Add concept of "verified" to statements/theorems.
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\documentclass{article}
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\usepackage{amsmath}
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\usepackage{graphicx}
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\usepackage{mathrsfs}
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\input{../../preamble}
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\graphicspath{{./images/}}
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\newcommand{\larea}[2]{\lean{../..}{Common/Real/Geometry/Area}{#1}{#2}}
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@ -11,16 +11,18 @@
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\begin{document}
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\header{Exercises 1.7}{Tom M. Apostol}
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The properties of area in this set of exercises are to be deduced from the
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axioms for area stated in the foregoing section.
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\section{Exercise 1}%
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\label{sec:exercise-1}
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\section*{Exercise 1}%
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\hyperlabel{sec:exercise-1}%
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Prove that each of the following sets is measurable and has zero area:
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\subsection{Exercise 1a}%
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\label{sub:exercise-1a}
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\subsection*{\proceeding{Exercise 1a}}%
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\hyperlabel{sub:exercise-1a}%
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A set consisting of a single point.
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@ -36,8 +38,8 @@ A set consisting of a single point.
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\end{proof}
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\subsection{Exercise 1b}%
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\label{sub:exercise-1b}
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\subsection*{\proceeding{Exercise 1b}}%
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\hyperlabel{sub:exercise-1b}%
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A set consisting of a finite number of points in a plane.
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@ -50,7 +52,7 @@ A set consisting of a finite number of points in a plane.
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\paragraph{Base Case}%
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Consider a set $S$ consisting of a single point in a plane.
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By \eqref{sub:exercise-1a}, $S$ is measurable with area $0$.
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By \nameref{sub:exercise-1a}, $S$ is measurable with area $0$.
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Thus $P(1)$ holds.
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\paragraph{Induction Step}%
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@ -62,7 +64,7 @@ A set consisting of a finite number of points in a plane.
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Denote the remaining set of points as $S_k$.
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By construction, $S_{k+1} = S_k \cup T$.
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By the induction hypothesis, $S_k$ is measurable with area $0$.
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By \eqref{sub:exercise-1a}, $T$ is measurable with area $0$.
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By \nameref{sub:exercise-1a}, $T$ is measurable with area $0$.
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By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is
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measurable, $S_k \cap T$ is measurable, and
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\begin{align}
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@ -82,7 +84,7 @@ A set consisting of a finite number of points in a plane.
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$S_k \cap T \neq \emptyset$.
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Since $T$ consists of a single point, $S_k \cap T = T$.
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By \eqref{sub:exercise-1a}, $a(S_k \cap T) = a(T) = 0$.
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By \nameref{sub:exercise-1a}, $a(S_k \cap T) = a(T) = 0$.
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\vspace{8pt}
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\noindent
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@ -95,8 +97,8 @@ A set consisting of a finite number of points in a plane.
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\end{proof}
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\subsection{Exercise 1c}%
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\label{sub:exercise-1c}
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\subsection*{\proceeding{Exercise 1c}}%
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\hyperlabel{sub:exercise-1c}%
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The union of a finite collection of line segments in a plane.
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@ -158,8 +160,8 @@ The union of a finite collection of line segments in a plane.
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\end{proof}
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\section{Exercise 2}%
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\label{sec:exercise-2}
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\section*{\unverified{Exercise 2}}%
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\hyperlabel{sec:exercise-2}%
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Every right triangular region is measurable because it can be obtained as the
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intersection of two rectangles.
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@ -209,8 +211,8 @@ Prove that every triangular region is measurable and that its area is one half
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\end{proof}
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\section{Exercise 3}%
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\label{sec:exercise-3}
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\section*{\unverified{Exercise 3}}%
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\hyperlabel{sec:exercise-3}%
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Prove that every trapezoid and every parallelogram is measurable and derive the
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usual formulas for their areas.
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@ -232,7 +234,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
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Then $S$ is the union of non-overlapping rectangle $R$ of width $b_1$ and
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height $h$ with right triangle $T$ of base $b_2 - b_1$ and height $h$.
