X-Git-Url: https://git.enpas.org/?a=blobdiff_plain;f=presentation.tex;h=2184ae49c31d3e0d65748257619744bf7538f3c5;hb=HEAD;hp=f3381bc273d653487bdebbfccb6fd9ca20852406;hpb=ebdf439a201d6fda0780284fc7f49b1f8deb88ab;p=bitonic-mengthesis.git

diff --git a/presentation.tex b/presentation.tex
index f3381bc..2184ae4 100644
--- a/presentation.tex
+++ b/presentation.tex
@@ -10,6 +10,16 @@
 \AtBeginSubsubsection{\frame{\subsubsectionpage}}
 \beamertemplatenavigationsymbolsempty
 
+\usepackage{centernot}
+\usepackage{cancel}
+\usepackage{tikz}
+\usetikzlibrary{shapes,arrows,positioning}
+\usetikzlibrary{intersections}
+\pgfdeclarelayer{background}
+\pgfdeclarelayer{foreground}
+\pgfsetlayers{background,main,foreground}
+
+
 \setlength{\parindent}{0pt}
 \setlength{\parskip}{6pt plus 2pt minus 1pt}
 \setlength{\emergencystretch}{3em}  % prevent overfull lines
@@ -39,7 +49,7 @@
 \newcommand{\myb}[1]{\AgdaBound{$#1$}}
 \newcommand{\myfield}{\AgdaField}
 \newcommand{\myind}{\AgdaIndent}
-\newcommand{\mykant}{\textmd{\textsc{Bertus}}}
+\newcommand{\mykant}{{\rmfamily\scshape Bertus}}
 \newcommand{\mysynel}[1]{#1}
 \newcommand{\myse}{\mysynel}
 \newcommand{\mytmsyn}{\langle t \rangle}
@@ -204,10 +214,12 @@
 \newcommand{\mytermi}[1]{\text{\texttt{#1}}}
 \newcommand{\mysee}[1]{\langle\myse{#1}\rangle}
 \newcommand{\mycomp}{\mathbin{\myfun{$\circ$}}}
-\newcommand{\mylist}[1]{\mathopen{\mytyc{$[$}} #1 \mathclose{\mytyc{$]$}}}
+\newcommand{\mylist}[1]{\mytyc{List}\myappsp #1}
 \newcommand{\mylistt}[1]{\mathopen{\mydc{$[$}} #1 \mathclose{\mydc{$]$}}}
 \newcommand{\myplus}{\mathbin{\myfun{$+$}}}
 \newcommand{\mytimes}{\mathbin{\myfun{$*$}}}
+\newcommand{\mysuc}{\mydc{suc}}
+\newcommand{\myzero}{\mydc{zero}}
 
 \renewcommand{\[}{\begin{equation*}}
 \renewcommand{\]}{\end{equation*}}
@@ -220,6 +232,19 @@
 \begin{document}
 \frame{\titlepage}
 
+\begin{frame}
+  \frametitle{\mykant?}
+
+  \mykant\ is an \emph{interactive theorem prover}/\emph{functional
+    programming language}, implemented in Haskell.
+
+  It is in the tradition of Agda and Epigram (and to a lesser extent
+  Coq), but with a more powerful notion of \emph{equality}.
+
+  We have figured out theory of \mykant, and have a near-complete
+  implementation.
+\end{frame}
+
 \begin{frame}
   \frametitle{Theorem provers are short-sighted}
 
@@ -253,254 +278,230 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{Observational equality and \mykant}
+  \frametitle{\mykant\ and observational equality}
+
+  \emph{Observational} equality solves this and other annoyances.
 
-  \emph{Observational equality} is a solution to this and other equality
-  annoyances.
+  \mykant\ is a system aiming at making observational equality more
+  usable.
+\end{frame}
 
-  \mykant\ is a system making observational equality more usable.
+\begin{frame}
+  \frametitle{Theorem provers, dependent types}
+  \begin{center}
+    \Huge
+    types $\leftrightarrow$ propositions
 
