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+
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\title{The Paths Towards Observational Equality}
-\author{Francesco Mazzoli \url{<fm2209@ic.ac.uk>}}
+\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.
+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 this theories,
-and talk about the various attempts have been made. 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.
+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. This early programming language encodes
-computation with a minimal sintax and most notably no ``data'' in the
-traditional sense, but just functions.
+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{\appspace}{\hspace{0.07cm}}
\newcommand{\app}[2]{#1\appspace#2}
-\newcommand{\abs}[2]{\lambda #1. #2}
-\newcommand{\termt}{\mathrm{T}}
-\newcommand{\termm}{\mathrm{M}}
-\newcommand{\termn}{\mathrm{N}}
-\newcommand{\termp}{\mathrm{P}}
-\newcommand{\separ}{\ |\ }
+\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{eqnarray*}
- \termt & ::= & x \separ (\abs{x}{\termt}) \separ (\app{\termt}{\termt}) \\
- x & \in & \text{Some enumerable set of symbols, e.g.}\ \{x, y, z, \dots , x_1, x_2, \dots\}
+ \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{eqnarray*}
+\end{center}
+
% I will omit parethesis in the usual manner. %TODO explain how
-Intuitively, abstractions ($\abs{x}{\termt}$) introduce functions with a named
-parameter ($x$), and applications ($\app{\termt}{\termm}$) apply a function
-($\termt$) to an argument ($\termm$).
+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:
+The `applying' is more formally explained with a reduction rule:
-\newcommand{\bred}{\to_{\beta}}
+\newcommand{\bred}{\leadsto}
+\newcommand{\bredc}{\bred^*}
-\begin{eqnarray*}
- \app{(\abs{x}{\termt})}{\termm} & \bred & \termt[\termm / x] \\
- \termt \bred \termm & \Rightarrow & \left \{
- \begin{array}{l}
- \app{\termt}{\termn} \bred \app{\termm}{\termn} \\
- \app{\termn}{\termt} \bred \app{\termn}{\termm} \\
- \abs{x}{\termt} \bred \abs{x}{\termm}
- \end{array}
- \right.
-\end{eqnarray*}
+\begin{center}
+\axdesc{reduction}{\termsyn \bred \termsyn}
+$$\app{(\abs{x}{\termt})}{\termm} \bred \termt[\termm ]$$
+\end{center}
-Where $\termt[\termm / x]$ expresses the operation that substitutes all
-occurrences of $x$ with $\termm$ in $\termt$.
+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):
+(`loops' in imperative terms):
\begin{equation*}
- \app{(\abs{x}{\app{x}{x}})}{(\abs{x}{\app{x}{x}})} \bred \app{(\abs{x}{\app{x}{x}})}{(\abs{x}{\app{x}{x}})} \bred \dots
+ \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
\end{equation*}
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
+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}{\mathrm{A}}
-\newcommand{\tyb}{\mathrm{B}}
-\newcommand{\tyc}{\mathrm{C}}
+\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.
+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 $\tya$''.
+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{equation*}
- \tya ::= x \separ \tya \tyarr \tya
-\end{equation*}
+\begin{center}
+ \axname{syntax}
+ $$\tysyn ::= x \separ \tysyn \tyarr \tysyn$$
+\end{center}
-The $x$ represents all the primitive types that we might want to add to our
-calculus, for example $\mathbb{N}$ or $\mathsf{Bool}$.
+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. Note that, being a map, no variable can
-appear twice as a subject in a context.
