 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
\lean{Bookshelf/Apostol/Chapter\_I\_03}
{Apostol.Chapter\_I\_03.sup\_minkowski\_sum\_eq\_sup\_add\_sup}
\divider
We prove (i) $\sup{A} + \sup{B}$ is an upper bound of $C$ and (ii)
$\sup{A} + \sup{B}$ is the \textit{least} upper bound of $C$.
@ -568,7 +546,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
\subsection{\verified{Theorem I.33b}}%
\label{sub:theorem-i.33b}
\begin{theorem}{I.33b}
\begin{theorem}[I.33b]
If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
$$\inf{C} = \inf{A} + \inf{B}.$$
@ -580,8 +558,6 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
\lean{Bookshelf/Apostol/Chapter\_I\_03}
{Apostol.Chapter\_I\_03.inf\_minkowski\_sum\_eq\_inf\_add\_inf}
\divider
We prove (i) $\inf{A} + \inf{B}$ is a lower bound of $C$ and (ii)
$\inf{A} + \inf{B}$ is the \textit{greatest} lower bound of $C$.
@ -641,7 +617,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
\section{\verified{Theorem I.34}}%
\label{sec:theorem-i.34}
\begin{theorem}{I.34}
\begin{theorem}[I.34]
Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that $$s \leq t$$
for every $s$ in $S$ and every $t$ in $T$. Then $S$ has a supremum, and $T$
@ -654,8 +630,6 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
\lean{Bookshelf/Apostol/Chapter\_I\_03}
{Apostol.Chapter\_I\_03.forall\_mem\_le\_forall\_mem\_imp\_sup\_le\_inf}
\divider
By hypothesis, $S$ and $T$ are nonempty sets.
Let $s \in S$ and $t \in T$.
Then $t$ is an upper bound of $S$ and $s$ is a lower bound of $T$.
@ -689,7 +663,7 @@ For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
\begin{axiom}
\leanPretty{Common/Geometry/Area}{Nonnegative-Property}
\leanp{Common/Geometry/Area}{Nonnegative-Property}
{Nonnegative Property}
\end{axiom}
@ -702,7 +676,7 @@ If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
\begin{axiom}
\leanPretty{Common/Geometry/Area}{Additive-Property}
\leanp{Common/Geometry/Area}{Additive-Property}
{Additive Property}
\end{axiom}
@ -715,7 +689,7 @@ If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
\begin{axiom}
\leanPretty{Common/Geometry/Area}{Difference-Property}
\leanp{Common/Geometry/Area}{Difference-Property}
{Difference Property}
\end{axiom}
@ -728,7 +702,7 @@ If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is
\begin{axiom}
\leanPretty{Common/Geometry/Area}{Invariance-Under-Congruence}
\leanp{Common/Geometry/Area}{Invariance-Under-Congruence}
{Invariance Under Congruence}
\end{axiom}
@ -741,7 +715,7 @@ If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
\begin{axiom}
\leanPretty{Common/Geometry/Area}{Choice-of-Scale}
\leanp{Common/Geometry/Area}{Choice-of-Scale}
{Choice of Scale}
\end{axiom}
@ -761,7 +735,7 @@ If there is one and only one number $c$ which satisfies the inequalities
\begin{axiom}
\leanPretty{Common/Geometry/Area}{Exhaustion-Property}
\leanp{Common/Geometry/Area}{Exhaustion-Property}
{Exhaustion Property}
\end{axiom}
@ -1045,7 +1019,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
These cases are exhaustive and in agreement with one another.
Thus $S$ is measurable and $$a(S) = \frac{b_1 + b_2}{2}h.$$
\divider
\suitdivider
Let $P$ be a parallelogram with base $b$ and height $h$.
Then $P$ is the union of non-overlapping triangle $T$ and right trapezoid $R$.
@ -1355,8 +1329,6 @@ $\floor{x + n} = \floor{x} + n$ for every integer $n$.
\lean{Bookshelf/Apostol/Chapter\_1\_11}
{Apostol.Chapter\_1\_11.exercise\_4a}
\divider
Let $x$ be a real number and $n$ an integer.
Let $m = \floor{x + n}$.
By definition of the floor function, $m$ is the unique integer such that
@ -1380,14 +1352,12 @@ $\floor{-x} =
\ \vspace{6pt}
\lean{Bookshelf/Apostol/Chapter\_1\_11}
\lean*{Bookshelf/Apostol/Chapter\_1\_11}
{Apostol.Chapter\_1\_11.exercise\_4b\_1}
\lean{Bookshelf/Apostol/Chapter\_1\_11}
{Apostol.Chapter\_1\_11.exercise\_4b\_2}
\divider
There are two cases to consider:
\paragraph{Case 1}%
@ -1428,8 +1398,6 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
\lean{Bookshelf/Apostol/Chapter\_1\_11}
{Apostol.Chapter\_1\_11.exercise\_4c}
\divider
Rewrite $x$ and $y$ as the sum of their floor and fractional components:
$x = \floor{x} + \{x\}$ and $y = \floor{y} + \{y\}$.
