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pdfauthor={Francesco Mazzoli <fm2209@ic.ac.uk>},
- pdftitle={The Path Towards Observational Equality},
+ pdftitle={The Paths Towards Observational Equality},
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-\title{The Path Towards Observational Equality}
-\author{Francesco Mazzoli \url{<fm2209@ic.ac.uk>}}
+\title{The Paths Towards 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}
+
+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.
+
+In the next sections I will give a very brief overview of STLC, and then
+describe how to augment it to reach the theory I am interested in,
+Inutitionistic Type Theory (ITT), also known as Martin-L\"{o}f Type Theory after
+its inventor. 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}.
+
+I will then 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.
+
+\section{Simple and not-so-simple types}
+
+\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 sintax and most notably no `data'
+in the traditional sense, but just functions.
+
+The syntax of $\lambda$-terms consists of just three things: variables,
+abstractions, and applications:
+
+\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}}}
+
+\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}
+$$\app{(\abs{x}{\termt})}{\termm} \bred \termt[\termm ]$$
+\end{center}
+
+Where $\termt[\termm ]$ expresses the operation that substitutes all
+occurrences of $x$ with $\termm$ in $\termt$. In the future, I will use
+$[\termt]$ as an abbreviation for $[\termt ]$. 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.
+
+% % TODO put the trans closure
+
+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 is called \emph{normal
+ form}. These concepts (reduction and normal forms) will run through all the
+material analysed.
+
+\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:
+
+\newcommand{\tyarr}{\to}
+\newcommand{\tysyn}{\mathit{type}}
+\newcommand{\ctxsyn}{\mathit{context}}
+\newcommand{\emptyctx}{\cdot}
+
+\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. We 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{}
+ \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: %TODO add credit to pierce
+\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]{\mathsf{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:
+$$
+ \termsyn ::= \dots \separ \lcfix{x : \tysyn}{\termsyn}
+$$
+\begin{center}
+ \begin{prooftree}
+ \AxiomC{$\Gamma;x : \tya \vdash \termt : \tya$}
+ \UnaryInfC{$\Gamma \vdash \lcfix{x : \tya}{\termt} : \tya$}
+ \end{prooftree}
+\end{center}
+$$
+ \lcfix{x : \tya}{\termt} \bred \termt[(\lcfix{x : \tya}{\termt}) ]
+$$
+
+However, we will keep STLC without such a facility. In the next section we shall
+see why that is preferable for our needs.
+
+\subsection{The Curry-Howard correspondence}
+\label{sec:curry-howard}
+
+\newcommand{\lcunit}{\mathsf{\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 wished 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 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, and
+that is 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, $\top$ is the type with just one trivial element, $\lcunit$.
+
+\newcommand{\lcinl}{\mathsf{inl}\appsp}
+\newcommand{\lcinr}{\mathsf{inr}\appsp}
+\newcommand{\lccase}[3]{\lcsyn{case}\appsp#1\appsp\lcsyn{of}\appsp#2\appsp#3}
+\newcommand{\lcfst}{\mathsf{fst}\appsp}
+\newcommand{\lcsnd}{\mathsf{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}{\mathsf{absurd}\appsp}
+\newcommand{\lcabsurdd}[1]{\mathsf{absurd}_{#1}\appsp}
+
+
+\begin{center}
+ \axname{syntax}
+ $$
+ \begin{array}{rcl}
+ \termsyn & ::= & \dots \\
+ & | & \lcinl \termsyn \separ \lcinr \termsyn \separ \lccase{\termsyn}{\termsyn}{\termsyn} \\
+ & | & (\termsyn , \termsyn) \separ \lcfst \termsyn \separ \lcsnd \termsyn \\
+ & | & \lcunit \\
+ \tysyn & ::= & \dots \separ \tysyn \vee \tysyn \separ \tysyn \wedge \tysyn \separ \bot \separ \top
+ \end{array}
+ $$
+\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 we will not touch those subjects.}.
+
+Finally, going back to our $\mathsf{fix}$ 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}{\mathsf{Type}}
+\newcommand{\lcite}[3]{\lcsyn{if}\appsp#1\appsp\lcsyn{then}\appsp#2\appsp\lcsyn{else}\appsp#3}
+\newcommand{\lcbool}{\mathsf{Bool}}
+\newcommand{\lcforallz}[2]{\forall #1 \absspace.\absspace #2}
+\newcommand{\lcforall}[3]{(#1 : #2) \tyarr #3}
+\newcommand{\lcexists}[3]{\exists #1 : #2 \absspace.\absspace #3}
+
+The STLC can be made more expressive in various ways. Henk Barendregt
+succinctly expressed geometrically how we can expand our type system:
+
+\begin{equation*}
+\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]
+}
+\end{equation*}
+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 $\lambda{2}$)] In other words, 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 $\lambda{\underline{\omega}}$)] In other
+ words, 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 (some of
+which I haven't mentioned yet). 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' above.