+$\Gamma; x : \tya$ 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{equation*}
- \termt ::= \dots \separ (\abs{x : \tya}{\termt})
-\end{equation*}
+\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}
- \begin{prooftree}
+ \axdesc{typing}{\Gamma \vdash \termsyn : \tysyn}
+
+ \vspace{0.5cm}
+
+ \begin{tabular}{c c c}
\AxiomC{}
- \UnaryInfC{$\Gamma, x : \tya \vdash x : \tya$}
- \end{prooftree}
- \begin{prooftree}
- \AxiomC{$\Gamma, x : \tya \vdash \termt : \tyb$}
+ \UnaryInfC{$\Gamma; x : \tya \vdash x : \tya$}
+ \DisplayProof
+ &
+ \AxiomC{$\Gamma; x : \tya \vdash \termt : \tyb$}
\UnaryInfC{$\Gamma \vdash \abs{x : \tya}{\termt} : \tya \tyarr \tyb$}
- \end{prooftree}
- \begin{prooftree}
+ \DisplayProof
+ \end{tabular}
+
+ \vspace{0.5cm}
+
+ \begin{tabular}{c}
\AxiomC{$\Gamma \vdash \termt : \tya \tyarr \tyb$}
\AxiomC{$\Gamma \vdash \termm : \tya$}
\BinaryInfC{$\Gamma \vdash \app{\termt}{\termm} : \tyb$}
- \end{prooftree}
+ \DisplayProof
+ \end{tabular}
\end{center}
-This typing system takes the name of ``simply typed lambda calculus'' (STLC),
+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}
- % TODO the definition of "stuck" thing is wrong
-\item[Progress] A well-typed term is not stuck. With stuck, we mean a compound
- term (not a variable or a value) that cannot be reduced further. In the raw
- $\lambda$-calculus all we have is functions, but if we add other primitive
- types and constructors it's easy to see how things can go bad - for example
- trying to apply a boolean to something.
+\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}
\app{(\abs{x : ?}{\app{x}{x}})}{(\abs{x : ?}{\app{x}{x}})}
\end{equation*}
-\newcommand{\lcfix}[2]{\mathsf{fix} \appspace #1. #2}
+\newcommand{\lcfix}[2]{\mathsf{fix} \appspace #1\absspace.\absspace #2}
This makes the STLC Turing incomplete. We can recover the ability to loop by
adding a combinator that recurses:
\begin{equation*}
- \termt ::= \dots \separ \lcfix{x : \tya}{\termt}
-\end{equation*}
-\begin{equation*}
- \lcfix{x : \tya}{\termt} \bred \termt[(\lcfix{x : \tya}{\termt}) / x]
+ \termsyn ::= \dots \separ \lcfix{x : \tysyn}{\termsyn}
\end{equation*}
\begin{center}
\begin{prooftree}
- \AxiomC{$\Gamma,x : \tya \vdash \termt : \tya$}
+ \AxiomC{$\Gamma;x : \tya \vdash \termt : \tya$}
\UnaryInfC{$\Gamma \vdash \lcfix{x : \tya}{\termt} : \tya$}
\end{prooftree}
\end{center}
+\begin{equation*}
+ \lcfix{x : \tya}{\termt} \bred \termt[(\lcfix{x : \tya}{\termt}) ]
+\end{equation*}
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. Terms are
-proofs, and their types are the propositions they prove. This remarkable fact
-is known as the Curry-Howard isomorphism.
+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 $(\tya \tyarr \tyb) \tyarr (\tyb \tyarr \tyc) \tyarr (\tyc
+The `arrow' ($\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*}
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
-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.
