From 9a89974f293aa53451cad42f07e54e5bca14af4f Mon Sep 17 00:00:00 2001
From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com>
Date: Tue, 25 Jun 2024 00:00:38 +0200
Subject: definition of equivalence_from_partition can proof false in
 everything.tex

---
 library/relation/equivalence.tex | 101 ++++++++++++++++++++-------------------
 1 file changed, 51 insertions(+), 50 deletions(-)

(limited to 'library/relation/equivalence.tex')

diff --git a/library/relation/equivalence.tex b/library/relation/equivalence.tex
index bda8486..87f70af 100644
--- a/library/relation/equivalence.tex
+++ b/library/relation/equivalence.tex
@@ -219,53 +219,54 @@
 \end{proof}
 
 
-
-\begin{definition}\label{equivalence_from_partition}
-    $\equivfrompartition{P} = \{(a, b)\mid a\in A, b\in A\mid \exists C\in P.\ a, b\in C\}$.
-\end{definition}
-
-\begin{proposition}\label{equivalence_from_partition_intro}
-    Let $P$ be a partition of $A$.
-    Let $a,b\in A$.
-    Suppose $a,b\in C\in P$.
-    Then $a\mathrel{\equivfrompartition{P}} b$.
-\end{proposition}
-
-\begin{proposition}\label{equivalence_from_partition_reflexive}
-    Let $P$ be a partition of $A$.
-    $\equivfrompartition{P}$ is reflexive on $A$.
-\end{proposition}
-
-\begin{proposition}\label{equivalence_from_partition_symmetric}
-    Let $P$ be a partition.
-    $\equivfrompartition{P}$ is symmetric.
-\end{proposition}
-\begin{proof}
-    Follows by \cref{symmetric,equivalence_from_partition,notin_emptyset}.
-\end{proof}
-
-\begin{proposition}\label{equivalence_from_partition_transitive}
-    Let $P$ be a partition.
-    $\equivfrompartition{P}$ is transitive.
-\end{proposition}
-
-\begin{proposition}\label{equivalence_from_partition_is_equivalence}
-    Let $P$ be a partition of $A$.
-    $\equivfrompartition{P}$ is an equivalence on $A$.
-\end{proposition}
-
-\begin{proposition}\label{equivalence_from_quotient}
-    Let $E$ be an equivalence on $A$.
-    Then $\equivfrompartition{\quotient{A}{E}} = E$.
-\end{proposition}
-\begin{proof}
-    Follows by set extensionality.
-\end{proof}
-
-\begin{proposition}\label{partition_eq_quotient_by_equivalence_from_partition}
-    Let $P$ be a partition of $A$.
-    Then $\quotient{A}{\equivfrompartition{P}} = P$.
-\end{proposition}
-\begin{proof}
-    Follows by set extensionality.
-\end{proof}
+%
+%\begin{definition}\label{equivalence_from_partition}
+%    $\equivfrompartition{P} = \{(a, b)\mid a\in A, b\in A\mid \exists C\in P.\ a, b\in C\}$.
+%\end{definition}
+%
+%\begin{proposition}\label{equivalence_from_partition_intro}
+%    Let $P$ be a partition of $A$.
+%    Let $a,b\in A$.
+%    Suppose $a,b\in C\in P$.
+%    Then $a\mathrel{\equivfrompartition{P}} b$.
+%\end{proposition}
+%
+%\begin{proposition}\label{equivalence_from_partition_reflexive}
+%    Let $P$ be a partition of $A$.
+%    $\equivfrompartition{P}$ is reflexive on $A$.
+%\end{proposition}
+%
+%\begin{proposition}\label{equivalence_from_partition_symmetric}
+%    Let $P$ be a partition.
+%    $\equivfrompartition{P}$ is symmetric.
+%\end{proposition}
+%\begin{proof}
+%    Follows by \cref{symmetric,equivalence_from_partition,notin_emptyset}.
+%\end{proof}
+%
+%\begin{proposition}\label{equivalence_from_partition_transitive}
+%    Let $P$ be a partition.
+%    $\equivfrompartition{P}$ is transitive.
+%\end{proposition}
+%
+%\begin{proposition}\label{equivalence_from_partition_is_equivalence}
+%    Let $P$ be a partition of $A$.
+%    $\equivfrompartition{P}$ is an equivalence on $A$.
+%\end{proposition}
+%
+%\begin{proposition}\label{equivalence_from_quotient}
+%    Let $E$ be an equivalence on $A$.
+%    Then $\equivfrompartition{\quotient{A}{E}} = E$.
+%\end{proposition}
+%\begin{proof}
+%    Follows by set extensionality.
+%\end{proof}
+%
+%\begin{proposition}\label{partition_eq_quotient_by_equivalence_from_partition}
+%    Let $P$ be a partition of $A$.
+%    Then $\quotient{A}{\equivfrompartition{P}} = P$.
+%\end{proposition}
+%\begin{proof}
+%    Follows by set extensionality.
+%\end{proof}
+%
\ No newline at end of file
-- 
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