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  breaklinks=true,
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  pdfauthor={Francesco Mazzoli <fm2209@ic.ac.uk>},
  pdftitle={Observational Equality},
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\title{Observational Equality}
\author{Francesco Mazzoli \href{mailto:fm2209@ic.ac.uk}{\nolinkurl{<fm2209@ic.ac.uk>}}}
\date{December 2012}

\begin{document}

\maketitle

\setlength{\tabcolsep}{12pt}

\begin{abstract}
The marriage between programming and logic has been a very fertile one.  In
particular, since the simply typed lambda calculus (STLC), a number of type
systems have been devised with increasing expressive power.

Section \ref{sec:types} I will give a very brief overview of STLC, and then
illustrate how it can be interpreted as a natural deduction system.  Section
\ref{sec:itt} will introduce Inutitionistic Type Theory (ITT), which expands on
this concept, employing a more expressive logic.  The exposition is quite dense
since there is a lot of material to cover; for a more complete treatment of the
material the reader can refer to \citep{Thompson1991, Pierce2002}. Section
\ref{sec:practical} will describe additions common in practical programming
languages based on ITT.

Finally, I will explain why equality has been a tricky business in these
theories, and talk about the various attempts have been made to make the
situation better.  One interesting development has recently emerged:
Observational Type theory.  I propose to explore the ways to turn these ideas
into useful practices for programming and theorem proving.
\end{abstract}

\tableofcontents

\section{Simple and not-so-simple types}
\label{sec:types}

\subsection{A note on notation}

\newcommand{\appsp}{\hspace{0.07cm}}
\newcommand{\app}[2]{#1\appsp#2}
\newcommand{\absspace}{\hspace{0.03cm}}
\newcommand{\abs}[2]{\lambda #1\absspace.\absspace#2}
\newcommand{\termt}{t}
\newcommand{\termm}{m}
\newcommand{\termn}{n}
\newcommand{\termp}{p}
\newcommand{\termf}{f}
\newcommand{\separ}{\ \ |\ \ }
\newcommand{\termsyn}{\mathit{term}}
\newcommand{\axname}[1]{\textbf{#1}}
\newcommand{\axdesc}[2]{\axname{#1} \fbox{$#2$}}
\newcommand{\lcsyn}[1]{\mathrm{\underline{#1}}}
\newcommand{\lccon}[1]{\mathsf{#1}}
\newcommand{\lcfun}[1]{\mathbf{#1}}
\newcommand{\tyarr}{\to}
\newcommand{\tysyn}{\mathit{type}}
\newcommand{\ctxsyn}{\mathit{context}}
\newcommand{\emptyctx}{\cdot}

In the following sections I will introduce a lot of syntax, typing, and
reduction rules, along with examples.  To make the exposition clearer, usually
the rules are preceded by what the rule looks like and what it shows (for
example \axdesc{typing}{\Gamma \vdash \termsyn : \tysyn}).

In the languages presented I will also use different fonts for different things:
\begin{tabular}{c | l}
  $\lccon{Sans}$  & Sans serif, capitalised, for type constructors. \\
  $\lccon{sans}$  & Sans serif, not capitalised, for data constructors. \\
  $\lcsyn{roman}$ & Roman, underlined, for the syntax of the language. \\
  $\lcfun{roman}$ & Roman, bold, for defined functions and values. \\
  $math$          & Math mode font for quantified variables and syntax elements.
\end{tabular}

Moreover, I will from time to time give examples in the Haskell programming
language as defined in \citep{Haskell2010}, which I will typeset in
\texttt{typewriter} font.  I assume that the reader is already familiar with
Haskell, plenty of good introductions are available \citep{LYAH,ProgInHask}.

\subsection{Untyped $\lambda$-calculus}

Along with Turing's machines, the earliest attempts to formalise computation
lead to the $\lambda$-calculus \citep{Church1936}.  This early programming
language encodes computation with a minimal syntax and no `data' in the
traditional sense, but just functions.

The syntax of $\lambda$-terms consists of just three things: variables,
abstractions, and applications:

\begin{center}
\axname{syntax}
$$
\begin{array}{rcl}
  \termsyn & ::= & x \separ (\abs{x}{\termsyn}) \separ (\app{\termsyn}{\termsyn}) \\
         x & \in & \text{Some enumerable set of symbols, e.g.}\ \{x, y, z, \dots , x_1, x_2, \dots\}
\end{array}
$$
\end{center}


% I will omit parethesis in the usual manner. %TODO explain how

I will use $\termt,\termm,\termn,\dots$ to indicate a generic term, and $x,y$
for variables.  I will also assume that all variable names in a term are unique
to avoid problems with name capturing.  Intuitively, abstractions
($\abs{x}{\termt}$) introduce functions with a named parameter ($x$), and
applications ($\app{\termt}{\termm}$) apply a function ($\termt$) to an argument
($\termm$).

The `applying' is more formally explained with a reduction rule:

\newcommand{\bred}{\leadsto}
\newcommand{\bredc}{\bred^*}

\begin{center}
\axdesc{reduction}{\termsyn \bred \termsyn}
$$\app{(\abs{x}{\termt})}{\termm} \bred \termt[\termm / x]$$
\end{center}

Where $\termt[\termm / x]$ expresses the operation that substitutes all
occurrences of $x$ with $\termm$ in $\termt$.  In the future, I will use
$[\termt]$ as an abbreviation for $[\termt / x]$.  In the systems presented, the
$\bred$ relation also includes reduction of subterms, for example if $\termt
\bred \termm$ then $\app{\termt}{\termn} \bred \app{\termm}{\termn}$, and so on.

These few elements are of remarkable expressiveness, and in fact Turing
complete.  As a corollary, we must be able to devise a term that reduces forever
(`loops' in imperative terms):
\[
  \app{(\abs{x}{\app{x}{x}})}{(\abs{x}{\app{x}{x}})} \bred \app{(\abs{x}{\app{x}{x}})}{(\abs{x}{\app{x}{x}})} \bred \dotsb
\]
Terms that can be reduced only a finite number of times (the non-looping ones)
are said to be \emph{normalising}, and the `final' term (where no reductions are
possible on the term or on its subterms) is called \emph{normal form}.

\subsection{The simply typed $\lambda$-calculus}

\newcommand{\tya}{A}
\newcommand{\tyb}{B}
\newcommand{\tyc}{C}

One way to `discipline' $\lambda$-terms is to assign \emph{types} to them, and
then check that the terms that we are forming make sense given our typing rules
\citep{Curry1934}.

We wish to introduce rules of the form $\Gamma \vdash \termt : \tya$, which
reads `in context $\Gamma$, term $\termt$ has type $\tya$'.

The syntax for types is as follows:

\begin{center}
  \axname{syntax}
   $$\tysyn ::= x \separ \tysyn \tyarr \tysyn$$
\end{center}

I will use $\tya,\tyb,\dots$ to indicate a generic type.

A context $\Gamma$ is a map from variables to types.  I will use the notation
$\Gamma; x : \tya$ to augment it and to `extract' pairs from it.