+(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}\appspace}
\newcommand{\lcinr}{\mathsf{inr}\appspace}
-\newcommand{\lccase}[3]{\mathsf{case}\appspace#1\appspace#2\appspace#3}
+\newcommand{\lccase}[3]{\lcsyn{case}\appspace#1\appspace\lcsyn{of}\appspace#2\appspace#3}
\newcommand{\lcfst}{\mathsf{fst}\appspace}
\newcommand{\lcsnd}{\mathsf{snd}\appspace}
\newcommand{\orint}{\vee I_{1,2}}
\newcommand{\andel}{\wedge E_{1,2}}
\newcommand{\botel}{\bot E}
\newcommand{\lcabsurd}{\mathsf{absurd}\appspace}
+\newcommand{\lcabsurdd}[1]{\mathsf{absurd}_{#1}\appspace}
-\begin{eqnarray*}
- \termt & ::= & \dots \\
- & | & \lcinl \termt \separ \lcinr \termt \separ \lccase{\termt}{\termt}{\termt} \\
- & | & (\termt , \termt) \separ \lcfst \termt \separ \lcsnd \termt
-\end{eqnarray*}
-\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*}
-\begin{equation*}
- \tya ::= \dots \separ \tya \vee \tya \separ \tya \wedge \tya \separ \bot
-\end{equation*}
\begin{center}
+ \axname{syntax}
+ \begin{eqnarray*}
+ \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{eqnarray*}
+\end{center}
+\begin{center}
+ \axdesc{typing}{\Gamma \vdash \termsyn : \tysyn}
\begin{prooftree}
\AxiomC{$\Gamma \vdash \termt : \tya$}
\RightLabel{$\orint$}
\RightLabel{$\orel$}
\TrinaryInfC{$\Gamma \vdash \lccase{\termt}{\termm}{\termn} : \tyc$}
\end{prooftree}
- \begin{prooftree}
+
+ \begin{tabular}{c c}
\AxiomC{$\Gamma \vdash \termt : \tya$}
\AxiomC{$\Gamma \vdash \termm : \tyb$}
\RightLabel{$\andint$}
\BinaryInfC{$\Gamma \vdash (\tya , \tyb) : \tya \wedge \tyb$}
- \end{prooftree}
- \begin{prooftree}
+ \DisplayProof
+ &
\AxiomC{$\Gamma \vdash \termt : \tya \wedge \tyb$}
\RightLabel{$\andel$}
\UnaryInfC{$\Gamma \vdash \lcfst \termt : \tya$}
\noLine
\UnaryInfC{$\Gamma \vdash \lcsnd \termt : \tyb$}
- \end{prooftree}
- \begin{prooftree}
+ \DisplayProof
+ \end{tabular}
+
+ \vspace{0.5cm}
+
+ \begin{tabular}{c c}
\AxiomC{$\Gamma \vdash \termt : \bot$}
\RightLabel{$\botel$}
- \UnaryInfC{$\Gamma \vdash \lcabsurd \termt : \tya$}
- \end{prooftree}
+ \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).
+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.
+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]{\mathsf{if}\appspace#1\appspace\mathsf{then}\appspace#2\appspace\mathsf{else}\appspace#3}
+\newcommand{\lcite}[3]{\lcsyn{if}\appspace#1\appspace\lcsyn{then}\appspace#2\appspace\lcsyn{else}\appspace#3}
\newcommand{\lcbool}{\mathsf{Bool}}
+\newcommand{\lcforallz}[2]{\forall #1 \absspace.\absspace #2}
+\newcommand{\lcforall}[3]{\forall #1 : #2 \absspace.\absspace #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:
3 dimensions:
\begin{description}
\item[Terms depending on types (towards $\lambda{2}$)] In other words, we can
- quantify over types in our type signatures: $(\lambda A : \lctype. \lambda x :
- A. x) : \forall A. A \to A$. The first and most famous instance of this idea
+ 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: $(\lambda A : \lctype. \lambda R : \lctype. (A \to R) \to R) : \lctype \to \lctype \to \lctype$.
-\item[Types depending on terms (towards $\lambda{P}$)] Also known as ``dependent
- types'', give great expressive power: $(\lambda x :
- \lcbool. \lcite{x}{\mathbb{N}}{\mathbb{Q}}) : \lcbool \to \lctype$.
+ 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.
+where $\lambda{C}$ sits in the `$\lambda$-cube' above.
\section{Intuitionistic Type Theory}
-In this section I will describe
+\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 $\forall$ 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}.