Now
@ -1474,8 +1442,6 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
\lean{Bookshelf/Apostol/Chapter\_1\_11}
{Apostol.Chapter\_1\_11.exercise\_4d}
\divider
This is immediately proven by applying \nameref{sec:hermites-identity}.
\end{proof}
@ -1490,8 +1456,6 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
\lean{Bookshelf/Apostol/Chapter\_1\_11}
{Apostol.Chapter\_1\_11.exercise\_4e}
\divider
This is immediately proven by applying \nameref{sec:hermites-identity}.
\end{proof}
@ -1508,8 +1472,6 @@ State and prove such a generalization.
\lean{Bookshelf/Apostol/Chapter\_1\_11}
{Apostol.Chapter\_1\_11.exercise\_5}
\divider
We prove that for all natural numbers $n$ and real numbers $x$, the following
identity holds:
\begin{equation}
@ -1734,8 +1696,6 @@ Now apply Exercises 4(a) and (b) to the bracket on the right.
\lean{Bookshelf/Apostol/Chapter\_1\_11}
{Apostol.Chapter\_1\_11.exercise\_7b}
\divider
Let $n = 1, \ldots, b - 1$.
By hypothesis, $a$ and $b$ are coprime.
Furthermore, $n < b$ for all values of $n$.
@ -1807,7 +1767,7 @@ This property is described by saying that every step function is a linear
\label{sec:step-additive-property}
\label{sec:theorem-1.2}
\begin{theorem}{1.2}
\begin{theorem}[1.2]
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval
$[a, b]$.
@ -1850,7 +1810,7 @@ This property is described by saying that every step function is a linear
\label{sec:step-homogeneous-property}
\label{sec:theorem-1.3}
\begin{theorem}{1.3}
\begin{theorem}[1.3]
Let $s$ be a \nameref{sec:def-step-function} on closed interval $[a, b]$.
For every real number $c$, we have
@ -1881,7 +1841,7 @@ This property is described by saying that every step function is a linear
\label{sec:step-linearity-property}
\label{sec:theorem-1.4}
\begin{theorem}{1.4}
\begin{theorem}[1.4]
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval
$[a, b]$.
@ -1911,7 +1871,7 @@ This property is described by saying that every step function is a linear
\label{sec:step-comparison-theorem}
\label{sec:theorem-1.5}
\begin{theorem}{1.5}
\begin{theorem}[1.5]
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval
$[a, b]$.
@ -1950,7 +1910,7 @@ This property is described by saying that every step function is a linear
\label{sec:step-additivity-with-respect-interval-integration}
\label{sec:theorem-1.6}
\begin{theorem}{1.6}
\begin{theorem}[1.6]
Let $a, b, c \in \mathbb{R}$ and $s$ a \nameref{sec:def-step-function} on the
smallest closed interval containing them.
@ -1996,7 +1956,7 @@ This property is described by saying that every step function is a linear
\label{sec:step-invariance-under-translation}
\label{sec:theorem-1.7}
\begin{theorem}{1.7}
\begin{theorem}[1.7]
Let $s$ be a step function on closed interval $[a, b]$.
Then
@ -2037,7 +1997,7 @@ This property is described by saying that every step function is a linear
\label{sec:step-expansion-contraction-interval-integration}
\label{sec:theorem-1.8}
\begin{theorem}{1.8}
\begin{theorem}[1.8]
Let $s$ be a step function on closed interval $[a, b]$.
Then
@ -2425,13 +2385,13 @@ Which of the following properties would remain valid in this new theory?
$\int_a^b s + \int_b^c s = \int_a^c s$.
\note{This property mirrors
\nameref{sec:step-additivity-with-respect-interval-integration}}
\nameref{sec:step-additivity-with-respect-interval-integration}.}
\begin{proof}
The above property is valid.
The above property is \textbf{valid}.
\divider
\vspace{6pt}
WLOG, suppose $a < b < c$.
Let $s$ be a step function defined on closed interval $[a, c]$.
@ -2464,9 +2424,9 @@ $\int_a^b (s + t) = \int_a^b s + \int_a^b t$.
\begin{proof}
The above property is invalid.
The above property is \textbf{invalid}.