Predictably, $\tya \tyarr \tyb$ is the type of a function from $\tya$ to
$\tyb$.  We need to be able to decorate our abstractions with
types\footnote{Actually, we don't need to: computers can infer the right type
  easily, but that is another story.}:
\begin{center}
  \axname{syntax}
   $$\termsyn ::= x \separ (\abs{x : \tysyn}{\termsyn}) \separ (\app{\termsyn}{\termsyn})$$
\end{center}
Now we are ready to give the typing judgements:

\begin{center}
  \axdesc{typing}{\Gamma \vdash \termsyn : \tysyn}

  \vspace{0.5cm}

  \begin{tabular}{c c c}
    \AxiomC{\phantom{1em}}
    \UnaryInfC{$\Gamma; x : \tya \vdash x : \tya$}
    \DisplayProof
    &
    \AxiomC{$\Gamma; x : \tya \vdash \termt : \tyb$}
    \UnaryInfC{$\Gamma \vdash \abs{x : \tya}{\termt} : \tya \tyarr \tyb$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \begin{tabular}{c}
    \AxiomC{$\Gamma \vdash \termt : \tya \tyarr \tyb$}
    \AxiomC{$\Gamma \vdash \termm : \tya$}
    \BinaryInfC{$\Gamma \vdash \app{\termt}{\termm} : \tyb$}
    \DisplayProof
  \end{tabular}
\end{center}

This typing system takes the name of `simply typed lambda calculus' (STLC),
and enjoys a number of properties.  Two of them are expected in most type
systems \citep{Pierce2002}:
\begin{description}
\item[Progress] A well-typed term is not stuck - either it is a value or it can
  take a step according to the evaluation rules.  With `value' we mean a term
  whose subterms (including itself) don't appear to the left of the $\bred$
  relation.
\item[Preservation] If a well-typed term takes a step of evaluation, then the
  resulting term is also well typed.
\end{description}

However, STLC buys us much more: every well-typed term
is normalising.  It is easy to see that we can't fill the blanks if we want to
give types to the non-normalising term shown before:
\begin{equation*}
  \app{(\abs{x : ?}{\app{x}{x}})}{(\abs{x : ?}{\app{x}{x}})}
\end{equation*}

\newcommand{\lcfix}[2]{\lcsyn{fix} \appsp #1\absspace.\absspace #2}

This makes the STLC Turing incomplete.  We can recover the ability to loop by
adding a combinator that recurses:
$$
  \termsyn ::= \dotsb \separ  \lcfix{x : \tysyn}{\termsyn}
$$
\begin{center}
  \begin{prooftree}
    \AxiomC{$\Gamma;x : \tya \vdash \termt : \tya$}
    \UnaryInfC{$\Gamma \vdash \lcfix{x : \tya}{\termt} : \tya$}
  \end{prooftree}
\end{center}
$$
  \lcfix{x : \tya}{\termt} \bred \termt[\lcfix{x : \tya}{\termt}]
$$
Which will deprive us of normalisation.  There is however a price to pay, which
will be made clear in the next section.

\subsection{The Curry-Howard correspondence}
\label{sec:curry-howard}

\newcommand{\lcunit}{\lcfun{\langle\rangle}}

It turns out that the STLC can be seen a natural deduction system for
propositional logic.  Terms are proofs, and their types are the propositions
they prove.  This remarkable fact is known as the Curry-Howard correspondence,
or isomorphism.

The `arrow' ($\to$) type corresponds to implication.  If we wish to prove that
$(\tya \tyarr \tyb) \tyarr (\tyb \tyarr \tyc) \tyarr (\tya \tyarr \tyc)$, all we
need to do is to devise a $\lambda$-term that has the correct type:
\begin{equation*}
  \abs{f : (\tya \tyarr \tyb)}{\abs{g : (\tyb \tyarr \tyc)}{\abs{x : \tya}{\app{g}{(\app{f}{x})}}}}
\end{equation*}
That is, function composition.  We might want extend our bare lambda calculus
with a couple of terms former to make our natural deduction more pleasant to
use.  For example, tagged unions (\texttt{Either} in Haskell) are disjunctions,
and tuples (or products) are conjunctions.  We also want to be able to express
falsity ($\bot$): that can done by introducing a type inhabited by no terms.  If
evidence of such a type is presented, then we can derive any type, which
expresses absurdity.  Conversely, truth ($\top$) is the type with just one
trivial element ($\lcunit$).

\newcommand{\lcinl}{\lccon{inl}\appsp}
\newcommand{\lcinr}{\lccon{inr}\appsp}
\newcommand{\lccase}[3]{\lcsyn{case}\appsp#1\appsp\lcsyn{of}\appsp#2\appsp#3}
\newcommand{\lcfst}{\lcfun{fst}\appsp}
\newcommand{\lcsnd}{\lcfun{snd}\appsp}
\newcommand{\orint}{\vee I_{1,2}}
\newcommand{\orintl}{\vee I_{1}}
\newcommand{\orintr}{\vee I_{2}}
\newcommand{\orel}{\vee E}
\newcommand{\andint}{\wedge I}
\newcommand{\andel}{\wedge E_{1,2}}
\newcommand{\botel}{\bot E}
\newcommand{\lcabsurd}{\lcfun{absurd}\appsp}
\newcommand{\lcabsurdd}[1]{\lcfun{absurd}_{#1}\appsp}


\begin{center}
  \axname{syntax}
  $$
  \begin{array}{rcl}
    \termsyn & ::= & \dotsb \\
             &  |  & \lcinl \termsyn \separ \lcinr \termsyn \separ \lccase{\termsyn}{\termsyn}{\termsyn} \\
             &  |  & (\termsyn , \termsyn) \separ \lcfst \termsyn \separ \lcsnd \termsyn \\
             &  |  & \lcunit \\
    \tysyn & ::= & \dotsb \separ \tysyn \vee \tysyn \separ \tysyn \wedge \tysyn \separ \bot \separ \top
  \end{array}
  $$
\end{center}
\begin{center}
  \axdesc{typing}{\Gamma \vdash \termsyn : \tysyn}
  \begin{prooftree}
    \AxiomC{$\Gamma \vdash \termt : \tya$}
    \RightLabel{$\orint$}
    \UnaryInfC{$\Gamma \vdash \lcinl \termt : \tya \vee \tyb$}
    \noLine
    \UnaryInfC{$\Gamma \vdash \lcinr \termt : \tyb \vee \tya$}
  \end{prooftree}
  \begin{prooftree}
    \AxiomC{$\Gamma \vdash \termt : \tya \vee \tyb$}
    \AxiomC{$\Gamma \vdash \termm : \tya \tyarr \tyc$}
    \AxiomC{$\Gamma \vdash \termn : \tyb \tyarr \tyc$}
    \RightLabel{$\orel$}
    \TrinaryInfC{$\Gamma \vdash \lccase{\termt}{\termm}{\termn} : \tyc$}
  \end{prooftree}

  \begin{tabular}{c c}
    \AxiomC{$\Gamma \vdash \termt : \tya$}
    \AxiomC{$\Gamma \vdash \termm : \tyb$}
    \RightLabel{$\andint$}
    \BinaryInfC{$\Gamma \vdash (\tya , \tyb) : \tya \wedge \tyb$}
    \DisplayProof
    &
    \AxiomC{$\Gamma \vdash \termt : \tya \wedge \tyb$}
    \RightLabel{$\andel$}
    \UnaryInfC{$\Gamma \vdash \lcfst \termt : \tya$}
    \noLine
    \UnaryInfC{$\Gamma \vdash \lcsnd \termt : \tyb$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \begin{tabular}{c c}
    \AxiomC{$\Gamma \vdash \termt : \bot$}
    \RightLabel{$\botel$}
    \UnaryInfC{$\Gamma \vdash \lcabsurdd{\tya} \termt : \tya$}
    \DisplayProof
    &
    \AxiomC{}
    \RightLabel{$\top I$}
    \UnaryInfC{$\Gamma \vdash \lcunit : \top$}
    \DisplayProof
  \end{tabular}
\end{center}
\begin{center}
  \axdesc{reduction}{\termsyn \bred \termsyn}
  \begin{eqnarray*}
    \lccase{(\lcinl \termt)}{\termm}{\termn} & \bred & \app{\termm}{\termt} \\
    \lccase{(\lcinr \termt)}{\termm}{\termn} & \bred & \app{\termn}{\termt} \\
    \lcfst (\termt , \termm)                 & \bred & \termt \\
    \lcsnd (\termt , \termm)                 & \bred & \termm
  \end{eqnarray*}
\end{center}

With these rules, our STLC now looks remarkably similar in power and use to the
natural deduction we already know.  $\neg A$ can be expressed as $A \tyarr
\bot$.  However, there is an important omission: there is no term of the type $A
\vee \neg A$ (excluded middle), or equivalently $\neg \neg A \tyarr A$ (double
negation), or indeed any term with a type equivalent to those.

