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%% -----------------------------------------------------------------------------
\begin{document}
-\begin{titlepage}
- \centering
+\pagenumbering{gobble}
+
+\begin{center}
+
+
+% Upper part of the page. The '~' is needed because \\
+% only works if a paragraph has started.
+\includegraphics[width=0.4\textwidth]{brouwer-cropped.png}~\\[1cm]
- \maketitle
- \thispagestyle{empty}
+\textsc{\Large Final year project}\\[0.5cm]
+
+% Title
+{ \huge \mykant: Implementing Observational Equality}\\[1.5cm]
+
+{\Large Francesco \textsc{Mazzoli} \href{mailto:fm2209@ic.ac.uk}{\nolinkurl{<fm2209@ic.ac.uk>}}}\\[0.8cm]
\begin{minipage}{0.4\textwidth}
\begin{flushleft} \large
\emph{Supervisor:}\\
- Dr. Steffen \textsc{van Backel}
+ Dr. Steffen \textsc{van Bakel}
\end{flushleft}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\begin{flushright} \large
- \emph{Co-marker:} \\
+ \emph{Second marker:} \\
Dr. Philippa \textsc{Gardner}
\end{flushright}
\end{minipage}
+\vfill
+
+% Bottom of the page
+{\large \today}
+
+\end{center}
+
+\clearpage
-\end{titlepage}
+\mbox{}
+\clearpage
\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.
+ The marriage between programming and logic has been a fertile one. In
+ particular, since the definition of the simply typed
+ $\lambda$-calculus, a number of type systems have been devised with
+ increasing expressive power.
- Among this systems, Inutitionistic Type Theory (ITT) has been a very
+ Among this systems, Intuitionistic Type Theory (ITT) has been a
popular framework for theorem provers and programming languages.
- However, equality has always been a tricky business in ITT and related
- theories.
-
- In these thesis we will explain why this is the case, and present
- Observational Type Theory (OTT), a solution to some of the problems
- with equality. We then describe $\mykant$, a theorem prover featuring
- OTT in a setting more close to the one found in current systems.
- Having implemented part of $\mykant$ as a Haskell program, we describe
- some of the implementation issues faced.
+ However, reasoning about equality has always been a tricky business in
+ ITT and related theories. In this thesis we shall explain why this is
+ the case, and present Observational Type Theory (OTT), a solution to
+ some of the problems with equality.
+
+ To bring OTT closer to the current practice of interactive theorem
+ provers, we describe \mykant, a system featuring OTT in a setting more
+ close to the one found in widely used provers such as Agda and Coq.
+ Most notably, we feature used defined inductive and record types and a
+ cumulative, implicit type hierarchy. Having implemented part of
+ $\mykant$ as a Haskell program, we describe some of the implementation
+ issues faced.
\end{abstract}
\clearpage
+\mbox{}
+\clearpage
+
\renewcommand{\abstractname}{Acknowledgements}
\begin{abstract}
- I would like to thank Steffen van Backel, my supervisor, who was brave
- enough to believe in my project and who provided much advice and
- support.
+ I would like to thank Steffen van Bakel, my supervisor, who was brave
+ enough to believe in my project and who provided support and
+ invaluable advice.
I would also like to thank the Haskell and Agda community on
\texttt{IRC}, which guided me through the strange world of types; and
in particular Andrea Vezzosi and James Deikun, with whom I entertained
- countless insightful discussions in the past year. Andrea suggested
+ countless insightful discussions over the past year. Andrea suggested
Observational Type Theory as a topic of study: this thesis would not
- exist without him. Before them, Tony Fields introduced me to Haskell,
+ exist without him. Before them, Tony Field introduced me to Haskell,
unknowingly filling most of my free time from that time on.
- Finally, much of the work stems from the research of Conor McBride,
+ Finally, most of the work stems from the research of Conor McBride,
who answered many of my doubts through these months. I also owe him
the colours.
\end{abstract}
+\clearpage
+\mbox{}
\clearpage
\tableofcontents
-\clearpage
+\section{Introduction}
+
+\pagenumbering{arabic}
+
+Functional programming is in good shape. In particular the `well-typed'
+line of work originating from Milner's ML has been extremely fruitful,
+in various directions. Nowadays functional, well-typed programming
+languages like Haskell or OCaml are slowly being absorbed by the
+mainstream. An important related development---and in fact the original
+motivator for ML's existence---is the advancement of the practice of
+\emph{interactive theorem provers}.
+
+
+An interactive theorem prover, or proof assistant, is a tool that lets
+the user develop formal proofs with the confidence of the machine
+checking them for correctness. While the effort towards a full
+formalisation of mathematics has been ongoing for more than a century,
+theorem provers have been the first class of software whose
+implementation depends directly on these theories.
+
+In a fortunate turn of events, it was discovered that well-typed
+functional programming and proving theorems in an \emph{intuitionistic}
+logic are the same activity. Under this discipline, the types in our
+programming language can be interpreted as proposition in our logic; and
+the programs implementing the specification given by the types as their
+proofs. This fact stimulated an active transfer of techniques and
+knowledge between logic and programming language theory, in both
+directions.
+
+Mathematics could provide programming with a wealth of abstractions and
+constructs developed over centuries. Moreover, identifying our types
+with a logic lets us focus on foundational questions regarding
+programming with a much more solid approach, given the years of rigorous
+study of logic. Programmers, on the other hand, had already developed a
+number of approaches to effectively collaborate with computers, through
+the study of programming languages.
+
+In this space, we shall follow the discipline of Intuitionistic Type
+Theory, or Martin-L\"{o}f Type Theory, after its inventor. First
+formulated in the 70s and then adjusted through a series of revisions,
+it has endured as the core of many practical systems widely in use
+today, and it is the most prominent instance of the proposition-as-types
+and proofs-as-programs paradigm. One of the most debated subjects in
+this field has been regarding what notion of equality should be
+exposed to the user.
+
+The tension when studying equality in type theory springs from the fact
+that there is a divide between what the user can prove equal
+\emph{inside} the theory---what is \emph{propositionally} equal---and
+what the theorem prover identifies as equal in its meta-theory---what is
+\emph{definitionally} equal. If we want our system to be well behaved
+(mostly if we want to keep type checking decidable) we must keep the two
+notions separate, with definitional equality inducing propositional
+equality, but not the reverse. However in this scenario propositional
+equality is weaker than we would like: we can only prove terms equal
+based on their syntactical structure, and not based on their behaviour.
+
+This thesis is concerned with exploring a new approach in this area,
+\emph{observational} equality. Promising to provide a more adequate
+propositional equality while retaining well-behavedness, it still is a
+relatively unexplored notion. We set ourselves to change that by
+studying it in a setting more akin to the one found in currently
+available theorem provers.
+
+\subsection{Structure}
+
+Section \ref{sec:types} will give a brief overview of the
+$\lambda$-calculus, both typed and untyped. This will give us the
+chance to introduce most of the concepts mentioned above rigorously, and
+gain some intuition about them. An excellent introduction to types in
+general can be found in \cite{Pierce2002}, although not from the
+perspective of theorem proving.
+
+Section \ref{sec:itt} will describe a set of basic construct that form a
+`baseline' Intuitionistic Type Theory. The goal is to familiarise with
+the main concept of ITT before attacking the problem of equality. Given
+the wealth of material covered the exposition is quite dense. Good
+introductions can be found in \cite{Thompson1991}, \cite{Nordstrom1990},
+and \cite{Martin-Lof1984} himself.
+
+Section \ref{sec:equality} will introduce propositional equality. The
+properties of propositional equality will be discussed along with its
+limitations. After reviewing some extensions to propositional equality,
+we will explain why identifying definitional equality with propositional
+equality causes problems.
+
+Section \ref{sec:ott} will introduce observational equality, following
+closely the original exposition by \cite{Altenkirch2007}. The
+presentation is free-standing but glosses over the meta-theoretic
+properties of OTT, focusing on the mechanisms that make it work.
+
+Section \ref{sec:kant-theory} is the central part of the thesis and will
+describe \mykant, a system we have developed incorporating OTT along
+constructs usually present in modern theorem provers. Along the way, we
+discuss these additional features and their trade-offs. Section
+\ref{sec:kant-practice} will describe an implementation implementing
+part of \mykant. A high level design of the software is given, along
+with a few specific implementation issues.
+
+Finally, Section \ref{sec:evaluation} will asses the decisions made in
+designing and implementing \mykant and the results achieved; and Section
+\ref{sec:future-work} will give a roadmap to bring \mykant\ on par and
+beyond the competition.
+
+\subsection{Contributions}
+\label{sec:contributions}
+
+The contribution of this thesis is threefold:
-\section{Simple and not-so-simple types}
-\label{sec:types}
+\begin{itemize}
+\item Provide a description of observational equality `in context', to
+ make the subject more accessible. Considering the possibilities that
+ OTT brings to the table, we think that introducing it to a wider
+ audience can only be beneficial.
+
+\item Fill in the gaps needed to make OTT work with user-defined
+ inductive types and a type hierarchy. We show how one notion of
+ equality is enough, instead of separate notions of value- and
+ type-equality as presented in the original paper. We are able to keep
+ the type equalities `small' while preserving subject reduction by
+ exploiting the fact that we work within a cumulative theory.
+ Incidentally, we also describe a generalised version of bidirectional
+ type checking for user defined types.
+
+\item Provide an implementation to probe the possibilities of OTT in a
+ more realistic setting. We have implemented an ITT with user defined
+ types but due to the limited time constraints we were not able to
+ complete the implementation of observational equality. Nonetheless,
+ we describe some interesting implementation issues faced by the type
+ theory implementor.
+\end{itemize}
-\subsection{The untyped $\lambda$-calculus}
+The system developed as part of this thesis, \mykant, incorporates OTT
+with features that are familiar to users of existing theorem provers
+adopting the proofs-as-programs mantra. The defining features of
+\mykant\ are:
+
+\begin{description}
+\item[Full dependent types] In ITT, types are very `first class' notion
+ and can be the result of computation---they can \emph{depend} on
+ values, thus the name \emph{dependent types}. \mykant\ espouses this
+ notion to its full consequences.
+
+\item[User defined data types and records] Instead of forcing the user
+ to choose from a restricted toolbox, we let her define types for
+ greater flexibility. We have two kinds of user defined types:
+ inductive data types, formed by various data constructors whose type
+ signatures can contain recursive occurrences of the type being
+ defined; and records, where we have just one data constructor and
+ projections to extract each each field in said constructor.
+
+\item[Consistency] Our system is meant to be consistent with respects to
+ the logic it embodies. For this reason, we restrict recursion to
+ \emph{structural} recursion on the defined inductive types, through
+ the use of operators (destructors) computing on each type. Following
+ the types-as-propositions interpretation, each destructor expresses an
+ induction principle on the data type it operates on. To achieve the
+ consistency of these operations we make sure that our recursive data
+ types are \emph{strictly positive}.
+
+\item[Bidirectional type checking] We take advantage of a
+ \emph{bidirectional} type inference system in the style of
+ \cite{Pierce2000}. This cuts down the type annotations by a
+ considerable amount in an elegant way and at a very low cost.
+ Bidirectional type checking is usually employed in core calculi, but
+ in \mykant\ we extend the concept to user defined data types.
+
+\item[Type hierarchy] In set theory we have to take treat powerset-like
+ objects with care, if we want to avoid paradoxes. However, the
+ working mathematician is rarely concerned by this, and the consistency
+ in this regard is implicitly assumed. In the tradition of
+ \cite{Russell1927}, in \mykant\ we employ a \emph{type hierarchy} to
+ make sure that these size issues are taken care of; and we employ
+ system so that the user will be free from thinking about the
+ hierarchy, just like the mathematician is.
+
+\item[Observational equality] The motivator of this thesis, \mykant\
+ incorporates a notion of observational equality, modifying the
+ original presentation by \cite{Altenkirch2007} to fit our more
+ expressive system. As mentioned, we reconcile OTT with user defined
+ types and a type hierarchy.
+
+\item[Type holes] When building up programs interactively, it is useful
+ to leave parts unfinished while exploring the current context. This
+ is what type holes are for.
+\end{description}
+
+\subsection{Notation and syntax}
-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. Here we give a brief overview of the
-language, which will give the chance to introduce concepts central to the
-analysis of all the following calculi. The exposition follows the one found in
-chapter 5 of \cite{Queinnec2003}.
+Appendix \ref{app:notation} describes the notation and syntax used in
+this thesis.
-The syntax of $\lambda$-terms consists of three things: variables, abstractions,
-and applications:
+\section{Simple and not-so-simple types}
+\label{sec:types}
+\epigraph{\emph{Well typed programs can't go wrong.}}{Robin Milner}
+
+\subsection{The untyped $\lambda$-calculus}
+\label{sec:untyped}
+
+Along with Turing's machines, the earliest attempts to formalise
+computation lead to the definition of 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. Here we give a brief overview of the language, which will
+give the chance to introduce concepts central to the analysis of all the
+following calculi. The exposition follows the one found in Chapter 5 of
+\cite{Queinnec2003}.
+
+\begin{mydef}[$\lambda$-terms]
+ Syntax of the $\lambda$-calculus: variables, abstractions, and
+ applications.
+\end{mydef}
+\mynegder
\mydesc{syntax}{ }{
$
\begin{array}{r@{\ }c@{\ }l}
$
}
-Parenthesis will be omitted in the usual way:
-$\myapp{\myapp{\mytmt}{\mytmm}}{\mytmn} =
-\myapp{(\myapp{\mytmt}{\mytmm})}{\mytmn}$.
+Parenthesis will be omitted in the usual way, with application being
+left associative.
Abstractions roughly corresponds to functions, and their semantics is more
-formally explained by the $\beta$-reduction rule:
+formally explained by the $\beta$-reduction rule.
+\begin{mydef}[$\beta$-reduction]
+$\beta$-reduction and substitution for the $\lambda$-calculus.
+\end{mydef}
+\mynegder
\mydesc{reduction:}{\mytmsyn \myred \mytmsyn}{
$
\begin{array}{l}
- \myapp{(\myabs{\myb{x}}{\mytmm})}{\mytmn} \myred \mysub{\mytmm}{\myb{x}}{\mytmn}\text{, where} \\
+ \myapp{(\myabs{\myb{x}}{\mytmm})}{\mytmn} \myred \mysub{\mytmm}{\myb{x}}{\mytmn}\text{ \textbf{where}} \\
\myind{2}
\begin{array}{l@{\ }c@{\ }l}
- \mysub{\myb{x}}{\myb{x}}{\mytmn} & = & \mytmn \\
- \mysub{\myb{y}}{\myb{x}}{\mytmn} & = & y\text{, with } \myb{x} \neq y \\
- \mysub{(\myapp{\mytmt}{\mytmm})}{\myb{x}}{\mytmn} & = & (\myapp{\mysub{\mytmt}{\myb{x}}{\mytmn}}{\mysub{\mytmm}{\myb{x}}{\mytmn}}) \\
- \mysub{(\myabs{\myb{x}}{\mytmm})}{\myb{x}}{\mytmn} & = & \myabs{\myb{x}}{\mytmm} \\
- \mysub{(\myabs{\myb{y}}{\mytmm})}{\myb{x}}{\mytmn} & = & \myabs{\myb{z}}{\mysub{\mysub{\mytmm}{\myb{y}}{\myb{z}}}{\myb{x}}{\mytmn}}, \\
- \multicolumn{3}{l}{\myind{2} \text{with $\myb{x} \neq \myb{y}$ and $\myb{z}$ not free in $\myapp{\mytmm}{\mytmn}$}}
+ \mysub{\myb{y}}{\myb{x}}{\mytmn} \mymetaguard \myb{x} = \myb{y} & \mymetagoes & \mytmn \\
+ \mysub{\myb{y}}{\myb{x}}{\mytmn} & \mymetagoes & \myb{y} \\
+ \mysub{(\myapp{\mytmt}{\mytmm})}{\myb{x}}{\mytmn} & \mymetagoes & (\myapp{\mysub{\mytmt}{\myb{x}}{\mytmn}}{\mysub{\mytmm}{\myb{x}}{\mytmn}}) \\
+ \mysub{(\myabs{\myb{x}}{\mytmm})}{\myb{x}}{\mytmn} & \mymetagoes & \myabs{\myb{x}}{\mytmm} \\
+ \mysub{(\myabs{\myb{y}}{\mytmm})}{\myb{x}}{\mytmn} & \mymetagoes & \myabs{\myb{z}}{\mysub{\mysub{\mytmm}{\myb{y}}{\myb{z}}}{\myb{x}}{\mytmn}} \\
+ \multicolumn{3}{l}{\myind{2} \text{\textbf{with} $\myb{x} \neq \myb{y}$ and $\myb{z}$ not free in $\myapp{\mytmm}{\mytmn}$}}
\end{array}
\end{array}
$
}
-The care required during substituting variables for terms is required to avoid
-name capturing. We will use substitution in the future for other name-binding
-constructs assuming similar precautions.
+The care required during substituting variables for terms is to avoid
+name capturing. We will use substitution in the future for other
+name-binding constructs assuming similar precautions.
-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):
-{\small
+These few elements have a remarkable expressiveness, and are in fact
+Turing complete. As a corollary, we must be able to devise a term that
+reduces forever (`loops' in imperative terms):
\[
- (\myapp{\omega}{\omega}) \myred (\myapp{\omega}{\omega}) \myred \cdots \text{, with $\omega = \myabs{x}{\myapp{x}{x}}$}
+ (\myapp{\omega}{\omega}) \myred (\myapp{\omega}{\omega}) \myred \cdots \text{, \textbf{where} $\omega = \myabs{x}{\myapp{x}{x}}$}
\]
-}
-A \emph{redex} is a term that can be reduced. In the untyped $\lambda$-calculus
-this will be the case for an application in which the first term is an
-abstraction, but in general we call aterm reducible if it appears to the left of
-a reduction rule. When a term contains no redexes it's said to be in
-\emph{normal form}. Given the observation above, not all terms reduce to a
-normal forms: we call the ones that do \emph{normalising}, and the ones that
-don't \emph{non-normalising}.
-
-The reduction rule presented is not syntax directed, but \emph{evaluation
- strategies} can be employed to reduce term systematically. Common evaluation
-strategies include \emph{call by value} (or \emph{strict}), where arguments of
-abstractions are reduced before being applied to the abstraction; and conversely
-\emph{call by name} (or \emph{lazy}), where we reduce only when we need to do so
-to proceed---in other words when we have an application where the function is
-still not a $\lambda$. In both these reduction strategies we never reduce under
-an abstraction: for this reason a weaker form of normalisation is used, where
-both abstractions and normal forms are said to be in \emph{weak head normal
- form}.
+\begin{mydef}[redex]
+ A \emph{redex} is a term that can be reduced.
+\end{mydef}
+In the untyped $\lambda$-calculus this will be the case for an
+application in which the first term is an abstraction, but in general we
+call a term reducible if it appears to the left of a reduction rule.
+\begin{mydef}[normal form]
+ A term that contains no redexes is said to be in \emph{normal form}.
+\end{mydef}
+\begin{mydef}[normalising terms and systems]
+ Terms that reduce in a finite number of reduction steps to a normal
+ form are \emph{normalising}. A system in which all terms are
+ normalising is said to have the \emph{normalisation property}, or
+ to be \emph{normalising}.
+\end{mydef}
+Given the reduction behaviour of $(\myapp{\omega}{\omega})$, it is clear
+that the untyped $\lambda$-calculus does not have the normalisation
+property.
+
+We have not presented reduction in an algorithmic way, but
+\emph{evaluation strategies} can be employed to reduce term
+systematically. Common evaluation strategies include \emph{call by
+ value} (or \emph{strict}), where arguments of abstractions are reduced
+before being applied to the abstraction; and conversely \emph{call by
+ name} (or \emph{lazy}), where we reduce only when we need to do so to
+proceed---in other words when we have an application where the function
+is still not a $\lambda$. In both these strategies we never
+reduce under an abstraction. For this reason a weaker form of
+normalisation is used, where all abstractions are said to be in
+\emph{weak head normal form} even if their body is not.
\subsection{The simply typed $\lambda$-calculus}
-A convenient way to `discipline' and reason about $\lambda$-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}. The first most basic instance
-of this idea takes the name of \emph{simply typed $\lambda$ calculus}, whose
-rules are shown in figure \ref{fig:stlc}.
+A convenient way to `discipline' and reason about $\lambda$-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}. The first
+most basic instance of this idea takes the name of \emph{simply typed
+ $\lambda$-calculus} (STLC).
+\begin{mydef}[Simply typed $\lambda$-calculus]
+ The syntax and typing rules for the STLC are given in Figure \ref{fig:stlc}.
+\end{mydef}
Our types contain a set of \emph{type variables} $\Phi$, which might
correspond to some `primitive' types; and $\myarr$, the type former for
\label{fig:stlc}
\end{figure}
-In the typing rules, a context $\myctx$ is used to store the types of bound
-variables: $\myctx; \myb{x} : \mytya$ adds a variable to the context and
-$\myctx(x)$ returns the type of the rightmost occurrence of $x$.
+In the typing rules, a context $\myctx$ is used to store the types of
+bound variables: $\myemptyctx$ is the empty context, and $\myctx;
+\myb{x} : \mytya$ adds a variable to the context. $\myctx(x)$ extracts
+the type of the rightmost occurrence of $x$.
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---it is either a variable, or its
- constructor does not appear on the left of the $\myred$ relation (currently
- only $\lambda$), or it can take a step according to the evaluation rules.
-\item[Preservation] If a well-typed term takes a step of evaluation, then the
- resulting term is also well-typed, and preserves the previous type. Also
- known as \emph{subject reduction}.
-\end{description}
+\begin{mydef}[Progress]
+ A well-typed term is not stuck---it is either a variable, or it does
+ not appear on the left of the $\myred$ relation , or it can take a
+ step according to the evaluation rules.
+\end{mydef}
+\begin{mydef}[Subject reduction]
+ If a well-typed term takes a step of evaluation, then the
+ resulting term is also well-typed, and preserves the previous type.
+\end{mydef}
However, STLC buys us much more: every well-typed term is normalising
-\citep{Tait1967}. It is easy to see that we can't fill the blanks if we want to
+\citep{Tait1967}. It is easy to see that we cannot fill the blanks if we want to
give types to the non-normalising term shown before:
-\begin{equation*}
+\[
\myapp{(\myabss{\myb{x}}{\myhole{?}}{\myapp{\myb{x}}{\myb{x}}})}{(\myabss{\myb{x}}{\myhole{?}}{\myapp{\myb{x}}{\myb{x}}})}
-\end{equation*}
-
+\]
This makes the STLC Turing incomplete. We can recover the ability to loop by
adding a combinator that recurses:
-
+\begin{mydef}[Fixed-point combinator]\end{mydef}
+\mynegder
\noindent
\begin{minipage}{0.5\textwidth}
\mydesc{syntax}{ } {
$ \mytmsyn ::= \cdots b \mysynsep \myfix{\myb{x}}{\mytysyn}{\mytmsyn} $
- \vspace{0.4cm}
+ \vspace{0.5cm}
}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\DisplayProof
}
\end{minipage}
-
+\mynegder
\mydesc{reduction:}{\myjud{\mytmsyn}{\mytmsyn}}{
$ \myfix{\myb{x}}{\mytya}{\mytmt} \myred \mysub{\mytmt}{\myb{x}}{(\myfix{\myb{x}}{\mytya}{\mytmt})}$
}
This will deprive us of normalisation, which is a particularly bad thing if we
want to use the STLC as described in the next section.
+Another important property of the STLC is the Church-Rosser property:
+\begin{mydef}[Church-Rosser property]
+ A system is said to have the \emph{Church-Rosser} property, or to be
+ \emph{confluent}, if given any two reductions $\mytmm$ and $\mytmn$ of
+ a given term $\mytmt$, there is exist a term to which both $\mytmm$
+ and $\mytmn$ can be reduced.
+\end{mydef}
+Given that the STLC has the normalisation property and the Church-Rosser
+property, each term has a \emph{unique} normal form.
+
\subsection{The Curry-Howard correspondence}
-It turns out that the STLC can be seen a natural deduction system for
-intuitionistic propositional logic. Terms are proofs, and their types are the
-propositions they prove. This remarkable fact is known as the Curry-Howard
+As hinted in the introduction, it turns out that the STLC can be seen a
+natural deduction system for intuitionistic propositional logic. Terms
+correspond to proofs, and their types correspond to the propositions
+they prove. This remarkable fact is known as the Curry-Howard
correspondence, or isomorphism.
The arrow ($\myarr$) type corresponds to implication. If we wish to prove that
that $(\mytya \myarr \mytyb) \myarr (\mytyb \myarr \mytycc) \myarr (\mytya
\myarr \mytycc)$, all we need to do is to devise a $\lambda$-term that has the
correct type:
-{\small\[
+\[
\myabss{\myb{f}}{(\mytya \myarr \mytyb)}{\myabss{\myb{g}}{(\mytyb \myarr \mytycc)}{\myabss{\myb{x}}{\mytya}{\myapp{\myb{g}}{(\myapp{\myb{f}}{\myb{x}})}}}}
-\]}
-That is, function composition. Going beyond arrow types, we can extend our bare
-lambda calculus with useful types to represent other logical constructs, as
-shown in figure \ref{fig:natded}.
+\]
+Which is known to functional programmers as function composition. Going
+beyond arrow types, we can extend our bare lambda calculus with useful
+types to represent other logical constructs.
+\begin{mydef}[The extended STLC]
+ Figure \ref{fig:natded} shows syntax, reduction, and typing rules for
+ the \emph{extended simply typed $\lambda$-calculus}.