\divider
\vspace{6pt}
Let $s$ and $t$ be step functions on closed interval $[a, b]$.
By definition of a step function, there exists a \nameref{sec:def-partition}
@ -2511,9 +2471,9 @@ $\int_a^b c \cdot s = c \int_a^b s$.
\begin{proof}
The above property is invalid.
The above property is \textbf{invalid}.
\divider
\vspace{6pt}
Let $s$ be a step function on closed interval $[a, b]$.
By definition of a step function, there exists a \nameref{sec:def-partition}
@ -2543,9 +2503,9 @@ $\int_{a+c}^{b+c} s(x) \mathop{dx} = \int_a^b s(x + c) \mathop{dx}$.
\begin{proof}
The above property is valid.
The above property is \textbf{valid}.
\divider
\vspace{6pt}
Let $s$ be a step function on closed interval $[a + c, b + c]$.
By definition of a \nameref{sec:def-step-function}, there exists a \nameref{sec:def-partition} $P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is constant on each open subinterval of $P$.
@ -2577,9 +2537,9 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\begin{proof}
The above property is valid.
The above property is \textbf{valid}.
\divider
\vspace{6pt}
Let $s$ and $t$ be step functions on closed interval $[a, b]$.
By definition of a \nameref{sec:def-step-function}, there exists a \nameref{sec:def-partition}
@ -2611,7 +2571,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\section{\partial{Theorem 1.9}}%
\label{sec:theorem-1.9}
\begin{theorem}{1.9}
\begin{theorem}[1.9]
Every function $f$ which is bounded on $[a, b]$ has a lower integral
$\ubar{I}(f)$ and an upper integral $\overline{I}(f)$ satisfying the
@ -2677,7 +2637,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\label{sec:measurability-ordinate-sets}
\label{sec:theorem-1.10}
\begin{theorem}{1.10}
\begin{theorem}[1.10]
Let $f$ be a nonnegative function, \nameref{sec:def-integrable} on an interval
$[a, b]$, and let $Q$ denote the ordinate set of $f$ over $[a, b]$.
@ -2705,7 +2665,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\label{sec:measurability-graph-nonnegative-function}
\label{sec:theorem-1.11}
\begin{theorem}{1.11}
\begin{theorem}[1.11]
Let $f$ be a nonnegative function, integrable on an interval $[a, b]$.
Then the graph of $f$, that is, the set
@ -2761,7 +2721,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\label{sec:integrability-bounded-monotonic-functions}
\label{sec:theorem-1.12}
\begin{theorem}{1.12}
\begin{theorem}[1.12]
If $f$ is \nameref{sec:def-monotonic} on a closed interval $[a, b]$, then $f$
is \nameref{sec:def-integrable} on $[a, b]$.
@ -2867,7 +2827,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
\label{sec:calculation-integral-bounded-monotonic-function}
\label{sec:theorem-1.13}
\begin{theorem}{1.13}
\begin{theorem}[1.13]
Assume $f$ is increasing on a closed interval $[a, b]$.
Let $x_k = a + k(b - a) / n$ for $k = 0, 1, \ldots, n$.

4
Bookshelf/Enderton.tex
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@ -1,9 +1,7 @@
\documentclass{report}
\input{../preamble}
\newcommand{\lean}[2]{\leanref{../#1.html\##2}{#2}}
\newcommand{\leanPretty}[3]{\leanref{../#1.html\##2}{#3}}
\makeleancommands
\begin{document}

7
Common/Real/Sequence.tex
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@ -1,8 +1,7 @@
\documentclass{article}
\input{../../preamble}
\newcommand{\lean}[2]{\leanref{../../#1.html\##2}{#2}}
\makeleancommands{../..}
\begin{document}
@ -28,8 +27,6 @@ Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
\lean{Common/Real/Sequence/Arithmetic}
{Real.Arithmetic.sum\_recursive\_closed}
\divider
Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
By definition, for all $k \in \mathbb{N}$,
\begin{equation}
@ -90,8 +87,6 @@ Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
\lean{Common/Real/Sequence/Geometric}
{Real.Geometric.sum\_recursive\_closed}
\divider
Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
By definition, for all $k \in \mathbb{N}$,
\begin{equation}

97
preamble.tex
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@ -1,6 +1,8 @@
\usepackage{amsfonts, amsmath, amsthm}
\usepackage{bigfoot}
\usepackage{comment}
\usepackage[shortlabels]{enumitem}
\usepackage{etoolbox}
\usepackage{environ}
\usepackage{fancybox}
\usepackage{fontawesome5}
@ -11,23 +13,63 @@
\usepackage{xr-hyper}
\usepackage{hyperref}
% Open "private" namespace.