This has a considerable effect on our logic and it's no coincidence, since there
is no obvious computational behaviour for laws like the excluded middle.
Theories of this kind are called \emph{intuitionistic}, or \emph{constructive},
and all the systems analysed will have this characteristic since they build on
the foundation of the STLC\footnote{There is research to give computational
  behaviour to classical logic, but I will not touch those subjects.}.

Finally, going back to our $\lcsyn{fix}$ combinator, it's now easy to see how
we would want to exclude such a thing if we want to use STLC as a logic, since
it allows us to prove everything: $(\lcfix{x : \tya}{x}) : \tya$ clearly works
for any $A$!  This is a crucial point: in general we wish to have systems that
do not let the user devise a term of type $\bot$, otherwise our logic will be
unsound\footnote{Obviously such a term can be present under a $\lambda$.}.

\subsection{Extending the STLC}

\newcommand{\lctype}{\lccon{Type}}
\newcommand{\lcite}[3]{\lcsyn{if}\appsp#1\appsp\lcsyn{then}\appsp#2\appsp\lcsyn{else}\appsp#3}
\newcommand{\lcbool}{\lccon{Bool}}
\newcommand{\lcforallz}[2]{\forall #1 \absspace.\absspace #2}
\newcommand{\lcforall}[3]{(#1 : #2) \tyarr #3}
\newcommand{\lcexists}[3]{(#1 : #2) \times #3}

The STLC can be made more expressive in various ways.  Henk Barendregt
succinctly expressed geometrically how we can expand our type system:

$$
\xymatrix@!0@=1.5cm{
  & \lambda\omega \ar@{-}[rr]\ar@{-}'[d][dd]
  & & \lambda C \ar@{-}[dd]
  \\
  \lambda2 \ar@{-}[ur]\ar@{-}[rr]\ar@{-}[dd]
  & & \lambda P2 \ar@{-}[ur]\ar@{-}[dd]
  \\
  & \lambda\underline\omega \ar@{-}'[r][rr]
  & & \lambda P\underline\omega
  \\
  \lambda{\to} \ar@{-}[rr]\ar@{-}[ur]
  & & \lambda P \ar@{-}[ur]
}
$$
Here $\lambda{\to}$, in the bottom left, is the STLC.  From there can move along
3 dimensions:
\begin{description}
\item[Terms depending on types (towards $\lambda{2}$)] We can quantify over
  types in our type signatures: $(\abs{A : \lctype}{\abs{x : A}{x}}) :
  \lcforallz{A}{A \tyarr A}$.  The first and most famous instance of this idea
  has been System F.  This gives us a form of polymorphism and has been wildly
  successful, also thanks to a well known inference algorithm for a restricted
  version of System F known as Hindley-Milner.  Languages like Haskell and SML
  are based on this discipline.
\item[Types depending on types (towards $\lambda{\underline{\omega}}$)] We have
  type operators: $(\abs{A : \lctype}{\abs{R : \lctype}{(A \to R) \to R}}) :
  \lctype \to \lctype \to \lctype$.
\item[Types depending on terms (towards $\lambda{P}$)] Also known as `dependent
  types', give great expressive power: $(\abs{x : \lcbool}{\lcite{x}{\mathbb{N}}{\mathbb{Q}}}) : \lcbool \to \lctype$.
\end{description}

All the systems preserve the properties that make the STLC well behaved.  The
system we are going to focus on, Intuitionistic Type Theory, has all of the
above additions, and thus would sit where $\lambda{C}$ sits in the
`$\lambda$-cube'.

\section{Intuitionistic Type Theory and dependent types}
\label{sec:itt}

\newcommand{\lcset}[1]{\lccon{Type}_{#1}}
\newcommand{\lcsetz}{\lccon{Type}}
\newcommand{\defeq}{\cong}

\subsection{A Bit of History}

Logic frameworks and programming languages based on type theory have a long
history.  Per Martin-L\"{o}f described the first version of his theory in 1971,
but then revised it since the original version was too impredicative and thus
inconsistent\footnote{In the early version $\lcsetz : \lcsetz$, see section
  \ref{sec:core-tt} for an explanation on why this causes problems.}.  For this
reason he gave a revised and consistent definition later \citep{Martin-Lof1984}.

A related development is the polymorphic $\lambda$-calculus, and specifically
the previously mentioned System F, which was developed independently by Girard
and Reynolds.  An overview can be found in \citep{Reynolds1994}.  The surprising
fact is that while System F is impredicative it is still consistent and strongly
normalising.  \cite{Coquand1986} further extended this line of work with the
Calculus of Constructions (CoC).

\subsection{A Core Type Theory}
\label{sec:core-tt}

The calculus I present follows the exposition in \citep{Thompson1991}, and is
quite close to the original formulation of predicative ITT as found in
\citep{Martin-Lof1984}.

\begin{center}
  \axname{syntax}
  \begin{eqnarray*}
  \termsyn & ::= & x \\
         &  |  & \lcforall{x}{\termsyn}{\termsyn} \separ \abs{x : \termsyn}{\termsyn} \separ \app{\termsyn}{\termsyn} \\
         &  |  & \lcexists{x}{\termsyn}{\termsyn} \separ (\termsyn , \termsyn)_{x.\termsyn} \separ \lcfst \termsyn \separ \lcsnd \termsyn \\
         &  |  & \bot \separ \lcabsurd_{\termsyn} \termsyn \\
         &  |  & \lcset{n} \\
   n     & \in & \mathbb{N}
 \end{eqnarray*}

  \axdesc{typing}{\Gamma \vdash \termsyn : \termsyn}

  \vspace{0.5cm}

  \begin{tabular}{c c c}
    \AxiomC{\phantom{1em}}
    \UnaryInfC{$\Gamma;x : \tya \vdash x : \tya$}
    \DisplayProof
    &
    \AxiomC{$\Gamma \vdash \termt : \tya$}
    \AxiomC{$\tya \defeq \tyb$}
    \BinaryInfC{$\Gamma \vdash \termt : \tyb$}
    \DisplayProof
    &
    \AxiomC{$\Gamma \vdash \termt : \bot$}
    \UnaryInfC{$\Gamma \vdash \lcabsurdd{\tya} \termt : \tya$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \begin{tabular}{c c}
    \AxiomC{$\Gamma;x : \tya \vdash \termt : \tya$}
    \UnaryInfC{$\Gamma \vdash \abs{x : \tya}{\termt} : \lcforall{x}{\tya}{\tyb}$}
    \DisplayProof
    &
    \AxiomC{$\Gamma \vdash \termt : \lcforall{x}{\tya}{\tyb}$}
    \AxiomC{$\Gamma \vdash \termm : \tya$}
    \BinaryInfC{$\Gamma \vdash \app{\termt}{\termm} : \tyb[\termm ]$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \begin{tabular}{c c}
    \AxiomC{$\Gamma \vdash \termt : \tya$}
    \AxiomC{$\Gamma \vdash \termm : \tyb[\termt ]$}
    \BinaryInfC{$\Gamma \vdash (\termt, \termm)_{x.\tyb} : \lcexists{x}{\tya}{\tyb}$}
    \noLine
    \UnaryInfC{$\phantom{1em}$}
    \DisplayProof
    &
    \AxiomC{$\Gamma \vdash \termt: \lcexists{x}{\tya}{\tyb}$}
    \UnaryInfC{$\hspace{0.7cm} \Gamma \vdash \lcfst \termt : \tya \hspace{0.7cm}$}
    \noLine
    \UnaryInfC{$\Gamma \vdash \lcsnd \termt : \tyb[\lcfst \termt ]$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \begin{tabular}{c c}
    \AxiomC{$\phantom{1em}$}
    \UnaryInfC{$\Gamma \vdash \lcset{n} : \lcset{n + 1}$}
    \noLine
    \UnaryInfC{$\phantom{1em}$}
    \DisplayProof
    &
    \AxiomC{$\Gamma \vdash \tya : \lcset{n}$}
    \AxiomC{$\Gamma; x : \tya \vdash \tyb : \lcset{m}$}
    \BinaryInfC{$\Gamma \vdash \lcforall{x}{\tya}{\tyb} : \lcset{n \sqcup m}$}
    \noLine
    \UnaryInfC{$\Gamma \vdash \lcexists{x}{\tya}{\tyb} : \lcset{n \sqcup m}$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \axdesc{reduction}{\termsyn \bred \termsyn}
  \begin{eqnarray*}
    \app{(\abs{x}{\termt})}{\termm} & \bred & \termt[\termm ] \\
    \lcfst (\termt, \termm) & \bred & \termt \\
    \lcsnd (\termt, \termm) & \bred & \termm
  \end{eqnarray*}
\end{center}

When showing examples types will be omitted when this can be done without loss
of clarity, for example $\abs{x}{x}$ in place of $\abs{A : \lcsetz}{\abs{x :
    A}{x}}$.  I will also use $\lcsetz$ as $\lcset{0}$.

