Restore exact cases and contradiction

1 files changed, 45 insertions, 0 deletions

diff --git a/test/phase5/exact-cases-contradiction.tex b/test/phase5/exact-cases-contradiction.tex
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+\begin{proposition}\label{cases_exact}
+    Let $A$ be a set.
+    $A=A$.
+\end{proposition}
+\begin{proof}
+    \begin{byCase}
+    \caseOf{$A=\emptyset$.}
+        Follows.
+    \caseOf{$A=\{A\}$.}
+        Follows.
+    \caseOf{$A=A$.}
+        Follows.
+    \end{byCase}
+\end{proof}
+
+\begin{proposition}\label{by_contradiction_exact}
+    Let $A$ be a set.
+    $A=A$.
+\end{proposition}
+\begin{proof}
+    Suppose not.
+    Contradiction.
+\end{proof}
+
+\begin{proposition}\label{arbitrary_contradiction_exact}
+    Let $A$ be a set.
+    Suppose $A\neq A$.
+    Then $A=A$.
+\end{proposition}
+\begin{proof}
+    Contradiction.
+\end{proof}
+
+\begin{proposition}\label{omitted_case_exact}
+    Let $A$ be a set.
+    $A=A$.
+\end{proposition}
+\begin{proof}
+    \begin{byCase}
+    \caseOf{$A=A$.}
+        Omitted.
+    \caseOf{$A\neq A$.}
+        Follows.
+    \end{byCase}
+\end{proof}