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\begin{document}
\begin{center}\large\bf
CIS 705 --- Programming Languages --- Spring 2009
\end{center}
\begin{center}\Large\bf
Assignment 6
\end{center}
\begin{center}\large\bf
Due by 2:30 p.m.\ on Thursday, May 7
\end{center}
\begin{center}
The context for this assignment is Chapters 11--12 of \emph{TAPL}.
\end{center}

\section*{Reformulating the Simply Typed Lambda Calculus with
  Unit and Products}

If $f$ is a function and $x,y$ are elements of our universe, we define
the function $f[x\mapsto y]$ from $\dom(f)\cup\{x\}$ to $\ran(f)\cup\{y\}$
by, for all $z\in\dom(f)\cup\{x\}$,
\begin{displaymath}
f[x\mapsto y](z) =
\left\{ \begin{array}{ll}
y,    & \eqtxtr{if} z = x, \\
f(z), & \eqtxtr{if} z\neq x .
\end{array} \right.
\end{displaymath}

If $f$ is a function and $X$ is a set, then $f/X$ is
the function from $\dom(f)\setminus X$ to $\ran(f)$ such that, for
all $z\in\dom(f)\setminus X$, $(f/X)(z) = f(z)$.

We reformulate the syntax of the simply typed lambda calculus with
unit and products as follows, where variables $x$ are as usual.  Our
\emph{types} are defined by:
\begin{alignat*}{2}
T &\goesto &\qquad \eqtxt{types:}\\
&\quad\Unit &\qquad \eqtxt{unit type}\\
&\quad T\times T &\qquad \eqtxt{product type} \\
&\quad T\fun T &\qquad \eqtxt{function type}
\end{alignat*}
As usual, $\fun$ associates to the right, and $\times$ has
higher precedence than $\fun$.
Our \emph{terms} are defined by:
\begin{alignat*}{2}
t &\goesto &\qquad \eqtxt{terms:}\\
&\quad \unit &\qquad \eqtxt{unit constant}\\
&\quad (t,t) &\qquad \eqtxt{pair} \\
&\quad \fst\,t &\qquad \eqtxt{first projection} \\
&\quad \snd\,t &\qquad \eqtxt{second projection} \\
&\quad x &\qquad \eqtxt{variable}\\
&\quad \lambda x\typrel T.\,t &\qquad \eqtxt{abstraction}\\
&\quad t\,t &\qquad \eqtxt{application}
\end{alignat*}
And our \emph{values} are defined by:
\begin{alignat*}{2}
v &\goesto &\qquad \eqtxt{values:}\\
&\quad\unit &\qquad \eqtxt{unit constant}\\
&\quad(v,v) &\quad \eqtxt{pair value} \\
&\quad \lambda x\typrel T.\,t &\qquad \eqtxt{abstraction value}
\end{alignat*}
As usual, application associates to the left and abstractions
extend as far as possible.  And $\fst$ and $\snd$ have higher
precedence than application.

In contrast to TAPL's approach, we do \emph{not} identify abstractions
up to the renaming of bound variables, so that, e.g.,
$\lambda x\typrel\Unit.\,x = \lambda y\typrel\Unit.\,y$ iff $x=y$.
The \emph{free variables} of a term $t$ ($\FV(t)$)
is defined recursively as follows:
\begin{align*}
\FV(\unit) &= \emptyset , \\
\FV((t_1,t_2)) &= \FV(t_1)\cup\FV(t_2) , \\
\FV(\fst\,t) &= \FV(t) , \\
\FV(\snd\,t) &= \FV(t) , \\
\FV(x) &= \{x\} , \\
\FV(\lambda x\typrel T.\, t) &= \FV(t)\setminus \{x\} , \\
\FV(t_1\,t_2) &= \FV(t_1)\cup\FV(t_2) .
\end{align*}
A term is \emph{closed} iff it has no free variables; otherwise it is
\emph{open}.

A \emph{simultaneous substitution} (or just \emph{substitution})
$\sigma$ is a function such that $\dom(\sigma)$ is a finite subset of
the variables, and $\ran(\sigma)$ is a subset of the closed values.
The result of \emph{applying} a substitution $\sigma$ \emph{to} a term
$t$ ($t\,\sigma$) is defined recursively by:
\begin{align*}
\unit\,\sigma &= \unit , \\
(t_1,t_2)\,\sigma &= (t_1\,\sigma, t_2\,\sigma) , \\
(\fst\,t)\,\sigma &= \fst(t\,\sigma) , \\
(\snd\,t)\,\sigma &= \snd(t\,\sigma) , \\
y\,\sigma &=
\left\{ \begin{array}{ll}
\sigma(y), & \eqtxtr{if} y\in\dom(\sigma), \\
y, & \eqtxtr{if} y\not\in\dom(\sigma),
\end{array} \right. \\
(\lambda x\typrel T.\,t)\,\sigma &=
\lambda x\typrel T.\,t\,(\sigma/\{x\}) , \\
(t_1\,t_2)\,\sigma &= (t_1\,\sigma)(t_2\,\sigma) .
\end{align*}