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By \larea{Choice-of-Scale}{Choice of Scale}, $R$ is measurable.
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By \eqref{sec:exercise-2}, $T$ is measurable.
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By \nameref{sec:exercise-2}, $T$ is measurable.
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By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$
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are both measurable and
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\begin{align*}
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@ -241,7 +243,8 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
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& = a(R) + a(T) - a(R \cap T) \\
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& = a(R) + a(T) & \text{by construction} \\
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& = b_1h + a(T) & \text{Choice of Scale} \\
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& = b_1h + \frac{1}{2}(b_2 - b_1)h & \eqref{sec:exercise-2} \\
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& = b_1h + \frac{1}{2}(b_2 - b_1)h
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& \text{\nameref{sec:exercise-2}} \\
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& = \frac{b_1 + b_2}{2}h.
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\end{align*}
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@ -251,7 +254,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
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Then $S$ is the union of non-overlapping triangle $T$ and right trapezoid $R$.
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Let $c$ denote the length of base $T$.
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Then $R$ has longer base edge of length $b_2 - c$.
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By \eqref{sec:exercise-2}, $T$ is measurable.
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By \nameref{sec:exercise-2}, $T$ is measurable.
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By Case 1, $R$ is measurable.
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By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$
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are both measurable and
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@ -259,7 +262,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
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a(S)
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& = a(T) + a(R) - a(R \cap T) \\
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& = a(T) + a(R) & \text{by construction} \\
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& = \frac{1}{2}ch + a(R) & \eqref{sec:exercise-2} \\
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& = \frac{1}{2}ch + a(R) & \text{\nameref{sec:exercise-2}} \\
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& = \frac{1}{2}ch + \frac{b_1 + b_2 - c}{2}h & \text{Case 1} \\
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& = \frac{b_1 + b_2}{2}h.
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\end{align*}
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@ -291,9 +294,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
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These cases are exhaustive and in agreement with one another.
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Thus $S$ is measurable and $$a(S) = \frac{b_1 + b_2}{2}h.$$
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\vspace{4pt}
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\hrule
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\vspace{10pt}
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\divider
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Let $P$ be a parallelogram with base $b$ and height $h$.
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Then $P$ is the union of non-overlapping triangle $T$ and right trapezoid $R$.
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@ -317,15 +318,15 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
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\end{proof}
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\section{Exercise 4}%
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\label{sec:exercise-4}
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\section*{Exercise 4}%
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\hyperlabel{sec:exercise-4}%
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Let $P$ be a polygon whose vertices are lattice points.
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The area of $P$ is $I + \frac{1}{2}B - 1$, where $I$ denotes the number of
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lattice points inside the polygon and $B$ denotes the number on the boundary.
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\subsection{Exercise 4a}%
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\label{sub:exercise-4a}
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\subsection*{\unverified{Exercise 4a}}%
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\hyperlabel{sub:exercise-4a}%
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Prove that the formula is valid for rectangles with sides parallel to the
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coordinate axes.
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@ -353,8 +354,8 @@ Prove that the formula is valid for rectangles with sides parallel to the
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\end{proof}
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\subsection{Exercise 4b}%
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\label{sub:exercise-4b}
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\subsection*{\unverified{Exercise 4b}}%
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\hyperlabel{sub:exercise-4b}%
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Prove that the formula is valid for right triangles and parallelograms.
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@ -396,19 +397,19 @@ Prove that the formula is valid for right triangles and parallelograms.
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& \eqref{sub:exercise-4b-eq1} \\
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& = \frac{1}{2}\left[ I_R + \frac{1}{2}B_R - 1 \right] \\
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& = \frac{1}{2}\left[ (w - 1)(h - 1) + \frac{1}{2}(2(w + h)) - 1 \right]
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& \eqref{sub:exercise-4a} \\
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& \text{\nameref{sub:exercise-4a}} \\
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& = \frac{1}{2}\left[ (w - 1)(h - 1) + w + h - 1 \right] \\
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& = \frac{1}{2}\left[ wh - w - h + 1 + w + h - 1 \right] \\
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& = \frac{wh}{2}.
