From e8f348f060dd7e505f0f8a8dbeb14e1a3cadc96e Mon Sep 17 00:00:00 2001
From: Spencer Killen <sjkillen@ualberta.ca>
Date: Mon, 4 Nov 2024 11:03:36 -0700
Subject: [PATCH] .

---
 glossary.tex | 14 +++++++++++++-
 notation.tex |  3 +++
 note.tex     | 15 +++++++++++++--
 3 files changed, 29 insertions(+), 3 deletions(-)

diff --git a/glossary.tex b/glossary.tex
index 335d418..5641d24 100644
--- a/glossary.tex
+++ b/glossary.tex
@@ -1 +1,13 @@
-\newcommand{\glsMonotone}{\hyperlink{glossary:monotone}{monotone}}
+\newcommand{\definition}[2]{\hypertarget{glossary:#1}{#2}}
+\newcommand{\definitionLink}[2]{\hyperlink{glossary:#1}{#2}\xspace}
+
+
+\newcommand{\Monotone}{\definitionLink{monotone}{monotone}}
+\newcommand{\Image}{\definitionLink{image}{monotone}}
+\let\imageNoLink\image
+\renewcommand{\image}[1]{\imageNoLink{#1}}
+\newcommand{\CompleteLattice}{\definitionLink{completelattice}{complete lattice}}
+
+
+\let\fixpointsOfNoLink\fixpointsOf{}
+\renewcommand{\fixpointsOf}{\definitionLink{fixpointsOf}{\fixpointsOfNoLink}}
diff --git a/notation.tex b/notation.tex
index 98af044..ba93611 100644
--- a/notation.tex
+++ b/notation.tex
@@ -1 +1,4 @@
 \newcommand{\fixpointsOf}{\textbf{\textit{fix}}}
+\renewcommand{\L}{\mathcal{L}}
+\newcommand{\lte}{\preceq}
+\newcommand{\image}[1]{[#1]}
\ No newline at end of file
diff --git a/note.tex b/note.tex
index 3a211b1..88112ad 100644
--- a/note.tex
+++ b/note.tex
@@ -1,8 +1,10 @@
 \documentclass{article}
 
+\usepackage{xspace}
 \usepackage{hyperref}
 \usepackage{bm}
 \usepackage[english]{babel}
+\usepackage{amsthm}
 \newcommand{\jh}[1]{{\leavevmode\color{blue!50!red}#1}}
 
 \input{notation.tex}
@@ -16,7 +18,9 @@
 
 % \maketitle
 
-\hypertarget{glossary:monotone}{define monotone}
+\definition{monotone}{define monotone}
+\definition{image}{define set image}
+\definition{completelattice}{define complete lattice}
 
 Hello world\cite{tarskilatticetheoretical1955}
 First, we generalize Knaster-Tarski Fixpoint Theorem.
@@ -25,9 +29,16 @@ $$\fixpointsOf(S)$$
 
 
 \begin{theorem}[Tarski-Knaster Fixpoint Theorem~\cite{tarskilatticetheoretical1955}]
-    For a \glsMonotone function
+    For a \Monotone function $o$ over a \CompleteLattice $\langle \L, \lte \rangle$, we have that $\langle \fixpointsOf(o), \lte \rangle$ is a \CompleteLattice.
 \end{theorem}
 
+\begin{theorem}
+    For a \Monotone function $o$ over a \CompleteLattice $\langle \L, \lte \rangle$, we have that $\langle o\imageNoLink{\L}, \lte \rangle$ is a \CompleteLattice.
+\end{theorem}
+\begin{proof}
+    foo
+\end{proof}
+
 \bibliographystyle{plain}
 \bibliography{references}