+\end{mydef}
\begin{figure}[t]
\mydesc{syntax}{ }{
\DisplayProof
\end{tabular}
- \myderivsp
+ \myderivspp
\begin{tabular}{cc}
\AxiomC{$\myjud{\mytmt}{\mytya}$}
\end{tabular}
- \myderivsp
+ \myderivspp
\begin{tabular}{cc}
\AxiomC{$\myjud{\mytmm}{\mytya \myarr \mytyb}$}
\DisplayProof
\end{tabular}
- \myderivsp
+ \myderivspp
\begin{tabular}{ccc}
\AxiomC{$\myjud{\mytmm}{\mytya}$}
\DisplayProof
\end{tabular}
}
-\caption{Rules for the extendend STLC. Only the new features are shown, all the
+\caption{Rules for the extended STLC. Only the new features are shown, all the
rules and syntax for the STLC apply here too.}
\label{fig:natded}
\end{figure}
$\mypair{\myarg}{\myarg}$ corresponds to $\wedge$ introduction, $\myfst$
and $\mysnd$ to $\wedge$ elimination.
-The trivial type $\myunit$ corresponds to the logical $\top$, and dually
-$\myempty$ corresponds to the logical $\bot$. $\myunit$ has one introduction
-rule ($\mytt$), and thus one inhabitant; and no eliminators. $\myempty$ has no
-introduction rules, and thus no inhabitants; and one eliminator ($\myabsurd{
-}$), corresponding to the logical \emph{ex falso quodlibet}.
+The trivial type $\myunit$ corresponds to the logical $\top$ (true), and
+dually $\myempty$ corresponds to the logical $\bot$ (false). $\myunit$
+has one introduction rule ($\mytt$), and thus one inhabitant; and no
+eliminators---we cannot gain any information from a witness of the
+single member of $\myunit$. $\myempty$ has no introduction rules, and
+thus no inhabitants; and one eliminator ($\myabsurd{ }$), corresponding
+to the logical \emph{ex falso quodlibet}.
With these rules, our STLC now looks remarkably similar in power and use to the
-natural deduction we already know. $\myneg \mytya$ can be expressed as $\mytya
-\myarr \myempty$. However, there is an important omission: there is no term of
+natural deduction we already know.
+\begin{mydef}[Negation]
+ $\myneg \mytya$ can be expressed as $\mytya \myarr \myempty$.
+\end{mydef}
+However, there is an important omission: there is no term of
the type $\mytya \mysum \myneg \mytya$ (excluded middle), or equivalently
$\myneg \myneg \mytya \myarr \mytya$ (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
+This has a considerable effect on our logic and it is 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},
+Logics 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.}.
+the foundation of the STLC.\footnote{There is research to give computational
+ behaviour to classical logic, but I will not touch those subjects.}
As in logic, if we want to keep our system consistent, we must make sure that no
closed terms (in other words terms not under a $\lambda$) inhabit $\myempty$.
The variant of STLC presented here is indeed
consistent, a result that follows from the fact that it is
-normalising. % TODO explain
+normalising.
Going back to our $\mysyn{fix}$ combinator, it is easy to see how it ruins our
desire for consistency. The following term works for every type $\mytya$,
including bottom:
-{\small\[
-(\myfix{\myb{x}}{\mytya}{\myb{x}}) : \mytya
-\]}
+\[(\myfix{\myb{x}}{\mytya}{\myb{x}}) : \mytya\]
\subsection{Inductive data}
\label{sec:ind-data}
of a type former, various constructors, and an eliminator (or destructor) that
serves as primitive recursor.
-For example, we might add a $\mylist$ type constructor, along with an `empty
+\begin{mydef}[Finite lists for the STLC]
+We add a $\mylist$ type constructor, along with an `empty
list' ($\mynil{ }$) and `cons cell' ($\mycons$) constructor. The eliminator for
-lists will be the usual folding operation ($\myfoldr$). See figure
+lists will be the usual folding operation ($\myfoldr$). Full rules in Figure
\ref{fig:list}.
-
+\end{mydef}
+\mynegder
\begin{figure}[h]
\mydesc{syntax}{ }{
$
\BinaryInfC{$\myjud{\mytmm \mycons \mytmn}{\myapp{\mylist}{\mytya}}$}
\DisplayProof
\end{tabular}
- \myderivsp
+ \myderivspp
\AxiomC{$\myjud{\mysynel{f}}{\mytya \myarr \mytyb \myarr \mytyb}$}
\AxiomC{$\myjud{\mytmm}{\mytyb}$}
\label{fig:list}
\end{figure}
-In section \ref{sec:well-order} we will see how to give a general account of
-inductive data. %TODO does this make sense to have here?
+In Section \ref{sec:well-order} we will see how to give a general account of
+inductive data.
\section{Intuitionistic Type Theory}
\label{sec:itt}
-\subsection{Extending the STLC}
+\epigraph{\emph{Martin-L{\"o}f's type theory is a well established and
+ convenient arena in which computational Christians are regularly
+ fed to logical lions.}}{Conor McBride}
-The STLC can be made more expressive in various ways. \cite{Barendregt1991}
-succinctly expressed geometrically how we can add expressivity:
+\subsection{Extending the STLC}
+\cite{Barendregt1991} succinctly expressed geometrically how we can add
+expressively to the STLC:
$$
\xymatrix@!0@=1.5cm{
& \lambda\omega \ar@{-}[rr]\ar@{-}'[d][dd]
\begin{description}
\item[Terms depending on types (towards $\lambda{2}$)] We can quantify over
types in our type signatures. For example, we can define a polymorphic
- identity function:
- {\small\[\displaystyle
- (\myabss{\myb{A}}{\mytyp}{\myabss{\myb{x}}{\myb{A}}{\myb{x}}}) : (\myb{A} : \mytyp) \myarr \myb{A} \myarr \myb{A}
- \]}
- The first and most famous instance of this idea has been System F. This 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.
+ identity function, where $\mytyp$ denotes the `type of types':
+ \[\displaystyle
+ (\myabss{\myb{A}}{\mytyp}{\myabss{\myb{x}}{\myb{A}}{\myb{x}}}) : (\myb{A} {:} \mytyp) \myarr \myb{A} \myarr \myb{A}
+ \]
+ The first and most famous instance of this idea has been System F.
+ This 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 \citep{milner1978theory}. Languages like
+ Haskell and SML are based on this discipline. In Haskell the above
+ example would be
+ \begin{Verbatim}
+id :: a -> a
+id x = x
+ \end{Verbatim}
+ Where \texttt{a} implicitly quantifies over a type, and will be
+ instantiated automatically thanks to the inference.
\item[Types depending on types (towards $\lambda{\underline{\omega}}$)] We have
type operators. For example we could define a function that given types $R$
and $\mytya$ forms the type that represents a value of type $\mytya$ in
- continuation passing style: {\small\[\displaystyle(\myabss{\myb{A} \myar \myb{R}}{\mytyp}{(\myb{A}
- \myarr \myb{R}) \myarr \myb{R}}) : \mytyp \myarr \mytyp \myarr \mytyp\]}
+ continuation passing style:
+ \[\displaystyle(\myabss{\myb{R} \myarr \myb{A}}{\mytyp}{(\myb{A}
+ \myarr \myb{R}) \myarr \myb{R}}) : \mytyp \myarr \mytyp \myarr \mytyp
+ \]
+ In Haskell we can define type operator of sorts, although we must
+ pair them with data constructors, to keep inference manageable:
+ \begin{Verbatim}
+newtype Cont r a = Cont ((a -> r) -> r)
+ \end{Verbatim}
+ Where the `type' (kind in Haskell parlance) of \texttt{Cont} will be
+ \texttt{* -> * -> *}, with \texttt{*} signifying the type of types in
+ Haskell.
\item[Types depending on terms (towards $\lambda{P}$)] Also known as `dependent
types', give great expressive power. For example, we can have values of whose
type depend on a boolean:
- {\small\[\displaystyle(\myabss{\myb{x}}{\mybool}{\myite{\myb{x}}{\mynat}{\myrat}}) : \mybool
- \myarr \mytyp\]}
+ \[\displaystyle(\myabss{\myb{x}}{\mybool}{\myite{\myb{x}}{\mynat}{\myrat}}) : \mybool
+ \myarr \mytyp\] We cannot give an Haskell example that expresses this
+ concept since Haskell does not support dependent types---it would be a
+ very different language if it did.
\end{description}
All the systems preserve the properties that make the STLC well behaved. The
\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 inconsistent due to its
-impredicativity\footnote{In the early version there was only one universe
- $\mytyp$ and $\mytyp : \mytyp$, see section \ref{sec:term-types} for an
- explanation on why this causes problems.}. For this reason he gave a revised
-and consistent definition later \citep{Martin-Lof1984}.
+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
+inconsistent due to its impredicativity.\footnote{In the early version
+ there was only one universe $\mytyp$ and $\mytyp : \mytyp$; see
+ Section \ref{sec:term-types} for an explanation on why this causes
+ problems.} For this reason he later gave a revised and consistent
+definition \citep{Martin-Lof1984}.
A related development is the polymorphic $\lambda$-calculus, and specifically
the previously mentioned System F, which was developed independently by Girard
normalising. \cite{Coquand1986} further extended this line of work with the
Calculus of Constructions (CoC).
-Most widely used interactive theorem provers are based on ITT. Popular ones
-include Agda \citep{Norell2007, Bove2009}, Coq \citep{Coq}, and Epigram
-\citep{McBride2004, EpigramTut}.
-
-\subsection{A note on inference}
-
-% TODO do this, adding links to the sections about bidi type checking and
-% implicit universes.
-In the following text I will often omit explicit typing for abstractions or
-
-Moreover, I will use $\mytyp$ without bothering to specify a
-universe, with the silent assumption that the definition is consistent
-regarding to the hierarchy.
+Most widely used interactive theorem provers are based on ITT. Popular
+ones include Agda \citep{Norell2007}, Coq \citep{Coq}, Epigram
+\citep{McBride2004, EpigramTut}, Isabelle \citep{Paulson1990}, and many
+others.
\subsection{A simple 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}. The system's syntax and reduction
-rules are presented in their entirety in figure \ref{fig:core-tt-syn}.
-The typing rules are presented piece by piece. Agda and \mykant\
-renditions of the presented theory and all the examples is reproduced in
-appendix \ref{app:itt-code}.
+The calculus I present follows the exposition in \cite{Thompson1991},
+and is quite close to the original formulation of \cite{Martin-Lof1984}.
+Agda and \mykant\ renditions of the presented theory and all the
+examples (even the ones presented only as type signatures) are
+reproduced in Appendix \ref{app:itt-code}.
+\begin{mydef}[Intuitionistic Type Theory (ITT)]
+The syntax and reduction rules are shown in Figure \ref{fig:core-tt-syn}.
+The typing rules are presented piece by piece in the following sections.
+\end{mydef}
\begin{figure}[t]
\mydesc{syntax}{ }{
$
\begin{array}{r@{\ }c@{\ }l}
\mytmsyn & ::= & \myb{x} \mysynsep
- \mytyp_{l} \mysynsep
+ \mytyp_{level} \mysynsep
\myunit \mysynsep \mytt \mysynsep
\myempty \mysynsep \myapp{\myabsurd{\mytmsyn}}{\mytmsyn} \\
& | & \mybool \mysynsep \mytrue \mysynsep \myfalse \mysynsep
& | & \myw{\myb{x}}{\mytmsyn}{\mytmsyn} \mysynsep
\mytmsyn \mynode{\myb{x}}{\mytmsyn} \mytmsyn \\
& | & \myrec{\mytmsyn}{\myb{x}}{\mytmsyn}{\mytmsyn} \\
- l & \in & \mathbb{N}
+ level & \in & \mathbb{N}
\end{array}
$
}
$
\end{tabular}
- \myderivsp
+ \myderivspp
$
\myrec{(\myse{s} \mynode{\myb{x}}{\myse{T}} \myse{f})}{\myb{y}}{\myse{P}}{\myse{p}} \myred
While the usefulness of doing this will become clear soon, a consequence is
that since types can be the result of computation, deciding type equality is
-not immediate as in the STLC. For this reason we define \emph{definitional
+not immediate as in the STLC.
+\begin{mydef}[Definitional equality]
+ We define \emph{definitional
equality}, $\mydefeq$, as the congruence relation extending
-$\myred$---moreover, when comparing types syntactically we do it up to
-renaming of bound names ($\alpha$-renaming). For example under this
-discipline we will find that
-{\small\[
-\myabss{\myb{x}}{\mytya}{\myb{x}} \mydefeq \myabss{\myb{y}}{\mytya}{\myb{y}}
-\]}
+$\myred$. Moreover, when comparing types syntactically we do it up to
+renaming of bound names ($\alpha$-renaming)
+\end{mydef}
+For example under this discipline we will find that
+\[
+\begin{array}{@{}l}
+ \myabss{\myb{x}}{\mytya}{\myb{x}} \mydefeq \myabss{\myb{y}}{\mytya}{\myb{y}} \\
+ \myapp{(\myabss{\myb{f}}{\mytya \myarr \mytya}{\myb{f}})}{(\myabss{\myb{y}}{\mytya}{\myb{y}})} \mydefeq \myabss{\myb{quux}}{\mytya}{\myb{quux}}
+\end{array}
+\]
Types that are definitionally equal can be used interchangeably. Here
the `conversion' rule is not syntax directed, but it is possible to
-employ $\myred$ to decide term equality in a systematic way, by always
-reducing terms to their normal forms before comparing them, so that a
-separate conversion rule is not needed. % TODO add section
-Another thing to notice is that considering the need to reduce terms to
-decide equality, it is essential for a dependently type system to be
-terminating and confluent for type checking to be decidable.
+employ $\myred$ to decide term equality in a systematic way, comparing
+terms by reducing to their normal forms and then comparing them
+syntactically; so that a separate conversion rule is not needed.
+Another thing to notice is that, considering the need to reduce terms to
+decide equality, for type checking to be decidable a dependently typed
+must be terminating and confluent; since every type needs to have a
+unique normal form for definitional equality to be decidable.
Moreover, we specify a \emph{type hierarchy} to talk about `large'
types: $\mytyp_0$ will be the type of types inhabited by data:
inconsistent for much the same reason that impredicative na\"{\i}ve set
theory is \citep{Hurkens1995}. However various techniques can be
employed to lift the burden of explicitly handling universes, as we will
-see in section \ref{sec:term-hierarchy}.
+see in Section \ref{sec:term-hierarchy}.
\subsubsection{Contexts}
\UnaryInfC{$\myjud{\myfalse}{\mybool}$}
\DisplayProof
\end{tabular}
- \myderivsp
+ \myderivspp
\AxiomC{$\myjud{\mytmt}{\mybool}$}
\AxiomC{$\myjudd{\myctx : \mybool}{\mytya}{\mytyp_l}$}
}
With booleans we get the first taste of the `dependent' in `dependent
-types'. While the two introduction rules ($\mytrue$ and $\myfalse$) are
-not surprising, the typing rules for $\myfun{if}$ are. In most strongly
-typed languages we expect the branches of an $\myfun{if}$ statements to
-be of the same type, to preserve subject reduction, since execution
-could take both paths. This is a pity, since the type system does not
-reflect the fact that in each branch we gain knowledge on the term we
-are branching on. Which means that programs along the lines of
-{\small\[\text{\texttt{if null xs then head xs else 0}}\]}
-are a necessary, well typed, danger.
+types'. While the two introduction rules for $\mytrue$ and $\myfalse$
+are not surprising, the typing rules for $\myfun{if}$ are. In most
+strongly typed languages we expect the branches of an $\myfun{if}$
+statements to be of the same type, to preserve subject reduction, since
+execution could take both paths. This is a pity, since the type system
+does not reflect the fact that in each branch we gain knowledge on the
+term we are branching on. Which means that programs along the lines of
+\begin{Verbatim}
+if null xs then head xs else 0
+\end{Verbatim}
+are a necessary, well-typed, danger.
However, in a more expressive system, we can do better: the branches' type can
depend on the value of the scrutinised boolean. This is what the typing rule
expresses: the user provides a type $\mytya$ ranging over an $\myb{x}$
-representing the scrutinised boolean type, and the branches are typechecked with
-the updated knowledge on the value of $\myb{x}$.
+representing the scrutinised boolean type, and the branches are type checked with
+the updated knowledge of the value of $\myb{x}$.
\subsubsection{$\myarr$, or dependent function}
+\label{sec:depprod}
\mydesc{typing:}{\myjud{\mytmsyn}{\mytmsyn}}{
\AxiomC{$\myjud{\mytya}{\mytyp_{l_1}}$}
\BinaryInfC{$\myjud{\myfora{\myb{x}}{\mytya}{\mytyb}}{\mytyp_{l_1 \mylub l_2}}$}
\DisplayProof
- \myderivsp
+ \myderivspp
\begin{tabular}{cc}
\AxiomC{$\myjudd{\myctx; \myb{x} : \mytya}{\mytmt}{\mytyb}$}
\end{tabular}
}
-Dependent functions are one of the two key features that perhaps most
-characterise dependent types---the other being dependent products. With
-dependent functions, the result type can depend on the value of the
-argument. This feature, together with the fact that the result type
-might be a type itself, brings a lot of interesting possibilities.
-Following this intuition, in the introduction rule, the return type is
-typechecked in a context with an abstracted variable of lhs' type, and
-in the elimination rule the actual argument is substituted in the return
-type. Keeping the correspondence with logic alive, dependent functions
-are much like universal quantifiers ($\forall$) in logic.
+Dependent functions are one of the two key features that characterise
+dependent types---the other being dependent products. With dependent
+functions, the result type can depend on the value of the argument.
+This feature, together with the fact that the result type might be a
+type itself, brings a lot of interesting possibilities. In the
+introduction rule, the return type is type checked in a context with an
+abstracted variable of domain's type; and in the elimination rule the
+actual argument is substituted in the return type. Keeping the
+correspondence with logic alive, dependent functions are much like
+universal quantifiers ($\forall$) in logic.
For example, assuming that we have lists and natural numbers in our
-language, using dependent functions we would be able to
-write:
-{\small\[
+language, using dependent functions we can write functions of
+types
+\[
\begin{array}{l}
\myfun{length} : (\myb{A} {:} \mytyp_0) \myarr \myapp{\mylist}{\myb{A}} \myarr \mynat \\
\myarg \myfun{$>$} \myarg : \mynat \myarr \mynat \myarr \mytyp_0 \\
\myfun{head} : (\myb{A} {:} \mytyp_0) \myarr (\myb{l} {:} \myapp{\mylist}{\myb{A}})
- \myarr \myapp{\myapp{\myfun{length}}{\myb{A}}}{\myb{l}} \mathrel{\myfun{>}} 0 \myarr
+ \myarr \myapp{\myapp{\myfun{length}}{\myb{A}}}{\myb{l}} \mathrel{\myfun{$>$}} 0 \myarr
\myb{A}
\end{array}
-\]}
+\]
-\myfun{length} is the usual polymorphic length function. $\myfun{>}$ is
-a function that takes two naturals and returns a type: if the lhs is
-greater then the rhs, $\myunit$ is returned, $\myempty$ otherwise. This
-way, we can express a `non-emptyness' condition in $\myfun{head}$, by
-including a proof that the length of the list argument is non-zero.
-This allows us to rule out the `empty list' case, so that we can safely
-return the first element.
+\myfun{length} is the usual polymorphic length
+function. $\myarg\myfun{$>$}\myarg$ is a function that takes two naturals
+and returns a type: if the lhs is greater then the rhs, $\myunit$ is
+returned, $\myempty$ otherwise. This way, we can express a
+`non-emptiness' condition in $\myfun{head}$, by including a proof that
+the length of the list argument is non-zero. This allows us to rule out
+the `empty list' case, so that we can safely return the first element.
-Again, we need to make sure that the type hierarchy is respected, which is the
-reason why a type formed by $\myarr$ will live in the least upper bound of the
-levels of argument and return type. This trend will continue with the other
-type-level binders, $\myprod$ and $\mytyc{W}$.
+Finally, we need to make sure that the type hierarchy is respected, which
+is the reason why a type formed by $\myarr$ will live in the least upper
+bound of the levels of argument and return type.
\subsubsection{$\myprod$, or dependent product}
\label{sec:disju}
\BinaryInfC{$\myjud{\myexi{\myb{x}}{\mytya}{\mytyb}}{\mytyp_{l_1 \mylub l_2}}$}
\DisplayProof
- \myderivsp
+ \myderivspp
\begin{tabular}{cc}
\AxiomC{$\myjud{\mytmm}{\mytya}$}
If dependent functions are a generalisation of $\myarr$ in the STLC,
dependent products are a generalisation of $\myprod$ in the STLC. The
improvement is that the second element's type can depend on the value of
-the first element. The corrispondence with logic is through the
+the first element. The correspondence with logic is through the
existential quantifier: $\exists x \in \mathbb{N}. even(x)$ can be
expressed as $\myexi{\myb{x}}{\mynat}{\myapp{\myfun{even}}{\myb{x}}}$.
The first element will be a number, and the second evidence that the
number is even. This highlights the fact that we are working in a
constructive logic: if we have an existence proof, we can always ask for
a witness. This means, for instance, that $\neg \forall \neg$ is not
-equivalent to $\exists$.
+equivalent to $\exists$. Additionally, we need to specify the type of
+the second element (ranging over the first element) explicitly when
+using $\mypair{\myarg}{\myarg}$.
Another perhaps more `dependent' application of products, paired with
$\mybool$, is to offer choice between different types. For example we
can easily recover disjunctions:
-{\small\[
+\[
\begin{array}{l}
\myarg\myfun{$\vee$}\myarg : \mytyp_0 \myarr \mytyp_0 \myarr \mytyp_0 \\
\myb{A} \mathrel{\myfun{$\vee$}} \myb{B} \mapsto \myexi{\myb{x}}{\mybool}{\myite{\myb{x}}{\myb{A}}{\myb{B}}} \\ \ \\
\myfun{case} : (\myb{A}\ \myb{B}\ \myb{C} {:} \mytyp_0) \myarr (\myb{A} \myarr \myb{C}) \myarr (\myb{B} \myarr \myb{C}) \myarr \myb{A} \mathrel{\myfun{$\vee$}} \myb{B} \myarr \myb{C} \\
\myfun{case} \myappsp \myb{A} \myappsp \myb{B} \myappsp \myb{C} \myappsp \myb{f} \myappsp \myb{g} \myappsp \myb{x} \mapsto \\
- \myind{2} \myapp{(\myitee{\myapp{\myfst}{\myb{b}}}{\myb{x}}{(\myite{\myb{b}}{\myb{A}}{\myb{B}})}{\myb{f}}{\myb{g}})}{(\myapp{\mysnd}{\myb{x}})}
+ \myind{2} \myapp{(\myitee{\myapp{\myfst}{\myb{x}}}{\myb{b}}{(\myite{\myb{b}}{\myb{A}}{\myb{B}})}{\myb{f}}{\myb{g}})}{(\myapp{\mysnd}{\myb{x}})}
\end{array}
-\]}
+\]
\subsubsection{$\mytyc{W}$, or well-order}
\label{sec:well-order}
\mydesc{typing:}{\myjud{\mytmsyn}{\mytmsyn}}{
+ \begin{tabular}{cc}
\AxiomC{$\myjud{\mytya}{\mytyp_{l_1}}$}
\AxiomC{$\myjudd{\myctx;\myb{x} : \mytya}{\mytyb}{\mytyp_{l_2}}$}
\BinaryInfC{$\myjud{\myw{\myb{x}}{\mytya}{\mytyb}}{\mytyp_{l_1 \mylub l_2}}$}
\DisplayProof
- \myderivsp
+ &
\AxiomC{$\myjud{\mytmt}{\mytya}$}
\AxiomC{$\myjud{\mysynel{f}}{\mysub{\mytyb}{\myb{x}}{\mytmt} \myarr \myw{\myb{x}}{\mytya}{\mytyb}}$}
\BinaryInfC{$\myjud{\mytmt \mynode{\myb{x}}{\mytyb} \myse{f}}{\myw{\myb{x}}{\mytya}{\mytyb}}$}
\DisplayProof
+ \end{tabular}
- \myderivsp
+ \myderivspp
\AxiomC{$\myjud{\myse{u}}{\myw{\myb{x}}{\myse{S}}{\myse{T}}}$}
\AxiomC{$\myjudd{\myctx; \myb{w} : \myw{\myb{x}}{\myse{S}}{\myse{T}}}{\myse{P}}{\mytyp_l}$}
}
Finally, the well-order type, or in short $\mytyc{W}$-type, which will
-let us represent inductive data in a general (but clumsy) way. The core
-idea is to
-
-% TODO finish
-
+let us represent inductive data in a general way. We can form `nodes'
+of the shape \[\mytmt \mynode{\myb{x}}{\mytyb} \myse{f} :
+\myw{\myb{x}}{\mytya}{\mytyb}\] where $\mytmt$ is of type $\mytya$ and
+is the data present in the node, and $\myse{f}$ specifies a `child' of
+the node for each member of $\mysub{\mytyb}{\myb{x}}{\mytmt}$. The
+$\myfun{rec}\ \myfun{with}$ acts as an induction principle on
+$\mytyc{W}$, given a predicate and a function dealing with the inductive
+case---we will gain more intuition about inductive data in Section
+\ref{sec:user-type}.