-  The theory of \mykant\ is complete, the practice, not quite.
+    programs $\leftrightarrow$ proofs
+  \end{center}
 \end{frame}
 
 \begin{frame}
-  \frametitle{Theorem provers, dependent types} First class types: we
-  can return them, have them as arguments, etc.
+  \frametitle{Theorem provers, dependent types}
   \[
-  \begin{array}{@{}l@{\ }l@{\ \ \ }l}
-    \mysyn{data}\ \myempty & & \text{No members.} \\
-    \mysyn{data}\ \myunit  & = \mytt & \text{One member.} \\
-    \mysyn{data}\ \mynat   & = \mydc{zero} \mydcsep \mydc{suc}\myappsp\mynat & \text{Natural numbers.}
+  \begin{array}{@{}l@{\ \ \ }l}
+    \mysyn{data}\ \myempty & \text{No members.} \\
+    \mysyn{data}\ \myunit \mapsto \mytt & \text{One member.}
   \end{array}
   \]
-  $\myempty : \mytyp$, $\myunit : \mytyp$, $\mynat : \mytyp$.
 
   $\myunit$ is trivially inhabitable: it corresponds to $\top$ in
   logic.
-
+  \[
+  \mytt : \myunit
+  \]
   $\myempty$ is \emph{not} inhabitable: it corresponds to $\bot$.
+  \[
+  \myfun{absurd} : \myempty \myarr \myb{A}
+  \]
 
+  First class types: we can return them, have them as arguments, etc:
+  $\myempty : \mytyp$, $\myunit : \mytyp$.
 \end{frame}
 
 \begin{frame}
   \frametitle{Theorem provers, dependent types}
-
-  We can express relations:
+  \[ \mysyn{data}\ \mylist{\myb{A}} \mapsto \mynil \mydcsep \myb{A} \mycons \mylist{\myb{A}} \]
+  We want to express a `non-emptiness' property for lists:
   \[
   \begin{array}{@{}l}
-    (\myfun{${>}$}) : \mynat \myarr \mynat \myarr \mytyp \\
-    \begin{array}{@{}c@{\,}c@{\,}c@{\ }l}
-      \mydc{zero} & \mathrel{\myfun{$>$}} & \myb{m} & = \myempty \\
-      \myb{n}     & \mathrel{\myfun{$>$}} & \mydc{zero} & = \myunit \\
-      (\mydc{suc} \myappsp \myb{n}) & \mathrel{\myfun{$>$}} & (\mydc{suc} \myappsp \myb{m}) & = \myb{n} \mathrel{\myfun{$>$}} \myb{m}
+    \myfun{non-empty} : \mylist{\myb{A}} \myarr \mytyp \\
+    \begin{array}{@{}l@{\myappsp}c@{\ }l}    
+    \myfun{non-empty} & \mynil & \mapsto \myempty \\
+    \myfun{non-empty} & (\myb{x} \mycons \myb{l}) & \mapsto \myunit
     \end{array}
   \end{array}
   \]
 
-  A term of type $\myb{m} \mathrel{\myfun{$>$}} \myb{n}$ represents a
-  \emph{proof} that $\myb{n}$ is indeed greater than $\myb{n}$.
+  A term of type $\myfun{non-empty} \myappsp \myb{l}$ represents a
+  \emph{proof} that $\myb{l}$ is indeed not empty.
   \[
-  \begin{array}{@{}l}
-    3 \mathrel{\myfun{$>$}} 1 \myred \myunit \\
-    2 \mathrel{\myfun{$>$}} 2 \myred \myempty \\
-    0 \mathrel{\myfun{$>$}} \myb{m} \myred \myempty
+  \begin{array}{@{}l@{\ \ \ }l}
+    \text{Can't prove} & \myfun{non-empty}\myappsp \mynil \myred \myempty \\
+    \text{Trivial to prove}  & \myfun{non-empty}\myappsp(2 \mycons \mynil) \myred \myunit
   \end{array}
   \]
-
-  Thus, proving that $2 \mathrel{\myfun{$>$}} 2$ corresponds to proving
-  falsity, while $3 \mathrel{\myfun{$>$}} 1$ is fine.
 \end{frame}
 
