Add concept of glossary.

Bookshelf/Apostol/Glossary.tex

@@ -0,0 +1,57 @@
+\documentclass{article}
+
+\input{../../preamble}
+
+\newcommand{\lean}[2]{\leanref{../../#1.html\##2}{#2}}
+
+\begin{document}
+
+\tableofcontents
+
+\section{The Concepts of Integral Calculus}%
+\label{sec:concepts-integral-calculus}
+
+\subsection{\defined{Partition}}%
+\label{sub:partition}
+
+Let $[a, b]$ be a closed interval decomposed into $n$ subintervals by inserting
+  $n - 1$ points of subdivision, say $x_1$, $x_2$, $\ldots$, $x_{n-1}$, subject
+  only to the restriction
+  \begin{equation}
+    \label{sec:partition-eq1}
+    a < x_1 < x_2 < \cdots < x_{n-1} < b.
+  \end{equation}
+It is convenient to denote the point $a$ itself by $x_0$ and the point $b$ by
+  $x_n$.
+A collection of points satisfying \eqref{sec:partition-eq1} is called a
+  \textbf{partition} $P$ of $[a, b]$, and we use the symbol
+  $$P = \{x_0, x_1, \ldots, x_n\}$$ to designate this partition.
+
+\begin{definition}
+
+  \lean{Common/Set/Intervals/Partition}{Set.Intervals.Partition}
+
+\end{definition}
+
+\subsection{\defined{Step Function}}%
+\label{sub:step-function}
+
+A function $s$, whose domain is a closed interval $[a, b]$, is called a step
+  function if there is a \nameref{sub:partition} $P = \{x_0, x_1, \ldots, x_n\}$
+  of $[a b]$ such that $s$ is constant on each open subinterval of $P$.
+That is to say, for each $k = 1, 2, \ldots, n$, there is a real number $s_k$
+  such that $$s(x) = s_k \quad\text{if}\quad x_{k-1} < x < x_k.$$
+Step functions are sometimes called piecewise constant functions.
+
+\vspace{8pt}
+\noindent
+\textit{Note:} At each of the endpoints $x_{k-1}$ and $x_k$ the function must
+  have some well-defined value, but this need not be the same as $s_k$.
+
+\begin{definition}
+
+  \lean{Common/Set/Intervals/StepFunction}{Set.Intervals.StepFunction}
+
+\end{definition}
+
+\end{document}

Common/Set/Intervals/Partition.lean

@@ -69,6 +69,12 @@ theorem right_ge_mem_self [Preorder α] [@DecidableRel α LT.lt]
   (p : Partition α) : ∀ x ∈ p, x ≤ p.b := by
   sorry
 
+/--
+The closed interval determined by the endpoints of the `Partition`.
+-/
+abbrev toIcc [Preorder α] [@DecidableRel α LT.lt]
+  (p : Partition α) := Set.Icc p.a p.b
+
 /-
 Return the closed subintervals determined by the `Partition`.
 -/
@@ -76,6 +82,12 @@ def closedSubintervals [Preorder α] [@DecidableRel α LT.lt]
   (p : Partition α) : List (Set α) :=
   p.toList.pairwise (fun x₁ x₂ => Icc x₁ x₂)
 
+/--
+The open interval determined by the endpoints of the `Partition`.
+-/
+abbrev toIoo [Preorder α] [@DecidableRel α LT.lt]
+  (p : Partition α) := Set.Ioo p.a p.b
+
 /-
 Return the open subintervals determined by the `Partition`.
 -/

Common/Set/Intervals/StepFunction.lean

@@ -8,20 +8,19 @@ Characterization of step functions.
 
