fixpoint-theory-nov24/note.tex

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\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.

$$\fixpointsOf(S)$$


\begin{theorem}[Tarski-Knaster Fixpoint Theorem~\cite{tarskilatticetheoretical1955}]
    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}

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