diff --git a/Bookshelf/Apostol/Chapter_1_07.tex b/Bookshelf/Apostol/Chapter_1_07.tex index e798ebf..53555f5 100644 --- a/Bookshelf/Apostol/Chapter_1_07.tex +++ b/Bookshelf/Apostol/Chapter_1_07.tex @@ -1,9 +1,9 @@ \documentclass{article} -\usepackage{amsmath} + \usepackage{graphicx} -\usepackage{mathrsfs} \input{../../preamble} + \graphicspath{{./images/}} \newcommand{\larea}[2]{\lean{../..}{Common/Real/Geometry/Area}{#1}{#2}} @@ -11,16 +11,18 @@ \begin{document} +\header{Exercises 1.7}{Tom M. Apostol} + The properties of area in this set of exercises are to be deduced from the axioms for area stated in the foregoing section. -\section{Exercise 1}% -\label{sec:exercise-1} +\section*{Exercise 1}% +\hyperlabel{sec:exercise-1}% Prove that each of the following sets is measurable and has zero area: -\subsection{Exercise 1a}% -\label{sub:exercise-1a} +\subsection*{\proceeding{Exercise 1a}}% +\hyperlabel{sub:exercise-1a}% A set consisting of a single point. @@ -36,8 +38,8 @@ A set consisting of a single point. \end{proof} -\subsection{Exercise 1b}% -\label{sub:exercise-1b} +\subsection*{\proceeding{Exercise 1b}}% +\hyperlabel{sub:exercise-1b}% A set consisting of a finite number of points in a plane. @@ -50,7 +52,7 @@ A set consisting of a finite number of points in a plane. \paragraph{Base Case}% Consider a set $S$ consisting of a single point in a plane. - By \eqref{sub:exercise-1a}, $S$ is measurable with area $0$. + By \nameref{sub:exercise-1a}, $S$ is measurable with area $0$. Thus $P(1)$ holds. \paragraph{Induction Step}% @@ -62,7 +64,7 @@ A set consisting of a finite number of points in a plane. Denote the remaining set of points as $S_k$. By construction, $S_{k+1} = S_k \cup T$. By the induction hypothesis, $S_k$ is measurable with area $0$. - By \eqref{sub:exercise-1a}, $T$ is measurable with area $0$. + By \nameref{sub:exercise-1a}, $T$ is measurable with area $0$. By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is measurable, $S_k \cap T$ is measurable, and \begin{align} @@ -82,7 +84,7 @@ A set consisting of a finite number of points in a plane. $S_k \cap T \neq \emptyset$. Since $T$ consists of a single point, $S_k \cap T = T$. - By \eqref{sub:exercise-1a}, $a(S_k \cap T) = a(T) = 0$. + By \nameref{sub:exercise-1a}, $a(S_k \cap T) = a(T) = 0$. \vspace{8pt} \noindent @@ -95,8 +97,8 @@ A set consisting of a finite number of points in a plane. \end{proof} -\subsection{Exercise 1c}% -\label{sub:exercise-1c} +\subsection*{\proceeding{Exercise 1c}}% +\hyperlabel{sub:exercise-1c}% The union of a finite collection of line segments in a plane. @@ -158,8 +160,8 @@ The union of a finite collection of line segments in a plane. \end{proof} -\section{Exercise 2}% -\label{sec:exercise-2} +\section*{\unverified{Exercise 2}}% +\hyperlabel{sec:exercise-2}% Every right triangular region is measurable because it can be obtained as the intersection of two rectangles. @@ -209,8 +211,8 @@ Prove that every triangular region is measurable and that its area is one half \end{proof} -\section{Exercise 3}% -\label{sec:exercise-3} +\section*{\unverified{Exercise 3}}% +\hyperlabel{sec:exercise-3}% Prove that every trapezoid and every parallelogram is measurable and derive the usual formulas for their areas. @@ -232,7 +234,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the Then $S$ is the union of non-overlapping rectangle $R$ of width $b_1$ and height $h$ with right triangle $T$ of base $b_2 - b_1$ and height $h$. By \larea{Choice-of-Scale}{Choice of Scale}, $R$ is measurable. - By \eqref{sec:exercise-2}, $T$ is measurable. + By \nameref{sec:exercise-2}, $T$ is measurable. By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$ are both measurable and \begin{align*} @@ -241,7 +243,8 @@ Prove that every trapezoid and every parallelogram is measurable and derive the & = a(R) + a(T) - a(R \cap T) \\ & = a(R) + a(T) & \text{by construction} \\ & = b_1h + a(T) & \text{Choice of Scale} \\ - & = b_1h + \frac{1}{2}(b_2 - b_1)h & \eqref{sec:exercise-2} \\ + & = b_1h + \frac{1}{2}(b_2 - b_1)h + & \text{\nameref{sec:exercise-2}} \\ & = \frac{b_1 + b_2}{2}h. \end{align*} @@ -251,7 +254,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the Then $S$ is the union of non-overlapping triangle $T$ and right trapezoid $R$. Let $c$ denote the length of base $T$. Then $R$ has longer base edge of length $b_2 - c$. - By \eqref{sec:exercise-2}, $T$ is measurable. + By \nameref{sec:exercise-2}, $T$ is measurable. By Case 1, $R$ is measurable. By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$ are both measurable and @@ -259,7 +262,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the a(S) & = a(T) + a(R) - a(R \cap T) \\ & = a(T) + a(R) & \text{by construction} \\ - & = \frac{1}{2}ch + a(R) & \eqref{sec:exercise-2} \\ + & = \frac{1}{2}ch + a(R) & \text{\nameref{sec:exercise-2}} \\ & = \frac{1}{2}ch + \frac{b_1 + b_2 - c}{2}h & \text{Case 1} \\ & = \frac{b_1 + b_2}{2}h. \end{align*} @@ -291,9 +294,7 @@ 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 $$a(S) = \frac{b_1 + b_2}{2}h.$$ - \vspace{4pt} - \hrule - \vspace{10pt} + \divider Let $P$ be a parallelogram with base $b$ and height $h$. Then $P$ is the union of non-overlapping triangle $T$ and right trapezoid $R$. @@ -317,15 +318,15 @@ Prove that every trapezoid and every parallelogram is measurable and derive the \end{proof} -\section{Exercise 4}% -\label{sec:exercise-4} +\section*{Exercise 4}% +\hyperlabel{sec:exercise-4}% Let $P$ be a polygon whose vertices are lattice points. The area of $P$ is $I + \frac{1}{2}B - 1$, where $I$ denotes the number of lattice points inside the polygon and $B$ denotes the number on the boundary. -\subsection{Exercise 4a}% -\label{sub:exercise-4a} +\subsection*{\unverified{Exercise 4a}}% +\hyperlabel{sub:exercise-4a}% Prove that the formula is valid for rectangles with sides parallel to the coordinate axes. @@ -353,8 +354,8 @@ Prove that the formula is valid for rectangles with sides parallel to the \end{proof} -\subsection{Exercise 4b}% -\label{sub:exercise-4b} +\subsection*{\unverified{Exercise 4b}}% +\hyperlabel{sub:exercise-4b}% Prove that the formula is valid for right triangles and parallelograms. @@ -396,19 +397,19 @@ Prove that the formula is valid for right triangles and parallelograms. & \eqref{sub:exercise-4b-eq1} \\ & = \frac{1}{2}\left[ I_R + \frac{1}{2}B_R - 1 \right] \\ & = \frac{1}{2}\left[ (w - 1)(h - 1) + \frac{1}{2}(2(w + h)) - 1 \right] - & \eqref{sub:exercise-4a} \\ + & \text{\nameref{sub:exercise-4a}} \\ & = \frac{1}{2}\left[ (w - 1)(h - 1) + w + h - 1 \right] \\ & = \frac{1}{2}\left[ wh - w - h + 1 + w + h - 1 \right] \\ & = \frac{wh}{2}. \end{align*} We do not prove this formula is valid for parallelograms here. - Instead, refer to \eqref{sub:exercise-4c} below. + Instead, refer to \nameref{sub:exercise-4c} below. \end{proof} -\subsection{Exercise 4c}% -\label{sub:exercise-4c} +\subsection*{\unverified{Exercise 