 namespace Set.Intervals
 
+open Partition
+
 /--
 A function `f`, whose domain is a closed interval `[a, b]`, is a `StepFunction`
 if there exists a `Partition` `P = {x₀, x₁, …, xₙ}` of `[a, b]` such that `f` is
 constant on each open subinterval of `P`.
 -/
 structure StepFunction (α : Type _) [Preorder α] [@DecidableRel α LT.lt] where
-  /- A partition of some closed interval `[a, b]`. -/
+  p : Partition α
-  partition : Partition α
+  toFun : ∀ x ∈ p.toIcc, α
-  /-- A function whose domain is a closed interval `[a, b]`. -/
-  function : ∀ x ∈ Icc partition.a partition.b, α
-  /-- Ensure the function is constant on each open subinterval of `p`. -/
   const_open_subintervals :
-    ∀ (hI : I ∈ partition.openSubintervals), ∃ c : α, ∀ (hy : y ∈ I),
+    ∀ (hI : I ∈ p.openSubintervals), ∃ c : α, ∀ (hy : y ∈ I),
-      function y (Partition.mem_open_subinterval_mem_closed_interval hI hy) = c
+      toFun y (mem_open_subinterval_mem_closed_interval hI hy) = c
 
 namespace StepFunction
 

DocGen4/Output/Index.lean

@@ -37,13 +37,20 @@ def index : BaseHtmlM Html := do templateExtends (baseHtml "Index") <|
         status:
         <ul>
           <li>
-            <span style="color:teal">Teal statements </span> are those that
+            <span style="color:darkgray">Dark gray statements </span> are
-            have been proven or encoded in both LaTeX and Lean.
+            reserved for definitions and axioms that have been encoded in both
+            LaTeX and Lean.
           </li>
           <li>
-            <span style="color:magenta">Magenta statements </span> are those
+            <span style="color:teal">Teal statements </span> are reserved for
-            that have been proven or encoded in LaTeX but not yet verified in
+            statements, theorems, lemmas, etc. that have been proven in both
-            Lean.
+            LaTeX and Lean.
+          </li>
+          <li>
+            <span style="color:magenta">Magenta statements </span> are reserved
+            for definitions, axioms, statements, theorems, lemmas, etc. that
+            have been proven or encoded in LaTeX but not yet proven or encoded
+            in Lean.
           </li>
           <li>
             <span style="color:red">Red </span> serves as a catch-all for all

preamble.tex

@@ -14,8 +14,8 @@
 % Linking
 % ========================================
 
-\hypersetup{colorlinks=true, urlcolor=blue}
+\hypersetup{colorlinks=true, linkcolor=blue, urlcolor=blue}
-\newcommand{\leanref}[2]{{\color{blue}$\pmb{\exists}\;{-}\;$}\href{#1}{#2}}
+\newcommand{\leanref}[2]{\textcolor{blue}{$\pmb{\exists}\;{-}\;$}\href{#1}{#2}}
 
 % ========================================
 % Environments
@@ -24,6 +24,9 @@
 \newenvironment{axiom}{%
   \paragraph{\normalfont\normalsize\textit{Axiom.}}}
   {\hfill$\square$}
+\newenvironment{definition}{%
+  \paragraph{\normalfont\normalsize\textit{Definition.}}}
+  {\hfill$\square$}
 \newcommand{\divider}{%
   \vspace{10pt}
   \hrule
@@ -47,20 +50,12 @@
 % Status
 % ========================================
 
-% Used for statements encoded in both LaTeX and Lean.
 \DeclareRobustCommand{\defined}[1]{%
-  \texorpdfstring{\color{teal}\faParagraph\ #1}{#1}}
+  \texorpdfstring{\color{darkgray}\faParagraph\ #1}{#1}}
-
-% Used for statements proven in both LaTeX and Lean.
 \DeclareRobustCommand{\verified}[1]{%
   \texorpdfstring{\color{teal}\faCheckCircle\ #1}{#1}}
-
-% Used for statements proven or encoded in LaTeX but not yet in Lean.
 \DeclareRobustCommand{\partial}[1]{%
   \texorpdfstring{\color{magenta}\faPencil*\ #1}{#1}}
-
-% A catch-all for anything that doesn't fit the above categories. Incomplete
-% definitions, statements without proof, etc. belong here.
 \DeclareRobustCommand{\unverified}[1]{%
   \texorpdfstring{\color{red}\faExclamationCircle\ #1}{#1}}