There are a lot of new factors at play here. The first thing to notice is that
the separation between types and terms is gone.  All we have is terms, that
include both values (terms of type $\lcset{0}$) and types (terms of type
$\lcset{n}$, with $n > 0$).  This change is reflected in the typing rules.
While in the STLC values and types are kept well separated (values never go
`right of the colon'), in ITT types can freely depend on values.

This relation is expressed in the typing rules for $\tyarr$ and $\times$: if a
function has type $\lcforall{x}{\tya}{\tyb}$, $\tyb$ can depend on $x$.
Examples will make this clearer once some base types are added in section
\ref{sec:base-types}.

$\tyarr$ and $\times$ are at the core of the machinery of ITT:

\begin{description}
\item[`forall' ($\tyarr$)] is a generalisation of $\tyarr$ in the STLC and
  expresses universal quantification in our logic.  It is often written with a
  syntax closer to logic, e.g. $\forall x : \tya. \tyb$.  In the
  literature this is also known as `dependent product' and shown as $\Pi$,
  following the interpretation of functions as infinitary products. I will just
  call it `dependent function', reserving `product' for $\exists$.

\item[`exists' ($\times$)] is a generalisation of $\wedge$ in the extended STLC
  of section \ref{sec:curry-howard}, and thus I will call it `dependent
  product'.  Like $\wedge$, it is formed by providing a pair of things.  In our
  logic, it represents existential quantification.  Similarly to $\tyarr$, it is
  always written as $\exists x : \tya. \tyb$.

  For added confusion, in the literature that calls forall $\Pi$, $\exists$ is
  often named `dependent sum' and shown as $\Sigma$.  This is following the
  interpretation of $\exists$ as a generalised, infinitary $\vee$, where the
  first element of the pair is the `tag' that determines which type the second
  element will have.
\end{description}

Another thing to notice is that types are very `first class': we are free to
create functions that accept and return types.  For this reason we define
$\defeq$ as the smallest equivalence relation extending $\bredc$, where $\bredc$
is the reflexive transitive closure of $\bred$; and we treat types that are
relate according to $\defeq$ as the same.  Another way of seeing $\defeq$ is
this: when we want to compare two types for equality, we reduce them as far as
possible and then check if they are equal\footnote{Note that when comparing
  terms we do it up to $\alpha$-renaming.  That is, we do not consider
  relabelling of variables as a difference - for example $\abs{x : A}{x} \defeq
  \abs{y : A}{y}$.}.  This works since not only each term has a normal form (ITT
is strongly normalising), but the normal form is also unique; or in other words
$\bred$ is confluent (if $\termt \bredc \termm$ and $\termt \bredc \termn$, then
there is a $p$ such that $\termm \bredc \termp$ and $\termn \bredc \termp$).
This measure makes sure that, for instance, $\app{(\abs{x :
    \lctype}{x})}{\lcbool} \defeq \lcbool$.  The theme of equality is central
and will be analysed better later.

The theory presented is \emph{stratified}.  We have a hierarchy of types
$\lcset{0} : \lcset{1} : \lcset{2} : \dots$, so that there is no `type of all
types', and our theory is predicative.  Type-formers like $\tyarr$ and $\times$
take the least upper bound $\sqcup$ of the contained types. The layers of the
hierarchy are called `universes'.  Theories where $\lcsetz : \lcsetz$ are
inconsistent due to Girard's paradox \citep{Hurkens1995}, and thus lose their
well-behavedness.  Some impredicativity sometimes has its place, either because
the theory retain good properties (normalization, consistency, etc.) anyway,
like in System F and CoC; or because we are at a stage at which we do not care -
we will see instances in section \ref{foo}
% TODO put citation here

Note that the Curry-Howard correspondence runs through ITT as it did with the
STLC with the difference that ITT corresponds to an higher order propositional
logic.

\subsection{Base Types}
\label{sec:base-types}

\newcommand{\lctrue}{\lccon{true}}
\newcommand{\lcfalse}{\lccon{false}}
\newcommand{\lcw}[3]{\lccon{W} #1 : #2 \absspace.\absspace #3}
\newcommand{\lcnode}[4]{#1 \lhd_{#2 . #3} #4}
\newcommand{\lcnodez}[2]{#1 \lhd #2}
\newcommand{\lcited}[5]{\lcsyn{if}\appsp#1/#2\appsp.\appsp#3\appsp\lcsyn{then}\appsp#4\appsp\lcsyn{else}\appsp#5}
\newcommand{\lcrec}[4]{\lcsyn{rec}\appsp#1/#2\appsp.\appsp#3\appsp\lcsyn{with}\appsp#4}
\newcommand{\lcrecz}[2]{\lcsyn{rec}\appsp#1\appsp\lcsyn{with}\appsp#2}
\newcommand{\AxiomL}[1]{\Axiom$\fCenter #1$}
\newcommand{\UnaryInfL}[1]{\UnaryInf$\fCenter #1$}

While the ITT presented is a fairly complete logic, it is not that useful for
programming.  If we wish to make it better, we can add some base types to
represent the data structures we know and love, such as numbers, lists, and
trees.  Apart from some unsurprising data types, we introduce $\lccon{W}$, a
very general tree-like structure useful to represent inductively defined types
and that will allow us to express structural recursion in an equally general
manner.

\begin{center}
  \axname{syntax}
  \begin{eqnarray*}
  \termsyn & ::= & \dots \\
           &  |  & \top \separ \lcunit \\
           &  |  & \lcbool \separ \lctrue \separ \lcfalse \separ \lcited{\termsyn}{x}{\termsyn}{\termsyn}{\termsyn} \\
           &  |  & \lcw{x}{\termsyn}{\termsyn} \separ \lcnode{\termsyn}{x}{\termsyn}{\termsyn} \separ \lcrec{\termsyn}{x}{\termsyn}{\termsyn}
 \end{eqnarray*}

