Add concept of glossary.
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\documentclass{article}
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\input{../../preamble}
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\newcommand{\lean}[2]{\leanref{../../#1.html\##2}{#2}}
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\begin{document}
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\tableofcontents
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\section{The Concepts of Integral Calculus}%
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\label{sec:concepts-integral-calculus}
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\subsection{\defined{Partition}}%
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\label{sub:partition}
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Let $[a, b]$ be a closed interval decomposed into $n$ subintervals by inserting
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$n - 1$ points of subdivision, say $x_1$, $x_2$, $\ldots$, $x_{n-1}$, subject
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only to the restriction
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\begin{equation}
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\label{sec:partition-eq1}
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a < x_1 < x_2 < \cdots < x_{n-1} < b.
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\end{equation}
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It is convenient to denote the point $a$ itself by $x_0$ and the point $b$ by
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$x_n$.
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A collection of points satisfying \eqref{sec:partition-eq1} is called a
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\textbf{partition} $P$ of $[a, b]$, and we use the symbol
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$$P = \{x_0, x_1, \ldots, x_n\}$$ to designate this partition.
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\begin{definition}
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\lean{Common/Set/Intervals/Partition}{Set.Intervals.Partition}
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\end{definition}
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\subsection{\defined{Step Function}}%
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\label{sub:step-function}
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A function $s$, whose domain is a closed interval $[a, b]$, is called a step
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function if there is a \nameref{sub:partition} $P = \{x_0, x_1, \ldots, x_n\}$
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of $[a b]$ such that $s$ is constant on each open subinterval of $P$.
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That is to say, for each $k = 1, 2, \ldots, n$, there is a real number $s_k$
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such that $$s(x) = s_k \quad\text{if}\quad x_{k-1} < x < x_k.$$
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Step functions are sometimes called piecewise constant functions.
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\vspace{8pt}
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\noindent
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\textit{Note:} At each of the endpoints $x_{k-1}$ and $x_k$ the function must
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have some well-defined value, but this need not be the same as $s_k$.
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\begin{definition}
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\lean{Common/Set/Intervals/StepFunction}{Set.Intervals.StepFunction}
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\end{definition}
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\end{document}
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@ -69,6 +69,12 @@ theorem right_ge_mem_self [Preorder α] [@DecidableRel α LT.lt]
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(p : Partition α) : ∀ x ∈ p, x ≤ p.b := by
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sorry
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/--
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The closed interval determined by the endpoints of the `Partition`.
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-/
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abbrev toIcc [Preorder α] [@DecidableRel α LT.lt]
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(p : Partition α) := Set.Icc p.a p.b
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/-
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Return the closed subintervals determined by the `Partition`.
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-/
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@ -76,6 +82,12 @@ def closedSubintervals [Preorder α] [@DecidableRel α LT.lt]
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(p : Partition α) : List (Set α) :=
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p.toList.pairwise (fun x₁ x₂ => Icc x₁ x₂)
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/--
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The open interval determined by the endpoints of the `Partition`.
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-/
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abbrev toIoo [Preorder α] [@DecidableRel α LT.lt]
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(p : Partition α) := Set.Ioo p.a p.b
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/-
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Return the open subintervals determined by the `Partition`.
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-/
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@ -8,20 +8,19 @@ Characterization of step functions.
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namespace Set.Intervals
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open Partition
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/--
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A function `f`, whose domain is a closed interval `[a, b]`, is a `StepFunction`
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if there exists a `Partition` `P = {x₀, x₁, …, xₙ}` of `[a, b]` such that `f` is
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constant on each open subinterval of `P`.
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-/
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structure StepFunction (α : Type _) [Preorder α] [@DecidableRel α LT.lt] where
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/- A partition of some closed interval `[a, b]`. -/
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partition : Partition α
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/-- A function whose domain is a closed interval `[a, b]`. -/
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function : ∀ x ∈ Icc partition.a partition.b, α
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/-- Ensure the function is constant on each open subinterval of `p`. -/
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p : Partition α
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toFun : ∀ x ∈ p.toIcc, α
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const_open_subintervals :
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∀ (hI : I ∈ partition.openSubintervals), ∃ c : α, ∀ (hy : y ∈ I),
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function y (Partition.mem_open_subinterval_mem_closed_interval hI hy) = c
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∀ (hI : I ∈ p.openSubintervals), ∃ c : α, ∀ (hy : y ∈ I),
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toFun y (mem_open_subinterval_mem_closed_interval hI hy) = c
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namespace StepFunction
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@ -37,13 +37,20 @@ def index : BaseHtmlM Html := do templateExtends (baseHtml "Index") <|
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status:
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<ul>
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<li>
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<span style="color:teal">Teal statements </span> are those that
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have been proven or encoded in both LaTeX and Lean.
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<span style="color:darkgray">Dark gray statements </span> are
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reserved for definitions and axioms that have been encoded in both
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LaTeX and Lean.
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</li>
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<li>
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<span style="color:magenta">Magenta statements </span> are those
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that have been proven or encoded in LaTeX but not yet verified in
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Lean.
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<span style="color:teal">Teal statements </span> are reserved for
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statements, theorems, lemmas, etc. that have been proven in both
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LaTeX and Lean.
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</li>
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<li>
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<span style="color:magenta">Magenta statements </span> are reserved
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for definitions, axioms, statements, theorems, lemmas, etc. that
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have been proven or encoded in LaTeX but not yet proven or encoded
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in Lean.
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</li>
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<li>
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<span style="color:red">Red </span> serves as a catch-all for all
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17
preamble.tex
17
preamble.tex
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@ -14,8 +14,8 @@
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% Linking
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% ========================================
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\hypersetup{colorlinks=true, urlcolor=blue}
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\newcommand{\leanref}[2]{{\color{blue}$\pmb{\exists}\;{-}\;$}\href{#1}{#2}}
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\hypersetup{colorlinks=true, linkcolor=blue, urlcolor=blue}
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\newcommand{\leanref}[2]{\textcolor{blue}{$\pmb{\exists}\;{-}\;$}\href{#1}{#2}}
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% ========================================
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% Environments
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@ -24,6 +24,9 @@
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\newenvironment{axiom}{%
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\paragraph{\normalfont\normalsize\textit{Axiom.}}}
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{\hfill$\square$}
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\newenvironment{definition}{%
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\paragraph{\normalfont\normalsize\textit{Definition.}}}
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{\hfill$\square$}
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\newcommand{\divider}{%
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\vspace{10pt}
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\hrule
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@ -47,20 +50,12 @@
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% Status
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% ========================================
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% Used for statements encoded in both LaTeX and Lean.
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\DeclareRobustCommand{\defined}[1]{%
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\texorpdfstring{\color{teal}\faParagraph\ #1}{#1}}
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% Used for statements proven in both LaTeX and Lean.
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\texorpdfstring{\color{darkgray}\faParagraph\ #1}{#1}}
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\DeclareRobustCommand{\verified}[1]{%
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\texorpdfstring{\color{teal}\faCheckCircle\ #1}{#1}}
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% Used for statements proven or encoded in LaTeX but not yet in Lean.
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\DeclareRobustCommand{\partial}[1]{%
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\texorpdfstring{\color{magenta}\faPencil*\ #1}{#1}}
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% A catch-all for anything that doesn't fit the above categories. Incomplete
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% definitions, statements without proof, etc. belong here.
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\DeclareRobustCommand{\unverified}[1]{%
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\texorpdfstring{\color{red}\faExclamationCircle\ #1}{#1}}
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