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\end{align*}
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We do not prove this formula is valid for parallelograms here.
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Instead, refer to \eqref{sub:exercise-4c} below.
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Instead, refer to \nameref{sub:exercise-4c} below.
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\end{proof}
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\subsection{Exercise 4c}%
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\label{sub:exercise-4c}
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\subsection*{\unverified{Exercise 4c}}%
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\hyperlabel{sub:exercise-4c}%
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Use induction on the number of edges to construct a proof for general polygons.
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@ -421,7 +422,7 @@ Use induction on the number of edges to construct a proof for general polygons.
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\paragraph{Base Case}%
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A $3$-polygon is a triangle.
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By \eqref{sub:exercise-4b}, the lattice point area formula holds.
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By \nameref{sub:exercise-4b}, the lattice point area formula holds.
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Thus $P(3)$ holds.
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\paragraph{Induction Step}%
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@ -473,8 +474,8 @@ Use induction on the number of edges to construct a proof for general polygons.
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\end{proof}
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\section{Exercise 5}%
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\label{sec:exercise-5}
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\section*{\unverified{Exercise 5}}%
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\hyperlabel{sec:exercise-5}%
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Prove that a triangle whose vertices are lattice points cannot be equilateral.
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@ -487,7 +488,7 @@ ways, using Exercises 2 and 4.]
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Let $T$ be an equilateral triangle whose vertices are lattice points.
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Assume each side of $T$ has length $a$.
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Then $T$ has height $h = (a\sqrt{3}) / 2$.
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By \eqref{sec:exercise-2},
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By \nameref{sec:exercise-2},
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\begin{equation}
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\label{sub:exercise-5-eq1}
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\tag{5.1}
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@ -495,7 +496,7 @@ ways, using Exercises 2 and 4.]
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\end{equation}
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Let $I$ and $B$ denote the number of interior and boundary lattice points of
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$T$ respectively.
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By \eqref{sec:exercise-4},
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By \nameref{sec:exercise-4},
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\begin{equation}
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\label{sub:exercise-5-eq2}
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\tag{5.2}
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@ -509,8 +510,8 @@ ways, using Exercises 2 and 4.]
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\end{proof}
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\section{Exercise 6}%
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\label{sec:exercise-6}
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\section*{\unverified{Exercise 6}}%
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\hyperlabel{sec:exercise-6}%
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Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all
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subsets of $A$.
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@ -1,25 +1,24 @@
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\documentclass{article}
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\usepackage{amsmath}
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\usepackage[shortlabels]{enumitem}
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\usepackage{soul, xcolor}
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\input{../../preamble}
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\newcommand{\link}[1]{\lean{../..}
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{Bookshelf/Apostol/Chapter\_1\_11}
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{Apostol.Chapter\_1\_11.#1}
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{Chapter\_1\_11.#1}
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{Bookshelf/Apostol/Chapter\_1\_11} % Location
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{Apostol.Chapter\_1\_11.#1} % Namespace
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{Chapter\_1\_11.#1} % Presentation
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}
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\begin{document}
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\section{Exercise 4}%
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\label{sec:exercise-4}
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\header{Exercises 1.11}{Tom M. Apostol}
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\section*{Exercise 4}%
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\hyperlabel{sec:exercise-4}%
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Prove that the greatest-integer function has the properties indicated:
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\subsection{Exercise 4a}%
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\label{sub:exercise-4a}
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\subsection*{\proceeding{Exercise 4a}}%
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\hyperlabel{sub:exercise-4a}%
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$\floor{x + n} = \floor{x} + n$ for every integer $n$.
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@ -29,8 +28,8 @@ $\floor{x + n} = \floor{x} + n$ for every integer $n$.