  \axdesc{typing}{\Gamma \vdash \termsyn : \termsyn}

  \vspace{0.5cm}

  \begin{tabular}{c c c}
    \AxiomC{}
    \UnaryInfC{$\hspace{0.2cm}\Gamma \vdash \top : \lcset{0} \hspace{0.2cm}$}
    \noLine
    \UnaryInfC{$\Gamma \vdash \lcbool : \lcset{0}$}
    \DisplayProof
    &
    \AxiomC{}
    \UnaryInfC{$\Gamma \vdash \lcunit : \top$}
    \noLine
    \UnaryInfC{\phantom{1em}}
    \DisplayProof
    &
    \AxiomC{}
    \UnaryInfC{$\Gamma \vdash \lctrue : \lcbool$}
    \noLine
    \UnaryInfC{$\Gamma \vdash \lcfalse : \lcbool$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \begin{tabular}{c}
    \AxiomC{$\Gamma \vdash \termt : \lcbool$}
    \AxiomC{$\Gamma \vdash \termm : \tya[\lctrue]$}
    \AxiomC{$\Gamma \vdash \termn : \tya[\lcfalse]$}
    \TrinaryInfC{$\Gamma \vdash \lcited{\termt}{x}{\tya}{\termm}{\termn} : \tya[\termt]$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \begin{tabular}{c}
    \AxiomC{$\Gamma \vdash \tya : \lcset{n}$}
    \AxiomC{$\Gamma; x : \tya \vdash \tyb : \lcset{m}$}
    \BinaryInfC{$\Gamma \vdash \lcw{x}{\tya}{\tyb} : \lcset{n \sqcup m}$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \begin{tabular}{c}
    \AxiomC{$\Gamma \vdash \termt : \tya$}
    \AxiomC{$\Gamma \vdash \termf : \tyb[\termt ] \tyarr \lcw{x}{\tya}{\tyb}$}
    \BinaryInfC{$\Gamma \vdash \lcnode{\termt}{x}{\tyb}{\termf} : \lcw{x}{\tya}{\tyb}$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \begin{tabular}{c}
    \AxiomC{$\Gamma \vdash \termt: \lcw{x}{\tya}{\tyb}$}
    \noLine
    \UnaryInfC{$\Gamma \vdash \lcforall{\termm}{\tya}{\lcforall{\termf}{\tyb[\termm] \tyarr \lcw{x}{\tya}{\tyb}}{(\lcforall{\termn}{\tyb[\termm]}{\tyc[\app{\termf}{\termn}]}) \tyarr \tyc[\lcnodez{\termm}{\termf}]}}$}
    \UnaryInfC{$\Gamma \vdash \lcrec{\termt}{x}{\tyc}{\termp} : \tyc[\termt]$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \axdesc{reduction}{\termsyn \bred \termsyn}
  \begin{eqnarray*}
    \lcited{\lctrue}{x}{\tya}{\termt}{\termm} & \bred & \termt \\
    \lcited{\lcfalse}{x}{\tya}{\termt}{\termm} & \bred & \termm \\
    \lcrec{\lcnodez{\termt}{\termf}}{x}{\tya}{\termp} & \bred & \app{\app{\app{\termp}{\termt}}{\termf}}{(\abs{\termm}{\lcrec{\app{f}{\termm}}{x}{\tya}{\termp}})}
  \end{eqnarray*}
\end{center}

The introduction and elimination for $\top$ and $\lcbool$ are unsurprising.
Note that in the $\lcite{\dotsb}{\dotsb}{\dotsb}$ construct the type of the
branches are dependent on the value of the conditional.

% TODO: explain better here

The rules for $\lccon{W}$, on the other hand, are quite an eyesore.  The idea
behind $\lccon{W}$ types is to build up `trees' where the number of `children'
of each node is dependent on the value in the node.  This is captured by the
$\lhd$ constructor, where the argument on the left is the value, and the
argument on the right is a function that returns a child for each possible value
of $\tyb[\text{node value}]$, if $\lcw{x}{\tya}{\tyb}$.  The recursor
$\lcrec{\termt}{x}{\tyc}{\termp}$ uses $p$ to inductively prove that
$\tyc[\termt]$ holds.

\subsection{Some examples}

Now we can finally provide some meaningful examples.  Apart from omitting types,
I will also use some abbreviations:
\begin{itemize}
  \item $\_\mathit{operator}\_$ to define infix operators
  \item $\abs{x\appsp y\appsp z : \tya}{\dotsb}$ to define multiple abstractions at the same
    time
\end{itemize}

\subsubsection{Sum types}

We would like to recover our sum type, or disjunction, $\vee$.  This is easily
achieved with $\times$:
\[
\begin{array}{rcl}
  \_\vee\_ & = &  \abs{\tya\appsp\tyb : \lcsetz}{\lcexists{x}{\lcbool}{\lcite{x}{\tya}{\tyb}}} \\
  \lcinl   & = & \abs{x}{(\lctrue, x)} \\
  \lcinr   & = & \abs{x}{(\lcfalse, x)} \\
  \lcfun{case} & = & \abs{x : \tya \vee \tyb}{\abs{f : \tya \tyarr \tyc}{\abs{g : \tyb \tyarr \tyc}{ \\
           &   & \hspace{0.5cm} \app{(\lcited{\lcfst x}{b}{(\lcite{b}{A}{B}) \tyarr C}{f}{g})}{(\lcsnd x)}}}}
\end{array}
\]
What is going on here?  We are using $\exists$ with $\lcbool$ as a tag, so that
we can choose between one of two types in the second element.  In
$\lcfun{case}$ we use $\lcite{\lcfst x}{\dotsb}{\dotsb}$ to discriminate on the
tag, that is, the first element of $x : \tya \vee \tyb$.  If the tag is true,
then we know that the second element is of type $\tya$, and we will apply $f$.
The same applies to the other branch, with $\tyb$ and $g$.

\subsubsection{Naturals}

Now it's time to showcase the power of $\lccon{W}$ types.
\begin{eqnarray*}
  \lccon{Nat}  & = & \lcw{b}{\lcbool}{\lcite{b}{\top}{\bot}}  \\
  \lccon{zero} & = & \lcfalse \lhd \abs{z}{\lcabsurd z} \\
  \lccon{suc}  & = & \abs{n}{(\lctrue \lhd \abs{\_}{n})}  \\
  \lcfun{plus} & = & \abs{x\appsp y}{\lcrecz{x}{(\abs{b}{\lcite{b}{\abs{\_\appsp f}{\app{\lccon{suc}}{(\app{f}{\lcunit})}}}{\abs{\_\appsp\_}{y}}})}}
\end{eqnarray*}
Here we encode natural numbers through $\lccon{W}$ types.  Each node contains a
$\lcbool$.  If the $\lcbool$ is $\lcfalse$, then there are no children, since we
have a function $\bot \tyarr \lccon{Nat}$: this is our $0$.  If it's $\lctrue$,
we need one child only: the predecessor.  We recurse giving $\lcsyn{rec}$ a
function that handles the `base case' 0 when the $\lcbool$ is $\lctrue$, and the
inductive case otherwise.

We postpone more complex examples after the introduction of inductive families
and pattern matching, since $\lccon{W}$ types get unwieldy very quickly.

\subsection{Propositional Equality}
\label{sec:propeq}

We can finally introduce one of the central subjects of my project:
propositional equality.

\newcommand{\lceq}[3]{#2 =_{#1} #3}
\newcommand{\lceqz}[2]{#1 = #2}
\newcommand{\lcrefl}[2]{\lccon{refl}_{#1}\appsp #2}
\newcommand{\lcsubst}[3]{\lcfun{subst}\appsp#1\appsp#2\appsp#3}

\begin{center}
  \axname{syntax}
  \begin{eqnarray*}
  \termsyn & ::= & ... \\
           &  |  & \lceq{\termsyn}{\termsyn}{\termsyn} \separ \lcrefl{\termsyn}{\termsyn} \separ \lcsubst{\termsyn}{\termsyn}{\termsyn}
 \end{eqnarray*}

  \axdesc{typing}{\Gamma \vdash \termsyn : \termsyn}

  \vspace{0.5cm}

  \begin{tabular}{c c}
    \AxiomC{$\Gamma \vdash \termt : \tya$}
    \UnaryInfC{$\Gamma \vdash \lcrefl{\tya}{\termt} : \lceq{\tya}{\termt}{\termt}$}
    \DisplayProof
    &
    \AxiomC{$\Gamma \vdash \termp : \lceq{\tya}{\termt}{\termm}$}
    \AxiomC{$\Gamma \vdash \tyb : \tya \tyarr \lcsetz$}
    \AxiomC{$\Gamma \vdash \termn : \app{\tyb}{\termt}$}
    \TrinaryInfC{$\Gamma \vdash \lcsubst{\termp}{\tyb}{\termn} : \app{\tyb}{\termm}$}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \axdesc{reduction}{\termsyn \bred \termsyn}
  \begin{eqnarray*}
    \lcsubst{(\lcrefl{\tya}{\termt})}{\tyb}{\termm} \bred \termm
  \end{eqnarray*}
\end{center}

Propositional equality internalises equality as a type.  The only way to
introduce an equality proof is by reflexivity ($\lccon{refl}$).  The eliminator
is in the style of `Leibnitz's law' of equality: `equals can be substituted for
equals' ($\lcfun{subst}$).  The computation rule refers to the fact that every
proof of equality will be a $\lccon{refl}$.  We can use $\neg
(\lceq{\tya}{x}{y})$ to express inequality.