+
+For example, if we want to form natural numbers, we can take
+\[
+ \begin{array}{@{}l}
+ \mytyc{Tr} : \mybool \myarr \mytyp_0 \\
+ \mytyc{Tr} \myappsp \myb{b} \mapsto \myfun{if}\, \myb{b}\, \myfun{then}\, \myunit\, \myfun{else}\, \myempty \\
+ \ \\
+ \mynat : \mytyp_0 \\
+ \mynat \mapsto \myw{\myb{b}}{\mybool}{(\mytyc{Tr}\myappsp\myb{b})}
+ \end{array}
+\]
+Each node will contain a boolean. If $\mytrue$, the number is non-zero,
+and we will have one child representing its predecessor, given that
+$\mytyc{Tr}$ will return $\myunit$. If $\myfalse$, the number is zero,
+and we will have no predecessors (children), given the $\myempty$:
+\[
+ \begin{array}{@{}l}
+ \mydc{zero} : \mynat \\
+ \mydc{zero} \mapsto \myfalse \mynodee (\myabs{\myb{x}}{\myabsurd{\mynat} \myappsp \myb{x}}) \\
+ \ \\
+ \mydc{suc} : \mynat \myarr \mynat \\
+ \mydc{suc}\myappsp \myb{x} \mapsto \mytrue \mynodee (\myabs{\myarg}{\myb{x}})
+ \end{array}
+\]
+And with a bit of effort, we can recover addition:
+\[
+ \begin{array}{@{}l}
+ \myfun{plus} : \mynat \myarr \mynat \myarr \mynat \\
+ \myfun{plus} \myappsp \myb{x} \myappsp \myb{y} \mapsto \\
+ \myind{2} \myfun{rec}\, \myb{x} / \myb{b}.\mynat \, \\
+ \myind{2} \myfun{with}\, \myabs{\myb{b}}{\\
+ \myind{2}\myind{2}\myfun{if}\, \myb{b} / \myb{b'}.((\mytyc{Tr} \myappsp \myb{b'} \myarr \mynat) \myarr (\mytyc{Tr} \myappsp \myb{b'} \myarr \mynat) \myarr \mynat) \\
+ \myind{2}\myind{2}\myfun{then}\,(\myabs{\myarg\, \myb{f}}{\mydc{suc}\myappsp (\myapp{\myb{f}}{\mytt})})\, \myfun{else}\, (\myabs{\myarg\, \myarg}{\myb{y}})}
+ \end{array}
+ \]
+ Note how we explicitly have to type the branches to make them match
+ with the definition of $\mynat$. This gives a taste of the clumsiness
+ of $\mytyc{W}$-types but not the whole story. Well-orders are
+ inadequate not only because they are verbose, but also because they
+ face deeper problems due to the weakness of the notion of equality
+ present in most type theories, which we will present in the next
+ section \citep{dybjer1997representing}. The `better' equality we will
+ present in Section \ref{sec:ott} helps but does not fully resolve
+ these issues.\footnote{See \url{http://www.e-pig.org/epilogue/?p=324},
+ which concludes with `W-types are a powerful conceptual tool, but
+ they’re no basis for an implementation of recursive data types in
+ decidable type theories.'} For this reasons \mytyc{W}-types have
+ remained nothing more than a reasoning tool, and practical systems
+ must implement more expressive ways to represent data.
\section{The struggle for equality}
\label{sec:equality}
-In the previous section we saw how a type checker (or a human) needs a
-notion of \emph{definitional equality}. Beyond this meta-theoretic
+\epigraph{\emph{Half of my time spent doing research involves thinking up clever
+ schemes to avoid needing functional extensionality.}}{@larrytheliquid}
+
+In the previous section we learnt how a type checker for ITT needs
+a notion of \emph{definitional equality}. Beyond this meta-theoretic
notion, in this section we will explore the ways of expressing equality
\emph{inside} the theory, as a reasoning tool available to the user.
This area is the main concern of this thesis, and in general a very
active research topic, since we do not have a fully satisfactory
solution, yet. As in the previous section, everything presented is
-formalised in Agda in appendix \ref{app:agda-itt}.
+formalised in Agda in Appendix \ref{app:agda-itt}.
\subsection{Propositional equality}
+\begin{mydef}[Propositional equality] The syntax, reduction, and typing
+ rules for propositional equality and related constructs are defined
+ as:
+\end{mydef}
+\mynegder
\noindent
\begin{minipage}{0.5\textwidth}
\mydesc{syntax}{ }{
$
\begin{array}{r@{\ }c@{\ }l}
\mytmsyn & ::= & \cdots \\
- & | & \mytmsyn \mypeq{\mytmsyn} \mytmsyn \mysynsep
+ & | & \mypeq \myappsp \mytmsyn \myappsp \mytmsyn \myappsp \mytmsyn \mysynsep
\myapp{\myrefl}{\mytmsyn} \\
& | & \myjeq{\mytmsyn}{\mytmsyn}{\mytmsyn}
\end{array}
}
\end{minipage}
\begin{minipage}{0.5\textwidth}
-\mydesc{reduction:}{\mytmsyn \myred \mytmsyn}{
+\mydesc{\phantom{y}reduction:}{\mytmsyn \myred \mytmsyn}{
$
\myjeq{\myse{P}}{(\myapp{\myrefl}{\mytmm})}{\mytmn} \myred \mytmn
$
- \vspace{0.9cm}
+ \vspace{1.05cm}
}
\end{minipage}
-
+\mynegder
\mydesc{typing:}{\myjud{\mytmsyn}{\mytmsyn}}{
\AxiomC{$\myjud{\mytya}{\mytyp_l}$}
\AxiomC{$\myjud{\mytmm}{\mytya}$}
\AxiomC{$\myjud{\mytmn}{\mytya}$}
- \TrinaryInfC{$\myjud{\mytmm \mypeq{\mytya} \mytmn}{\mytyp_l}$}
+ \TrinaryInfC{$\myjud{\mypeq \myappsp \mytya \myappsp \mytmm \myappsp \mytmn}{\mytyp_l}$}
\DisplayProof
- \myderivsp
+ \myderivspp
\begin{tabular}{cc}
\AxiomC{$\begin{array}{c}\ \\\myjud{\mytmm}{\mytya}\hspace{1.1cm}\mytmm \mydefeq \mytmn\end{array}$}
- \UnaryInfC{$\myjud{\myapp{\myrefl}{\mytmm}}{\mytmm \mypeq{\mytya} \mytmn}$}
+ \UnaryInfC{$\myjud{\myapp{\myrefl}{\mytmm}}{\mypeq \myappsp \mytya \myappsp \mytmm \myappsp \mytmn}$}
\DisplayProof
&
\AxiomC{$
\begin{array}{c}
- \myjud{\myse{P}}{\myfora{\myb{x}\ \myb{y}}{\mytya}{\myfora{q}{\myb{x} \mypeq{\mytya} \myb{y}}{\mytyp_l}}} \\
- \myjud{\myse{q}}{\mytmm \mypeq{\mytya} \mytmn}\hspace{1.1cm}\myjud{\myse{p}}{\myapp{\myapp{\myapp{\myse{P}}{\mytmm}}{\mytmm}}{(\myapp{\myrefl}{\mytmm})}}
+ \myjud{\myse{P}}{\myfora{\myb{x}\ \myb{y}}{\mytya}{\myfora{q}{\mypeq \myappsp \mytya \myappsp \myb{x} \myappsp \myb{y}}{\mytyp_l}}} \\
+ \myjud{\myse{q}}{\mypeq \myappsp \mytya \myappsp \mytmm \myappsp \mytmn}\hspace{1.1cm}\myjud{\myse{p}}{\myapp{\myapp{\myapp{\myse{P}}{\mytmm}}{\mytmm}}{(\myapp{\myrefl}{\mytmm})}}
\end{array}
$}
\UnaryInfC{$\myjud{\myjeq{\myse{P}}{\myse{q}}{\myse{p}}}{\myapp{\myapp{\myapp{\myse{P}}{\mytmm}}{\mytmn}}{q}}$}
\DisplayProof
\end{tabular}
}
+\ \\
-To express equality between two terms inside ITT, the obvious way to do so is
-to have the equality construction to be a type-former. Here we present what
-has survived as the dominating form of equality in systems based on ITT up to
-the present day.
+To express equality between two terms inside ITT, the obvious way to do
+so is to have equality to be a type. Here we present what has survived
+as the dominating form of equality in systems based on ITT up since
+\cite{Martin-Lof1984} up to the present day.
-Our type former is $\mypeq{\mytya}$, which given a type (in this case
-$\mytya$) relates equal terms of that type. $\mypeq{}$ has one introduction
-rule, $\myrefl$, which introduces an equality relation between definitionally
-equal terms.
+Our type former is $\mypeq$, which given a type relates equal terms of
+that type. $\mypeq$ has one introduction rule, $\myrefl$, which
+introduces an equality relation between definitionally equal terms.
-Finally, we have one eliminator for $\mypeq{}$, $\myjeqq$. $\myjeq{\myse{P}}{\myse{q}}{\myse{p}}$ takes
+Finally, we have one eliminator for $\mypeq$ , $\myjeqq$ (also known as
+`\myfun{J} axiom' in the literature).
+$\myjeq{\myse{P}}{\myse{q}}{\myse{p}}$ takes
\begin{itemize}
\item $\myse{P}$, a predicate working with two terms of a certain type (say
- $\mytya$) and a proof of their equality
+ $\mytya$) and a proof of their equality;
\item $\myse{q}$, a proof that two terms in $\mytya$ (say $\myse{m}$ and
- $\myse{n}$) are equal
-\item and $\myse{p}$, an inhabitant of $\myse{P}$ applied to $\myse{m}$, plus
- the trivial proof by reflexivity showing that $\myse{m}$ is equal to itself
+ $\myse{n}$) are equal;
+\item and $\myse{p}$, an inhabitant of $\myse{P}$ applied to $\myse{m}$
+ twice, plus the trivial proof by reflexivity showing that $\myse{m}$
+ is equal to itself.
\end{itemize}
-Given these ingredients, $\myjeqq$ retuns a member of $\myse{P}$ applied to
-$\mytmm$, $\mytmn$, and $\myse{q}$. In other words $\myjeqq$ takes a
-witness that $\myse{P}$ works with \emph{definitionally equal} terms, and
-returns a witness of $\myse{P}$ working with \emph{propositionally equal}
-terms. Invokations of $\myjeqq$ will vanish when the equality proofs will
-reduce to invocations to reflexivity, at which point the arguments must be
-definitionally equal, and thus the provided
+Given these ingredients, $\myjeqq$ returns a member of $\myse{P}$
+applied to $\mytmm$, $\mytmn$, and $\myse{q}$. In other words $\myjeqq$
+takes a witness that $\myse{P}$ works with \emph{definitionally equal}
+terms, and returns a witness of $\myse{P}$ working with
+\emph{propositionally equal} terms. Given its reduction rules,
+invocations of $\myjeqq$ will vanish when the equality proofs will
+reduce to invocations to reflexivity, at which point the arguments must
+be definitionally equal, and thus the provided
$\myapp{\myapp{\myapp{\myse{P}}{\mytmm}}{\mytmm}}{(\myapp{\myrefl}{\mytmm})}$
-can be returned.
+can be returned. This means that $\myjeqq$ will not compute with
+hypothetical proofs, which makes sense given that they might be false.
-While the $\myjeqq$ rule is slightly convoluted, ve can derive many more
+While the $\myjeqq$ rule is slightly convoluted, we can derive many more
`friendly' rules from it, for example a more obvious `substitution' rule, that
replaces equal for equal in predicates:
-{\small\[
+\[
\begin{array}{l}
-\myfun{subst} : \myfora{\myb{A}}{\mytyp}{\myfora{\myb{P}}{\myb{A} \myarr \mytyp}{\myfora{\myb{x}\ \myb{y}}{\myb{A}}{\myb{x} \mypeq{\myb{A}} \myb{y} \myarr \myapp{\myb{P}}{\myb{x}} \myarr \myapp{\myb{P}}{\myb{y}}}}} \\
+\myfun{subst} : \myfora{\myb{A}}{\mytyp}{\myfora{\myb{P}}{\myb{A} \myarr \mytyp}{\myfora{\myb{x}\ \myb{y}}{\myb{A}}{\mypeq \myappsp \myb{A} \myappsp \myb{x} \myappsp \myb{y} \myarr \myapp{\myb{P}}{\myb{x}} \myarr \myapp{\myb{P}}{\myb{y}}}}} \\
\myfun{subst}\myappsp \myb{A}\myappsp\myb{P}\myappsp\myb{x}\myappsp\myb{y}\myappsp\myb{q}\myappsp\myb{p} \mapsto
\myjeq{(\myabs{\myb{x}\ \myb{y}\ \myb{q}}{\myapp{\myb{P}}{\myb{y}}})}{\myb{p}}{\myb{q}}
\end{array}
-\]}
-Once we have $\myfun{subst}$, we can easily prove more familiar laws regarding
-equality, such as symmetry, transitivity, and a congruence law.
-
-% TODO finish this
+\]
+Once we have $\myfun{subst}$, we can easily prove more familiar laws
+regarding equality, such as symmetry, transitivity, congruence laws,
+etc.\footnote{For definitions of these functions, refer to Appendix \ref{app:itt-code}.}
\subsection{Common extensions}
-Our definitional equality can be made larger in various ways, here we
-review some common extensions.
-
-\subsubsection{Congruence laws and $\eta$-expansion}
-
-A simple type-directed check that we can do on functions and records is
-$\eta$-expansion. We can then have
-
+Our definitional and propositional equalities can be enhanced in various
+ways. Obviously if we extend the definitional equality we are also
+automatically extend propositional equality, given how $\myrefl$ works.
+
+\subsubsection{$\eta$-expansion}
+\label{sec:eta-expand}
+
+A simple extension to our definitional equality is achieved by $\eta$-expansion.
+Given an abstract variable $\myb{f} : \mytya \myarr \mytyb$ the aim is
+to have that $\myb{f} \mydefeq
+\myabss{\myb{x}}{\mytya}{\myapp{\myb{f}}{\myb{x}}}$. We can achieve
+this by `expanding' terms depending on their types, a process known as
+\emph{quotation}---a term borrowed from the practice of
+\emph{normalisation by evaluation}, where we embed terms in some host
+language with an existing notion of computation, and then reify them
+back into terms, which will `smooth out' differences like the one above
+\citep{Abel2007}.
+
+The same concept applies to $\myprod$, where we expand each inhabitant
+reconstructing it by getting its projections, so that $\myb{x}
+\mydefeq \mypair{\myfst \myappsp \myb{x}}{\mysnd \myappsp \myb{x}}$.
+Similarly, all one inhabitants of $\myunit$ and all zero inhabitants of
+$\myempty$ can be considered equal. Quotation can be performed in a
+type-directed way, as we will witness in Section \ref{sec:kant-irr}.
+
+\begin{mydef}[Congruence and $\eta$-laws]
+ To justify quotation in our type system we add a congruence law for
+ abstractions and a similar law for products, plus the fact that all
+ elements of $\myunit$ or $\myempty$ are equal.
+\end{mydef}
+\mynegder
\mydesc{definitional equality:}{\myjud{\mytmm \mydefeq \mytmn}{\mytmsyn}}{
\begin{tabular}{cc}
- \AxiomC{$\myjud{f \mydefeq (\myabss{\myb{x}}{\mytya}{\myapp{\myse{g}}{\myb{x}}})}{\myfora{\myb{x}}{\mytya}{\mytyb}}$}
+ \AxiomC{$\myjudd{\myctx; \myb{y} : \mytya}{\myapp{\myse{f}}{\myb{x}} \mydefeq \myapp{\myse{g}}{\myb{x}}}{\mysub{\mytyb}{\myb{x}}{\myb{y}}}$}
\UnaryInfC{$\myjud{\myse{f} \mydefeq \myse{g}}{\myfora{\myb{x}}{\mytya}{\mytyb}}$}
\DisplayProof
&
- \AxiomC{$\myjud{\mytmm \mydefeq \mypair{\myapp{\myfst}{\mytmn}}{\myapp{\mysnd}{\mytmn}}}{\myexi{\myb{x}}{\mytya}{\mytyb}}$}
+ \AxiomC{$\myjud{\mypair{\myapp{\myfst}{\mytmm}}{\myapp{\mysnd}{\mytmm}} \mydefeq \mypair{\myapp{\myfst}{\mytmn}}{\myapp{\mysnd}{\mytmn}}}{\myexi{\myb{x}}{\mytya}{\mytyb}}$}
\UnaryInfC{$\myjud{\mytmm \mydefeq \mytmn}{\myexi{\myb{x}}{\mytya}{\mytyb}}$}
\DisplayProof
\end{tabular}
- \myderivsp
+ \myderivspp
+ \begin{tabular}{cc}
\AxiomC{$\myjud{\mytmm}{\myunit}$}
\AxiomC{$\myjud{\mytmn}{\myunit}$}
\BinaryInfC{$\myjud{\mytmm \mydefeq \mytmn}{\myunit}$}
\DisplayProof
+ &
+ \AxiomC{$\myjud{\mytmm}{\myempty}$}
+ \AxiomC{$\myjud{\mytmn}{\myempty}$}
+ \BinaryInfC{$\myjud{\mytmm \mydefeq \mytmn}{\myempty}$}
+ \DisplayProof
+ \end{tabular}
}
-% TODO finish
-
\subsubsection{Uniqueness of identity proofs}
-% TODO finish
-% TODO reference the fact that J does not imply J
-% TODO mention univalence
+Another common but controversial addition to propositional equality is
+the $\myfun{K}$ axiom, which essentially states that all equality proofs
+are by reflexivity.
-
-\mydesc{definitional equality:}{\myjud{\mytmm \mydefeq \mytmn}{\mytmsyn}}{
+\begin{mydef}[$\myfun{K}$ axiom]\end{mydef}
+\mydesc{typing:}{\myjud{\mytmm \mydefeq \mytmn}{\mytmsyn}}{
\AxiomC{$
\begin{array}{@{}c}
- \myjud{\myse{P}}{\myfora{\myb{x}}{\mytya}{\myb{x} \mypeq{\mytya} \myb{x} \myarr \mytyp}} \\\
- \myjud{\myse{p}}{\myfora{\myb{x}}{\mytya}{\myse{P} \myappsp \myb{x} \myappsp \myb{x} \myappsp (\myrefl \myapp \myb{x})}} \hspace{1cm}
+ \myjud{\myse{P}}{\myfora{\myb{x}}{\mytya}{\mypeq \myappsp \mytya \myappsp \myb{x}\myappsp \myb{x} \myarr \mytyp}} \\\
\myjud{\mytmt}{\mytya} \hspace{1cm}
+ \myjud{\myse{p}}{\myse{P} \myappsp \mytmt \myappsp (\myrefl \myappsp \mytmt)} \hspace{1cm}
\myjud{\myse{q}}{\mytmt \mypeq{\mytya} \mytmt}
\end{array}
$}
- \UnaryInfC{$\myjud{\myfun{K} \myappsp \myse{P} \myappsp \myse{p} \myappsp \myse{t} \myappsp \myse{q}}{\myse{P} \myappsp \mytmt \myappsp \myse{q}}$}
+ \UnaryInfC{$\myjud{\myfun{K} \myappsp \myse{P} \myappsp \myse{t} \myappsp \myse{p} \myappsp \myse{q}}{\myse{P} \myappsp \mytmt \myappsp \myse{q}}$}
\DisplayProof
}
-\subsection{Limitations}
+\cite{Hofmann1994} showed that $\myfun{K}$ is not derivable from
+$\myjeqq$, and \cite{McBride2004} showed that it is needed to implement
+`dependent pattern matching', as first proposed by \cite{Coquand1992}.\footnote{See Section \ref{sec:future-work} for more on dependent pattern matching.}
+Thus, $\myfun{K}$ is derivable in the systems that implement dependent
+pattern matching, such as Epigram and Agda; but for example not in Coq.
+
+$\myfun{K}$ is controversial mainly because it is at odds with
+equalities that include computational behaviour, most notably
+Voevodsky's \emph{Univalent Foundations}, which feature a \emph{univalence}
+axiom that identifies isomorphisms between types with propositional
+equality. For example we would have two isomorphisms, and thus two
+equalities, between $\mybool$ and $\mybool$, corresponding to the two
+permutations---one is the identity, and one swaps the elements. Given
+this, $\myfun{K}$ and univalence are inconsistent, and thus a form of
+dependent pattern matching that does not imply $\myfun{K}$ is subject of
+research.\footnote{More information about univalence can be found at
+ \url{http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations.html}.}
-\epigraph{\emph{Half of my time spent doing research involves thinking up clever
- schemes to avoid needing functional extensionality.}}{@larrytheliquid}
+\subsection{Limitations}
-However, propositional equality as described is quite restricted when
+Propositional equality as described is quite restricted when
reasoning about equality beyond the term structure, which is what definitional
-equality gives us (extension notwithstanding).
+equality gives us (extensions notwithstanding).
+
+\begin{mydef}[Extensional equality]
+Given two functions $\myse{f}$ and $\myse{g}$ of type $\mytya \myarr \mytyb$, they are are said to be \emph{extensionally equal} if
+\[ (\myb{x} {:} \mytya) \myarr \mypeq \myappsp \mytyb \myappsp (\myse{f} \myappsp \myb{x}) \myappsp (\myse{g} \myappsp \myb{x}) \]
+\end{mydef}
The problem is best exemplified by \emph{function extensionality}. In
-mathematics, we would expect to be able to treat functions that give equal
-output for equal input as the same. When reasoning in a mechanised framework
-we ought to be able to do the same: in the end, without considering the
-operational behaviour, all functions equal extensionally are going to be
-replaceable with one another.
-
-However this is not the case, or in other words with the tools we have we have
-no term of type
-{\small\[
+mathematics, we would expect to be able to treat functions that give
+equal output for equal input as equal. When reasoning in a mechanised
+framework we ought to be able to do the same: in the end, without
+considering the operational behaviour, all functions equal extensionally
+are going to be replaceable with one another.
+
+However this is not the case, or in other words with the tools we have there is no closed term of type
+\[
\myfun{ext} : \myfora{\myb{A}\ \myb{B}}{\mytyp}{\myfora{\myb{f}\ \myb{g}}{
\myb{A} \myarr \myb{B}}{
- (\myfora{\myb{x}}{\myb{A}}{\myapp{\myb{f}}{\myb{x}} \mypeq{\myb{B}} \myapp{\myb{g}}{\myb{x}}}) \myarr
- \myb{f} \mypeq{\myb{A} \myarr \myb{B}} \myb{g}
+ (\myfora{\myb{x}}{\myb{A}}{\mypeq \myappsp \myb{B} \myappsp (\myapp{\myb{f}}{\myb{x}}) \myappsp (\myapp{\myb{g}}{\myb{x}})}) \myarr
+ \mypeq \myappsp (\myb{A} \myarr \myb{B}) \myappsp \myb{f} \myappsp \myb{g}
}
}
-\]}
+\]
To see why this is the case, consider the functions
-{\small\[\myabs{\myb{x}}{0 \mathrel{\myfun{+}} \myb{x}}$ and $\myabs{\myb{x}}{\myb{x} \mathrel{\myfun{+}} 0}\]}
-where $\myfun{+}$ is defined by recursion on the first argument,
+\[\myabs{\myb{x}}{0 \mathrel{\myfun{$+$}} \myb{x}}$ and $\myabs{\myb{x}}{\myb{x} \mathrel{\myfun{$+$}} 0}\]
+where $\myfun{$+$}$ is defined by recursion on the first argument,
gradually destructing it to build up successors of the second argument.
The two functions are clearly extensionally equal, and we can in fact
prove that
-{\small\[
-\myfora{\myb{x}}{\mynat}{(0 \mathrel{\myfun{+}} \myb{x}) \mypeq{\mynat} (\myb{x} \mathrel{\myfun{+}} 0)}
-\]}
-By analysis on the $\myb{x}$. However, the two functions are not
-definitionally equal, and thus we won't be able to get rid of the
-quantification.
-
-For the reasons above, theories that offer a propositional equality
-similar to what we presented are called \emph{intensional}, as opposed
-to \emph{extensional}. Most systems in wide use today (such as Agda,
-Coq, and Epigram) are of this kind.
+\[
+\myfora{\myb{x}}{\mynat}{\mypeq \myappsp \mynat \myappsp (0 \mathrel{\myfun{$+$}} \myb{x}) \myappsp (\myb{x} \mathrel{\myfun{$+$}} 0)}
+\]
+By induction on $\mynat$ applied to $\myb{x}$. However, the two
+functions are not definitionally equal, and thus we will not be able to get
+rid of the quantification.
-This is quite an annoyance that often makes reasoning awkward to execute. It
-also extends to other fields, for example proving bisimulation between
-processes specified by coinduction, or in general proving equivalences based
-on the behaviour on a term.
+For the reasons given above, theories that offer a propositional equality
+similar to what we presented are called \emph{intensional}, as opposed
+to \emph{extensional}. Most systems widely used today (such as Agda,
+Coq, and Epigram) are of the former kind.
+
+This is quite an annoyance that often makes reasoning awkward or
+impossible to execute. For example, we might want to represent terms of
+some language in Agda and give their denotation by embedding them in
+Agda---if we had $\lambda$-terms, functions will be Agda functions,
+application will be Agda's function application, and so on. Then we
+would like to perform optimisation passes on the terms, and verify that
+they are sound by proving that the denotation of the optimised version
+is equal to the denotation of the starting term.
+
+But if the embedding uses functions---and it probably will---we are
+stuck with an equality that identifies as equal only syntactically equal
+functions! Since the point of optimising is about preserving the
+denotational but changing the operational behaviour of terms, our
+equality falls short of our needs. Moreover, the problem extends to
+other fields beyond functions, such as bisimulation between processes
+specified by coinduction, or in general proving equivalences based on
+the behaviour of a term.
