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%
% Proof Systems, etc.
%
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\def\FO{{\mathrm{FO}}}
\def\PA{{\mathrm{PA}}}

%
% Logical syntax and symbols
%
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\def\limplies{\rightarrow}
\def\tautEq{\vDash\Dashv}
\def\True{{\mathrm T}}
\def\False{{\mathrm F}}

\def\bigand{\bigwedge}
\def\bigor{\bigvee}

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\begin{document}


\begin{center}
{\bf Math 160A - Fall 2021 -
\bf Homework 8} \\  {\bf Due Friday, December 3, 11:00pm} \\
(Hand in by uploading to Gradescope)
\end{center}

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Your answers may use results proved in class, and results in the (forthcoming) PDF drafts.

\noindent
{\bf 1.} 
Suppose that $y$ and $t$ are each substitutable for $x$ in~$A$.
\begin{enumerate}[nosep]
\item Show that $\forall x\, A \vdash A(t/x)$ by giving
an explicit $\FO$-proof of $A(t/x)$ from $\forall x\, A$.
(An ``explicit'' proof means writing out all the formulas
in the proof, indicating if they are a hypothesis, if they are an axiom
or if they are inferred by Modus Ponens or Generalization.)
\item Show that $\forall x \, A \vdash \forall y \, A(y/x)$ by
giving an explicit proof of $\forall y \, A(y/x)$ from $\forall x \, A$.
\end{enumerate}

\noindent
{\bf 2.} 
Prove the following:
\begin{enumerate}[nosep]
\item $\vdash f(x) = f(x)$.
\item $\vdash x=y \limplies u=v \limplies g(f(x),f(u)) = g(f(y), f(v))$.
\item $\vdash x=y \limplies P(y) \limplies P(x)$.
\end{enumerate}

\noindent
{\bf 3.}
Prove that $\vdash \forall x\, A \limplies \exists x\, A$.

\noindent
{\bf 4.}
Suppose that $\forall x\, A \limplies C$ is a sentence. In other words,
$x$ does not occur free in~$C$ and $x$~is the only variable
which appears free in~$A$.
Prove:
\begin{enumerate}[nosep]
\item $\forall x\, (A \limplies C) \vdash \exists x\, A \limplies C$.
\item $\exists x\, A \limplies C \vdash \forall x\, (A \limplies C)$.
\end{enumerate}
(This exercise also holds under just the assumption that $x$ is not free in~$C$.)

\noindent
{\bf 5.}
The Deduction Theorem for $\FO$ requires
$A$~to be a sentence. Show that this hypothesis cannot be eliminated
by giving an example of formulas $A$ and~$B$ such that
$A \vdash B$ but $\nvdash A \limplies B$. (You can use the
Soundness Theorem to prove $\nvdash A  \limplies B$ by showing that $\nvDash A \limplies B$.)


\end{document}
