\documentclass[12pt]{article}
\input{hw1-macros}
\usepackage{palatino}
\title{\Large\textbf{
Homework 1: Heyting Algebra and IPL}
\normalsize\\
15-819 Homotopy Type Theory}
\author{}
\date{%
Out: 19/Sep/13\\
Due: 3/Oct/13
}
\begin{document}
\maketitle
This is 15-819's first homework assignment!
\section{Heyting Meets Boole}
\begin{task}
Show that $A \meet (B \join C) \leq (A \meet B) \join (A \meet C)$
in any Heyting algebra.
\begin{hint}
You might find Yoneda's Lemma useful, which says (in this particular context)
$A \leq B$ iff for all $C$, $B \leq C$ implies $A \leq C$.
There is a short proof with Yoneda's, and another short proof without.
\end{hint}
\end{task}
\begin{task}
Show that in any Heyting algebra,
$A \exp \init$ is one of the largest elements inconsistent with $A$,
and is equivalent to any largest inconsistent one.
\end{task}
\begin{task}
Show that in any Boolean algebra (complemented distributive lattice),
$\comp A \join B$ is a valid implementation of $A \exp B$.
That is, it satisfies all properties of $A \exp B$.
\end{task}
\section{IPL Structural Engineering}
\begin{task}
Show that IPL is transitive, which is to say
\[
\infer{
\ctx, \ctx' \entails P \true
\\
\ctx, P \true, \ctx' \entails Q \true
}{
\ctx, \ctx' \entails Q \true
}
\]
is admissible.
You only have to consider the case that the last rule applied in the right sub-derivation
(of $\ctx, P \true, \ctx' \entails Q \true$)
is either the primitive reflexivity or rules in the negative fragment.
You may assume weakening and exchange as admissible rules.
\end{task}
\section{Semantical Analysis of IPL}
\begin{task}
Show that for any Heyting algebra and any evaluation function on atoms,
if $\ctx \entails P \true$ then $\ctx^+ \leq P^*$.
You only have to consider the cases in which the last rule applied is ($\imp$I) or ($\imp$E).
\end{task}
\begin{task}
Consider the Lindenbaum algebra of IPL where
the elements are all propositions in IPL
(with the translation $(\text{--})^*$ being the identity function)
and the relationship $\leq$ is defined by provability in IPL.%
\footnote{To simplify the problem, we avoid taking quotients by interprovability,
but one must consider that if a partial order is desired.}
That is, $A \leq B$ iff $A \true \entails B \true$.
Show that this is a Heyting algebra.
You only have to prove the transitivity.
You may assume weakening and exchange of IPL, or cite previous tasks as lemmas.
\end{task}
\end{document}