\makeatletter
% ========================================
% General
% ========================================
\newcommand{\header}[2]{\title{#1}\author{#2}\date{}\maketitle}
% ========================================
% Dividers
% ========================================
\newcommand\@linespace{\vspace{10pt}}
\newcommand\linedivider{\@linespace\hrule\@linespace}
\WithSuffix\newcommand\linedivider*{\@linespace\hrule}
\newcommand\suitdivider{$$\spadesuit\spadesuit\spadesuit$$}
% ========================================
% Linking
% ========================================
\hypersetup{colorlinks=true, linkcolor=blue, urlcolor=blue}
\newcommand{\leanref}[2]{\textcolor{blue}{$\pmb{\exists}\;{-}\;$}\href{#1}{#2}}
\newcommand{\textref}[1]{\text{\nameref{#1}}}
\newcommand\@leanlink[4]{%
\textcolor{blue}{$\pmb{\exists}\;{-}\;$}\href{#1/#2.html\##3}{#4}}
% Reference to an anchor of Lean documentation.
\newcommand\leanref[3]{%
\@leanlink{#1}{#2}{#3}{#3}\vspace{10pt}}
\WithSuffix\newcommand\leanref*[3]{%
\@leanlink{#1}{#2}{#3}{#3}}
% Variant that allows customizing display text.
\newcommand\leanpref[4]{%
\@leanlink{#1}{#2}{#3}{#4}\vspace{10pt}}
\WithSuffix\newcommand\leanpref*[4]{%
\@leanlink{#1}{#2}{#3}{#4}}
% Macro to build all Lean related commands relative to a specified directory.
\newcommand\makeleancommands[1]{%
\newcommand\lean[2]{%
\leanref{#1}{##1}{##2}}
\WithSuffix\newcommand\lean*[2]{%
\leanref*{#1}{##1}{##2}}
\newcommand\leanp[3]{%
\leanpref{#1}{##1}{##2}{##3}}
\WithSuffix\newcommand\leanp*[3]{%
\leanpref*{#1}{##1}{##2}{##3}}
}
% ========================================
% Environments
% Admonitions
% ========================================
\newcommand{\divider}{\vspace{10pt}\hrule\vspace{10pt}}
\newcommand{\header}[2]{\title{#1}\author{#2}\date{}\maketitle}
% Admonitions.
\newcommand{\admonition}[2]{%
\newcommand{\@admonition}[2]{%
\begin{center}
\doublebox{
\begin{minipage}{0.95\textwidth}
@ -36,26 +78,33 @@
\vspace{2pt}
\end{minipage}}
\end{center}}
\newcommand{\note}[1]{\admonition{Note:}{#1}}
\newcommand{\todo}[1]{\admonition{TODO:}{#1}}
% Statements.
\newenvironment{axiom}{%
\paragraph{\normalfont\normalsize\textit{Axiom.}}}
{\hfill$\square$}
\newenvironment{definition}{%
\paragraph{\normalfont\normalsize\textit{Definition.}}}
{\hfill$\square$}
\newcommand{\note}[1]{\@admonition{Note:}{#1}}
\newcommand{\todo}[1]{\@admonition{TODO:}{#1}}
% ========================================
% Statements
% ========================================
\newcommand\@statement[1]{%
\linedivider*\paragraph{\normalfont\normalsize\textit{#1.}}}
\newenvironment{axiom}{\@statement{Axiom}}{\hfill$\square$}
\newenvironment{definition}{\@statement{Definition}}{\hfill$\square$}
\renewenvironment{proof}{\@statement{Proof}}{\hfill$\square$}
\newtheorem{lemmainner}{Lemma}
\newenvironment{lemma}[1]{%
\renewcommand\thelemmainner{#1}%
\lemmainner
\newenvironment{lemma}[1][]{%
\ifstrempty{#1}
{\lemmainner}
{\renewcommand\thelemmainner{#1}\lemmainner}
}{\endlemmainner}
\newtheorem{theoreminner}{Theorem}
\newenvironment{theorem}[1]{%
\renewcommand\thetheoreminner{#1}%
\theoreminner
\newenvironment{theorem}[1][]{%
\ifstrempty{#1}
{\theoreminner}
{\renewcommand\thetheoreminner{#1}\theoreminner}
}{\endtheoreminner}
% ========================================
@ -82,3 +131,7 @@
\newcommand{\ico}[2]{\left[#1, #2\right)}
\newcommand{\ioc}[2]{\left(#1, #2\right]}
\newcommand{\ioo}[2]{\left(#1, #2\right)}
\newcommand{\ubar}[1]{\text{\b{$#1$}}}
% Close off "private" namespace.
\makeatother