\subsection{Equality reflection}
-One way to `solve' this problem is by identifying propositional equality with
-definitional equality:
+One way to `solve' this problem is by identifying propositional equality
+with definitional equality.
+\begin{mydef}[Equality reflection]\end{mydef}
\mydesc{typing:}{\myjud{\mytmsyn}{\mytmsyn}}{
- \AxiomC{$\myjud{\myse{q}}{\mytmm \mypeq{\mytya} \mytmn}$}
+ \AxiomC{$\myjud{\myse{q}}{\mypeq \myappsp \mytya \myappsp \mytmm \myappsp \mytmn}$}
\UnaryInfC{$\myjud{\mytmm \mydefeq \mytmn}{\mytya}$}
\DisplayProof
}
-This rule takes the name of \emph{equality reflection}, and is a very
-different rule from the ones we saw up to now: it links a typing judgement
-internal to the type theory to a meta-theoretic judgement that the type
-checker uses to work with terms. It is easy to see the dangerous consequences
-that this causes:
+The \emph{equality reflection} rule is a very different rule from the
+ones we saw up to now: it links a typing judgement internal to the type
+theory to a meta-theoretic judgement that the type checker uses to work
+with terms. It is easy to see the dangerous consequences that this
+causes:
\begin{itemize}
-\item The rule is syntax directed, and the type checker is presumably expected
- to come up with equality proofs when needed.
+\item The rule is not syntax directed, and the type checker is
+ presumably expected to come up with equality proofs when needed.
\item More worryingly, type checking becomes undecidable also because
- computing under false assumptions becomes unsafe, since we can use
- equality reflection and the conversion rule to have terms of any
- type.
- Consider for example {\small\[ \myabss{\myb{q}}{\mytya
- \mypeq{\mytyp} (\mytya \myarr \mytya)}{\myhole{?}}
- \]}
-Using the assumed proof in tandem with equality reflection we
-could easily write a classic Y combinator, sending the compiler into a
-loop. In general, we using the conversion rule
-% TODO check that this makes sense
+ computing under false assumptions becomes unsafe, since we derive any
+ equality proof and then use equality reflection and the conversion
+ rule to have terms of any type.
\end{itemize}
Given these facts theories employing equality reflection, like NuPRL
\citep{NuPRL}, carry the derivations that gave rise to each typing judgement
-to keep the systems manageable. % TODO more info, problems with that.
+to keep the systems manageable.
For all its faults, equality reflection does allow us to prove extensionality,
using the extensions we gave above. Assuming that $\myctx$ contains
-{\small\[\myb{A}, \myb{B} : \mytyp; \myb{f}, \myb{g} : \myb{A} \myarr \myb{B}; \myb{q} : \myfora{\myb{x}}{\myb{A}}{\myapp{\myb{f}}{\myb{x}} \mypeq{} \myapp{\myb{g}}{\myb{x}}}\]}
+\[\myb{A}, \myb{B} : \mytyp; \myb{f}, \myb{g} : \myb{A} \myarr \myb{B}; \myb{q} : \myfora{\myb{x}}{\myb{A}}{\myapp{\myb{f}}{\myb{x}} \mypeq{} \myapp{\myb{g}}{\myb{x}}}\]
We can then derive
\begin{prooftree}
- \small
- \AxiomC{$\hspace{1.1cm}\myjudd{\myctx; \myb{x} : \myb{A}}{\myapp{\myb{q}}{\myb{x}}}{\myapp{\myb{f}}{\myb{x}} \mypeq{} \myapp{\myb{g}}{\myb{x}}}\hspace{1.1cm}$}
+ \mysmall
+ \AxiomC{$\myjudd{\myctx; \myb{x} : \myb{A}}{\myb{q}}{\mypeq \myappsp \myb{A} \myappsp (\myapp{\myb{f}}{\myb{x}}) \myappsp (\myapp{\myb{g}}{\myb{x}})}$}
\RightLabel{equality reflection}
\UnaryInfC{$\myjudd{\myctx; \myb{x} : \myb{A}}{\myapp{\myb{f}}{\myb{x}} \mydefeq \myapp{\myb{g}}{\myb{x}}}{\myb{B}}$}
\RightLabel{congruence for $\lambda$s}
\UnaryInfC{$\myjud{(\myabs{\myb{x}}{\myapp{\myb{f}}{\myb{x}}}) \mydefeq (\myabs{\myb{x}}{\myapp{\myb{g}}{\myb{x}}})}{\myb{A} \myarr \myb{B}}$}
\RightLabel{$\eta$-law for $\lambda$}
- \UnaryInfC{$\hspace{1.45cm}\myjud{\myb{f} \mydefeq \myb{g}}{\myb{A} \myarr \myb{B}}\hspace{1.45cm}$}
+ \UnaryInfC{$\myjud{\myb{f} \mydefeq \myb{g}}{\myb{A} \myarr \myb{B}}$}
\RightLabel{$\myrefl$}
- \UnaryInfC{$\myjud{\myapp{\myrefl}{\myb{f}}}{\myb{f} \mypeq{} \myb{g}}$}
+ \UnaryInfC{$\myjud{\myapp{\myrefl}{\myb{f}}}{\mypeq \myappsp (\myb{A} \myarr \myb{B}) \myappsp \myb{f} \myappsp \myb{g}}$}
\end{prooftree}
-
-Now, the question is: do we need to give up well-behavedness of our theory to
+For this reason, theories employing equality reflection are often
+grouped under the name of \emph{Extensional Type Theory} (ETT). Now,
+the question is: do we need to give up well-behavedness of our theory to
gain extensionality?
-\subsection{Some alternatives}
-
-% TODO finish
-% TODO add `extentional axioms' (Hoffman), setoid models (Thorsten)
-
-\section{Observational equality}
+\section{The observational approach}
\label{sec:ott}
A recent development by \citet{Altenkirch2007}, \emph{Observational Type
Theory} (OTT), promises to keep the well behavedness of ITT while
-being able to gain many useful equality proofs\footnote{It is suspected
+being able to gain many useful equality proofs,\footnote{It is suspected
that OTT gains \emph{all} the equality proofs of ETT, but no proof
- exists yet.}, including function extensionality. The main idea is to
+ exists yet.} including function extensionality. The main idea is to
give the user the possibility to \emph{coerce} (or transport) values
from a type $\mytya$ to a type $\mytyb$, if the type checker can prove
-structurally that $\mytya$ and $\mytya$ are equal; and providing a
+structurally that $\mytya$ and $\mytyb$ are equal; and providing a
value-level equality based on similar principles. Here we give an
exposition which follows closely the original paper.
\subsection{A simpler theory, a propositional fragment}
+\begin{mydef}[OTT's simple theory, with propositions]\ \end{mydef}
+\mynegder
\mydesc{syntax}{ }{
$\mytyp_l$ is replaced by $\mytyp$. \\\ \\
$
$
}
+\mynegder
+
+\mydesc{reduction:}{\mytmsyn \myred \mytmsyn}{
+ $
+ \begin{array}{l@{}l@{\ }c@{\ }l}
+ \myITE{\mytrue &}{\mytya}{\mytyb} & \myred & \mytya \\
+ \myITE{\myfalse &}{\mytya}{\mytyb} & \myred & \mytyb
+ \end{array}
+ $
+}
+
+\mynegder
+
\mydesc{typing:}{\myjud{\mytmsyn}{\mytmsyn}}{
\begin{tabular}{cc}
\AxiomC{$\myjud{\myse{P}}{\myprop}$}
\end{tabular}
}
+\mynegder
+
\mydesc{propositions:}{\myjud{\myprsyn}{\myprop}}{
\begin{tabular}{ccc}
\AxiomC{\phantom{$\myjud{\myse{P}}{\myprop}$}}
\end{tabular}
}
-Our foundation will be a type theory like the one of section
+Our foundation will be a type theory like the one of Section
\ref{sec:itt}, with only one level: $\mytyp_0$. In this context we will
drop the $0$ and call $\mytyp_0$ $\mytyp$. Moreover, since the old
$\myfun{if}\myarg\myfun{then}\myarg\myfun{else}$ was able to return
one.
However, we have an addition: a universe of \emph{propositions},
-$\myprop$. $\myprop$ isolates a fragment of types at large, and
-indeed we can `inject' any $\myprop$ back in $\mytyp$ with $\myprdec{\myarg}$: \\
+$\myprop$.\footnote{Note that we do not need syntax for the type of props, $\myprop$, since the user cannot abstract over them. In fact, we do not not need syntax for $\mytyp$ either, for the same reason.} $\myprop$ isolates a fragment of types at large, and
+indeed we can `inject' any $\myprop$ back in $\mytyp$ with $\myprdec{\myarg}$.
+\begin{mydef}[Proposition decoding]\ \end{mydef}
\mydesc{proposition decoding:}{\myprdec{\mytmsyn} \myred \mytmsyn}{
\begin{tabular}{cc}
$
\end{tabular}
} \\
Propositions are what we call the types of \emph{proofs}, or types
- whose inhabitants contain no `data', much like $\myunit$. The goal of
- doing this is twofold: erasing all top-level propositions when
- compiling; and to identify all equivalent propositions as the same, as
- we will see later.
+ whose inhabitants contain no `data', much like $\myunit$. The goal
+ when isolating \mytyc{Prop} is twofold: erasing all top-level
+ propositions when compiling; and identifying all equivalent
+ propositions as the same, as we will see later.
Why did we choose what we have in $\myprop$? Given the above
- criteria, $\mytop$ obviously fits the bill. A pair of propositions
- $\myse{P} \myand \myse{Q}$ still won't get us data. Finally, if
- $\myse{P}$ is a proposition and we have
- $\myprfora{\myb{x}}{\mytya}{\myse{P}}$ , the decoding will be a
- function which returns propositional content. The only threat is
- $\mybot$, by which we can fabricate anything we want: however if we
- are consistent there will be nothing of type $\mybot$ at the top
- level, which is what we care about regarding proof erasure.
+ criteria, $\mytop$ obviously fits the bill, since it has one element.
+ A pair of propositions $\myse{P} \myand \myse{Q}$ still won't get us
+ data, since if they both have one element the only possible pair is
+ the one formed by said elements. Finally, if $\myse{P}$ is a
+ proposition and we have $\myprfora{\myb{x}}{\mytya}{\myse{P}}$, the
+ decoding will be a constant function for propositional content. The
+ only threat is $\mybot$, by which we can fabricate anything we want:
+ however if we are consistent there will be no closed term of type
+ $\mybot$ at, which is enough regarding proof erasure and
+ term equality.
+
+ As an example of types that are \emph{not} propositional, consider
+ $\mydc{Bool}$eans, which are the quintessential `relevant' data, since
+ they are often used to decide the execution path of a program through
+ $\myfun{if}\myarg\myfun{then}\myarg\myfun{else}\myarg$ constructs.
\subsection{Equality proofs}
+\begin{mydef}[Equality proofs and related operations]\ \end{mydef}
+\mynegder
\mydesc{syntax}{ }{
$
\begin{array}{r@{\ }c@{\ }l}
While isolating a propositional universe as presented can be a useful
exercises on its own, what we are really after is a useful notion of
-equality. In OTT we want to maintain the notion that things judged to
-be equal are still always repleaceable for one another with no
-additional changes. Note that this is not the same as saying that they
-are definitionally equal, since as we saw extensionally equal functions,
-while satisfying the above requirement, are not definitionally equal.
+equality. In OTT we want to maintain that things judged to be equal are
+still always replaceable for one another with no additional
+changes. Note that this is not the same as saying that they are
+definitionally equal, since as we saw extensionally equal functions,
+while satisfying the above requirement, are not.
Towards this goal we introduce two equality constructs in
$\myprop$---the fact that they are in $\myprop$ indicates that they
will still be the same.
Before introducing the core ideas that make OTT work, let us distinguish
-between \emph{canonical} and \emph{neutral} types. Canonical types are
-those arising from the ground types ($\myempty$, $\myunit$, $\mybool$)
-and the three type formers ($\myarr$, $\myprod$, $\mytyc{W}$). Neutral
-types are those formed by
-$\myfun{If}\myarg\myfun{Then}\myarg\myfun{Else}\myarg$.
-Correspondingly, canonical terms are those inhabiting canonical types
-($\mytt$, $\mytrue$, $\myfalse$, $\myabss{\myb{x}}{\mytya}{\mytmt}$,
-...), and neutral terms those formed by eliminators\footnote{Using the
- terminology from section \ref{sec:types}, we'd say that canonical
- terms are in \emph{weak head normal form}.}. In the current system
-(and hopefully in well-behaved systems), all closed terms reduce to a
-canonical term, and all canonical types are inhabited by canonical
-terms.
+between \emph{canonical} and \emph{neutral} terms and types.
+
+\begin{mydef}[Canonical and neutral types and terms]
+ In a type theory, \emph{neutral} terms are those formed by an
+ abstracted variable or by an eliminator (including function
+ application). Everything else is \emph{canonical}.
+
+ In the current system, data constructors ($\mytt$, $\mytrue$,
+ $\myfalse$, $\myabss{\myb{x}}{\mytya}{\mytmt}$, ...) will be
+ canonical, the rest neutral. Correspondingly, canonical types are
+ those arising from the ground types ($\myempty$, $\myunit$, $\mybool$)
+ and the three type formers ($\myarr$, $\myprod$, $\mytyc{W}$).
+ Neutral types are those formed by
+ $\myfun{If}\myarg\myfun{Then}\myarg\myfun{Else}\myarg$.
+\end{mydef}
+\begin{mydef}[Canonicity]
+ If in a system all canonical types are inhabited by canonical terms
+ the system is said to have the \emph{canonicity} property.
+\end{mydef}
+The current system, and well-behaved systems in general, has the
+canonicity property. Another consequence of normalisation is that all
+closed terms will reduce to a canonical term.
\subsubsection{Type equality, and coercions}
The plan is to decompose type-level equalities between canonical types
into decodable propositions containing equalities regarding the
-subterms, and to use coerce recursively on the subterms using the
-generated equalities. This interplay between type equalities and
-\myfun{coe} ensures that invocations of $\myfun{coe}$ will vanish when
-we have evidence of the structural equality of the types we are
-transporting terms across. If the type is neutral, the equality won't
-reduce and thus $\myfun{coe}$ won't reduce either. If we come an
-equality between different canonical types, then we reduce the equality
-to bottom, making sure that no such proof can exist, and providing an
-`escape hatch' in $\myfun{coe}$.
+subterms. So if are equating two product types, the equality will
+reduce to two subequalities regarding the first and second type. Then,
+we can \myfun{coe}rce to transport values between equal types.
+Following the subequalities, \myfun{coe} will procede recursively on the
+subterms.
+
+This interplay between the canonicity of equated types, type
+equalities, and \myfun{coe}, ensures that invocations of $\myfun{coe}$
+will vanish when we have evidence of the structural equality of the
+types we are transporting terms across. If the type is neutral, the
+equality will not reduce and thus $\myfun{coe}$ will not reduce either.
+If we come across an equality between different canonical types, then we
+reduce the equality to bottom, making sure that no such proof can exist,
+and providing an `escape hatch' in $\myfun{coe}$.
\begin{figure}[t]
\mybool & \myeq & \mybool & \myred \mytop \\
\myexi{\myb{x_1}}{\mytya_1}{\mytyb_1} & \myeq & \myexi{\myb{x_2}}{\mytya_2}{\mytya_2} & \myred \\
\multicolumn{4}{l}{
- \myind{2} \mytya_1 \myeq \mytyb_1 \myand
+ \myind{2} \mytya_1 \myeq \mytya_2 \myand
\myprfora{\myb{x_1}}{\mytya_1}{\myprfora{\myb{x_2}}{\mytya_2}{\myjm{\myb{x_1}}{\mytya_1}{\myb{x_2}}{\mytya_2}} \myimpl \mytyb_1[\myb{x_1}] \myeq \mytyb_2[\myb{x_2}]}
} \\
\myfora{\myb{x_1}}{\mytya_1}{\mytyb_1} & \myeq & \myfora{\myb{x_2}}{\mytya_2}{\mytyb_2} & \myred \cdots \\
$
\begin{array}[t]{@{}l@{\ }l@{\ }l@{\ }l@{\ }l@{\ }c@{\ }l@{\ }}
\mycoe & \myempty & \myempty & \myse{Q} & \myse{t} & \myred & \myse{t} \\
- \mycoe & \myunit & \myunit & \myse{Q} & \mytt & \myred & \mytt \\
+ \mycoe & \myunit & \myunit & \myse{Q} & \myse{t} & \myred & \mytt \\
\mycoe & \mybool & \mybool & \myse{Q} & \mytrue & \myred & \mytrue \\
\mycoe & \mybool & \mybool & \myse{Q} & \myfalse & \myred & \myfalse \\
\mycoe & (\myexi{\myb{x_1}}{\mytya_1}{\mytyb_1}) &
& \myb{\mytmn_1} & \mapsto & \myapp{\mysnd}{\mytmt_1} : \mysub{\mytyb_1}{\myb{x_1}}{\myb{\mytmm_1}} \\
& \myb{Q_A} & \mapsto & \myapp{\myfst}{\myse{Q}} : \mytya_1 \myeq \mytya_2 \\
& \myb{\mytmm_2} & \mapsto & \mycoee{\mytya_1}{\mytya_2}{\myb{Q_A}}{\myb{\mytmm_1}} : \mytya_2 \\
- & \myb{Q_B} & \mapsto & (\myapp{\mysnd}{\myse{Q}}) \myappsp \myb{\mytmm_1} \myappsp \myb{\mytmm_2} \myappsp (\mycohh{\mytya_1}{\mytya_2}{\myb{Q_A}}{\myb{\mytmm_1}}) : \\ & & & \myprdec{\mysub{\mytyb_1}{\myb{x_1}}{\myb{\mytmm_1}} \myeq \mysub{\mytyb_2}{\myb{x_2}}{\myb{\mytmm_2}}} \\
+ & \myb{Q_B} & \mapsto & (\myapp{\mysnd}{\myse{Q}}) \myappsp \myb{\mytmm_1} \myappsp \myb{\mytmm_2} \myappsp (\mycohh{\mytya_1}{\mytya_2}{\myb{Q_A}}{\myb{\mytmm_1}}) : \myprdec{\mysub{\mytyb_1}{\myb{x_1}}{\myb{\mytmm_1}} \myeq \mysub{\mytyb_2}{\myb{x_2}}{\myb{\mytmm_2}}} \\
& \myb{\mytmn_2} & \mapsto & \mycoee{\mysub{\mytyb_1}{\myb{x_1}}{\myb{\mytmm_1}}}{\mysub{\mytyb_2}{\myb{x_2}}{\myb{\mytmm_2}}}{\myb{Q_B}}{\myb{\mytmn_1}} : \mysub{\mytyb_2}{\myb{x_2}}{\myb{\mytmm_2}} \\
\mysyn{in} & \multicolumn{3}{@{}l}{\mypair{\myb{\mytmm_2}}{\myb{\mytmn_2}}}
\end{array}} \\
\mytmt & \myred &
\cdots \\
- \mycoe & \mytya & \mytyb & \myse{Q} & \mytmt & \myred & \\
- \multicolumn{7}{l}{
- \myind{2}\myapp{\myabsurd{\mytyb}}{\myse{Q}}\ \text{if $\mytya$ and $\mytyb$ are canonical.}
- }
+ \mycoe & \mytya & \mytyb & \myse{Q} & \mytmt & \myred & \myapp{\myabsurd{\mytyb}}{\myse{Q}}\ \text{if $\mytya$ and $\mytyb$ are canonical.}
\end{array}
$
}
\label{fig:eqred}
\end{figure}
-Figure \ref{fig:eqred} illustrates this idea in practice. For ground
-types, the proof is the trivial element, and \myfun{coe} is the
-identity. For the three type binders, things are similar but subtly
-different---the choices we make in the type equality are dictated by
-the desire of writing the $\myfun{coe}$ in a natural way.
+\begin{mydef}[Type equalities reduction, and \myfun{coe}rcions] Figure
+ \ref{fig:eqred} illustrates the rules to reduce equalities and to
+ coerce terms. We use a $\mysyn{let}$ syntax for legibility.
+\end{mydef}
+For ground types, the proof is the trivial element, and \myfun{coe} is
+the identity. For $\myunit$, we can do better: we return its only
+member without matching on the term. For the three type binders the
+choices we make in the type equality are dictated by the desire of
+writing the $\myfun{coe}$ in a natural way.
$\myprod$ is the easiest case: we decompose the proof into proofs that
the first element's types are equal ($\mytya_1 \myeq \mytya_2$), and a
proof that given equal values in the first element, the types of the
second elements are equal too
($\myprfora{\myb{x_1}}{\mytya_1}{\myprfora{\myb{x_2}}{\mytya_2}{\myjm{\myb{x_1}}{\mytya_1}{\myb{x_2}}{\mytya_2}}
- \myimpl \mytyb_1 \myeq \mytyb_2}$)\footnote{We are using $\myimpl$ to
- indicate a $\forall$ where we discard the first value. We write
- $\mytyb_1[\myb{x_1}]$ to indicate that the $\myb{x_1}$ in $\mytyb_1$
- is re-bound to the $\myb{x_1}$ quantified by the $\forall$, and
- similarly for $\myb{x_2}$ and $\mytyb_2$.}. This also explains the
-need for heterogeneous equality, since in the second proof it would be
-awkward to express the fact that $\myb{A_1}$ is the same as $\myb{A_2}$.
-In the respective $\myfun{coe}$ case, since the types are canonical, we
-know at this point that the proof of equality is a pair of the shape
-described above. Thus, we can immediately coerce the first element of
-the pair using the first element of the proof, and then instantiate the
-second element with the two first elements and a proof by coherence of
-their equality, since we know that the types are equal. The cases for
-the other binders are omitted for brevity, but they follow the same
-principle.
+ \myimpl \mytyb_1[\myb{x_1}] \myeq \mytyb_2[\myb{x_2}]}$).\footnote{We
+ are using $\myimpl$ to indicate a $\forall$ where we discard the
+ quantified value. We write $\mytyb_1[\myb{x_1}]$ to indicate that the
+ $\myb{x_1}$ in $\mytyb_1$ is re-bound to the $\myb{x_1}$ quantified by
+ the $\forall$, and similarly for $\myb{x_2}$ and $\mytyb_2$.} This
+also explains the need for heterogeneous equality, since in the second
+proof we need to equate terms of possibly different types. In the
+respective $\myfun{coe}$ case, since the types are canonical, we know at
+this point that the proof of equality is a pair of the shape described
+above. Thus, we can immediately coerce the first element of the pair
+using the first element of the proof, and then instantiate the second
+element with the two first elements and a proof by coherence of their
+equality, since we know that the types are equal.
+
+The cases for the other binders are omitted for brevity, but they follow
+the same principle with some twists to make $\myfun{coe}$ work with the
+generated proofs; the reader can refer to the paper for details.
\subsubsection{$\myfun{coe}$, laziness, and $\myfun{coh}$erence}
+\label{sec:lazy}
It is important to notice that in the reduction rules for $\myfun{coe}$
-are never obstructed by the proofs: with the exception of comparisons
-between different canonical types we never pattern match on the pairs,
-but always look at the projections. This means that, as long as we are
-consistent, and thus as long as we don't have $\mybot$-inducing proofs,
-we can add propositional axioms for equality and $\myfun{coe}$ will
-still compute. Thus, we can take $\myfun{coh}$ as axiomatic, and we can
-add back familiar useful equality rules:
+are never obstructed by the structure of the proofs. With the exception
+of comparisons between different canonical types we never `pattern
+match' on the proof pairs, but always look at the projections. This
+means that, as long as we are consistent, and thus as long as we don't
+have $\mybot$-inducing proofs, we can add propositional axioms for
+equality and $\myfun{coe}$ will still compute. Thus, we can take
+$\myfun{coh}$ as axiomatic, and we can add back familiar useful equality
+rules:
\mydesc{typing:}{\myjud{\mytmsyn}{\mytmsyn}}{
- \begin{tabular}{cc}
\AxiomC{$\myjud{\mytmt}{\mytya}$}
- \UnaryInfC{$\myjud{\myapp{\myrefl}{\mytmt}}{\myprdec{\myjm{\myb{x}}{\myb{\mytya}}{\myb{x}}{\myb{\mytya}}}}$}
+ \UnaryInfC{$\myjud{\myapp{\myrefl}{\mytmt}}{\myprdec{\myjm{\mytmt}{\mytya}{\mytmt}{\mytya}}}$}
\DisplayProof
- &
+
+ \myderivspp
+
\AxiomC{$\myjud{\mytya}{\mytyp}$}
\AxiomC{$\myjudd{\myctx; \myb{x} : \mytya}{\mytyb}{\mytyp}$}
\BinaryInfC{$\myjud{\mytyc{R} \myappsp (\myb{x} {:} \mytya) \myappsp \mytyb}{\myfora{\myb{y}\, \myb{z}}{\mytya}{\myprdec{\myjm{\myb{y}}{\mytya}{\myb{z}}{\mytya} \myimpl \mysub{\mytyb}{\myb{x}}{\myb{y}} \myeq \mysub{\mytyb}{\myb{x}}{\myb{z}}}}}$}
\DisplayProof
- \end{tabular}
}
$\myrefl$ is the equivalent of the reflexivity rule in propositional
us recover $\myfun{subst}$. Note that while we need to provide ad-hoc
rules in the restricted, non-hierarchical theory that we have, if our
theory supports abstraction over $\mytyp$s we can easily add these
-axioms as abstracted variables.