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\end{proof}
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\subsection{Exercise 4b}%
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\label{sub:exercise-4b}
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\subsection*{\proceeding{Exercise 4b}}%
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\hyperlabel{sub:exercise-4b}%
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$\floor{-x} =
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\begin{cases}
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@ -49,8 +48,8 @@ $\floor{-x} =
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\end{proof}
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\subsection{Exercise 4c}%
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\label{sub:exercise-4c}
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\subsection*{\proceeding{Exercise 4c}}%
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\hyperlabel{sub:exercise-4c}%
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$\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
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@ -60,8 +59,8 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
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\end{proof}
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\subsection{Exercise 4d}%
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\label{sub:exercise-4d}
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\subsection*{\proceeding{Exercise 4d}}%
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\hyperlabel{sub:exercise-4d}%
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$\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
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@ -71,8 +70,8 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
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\end{proof}
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\subsection{Exercise 4e}%
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\label{sub:exercise-4e}
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\subsection*{\proceeding{Exercise 4e}}%
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\hyperlabel{sub:exercise-4e}%
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$\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
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@ -82,24 +81,20 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
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\end{proof}
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\section{Exercise 5}%
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\label{sec:exercise-5}
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\section*{\proceeding{Exercise 5}}%
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\hyperlabel{sec:exercise-5}%
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The formulas in Exercises 4(d) and 4(e) suggest a generalization for
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$\floor{nx}$.
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State and prove such a generalization.
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\vspace{6pt}
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\noindent
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\hl{Note}: The stated generalization is known as "Hermite's Identity."
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\note{The stated generalization is known as "Hermite's Identity."}
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\begin{proof}
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\link{exercise\_5}
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\vspace{10pt}
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\hrule
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\vspace{10pt}
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\divider
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We prove that for all natural numbers $n$ and real numbers $x$, the following
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identity holds:
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@ -132,7 +127,7 @@ State and prove such a generalization.
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& = \floor{n(\floor{x} + r)} \nonumber \\
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& = \floor{n\floor{x} + nr} \nonumber \\
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& = \floor{n\floor{x}} + \floor{nr}. \nonumber
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& \eqref{sub:exercise-4a} \\
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& \text{\nameref{sub:exercise-4a}} \\
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& = \floor{n\floor{x}} + j \nonumber \\
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& = n\floor{x} + j. \label{sec:exercise-5-eq3}
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\end{align}
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\end{proof}
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\section{Exercise 6}%
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\label{sec:exercise-6}
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\section*{\unverified{Exercise 6}}%
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\hyperlabel{sec:exercise-6}%
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Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are
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integers.
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@ -199,15 +194,15 @@ Prove that the number of lattice points in $S$ is equal to the sum
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\end{proof}
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\section{Exercise 7}%
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\label{sec:exercise-7}
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\section*{Exercise 7}%
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\hyperlabel{sec:exercise-7}%
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If $a$ and $b$ are positive integers with no common factor, we have the formula
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$$\sum_{n=1}^{b-1} \floor{\frac{na}{b}} = \frac{(a - 1)(b - 1)}{2}.$$
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When $b = 1$, the sum on the left is understood to be $0$.
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\subsection{Exercise 7a}%
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\label{sub:exercise-7a}
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\subsection*{\unverified{Exercise 7a}}%
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\hyperlabel{sub:exercise-7a}%
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Derive this result by a geometric argument, counting lattice points in a right
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triangle.
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@ -218,8 +213,8 @@ Derive this result by a geometric argument, counting lattice points in a right
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\end{proof}
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\subsection{Exercise 7b}%
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\label{sub:exercise-7b}
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\subsection*{\proceeding{Exercise 7b}}%
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\hyperlabel{sub:exercise-7b}%
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Derive the result analytically as follows:
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By changing the index of summation, note that
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@ -232,8 +227,8 @@ Now apply Exercises 4(a) and (b) to the bracket on the right.
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\end{proof}
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\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
|
||||
|
|
|
|||
|
|
@ -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
|
||||
$$s \leq t$$
|
||||
|
|
|
|||
|
|
@ -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}
|
||||
|
|
|
|||
|
|
@ -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 $n \in \mathbb{N}$,
|
||||
|
|
|
|||
|
|
@ -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 $n \in \mathbb{N}$,
|
||||
|
|
|
|||
68
preamble.tex
68
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}}
|
||||
|
|
|
|||
Loading…
Reference in New Issue