Setup for local navigation between Lean index and LaTeX.
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\documentclass{article}
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\input{../../preamble}
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\input{preamble}
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\newcommand{\linkA}[1]{\href{/doc/Bookshelf/Real/Sequence/Arithmetic.html\##1}{#1}}
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\newcommand{\linkG}[1]{\href{/doc/Bookshelf/Real/Sequence/Geometric.html\##1}{#1}}
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\begin{document}
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\begin{theorem}[Sum of Arithmetic Series]
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\section*{Sum of Arithmetic Series}%
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\label{sec:sum-arithmetic-series}
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Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
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Then for some $n \in \mathbb{N}$,
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$$\sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}.$$
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\end{theorem}
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\begin{proof}
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\href{Sequence/Arithmetic.lean}{Bookshelf.Real.Sequence.Arithmetic.sum_recursive_closed}
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\linkA{Real.Arithmetic.sum\_recursive\_closed}
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\end{proof}
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\begin{theorem}[Sum of Geometric Series]
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\section*{Sum of Geometric Series}%
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\label{sec:sum-geometric-series}
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Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
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Then for some $n \in \mathbb{N}$,
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$$\sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}.$$
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\end{theorem}
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\begin{proof}
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\href{Sequence/Geometric.lean}{Bookshelf.Real.Sequence.Geometric.sum_recursive_closed}
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\linkG{Real.Geometric.sum\_recursive\_closed}
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\end{proof}
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@ -1,19 +1,20 @@
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\documentclass{article}
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\input{../../preamble}
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\input{preamble}
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\newcommand{\link}[1]{\href{/doc/MathematicalIntroductionLogic/Enderton/Chapter0.html\##1}{#1}}
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\begin{document}
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\begin{theorem}[Lemma 0A]
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\section*{Lemma 0A}%
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\label{sec:lemma-0a}
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Assume that $\langle x_1, \ldots, x_m \rangle = \langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$.
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Then $x_1 = \langle y_1, \ldots, y_{k+1} \rangle$.
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\end{theorem}
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\begin{proof}
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\href{Chapter0.lean}{Enderton.Chapter0.lemma_0a}
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\link{lemma\_0a}
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\end{proof}
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@ -1,14 +1,16 @@
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\documentclass{article}
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\input{../preamble}
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\input{preamble}
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\begin{document}
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\newcommand{\bird}[1]{\item{\makebox[5cm][l]{\textbf{#1:}}}}
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A list of birds as defined in \textit{To Mock a Mockingbird}.
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\section*{Aviary}%
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\label{sec:aviary}
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Refer to \href{Aviary.lean}{Smullyan/Aviary.lean} for implementation examples.
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A list of birds as defined in \textit{To Mock a Mockingbird}.
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Refer to \href{/doc/MockMockingbird/Aviary.html}{MockMockingbird/Aviary} for implementation examples.
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\begin{itemize}
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\bird{Bald Eagle} $\hat{E}xy_1y_2y_3z_1z_2z_3 = x(y_1y_2y_3)(z_1z_2z_3)$
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@ -1,63 +1,62 @@
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\documentclass{article}
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\usepackage[shortlabels]{enumitem}
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\input{../../../preamble}
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\input{preamble}
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\newcommand{\link}[1]{\href{/doc/OneVariableCalculus/Apostol/Chapter_I_3.html\##1}{#1}}
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\begin{document}
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\begin{xtheorem}{I.27}
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\section*{Theorem I.27}%
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\label{sec:theorem-i.27}
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Every nonempty set $S$ that is bounded below has a greatest lower bound; that
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is, there is a real number $L$ such that $L = \inf{S}$.
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\end{xtheorem}
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\begin{proof}
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\href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.exists_isGLB}
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\link{Real.exists\_isGLB}
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\end{proof}
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\begin{xtheorem}{I.29}
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\section*{Theorem I.29}%
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\label{sec:theorem-i.29}
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For every real $x$ there exists a positive integer $n$ such that $n > x$.
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\end{xtheorem}
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\begin{proof}
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\href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.exists_pnat_geq_self}
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\link{Real.exists\_pnat\_geq\_self}
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\end{proof}
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\begin{xtheorem}{I.30}[Archimedean Property of the Reals]
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\section*{Theorem I.30 (Archimedean Property of the Reals)}%
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\label{sec:theorem-i.30}
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If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
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integer $n$ such that $nx > y$.
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\end{xtheorem}
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\begin{proof}
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\href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.exists_pnat_mul_self_geq_of_pos}
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\link{Real.exists\_pnat\_mul\_self\_geq\_of\_pos}
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\end{proof}
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\begin{xtheorem}{I.31}
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\section*{Theorem I.31}%
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\label{sec:theorem-i.31}
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If three real numbers $a$, $x$, and $y$ satisfy the inequalities
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$$a \leq x \leq a + \frac{y}{n}$$
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for every integer $n \geq 1$, then $x = a$.