+axioms as top-level abstracted variables.
\subsubsection{Value-level equality}
+\begin{mydef}[Value-level equality]\ \end{mydef}
+\mynegder
\mydesc{equality reduction:}{\myprsyn \myred \myprsyn}{
$
\begin{array}{r@{ }c@{\ }c@{\ }c@{}l@{\ }c@{\ }r@{}c@{\ }c@{\ }c@{}l@{\ }l}
(&\mytmt_1 & : & \myempty&) & \myeq & (&\mytmt_2 & : & \myempty &) & \myred \mytop \\
- (&\mytmt_1 & : & \myempty&) & \myeq & (&\mytmt_2 & : & \myempty&) & \myred \mytop \\
+ (&\mytmt_1 & : & \myunit&) & \myeq & (&\mytmt_2 & : & \myunit&) & \myred \mytop \\
(&\mytrue & : & \mybool&) & \myeq & (&\mytrue & : & \mybool&) & \myred \mytop \\
(&\myfalse & : & \mybool&) & \myeq & (&\myfalse & : & \mybool&) & \myred \mytop \\
- (&\mytmt_1 & : & \mybool&) & \myeq & (&\mytmt_1 & : & \mybool&) & \myred \mybot \\
+ (&\mytrue & : & \mybool&) & \myeq & (&\myfalse & : & \mybool&) & \myred \mybot \\
+ (&\myfalse & : & \mybool&) & \myeq & (&\mytrue & : & \mybool&) & \myred \mybot \\
(&\mytmt_1 & : & \myexi{\mytya_1}{\myb{x_1}}{\mytyb_1}&) & \myeq & (&\mytmt_2 & : & \myexi{\mytya_2}{\myb{x_2}}{\mytyb_2}&) & \myred \\
& \multicolumn{11}{@{}l}{
\myind{2} \myjm{\myapp{\myfst}{\mytmt_1}}{\mytya_1}{\myapp{\myfst}{\mytmt_2}}{\mytya_2} \myand
}}
} \\
(&\mytmt_1 \mynodee \myse{f}_1 & : & \myw{\mytya_1}{\myb{x_1}}{\mytyb_1}&) & \myeq & (&\mytmt_1 \mynodee \myse{f}_1 & : & \myw{\mytya_2}{\myb{x_2}}{\mytyb_2}&) & \myred \cdots \\
- (&\mytmt_1 & : & \mytya_1&) & \myeq & (&\mytmt_2 & : & \mytya_2 &) & \myred \\
- & \multicolumn{11}{@{}l}{
- \myind{2} \mybot\ \text{if $\mytya_1$ and $\mytya_2$ are canonical.}
- }
+ (&\mytmt_1 & : & \mytya_1&) & \myeq & (&\mytmt_2 & : & \mytya_2 &) & \myred \mybot\ \text{if $\mytya_1$ and $\mytya_2$ are canonical.}
\end{array}
$
}
As with type-level equality, we want value-level equality to reduce
-based on the structure of the compared terms.
-
-\subsection{Proof irrelevance}
-
-% \section{Augmenting ITT}
-% \label{sec:practical}
-
-% \subsection{A more liberal hierarchy}
-
-% \subsection{Type inference}
-
-% \subsubsection{Bidirectional type checking}
-
-% \subsubsection{Pattern unification}
-
-% \subsection{Pattern matching and explicit fixpoints}
-
-% \subsection{Induction-recursion}
-
-% \subsection{Coinduction}
-
-% \subsection{Dealing with partiality}
-
-% \subsection{Type holes}
+based on the structure of the compared terms. When matching
+propositional data, such as $\myempty$ and $\myunit$, we automatically
+return the trivial type, since if a type has zero one members, all
+members will be equal. When matching on data-bearing types, such as
+$\mybool$, we check that such data matches, and return bottom otherwise.
+When matching on records and functions, we rebuild the records to
+achieve $\eta$-expansion, and relate functions if they are extensionally
+equal---exactly what we wanted. The case for \mytyc{W} is omitted but
+unsurprising, it checks that equal data in the nodes will bring equal
+children.
+
+\subsection{Proof irrelevance and stuck coercions}
+\label{sec:ott-quot}
+
+The last effort is required to make sure that proofs (members of
+$\myprop$) are \emph{irrelevant}. Since they are devoid of
+computational content, we would like to identify all equivalent
+propositions as the same, in a similar way as we identified all
+$\myempty$ and all $\myunit$ as the same in section
+\ref{sec:eta-expand}.
+
+Thus we will have a quotation that will not only perform
+$\eta$-expansion, but will also identify and mark proofs that could not
+be decoded (that is, equalities on neutral types). Then, when
+comparing terms, marked proofs will be considered equal without
+analysing their contents, thus gaining irrelevance.
+
+Moreover we can safely advance `stuck' $\myfun{coe}$rcions between
+non-canonical but definitionally equal types. Consider for example
+\[
+\mycoee{(\myITE{\myb{b}}{\mynat}{\mybool})}{(\myITE{\myb{b}}{\mynat}{\mybool})}{\myb{x}}
+\]
+Where $\myb{b}$ and $\myb{x}$ are abstracted variables. This
+$\myfun{coe}$ will not advance, since the types are not canonical.
+However they are definitionally equal, and thus we can safely remove the
+coerce and return $\myb{x}$ as it is.
-\section{\mykant : the theory}
+\section{\mykant: the theory}
\label{sec:kant-theory}
+\epigraph{\emph{The construction itself is an art, its application to the world an evil parasite.}}{Luitzen Egbertus Jan `Bertus' Brouwer}
+
\mykant\ is an interactive theorem prover developed as part of this thesis.
The plan is to present a core language which would be capable of serving as
the basis for a more featureful system, while still presenting interesting
features and more importantly observational equality.
-The author learnt the hard way the implementations challenges for such a
-project, and while there is a solid and working base to work on, observational
-equality is not currently implemented. However, a detailed plan on how to add
-it this functionality is provided, and should not prove to be too much work.
-
-The features currently implemented in \mykant\ are:
-
-\begin{description}
-\item[Full dependent types] As we would expect, we have dependent a system
- which is as expressive as the `best' corner in the lambda cube described in
- section \ref{sec:itt}.
-
-\item[Implicit, cumulative universe hierarchy] The user does not need to
- specify universe level explicitly, and universes are \emph{cumulative}.
-
-\item[User defined data types and records] Instead of forcing the user to
- choose from a restricted toolbox, we let her define inductive data types,
- with associated primitive recursion operators; or records, with associated
- projections for each field.
-
-\item[Bidirectional type checking] While no `fancy' inference via unification
- is present, we take advantage of an type synthesis system in the style of
- \cite{Pierce2000}, extending the concept for user defined data types.
-
-\item[Type holes] When building up programs interactively, it is useful to
- leave parts unfinished while exploring the current context. This is what
- type holes are for.
-\end{description}
-
-The planned features are:
-
-\begin{description}
-\item[Observational equality] As described in section \ref{sec:ott} but
- extended to work with the type hierarchy and to admit equality between
- arbitrary data types.
-
-\item[Coinductive data] ...
-\end{description}
-
-We will analyse the features one by one, along with motivations and tradeoffs
-for the design decisions made.
+We will first present the features of the system, along with motivations
+and trade-offs for the design decisions made. Then we describe the
+implementation we have developed in Section \ref{sec:kant-practice}.
+For an overview of the features of \mykant, see
+Section \ref{sec:contributions}, here we present them one by one. The
+exception is type holes, which we do not describe holes rigorously, but
+provide more information about them in Section \ref{sec:type-holes}.
+
+Note that in this section we will present \mykant\ terms in a fancy
+\LaTeX\ dress too keep up with the presentation, but every term, with its
+syntax reduced to the concrete syntax, is a valid \mykant\ term accepted
+by \mykant\ the software, and not only \mykant\ the theory. Appendix
+\ref{app:kant-examples} displays most of the terms in this section in
+their concrete syntax.
\subsection{Bidirectional type checking}
-We start by describing bidirectional type checking since it calls for fairly
-different typing rules that what we have seen up to now. The idea is to have
-two kind of terms: terms for which a type can always be inferred, and terms
-that need to be checked against a type. A nice observation is that this
-duality runs through the semantics of the terms: data destructors (function
-application, record projections, primitive re cursors) \emph{infer} types,
-while data constructors (abstractions, record/data types data constructors)
-need to be checked. In the literature these terms are respectively known as
+We start by describing bidirectional type checking since it calls for
+fairly different typing rules that what we have seen up to now. The
+idea is to have two kinds of terms: terms for which a type can always be
+inferred, and terms that need to be checked against a type. A nice
+observation is that this duality is in correspondence with the notion of
+canonical and neutral terms: neutral terms
+(abstracted or defined variables, function application, record
+projections, primitive recursors, etc.) \emph{infer} types, canonical
+terms (abstractions, record/data types data constructors, etc.) need to
+be \emph{checked}.
To introduce the concept and notation, we will revisit the STLC in a
-bidirectional style. The presentation follows \cite{Loh2010}.
+bidirectional style. The presentation follows \cite{Loh2010}. The
+syntax for our bidirectional STLC is the same as the untyped
+$\lambda$-calculus, but with an extra construct to annotate terms
+explicitly---this will be necessary when dealing with top-level
+canonical terms. The types are the same as those found in the normal
+STLC.
+
+\begin{mydef}[Syntax for the annotated $\lambda$-calculus]\ \end{mydef}
+\mynegder
+\mydesc{syntax}{ }{
+ $
+ \begin{array}{r@{\ }c@{\ }l}
+ \mytmsyn & ::= & \myb{x} \mysynsep \myabs{\myb{x}}{\mytmsyn} \mysynsep (\myapp{\mytmsyn}{\mytmsyn}) \mysynsep (\mytmsyn : \mytysyn)
+ \end{array}
+ $
+}
+
+We will have two kinds of typing judgements: \emph{inference} and
+\emph{checking}. $\myinf{\mytmt}{\mytya}$ indicates that $\mytmt$
+infers the type $\mytya$, while $\mychk{\mytmt}{\mytya}$ can be checked
+against type $\mytya$. The arrows signify the direction of the type
+checking---inference pushes types up, checking propagates types
+down.
+
+The type of variables in context is inferred, and so are annotate terms.
+The type of applications is inferred too, propagating types down the
+applied term. Abstractions are checked. Finally, we have a rule to
+check the type of an inferrable term.
+
+\begin{mydef}[Bidirectional type checking for the STLC]\ \end{mydef}
+\mynegder
+\mydesc{typing:}{\myctx \vdash \mytmsyn \Updownarrow \mytmsyn}{
+ \begin{tabular}{cc}
+ \AxiomC{$\myctx(x) = A$}
+ \UnaryInfC{$\myinf{\myb{x}}{A}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myjudd{\myctx;\myb{x} : A}{\mytmt}{\mytyb}$}
+ \UnaryInfC{$\mychk{\myabs{x}{\mytmt}}{(\myb{x} {:} \mytya) \myarr \mytyb}$}
+ \DisplayProof
+ \end{tabular}
-% TODO do this --- is it even necessary
+ \myderivspp
-% The syntax of
+ \begin{tabular}{ccc}
+ \AxiomC{$\myinf{\mytmm}{\mytya \myarr \mytyb}$}
+ \AxiomC{$\mychk{\mytmn}{\mytya}$}
+ \BinaryInfC{$\myjud{\myapp{\mytmm}{\mytmn}}{\mytyb}$}
+ \DisplayProof
+ &
+ \AxiomC{$\mychk{\mytmt}{\mytya}$}
+ \UnaryInfC{$\myinf{\myann{\mytmt}{\mytya}}{\mytya}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myinf{\mytmt}{\mytya}$}
+ \UnaryInfC{$\mychk{\mytmt}{\mytya}$}
+ \DisplayProof
+ \end{tabular}
+}
+
+For example, if we wanted to type function composition (in this case for
+naturals), we would have to annotate the term:
+\[
+\begin{array}{@{}l}
+ \myfun{comp} : (\mynat \myarr \mynat) \myarr (\mynat \myarr \mynat) \myarr \mynat \myarr \mynat \\
+ \myfun{comp} \mapsto (\myabs{\myb{f}\, \myb{g}\, \myb{x}}{\myb{f}\myappsp(\myb{g}\myappsp\myb{x})})
+\end{array}
+\]
+But we would not have to annotate functions passed to it, since the type would be propagated to the arguments:
+\[
+ \myfun{comp}\myappsp (\myabs{\myb{x}}{\myb{x} \mathrel{\myfun{$+$}} 3}) \myappsp (\myabs{\myb{x}}{\myb{x} \mathrel{\myfun{$*$}} 4}) \myappsp 42 : \mynat
+\]
\subsection{Base terms and types}
variables and substitution is left unspecified, and explained in section
\ref{sec:term-repr}, along with other implementation issues. We are
also going to give an account of the implicit type hierarchy separately
-in section \ref{sec:term-hierarchy}, so as not to clutter derivation
+in Section \ref{sec:term-hierarchy}, so as not to clutter derivation
rules too much, and just treat types as impredicative for the time
being.
+\begin{mydef}[Syntax for base types in \mykant]\ \end{mydef}
+\mynegder
\mydesc{syntax}{ }{
$
\begin{array}{r@{\ }c@{\ }l}
}
The syntax for our calculus includes just two basic constructs:
-abstractions and $\mytyp$s. Everything else will be provided by
-user-definable constructs. Since we let the user define values, we will
-need a context capable of carrying the body of variables along with
-their type. Bound names and defined names are treated separately in the
-syntax, and while both can be associated to a type in the context, only
-defined names can be associated with a body:
-
+abstractions and $\mytyp$s. Everything else will be user-defined.
+Since we let the user define values too, we will need a context capable
+of carrying the body of variables along with their type.
+
+\begin{mydef}[Context validity]
+Bound names and defined names are treated separately in the syntax, and
+while both can be associated to a type in the context, only defined
+names can be associated with a body.
+\end{mydef}
+\mynegder
\mydesc{context validity:}{\myvalid{\myctx}}{
\begin{tabular}{ccc}
\AxiomC{\phantom{$\myjud{\mytya}{\mytyp_l}$}}
\UnaryInfC{$\myvalid{\myemptyctx}$}
\DisplayProof
&
- \AxiomC{$\myjud{\mytya}{\mytyp}$}
+ \AxiomC{$\mychk{\mytya}{\mytyp}$}
\AxiomC{$\mynamesyn \not\in \myctx$}
\BinaryInfC{$\myvalid{\myctx ; \mynamesyn : \mytya}$}
\DisplayProof
&
- \AxiomC{$\myjud{\mytmt}{\mytya}$}
+ \AxiomC{$\mychk{\mytmt}{\mytya}$}
\AxiomC{$\myfun{f} \not\in \myctx$}
\BinaryInfC{$\myvalid{\myctx ; \myfun{f} \mapsto \mytmt : \mytya}$}
\DisplayProof
the usual function application ($\beta$-reduction), but also a rule to
replace names with their bodies ($\delta$-reduction), and one to discard
type annotations. For this reason reduction is done in-context, as
-opposed to what we have seen in the past:
+opposed to what we have seen in the past.
+\begin{mydef}[Reduction rules for base types in \mykant]\ \end{mydef}
+\mynegder
\mydesc{reduction:}{\myctx \vdash \mytmsyn \myred \mytmsyn}{
\begin{tabular}{ccc}
\AxiomC{\phantom{$\myb{x} \mapsto \mytmt : \mytya \in \myctx$}}
\end{tabular}
}
-We can now give types to our terms. The type of names, both defined and
-abstract, is inferred. The type of applications is inferred too,
-propagating types down the applied term. Abstractions are checked.
-Finally, we have a rule to check the type of an inferrable term. We
-defer the question of term equality (which is needed for type checking)
-to section \label{sec:kant-irr}.
+We can now give types to our terms. Although we include the usual
+conversion rule, we defer a detailed account of definitional equality to
+Section \ref{sec:kant-irr}.
-\mydesc{typing:}{\myctx \vdash \mytmsyn \Leftrightarrow \mytmsyn}{
- \begin{tabular}{ccc}
+\begin{mydef}[Bidirectional type checking for base types in \mykant]\ \end{mydef}
+\mynegder
+\mydesc{typing:}{\myctx \vdash \mytmsyn \Updownarrow \mytmsyn}{
+ \begin{tabular}{cccc}
\AxiomC{$\myse{name} : A \in \myctx$}
\UnaryInfC{$\myinf{\myse{name}}{A}$}
\DisplayProof
\UnaryInfC{$\myinf{\myfun{f}}{A}$}
\DisplayProof
&
+ \AxiomC{$\mychk{\mytmt}{\mytya}$}
+ \UnaryInfC{$\myinf{\myann{\mytmt}{\mytya}}{\mytya}$}
+ \DisplayProof
+ &
\AxiomC{$\myinf{\mytmt}{\mytya}$}
- \UnaryInfC{$\mychk{\myann{\mytmt}{\mytya}}{\mytya}$}
+ \AxiomC{$\myctx \vdash \mytya \mydefeq \mytyb$}
+ \BinaryInfC{$\mychk{\mytmt}{\mytyb}$}
\DisplayProof
\end{tabular}
- \myderivsp
- \begin{tabular}{ccc}
+ \myderivspp
+
+ \begin{tabular}{cc}
+
+ \AxiomC{\phantom{$\mychkk{\myctx; \myb{x}: \mytya}{\mytmt}{\mytyb}$}}
+ \UnaryInfC{$\myinf{\mytyp}{\mytyp}$}
+ \DisplayProof
+ &
+ \AxiomC{$\mychk{\mytya}{\mytyp}$}
+ \AxiomC{$\mychkk{\myctx; \myb{x} : \mytya}{\mytyb}{\mytyp}$}
+ \BinaryInfC{$\myinf{(\myb{x} {:} \mytya) \myarr \mytyb}{\mytyp}$}
+ \DisplayProof
+
+ \end{tabular}
+
+
+ \myderivspp
+
+ \begin{tabular}{cc}
\AxiomC{$\myinf{\mytmm}{\myfora{\myb{x}}{\mytya}{\mytyb}}$}
\AxiomC{$\mychk{\mytmn}{\mytya}$}
\BinaryInfC{$\myinf{\myapp{\mytmm}{\mytmn}}{\mysub{\mytyb}{\myb{x}}{\mytmn}}$}
\UnaryInfC{$\mychk{\myabs{\myb{x}}{\mytmt}}{\myfora{\myb{x}}{\mytyb}{\mytyb}}$}
\DisplayProof
\end{tabular}
+
}
\subsection{Elaboration}
but also custom data types and records. \emph{Elaboration} consists of
turning these declarations into workable syntax, types, and reduction
rules. The treatment of custom types in $\mykant$\ is heavily inspired
-by McBride and McKinna early work on Epigram \citep{McBride2004},
+by McBride's and McKinna's early work on Epigram \citep{McBride2004},
although with some differences.
\subsubsection{Term vectors, telescopes, and assorted notation}
-We use a vector notation to refer to a series of term applied to
-another, for example $\mytyc{D} \myappsp \vec{A}$ is a shorthand for
-$\mytyc{D} \myappsp \mytya_1 \cdots \mytya_n$, for some $n$. $n$ is
-consistently used to refer to the length of such vectors, and $i$ to
-refer to an index in such vectors. We also often need to `build up'
-terms vectors, in which case we use $\myemptyctx$ for an empty vector
-and add elements to an existing vector with $\myarg ; \myarg$, similarly
-to what we do for context.
+\begin{mydef}[Term vector]
+ A \emph{term vector} is a series of terms. The empty vector is
+ represented by $\myemptyctx$, and a new element is added with
+ $\myarg;\myarg$, similarly to contexts---$\vec{t};\mytmm$.
+\end{mydef}
+
+We denote term vectors with the usual arrow notation,
+e.g. $vec{\mytmt}$, $\myvec{\mytmt};\mytmm$, etc. We often use term
+vectors to refer to a series of term applied to another. For example
+$\mytyc{D} \myappsp \vec{A}$ is a shorthand for $\mytyc{D} \myappsp
+\mytya_1 \cdots \mytya_n$, for some $n$. $n$ is consistently used to
+refer to the length of such vectors, and $i$ to refer to an index such
+that $1 \le i \le n$.
+
+\begin{mydef}[Telescope]
+ A \emph{telescope} is a series of typed bindings. The empty telescope
+ is represented by $\myemptyctx$, and a binding is added via
+ $\myarg;\myarg$.
+\end{mydef}
To present the elaboration and operations on user defined data types, we
-frequently make use what de Bruijn called \emph{telescopes}
-\citep{Bruijn91}, a construct that will prove useful when dealing with
-the types of type and data constructors. A telescope is a series of
-nested typed bindings, such as $(\myb{x} {:} \mynat); (\myb{p} {:}
-\myapp{\myfun{even}}{\myb{x}})$. Consistently with the notation for
-contexts and term vectors, we use $\myemptyctx$ to denote an empty
-telescope and $\myarg ; \myarg$ to add a new binding to an existing
-telescope.
-
-We refer to telescopes with $\mytele$, $\mytele'$, $\mytele_i$, etc. If
-$\mytele$ refers to a telescope, $\mytelee$ refers to the term vector
-made up of all the variables bound by $\mytele$. $\mytele \myarr
-\mytya$ refers to the type made by turning the telescope into a series
-of $\myarr$. Returning to the examples above, we have that
-{\small\[
+frequently make use what \cite{Bruijn91} called \emph{telescopes}, a
+construct that will prove useful when dealing with the types of type and
+data constructors. We refer to telescopes with $\mytele$, $\mytele'$,
+$\mytele_i$, etc. If $\mytele$ refers to a telescope, $\mytelee$ refers
+to the term vector made up of all the variables bound by $\mytele$.
+$\mytele \myarr \mytya$ refers to the type made by turning the telescope
+into a series of $\myarr$. For example we have that
+\[
(\myb{x} {:} \mynat); (\myb{p} : \myapp{\myfun{even}}{\myb{x}}) \myarr \mynat =
(\myb{x} {:} \mynat) \myarr (\myb{p} : \myapp{\myfun{even}}{\myb{x}}) \myarr \mynat
-\]}
+\]
We make use of various operations to manipulate telescopes:
\begin{itemize}
(1-indexed).
\item $\mytake_i(\mytele)$ refers to the telescope created by taking the
first $i$ elements of $\mytele$: $\mytake_1((\myb{x} {:} \mynat); (\myb{p} :
- \myapp{\myfun{even}}{\myb{x}})) = (\myb{x} {:} \mynat)$
+ \myapp{\myfun{even}}{\myb{x}})) = (\myb{x} {:} \mynat)$.
\item $\mytele \vec{A}$ refers to the telescope made by `applying' the
terms in $\vec{A}$ on $\mytele$: $((\myb{x} {:} \mynat); (\myb{p} :
\myapp{\myfun{even}}{\myb{x}}))42 = (\myb{p} :
\myapp{\myfun{even}}{42})$.
\end{itemize}
+Additionally, when presenting syntax elaboration, We use $\mytmsyn^n$ to
+indicate a term vector composed of $n$ elements. When clear from the
+context, we use term vectors to signify their length,
+e.g. $\mytmsyn^{\mytele}$, or $1 \le i \le \mytele$.
+
\subsubsection{Declarations syntax}
+\begin{mydef}[Syntax of declarations in \mykant]\ \end{mydef}
+\mynegder
\mydesc{syntax}{ }{
$
\begin{array}{r@{\ }c@{\ }l}
\mydeclsyn & ::= & \myval{\myb{x}}{\mytmsyn}{\mytmsyn} \\
& | & \mypost{\myb{x}}{\mytmsyn} \\
- & | & \myadt{\mytyc{D}}{\mytelesyn}{}{\mydc{c} : \mytelesyn\ |\ \cdots } \\
- & | & \myreco{\mytyc{D}}{\mytelesyn}{}{\myfun{f} : \mytmsyn,\ \cdots } \\
+ & | & \myadt{\mytyc{D}}{\myappsp \mytelesyn}{}{\mydc{c} : \mytelesyn\ |\ \cdots } \\
+ & | & \myreco{\mytyc{D}}{\myappsp \mytelesyn}{}{\myfun{f} : \mytmsyn,\ \cdots } \\
\mytelesyn & ::= & \myemptytele \mysynsep \mytelesyn \mycc (\myb{x} {:} \mytmsyn) \\
\mynamesyn & ::= & \cdots \mysynsep \mytyc{D} \mysynsep \mytyc{D}.\mydc{c} \mysynsep \mytyc{D}.\myfun{f}
\end{array}
$
}
-
In \mykant\ we have four kind of declarations:
\begin{description}
\item[Defined value] A variable, together with a type and a body.
\item[Abstract variable] An abstract variable, with a type but no body.
-\item[Inductive data] A datatype, with a type constructor and various data
- constructors---somewhat similar to what we find in Haskell. A primitive
- recursor (or `destructor') will be generated automatically.
-\item[Record] A record, which consists of one data constructor and various
- fields, with no recursive occurrences.
+\item[Inductive data] A \emph{data type}, with a \emph{type constructor}
+ (denoted in blue, capitalised, sans serif: $\mytyc{D}$) various
+ \emph{data constructors} (denoted in red, lowercase, sans serif:
+ $\mydc{c}$), quite similar to what we find in Haskell. A primitive
+ \emph{eliminator} (or \emph{destructor}, or \emph{recursor}; denoted
+ by green, lowercase, roman: \myfun{elim}) will be used to compute with
+ each data type.