4c}}% +\hyperlabel{sub:exercise-4c}% Use induction on the number of edges to construct a proof for general polygons. @@ -421,7 +422,7 @@ Use induction on the number of edges to construct a proof for general polygons. \paragraph{Base Case}% A $3$-polygon is a triangle. - By \eqref{sub:exercise-4b}, the lattice point area formula holds. + By \nameref{sub:exercise-4b}, the lattice point area formula holds. Thus $P(3)$ holds. \paragraph{Induction Step}% @@ -473,8 +474,8 @@ Use induction on the number of edges to construct a proof for general polygons. \end{proof} -\section{Exercise 5}% -\label{sec:exercise-5} +\section*{\unverified{Exercise 5}}% +\hyperlabel{sec:exercise-5}% Prove that a triangle whose vertices are lattice points cannot be equilateral. @@ -487,7 +488,7 @@ ways, using Exercises 2 and 4.] Let $T$ be an equilateral triangle whose vertices are lattice points. Assume each side of $T$ has length $a$. Then $T$ has height $h = (a\sqrt{3}) / 2$. - By \eqref{sec:exercise-2}, + By \nameref{sec:exercise-2}, \begin{equation} \label{sub:exercise-5-eq1} \tag{5.1} @@ -495,7 +496,7 @@ ways, using Exercises 2 and 4.] \end{equation} Let $I$ and $B$ denote the number of interior and boundary lattice points of $T$ respectively. - By \eqref{sec:exercise-4}, + By \nameref{sec:exercise-4}, \begin{equation} \label{sub:exercise-5-eq2} \tag{5.2} @@ -509,8 +510,8 @@ ways, using Exercises 2 and 4.] \end{proof} -\section{Exercise 6}% -\label{sec:exercise-6} +\section*{\unverified{Exercise 6}}% +\hyperlabel{sec:exercise-6}% Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all subsets of $A$. diff --git a/Bookshelf/Apostol/Chapter_1_11.tex b/Bookshelf/Apostol/Chapter_1_11.tex index 8dfd3e2..6fc243b 100644 --- a/Bookshelf/Apostol/Chapter_1_11.tex +++ b/Bookshelf/Apostol/Chapter_1_11.tex @@ -1,25 +1,24 @@ \documentclass{article} -\usepackage{amsmath} -\usepackage[shortlabels]{enumitem} -\usepackage{soul, xcolor} \input{../../preamble} \newcommand{\link}[1]{\lean{../..} - {Bookshelf/Apostol/Chapter\_1\_11} - {Apostol.Chapter\_1\_11.#1} - {Chapter\_1\_11.#1} + {Bookshelf/Apostol/Chapter\_1\_11} % Location + {Apostol.Chapter\_1\_11.#1} % Namespace + {Chapter\_1\_11.#1} % Presentation } \begin{document} -\section{Exercise 4}% -\label{sec:exercise-4} +\header{Exercises 1.11}{Tom M. Apostol} + +\section*{Exercise 4}% +\hyperlabel{sec:exercise-4}% Prove that the greatest-integer function has the properties indicated: -\subsection{Exercise 4a}% -\label{sub:exercise-4a} +\subsection*{\proceeding{Exercise 4a}}% +\hyperlabel{sub:exercise-4a}% $\floor{x + n} = \floor{x} + n$ for every integer $n$. @@ -29,8 +28,8 @@ $\floor{x + n} = \floor{x} + n$ for every integer $n$. \end{proof} -\subsection{Exercise 4b}% -\label{sub:exercise-4b} +\subsection*{\proceeding{Exercise 4b}}% +\hyperlabel{sub:exercise-4b}% $\floor{-x} = \begin{cases} @@ -49,8 +48,8 @@ $\floor{-x} = \end{proof} -\subsection{Exercise 4c}% -\label{sub:exercise-4c} +\subsection*{\proceeding{Exercise 4c}}% +\hyperlabel{sub:exercise-4c}% $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$. @@ -60,8 +59,8 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$. \end{proof} -\subsection{Exercise 4d}% -\label{sub:exercise-4d} +\subsection*{\proceeding{Exercise 4d}}% +\hyperlabel{sub:exercise-4d}% $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$ @@ -71,8 +70,8 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$ \end{proof} -\subsection{Exercise 