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+\begin{definition}\label{bar}
+    $x$ is a bar iff $x = x$.
+\end{definition}
+
+\begin{definition}\label{foo}
+    $x$ is foo iff $x = x$.
+\end{definition}
+
+\begin{definition}\label{baz}
+    $x$ is baz iff $x = x$.
+\end{definition}
+
+\begin{proposition}\label{nouns}
+    Let $x, y$ be bars.
+    Then $x = x$.
+\end{proposition}
+
+\begin{proposition}\label{adj_nouns}
+    Let $x, y$ be foo bars.
+    Then $x = x$.
+\end{proposition}
+
+
+\begin{proposition}\label{nouns_suchthat}
+    Let $x, y$ be bars such that $x$ is foo and $y$ is baz.
+    Then $x = x$.
+\end{proposition}
+
+
+\begin{proposition}\label{noun_verb}
+    $x = y$ iff $x$ is a bar equal to $y$.
+\end{proposition}
+
+
+\begin{proposition}\label{adjs}
+    $x$ is foo and baz.
+\end{proposition}