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\noindent
\parbox{2.25in}{\bf Math 2110}
\hfill {\Large \bf Homework \#4} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Sept. $21^\mathrm{st}$, 2020}

\vspace{0.1in}

\noindent {\large\bf Name:} \underline{\Large\quad My Name Goes Here \quad} \hfill Please use \LaTeX\  to type up this homework set!

\vspace{0.2in}

\begin{description}
\item[\#1] I have provided the proof of Lemma K19 - it's just missing justifications for each line. Please fill in the justifications.

{\noindent {\fbox{\bf Lemma K19:}} \quad}   $\neg\forall x\, A(x) \vdash_K \exists x\, \neg A(x)$

\begin{enumerate}[1:]
   \item $\neg\forall x\, A(x)$  \hfill \parbox{2.8in}{\underline{\arabic{enumi}: Given}}
   \item $\neg\neg A(x) \to A(x)$  \hfill \parbox{2.8in}{\underline{\arabic{enumi}: ?????}}
   \item $\forall x\, (\neg\neg A(x) \to A(x))$  \hfill \parbox{2.8in}{\underline{\arabic{enumi}: ?????}}
   \item $\forall x\, (\neg\neg A(x) \to A(x)) \to (\forall x\, \neg\neg A(x) \to \forall x\, A(x))$ \hfill \parbox{2.8in}{\underline{\arabic{enumi}: ?????}}
   \item $\forall x\, \neg\neg A(x) \to \forall x\, A(x)$  \hfill \parbox{2.8in}{\underline{\arabic{enumi}: ?????}}
   \item $(\forall x\, \neg\neg A(x) \to \forall x\, A(x)) \to (\neg \forall x\, A(x) \to \neg \forall x\, \neg\neg A(x))$  \hfill \parbox{2.8in}{\underline{\arabic{enumi}: ?????}}
   \item $\neg \forall x\, A(x) \to \neg \forall x\, \neg\neg A(x)$  \hfill \parbox{2.8in}{\underline{\arabic{enumi}: ?????}}
   \item $\neg \forall x\, \neg\neg A(x)$  \hfill \parbox{2.8in}{\underline{\arabic{enumi}: ?????}}
   \item $\exists x\, \neg A(x)$  \hfill \parbox{2.8in}{\underline{\arabic{enumi}: ?????}}
\end{enumerate}

\vspace{0.10in}

\item[\#2] Prove the following lemmas: K7, K13, K16, and K32.

{\noindent {\fbox{\bf Lemma K7:}} \quad}  $ \vdash_K \forall x\,(A(x)\to B(x)) \to (\forall x\, A(x) \to \forall x\, B(x))$

% PROOF GOES HERE 

\vspace{0.1in}

{\noindent {\fbox{\bf Lemma K13:}} \quad} $ \vdash_K \forall x\, A(x) \to \exists x\, A(x)$

% PROOF GOES HERE 

\vspace{0.1in}

{\noindent {\fbox{\bf Lemma K16:}} \quad}  $\neg \exists x\, A(x) \vdash_K \forall x\, \neg A(x)$

% PROOF GOES HERE 

\vspace{0.1in}

{\noindent {\fbox{\bf Lemma K32:}} \quad} $\vdash_K (\forall x\, A(x) \wedge \exists x\, B(x)) \to \exists x\,(A(x) \wedge B(x))$

% PROOF GOES HERE 

\vspace{0.1in}


\item[Template Proof Code:]

If we use the Deduction Theorem our proof looks something like this\dots
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{0.1in}

Consider $stuff \vdash_L stuff$.

\noindent {\bf Proof:}

\begin{enumerate}[1:]
   \item $???$                                                     \hfill \parbox{3.2in}{\underline{\arabic{enumi}: ???}}
   \item $???$                                                     \hfill \parbox{3.2in}{\underline{\arabic{enumi}: ???}}
\end{enumerate}

\vspace{0.1in}

By the Deduction Theorem blah blah blah.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.2in}

Otherwise, our proof could just look like this\dots
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{enumerate}[1:]
   \item $???$                                                     \hfill \parbox{3.2in}{\underline{\arabic{enumi}: ???}}
   \item $???$                                                     \hfill \parbox{3.2in}{\underline{\arabic{enumi}: ???}}
\end{enumerate}



\end{description}


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