 \begin{frame}
   \frametitle{Example: safe $\myfun{head}$ function}
-
+  \only<3>{We can eliminate the `empty list' case:}
   \[
   \begin{array}{@{}l}
-    \mysyn{data}\ \mylistt{\myb{A}} = \mynil \mydcsep \myb{A} \mycons \mylistt{\myb{A}} \\
-    \ \\
-    \myfun{length} : \mylistt{\myb{A}} \myarr \mynat \\
-    \begin{array}{@{}l@{\myappsp}c@{\ }c@{\ }l}
-      \myfun{length} & \mynil & = & \mydc{zero} \\
-      \myfun{length} & (\myb{x} \mycons \myb{xs}) & = & \mydc{suc} \myappsp (\myfun{length} \myappsp \myb{xs})
-    \end{array} \\
-    \ \\
-    \myfun{head} : \myfora{\myb{l}}{\mytyc{List}\myappsp\myb{A}}{ \myfun{length}\myappsp\myb{l} \mathrel{\myfun{$>$}} 0 \myarr \myb{A}} \\
+    \myfun{head} : \myfora{\myb{l}}{\mytyc{List}\myappsp\myb{A}}{ \myfun{non-empty}\myappsp\myb{l} \myarr \myb{A}} \\
     \begin{array}{@{}l@{\myappsp}c@{\myappsp}c@{\ }c@{\ }l}
-      \myfun{head} & \mynil & \myb{p} & = & \myhole{?} \\
-      \myfun{head} & (\myb{x} \mycons \myb{xs}) & \myb{p} & = & \myb{x}
+      \myfun{head} & \mynil & \myb{p} & \mapsto & \only<1,2>{\myhole{?}}\only<3>{\myabsurd\myappsp\myb{p}} \\
+      \myfun{head} & (\myb{x} \mycons \myb{xs}) & \myb{p} & \mapsto & \myb{x}
     \end{array}
   \end{array}
   \]
 
+  \only<1>{
+    The logic equivalent would be
+    \[
+    \forall \myb{l} {:} \mylist{\myb{A}}.\ \myfun{non-empty}\myappsp\myb{l} \myarr \myb{A}
+    \]
+    `For all non-empty lists of type $\myb{A}$, we can get an element of $\myb{A}$.'
+  }
+  \only<2>{
   The type of $\myb{p}$ in the $\myhole{?}$ is $\myempty$, since
-  \[\myfun{length} \myappsp \mynil \mathrel{\myfun{$>$}} 0 \myred 0 \mathrel{\myfun{$>$}} 0 \myred \myempty \]
-\end{frame}
-
-
-\begin{frame}
-  \frametitle{Example: safe $\myfun{head}$ function}
-
-  \[
-  \begin{array}{@{}l}
-    \mysyn{data}\ \mylistt{\myb{A}} = \mynil \mydcsep \myb{A} \mycons \mylistt{\myb{A}} \\
-    \ \\
-    \myfun{length} : \mytyc{List}\myappsp\myb{A} \myarr \mynat \\
-    \begin{array}{@{}l@{\myappsp}c@{\ }c@{\ }l}
-      \myfun{length} & \mynil & = & \mydc{zero} \\
-      \myfun{length} & (\myb{x} \mycons \myb{xs}) & = & \mydc{suc} \myappsp (\myfun{length} \myappsp \myb{xs})
-    \end{array} \\
-    \ \\
-    \myfun{head} : \myfora{\myb{l}}{\mytyc{List}\myappsp\myb{A}}{ \myfun{length}\myappsp\myb{l} \mathrel{\myfun{$>$}} 0 \myarr \myb{A}} \\
-    \begin{array}{@{}l@{\myappsp}c@{\myappsp}c@{\ }c@{\ }l}
-      \myfun{head} & \mynil & \myb{p} & = & \myabsurd \myappsp \myb{p} \\
-      \myfun{head} & (\myb{x} \mycons \myb{xs}) & \myb{p} & = & \myb{x}
-    \end{array}
-  \end{array}
-  \]
-
-  Where $\myfun{absurd}$ corresponds to the logical \emph{ex falso
-    quodlibet}---given $\myempty$, we can get anything:
-  \[
-  \myfun{absurd} : \myempty \myarr \myb{A}
-  \]
+  \[\myfun{non-empty}\myappsp\mynil \myred \myempty \]}
+  \only<3>{
+    Remember:
+    \[ \myfun{absurd} : \myempty \myarr \myb{A} \]
+  }
 \end{frame}
 
 \begin{frame}
   \frametitle{How do we type check this thing?}
+  \[\text{Is\ } \myfun{non-empty}\myappsp(3 \mycons \mynil) \text{\ the same as\ } \myunit\text{?}\]
+  Or in other words, is
+  \[ \mytt : \myunit \]
+  A valid argument to
   \[
-  \myfun{head} \myappsp \mylistt{3} : \myfun{length} \myappsp \mylistt{3} \mathrel{\myfun{$>$}} 0 \myarr \mynat
+  \myfun{head} \myappsp (3 \mycons \mynil) : \myfun{non-empty}\myappsp(3 \mycons \mynil) \myarr \mynat
   \]
 