+
+$\forall$ and $\exists$ are at the core of the machinery of ITT:
+
+\begin{description}
+\item[`forall' ($\forall$)] is a generalisation of $\tyarr$ in the STLC and
+ expresses universal quantification in our logic. 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 \citep{Pierce2000} will have to be deployed in a
+practical system. When showing examples obvious types will be omitted when this
+can be done without loss of clarity.
+
+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 $(
+
+\subsection{Base Types}
+\label{sec:base-types}
+
+\newcommand{\lctrue}{\mathsf{true}}
+\newcommand{\lcfalse}{\mathsf{false}}
+\newcommand{\lcw}[3]{\mathsf{W} #1 : #2 \absspace.\absspace #3}
+\newcommand{\lcnode}[4]{#1 \lhd_{#2 . #3} #4}
+\newcommand{\lcnodez}[2]{#1 \lhd #2}
+\newcommand{\lcited}[5]{\lcsyn{if}\appspace#1/#2\appspace.\appspace#3\appspace\lcsyn{then}\appspace#4\appspace\lcsyn{else}\appspace#5}
+\newcommand{\lcrec}[4]{\lcsyn{rec}\appspace#1/#2\appspace.\appspace#3\appspace\lcsyn{with}\appspace#4}
+\newcommand{\lcrecz}[2]{\lcsyn{rec}\appspace#1\appspace\lcsyn{with}\appspace#2}
+\newcommand{\AxiomL}[1]{\Axiom$\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 & ::= & ... \\
+ & | & \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{}
+ \RightLabel{$\lcbool I_{1,2}$}
+ \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. I will use some
+abbreviations and convenient syntax:
+\begin{itemize}
+ \item $\_\mathit{operator}\_$ to define infix operators
+ \item $\abs{\{x : \tya\}}{\dotsb}$ to define an abstraction that I will not
+ explicitly apply since the $x$ can be inferred easily.
+ \item $\abs{x\appspace y\appspace z : \tya}{\dotsb}$ to define multiple abstractions at the same
+ time
+ \item I will omit the explicit typing when forming $\exists$ or $\mathsf{W}$,
+ and when eliminating $\lcbool$, since the types are almost always clear and
+ writing them each time is extremely cumbersome.
+\end{itemize}
+
+\subsubsection{Sum types}
+
+We would like to recover our sum type, or disjunction, $\vee$. This is easily
+done with $\exists$:
+\begin{eqnarray*}
+ \_\vee\_ & = & \abs{\tya\appspace\tyb : \lcsetz}{\lcexists{x}{\lcbool}{\lcite{x}{\tya}{\tyb}}} \\
+ \lcinl & = & \abs{\{\tya\appspace\tyb : \lcsetz\}}{\abs{x : \tya \vee \tyb}{(\lctrue, x)}} \\
+ \lcinr & = & \abs{\{\tya\appspace\tyb : \lcsetz\}}{\abs{x : \tya \vee \tyb}{(\lcfalse, x)}} \\
+ \mathsf{case} & = & \abs{\{\tya\appspace\tyb\appspace\tyc : \lcsetz\}}{\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{eqnarray*}
+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 and similarly lists}
+
+Now it's time to showcase the power of $\mathsf{W}$ types.
+
+\begin{eqnarray*}
+ \mathsf{Nat} & = & \lcw{b}{\lcbool}{\abs{b}{\lcite{b}{\top}{\bot}}} \\
+ \mathsf{zero} & = & \lcfalse \lhd \abs{z}{\lcabsurd z} \\
+ \mathsf{suc} & = & \abs{n}{(\lctrue \lhd \abs{\_}{n})} \\
+ \mathsf{plus} & = & \abs{x\appspace y}{\lcrecz{x}{\abs{b}{\lcite{b}{\abs{\_\appspace f}{\app{\mathsf{suc}}{(\app{f}{\lcunit})}}}{\abs{\_\appspace\_}{y}}}}}
+\end{eqnarray*}
+
+\bibliographystyle{authordate1}
+\bibliography{background}
\end{document}
projects / bitonic-mengthesis.git / blobdiff