The \emph{evaluation relation} \fbox{$t\fun t'$} between \emph{closed}
terms is defined inductively by:
\begin{alignat*}{2}
&\frac{t_1\fun t'_1}%
{(t_1, t_2) \fun (t'_1,t_2)} &\quad\eqtxt{(E-Pair1)} \\[.2cm]
&\frac{t_2\fun t'_2}%
{(v_1, t_2) \fun (v_1,t'_2)} &\quad\eqtxt{(E-Pair2)} \\[.2cm]
&\frac{t\fun t'}%
{\fst\,t \fun \fst\,t'} &\quad\eqtxt{(E-Fst)} \\[.2cm]
&\fst(v_1,v_2)\fun v_1 &\quad\eqtxt{(E-FstVal)} \\[.2cm]
&\frac{t\fun t'}%
{\snd\,t \fun \snd\,t'} &\quad\eqtxt{(E-Snd)} \\[.2cm]
&\snd(v_1,v_2)\fun v_2 &\quad\eqtxt{(E-SndVal)} \\[.2cm]
&\frac{t_1\fun t'_1}%
{t_1\,t_2 \fun t'_1\,t_2} &\quad\eqtxt{(E-App1)} \\[.2cm]
&\frac{t_2\fun t'_2}%
{v_1\,t_2 \fun v_1\,t'_2} &\quad\eqtxt{(E-App2)} \\[.2cm]
&(\lambda x\typrel T.\,t)v\fun t\,\{(x,v)\} &\quad\eqtxt{(E-AppAbs)}
\end{alignat*}
So, in all but the last rule, $t$, $t'$, $t_1$, $t'_1$, $t_2$, $t'_2$,
$v$, $v_1$ and $v_2$ are \emph{closed}, whereas, in (E-AppAbs),
$v$ is closed but $\FV(t)\sub\{x\}$.  A closed term
$t$ is a \emph{normal form} iff there is no closed term $t'$ such that
$t\fun t'$.  A closed term $t$ is \emph{stuck} iff $t$ is a normal
form but $t$ is not a value.  A closed term $t$ \emph{converges} iff
there is a closed value $v$ such that $t\fun^*v$; otherwise, it
\emph{diverges}.

An easy induction on the evaluation relation suffices to show that
\textbf{evaluation is deterministic}: for all closed terms $t$, $t'$ and
$t''$, if $t\fun t'$ and $t\fun t''$, then $t'=t''$.

A \emph{typing context} (or just \emph{context}) $\Gamma$ is a function
such that $\dom(\Gamma)$ is a finite subset of the variables,
and $\ran(\Gamma)$ is a subset of the types.
The \emph{typing relation} \fbox{$\Gamma\vdash t\typrel T$} between
typing contexts, terms and types is defined inductively by:
\begin{alignat*}{2}
&\Gamma\vdash\unit:\Unit & \quad\eqtxt{(T-Unit)} \\[.2cm]
&\frac{\Gamma\vdash t_1\typrel T_1\qquad
\Gamma\vdash t_2\typrel T_2}%
{\Gamma\vdash (t_1,t_2) \typrel T_1\times T_2} & \quad\eqtxt{(T-Pair)} \\[.2cm]
&\frac{\Gamma\vdash t\typrel T_1\times T_2}
{\Gamma\vdash\fst\,t\typrel T_1} & \quad\eqtxt{(T-Fst)} \\[.2cm]
&\frac{\Gamma\vdash t\typrel T_1\times T_2}
{\Gamma\vdash\snd\,t\typrel T_2} & \quad\eqtxt{(T-Snd)} \\[.2cm]
&\frac{(x, T)\in\Gamma}%
{\Gamma\vdash x \typrel T} & \quad\eqtxt{(T-Var)} \\[.2cm]
&\frac{\Gamma[x\mapsto T_1]\vdash t\typrel T_2}
{\Gamma\vdash\lambda x\typrel T_1.\,t\typrel T_1\fun T_2} & \quad\eqtxt{(T-Abs)} \\[.2cm]
&\frac{\Gamma\vdash t_1\typrel T_1\fun T_2\qquad
\Gamma\vdash t_2\typrel T_1}%
{\Gamma\vdash t_1\,t_2 \typrel T_2} & \quad\eqtxt{(T-App)}
\end{alignat*}
We say that a closed term $t$ is \emph{well-typed} iff
$\emptyset\vdash t\typrel T$ for some type $T$.