+\item[Record] A \emph{record}, which like data types consists of a type
+ constructor but only one data constructor. The user can also define
+ various \emph{fields}, with no recursive occurrences of the type. The
+ functions extracting the fields' values from an instance of a record
+ are called \emph{projections} (denoted in the same way as destructors).
\end{description}
-Elaborating defined variables consists of type checking body against the
-given type, and updating the context to contain the new binding.
+Elaborating defined variables consists of type checking the body against
+the given type, and updating the context to contain the new binding.
Elaborating abstract variables and abstract variables consists of type
-checking the type, and updating the context with a new typed variable:
+checking the type, and updating the context with a new typed variable.
+\begin{mydef}[Elaboration of defined and abstract variables]\ \end{mydef}
+\mynegder
\mydesc{context elaboration:}{\myelab{\mydeclsyn}{\myctx}}{
\begin{tabular}{cc}
- \AxiomC{$\myjud{\mytmt}{\mytya}$}
+ \AxiomC{$\mychk{\mytmt}{\mytya}$}
\AxiomC{$\myfun{f} \not\in \myctx$}
\BinaryInfC{
$\myctx \myelabt \myval{\myfun{f}}{\mytya}{\mytmt} \ \ \myelabf\ \ \myctx; \myfun{f} \mapsto \mytmt : \mytya$
}
\DisplayProof
&
- \AxiomC{$\myjud{\mytya}{\mytyp}$}
+ \AxiomC{$\mychk{\mytya}{\mytyp}$}
\AxiomC{$\myfun{f} \not\in \myctx$}
\BinaryInfC{
$
\subsubsection{User defined types}
\label{sec:user-type}
-Elaborating user defined types is the real effort. First, let's explain
-what we can defined, with some examples.
+Elaborating user defined types is the real effort. First, we will
+explain what we can define, with some examples.
\begin{description}
\item[Natural numbers] To define natural numbers, we create a data type
with two constructors: one with zero arguments ($\mydc{zero}$) and one
with one recursive argument ($\mydc{suc}$):
- {\small\[
+ \[
\begin{array}{@{}l}
\myadt{\mynat}{ }{ }{
\mydc{zero} \mydcsep \mydc{suc} \myappsp \mynat
}
\end{array}
- \]}
+ \]
This is very similar to what we would write in Haskell:
- {\small\[\text{\texttt{data Nat = Zero | Suc Nat}}\]}
+ \begin{Verbatim}
+data Nat = Zero | Suc Nat
+ \end{Verbatim}
Once the data type is defined, $\mykant$\ will generate syntactic
constructs for the type and data constructors, so that we will have
\begin{center}
- \small
+ \mysmall
\begin{tabular}{ccc}
\AxiomC{\phantom{$\mychk{\mytmt}{\mynat}$}}
\UnaryInfC{$\myinf{\mynat}{\mytyp}$}
\end{center}
While in Haskell (or indeed in Agda or Coq) data constructors are
treated the same way as functions, in $\mykant$\ they are syntax, so
- for example using $\mytyc{\mynat}.\mydc{suc}$ on its own will be a
+ for example using $\mytyc{\mynat}.\mydc{suc}$ on its own will give a
syntax error. This is necessary so that we can easily infer the type
of polymorphic data constructors, as we will see later.
Moreover, each data constructor is prefixed by the type constructor
name, since we need to retrieve the type constructor of a data
constructor when type checking. This measure aids in the presentation
- of various features but it is not needed in the implementation, where
- we can have a dictionary to lookup the type constructor corresponding
+ of the theory but it is not needed in the implementation, where
+ we can have a dictionary to look up the type constructor corresponding
to each data constructor. When using data constructors in examples I
- will omit the type constructor prefix for brevity.
+ will omit the type constructor prefix for brevity, in this case
+ writing $\mydc{zero}$ instead of $\mynat.\mydc{zero}$ and $\mydc{suc}$ instead of
+ $\mynat.\mydc{suc}$.
Along with user defined constructors, $\mykant$\ automatically
generates an \emph{eliminator}, or \emph{destructor}, to compute with
natural numbers: If we have $\mytmt : \mynat$, we can destruct
$\mytmt$ using the generated eliminator `$\mynat.\myfun{elim}$':
\begin{prooftree}
- \small
+ \mysmall
\AxiomC{$\mychk{\mytmt}{\mynat}$}
\UnaryInfC{$
\myinf{\mytyc{\mynat}.\myfun{elim} \myappsp \mytmt}{
While the induction principle is usually seen as a mean to prove
properties about numbers, in the intuitionistic setting it is also a
- mean to compute. In this specific case we will $\mynat.\myfun{elim}$
- will return the base case if the provided number is $\mydc{zero}$, and
- recursively apply the inductive step if the number is a
+ mean to compute. In this specific case $\mynat.\myfun{elim}$
+ returns the base case if the provided number is $\mydc{zero}$, and
+ recursively applies the inductive step if the number is a
$\mydc{suc}$cessor:
- {\small\[
+ \[
\begin{array}{@{}l@{}l}
\mytyc{\mynat}.\myfun{elim} \myappsp \mydc{zero} & \myappsp \myse{P} \myappsp \myse{pz} \myappsp \myse{ps} \myred \myse{pz} \\
\mytyc{\mynat}.\myfun{elim} \myappsp (\mydc{suc} \myappsp \mytmt) & \myappsp \myse{P} \myappsp \myse{pz} \myappsp \myse{ps} \myred \myse{ps} \myappsp \mytmt \myappsp (\mynat.\myfun{elim} \myappsp \mytmt \myappsp \myse{P} \myappsp \myse{pz} \myappsp \myse{ps})
\end{array}
- \]}
+ \]
The Haskell equivalent would be
- {\small\[
- \begin{array}{@{}l}
- \text{\texttt{elim :: Nat -> a -> (Nat -> a -> a) -> a}}\\
- \text{\texttt{elim Zero pz ps = pz}}\\
- \text{\texttt{elim (Suc n) pz ps = ps n (elim n pz ps)}}
- \end{array}
- \]}
- Which buys us the computational behaviour, but not the reasoning power.
- % TODO maybe more examples, e.g. Haskell eliminator and fibonacci
+ \begin{Verbatim}
+elim :: Nat -> a -> (Nat -> a -> a) -> a
+elim Zero pz ps = pz
+elim (Suc n) pz ps = ps n (elim n pz ps)
+\end{Verbatim}
+Which buys us the computational behaviour, but not the reasoning power,
+since we cannot express the notion of a predicate depending on
+$\mynat$---the type system is far too weak.
\item[Binary trees] Now for a polymorphic data type: binary trees, since
lists are too similar to natural numbers to be interesting.
- {\small\[
+ \[
\begin{array}{@{}l}
\myadt{\mytree}{\myappsp (\myb{A} {:} \mytyp)}{ }{
\mydc{leaf} \mydcsep \mydc{node} \myappsp (\myapp{\mytree}{\myb{A}}) \myappsp \myb{A} \myappsp (\myapp{\mytree}{\myb{A}})
}
\end{array}
- \]}
- Now the purpose of constructors as syntax can be explained: what would
+ \]
+ Now the purpose of `constructors as syntax' can be explained: what would
the type of $\mydc{leaf}$ be? If we were to treat it as a `normal'
term, we would have to specify the type parameter of the tree each
time the constructor is applied:
- {\small\[
+ \[
\begin{array}{@{}l@{\ }l}
\mydc{leaf} & : \myfora{\myb{A}}{\mytyp}{\myapp{\mytree}{\myb{A}}} \\
\mydc{node} & : \myfora{\myb{A}}{\mytyp}{\myapp{\mytree}{\myb{A}} \myarr \myb{A} \myarr \myapp{\mytree}{\myb{A}} \myarr \myapp{\mytree}{\myb{A}}}
\end{array}
- \]}
+ \]
The problem with this approach is that creating terms is incredibly
- verbose and dull, since we would need to specify the type parameters
- each time. For example if we wished to create a $\mytree \myappsp
- \mynat$ with two nodes and three leaves, we would have to write
- {\small\[
+ verbose and dull, since we would need to specify the type parameter of
+ $\mytyc{Tree}$ each time. For example if we wished to create a
+ $\mytree \myappsp \mynat$ with two nodes and three leaves, we would
+ write
+ \[
\mydc{node} \myappsp \mynat \myappsp (\mydc{node} \myappsp \mynat \myappsp (\mydc{leaf} \myappsp \mynat) \myappsp (\myapp{\mydc{suc}}{\mydc{zero}}) \myappsp (\mydc{leaf} \myappsp \mynat)) \myappsp \mydc{zero} \myappsp (\mydc{leaf} \myappsp \mynat)
- \]}
+ \]
The redundancy of $\mynat$s is quite irritating. Instead, if we treat
constructors as syntactic elements, we can `extract' the type of the
parameter from the type that the term gets checked against, much like
- we get the type of abstraction arguments:
+ what we do to type abstractions:
\begin{center}
- \small
+ \mysmall
\begin{tabular}{cc}
\AxiomC{$\mychk{\mytya}{\mytyp}$}
\UnaryInfC{$\mychk{\mydc{leaf}}{\myapp{\mytree}{\mytya}}$}
\end{tabular}
\end{center}
Which enables us to write, much more concisely
- {\small\[
+ \[
\mydc{node} \myappsp (\mydc{node} \myappsp \mydc{leaf} \myappsp (\myapp{\mydc{suc}}{\mydc{zero}}) \myappsp \mydc{leaf}) \myappsp \mydc{zero} \myappsp \mydc{leaf} : \myapp{\mytree}{\mynat}
- \]}
+ \]
We gain an annotation, but we lose the myriad of types applied to the
constructors. Conversely, with the eliminator for $\mytree$, we can
infer the type of the arguments given the type of the destructed:
\begin{prooftree}
- \footnotesize
+ \small
\AxiomC{$\myinf{\mytmt}{\myapp{\mytree}{\mytya}}$}
\UnaryInfC{$
\myinf{\mytree.\myfun{elim} \myappsp \mytmt}{
$}
\end{prooftree}
As expected, the eliminator embodies structural induction on trees.
+ We have a base case for $\myb{P} \myappsp \mydc{leaf}$, and an
+ inductive step that given two subtrees and the predicate applied to
+ them needs to return the predicate applied to the tree formed by a
+ node with the two subtrees as children.
\item[Empty type] We have presented types that have at least one
constructors, but nothing prevents us from defining types with
\emph{no} constructors:
- {\small\[
- \myadt{\mytyc{Empty}}{ }{ }{ }
- \]}
+ \[\myadt{\mytyc{Empty}}{ }{ }{ }\]
What shall the `induction principle' on $\mytyc{Empty}$ be? Does it
even make sense to talk about induction on $\mytyc{Empty}$?
- $\mykant$\ does not care, and generates an eliminator with no `cases',
- and thus corresponding to the $\myfun{absurd}$ that we know and love:
+ $\mykant$\ does not care, and generates an eliminator with no `cases':
\begin{prooftree}
- \small
+ \mysmall
\AxiomC{$\myinf{\mytmt}{\mytyc{Empty}}$}
\UnaryInfC{$\myinf{\myempty.\myfun{elim} \myappsp \mytmt}{(\myb{P} {:} \mytmt \myarr \mytyp) \myarr \myapp{\myb{P}}{\mytmt}}$}
\end{prooftree}
+ which lets us write the $\myfun{absurd}$ that we know and love:
+ \[
+ \begin{array}{l@{}}
+ \myfun{absurd} : (\myb{A} {:} \mytyp) \myarr \myempty \myarr \myb{A} \\
+ \myfun{absurd}\myappsp \myb{A} \myappsp \myb{x} \mapsto \myempty.\myfun{elim} \myappsp \myb{x} \myappsp (\myabs{\myarg}{\myb{A}})
+ \end{array}
+ \]
\item[Ordered lists] Up to this point, the examples shown are nothing
new to the \{Haskell, SML, OCaml, functional\} programmer. However
dependent types let us express much more than that. A useful example
is the type of ordered lists. There are many ways to define such a
- thing, we will define our type to store the bounds of the list, making
+ thing, but we will define ours to store the bounds of the list, making
sure that $\mydc{cons}$ing respects that.
First, using $\myunit$ and $\myempty$, we define a type expressing the
ordering on natural numbers, $\myfun{le}$---`less or equal'.
$\myfun{le}\myappsp \mytmm \myappsp \mytmn$ will be inhabited only if
$\mytmm \le \mytmn$:
- {\small\[
+ \[
\begin{array}{@{}l}
- \myfun{le} : \mynat \myarr \mynat \myarr \mytyp \mapsto \\
- \myind{2} \myabs{\myb{n}}{\\
- \myind{2}\myind{2} \mynat.\myfun{elim} \\
- \myind{2}\myind{2}\myind{2} \myb{n} \\
- \myind{2}\myind{2}\myind{2} (\myabs{\myarg}{\mynat \myarr \mytyp}) \\
- \myind{2}\myind{2}\myind{2} (\myabs{\myarg}{\myunit}) \\
- \myind{2}\myind{2}\myind{2} (\myabs{\myb{n}\, \myb{f}\, \myb{m}}{
+ \myfun{le} : \mynat \myarr \mynat \myarr \mytyp \\
+ \myfun{le} \myappsp \myb{n} \mapsto \\
+ \myind{2} \mynat.\myfun{elim} \\
+ \myind{2}\myind{2} \myb{n} \\
+ \myind{2}\myind{2} (\myabs{\myarg}{\mynat \myarr \mytyp}) \\
+ \myind{2}\myind{2} (\myabs{\myarg}{\myunit}) \\
+ \myind{2}\myind{2} (\myabs{\myb{n}\, \myb{f}\, \myb{m}}{
\mynat.\myfun{elim} \myappsp \myb{m} \myappsp (\myabs{\myarg}{\mytyp}) \myappsp \myempty \myappsp (\myabs{\myb{m'}\, \myarg}{\myapp{\myb{f}}{\myb{m'}}})
})
- }
\end{array}
- \]} We return $\myunit$ if the scrutinised is $\mydc{zero}$ (every
+ \]
+ We return $\myunit$ if the scrutinised is $\mydc{zero}$ (every
number in less or equal than zero), $\myempty$ if the first number is
a $\mydc{suc}$cessor and the second a $\mydc{zero}$, and we recurse if
they are both successors. Since we want the list to have possibly
`open' bounds, for example for empty lists, we create a type for
- `lifted' naturals with a bottom (less than everything) and top
- (greater than everything) elements, along with an associated comparison
+ `lifted' naturals with a bottom ($\le$ everything but itself) and top
+ ($\ge$ everything but itself) elements, along with an associated comparison
function:
- {\small\[
+ \[
\begin{array}{@{}l}
\myadt{\mytyc{Lift}}{ }{ }{\mydc{bot} \mydcsep \mydc{lift} \myappsp \mynat \mydcsep \mydc{top}}\\
- \myfun{le'} : \mytyc{Lift} \myarr \mytyc{Lift} \myarr \mytyp \mapsto \cdots \\
+ \myfun{le'} : \mytyc{Lift} \myarr \mytyc{Lift} \myarr \mytyp\\
+ \myfun{le'} \myappsp \myb{l_1} \mapsto \\
+ \myind{2} \mytyc{Lift}.\myfun{elim} \\
+ \myind{2}\myind{2} \myb{l_1} \\
+ \myind{2}\myind{2} (\myabs{\myarg}{\mytyc{Lift} \myarr \mytyp}) \\
+ \myind{2}\myind{2} (\myabs{\myarg}{\myunit}) \\
+ \myind{2}\myind{2} (\myabs{\myb{n_1}\, \myb{l_2}}{
+ \mytyc{Lift}.\myfun{elim} \myappsp \myb{l_2} \myappsp (\myabs{\myarg}{\mytyp}) \myappsp \myempty \myappsp (\myabs{\myb{n_2}}{\myfun{le} \myappsp \myb{n_1} \myappsp \myb{n_2}}) \myappsp \myunit
+ }) \\
+ \myind{2}\myind{2} (\myabs{\myb{l_2}}{
+ \mytyc{Lift}.\myfun{elim} \myappsp \myb{l_2} \myappsp (\myabs{\myarg}{\mytyp}) \myappsp \myempty \myappsp (\myabs{\myarg}{\myempty}) \myappsp \myunit
+ })
\end{array}
- \]} Finally, we can defined a type of ordered lists. The type is
- parametrised over two values representing the lower and upper bounds
- of the elements, as opposed to the type parameters that we are used
- to. Then, an empty list will have to have evidence that the bounds
- are ordered, and each time we add an element we require the list to
- have a matching lower bound:
- {\small\[
+ \]
+ Finally, we can define a type of ordered lists. The type is
+ parametrised over two \emph{values} representing the lower and upper
+ bounds of the elements, as opposed to the \emph{type} parameters
+ that we are used to in Haskell or similar languages. An empty
+ list will have to have evidence that the bounds are ordered, and
+ each time we add an element we require the list to have a matching
+ lower bound:
+ \[
\begin{array}{@{}l}
\myadt{\mytyc{OList}}{\myappsp (\myb{low}\ \myb{upp} {:} \mytyc{Lift})}{\\ \myind{2}}{
- \mydc{nil} \myappsp (\myfun{le'} \myappsp \myb{low} \myappsp \myb{upp}) \mydcsep \mydc{cons} \myappsp (\myb{n} {:} \mynat) \myappsp \mytyc{OList} \myappsp (\myfun{lift} \myappsp \myb{n}) \myappsp (\myfun{le'} \myappsp \myb{low} \myappsp (\myfun{lift} \myappsp \myb{n})
+ \mydc{nil} \myappsp (\myfun{le'} \myappsp \myb{low} \myappsp \myb{upp}) \mydcsep \mydc{cons} \myappsp (\myb{n} {:} \mynat) \myappsp (\mytyc{OList} \myappsp (\myfun{lift} \myappsp \myb{n}) \myappsp \myb{upp}) \myappsp (\myfun{le'} \myappsp \myb{low} \myappsp (\myfun{lift} \myappsp \myb{n})
}
\end{array}
- \]} If we want we can then employ this structure to write and prove
- correct various sorting algorithms\footnote{See this presentation by
- Conor McBride:
+ \]
+ Note that in the $\mydc{cons}$ constructor we quantify over the first
+ argument, which will determine the type of the following
+ arguments---again something we cannot do in systems like Haskell. If
+ we want we can then employ this structure to write and prove correct
+ various sorting algorithms.\footnote{See this presentation by Conor
+ McBride:
\url{https://personal.cis.strath.ac.uk/conor.mcbride/Pivotal.pdf},
and this blog post by the author:
- \url{http://mazzo.li/posts/AgdaSort.html}.}.
-
- % TODO
+ \url{http://mazzo.li/posts/AgdaSort.html}.}
\item[Dependent products] Apart from $\mysyn{data}$, $\mykant$\ offers
us another way to define types: $\mysyn{record}$. A record is a
- datatype with one constructor and `projections' to extract specific
+ data type with one constructor and `projections' to extract specific
fields of the said constructor.
For example, we can recover dependent products:
- {\small\[
+ \[
\begin{array}{@{}l}
\myreco{\mytyc{Prod}}{\myappsp (\myb{A} {:} \mytyp) \myappsp (\myb{B} {:} \myb{A} \myarr \mytyp)}{\\ \myind{2}}{\myfst : \myb{A}, \mysnd : \myapp{\myb{B}}{\myb{fst}}}
\end{array}
- \]}
+ \]
Here $\myfst$ and $\mysnd$ are the projections, with their respective
- types. Note that each field can refer to the preceding fields. A
- constructor will be automatically generated, under the name of
- $\mytyc{Prod}.\mydc{constr}$. Dually to data types, we will omit the
- type constructor prefix for record projections.
+ types. Note that each field can refer to the preceding fields---in
+ this case we have the type of $\myfun{snd}$ depending on the value of
+ $\myfun{fst}$. A constructor will be automatically generated, under
+ the name of $\mytyc{Prod}.\mydc{constr}$. Dually to data types, we
+ will omit the type constructor prefix for record projections.
Following the bidirectionality of the system, we have that projections
(the destructors of the record) infer the type, while the constructor
gets checked:
\begin{center}
- \small
+ \mysmall
\begin{tabular}{cc}
\AxiomC{$\mychk{\mytmm}{\mytya}$}
\AxiomC{$\mychk{\mytmn}{\myapp{\mytyb}{\mytmm}}$}
\UnaryInfC{\phantom{$\myinf{\myfun{snd} \myappsp \mytmt}{\mytyb \myappsp (\myfst \myappsp \mytmt)}$}}
\DisplayProof
&
- \AxiomC{$\myinf{\mytmt}{\mytyc{Prod} \myappsp \mytya \myappsp \mytyb}$}
+ \AxiomC{$\hspace{0.2cm}\myinf{\mytmt}{\mytyc{Prod} \myappsp \mytya \myappsp \mytyb}\hspace{0.2cm}$}
\UnaryInfC{$\myinf{\myfun{fst} \myappsp \mytmt}{\mytya}$}
\noLine
\UnaryInfC{$\myinf{\myfun{snd} \myappsp \mytmt}{\mytyb \myappsp (\myfst \myappsp \mytmt)}$}
\DisplayProof
\end{tabular}
\end{center}
- What we have is equivalent to ITT's dependent products.
+ What we have defined here is equivalent to ITT's dependent products.
+
\end{description}
\begin{figure}[p]
- \begin{subfigure}[b]{\textwidth}
- \vspace{-1cm}
+ \vspace{-.5cm}
\mydesc{syntax}{ }{
- \footnotesize
+ \small
$
\begin{array}{l}
\mynamesyn ::= \cdots \mysynsep \mytyc{D} \mysynsep \mytyc{D}.\mydc{c} \mysynsep \mytyc{D}.\myfun{f}
$
}
+ \mynegder
+
\mydesc{syntax elaboration:}{\mydeclsyn \myelabf \mytmsyn ::= \cdots}{
- \footnotesize
+ \small
$
\begin{array}{r@{\ }l}
& \myadt{\mytyc{D}}{\mytele}{}{\cdots\ |\ \mydc{c}_n : \mytele_n } \\
$
}
+ \mynegder
+
\mydesc{context elaboration:}{\myelab{\mydeclsyn}{\myctx}}{
- \footnotesize
+ \small
\AxiomC{$
\begin{array}{c}
}
+ \mynegder
+
\mydesc{reduction elaboration:}{\mydeclsyn \myelabf \myctx \vdash \mytmsyn \myred \mytmsyn}{
- \footnotesize
+ \small
$\myadt{\mytyc{D}}{\mytele}{}{ \cdots \ |\ \mydc{c}_n : \mytele_n } \ \ \myelabf$
\AxiomC{$\mytyc{D} : \mytele \myarr \mytyp \in \myctx$}
\AxiomC{$\mytyc{D}.\mydc{c}_i : \mytele;\mytele_i \myarr \myapp{\mytyc{D}}{\mytelee} \in \myctx$}
\end{array}
$
}
- \end{subfigure}
- \begin{subfigure}[b]{\textwidth}
+ \mynegder
+
\mydesc{syntax elaboration:}{\myelab{\mydeclsyn}{\mytmsyn ::= \cdots}}{
- \footnotesize
+ \small
$
\begin{array}{r@{\ }c@{\ }l}
\myctx & \myelabt & \myreco{\mytyc{D}}{\mytele}{}{ \cdots, \myfun{f}_n : \myse{F}_n } \\
$
}
+ \mynegder
\mydesc{context elaboration:}{\myelab{\mydeclsyn}{\myctx}}{
- \footnotesize
+ \small
\AxiomC{$
\begin{array}{c}
\myinf{\mytele \myarr \mytyp}{\mytyp}\hspace{0.8cm}
\DisplayProof
}
+ \mynegder
+
\mydesc{reduction elaboration:}{\mydeclsyn \myelabf \myctx \vdash \mytmsyn \myred \mytmsyn}{
- \footnotesize
+ \small
$\myreco{\mytyc{D}}{\mytele}{}{ \cdots, \myfun{f}_n : \myse{F}_n } \ \ \myelabf$
\AxiomC{$\mytyc{D} \in \myctx$}
\UnaryInfC{$\myctx \vdash \myapp{\mytyc{D}.\myfun{f}_i}{(\mytyc{D}.\mydc{constr} \myappsp \vec{t})} \myred t_i$}
\DisplayProof
}
- \end{subfigure}
\caption{Elaboration for data types and records.}
\label{fig:elab}
\end{figure}
-Following the intuition given by the examples, the mechanised
-elaboration is presented in figure \ref{fig:elab}, which is essentially
-a modification of figure 9 of \citep{McBride2004}\footnote{However, our
- datatypes do not have indices, we do bidirectional typechecking by
- treating constructors/destructors as syntactic constructs, and we have
- records.}.
-
-In data types declarations we allow recursive occurrences as long as
-they are \emph{strictly positive}, employing a syntactic check to make
-sure that this is the case. See \cite{Dybjer1991} for a more formal
-treatment of inductive definitions in ITT.
+\begin{mydef}[Elaboration for user defined types]
+ Following the intuition given by the examples, the full elaboration
+ machinery is presented Figure \ref{fig:elab}.
+\end{mydef}
+Our elaboration is essentially a modification of Figure 9 of
+\cite{McBride2004}. However, our data types are not inductive
+families,\footnote{See Section \ref{sec:future-work} for a brief
+ description of inductive families.} we do bidirectional type checking
+by treating constructors/destructors as syntax, and we have records.
+
+\begin{mydef}[Strict positivity]
+ A inductive type declaration is \emph{strictly positive} if recursive
+ occurrences of the type we are defining do not appear embedded
+ anywhere in the domain part of any function in the types for the data
+ constructors.
