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\noindent
\parbox{2.25in}{\bf Math 2110}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Sept. $2^\mathrm{nd}$, 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 7 - it's just missing justifications for each line. Please fill in the justifications.

{\noindent {\fbox{\bf Lemma 7:}} \quad}  $A \to (B \to C) \vdash_L B \to (A \to C)$

\begin{enumerate}[1:]
   \item $A \to (B \to C)$                                                     \hfill \parbox{3.2in}{\underline{\arabic{enumi}: Given}}
   \item $B \to (A \to (B \to C))$                                          \hfill \parbox{3.2in}{\underline{\arabic{enumi}: ?????}}
   \item $(A \to (B \to C)) \to ((A \to B) \to (A \to C))$       \hfill \parbox{3.2in}{\underline{\arabic{enumi}: ?????}}
   \item $(A \to B) \to (A \to C)$                                          \hfill \parbox{3.2in}{\underline{\arabic{enumi}: ?????}}
   \item $B \to ((A \to B) \to (A \to C))$                              \hfill \parbox{3.2in}{\underline{\arabic{enumi}: ?????}}
   \item $B \to (A \to B)$                                                     \hfill \parbox{3.2in}{\underline{\arabic{enumi}: ?????}}
   \item $B \to (A \to C)$                                                    \hfill \parbox{3.2in}{\underline{\arabic{enumi}: ?????}}
\end{enumerate}

\vspace{0.10in}

\item[\#2] Prove the following lemmas (without using the deduction theorem): L8 and L12.

{\noindent {\fbox{\bf Lemma 8:}} \quad} $A \to B, B \to C \vdash_L A \to C$

% PROOF GOES HERE 

\vspace{0.1in}

{\noindent {\fbox{\bf Lemma 12:}} \quad} $ \vdash_L B \to \neg \neg B$

% PROOF GOES HERE 

\vspace{0.1in}

\item[\#3] Prove the following lemmas (you may use the deduction theorem if you want): L15, L22, and L24.

{\noindent {\fbox{\bf Lemma 15:}} \quad} $ \vdash_L \neg A \to (A \to B)$

% PROOF GOES HERE 

\vspace{0.1in}

{\noindent {\fbox{\bf Lemma 22:}} \quad} $B \vdash_L A \vee B$

% PROOF GOES HERE 

\vspace{0.1in}

{\noindent {\fbox{\bf Lemma 24:}} \quad}  $A \leftrightarrow B \vdash_L B \to A$

% PROOF GOES HERE 

\vspace{0.1in}

\item[\#4] Consider the statements: \quad $A \to B,B \vdash_L A$ \quad and \quad $A,\neg B \vdash_L \neg(A \wedge B)$.

One of these is actually a theorem of $L$. The other statement is not. Prove this is the case. 

{}[{\it Hint/suggestion:} Truth tables should help. Don't try to find System L proofs.]

\vspace{0.1in}

{\bf NOTE:} While you should create truth tables for yourself, I won't require that you type them up. You can merely summarize what they tell you. You can say something like ``The truth table of XXXX showed that this statement was a YYYY. Therefore, ZZZZ.''

\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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