-  Will $\mytt : \myunit$ do as an argument?  In other words, when type
-  checking, do we have that
-  \[
-  \begin{array}{@{}c@{\ }c@{\ }c}
-    \myunit & \mydefeq & \myfun{length} \myappsp \mylistt{3} \mathrel{\myfun{$>$}} 0 \\
-    \myfun{length} \myappsp \mynil \mathrel{\myfun{$>$}} 0 & \mydefeq & \myempty \\
-    (\myabs{\myb{x}\, \myb{y}}{\myb{y}}) \myappsp \myunit \myappsp \myappsp \mynat & \mydefeq & (\myabs{\myb{x}\, \myb{y}}{\myb{x}}) \myappsp \mynat \myappsp \myunit \\
-    & \vdots &
-  \end{array}
-  \]
-  ?
-\end{frame}
-
-\begin{frame}
-  \frametitle{Definitional equality}
-  
-  The type checker needs a notion of equality between types.
-
-  We reduce terms `as far as possible' (to their \emph{normal form}) and
-  then compare them syntactically:
+  Yes: to typecheck, we reduce terms fully (to their \emph{normal form})
+  before comparing:
   \[
   \begin{array}{@{}r@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }l}
-    \myunit & \myredd & \myunit & \mydefeq & \myunit & \myreddd & \myfun{length} \myappsp \mylistt{3} \mathrel{\myfun{$>$}} 0 \\
-    \myfun{length} \myappsp \mynil \mathrel{\myfun{$>$}} 0 & \myredd & \myempty & \mydefeq & \myempty & \myreddd & \myempty \\
+    \myunit & \myredd & \myunit & \mydefeq & \myunit & \myreddd & \myfun{non-empty}\myappsp(3 \mycons \mynil) \\
     (\myabs{\myb{x}\, \myb{y}}{\myb{y}}) \myappsp \myunit \myappsp \myappsp \mynat & \myredd & \mynat & \mydefeq & \mynat & \myreddd & (\myabs{\myb{x}\, \myb{y}}{\myb{x}}) \myappsp \mynat \myappsp \myunit \\
     & & & \vdots & & & 
   \end{array}
   \]
-
-  This equality, $\mydefeq$, takes the name of \emph{definitional} equality.
+  \[
+  \mydefeq\ \text{takes the name of \textbf{definitional equality}.}
+  \]
 \end{frame}
 
 \begin{frame}
   \frametitle{Propositional equality} Using definitional equality, we
   can give the user a type-level notion of term equality.
   \[
-  (\myeq) : \myb{A} \myarr \myb{A} \myarr \mytyp
+  (\myeq) : \myb{A} \myarr \myb{A} \myarr \mytyp\ \ \ \text{internalises equality \textbf{as a type}}
   \]
-  We introduce members of $\myeq$ by reflexivity:
+  We introduce members of $\myeq$ by reflexivity, for example
   \[
-  \myrefl\myappsp\mytmt : \mytmt \myeq \mytmt
+  \myrefl\myappsp5 : 5 \myeq 5
   \]
-  So that $\myrefl$ will relate definitionally equal terms:
+  But also
   \[
-  \myrefl\myappsp 5 : (3 + 2) \myeq (1 + 4)\ \text{since}\ (3 + 2) \myeq (1 + 4) \myredd 5 \myeq 5
+  \begin{array}{@{}l}
+  \myrefl\myappsp 5 : (3 + 2) \myeq (1 + 4)\text{, since}\\
+  (3 + 2) \myeq (1 + 4) \myredd 5 \myeq 5
+  \end{array}
   \]
   Then we can use a substitution law to derive other
   laws---transitivity, congruence, etc.
+  \[ \myeq\ \text{takes the name of \textbf{propositional equality}} \]
 \end{frame}
 
 \begin{frame}
 \frametitle{The problem with prop. equality}
 Going back to $\myfun{map}$, we can prove that
-\[    \forall \myb{f} {:} (\myb{b} \myarr \myb{c}), \myb{g} {:} (\myb{a} \myarr \myb{b}), \myb{l} {:} \mylist{\myb{a}}. (\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g}) \myappsp \myb{l} \myeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) \myappsp \myb{l}    \]
-Because we can prove, by induction on $\myb{l}$, that we will always get definitionally equal lists.
+\[    \forall \myb{f} {:} (\myb{b} \myarr \myb{c}), \myb{g} {:} (\myb{a} \myarr \myb{b}), \myb{l} {:} \mylist{\myb{a}}.\ (\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g}) \myappsp \myb{l} \myeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) \myappsp \myb{l}    \]
+Because
+\[
+(\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g})\myappsp \myb{l} \mydefeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) \myappsp \myb{l}
+\]
+By induction on $\myb{l}$.
 