+\end{mydef}
+In data type declarations we allow recursive occurrences as long as they
+are strictly positive, which ensures the consistency of the theory. To
+achieve that we employing a syntactic check to make sure that this is
+the case---in fact the check is stricter than necessary for simplicity,
+given that we allow recursive occurrences only at the top level of data
+constructor arguments. For example a definition of the $\mytyc{W}$ type
+is accepted in Agda but rejected in \mykant. This is to make the
+eliminator generation simpler, and in practice it is seldom an
+impediment.
+
+Without these precautions, we can easily derive any type with no
+recursion:
+\begin{Verbatim}
+data Fix a = Fix (Fix a -> a) -- Negative occurrence of `Fix a'
+-- Term inhabiting any type `a'
+boom :: a
+boom = (\f -> f (Fix f)) (\x -> (\(Fix f) -> f) x x)
+\end{Verbatim}
+See \cite{Dybjer1991} for a more formal treatment of inductive
+definitions in ITT.
For what concerns records, recursive occurrences are disallowed. The
reason for this choice is answered by the reason for the choice of
having records at all: we need records to give the user types with
-$\eta$-laws for equality, as we saw in section % TODO add section
-and in the treatment of OTT in section \ref{sec:ott}. If we tried to
+$\eta$-laws for equality, as we saw in Section \ref{sec:eta-expand}
+and in the treatment of OTT in Section \ref{sec:ott}. If we tried to
$\eta$-expand recursive data types, we would expand forever.
+\begin{mydef}[Bidirectional type checking for elaborated types]
To implement bidirectional type checking for constructors and
destructors, we store their types in full in the context, and then
-instantiate when due:
-
-\mydesc{typing:}{ }{
- \AxiomC{$
+instantiate when due.
+\end{mydef}
+\mynegder
+\mydesc{typing:}{\myctx
+ \vdash \mytmsyn \Updownarrow \mytmsyn}{ \AxiomC{$
\begin{array}{c}
\mytyc{D} : \mytele \myarr \mytyp \in \myctx \hspace{1cm}
\mytyc{D}.\mydc{c} : \mytele \mycc \mytele' \myarr
\UnaryInfC{$\mychk{\myapp{\mytyc{D}.\mydc{c}}{\vec{t}}}{\myapp{\mytyc{D}}{\vec{A}}}$}
\DisplayProof
- \myderivsp
+ \myderivspp
\AxiomC{$\mytyc{D} : \mytele \myarr \mytyp \in \myctx$}
\AxiomC{$\mytyc{D}.\myfun{f} : \mytele \mycc (\myb{x} {:}
\myse{F})(\vec{A};\mytmt)}$}
\DisplayProof
}
+Note that for 0-ary type constructors, like $\mynat$, we do not need to
+check canonical terms: we can automatically infer that $\mydc{zero}$ and
+$\mydc{suc}\myappsp n$ are of type $\mynat$. \mykant\ implements this measure, even
+if it is not shown in the typing rule for simplicity.
\subsubsection{Why user defined types? Why eliminators?}
-% TODO reference levitated theories, indexed containers
+The hardest design choice in developing $\mykant$\ was to decide whether
+user defined types should be included, and how to handle them. As we
+saw, while we can devise general structures like $\mytyc{W}$, they are
+unsuitable both for for direct usage and `mechanical' usage. Thus most
+theorem provers in the wild provide some means for the user to define
+structures tailored to specific uses.
-foobar
+Even if we take user defined types for granted, while there is not much
+debate on how to handle records, there are two broad schools of thought
+regarding the handling of data types:
+\begin{description}
+\item[Fixed points and pattern matching] The road chosen by Agda and Coq.
+ Functions are written like in Haskell---matching on the input and with
+ explicit recursion. An external check on the recursive arguments
+ ensures that they are decreasing, and thus that all functions
+ terminate. This approach is the best in terms of user usability, but
+ it is tricky to implement correctly.
+
+\item[Elaboration into eliminators] The road chose by \mykant, and
+ pioneered by the Epigram line of work. The advantage is that we can
+ reduce every data type to simple definitions which guarantee
+ termination and are simple to reduce and type. It is however more
+ cumbersome to use than pattern matching, although \cite{McBride2004}
+ has shown how to implement an expressive pattern matching interface on
+ top of a larger set of combinators of those provided by \mykant.
+
+ We can go ever further down this road and elaborate the declarations
+ for data types themselves to a small set of primitives, so that our `core'
+ language will be very small and manageable
+ \citep{dagand2012elaborating, chapman2010gentle}.
+\end{description}
+
+We chose the safer and easier to implement path, given the time
+constraints and the higher confidence of correctness. See also Section
+\ref{sec:future-work} for a brief overview of ways to extend or treat
+user defined types.
\subsection{Cumulative hierarchy and typical ambiguity}
\label{sec:term-hierarchy}
-A type hierarchy as presented in section \label{sec:itt} is a
+Having a well founded type hierarchy is crucial if we want to retain
+consistency, otherwise we can break our type systems by proving bottom,
+as shown in Appendix \ref{app:hurkens}.
+
+However, hierarchy as presented in section \ref{sec:itt} is a
considerable burden on the user, on various levels. Consider for
-example how we recovered disjunctions in section \ref{sec:disju}: we
+example how we recovered disjunctions in Section \ref{sec:disju}: we
have a function that takes two $\mytyp_0$ and forms a new $\mytyp_0$.
-What if we wanted to form a disjunction containing two $\mytyp_0$, or
-$\mytyp_{42}$? Our definition would fail us, since $\mytyp_0 :
-\mytyp_1$.
+What if we wanted to form a disjunction containing something a
+$\mytyp_1$, or $\mytyp_{42}$? Our definition would fail us, since
+$\mytyp_1 : \mytyp_2$.
+
+\begin{figure}[b!]
+
+\mydesc{cumulativity:}{\myctx \vdash \mytmsyn \mycumul \mytmsyn}{
+ \begin{tabular}{ccc}
+ \AxiomC{$\myctx \vdash \mytya \mydefeq \mytyb$}
+ \UnaryInfC{$\myctx \vdash \mytya \mycumul \mytyb$}
+ \DisplayProof
+ &
+ \AxiomC{\phantom{$\myctx \vdash \mytya \mydefeq \mytyb$}}
+ \UnaryInfC{$\myctx \vdash \mytyp_l \mycumul \mytyp_{l+1}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myctx \vdash \mytya \mycumul \mytyb$}
+ \AxiomC{$\myctx \vdash \mytyb \mycumul \myse{C}$}
+ \BinaryInfC{$\myctx \vdash \mytya \mycumul \myse{C}$}
+ \DisplayProof
+ \end{tabular}
+
+ \myderivspp
+
+ \begin{tabular}{ccc}
+ \AxiomC{$\myjud{\mytmt}{\mytya}$}
+ \AxiomC{$\myctx \vdash \mytya \mycumul \mytyb$}
+ \BinaryInfC{$\myjud{\mytmt}{\mytyb}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myctx \vdash \mytya_1 \mydefeq \mytya_2$}
+ \AxiomC{$\myctx; \myb{x} : \mytya_1 \vdash \mytyb_1 \mycumul \mytyb_2$}
+ \BinaryInfC{$\myctx (\myb{x} {:} \mytya_1) \myarr \mytyb_1 \mycumul (\myb{x} {:} \mytya_2) \myarr \mytyb_2$}
+ \DisplayProof
+ \end{tabular}
+}
+\caption{Cumulativity rules for base types in \mykant, plus a
+ `conversion' rule for cumulative types.}
+ \label{fig:cumulativity}
+\end{figure}
One way to solve this issue is a \emph{cumulative} hierarchy, where
$\mytyp_{l_1} : \mytyp_{l_2}$ iff $l_1 < l_2$. This way we retain
consistency, while allowing for `large' definitions that work on small
-types too. For example we might define our disjunction to be
-{\small\[
+types too.
+
+\begin{mydef}[Cumulativity for \mykant' base types]
+ Figure \ref{fig:cumulativity} gives a formal definition of
+ \emph{cumulativity} for the base types. Similar measures can be taken
+ for user defined types, withe the type living in the least upper bound
+ of the levels where the types contained data live.
+\end{mydef}
+For example we might define our disjunction to be
+\[
\myarg\myfun{$\vee$}\myarg : \mytyp_{100} \myarr \mytyp_{100} \myarr \mytyp_{100}
-\]}
+\]
And hope that $\mytyp_{100}$ will be large enough to fit all the types
that we want to use with our disjunction. However, there are two
-problems with this. First, there is the obvious clumsyness of having to
-manually specify the size of types. More importantly, if we want to use
+problems with this. First, clumsiness of having to manually specify the
+size of types is still there. More importantly, if we want to use
$\myfun{$\vee$}$ itself as an argument to other type-formers, we need to
make sure that those allow for types at least as large as
$\mytyp_{100}$.
A better option is to employ a mechanised version of what Russell called
\emph{typical ambiguity}: we let the user live under the illusion that
$\mytyp : \mytyp$, but check that the statements about types are
-consistent behind the hood. $\mykant$\ implements this following the
-lines of \cite{Huet1988}. See also \citep{Harper1991} for a published
-reference, although describing a more complex system allowing for both
-explicit and explicit hierarchy at the same time.
+consistent under the hood. $\mykant$\ implements this following the
+plan given by \cite{Huet1988}. See also \cite{Harper1991} for a
+published reference, although describing a more complex system allowing
+for both explicit and explicit hierarchy at the same time.
We define a partial ordering on the levels, with both weak ($\le$) and
-strong ($<$) constraints---the laws governing them being the same as the
+strong ($<$) constraints, the laws governing them being the same as the
ones governing $<$ and $\le$ for the natural numbers. Each occurrence
-of $\mytyp$ is decorated with a unique reference, and we keep a set of
-constraints and add new constraints as we type check, generating new
-references when needed.
+of $\mytyp$ is decorated with a unique reference. We keep a set of
+constraints regarding the ordering of each occurrence of $\mytyp$, each
+represented by its unique reference. We add new constraints as we type
+check, generating new references when needed.
For example, when type checking the type $\mytyp\, r_1$, where $r_1$
denotes the unique reference assigned to that term, we will generate a
-new fresh reference $\mytyp\, r_2$, and add the constraint $r_1 < r_2$
-to the set. When type checking $\myctx \vdash
+new fresh reference and return the type $\mytyp\, r_2$, adding the
+constraint $r_1 < r_2$ to the set. When type checking $\myctx \vdash
\myfora{\myb{x}}{\mytya}{\mytyb}$, if $\myctx \vdash \mytya : \mytyp\,
r_1$ and $\myctx; \myb{x} : \mytyb \vdash \mytyb : \mytyp\,r_2$; we will
generate new reference $r$ and add $r_1 \le r$ and $r_2 \le r$ to the
set.
If at any point the constraint set becomes inconsistent, type checking
-fails. Moreover, when comparing two $\mytyp$ terms we equate their
-respective references with two $\le$ constraints---the details are
-explained in section \ref{sec:hier-impl}.
+fails. Moreover, when comparing two $\mytyp$ terms---during the process
+of deciding definitional equality for two terms---we equate their
+respective references with two $\le$ constraints. Implementation
+details are given in Section \ref{sec:hier-impl}.
Another more flexible but also more verbose alternative is the one
chosen by Agda, where levels can be quantified so that the relationship
between arguments and result in type formers can be explicitly
expressed:
-{\small\[
+\[
\myarg\myfun{$\vee$}\myarg : (l_1\, l_2 : \mytyc{Level}) \myarr \mytyp_{l_1} \myarr \mytyp_{l_2} \myarr \mytyp_{l_1 \mylub l_2}
-\]}
+\]
Inference algorithms to automatically derive this kind of relationship
-are currently subject of research. We chose less flexible but more
+are currently subject of research. We choose a less flexible but more
concise way, since it is easier to implement and better understood.
\subsection{Observational equality, \mykant\ style}
There are two correlated differences between $\mykant$\ and the theory
used to present OTT. The first is that in $\mykant$ we have a type
hierarchy, which lets us, for example, abstract over types. The second
-is that we let the user define inductive types.
+is that we let the user define inductive types and records.
Reconciling propositions for OTT and a hierarchy had already been
-investigated by Conor McBride\footnote{See
- \url{http://www.e-pig.org/epilogue/index.html?p=1098.html}.}, and we
-follow his footsteps. Most of the work, as an extension of elaboration,
-is to generate reduction rules and coercions.
+investigated by Conor McBride,\footnote{See
+ \url{http://www.e-pig.org/epilogue/index.html?p=1098.html}.} and we
+follow some of his suggestions, with some innovation. Most of the dirty
+work, as an extension of elaboration, is to handle reduction rules and
+coercions for data types---both type constructors and data constructors.
\subsubsection{The \mykant\ prelude, and $\myprop$ositions}
Before defining $\myprop$, we define some basic types inside $\mykant$,
-as the target for the $\myprop$ decoder:
-\begin{framed}
-\small
-$
-\begin{array}{l}
+as the target for the $\myprop$ decoder.
+\begin{mydef}[\mykant' propositional prelude]\ \end{mydef}
+\[
+\begin{array}{@{}l}
\myadt{\mytyc{Empty}}{}{ }{ } \\
\myfun{absurd} : (\myb{A} {:} \mytyp) \myarr \mytyc{Empty} \myarr \myb{A} \mapsto \\
\myind{2} \myabs{\myb{A\ \myb{bot}}}{\mytyc{Empty}.\myfun{elim} \myappsp \myb{bot} \myappsp (\myabs{\_}{\myb{A}})} \\
\myreco{\mytyc{Prod}}{\myappsp (\myb{A}\ \myb{B} {:} \mytyp)}{ }{\myfun{fst} : \myb{A}, \myfun{snd} : \myb{B} }
\end{array}
-$
-\end{framed}
-When using $\mytyc{Prod}$, we shall use $\myprod$ to define `nested'
-products, and $\myproj{n}$ to project elements from them, so that
-{\small
-\[
-\begin{array}{@{}l}
-\mytya \myprod \mytyb = \mytyc{Prod} \myappsp \mytya \myappsp (\mytyc{Prod} \myappsp \mytyb \myappsp \myunit) \\
-\mytya \myprod \mytyb \myprod \myse{C} = \mytyc{Prod} \myappsp \mytya \myappsp (\mytyc{Prod} \myappsp \mytyb \myappsp (\mytyc{Prod} \myappsp \mytyc \myappsp \myunit)) \\
-\myind{2} \vdots \\
-\myproj{1} : \mytyc{Prod} \myappsp \mytya \myappsp \mytyb \myarr \mytya \\
-\myproj{2} : \mytyc{Prod} \myappsp \mytya \myappsp (\mytyc{Prod} \myappsp \mytyb \myappsp \myse{C}) \myarr \mytyb \\
-\myind{2} \vdots
-\end{array}
\]
-}
-And so on, so that $\myproj{n}$ will work with all products with at
-least than $n$ elements. Then we can define propositions, and decoding:
+\begin{mydef}[Propositions and decoding]\ \end{mydef}
+\mynegder
\mydesc{syntax}{ }{
$
\begin{array}{r@{\ }c@{\ }l}
\end{array}
$
}
-
+\mynegder
\mydesc{proposition decoding:}{\myprdec{\mytmsyn} \myred \mytmsyn}{
\begin{tabular}{cc}
$
&
$
\begin{array}{r@{ }c@{ }l@{\ }c@{\ }l}
- \myprdec{&\myse{P} \myand \myse{Q} &} & \myred & \myprdec{\myse{P}} \myprod \myprdec{\myse{Q}} \\
+ \myprdec{&\myse{P} \myand \myse{Q} &} & \myred & \mytyc{Prod} \myappsp \myprdec{\myse{P}} \myappsp \myprdec{\myse{Q}} \\
\myprdec{&\myprfora{\myb{x}}{\mytya}{\myse{P}} &} & \myred &
\myfora{\myb{x}}{\mytya}{\myprdec{\myse{P}}}
\end{array}
\end{tabular}
}
-\subsubsection{Why $\myprop$?}
-
-It is worth to ask if $\myprop$ is needed at all. It is perfectly
-possible to have the type checker identify propositional types
-automatically, and in fact that is what The author initially planned to
-identify the propositional fragment internally \cite{Jacobs1994}.
-
-% TODO finish
+We will overload the $\myand$ symbol to define `nested' products, and
+$\myproj{n}$ to project elements from them, so that
+\[
+\begin{array}{@{}l}
+\mytya \myand \mytyb = \mytya \myand (\mytyb \myand \mytop) \\
+\mytya \myand \mytyb \myand \myse{C} = \mytya \myand (\mytyb \myand (\myse{C} \myand \mytop)) \\
+\myind{2} \vdots \\
+\myproj{1} : \myprdec{\mytya \myand \mytyb} \myarr \myprdec{\mytya} \\
+\myproj{2} : \myprdec{\mytya \myand \mytyb \myand \myse{C}} \myarr \myprdec{\mytyb} \\
+\myind{2} \vdots
+\end{array}
+\]
+And so on, so that $\myproj{n}$ will work with all products with at
+least than $n$ elements. Logically a 0-ary $\myand$ will correspond to
+$\mytop$.
-\subsubsection{OTT constructs}
+\subsubsection{Some OTT examples}
-Before presenting the direction that $\mykant$\ takes, let's consider
-some examples of use-defined data types, and the result we would expect,
+Before presenting the direction that $\mykant$\ takes, let us consider
+two examples of use-defined data types, and the result we would expect
given what we already know about OTT, assuming the same propositional
equalities.
\begin{description}
-\item[Product types] Let's consider first the already mentioned
+\item[Product types] Let us consider first the already mentioned
dependent product, using the alternate name $\mysigma$\footnote{For
extra confusion, `dependent products' are often called `dependent
sums' in the literature, referring to the interpretation that
identifies the first element as a `tag' deciding the type of the
second element, which lets us recover sum types (disjuctions), as we
- saw in section \ref{sec:user-type}. Thus, $\mysigma$.} to
- avoid confusion with the $\mytyc{Prod}$ in the prelude: {\small\[
+ saw in Section \ref{sec:depprod}. Thus, $\mysigma$.} to
+ avoid confusion with the $\mytyc{Prod}$ in the prelude:
+ \[
\begin{array}{@{}l}
\myreco{\mysigma}{\myappsp (\myb{A} {:} \mytyp) \myappsp (\myb{B} {:} \myb{A} \myarr \mytyp)}{\\ \myind{2}}{\myfst : \myb{A}, \mysnd : \myapp{\myb{B}}{\myb{fst}}}
\end{array}
- \]} Let's start with type-level equality. The result we want is
- {\small\[
+ \]
+ First type-level equality. The result we want is
+ \[
\begin{array}{@{}l}
\mysigma \myappsp \mytya_1 \myappsp \mytyb_1 \myeq \mysigma \myappsp \mytya_2 \myappsp \mytyb_2 \myred \\
\myind{2} \mytya_1 \myeq \mytya_2 \myand \myprfora{\myb{x_1}}{\mytya_1}{\myprfora{\myb{x_2}}{\mytya_2}{\myjm{\myb{x_1}}{\mytya_1}{\myb{x_2}}{\mytya_2}} \myimpl \myapp{\mytyb_1}{\myb{x_1}} \myeq \myapp{\mytyb_2}{\myb{x_2}}}
\end{array}
- \]} The difference here is that in the original presentation of OTT
- the type binders are explicit, while here $\mytyb_1$ and $\mytyb_2$
+ \]
+ The difference here is that in the original presentation of OTT the
+ type binders are explicit, while here $\mytyb_1$ and $\mytyb_2$ are
functions returning types. We can do this thanks to the type
hierarchy, and this hints at the fact that heterogeneous equality will
- have to allow $\mytyp$ `to the right of the colon', and in fact this
- provides the solution to simplify the equality above.
+ have to allow $\mytyp$ `to the right of the colon'. Indeed,
+ heterogeneous equalities involving abstractions over types will
+ provide the solution to simplify the equality above.
If we take, just like we saw previously in OTT
- {\small\[
+ \[
\begin{array}{@{}l}
\myjm{\myse{f}_1}{\myfora{\mytya_1}{\myb{x_1}}{\mytyb_1}}{\myse{f}_2}{\myfora{\mytya_2}{\myb{x_2}}{\mytyb_2}} \myred \\
\myind{2} \myprfora{\myb{x_1}}{\mytya_1}{\myprfora{\myb{x_2}}{\mytya_2}{
\myjm{\myapp{\myse{f}_1}{\myb{x_1}}}{\mytyb_1[\myb{x_1}]}{\myapp{\myse{f}_2}{\myb{x_2}}}{\mytyb_2[\myb{x_2}]}
}}
\end{array}
- \]} Then we can simply take
- {\small\[
+ \]
+ Then we can simply have
+ \[
\begin{array}{@{}l}
\mysigma \myappsp \mytya_1 \myappsp \mytyb_1 \myeq \mysigma \myappsp \mytya_2 \myappsp \mytyb_2 \myred \\ \myind{2} \mytya_1 \myeq \mytya_2 \myand \myjm{\mytyb_1}{\mytya_1 \myarr \mytyp}{\mytyb_2}{\mytya_2 \myarr \mytyp}
\end{array}
- \]} Which will reduce to precisely what we desire. For what
- concerns coercions and quotation, things stay the same (apart from the
- fact that we apply to the second argument instead of substituting).
- We can recognise records such as $\mysigma$ as such and employ
- projections in value equality, coercions, and quotation; as to not
- impede progress if not necessary.
+ \]
+ Which will reduce to precisely what we desire, but with an
+ heterogeneous equalities relating types instead of values:
+ \[
+ \begin{array}{@{}l}
+ \mytya_1 \myeq \mytya_2 \myand \myjm{\mytyb_1}{\mytya_1 \myarr \mytyp}{\mytyb_2}{\mytya_2 \myarr \mytyp} \myred \\
+ \mytya_1 \myeq \mytya_2 \myand
+ \myprfora{\myb{x_1}}{\mytya_1}{\myprfora{\myb{x_2}}{\mytya_2}{
+ \myjm{\myb{x_1}}{\mytya_1}{\myb{x_2}}{\mytya_2} \myimpl
+ \myjm{\myapp{\mytyb_1}{\myb{x_1}}}{\mytyp}{\myapp{\mytyb_2}{\myb{x_2}}}{\mytyp}
+ }}
+ \end{array}
+ \]
+ If we pretend for the moment that those heterogeneous equalities were
+ type equalities, things run smoothly. For what concerns coercions and
+ quotation, things stay the same (apart from the fact that we apply to
+ the second argument instead of substituting). We can recognise
+ records such as $\mysigma$ as such and employ projections in value
+ equality and coercions; as to not impede progress if not necessary.
\item[Lists] Now for finite lists, which will give us a taste for data
constructors:
- {\small\[
+ \[
\begin{array}{@{}l}
\myadt{\mylist}{\myappsp (\myb{A} {:} \mytyp)}{ }{\mydc{nil} \mydcsep \mydc{cons} \myappsp \myb{A} \myappsp (\myapp{\mylist}{\myb{A}})}
\end{array}
- \]}
+ \]
Type equality is simple---we only need to compare the parameter:
- {\small\[
+ \[
\mylist \myappsp \mytya_1 \myeq \mylist \myappsp \mytya_2 \myred \mytya_1 \myeq \mytya_2
- \]} For coercions, we transport based on the constructor, recycling
- the proof for the inductive occurrence: {\small\[
+ \]
+ For coercions, we transport based on the constructor, recycling the
+ proof for the inductive occurrence:
+ \[
\begin{array}{@{}l@{\ }c@{\ }l}
\mycoe \myappsp (\mylist \myappsp \mytya_1) \myappsp (\mylist \myappsp \mytya_2) \myappsp \myse{Q} \myappsp \mydc{nil} & \myred & \mydc{nil} \\
\mycoe \myappsp (\mylist \myappsp \mytya_1) \myappsp (\mylist \myappsp \mytya_2) \myappsp \myse{Q} \myappsp (\mydc{cons} \myappsp \mytmm \myappsp \mytmn) & \myred & \\
\multicolumn{3}{l}{\myind{2} \mydc{cons} \myappsp (\mycoe \myappsp \mytya_1 \myappsp \mytya_2 \myappsp \myse{Q} \myappsp \mytmm) \myappsp (\mycoe \myappsp (\mylist \myappsp \mytya_1) \myappsp (\mylist \myappsp \mytya_2) \myappsp \myse{Q} \myappsp \mytmn)}
\end{array}
- \]} Value equality is unsurprising---we match the constructors, and
+ \]
+ Value equality is unsurprising---we match the constructors, and
return bottom for mismatches. However, we also need to equate the
- parameter in $\mydc{nil}$: {\small\[
+ parameter in $\mydc{nil}$:
+ \[
\begin{array}{r@{ }c@{\ }c@{\ }c@{}l@{\ }c@{\ }r@{}c@{\ }c@{\ }c@{}l@{\ }l}
(& \mydc{nil} & : & \myapp{\mylist}{\mytya_1} &) & \myeq & (& \mydc{nil} & : & \myapp{\mylist}{\mytya_2} &) \myred \mytya_1 \myeq \mytya_2 \\
(& \mydc{cons} \myappsp \mytmm_1 \myappsp \mytmn_1 & : & \myapp{\mylist}{\mytya_1} &) & \myeq & (& \mydc{cons} \myappsp \mytmm_2 \myappsp \mytmn_2 & : & \myapp{\mylist}{\mytya_2} &) \myred \\
(& \mydc{nil} & : & \myapp{\mylist}{\mytya_1} &) & \myeq & (& \mydc{cons} \myappsp \mytmm_2 \myappsp \mytmn_2 & : & \myapp{\mylist}{\mytya_2} &) \myred \mybot \\
(& \mydc{cons} \myappsp \mytmm_1 \myappsp \mytmn_1 & : & \myapp{\mylist}{\mytya_1} &) & \myeq & (& \mydc{nil} & : & \myapp{\mylist}{\mytya_2} &) \myred \mybot
\end{array}
- \]}
- Finally, quotation
- % TODO quotation
+ \]
+\end{description}
-
-
+\subsubsection{Only one equality}
-\end{description}
-
+Given the examples above, a more `flexible' heterogeneous equality must
+emerge, since of the fact that in $\mykant$ we re-gain the possibility
+of abstracting and in general handling types in a way that was not
+possible in the original OTT presentation. Moreover, we found that the
+rules for value equality work well if used with user defined type
+abstractions---for example in the case of dependent products we recover
+the original definition with explicit binders, in a natural manner.