+
+\section{Intuitionistic Type Theory and dependent types}
+
+\newcommand{\lcset}[1]{\mathsf{Type}_{#1}}
+\newcommand{\lcsetz}{\mathsf{Type}}
+\newcommand{\defeq}{\equiv}
+
+\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 one of the polymorphic $\lambda$-calculus, and
+specifically the previously mentioned System F, which was invented 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} Huet 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 as
+said previously 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{}
+ \RightLabel{var}
+ \UnaryInfC{$\Gamma;x : \tya \vdash x : \tya$}
+ \DisplayProof
+ &
+ \AxiomC{$\Gamma \vdash \termt : \bot$}
+ \UnaryInfC{$\Gamma \vdash \lcabsurdd{\tya} \termt : \tya$}
+ \DisplayProof
+ &
+ \AxiomC{$\Gamma \vdash \termt : \tya$}
+ \AxiomC{$\tya \defeq \tyb$}
+ \BinaryInfC{$\Gamma \vdash \termt : \tyb$}
+ \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{$\Gamma \vdash (\termt, \termm)_{x.\tyb} : \lcexists{x}{\tya}{\tyb}$}
+ \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{}
+ \UnaryInfC{$\Gamma \vdash \lcset{n} : \lcset{n + 1}$}
+ \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}
+
+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 $\exists$: 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 $\exists$ 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. $\forall x : \mathsf{Bool}. \dotsb$. In the
+ literature this is also known as `dependent product' and shown as $\Pi$,
+ following the interpretation of functions as infinitary products. We will just
+ call it `dependent function', reserving `product' for $\exists$.
+
+\item[`exists' ($\exists$)] is a generalisation of $\wedge$ in the extended
+ STLC of section \ref{sec:curry-howard}, and thus we will call it `dependent
+ product'. Like $\wedge$, it is formed by providing a pair of things. In our
+ logic, it represents existential quantification.
+
+ 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 decides 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 $\defeq$ as
+the smallest equivalence relation extending $\bredc$, where $\bredc$ is the
+reflexive transitive closure of $\bred$; and we treat types that are equal
+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 $\termm \bredc \termp$
+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. The layers of the hierarchy are called
+`universes'. $\lcsetz : \lcsetz$ ITT is inconsistent due to Girard's paradox
+\citep{Hurkens1995}, and thus loses its 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 of the last
+motivation later. Moreover, universes can be inferred mechanically
+\citep{Pollack1990}. It is also convenient to have a \emph{cumulative} theory,
+where $\lcset{n} : \lcset{m}$ iff $n < m$. We eschew these measures to keep the
+presentation simple.
+
+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. When showing examples types will be omitted when this can be
+done without loss of clarity, for example $\abs{x}{x}$ in place of $\abs{A :
+ \lcsetz}{\abs{x : A}{x}}$.
+
+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.
+
+% TODO describe abbreviations somewhere
+% I will use various abbreviations:
+% \begin{itemize}
+% \item $\lcsetz$ for $\lcset{0}$
+% \item $\tya \tyarr \tyb$ for $\lcforall{-}{\tya}{\tyb}$, when $\tyb$ does not
+% depend on the value of type $\tya$
+% \item $(
+
+\subsection{Base Types}
+\label{sec:base-types}
+
+\newcommand{\lctrue}{\mathsf{true}}
+\newcommand{\lcfalse}{\mathsf{false}}
+\newcommand{\lcw}[3]{\mathsf{W} #1 : #2 \absspace.\absspace #3}
+\newcommand{\lcnode}[4]{#1 \lhd_{#2 . #3} #4}
+\newcommand{\lcnodez}[2]{#1 \lhd #2}
+\newcommand{\lcited}[5]{\lcsyn{if}\appsp#1/#2\appsp.\appsp#3\appsp\lcsyn{then}\appsp#4\appsp\lcsyn{else}\appsp#5}
+\newcommand{\lcrec}[4]{\lcsyn{rec}\appsp#1/#2\appsp.\appsp#3\appsp\lcsyn{with}\appsp#4}
+\newcommand{\lcrecz}[2]{\lcsyn{rec}\appsp#1\appsp\lcsyn{with}\appsp#2}
+\newcommand{\AxiomL}[1]{\Axiom$\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 $\mathsf{W}$, a
+very general tree-like structure useful to represent inductively defined types.