-But without the $\myb{l}$, we cannot compute, so we are stuck with
+Without the $\myb{l}$ we cannot compute, so we are stuck with
 \[
 \myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g} \not\mydefeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g})
 \]
 \end{frame}
 
 \begin{frame}
-  \frametitle{The solution}
+  \frametitle{Observational equality}
 
-  \emph{Observational} equality, instead of basing its equality on
-  definitional equality, looks at the structure of the type to decide:
+  Instead of basing its equality on definitional equality, look at the
+  structure of the type to decide.
+
+  For functions this will mean that proving
   \[
-  \begin{array}{@{}l}
-    (\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g} : \mylistt{\myb{A_1}} \myarr \mylistt{\myb{C_1}}) \myeq (\myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) : \mylistt{\myb{A_2}} \myarr \mylistt{\myb{C_2}}) \myred \\
-    \myind{2} (\myb{l_1} : \myb{A_1}) \myarr (\myb{l_2} : \myb{A_2}) \myarr (\myb{l_1} : \myb{A_1}) \myeq (\myb{l_2} : \myb{A_2}) \myarr \\
-    \myind{2} ((\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g}) \myappsp \myb{l} : \mylistt{\myb{C_1}}) \myeq (\myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) \myappsp \myb{l} : \mylistt{\myb{C_2}})
-  \end{array}
+    \myfun{map}\myappsp \myse{f} \mycomp \myfun{map}\myappsp \myse{g} \myeq \myfun{map}\myappsp (\myse{f} \mycomp \myse{g})
+  \]
+  Will reduce to proving that
+  \[
+  (\myb{l} : \mylist{\myb{A}}) \myarr 
+    (\myfun{map}\myappsp \myse{f} \mycomp \myfun{map}\myappsp \myse{g})\myappsp\myb{l} \myeq \myfun{map}\myappsp (\myse{f} \mycomp \myse{g})\myappsp \myb{l}
   \]
-  This extends to other structures (tuples, inductive types, \dots).
-  Moreover, if we can deem two \emph{types} equal, we can \emph{coerce}
-  values from one to the other.
+  Which is what we want.  The interesting part is how to make the system
+  compute nicely.
 \end{frame}
 
 \begin{frame}
-  \frametitle{\mykant}
-
-  Observational equality was described in a very restricted theory.
-
-  \mykant\ aims to incorporate it in a more `practical' environment,
-  where we have:
-  \begin{itemize}
-    \item User defined data types (inductive data and records).
-    \item A type hierarchy.
-    \item Partial type inference (bidirectional type checking).
-    \item Type holes.
-  \end{itemize}
+  \begin{center}
+    {\huge \mykant' features}
+  \end{center}
 \end{frame}
 
