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}
    Consider $\langle \L', \lte' \rangle$ where
    \begin{align*}
        \L' \define \{ x' ~|~ x \in \L \} \\
        x' \lte' y' \iff x \lte y \textrm{ where } x, y \in \L
    \end{align*}
    Clearly, $\langle \L', \lte' \rangle$ is a \CompleteLattice.
    Combining $\L$ and $\L'$
    \begin{align*}
        \L^* \define \{ \top^*, \bot^* \} \union \L \union \L' \\
        \forall x^*, y^* \in \L^*, x^* \lte y^* \iff \begin{cases}
            (x^* = \bot^*) \lor (y^* = \top^*) \\
            two \\ 
            three
        \end{cases}
    \end{align*}
\end{proof}

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