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\end{xtheorem}
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\begin{proof}
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\href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.forall_pnat_leq_self_leq_frac_imp_eq}
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\link{Real.forall\_pnat\_leq\_self\_leq\_frac\_imp\_eq}
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\end{proof}
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\begin{xtheorem}{I.32}
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\section*{Theorem I.32}%
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\label{sec:theorem-i.32}
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Let $h$ be a given positive number and let $S$ be a set of real numbers.
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\begin{enumerate}[(a)]
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$$x < \inf{S} + h.$$
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\end{enumerate}
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\end{xtheorem}
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\begin{proof}
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\ % Force space prior to *Proof.*
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\begin{enumerate}[(a)]
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\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.sup_imp_exists_gt_sup_sub_delta}
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\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.inf_imp_exists_lt_inf_add_delta}
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\item \link{Real.sup\_imp\_exists\_gt\_sup\_sub\_delta}
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\item \link{Real.inf\_imp\_exists\_lt\_inf\_add\_delta}
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\end{enumerate}
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\end{proof}
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\begin{xtheorem}{I.33}[Additive Property]
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\section*{Theorem I.33 (Additive Property)}%
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\label{sec:theorem-i.33}
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Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
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$$C = \{a + b : a \in A, b \in B\}.$$
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$$\inf{C} = \inf{A} + \inf{B}.$$
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\end{enumerate}
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\end{xtheorem}
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\begin{proof}
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\ % Force space prior to *Proof.*
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\begin{enumerate}[(a)]
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\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.sup_minkowski_sum_eq_sup_add_sup}
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\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.inf_minkowski_sum_eq_inf_add_inf}
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\item \link{Real.sup\_minkowski\_sum\_eq\_sup\_add\_sup}
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\item \link{Real.inf\_minkowski\_sum\_eq\_inf\_add\_inf}
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\end{enumerate}
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\end{proof}
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\begin{xtheorem}{I.34}
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\section*{Theorem I.34}%
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\label{sec:theorem-i.34}
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Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that
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$$s \leq t$$
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has an infimum, and they satisfy the inequality
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$$\sup{S} \leq \inf{T}.$$
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\end{xtheorem}
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\begin{proof}
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\href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.forall_mem_le_forall_mem_imp_sup_le_inf}
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\link{Real.forall\_mem\_le\_forall\_mem\_imp\_sup\_le\_inf}
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\end{proof}
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13
README.md
13
README.md
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@ -26,4 +26,15 @@ Run the following to build and serve this:
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This assumes you have `python3` available in your `$PATH`. To change how the
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server behaves, refer to the `.env` file located in the root directory of this
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project.
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project. To also serve the corresponding LaTeX files scattered throughout this
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project, first install the following:
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- `tex4ht`
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- `make4ht`
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- `luaxml`
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Afterward, you can generate the necessary HTML via:
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```bash
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> find . -name '*.tex' | grep -v preamble | xargs -I {} make4ht -e build.mk4 {}
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```
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local mkutils = require 'mkutils'
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local function get_parent_dir(file)
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local handle = io.popen(assert('dirname ' .. file))
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local dir = handle:read('a')
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handle:close()
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return dir:gsub('^%s*(.-)%s*$', '%1')
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end
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local dir = get_parent_dir(arg[3])
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local outdir = 'build/tex/' .. dir .. '/' .. settings.input
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os.execute('mkdir -p ' .. outdir)
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settings.input = outdir .. '/' .. settings.input
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settings.latex_par = '-output-directory=' .. outdir
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Make:match('.css$', 'mv ${filename} ' .. outdir .. '/${filename}')
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Make:match('.png$', 'mv ${filename} ' .. outdir .. '/${filename}')
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Make:match('.tmp$', 'rm ${filename}')
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Make:match('tmp$', function(filename)
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local basename = mkutils.remove_extension(filename)
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for _, ext in ipairs { 'aux', 'xref', '4ct', '4tc', 'dvi', 'idv', 'lg', 'log', 'tmp' } do
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os.remove(outdir .. '/' .. basename .. '.' .. ext)
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end
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end)
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-/
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script «doc-server» (_args) do
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let ((), config) <- StateT.run readConfig {}
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IO.println s!"Running on `http://localhost:{config.port}`"
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IO.println s!"Running Lean on `http://localhost:{config.port}/doc`"
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IO.println s!"Running LaTeX on `http://localhost:{config.port}/tex`"
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_ <- IO.Process.run {
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cmd := "python3",
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args := #["-m", "http.server", toString config.port, "-d", "build/doc"],
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args := #["-m", "http.server", toString config.port, "-d", "build"],
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}
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return 0
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