 \begin{frame}
   \frametitle{Inductive data}
-  Good old Haskell data types:
   \[
-  \mysyn{data}\ \mytyc{List}\myappsp \myb{A} = \mynil \mydcsep \myb{A} \mycons \mytyc{List}\myappsp\myb{A}
+  \mysyn{data}\ \mytyc{List}\myappsp (\myb{A} : \mytyp) \mapsto \mynil \mydcsep \myb{A} \mycons \mytyc{List}\myappsp\myb{A}
   \]
-  But instead of general recursion and pattern matching, we have
-  structural induction:
+  Each with an induction principle:
   \[
   \begin{array}{@{}l@{\ }l}
-    \mytyc{List}.\myfun{elim} : & (\myb{P} : \mytyc{List}\myappsp\myb{A} \myarr \mytyp) \myarr \\
+    \mytyc{List}.\myfun{elim} : & \AgdaComment{-{}- The property that we want to prove:} \\
+     & (\myb{P} : \mytyc{List}\myappsp\myb{A} \myarr \mytyp) \myarr \\
+     & \AgdaComment{-{}- If it holds for the base case:} \\
     & \myb{P} \myappsp \mynil \myarr \\
+    & \AgdaComment{-{}- And for the inductive step:} \\
     & ((\myb{x} : \myb{A}) \myarr (\myb{l} : \mytyc{List}\myappsp \myb{A}) \myarr \myb{P} \myappsp \myb{l} \myarr \myb{P} \myappsp (\myb{x} \mycons \myb{l})) \myarr \\
+    & \AgdaComment{-{}- Then it holds for every list:} \\
     & (\myb{l} : \mytyc{List}\myappsp\myb{A}) \myarr \myb{P} \myappsp \myb{l}
   \end{array}
   \]
-  Reduction:
+  Induction is also computation, via structural recursion:
   \[
   \begin{array}{@{}l@{\ }l}
     \mytyc{List}.\myfun{elim} \myappsp \myse{P} \myappsp \myse{pn} \myappsp \myse{pc} \myappsp \mynil & \myred \myse{pn} \\
-    \mytyc{List}.\myfun{elim} \myappsp \myse{P} \myappsp \myse{pn} \myappsp \myse{pc} \myappsp (\mytmm \mycons \mytmn) & \myred \myse{pc} \myappsp \mytmm \myappsp \mytmn \myappsp (\mytyc{List}.\myfun{elim} \myappsp \myse{P} \myappsp \myse{pn} \myappsp \myse{ps} \myappsp \mytmt )
+    \mytyc{List}.\myfun{elim} \myappsp \myse{P} \myappsp \myse{pn} \myappsp \myse{pc} \myappsp (\mytmm \mycons \mytmn) & \myred \myse{pc} \myappsp \mytmm \myappsp \mytmn \myappsp (\mytyc{List}.\myfun{elim} \myappsp \myse{P} \myappsp \myse{pn} \myappsp \myse{ps} \myappsp \mytmn )
   \end{array}
   \]
 \end{frame}
 
 \begin{frame}
-  \frametitle{Records}
-  But also records:
+  \frametitle{Dependent defined types} \emph{Unlike} Haskell, we can
+  have fields of a data constructor to depend on earlier fields:
   \[
-  \mysyn{record}\ \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B} = \mydc{tuple}\ \{ \myfun{fst} : \myb{A}, \myfun{snd} : \myb{B} \}
+  \begin{array}{@{}l}
+    \mysyn{record}\ \mytyc{Tuple}\myappsp(\myb{A} : \mytyp)\myappsp\myhole{$(\myb{B} : \myb{A} \myarr \mytyp)$} \mapsto \\
+    \myind{2}\mydc{tuple}\ \{ \myfun{fst} : \myb{A}, \myfun{snd} : \myb{B}\myappsp\myb{fst} \}
+  \end{array}
   \]
-  Where each field defines a projection from instances of the record:
+
+  The \emph{type} of the second element depends on the \emph{value} of
+  the first:
   \[
-  \begin{array}{@{}l@{\ }c@{\ }l}
-    \myfun{fst} & : & \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B} \myarr \myb{A} \\
-    \myfun{snd} & : & \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B} \myarr \myb{B}
+  \begin{array}{@{}l@{\ }l}
+    \myfun{fst} & : \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B} \myarr \myb{A} \\
+    \myfun{snd} & : (\myb{x} : \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B}) \myarr \myb{B} \myappsp (\myfun{fst} \myappsp \myb{x})
   \end{array}
   \]
   Where the projection's reduction rules are predictably
@@ -513,39 +514,36 @@ But without the $\myb{l}$, we cannot compute, so we are stuck with
 \end{frame}
 
