\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}