These elements conclude our presentation of a `core' type theory.  For an
extended example of a similar theory in use see Section 6.2 of
\cite{Thompson1991}\footnote{Note that while I attempted to formalise the proof
  in Agda, I found a bug in the book!  See the errata for details:
  \url{http://www.cs.kent.ac.uk/people/staff/sjt/TTFP/errata.html}.}.  The above
language and examples have been codified in Agda\footnote{More on Agda in the
  next section.}, see appendix \ref{app:agda-code}.

\section{A more useful language}
\label{sec:practical}

While our core type theory equipped with $\lccon{W}$ types is very usefully
conceptually as a simple but complete language, things get messy very fast.  In
this section I will present the elements that are usually included in theorem
provers or programming languages to make them usable by mathematicians or
programmers.

All the features presented are present in the second version of the Agda system
\citep{Norell2007, Bove2009}.  Agda follows a tradition of theorem provers based
on ITT with inductive families and pattern matching.  The closest relative of
Agda, apart from previous versions, is Epigram \citep{McBride2004, EpigramTut}.
Coq is another notable theorem prover based on the same concepts, and the first
and most popular \citep{Coq}.

The main difference between Coq and Agda/Epigram is that the former focuses on
theorem proving, while the latter also want to be useful as functional
programming languages, while still offering good tools to express
mathematics\footnote{In fact, currently, Agda is used almost exclusively to
  express mathematics, rather than to program.}.  This is reflected in a series
of differences that I will not discuss here (most notably the absence of tactics
but better support for pattern matching in Agda/Epigram).  I will take the same
approach as Agda/Epigram and will give a perspective focused on functional
programming rather than theorem proving.  Every feature will be presented as it
is Agda.

\subsection{Bidirectional type checking}

Lastly, the theory I present is fully explicit in the sense that the user has to
specify every type when forming abstractions, products, etc.  This can be a
great burden if one wants to use the theory directly.  Complete inference is
undecidable (which is hardly surprising considering the role that types play)
but partial inference (also called `bidirectional type checking' in this
context) in the style of \cite{Pierce2000} will have to be deployed in a
practical system.  

\subsection{Inductive families}

\newcommand{\lcdata}[1]{\lcsyn{data}\appsp #1}
\newcommand{\lcdb}{\ |\ }
\newcommand{\lcind}{\hspace{0.2cm}}
\newcommand{\lcwhere}{\appsp \lcsyn{where}}
\newcommand{\lclist}[1]{\app{\lccon{List}}{#1}}
\newcommand{\lcnat}{\lccon{Nat}}
\newcommand{\lcvec}[2]{\app{\app{\lccon{Vec}}{#1}}{#2}}

Inductive families were first introduced by \cite{Dybjer1991}.  For the reader
familiar with the recent developments present in the GHC compiler for Haskell,
inductive families will look similar to GADTs (Generalised Abstract Data Types)
\citep[Section 7.4.7]{GHC}.
Haskell style data types provide \emph{parametric polymorphism}, so that we can
define types that range over type parameters:
\begin{lstlisting}
List a = Nil | Cons a (List a)
\end{lstlisting}
In this way we define the \texttt{List} type once while allowing elements to be
of any type.  In Haskell \texttt{List} will be a type constructor of kind
\texttt{* -> *}, while \texttt{Nil :: List a} and \texttt{Cons :: a -> List a -> List a}\footnote{Note that the \texttt{a}s are implicitly quantified type variables}.

Inductive families bring this concept one step further by allowing some of the
parameters to be constrained by constructors.  We call these `variable'
parameters `indices'.  For example we might define natural numbers
\[
\begin{array}{l}
\lcdata{\lcnat} : \lcsetz \lcwhere \\
  \begin{array}{l@{\ }l}
    \lcind \lccon{zero} &: \lcnat \\
    \lcind \lccon{suc}  &: \lcnat \tyarr \lcnat
  \end{array}
\end{array}
\]
And then define a type for lists indexed by length:
\[
\begin{array}{l}
\lcdata{\lccon{Vec}}\appsp (A : \lcsetz) : \lcnat \tyarr \lcsetz \lcwhere \\
  \begin{array}{l@{\ }l}
    \lcind \lccon{nil} &: \lcvec{A}{\lccon{zero}} \\
    \lcind \_::\_ &: \{n : \lccon{Nat}\} \tyarr A \tyarr \lcvec{A}{n} \tyarr \lcvec{A}{\app{(\lccon{suc}}{n})}
  \end{array}
\end{array}
\]
Note that contrary to the Haskell ADT notation, with inductive families we
explicitly list the types for the type and data constructors.  In $\lccon{Vec}$,
$A$ is a parameter (same across all constructors) while the $\lcnat$ is an
index.  In our syntax, in the $\lcsyn{data}$ declaration, things to the left of
the colon are parameters, while on the right we can define the type of indices.
Also note that the parameters' identifiers will be in scope across all
constructors, while indices' won't.

In this $\lccon{Vec}$ example, when we form a new list the length is
$\lccon{zero}$.  When we append a new element to an existing list of length $n$,
the new list is of length $\app{\lccon{suc}}{n}$, that is, one more than the
previous length.  Also note that we are using the $\{n : \lcnat\} \tyarr \dotsb$
syntax to indicate an argument that we will omit when using $\_::\_$.  In the
future I shall also omit the type of these implicit parameters, in line with how
Agda works.

Once we have $\lccon{Vec}$ we can do things much more safely than with normal
lists.  For example, we can define an $\lcfun{head}$ function that returns the
first element of the list:
\[
\begin{array}{l}
\lcfun{head} : \{A\ n\} \tyarr \lcvec{A}{(\app{\lccon{suc}}{n})} \tyarr A
\end{array}
\]
Note that we will not be able to provide this function with a
$\lcvec{A}{\lccon{zero}}$, since it needs an index which is a successor of some
number.

\newcommand{\lcfin}[1]{\app{\lccon{Fin}}{#1}}

If we wish to index a $\lccon{Vec}$ safely, we can define an inductive family
$\lcfin{n}$, which represents the family of numbers smaller than a given number
$n$:
\[
\begin{array}{l}
  \lcdata{\lccon{Fin}} : \lcnat \tyarr \lcsetz \lcwhere \\
  \begin{array}{l@{\ }l}
    \lcind \lccon{fzero} &: \{n\} \tyarr \lcfin{(\app{\lccon{suc}}{n})} \\
    \lcind \lccon{fsuc}  &: \{n\} \tyarr \lcfin{n} \tyarr \lcfin{(\app{\lccon{suc}}{n})}
  \end{array}
\end{array}
\]
$\lccon{fzero}$ is smaller than any number apart from $\lccon{zero}$.  Given
the family of numbers smaller than $n$, we can produce the family of numbers
smaller than $\app{\lccon{suc}}{n}$.  Now we can define an `indexing' operation
for our $\lccon{Vec}$:
\[
\begin{array}{l}
\_\lcfun{!!}\_ : \{A\ n\} \tyarr \lcvec{A}{n} \tyarr \lcfin{n} \tyarr A
\end{array}
\]
In this way we will be sure that the number provided as an index will be smaller
than the length of the $\lccon{Vec}$ provided.