 \begin{frame}
-  \frametitle{Dependend defined types} \emph{Unlike} Haskell, we can
-  have fields of a data constructor to depend on earlier fields:
+  \frametitle{Example: the type of even numbers}
+  For example we can define the type of even numbers:
+  % TODO fix
   \[
   \begin{array}{@{}l}
-    \mysyn{record}\ \mytyc{Tuple}\myappsp(\myb{A} : \mytyp)\myappsp(\myb{B} : \myb{A} \myarr \mytyp) = \\
-    \myind{2}\mydc{tuple}\ \{ \myfun{fst} : \myb{A}, \myfun{snd} : \myb{B}\myappsp\myb{fst} \}
-  \end{array}
-  \]
-  $\mytyc{Tuple}$ takes a $\mytyp$, $\myb{A}$, and a predicate from
-  elements of $\myb{A}$ to types, $\myb{B}$.
-
-  This way, the \emph{type} of the second element depends on the
-  \emph{value} of the first:
-  \[
-  \begin{array}{@{}l@{\ }l}
-    \myfun{fst} & : \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B} \myarr \myb{A} \\
-    \myfun{snd} & : (\myb{x} : \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B}) \myarr \myb{B} \myappsp (\myfun{fst} \myappsp \myb{x})
+    \mysyn{data}\ \mynat \mapsto \mydc{zero} \mydcsep \mydc{suc}\myappsp\mynat \\
+    \ \\
+    \myfun{even} : \mynat \myarr \mytyp \\
+    \begin{array}{@{}l@{\myappsp}c@{\ }l}
+      \myfun{even} & \myzero & \mapsto \myunit \\
+      \myfun{even} & (\mysuc \myappsp \myzero) & \mapsto \myempty \\
+      \myfun{even} & (\mysuc \myappsp (\mysuc \myappsp \myb{n})) & \mapsto \myfun{even} \myappsp \myb{n}
+    \end{array} \\
+    \ \\
+    \mytyc{Even} : \mytyp \\
+    \mytyc{Even} \mapsto \mytyc{Tuple}\ \mynat\ \myfun{even}
   \end{array}
   \]
 \end{frame}
 
 \begin{frame}
   \frametitle{Type hierarchy}
-  Up to now, we have thrown $\mytyp$ around, as `the type of types'.
-
-  But what is the type of $\mytyp$ itself?  The simple way out is
+  \[\{\mynat, \mybool, \mytyc{List}\myappsp\mynat, \cdots\} : \mytyp\]
+  What is the type of $\mytyp$?
   \[
-  \mytyp : \mytyp
+  \cancel{\mytyp : \mytyp}\ \ \ \text{\textbf{inconsistent}}
   \]
-  This solution is not only simple, but inconsistent, for the same
-  reason that the notion of a `powerset' in na{\"i}ve set theory is.
+  Similar to Russel's paradox in na{\"i}ve set theory.
 
-  Instead, following Russell, we shall have
+  Instead, we have a hierarchy:
   \[
   \{\mynat, \mybool, \mytyc{List}\myappsp\mynat, \cdots\} : \mytyp_0 : \mytyp_1 : \cdots
   \]
@@ -554,9 +552,142 @@ But without the $\myb{l}$, we cannot compute, so we are stuck with
 