+
+\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{$\Gamma \vdash \lcunit : \top$}
+ \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.
+
+The rules for $\mathsf{W}$, on the other hand, are quite an eyesore. The idea
+behind $\mathsf{W}$ types is to build up `trees' where shape of 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
+done with $\exists$:
+\[
+\begin{array}{rcl}
+ \_\vee\_ & = & \abs{\tya\appsp\tyb : \lcsetz}{\lcexists{x}{\lcbool}{\lcite{x}{\tya}{\tyb}}} \\
+ \lcinl & = & \abs{x}{(\lctrue, x)} \\
+ \lcinr & = & \abs{x}{(\lcfalse, x)} \\
+ \mathsf{case} & = & \abs{x : \tya \vee \tyb}{\abs{f : \tya \tyarr \tyc}{\abs{g : \tyb \tyarr \tyc}{ \\
+ & & \hspace{0.5cm} \app{(\lcited{\lcfst x}{b}{(\lcite{b}{A}{B}) \tyarr C}{f}{g})}{(\lcsnd x)}}}}
+\end{array}
+\]
+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
+$\mathsf{case}$ 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 $\mathsf{W}$ types.
+\begin{eqnarray*}
+ \mathsf{Nat} & = & \lcw{b}{\lcbool}{\lcite{b}{\top}{\bot}} \\
+ \mathsf{zero} & = & \lcfalse \lhd \abs{z}{\lcabsurd z} \\
+ \mathsf{suc} & = & \abs{n}{(\lctrue \lhd \abs{\_}{n})} \\
+ \mathsf{plus} & = & \abs{x\appsp y}{\lcrecz{x}{(\abs{b}{\lcite{b}{\abs{\_\appsp f}{\app{\mathsf{suc}}{(\app{f}{\lcunit})}}}{\abs{\_\appsp\_}{y}}})}}
+\end{eqnarray*}
+Here we encode natural numbers through $\mathsf{W}$ types. Each node contains a
+$\lcbool$. If the $\lcbool$ is $\lcfalse$, then there are no children, since we
+have a function $\bot \tyarr \mathsf{Nat}$: this is our $0$. If it's $\lctrue$,
+we need one child only: the predecessor. Similarly, we recurse giving to
+$\lcsyn{rec}$ 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 $\mathsf{W}$ types get unwieldy very quickly.
+
+\subsubsection{Propositional Equality}
+
+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]{\mathsf{refl}_{#1}\appsp #2}
+\newcommand{\lcsubst}[3]{\mathsf{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. This lets us prove
+useful things. The only way to introduce an equality proof is by reflexivity
+($\mathsf{refl}$). The eliminator is in the style of `Leibnitz's law' of
+equality: `equals can be substituted for equals' ($\mathsf{subst}$). The
+computation rule refers to the fact that every canonical proof of equality will
+be a $\mathsf{refl}$. 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 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}
+
+While our core type theory equipped with $\mathsf{W}$ types is very usefully
+conceptually as a simple but complete language, things get messy very fast. In
+this section we will present the elements that are usually include in theorem
+provers or programming languages to make them usable by mathematicians or
+programmers.
+
+All the features presented, except for quotient types, are present in the second
+version of the Agda system, which is described in the PhD thesis of the author
+\citep{Norell2007}. 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, whose type theory is described in
+\cite{McBride2004}. Coq is another notable theorem prover based on the same
+concepts, and the first and most popular \cite{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 present in Agda (if possible, since some features are not present in Agda
+yet).
+
+\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{\mathsf{List}}{#1}}
+\newcommand{\lcnat}{\app{\mathsf{Nat}}}
+\newcommand{\lcvec}[2]{\app{\app{\mathsf{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).
+% TODO: add citation
+Haskell style data types provide \emph{parametric polymorphism}, so that we can
+define types that range over type parameters:
+\[
+\begin{array}{l}
+\lcdata{\lclist{A}} = \mathsf{Nil} \lcdb A :: \lclist{A}
+\end{array}
+\]
+In this way we define the $\mathsf{List}$ type once while allowing elements to
+be of any type. $\mathsf{List}$ will be a type constructor of type $\lcsetz
+\tyarr \lcsetz$, while $\mathsf{nil} : \lcforall{A}{\lcsetz}{\lclist{A}}$ and
+$\_::\_ : \lcforall{A}{\lcsetz}{A \tyarr \lclist{A} \tyarr \lclist{A}}$.
