From ec3b6503e764daa3e163e6f959af0c239e65a89a Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Mon, 15 May 2023 00:08:40 +0100
Subject: [PATCH] Refine layout in epsilons lemma/dfn
---
main.tex | 33 +++++++++++++++------------------
1 file changed, 15 insertions(+), 18 deletions(-)
diff --git a/main.tex b/main.tex
index 77c2ced..489bee5 100644
--- a/main.tex
+++ b/main.tex
@@ -1041,36 +1041,33 @@ inverse, for each $q$ considered.
Consider the following tautology:
\begin{align}
- &\frac{ x }{ m }
+ \frac{ x }{ m }
- \frac{
(\aa r+2\bb)\aa
}{
2n^2
}
= \frac{ k }{ 2mn^2 }
- \quad \text{for some } x \in \ZZ, k \in \ZZ_{>0}
+ \quad \text{for some } x \in \ZZ
+ \span \span \span \span \span
\label{eqn:finding_better_eps_problem}
-\\ &\iff
+\\ &\Longleftrightarrow&
- (\aa r+2\bb)\aa m
- \equiv k
- \quad \mod 2n^2
- \quad \text{for some } k \in \ZZ_{>0}
-\\ &\iff
+ &\equiv k &&
+ \mod 2n^2
+\\ &\Longleftrightarrow&
- \aa^2 m r - 2\aa\bb m
- \equiv k
- \quad \mod 2n^2
- \quad \text{for some } k \in \ZZ_{>0}
-\\ &\Longrightarrow
+ &\equiv k &&
+ \mod 2n^2
+\\ &\Longrightarrow&
\aa^2 \aa^{'}\bb m - 2\aa\bb m
- \equiv k
- \quad \mod \gcd(2n^2, \aa^2 mn)
- \quad \text{for some } k \in \ZZ_{>0}
+ &\equiv k &&
+ \mod \gcd(2n^2, \aa^2 mn)
\label{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
-\\ &\Longrightarrow
+\\ &\Longrightarrow&
-\aa\bb m
- \equiv k
- \quad \mod n
- \quad \text{for some } k \in \ZZ_{>0}
+ &\equiv k &&
+ \mod n
\label{eqn:better_eps_problem_k_mod_n}
\end{align}
--
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