 \begin{frame}
   \frametitle{Cumulativity and typical ambiguity}
-  Is it OK to take
-  \[ \mytyp_0 : \mytyp_2 \]
-  ?
+  Instead of:
+  \[ \mytyp_0 : \mytyp_1 \ \ \ \text{but}\ \ \ \mytyp_0 \centernot{:} \mytyp_2\]
+  We have a cumulative hierarchy, so that
+  \[ \mytyp_n : \mytyp_m \ \ \ \text{iff}\ \ \ n < m \]
+  For example
+  \[ \mynat : \mytyp_0\ \ \ \text{and}\ \ \ \mynat : \mytyp_1\ \ \ \text{and}\ \ \ \mynat : \mytyp_{50} \]
+
+  But in \mykant, you can forget about levels: the consistency is
+  checked automatically---a mechanised version of what Russell called
+  \emph{typical ambiguity}.
+
+  \begin{center}
+    \begin{tikzpicture}[remember picture, node distance=1.5cm]
+      \begin{pgfonlayer}{foreground}
+      % Place nodes
+      \node (x) {$x$};
+      \node [right of=x] (y) {$y$};
+      \node [right of=y] (z) {$z$};
+      \node [below of=z] (k) {$k$};
+      \node [left  of=k] (j) {$j$};
+      %% Lines
+      \path[->]
+      (x) edge node [above] {$<$}   (y)
+      (y) edge node [above] {$\le$} (z)
+      (z) edge node [right] {$<$}   (k)
+      (k) edge node [below] {$\le$} (j)
+      (j) edge node [left ] {$\le$} (y);
+    \end{pgfonlayer}{foreground}
+    \end{tikzpicture}
+    \begin{tikzpicture}[remember picture, overlay]
+      \begin{pgfonlayer}{background}
+      \fill [red, opacity=0.3, rounded corners]
+      (-2.7,2.6) rectangle (-0.2,0.05)
+      (-4.1,2.4) rectangle (-3.3,1.6);
+    \end{pgfonlayer}{background}
+    \end{tikzpicture}
+  \end{center}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Bidirectional type checking}
+  \[
+  \mysyn{data}\ \mytyc{List}\myappsp (\myb{A} : \mytyp) \mapsto \mydc{nil} \mydcsep \mydc{cons} \myappsp \myb{A}\myappsp (\mytyc{List}\myappsp\myb{A})
+  \]
+
+  With no type inference, we have explicit types for the constructors:
+  \[
+  \begin{array}{@{}l@{\ }l}
+    \mydc{nil} & : (\myb{A} : \mytyp) \myarr \mytyc{List}\myappsp\myb{A} \\
+    \mydc{cons} &: (\myb{A} : \mytyp) \myarr \myb{A} \myarr \mytyc{List}\myappsp\myb{A} \myarr \mytyc{List}\myappsp\myb{A}\\
+  \end{array}
+  \]
+  ...ugh:
+  \[
+  \mydc{cons}\myappsp \mynat\myappsp 1 \myappsp (\mydc{cons}\myappsp \mynat \myappsp 2 \myappsp (\mydc{cons}\myappsp \mynat \myappsp 3 \myappsp (\mydc{nil}\myappsp \mynat)))
+  \]
+  Instead, annotate terms and propagate the type:
+  \[
+  \mydc{cons}\myappsp 1 \myappsp (\mydc{cons}\myappsp 2 \myappsp (\mydc{cons} \myappsp 3 \myappsp \mydc{nil})) : \mytyc{List}\myappsp\mynat
+  \]
+  Conversely, when we use eliminators the type can be inferred.
+\end{frame}
+
+\begin{frame}
+  \frametitle{OTT + user defined types}
+
+  For each type, we need to:
+  \begin{itemize}
+  \item Describe when two types formed by the defined type constructors
+    are equal;
+    \[ \mylist{\mytya_1} \myeq \mylist{\mytya_2} \]
+  \item Describe when two values of the defined type are equal;
+    \[
+    \begin{array}{@{}c@{\ \ \ \ \ \ }c}
+      \mynil \myeq \mynil & \mynil \myeq \mytmm \mycons \mytmn \\
+      \mytmm \mycons \mytmn \myeq \mynil & \mytmm_1 \mycons \mytmn_1 \myeq \mytmm_2 \mycons \mytmn_2
+    \end{array}
+    \]
+  \item Describe how to transport values of the defined type.
+  \end{itemize}
+\end{frame}
+
+\begin{frame}
+  \frametitle{OTT + hierarchy}
+
+  Since equalities reduce to functions abstracting over various things,
+  we need to make sure that the hierarchy is respected.
+
+  For example we have that
+  \[
+  \begin{array}{@{}l}
+    \myjm{(\myb{x_1} {:} \mytya_1) \myarr \mytyb_1}{\mytyp}{(\myb{x_2} {:} \mytya_2) \myarr \mytyb_2}{\mytyp} \myred \\ 
+    \myind{2} \myjm{\mytya_1}{\mytyp}{\mytya_2}{\mytyp} \myand  \\
+    \myind{2} ((\myb{x_1} : \mytya_1) \myarr (\myb{x_2} : \mytya_2) \myarr \myjm{\myb{x_1}}{\mytya_1}{\myb{x_2}}{\mytya_2} \myarr \mytyb_1[\myb{x_1}] \myeq \mytyb_2[\myb{x_2}])
+  \end{array}
+  \]
+
+  Subtle point---I discovered a problem in the theory after
+  submitting... but I have a fix.
+\end{frame}
+
+\begin{frame}
+  \frametitle{Bonus!  WebSocket prompt}
+  \url{http://bertus.mazzo.li}, go DDOS it!
+
+  \includegraphics[width=\textwidth]{web-prompt.png}
+\end{frame}
+
+\begin{frame}
+\begin{center}
+{\Huge Demo}
+\end{center}
+\end{frame}
+
+\begin{frame}
+  \frametitle{What have we done?}
+
+  \begin{itemize}
+    \item A small theorem prover for intuitionistic logic, featuring:
+    \item Inductive data and record types;
+    \item A cumulative, implicit type hierarchy;
+    \item Partial type inference---bidirectional type checking;
+    \item Type holes;
+    \item Observational equality---coming soon to the implementation.
+  \end{itemize}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Further work}
+
+  \begin{itemize}
+    \item Pattern matching and explicit recursion
+    \item More expressive data types
+    \item Inference via unification
+    \item Codata
+  \end{itemize}
 \end{frame}
 
 \begin{frame}