An easy induction on the typing relation suffices to show that, for
all contexts $\Gamma$, terms $t$ and types $T$, if $\Gamma\vdash
t\typrel T$, then $\FV(t)\sub\dom(\Gamma)$.  Another induction on the
typing relation shows the \textbf{weakening lemma}: for all contexts
$\Gamma$ and $\Gamma'$, terms $t$ and types $T$, if $\Gamma\vdash
t\typrel T$ and $\Gamma\sub\Gamma'$, then $\Gamma'\vdash t\typrel T$.
And another induction on the typing relation suffices to show the
\textbf{uniqueness of typing}: for all contexts $\Gamma$, terms $t$
and types $T$ and $T'$, if $\Gamma\vdash t\typrel T$ and $\Gamma\vdash
t\typrel T'$, then $T=T'$.

The \textbf{canonical forms lemma} holds:
\begin{itemize}
\item If $v$ is a value and $\emptyset\vdash v\typrel\Unit$, then
  $v=\unit$.

\item For all types $T_1$ and $T_2$, if $v$ is a value and
  $\emptyset\vdash v\typrel T_1\times T_2$, then
  $v=(v_1,v_2)$, where $\emptyset\vdash v_1\typrel T_1$ and
  $\emptyset\vdash v_2\typrel T_2$.

\item For all types $T_1$ and $T_2$, if $v$ is a value and
  $\emptyset\vdash v\typrel T_1\fun T_2$, then
  $v=\lambda x\typrel T_1.\,t$ for a variable $x$ and term $t$
  such that $\{(x,T_1)\}\vdash t\typrel T_2$.
\end{itemize}

The \textbf{progress theorem} holds: for all closed terms $t$, if $t$ is
well-typed, then $t$ is not stuck.  And the \textbf{preservation theorem}
holds: for all closed terms $t$ and $t'$ and types $T$, if
$\emptyset\vdash t\typrel T$ and $t\fun t'$, then $\emptyset\vdash
t'\typrel T$.

Define a function $R$ from types to sets of terms, by recursion
(where we write $R(T)$ as $R_T$):
\begin{itemize}
\item $R_\Unit = \setof{t}{\emptyset\vdash t\typrel\Unit\eqtxt{and} t
    \eqtxtl{converges}}$;

\item $R_{T_1\times T_2} = \setof{t}{\emptyset\vdash t\typrel T_1\times
    T_2\eqtxt{and} t\eqtxt{converges and} \fst\,t\in R_{T_1} \eqtxt{and}
    \snd\,t\in R_{T_2}}$;

\item $R_{T_1\fun T_2} = \setof{t}{\emptyset\vdash t\typrel T_1\fun
    T_2\eqtxt{and} t\eqtxt{converges and, for all terms} s, \eqtxt{if}
    s\in R_{T_1} , \eqtxt{then} t\,s\in R_{T_2}}$.
\end{itemize}
Note that, for all types $T$, $R_T\sub\setof{t}{\emptyset\vdash t\typrel T
\eqtxt{and} t \eqtxtl{converges}}$.
We often write $R_T(t)$ instead of $t\in R_T$.

\section*{Exercise 1 (10 Points)}

Prove that, for all terms $t$ and substitutions $\sigma$, if
$\FV(t)\cap\dom(\sigma)=\emptyset$, then $t\,\sigma = t$.

\section*{Exercise 2 (15 Points)}

Prove that, for all terms $t$ and substitutions $\sigma$,
$\FV(t\,\sigma) = \FV(t) \setminus \dom(\sigma)$.  (This result is
needed to see that the rule (E-AppAbs) of the definition of $\fun$ is
valid.)

\section*{Exercise 3 (15 Points)}

Prove that, for all contexts $\Gamma$, terms $t$, types $T$ and
substitutions $\sigma$, if $\Gamma\vdash t\typrel T$ and, for all
$x\in\dom(\Gamma)\cap\dom(\sigma)$,
$\emptyset\vdash\sigma(x)\typrel\Gamma(x)$, then
$\Gamma/\dom(\sigma)\vdash t\,\sigma\typrel T$.

\section*{Exercise 4 (25 Points)}

Prove that, for all types $T$, for all closed terms $t$ and $t'$, if
$\emptyset\vdash t\typrel T$ and $t\fun t'$, then $R_T(t)$ iff
$R_T(t')$.

\section*{Exercise 5 (30 Points)}

Prove that, for all contexts $\Gamma$, terms $t$, types $T$ and
substitutions $\sigma$, if $\Gamma\vdash t\typrel T$,
$\dom(\Gamma)\sub \dom(\sigma)$, and for all
$x\in\dom(\Gamma)$, $R_{\Gamma(x)}(\sigma(x))$, then
$R_T(t\,\sigma)$.

\section*{Exercise 6 (5 Points)}

Prove that all closed, well-typed terms converge.
\end{document}