\subsubsection{Computing with inductive families}

I have carefully avoided to define the functions that I mentioned as example,
since one question still is unanswered: `how do we work with inductive
families'?  The most basic approach is to work with eliminators, that can be
automatically provided by the programming language for each data type defined.
For example the induction principle on natural numbers will serve as an
eliminator for $\lcnat$:
\[
\begin{array}{l@{\ } c@{\ } l}
  \lcfun{NatInd} & : & (A : \lcsetz) \tyarr \\
                 &   & A \tyarr (\lcnat \tyarr A \tyarr A) \tyarr \\
                 &   & \lcnat \tyarr A
\end{array}
\]

That is, if we provide an $A$ (base case), and then given a number and an $A$ we
give the next $A$ (inductive step), then an $A$ can be computed for every
number.  This can be expressed easily in Haskell:
\begin{lstlisting}
data Nat = Zero | Suc Nat
natInd :: a -> (Nat -> a -> a) -> Nat -> a
natInd z _  Zero   = z
natInd z f (Suc n) = f n (natInd z f n)
\end{lstlisting}
However, in a dependent setting, we can do better:
\[
\begin{array}{l@{\ } c@{\ } l}
  \lcfun{NatInd} & : & (P : \lcnat \tyarr \lcsetz) \tyarr \\
                 &   & \app{P}{\lccon{zero}} \tyarr ((n : \lcnat) \tyarr \app{P}{n} \tyarr \app{P}{(\app{\lccon{suc}}{n})}) \tyarr \\
                 &   & (n : \lcnat) \tyarr \app{P}{n}
\end{array}
\]
This expresses the fact that the resulting type can be dependent on the number.
In other words, we are proving that $P$ holds for all $n : \lcnat$.

Naturally a reduction rule will be associated with each eliminator:
$$
\begin{array}{l c l}
\app{\app{\app{\app{\lcfun{NatInd}}{P}}{z}}{f}}{\lccon{zero}} & \bred & z \\
\app{\app{\app{\app{\lcfun{NatInd}}{P}}{z}}{f}}{(\app{\lccon{suc}}{n})} & \bred & \app{\app{f}{n}}{(\app{\app{\app{\app{\lcfun{NatInd}}{P}}{z}}{f}}{n})}
\end{array}
$$
Which echoes the \texttt{natInd} function defined in Haskell.  An extensive
account on combinators and inductive families can be found in \cite{McBride2004}.

\subsubsection{Pattern matching and guarded recursion}

However, combinators are far more cumbersome to use than the techniques usually
employed in functional programming: pattern matching and recursion.
\emph{General} recursion cannot be added if we want to keep our theory free of
$\bot$.  The common solution to this problem is to allow recursive calls only if
the arguments are structurally smaller than what the function received.  Pattern
matching on the other hand gains considerable power with inductive families,
since when we match a constructor we are gaining information on the indices of
the family.  Thus matching constructors will enable us to restrict patterns of
other arguments.

Following this discipline defining $\lcfun{head}$ becomes easy:
\[
\begin{array}{l}
\lcfun{head} : \{A\ n\} \tyarr \lcvec{A}{(\app{\lccon{suc}}{n})} \tyarr A \\
\lcfun{head}\appsp(x :: \_) = x
\end{array}
\]
We need no case for $\lccon{nil}$, since the type checker knows that the
equation $n = \lccon{zero} = \app{\lccon{suc}}{n'}$ cannot be satisfied.

More details on the implementations of inductive families can be found in
\citep{McBride2004, Norell2007}.

\subsection{Friendlier universes}

Universes as presented in section \ref{sec:core-tt} are quite cumbersome to work
with.  Consider the example where we define an inductive family for booleans and
then we want to define an `if then else' eliminator:
\[
\begin{array}{l}
\lcdata{\lcbool} : \lcsetz \lcwhere \\
  \begin{array}{l@{\ }l}
    \lcind \lccon{true}  &: \lcbool \\
    \lcind \lccon{false} &: \lcbool
  \end{array}\\
\\
\lcfun{ite} : \{A : \lcset{0}\} \tyarr \lcbool \tyarr A \tyarr A \tyarr A \\
\lcfun{ite} \appsp \lccon{true} \appsp x \appsp \_ = x \\
\lcfun{ite} \appsp \lccon{false} \appsp \_ \appsp x = x
\end{array}
\]

What if we want to use $\lcfun{ite}$ with types, for example $\lcfun{ite} \appsp
b \appsp \lcnat \appsp \lccon{Bool}$?  Clearly $\lcnat$ is not of type
$\lcset{0}$, so we'd have to redefine $\lcfun{ite}$ with exactly the same body,
but different type signature.  The problem is even more evident with type
families: for example our $\lccon{List}$ can only contain values, and if we want
to build lists of types we need to define another identical, but differently
typed, family.

One option is to have a \emph{cumulative} theory, where $\lcset{n} : \lcset{m}$
iff $n < m$.  Then we can have a sufficiently large level in our type signature
and forget about it.  Moreover, levels in this setting can be inferred
mechanically \citep{Pollack1990}, and thus we might lift the burden of universes
from the user.  This is the approach taken by Epigram.

Another more expressive (but currently more cumbersome) way is to expose
universes more, giving the user a way to quantify them and to take the least
upper bound of two levels.  This is the approach taken by Agda, where for
example we can define a level-polymorphic $\times$:
\[
\begin{array}{l}
\lcdata{\_\times\_}\ \{a\ b : \lccon{Level}\}\ (A : \lcset{a})\ (B : A \tyarr \lcset{b}) : \lcset{a \sqcup b} \lcwhere \\
  \lcind \_ , \_ : (x : A) \tyarr \app{B}{x} \tyarr A \times \app{B}{x}
\end{array}
\]
Levels can be made implicit as shown and can be almost always inferred.
However, having to decorate each type signature with quantified levels adds
quite a lot of noise.  An inference algorithm that automatically quantifies and
instantiates levels (much like Hindley-Milner for types) seems feasible, but
currently not implemented anywhere.

\subsection{Coinduction}

One thing that the Haskell programmer will miss in ITT as presented is the
possibility to work with infinite data.  The classic example is the manipulation
of infinite streams:
\begin{lstlisting}
zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]

fibs :: [Int]
fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
\end{lstlisting}
While we can clearly write useful programs of this kind, we need to be careful,
since \texttt{length fibs}, for example, does not make much sense\footnote{Note
  that if instead of machine \texttt{Int}s we used naturals as defined
  previously, getting the length of an infinite list would a productive
  definition.}.

In less informal terms, we need to distinguish between \emph{productive} and
non-productive definitions.  A productive definition is one for which pattern
matching on data will never diverge, which is the case for \texttt{fibs} but
not, for example, for \texttt{let x = x in x :: [Int]}.  It is very desirable to
recover \emph{only} the productive definition so that total programs working
with infinite data can be written.

This desire has lead to work on coindunction
% TODO finish

\section{Many equalities}

\subsection{Revision, and function extensionality}

\epigraph{\emph{Half of my time spent doing research involves thinking up clever
  schemes to avoid needing functional extensionality.}}{@larrytheliquid}

Up to this point, I have introduced two equalities: \emph{definitional} equality
($\defeq$) and \emph{propositional} equality ($=_{\tya}$).

Definitional equality relates terms that the type checker identifies as equal.
Our definition in section \ref{sec:core-tt} consisted of `reduce the two terms
to their normal forms, then check if they are equal (up to $\alpha$-renaming)'.
We can extend this definition in other ways so that more terms will be
identified as equal.  Common ones include doing $\eta$-expansion, which converts
partial appications of functions, and congruence laws for $\lambda$:
\begin{center}
  \begin{tabular}{c c c}
    \AxiomC{$\Gamma \vdash \termf : \tya \tyarr \tyb$}
    \UnaryInfC{$\Gamma \vdash \termf \defeq \abs{x}{\app{\termf}{x}}$}
    \DisplayProof
    &
    \AxiomC{$\Gamma; x : \tya \vdash \termf \defeq g$}
    \UnaryInfC{$\Gamma \vdash \abs{x}{\termf} \defeq \abs{x}{g}$}
    \DisplayProof
  \end{tabular}
\end{center}
Definitional equality is used in the type checking process, and this means that
in dependently typed languages type checking and evaluation are intertwined,
since we need to reduce types to normal forms to check if they are equal.  It
also means that we need be able to compute under binders, for example to
determine that $\abs{x}{\app{(\abs{y}{y})}{\tya}} \defeq \abs{x}{\tya}$.  This
process goes on `under the hood' and is outside the control of the user.