4e}% -\label{sub:exercise-4e} +\subsection*{\proceeding{Exercise 4e}}% +\hyperlabel{sub:exercise-4e}% $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$ @@ -82,24 +81,20 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$ \end{proof} -\section{Exercise 5}% -\label{sec:exercise-5} +\section*{\proceeding{Exercise 5}}% +\hyperlabel{sec:exercise-5}% The formulas in Exercises 4(d) and 4(e) suggest a generalization for $\floor{nx}$. State and prove such a generalization. -\vspace{6pt} -\noindent -\hl{Note}: The stated generalization is known as "Hermite's Identity." +\note{The stated generalization is known as "Hermite's Identity."} \begin{proof} \link{exercise\_5} - \vspace{10pt} - \hrule - \vspace{10pt} + \divider We prove that for all natural numbers $n$ and real numbers $x$, the following identity holds: @@ -132,7 +127,7 @@ State and prove such a generalization. & = \floor{n(\floor{x} + r)} \nonumber \\ & = \floor{n\floor{x} + nr} \nonumber \\ & = \floor{n\floor{x}} + \floor{nr}. \nonumber - & \eqref{sub:exercise-4a} \\ + & \text{\nameref{sub:exercise-4a}} \\ & = \floor{n\floor{x}} + j \nonumber \\ & = n\floor{x} + j. \label{sec:exercise-5-eq3} \end{align} @@ -175,8 +170,8 @@ State and prove such a generalization. \end{proof} -\section{Exercise 6}% -\label{sec:exercise-6} +\section*{\unverified{Exercise 6}}% +\hyperlabel{sec:exercise-6}% Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are integers. @@ -199,15 +194,15 @@ Prove that the number of lattice points in $S$ is equal to the sum \end{proof} -\section{Exercise 7}% -\label{sec:exercise-7} +\section*{Exercise 7}% +\hyperlabel{sec:exercise-7}% If $a$ and $b$ are positive integers with no common factor, we have the formula $$\sum_{n=1}^{b-1} \floor{\frac{na}{b}} = \frac{(a - 1)(b - 1)}{2}.$$ When $b = 1$, the sum on the left is understood to be $0$. -\subsection{Exercise 7a}% -\label{sub:exercise-7a} +\subsection*{\unverified{Exercise 7a}}% +\hyperlabel{sub:exercise-7a}% Derive this result by a geometric argument, counting lattice points in a right triangle. @@ -218,8 +213,8 @@ Derive this result by a geometric argument, counting lattice points in a right \end{proof} -\subsection{Exercise 7b}% -\label{sub:exercise-7b} +\subsection*{\proceeding{Exercise 7b}}% +\hyperlabel{sub:exercise-7b}% Derive the result analytically as follows: By changing the index of summation, note that @@ -232,8 +227,8 @@ Now apply Exercises 4(a) and (b) to the bracket on the right. \end{proof} -\section{Exercise 8}% -\label{sec:exercise-8} +\section*{\unverified{Exercise 8}}% +\hyperlabel{sec:exercise-8}% Let $S$ be a set of points on the real line. The \textit{characteristic function} of $S$ is, by definition, the function diff --git a/Bookshelf/Apostol/Chapter_I_03.tex b/Bookshelf/Apostol/Chapter_I_03.tex index 54f16d2..2f5a229 100644 --- a/Bookshelf/Apostol/Chapter_I_03.tex +++ b/Bookshelf/Apostol/Chapter_I_03.tex @@ -1,5 +1,4 @@ \documentclass{article} -\usepackage[shortlabels]{enumitem} \input{../../preamble} @@ -11,8 +10,10 @@ \begin{document} -\section*{Theorem I.27}% -\label{sec:theorem-i.27} +\header{A Set of Axioms for the Real-Number System}{Tom M. Apostol} + +\section*{\proceeding{Theorem I.27}}% +\hyperlabel{sec: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 $L = \inf{S}$. @@ -23,8 +24,8 @@ is, there is a real number $L$ such that $L = \inf{S}$. \end{proof} -\section*{Theorem I.29}% -\label{sec:theorem-i.29} +\section*{\proceeding{Theorem