+
+Inductive families bring this concept one step further by allowing some of the
+parameters to be constrained in constructors. We call these parameters
+`indices'. For example we might define natural numbers
+\[
+\begin{array}{l}
+\lcdata{\lcnat} : \lcsetz \lcwhere \\
+ \begin{array}{l@{\ }l}
+ \lcind \mathsf{zero} &: \lcnat \\
+ \lcind \mathsf{suc} &: \lcnat \tyarr \lcnat
+ \end{array}
+\end{array}
+\]
+And then define a type for lists indexed by length:
+\[
+\begin{array}{l}
+\lcdata{\mathsf{Vec}}\appsp (A : \lcsetz) : \lcnat \tyarr \lcsetz \lcwhere \\
+ \begin{array}{l@{\ }l}
+ \lcind \mathsf{nil} &: \lcvec{A}{\mathsf{zero}} \\
+ \lcind \_::\_ &: \{n : \mathsf{Nat}\} \tyarr A \tyarr \lcvec{A}{n} \tyarr \lcvec{A}{\app{(\mathsf{suc}}{n})}
+ \end{array}
+\end{array}
+\]
+Note that contrary to the Haskell ADT notation, with inductive families we
+explicitly list the types for the type constructor and data constructors. In
+$\mathsf{Vec}$, $A$ is a parameter (same across all constructors) while the
+$\lcnat$ is an integer. 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 can be named across all
+constructors, while indices can't.
+
+In this $\mathsf{Vec}$ example, when we form a new list the length is
+$\mathsf{zero}$. When we append a new element to an existing list of length
+$n$, the new list is of length $\app{\mathsf{suc}}{n}$, 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
+how Agda works.
+
+Once we have our $\mathsf{Vec}$ we can do things much more safely than with
+normal lists. For example, we can define an $\mathsf{head}$ function that
+returns the first element of the list:
+\[
+\begin{array}{l}
+\mathsf{head} : \{A\ n\} \tyarr \lcvec{A}{(\app{\mathsf{suc}}{n})} \tyarr A
+\end{array}
+\]
+Note that we will not be able to provide this function with a
+$\lcvec{A}{\mathsf{zero}}$, since it needs an index which is a successor of some
+number.
+
+\newcommand{\lcfin}[1]{\app{\mathsf{Fin}}{#1}}
+
+If we wish to index a $\mathsf{Vec}$ safely, we can define an inductive family
+$\lcfin{n}$, which represents the family of numbers smaller than a given number
+$n$:
+\[
+\begin{array}{l}
+ \lcdata{\mathsf{Fin}} : \lcnat \tyarr \lcsetz \lcwhere \\
+ \begin{array}{l@{\ }l}
+ \lcind \mathsf{fzero} &: \{n\} \tyarr \lcfin{n} \\
+ \lcind \mathsf{fsuc} &: \{n\} \tyarr (i : \lcfin{n}) \tyarr \lcfin{(\app{\mathsf{suc}}{n})}
+ \end{array}
+\end{array}
+\]
+$\mathsf{fzero}$ is smaller than any number apart from $\mathsf{zero}$. Given
+the family of numbers smaller than $i$, we can produce the family of numbers
+smaller than $\app{\mathsf{suc}}{n}$. Now we can define an `indexing' operation
+for our $\mathsf{Vec}$:
+\[
+\begin{array}{l}
+\_\mathsf{!!}\_ : \{A\ n\} \tyarr \lcvec{A}{n} \tyarr \lcfin{n} \tyarr A
+\end{array}
+\]
+In this way we will be sure that the number provided as an index will be smaller
+than the length of the $\mathsf{Vec}$ provided.
+
+
+\subsection{Pattern matching}
+
+\subsection{Guarded recursion}
+
+\subsection{Corecursion}
+
+\subsection{Quotient types}
+
+
+\section{Why dependent types?}
+
+\section{Many equalities}
+
+\subsection{Revision}
+
+\subsection{Extensional equality and its problems}
+
+\subsection{Observational equality}
+
+\section{What to do}
+
+
+\bibliographystyle{authordate1}
+\bibliography{background}
+
+\appendix
+\section{Agda code}
+\label{app:agda-code}
+
\end{document}
projects / bitonic-mengthesis.git / blobdiff