Propositional equality, on the other hand, is available to the user to reason
about equality, internalising it as a type.  As we have seen in section
\ref{sec:propeq} propositional equality is introduced by reflexivity and
eliminated with a `Leibnitz's law' style rule ($\lcfun{subst}$).  Now that we
have inductive families and dependent pattern matching we do not need hard coded
rules to express this concepts\footnote{Here I use Agda notation, and thus I
  cannot redefine $=$ and use subscripts, so I am forced to use $\equiv$ with
  implicit types.  After I will carry on using the old notation.}:
\[
\begin{array}{l}
  \lcdata{\lccon{\_\equiv\_}} : \{A : \lcsetz\} : A \tyarr A \tyarr \lcsetz \lcwhere \\
  \begin{array}{l@{\ }l}
    \lcind \lccon{refl} &: \{x : A\} \tyarr x \equiv x
  \end{array} \\
  \\
  \lcfun{subst} : \{A : \lcsetz\} \tyarr \{t\ m : A\} \tyarr t \equiv m \tyarr (B : A \tyarr \lcsetz) \tyarr \app{B}{t} \tyarr \app{B}{m} \\
  \lcfun{subst} \appsp \lccon{refl}\appsp B \appsp n = n
\end{array}
\]
Here matching $\lccon{refl}$ tells the type checker that $t \defeq m$, and thus
$\app{B}{t} \defeq \app{B}{m}$, so we can just return $n$.

While propositional equality is a very useful construct, we can prove less terms
equal than we would like to.  For example, if we have the usual functions
\[
\begin{array}{l@{\ } l}
\lcfun{id} & : \{A : \lcsetz\} \tyarr A \tyarr A \\
\lcfun{map} & : \{A\ B : \lcsetz\} \tyarr (A \tyarr B) \tyarr \app{\lccon{List}}{A} \tyarr \app{\lccon{List}}{B}
\end{array}
\]
we would certainly like to have a term of type
\[\{A\ B : \lcsetz\} \tyarr
\app{\lcfun{map}}{\lcfun{id}} =_{\lclist{A} \tyarr \lclist{B}} \lcfun{id}\]
We cannot do this in ITT, since the things on the sides of $=$ look too
different at the term level.  What we can have is
\[
\{A\ B : \lcsetz\} \tyarr (x : \lclist{A}) \tyarr \app{\app{\lcfun{map}}{\lcfun{id}}}{x}
=_{\lclist{B}} \app{\lcfun{id}}{x}
\]
Since the type checker has something to compute with (even if only a variable)
and reduce the two lists to equal normal forms.  The root of this problem is
that \emph{function extensionality} does not hold:
\[
\begin{array}{l@{\ } l}
\lcfun{ext} : & \{A\ B : \lcsetz\} \tyarr (f\ g : A \tyarr B) \tyarr \\
              & ((x : A) \tyarr \app{f}{x} =_{A} \app{g}{x}) \tyarr f =_{A \tyarr B} g
\end{array}
\]
Which is a long standing issue in ITT.

\subsection{Extensional Type Theory and its problems}

One way to `solve' the problem and gain extensionality is to allow the type
checker to derive definitional equality from propositional equality, introducing
what is known as the `equality reflection' rule:
\begin{center}
  \begin{prooftree}
    \AxiomC{$\Gamma \vdash t =_A m$}
    \UnaryInfC{$\Gamma \vdash t \defeq m$}
  \end{prooftree}
\end{center}
This jump from types to a metatheoretic relation has deep consequences.  But
firstly, let's get extensionality out of the way.  Given $\Gamma = \lcfun{eq} :
(x : A) \tyarr \app{f}{x} = \app{g}{x}$, we have:
\begin{center}
  \begin{prooftree}
    \AxiomC{$\Gamma; x : A \vdash \app{\lcfun{eq}}{x} : \app{f}{x} = \app{g}{x}$}
    \RightLabel{(equality reflection)}
    \UnaryInfC{$\Gamma; x : A \vdash \app{f}{x} \defeq \app{g}{x}$}
    \RightLabel{(congruence for $\lambda$)}
    \UnaryInfC{$\Gamma \vdash \abs{x : A}{\app{f}{x}} \defeq \abs{x : A}{\app{g}{x}}$}
    \RightLabel{($\eta$-law)}
    \UnaryInfC{$\Gamma \vdash f \defeq g$}
    \RightLabel{($\lccon{refl}$)}
    \UnaryInfC{$\Gamma \vdash f = g$}
  \end{prooftree}
\end{center}
Since the above is possible, theories that include the equality reflection rule
are often called `Extensional Type Theories', or ETTs.  A notable exponent of
this discipline is the NuPRL system \citep{Constable86}.  Moreover, equality
reflection simplifies $\lcfun{subst}$-like operations, since if we have $t = m$
and $\app{A}{t}$, then by equality reflection clearly $\app{A}{t} \defeq
\app{A}{m}$.

However equality reflection comes at a great price.  The first hit is that it is
now unsafe to compute under binders, since we can send the type checker into a
loop given that under binders we can have any equality proof we might want
(think about what happens with $\abs{x : \bot}{\dotsb}$), and thus we can
convince the type checker that a term has any type we might want.  Secondly,
equality reflection is not syntax directed, thus the type checker would need to
`invent' proofs of propositional equality to prove two terms definitionally
equal, rendering the type checking process undecidable.  Thus the type checker
needs to carry complete derivations for each typing judgement, instead of
relying on terms only.

\subsection{Observational equality}

A recent development by \cite{Altenkirch2007} promises to keep the well
behavedness of ITT while being able to gain many useful equality proofs,
including function extensionality.  Starting from a theory similar to the one
presented in section \ref{sec:itt} but with only $\lcset{0}$, a propositional
subuniverse of $\lcsetz$ is introduced, plus a `decoding' function:

\newcommand{\lcprop}{\lccon{Prop}}

\begin{center}
  \begin{tabular}{c c c}
    \AxiomC{\phantom{1em}}
    \UnaryInfC{$\Gamma \vdash \lcprop : \lcsetz$}
    \noLine
    \UnaryInfC{\phantom{1em}}
    \DisplayProof
    &
    \AxiomC{\phantom{1em}}
    \UnaryInfC{$\Gamma \vdash \bot : \lcprop$}
    \noLine
    \UnaryInfC{$\Gamma \vdash \top : \lcprop$}
    \DisplayProof
    &
    \AxiomC{$\Gamma \vdash P : \lcprop$}
    \AxiomC{$\Gamma \vdash Q : \lcprop$}
    \BinaryInfC{$\Gamma \vdash P \wedge Q : \lcprop$}
    \noLine
    \UnaryInfC{\phantom{1em}}
    \DisplayProof
  \end{tabular}

  \vspace{0.5cm}

  \begin{tabular}{c}
    \AxiomC{$\Gamma \vdash S : \lcsetz$}
    \AxiomC{$\Gamma \vdash q : \lcprop$}
    \BinaryInfC{$\Gamma \vdash p \wedge q : \lcprop$}
    \noLine
    \UnaryInfC{\phantom{1em}}
    \DisplayProof
  \end{tabular}
\end{center}

\section{What to do}


\bibliographystyle{authordate1}
\bibliography{background}

\appendix
\section{Agda code}
\label{app:agda-code}

\lstset{
  xleftmargin=0pt
}

\lstinputlisting{background.agda}

\end{document}