I.29}}% +\hyperlabel{sec:theorem-i.29} For every real $x$ there exists a positive integer $n$ such that $n > x$. @@ -34,20 +35,22 @@ For every real $x$ there exists a positive integer $n$ such that $n > x$. \end{proof} -\section*{Theorem I.30 (Archimedean Property of the Reals)}% -\label{sec:theorem-i.30} +\section*{\proceeding{Theorem I.30}}% +\hyperlabel{sec:theorem-i.30}% If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive integer $n$ such that $nx > y$. +\note{This is known as the "Archimedean Property of the Reals."} + \begin{proof} \link{exists\_pnat\_mul\_self\_geq\_of\_pos} \end{proof} -\section*{Theorem I.31}% -\label{sec:theorem-i.31} +\section*{\proceeding{Theorem I.31}}% +\hyperlabel{sec:theorem-i.31}% If three real numbers $a$, $x$, and $y$ satisfy the inequalities $$a \leq x \leq a + \frac{y}{n}$$ @@ -59,8 +62,8 @@ for every integer $n \geq 1$, then $x = a$. \end{proof} -\section*{Theorem I.32}% -\label{sec:theorem-i.32} +\section*{\proceeding{Theorem I.32}}% +\hyperlabel{sec:theorem-i.32}% Let $h$ be a given positive number and let $S$ be a set of real numbers. \begin{enumerate}[(a)] @@ -81,8 +84,8 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers. \end{proof} -\section*{Theorem I.33 (Additive Property)}% -\label{sec:theorem-i.33} +\section*{\proceeding{Theorem I.33}}% +\hyperlabel{sec:theorem-i.33}% Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set $$C = \{a + b : a \in A, b \in B\}.$$ @@ -94,6 +97,8 @@ $$C = \{a + b : a \in A, b \in B\}.$$ $$\inf{C} = \inf{A} + \inf{B}.$$ \end{enumerate} +\note{This is known as the "Additive Property."} + \begin{proof} \ % Force space prior to *Proof.* @@ -105,8 +110,8 @@ $$C = \{a + b : a \in A, b \in B\}.$$ \end{proof} -\section*{Theorem I.34}% -\label{sec:theorem-i.34} +\section*{\proceeding{Theorem I.34}}% +\hyperlabel{sec:theorem-i.34}% Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that $$s \leq t$$ diff --git a/Bookshelf/Enderton/Chapter0.tex b/Bookshelf/Enderton/Chapter0.tex index d25cd7b..d9a2caf 100644 --- a/Bookshelf/Enderton/Chapter0.tex +++ b/Bookshelf/Enderton/Chapter0.tex @@ -3,17 +3,20 @@ \input{../../preamble} \newcommand{\link}[1]{\lean{../..} - {Bookshelf/Enderton/Chapter0} - {Enderton.Chapter0.#1} - {Chapter0.#1} + {Bookshelf/Enderton/Chapter0} % Location + {Enderton.Chapter0.#1} % Namespace + {Chapter0.#1} % Presentation } \begin{document} -\section*{Lemma 0A}% -\label{sec:lemma-0a} +\header{Useful Facts About Sets}{Herbert B. Enderton} -Assume that $\langle x_1, \ldots, x_m \rangle = \langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$. +\section{\proceeding{Lemma 0A}}% +\hyperlabel{sec:lemma-0a}% + +Assume that $\langle x_1, \ldots, x_m \rangle = + \langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$. Then $x_1 = \langle y_1, \ldots, y_{k+1} \rangle$. \begin{proof} diff --git a/Common/Real/Sequence/Arithmetic.tex b/Common/Real/Sequence/Arithmetic.tex index f183efc..190d1f8 100644 --- a/Common/Real/Sequence/Arithmetic.tex +++ b/Common/Real/Sequence/Arithmetic.tex @@ -3,15 +3,15 @@ \input{../../../preamble} \newcommand{\link}[1]{\lean{../../..} - {Common/Real/Sequence/Arithmetic} - {Real.Arithmetic.#1} - {Real.Arithmetic.#1} + {Common/Real/Sequence/Arithmetic} % Location + {Real.Arithmetic.#1} % Namespace + {Real.Arithmetic.#1} % Presentation } \begin{document} -\section*{Sum of Arithmetic Series}% -\label{sec:sum-arithmetic-series} +\section{\proceeding{Sum of Arithmetic Series}}% +\hyperlabel{sec:sum-arithmetic-series}% Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$. Then for some $n \in \mathbb{N}$, diff --git a/Common/Real/Sequence/Geometric.tex b/Common/Real/Sequence/Geometric.tex index b048ef4..56b1a63 100644 --- a/Common/Real/Sequence/Geometric.tex +++ b/Common/Real/Sequence/Geometric.tex @@ -3,15 +3,15 @@ \input{../../../preamble} \newcommand{\link}[1]{\lean{../../..} - {Common/Real/Sequence/Geometric} - {Real.Geometric.#1} - {Real.Geometric.#1} + {Common/Real/Sequence/Geometric} % Location + {Real.Geometric.#1} % Namespace + {Real.Geometric.#1} % Presentation } \begin{document} -\section*{Sum of Geometric Series}% -\label{sec:sum-geometric-series} +\section{\proceeding{Sum of Geometric Series}}% +\hyperlabel{sec:sum-geometric-series}% Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$. Then for some $n \in \mathbb{N}$, diff --git a/preamble.tex b/preamble.tex index 855a02f..9aecbe7 100644 --- a/preamble.tex +++ b/preamble.tex @@ -1,15 +1,66 @@ -\usepackage{amsfonts, amsthm} +\usepackage{amsfonts, amsmath, amsthm} +\usepackage[shortlabels]{enumitem} +\usepackage{environ} +\usepackage{fancybox} +\usepackage{fontawesome5} \usepackage{hyperref} +\usepackage{mathrsfs} +\usepackage{soul, xcolor} -\newtheorem{theorem}{Theorem} -\newtheorem{xtheoreminner}{Theorem} -\newenvironment{xtheorem}[1]{% - \renewcommand\thextheoreminner{#1}% - \xtheoreminner -}{\endxtheoreminner} +% ======================================== +% Linking +% ======================================== \hypersetup{colorlinks=true, urlcolor=blue} +% The first argument refers to a relative path upward from a current file to +% the root of the workspace (i.e. where this `preamble.tex` file is located). +% #1 - Path to root +% #2 - Location +% #3 - Namespace +% #4 - Presentation +\newcommand{\lean}[4]{\href{#1/#2.html\##3}{#4}} +\newcommand{\hyperlabel}[1]{% + \label{#1}% + \hypertarget{#1}{}} + +% ======================================== +% Environments +% ======================================== + +\newcommand{\divider}{% + \vspace{10pt} + \hrule + \vspace{10pt}} +\newcommand{\header}[2]{% + \title{#1} + \author{#2} + \date{} + \maketitle} +\newcommand{\note}[1]{% + \begin{center} + \doublebox{ + \begin{minipage}{0.95\textwidth} + \vspace{2pt} + \hl{Note}: #1 + \vspace{2pt} + \end{minipage}} + \end{center}} + +% Status of a proof. A statement/theorem is verified if both a LaTeX proof +% and a corresponding Lean proof has been written. If a Lean proof is in +% progress, it's in a "proceeding" state. Otherwise it is unverified. +\DeclareRobustCommand{\verified}[1]{% + \texorpdfstring{\color{teal}#1\ \faCheckCircle}{#1}} +\DeclareRobustCommand{\proceeding}[1]{% + \texorpdfstring{\color{magenta}#1\ \faSpinner}{#1}} +\DeclareRobustCommand{\unverified}[1]{% + \texorpdfstring{\color{red}#1\ \faExclamationCircle}{#1}} + +% ======================================== +% Math +% ======================================== + \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\ceil}[1]{\left\lceil#1\right\rceil} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} @@ -17,6 +68,3 @@ \newcommand{\ico}[2]{\left[#1, #2\right)} \newcommand{\ioc}[2]{\left(#1, #2\right]} \newcommand{\ioo}[2]{\left(#1, #2\right)} -% The first argument refers to a relative path upward from a current file to -% the root of the workspace (i.e. where this `preamble.tex` file is located). -\newcommand{\lean}[4]{\href{#1/#2.html\##3}{#4}}