+
+\begin{mydef}[Propositions, coercions, coherence, equalities and
+ equality reduction for \mykant] See Figure \ref{fig:kant-eq-red}.
+\end{mydef}
+\begin{mydef}[Type equality in \mykant]
+ We define $\mytya \myeq \mytyb$ as an abbreviation for
+ $\myjm{\mytya}{\mytyp}{\mytyb}{\mytyp}$.
+\end{mydef}
+
+In fact, we can drop a separate notion of type-equality, which will
+simply be served by $\myjm{\mytya}{\mytyp}{\mytyb}{\mytyp}$. We shall
+still distinguish equalities relating types for hierarchical
+purposes. We exploit record to perform $\eta$-expansion. Moreover,
+given the nested $\myand$s, values of data types with zero constructors
+(such as $\myempty$) and records with zero destructors (such as
+$\myunit$) will be automatically always identified as equal. As in the
+original OTT, and for the same reasons, we can take $\myfun{coh}$ as
+axiomatic.
+
+
+\begin{figure}[p]
\mydesc{syntax}{ }{
+ \small
$
\begin{array}{r@{\ }c@{\ }l}
\mytmsyn & ::= & \cdots \mysynsep \mycoee{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn} \mysynsep
$
}
-\begin{figure}[p]
-\mydesc{typing:}{\myctx \vdash \myprsyn \myred \myprsyn}{
- \footnotesize
+\mynegder
+
+\mydesc{typing:}{\myctx \vdash \mytmsyn \Leftrightarrow \mytmsyn}{
+ \small
\begin{tabular}{cc}
- \AxiomC{$\myjud{\myse{P}}{\myprdec{\mytya \myeq \mytyb}}$}
- \AxiomC{$\myjud{\mytmt}{\mytya}$}
- \BinaryInfC{$\myjud{\mycoee{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\mytyb}$}
+ \AxiomC{$\mychk{\myse{P}}{\myprdec{\mytya \myeq \mytyb}}$}
+ \AxiomC{$\mychk{\mytmt}{\mytya}$}
+ \BinaryInfC{$\myinf{\mycoee{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\mytyb}$}
\DisplayProof
&
- \AxiomC{$\myjud{\myse{P}}{\myprdec{\mytya \myeq \mytyb}}$}
- \AxiomC{$\myjud{\mytmt}{\mytya}$}
- \BinaryInfC{$\myjud{\mycohh{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\myprdec{\myjm{\mytmt}{\mytya}{\mycoee{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\mytyb}}}$}
+ \AxiomC{$\mychk{\myse{P}}{\myprdec{\mytya \myeq \mytyb}}$}
+ \AxiomC{$\mychk{\mytmt}{\mytya}$}
+ \BinaryInfC{$\myinf{\mycohh{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\myprdec{\myjm{\mytmt}{\mytya}{\mycoee{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\mytyb}}}$}
\DisplayProof
\end{tabular}
}
+
+\mynegder
+
\mydesc{propositions:}{\myjud{\myprsyn}{\myprop}}{
- \footnotesize
+ \small
\begin{tabular}{cc}
\AxiomC{\phantom{$\myjud{\myse{P}}{\myprop}$}}
\UnaryInfC{$\myjud{\mytop}{\myprop}$}
\DisplayProof
\end{tabular}
- \myderivsp
+ \myderivspp
\begin{tabular}{cc}
\AxiomC{$
\DisplayProof
\end{tabular}
}
- % TODO equality for decodings
+
+\mynegder
+
\mydesc{equality reduction:}{\myctx \vdash \myprsyn \myred \myprsyn}{
- \footnotesize
+ \small
+ \begin{tabular}{cc}
\AxiomC{}
\UnaryInfC{$\myctx \vdash \myjm{\mytyp}{\mytyp}{\mytyp}{\mytyp} \myred \mytop$}
\DisplayProof
+ &
+ \AxiomC{}
+ \UnaryInfC{$\myctx \vdash \myjm{\myprdec{\myse{P}}}{\mytyp}{\myprdec{\myse{Q}}}{\mytyp} \myred \mytop$}
+ \DisplayProof
+ \end{tabular}
- \myderivsp
+ \myderivspp
\AxiomC{}
\UnaryInfC{$
\myctx \vdash &
\myjm{\myfora{\myb{x_1}}{\mytya_1}{\mytyb_1}}{\mytyp}{\myfora{\myb{x_2}}{\mytya_2}{\mytyb_2}}{\mytyp} \myred \\
& \myind{2} \mytya_2 \myeq \mytya_1 \myand \myprfora{\myb{x_2}}{\mytya_2}{\myprfora{\myb{x_1}}{\mytya_1}{
- \myjm{\myb{x_2}}{\mytya_2}{\myb{x_1}}{\mytya_1} \myimpl \mytyb_1 \myeq \mytyb_2
+ \myjm{\myb{x_2}}{\mytya_2}{\myb{x_1}}{\mytya_1} \myimpl \mytyb_1[\myb{x_1}] \myeq \mytyb_2[\myb{x_2}]
}}
\end{array}
$}
\DisplayProof
- \myderivsp
+ \myderivspp
\AxiomC{}
\UnaryInfC{$
\DisplayProof
- \myderivsp
+ \myderivspp
\AxiomC{$\mytyc{D} : \mytele \myarr \mytyp \in \myctx$}
\UnaryInfC{$
$}
\DisplayProof
- \myderivsp
+ \myderivspp
\AxiomC{$
\begin{array}{@{}c}
\mydataty(\mytyc{D}, \myctx)\hspace{0.8cm}
- \mytyc{D}.\mydc{c} : \mytele;\mytele' \myarr \mytyc{D} \myappsp \mytelee \in \myctx \\
+ \mytyc{D}.\mydc{c} : \mytele;\mytele' \myarr \mytyc{D} \myappsp \mytelee \in \myctx \hspace{0.8cm}
\mytele_A = (\mytele;\mytele')\vec{A}\hspace{0.8cm}
\mytele_B = (\mytele;\mytele')\vec{B}
\end{array}
$}
\DisplayProof
- \myderivsp
+ \myderivspp
\AxiomC{$\mydataty(\mytyc{D}, \myctx)$}
\UnaryInfC{$
$}
\DisplayProof
- \myderivsp
+ \myderivspp
\AxiomC{$
\begin{array}{@{}c}
$}
\DisplayProof
- \myderivsp
+ \myderivspp
\AxiomC{}
\UnaryInfC{$\myjm{\mytmm}{\mytya}{\mytmn}{\mytyb} \myred \mybot\ \text{if $\mytya$ and $\mytyb$ are canonical types.}$}
\DisplayProof
}
- \caption{Equality reduction for $\mykant$.}
+\caption{Propositions and equality reduction in $\mykant$. We assume
+ the presence of $\mydataty$ and $\myisreco$ as operations on the
+ context to recognise whether a user defined type is a data type or a
+ record.}
\label{fig:kant-eq-red}
\end{figure}
+\subsubsection{Coercions}
+
+For coercions the algorithm is messier and not reproduced here for lack
+of a decent notation---the details are hairy but uninteresting. To give
+an idea of the possible complications, let us conceive a type that
+showcases trouble not arising in the previous examples.
+\[
+\begin{array}{@{}l}
+\myadt{\mytyc{Max}}{\myappsp (\myb{A} {:} \mynat \myarr \mytyp) \myappsp (\myb{B} {:} (\myb{x} {:} \mynat) \myarr \myb{A} \myappsp \myb{x} \myarr \mytyp) \myappsp (\myb{k} {:} \mynat)}{ \\ \myind{2}}{
+ \mydc{max} \myappsp (\myb{A} \myappsp \myb{k}) \myappsp (\myb{x} {:} \mynat) \myappsp (\myb{a} {:} \myb{A} \myappsp \myb{x}) \myappsp (\myb{B} \myappsp \myb{x} \myappsp \myb{a})
+}
+\end{array}
+\]
+For type equalities we will have
+\[
+\begin{array}{@{}l@{\ }l}
+ \myjm{\mytyc{Max} \myappsp \mytya_1 \myappsp \mytyb_1 \myappsp \myse{k}_1}{\mytyp}{\mytyc{Max} \myappsp \mytya_2 \myappsp \myappsp \mytyb_2 \myappsp \myse{k}_2}{\mytyp} & \myred \\[0.2cm]
+ \begin{array}{@{}l}
+ \myjm{\mytya_1}{\mynat \myarr \mytyp}{\mytya_2}{\mynat \myarr \mytyp} \myand \\
+ \myjm{\mytyb_1}{(\myb{x} {:} \mynat) \myarr \mytya_1 \myappsp \myb{x} \myarr \mytyp}{\mytyb_2}{(\myb{x} {:} \mynat) \myarr \mytya_2 \myappsp \myb{x} \myarr \mytyp} \\
+ \myjm{\myse{k}_1}{\mynat}{\myse{k}_2}{\mynat}
+ \end{array} & \myred \\[0.7cm]
+ \begin{array}{@{}l}
+ (\mynat \myeq \mynat \myand (\myprfora{\myb{x_1}\, \myb{x_2}}{\mynat}{\myjm{\myb{x_1}}{\mynat}{\myb{x_2}}{\mynat} \myimpl \myapp{\mytya_1}{\myb{x_1}} \myeq \myapp{\mytya_2}{\myb{x_2}}})) \myand \\
+ (\mynat \myeq \mynat \myand \left(
+ \begin{array}{@{}l}
+ \myprfora{\myb{x_1}\, \myb{x_2}}{\mynat}{\myjm{\myb{x_1}}{\mynat}{\myb{x_2}}{\mynat} \myimpl \\ \myjm{\mytyb_1 \myappsp \myb{x_1}}{\mytya_1 \myappsp \myb{x_1} \myarr \mytyp}{\mytyb_2 \myappsp \myb{x_2}}{\mytya_2 \myappsp \myb{x_2} \myarr \mytyp}}
+ \end{array}
+ \right)) \myand \\
+ \myjm{\myse{k}_1}{\mynat}{\myse{k}_2}{\mynat}
+ \end{array} & \myred \\[0.9cm]
+ \begin{array}{@{}l}
+ (\mytop \myand (\myprfora{\myb{x_1}\, \myb{x_2}}{\mynat}{\myjm{\myb{x_1}}{\mynat}{\myb{x_2}}{\mynat} \myimpl \myapp{\mytya_1}{\myb{x_1}} \myeq \myapp{\mytya_2}{\myb{x_2}}})) \myand \\
+ (\mytop \myand \left(
+ \begin{array}{@{}l}
+ \myprfora{\myb{x_1}\, \myb{x_2}}{\mynat}{\myjm{\myb{x_1}}{\mynat}{\myb{x_2}}{\mynat} \myimpl \\
+ \myprfora{\myb{y_1}}{\mytya_1 \myappsp \myb{x_1}}{\myprfora{\myb{y_2}}{\mytya_2 \myappsp \myb{x_2}}{\myjm{\myb{y_1}}{\mytya_1 \myappsp \myb{x_1}}{\myb{y_2}}{\mytya_2 \myappsp \myb{x_2}} \myimpl \\
+ \mytyb_1 \myappsp \myb{x_1} \myappsp \myb{y_1} \myeq \mytyb_2 \myappsp \myb{x_2} \myappsp \myb{y_2}}}}
+ \end{array}
+ \right)) \myand \\
+ \myjm{\myse{k}_1}{\mynat}{\myse{k}_2}{\mynat}
+ \end{array} &
+\end{array}
+\]
+The result, while looking complicated, is actually saying something
+simple---given equal inputs, the parameters for $\mytyc{Max}$ will
+return equal types. Moreover, we have evidence that the two $\myb{k}$
+parameters are equal. When coercing, we need to mechanically generate
+one proof of equality for each argument, and then coerce:
+\[
+\begin{array}{@{}l}
+\mycoee{(\mytyc{Max} \myappsp \mytya_1 \myappsp \mytyb_1 \myappsp \myse{k}_1)}{(\mytyc{Max} \myappsp \mytya_2 \myappsp \mytyb_2 \myappsp \myse{k}_2)}{\myse{Q}}{(\mydc{max} \myappsp \myse{ak}_1 \myappsp \myse{n}_1 \myappsp \myse{a}_1 \myappsp \myse{b}_1)} \myred \\
+\myind{2}
+\begin{array}[t]{l@{\ }l@{\ }c@{\ }l}
+ \mysyn{let} & \myb{Q_{Ak}} & \mapsto & \myhole{?} : \myprdec{\mytya_1 \myappsp \myse{k}_1 \myeq \mytya_2 \myappsp \myse{k}_2} \\
+ & \myb{ak_2} & \mapsto & \mycoee{(\mytya_1 \myappsp \myse{k}_1)}{(\mytya_2 \myappsp \myse{k}_2)}{\myb{Q_{Ak}}}{\myse{ak_1}} : \mytya_1 \myappsp \myse{k}_2 \\
+ & \myb{Q_{\mathbb{N}}} & \mapsto & \myhole{?} : \myprdec{\mynat \myeq \mynat} \\
+ & \myb{n_2} & \mapsto & \mycoee{\mynat}{\mynat}{\myb{Q_{\mathbb{N}}}}{\myse{n_1}} : \mynat \\
+ & \myb{Q_A} & \mapsto & \myhole{?} : \myprdec{\mytya_1 \myappsp \myse{n_1} \myeq \mytya_2 \myappsp \myb{n_2}} \\
+ & \myb{a_2} & \mapsto & \mycoee{(\mytya_1 \myappsp \myse{n_1})}{(\mytya_2 \myappsp \myb{n_2})}{\myb{Q_A}} : \mytya_2 \myappsp \myb{n_2} \\
+ & \myb{Q_B} & \mapsto & \myhole{?} : \myprdec{\mytyb_1 \myappsp \myse{n_1} \myappsp \myse{a}_1 \myeq \mytyb_1 \myappsp \myb{n_2} \myappsp \myb{a_2}} \\
+ & \myb{b_2} & \mapsto & \mycoee{(\mytyb_1 \myappsp \myse{n_1} \myappsp \myse{a_1})}{(\mytyb_2 \myappsp \myb{n_2} \myappsp \myb{a_2})}{\myb{Q_B}} : \mytyb_2 \myappsp \myb{n_2} \myappsp \myb{a_2} \\
+ \mysyn{in} & \multicolumn{3}{@{}l}{\mydc{max} \myappsp \myb{ak_2} \myappsp \myb{n_2} \myappsp \myb{a_2} \myappsp \myb{b_2}}
+\end{array}
+\end{array}
+\]
+For equalities regarding types that are external to the data type we can
+derive a proof by reflexivity by invoking $\mydc{refl}$ as defined in
+Section \ref{sec:lazy}, and the instantiate arguments if we need too.
+In this case, for $\mynat$, we do not have any arguments. For
+equalities concerning arguments of the type constructor or already
+coerced arguments of the type constructor we have to refer to the right
+proof and use $\mycoh$erence when due, which is where the technical
+annoyance lies:
+\[
+\begin{array}{@{}l}
+\mycoee{(\mytyc{Max} \myappsp \mytya_1 \myappsp \mytyb_1 \myappsp \myse{k}_1)}{(\mytyc{Max} \myappsp \mytya_2 \myappsp \mytyb_2 \myappsp \myse{k}_2)}{\myse{Q}}{(\mydc{max} \myappsp \myse{ak}_1 \myappsp \myse{n}_1 \myappsp \myse{a}_1 \myappsp \myse{b}_1)} \myred \\
+\myind{2}
+\begin{array}[t]{l@{\ }l@{\ }c@{\ }l}
+ \mysyn{let} & \myb{Q_{Ak}} & \mapsto & (\myproj{2} \myappsp (\myproj{1} \myappsp \myse{Q})) \myappsp \myse{k_1} \myappsp \myse{k_2} \myappsp (\myproj{3} \myappsp \myse{Q}) : \myprdec{\mytya_1 \myappsp \myse{k}_1 \myeq \mytya_2 \myappsp \myse{k}_2} \\
+ & \myb{ak_2} & \mapsto & \mycoee{(\mytya_1 \myappsp \myse{k}_1)}{(\mytya_2 \myappsp \myse{k}_2)}{\myb{Q_{Ak}}}{\myse{ak_1}} : \mytya_1 \myappsp \myse{k}_2 \\
+ & \myb{Q_{\mathbb{N}}} & \mapsto & \mydc{refl} \myappsp \mynat : \myprdec{\mynat \myeq \mynat} \\
+ & \myb{n_2} & \mapsto & \mycoee{\mynat}{\mynat}{\myb{Q_{\mathbb{N}}}}{\myse{n_1}} : \mynat \\
+ & \myb{Q_A} & \mapsto & (\myproj{2} \myappsp (\myproj{1} \myappsp \myse{Q})) \myappsp \myse{n_1} \myappsp \myb{n_2} \myappsp (\mycohh{\mynat}{\mynat}{\myb{Q_{\mathbb{N}}}}{\myse{n_1}}) : \myprdec{\mytya_1 \myappsp \myse{n_1} \myeq \mytya_2 \myappsp \myb{n_2}} \\
+ & \myb{a_2} & \mapsto & \mycoee{(\mytya_1 \myappsp \myse{n_1})}{(\mytya_2 \myappsp \myb{n_2})}{\myb{Q_A}} : \mytya_2 \myappsp \myb{n_2} \\
+ & \myb{Q_B} & \mapsto & (\myproj{2} \myappsp (\myproj{2} \myappsp \myse{Q})) \myappsp \myse{n_1} \myappsp \myb{n_2} \myappsp \myb{Q_{\mathbb{N}}} \myappsp \myse{a_1} \myappsp \myb{a_2} \myappsp (\mycohh{(\mytya_1 \myappsp \myse{n_1})}{(\mytya_2 \myappsp \myse{n_2})}{\myb{Q_A}}{\myse{a_1}}) : \myprdec{\mytyb_1 \myappsp \myse{n_1} \myappsp \myse{a}_1 \myeq \mytyb_1 \myappsp \myb{n_2} \myappsp \myb{a_2}} \\
+ & \myb{b_2} & \mapsto & \mycoee{(\mytyb_1 \myappsp \myse{n_1} \myappsp \myse{a_1})}{(\mytyb_2 \myappsp \myb{n_2} \myappsp \myb{a_2})}{\myb{Q_B}} : \mytyb_2 \myappsp \myb{n_2} \myappsp \myb{a_2} \\
+ \mysyn{in} & \multicolumn{3}{@{}l}{\mydc{max} \myappsp \myb{ak_2} \myappsp \myb{n_2} \myappsp \myb{a_2} \myappsp \myb{b_2}}
+\end{array}
+\end{array}
+\]
+
\subsubsection{$\myprop$ and the hierarchy}
-Where is $\myprop$ placed in the $\mytyp$ hierarchy? At each universe
-level, we will have that
+We shall have, at each universe level, not only a $\mytyp_l$ but also a
+$\myprop_l$. Where will propositions placed in the type hierarchy? The
+main indicator is the decoding operator, since it converts into things
+that already live in the hierarchy. For example, if we have
+\[
+ \myprdec{\mynat \myarr \mybool \myeq \mynat \myarr \mybool} \myred
+ \mytop \myand ((\myb{x}\, \myb{y} : \mynat) \myarr \mytop \myarr \mytop)
+\]
+we will better make sure that the `to be decoded' is at level compatible
+(read: larger) with its reduction. In the example above, we will have
+that proposition to be at least as large as the type of $\mynat$, since
+the reduced proof will abstract over it. Pretending that we had
+explicit, non cumulative levels, it would be tempting to have
+\begin{center}
+\begin{tabular}{cc}
+ \AxiomC{$\myjud{\myse{Q}}{\myprop_l}$}
+ \UnaryInfC{$\myjud{\myprdec{\myse{Q}}}{\mytyp_l}$}
+ \DisplayProof
+&
+ \AxiomC{$\myjud{\mytya}{\mytyp_l}$}
+ \AxiomC{$\myjud{\mytyb}{\mytyp_l}$}
+ \BinaryInfC{$\myjud{\myjm{\mytya}{\mytyp_{l}}{\mytyb}{\mytyp_{l}}}{\myprop_l}$}
+ \DisplayProof
+\end{tabular}
+\end{center}
+$\mybot$ and $\mytop$ living at any level, $\myand$ and $\forall$
+following rules similar to the ones for $\myprod$ and $\myarr$ in
+Section \ref{sec:itt}. However, we need to be careful with value
+equality since for example we have that
+\[
+ \myprdec{\myjm{\myse{f}_1}{\myfora{\myb{x_1}}{\mytya_1}{\mytyb_1}}{\myse{f}_2}{\myfora{\myb{x_2}}{\mytya_2}{\mytyb_2}}}
+ \myred
+ \myfora{\myb{x_1}}{\mytya_1}{\myfora{\myb{x_2}}{\mytya_2}{\cdots}}
+\]
+where the proposition decodes into something of at least type $\mytyp_l$, where
+$\mytya_l : \mytyp_l$ and $\mytyb_l : \mytyp_l$. We can resolve this
+tension by making all equalities larger:
+\begin{prooftree}
+ \AxiomC{$\myjud{\mytmm}{\mytya}$}
+ \AxiomC{$\myjud{\mytya}{\mytyp_l}$}
+ \AxiomC{$\myjud{\mytmn}{\mytyb}$}
+ \AxiomC{$\myjud{\mytyb}{\mytyp_l}$}
+ \QuaternaryInfC{$\myjud{\myjm{\mytmm}{\mytya}{\mytmm}{\mytya}}{\myprop_l}$}
+\end{prooftree}
+This is disappointing, since type equalities will be needlessly large:
+$\myprdec{\myjm{\mytya}{\mytyp_l}{\mytyb}{\mytyp_l}} : \mytyp_{l + 1}$.
-\subsubsection{Quotation and irrelevance}
-\ref{sec:kant-irr}
+However, considering that our theory is cumulative, we can do better.
+Assuming rules for $\myprop$ cumulativity similar to the ones for
+$\mytyp$, we will have (with the conversion rule reproduced as a
+reminder):
+\begin{center}
+ \begin{tabular}{cc}
+ \AxiomC{$\myctx \vdash \mytya \mycumul \mytyb$}
+ \AxiomC{$\myjud{\mytmt}{\mytya}$}
+ \BinaryInfC{$\myjud{\mytmt}{\mytyb}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myjud{\mytya}{\mytyp_l}$}
+ \AxiomC{$\myjud{\mytyb}{\mytyp_l}$}
+ \BinaryInfC{$\myjud{\myjm{\mytya}{\mytyp_{l}}{\mytyb}{\mytyp_{l}}}{\myprop_l}$}
+ \DisplayProof
+ \end{tabular}
-foo
+ \myderivspp
-\section{\mykant : The practice}
-\label{sec:kant-practice}
+ \AxiomC{$\myjud{\mytmm}{\mytya}$}
+ \AxiomC{$\myjud{\mytya}{\mytyp_l}$}
+ \AxiomC{$\myjud{\mytmn}{\mytyb}$}
+ \AxiomC{$\myjud{\mytyb}{\mytyp_l}$}
+ \AxiomC{$\mytya$ and $\mytyb$ are not $\mytyp_{l'}$}
+ \QuinaryInfC{$\myjud{\myjm{\mytmm}{\mytya}{\mytmm}{\mytya}}{\myprop_l}$}
+ \DisplayProof
+\end{center}
-The codebase consists of around 2500 lines of Haskell, as reported by
-the \texttt{cloc} utility. The high level design is inspired by Conor
-McBride's work on various incarnations of Epigram, and specifically by
-the first version as described \citep{McBride2004} and the codebase for
-the new version \footnote{Available intermittently as a \texttt{darcs}
- repository at \url{http://sneezy.cs.nott.ac.uk/darcs/Pig09}.}. In
-many ways \mykant\ is something in between the first and second version
-of Epigram.
+That is, we are small when we can (type equalities) and large otherwise.
+This would not work in a non-cumulative theory because subject reduction
+would not hold. Consider for instance
+\[
+ \myjm{\mynat}{\myITE{\mytrue}{\mytyp_0}{\mytyp_0}}{\mybool}{\myITE{\mytrue}{\mytyp_0}{\mytyp_0}}
+ : \myprop_1
+\]
+which reduces to
+\[\myjm{\mynat}{\mytyp_0}{\mybool}{\mytyp_0} : \myprop_0 \]
+We need members of $\myprop_0$ to be members of $\myprop_1$ too, which
+will be the case with cumulativity. This buys us a cheap type level
+equality without having to replicate functionality with a dedicated
+construct.
-The interaction happens in a read-eval-print loop (REPL). The REPL is a
-available both as a commandline application and in a web interface,
-which is available at \url{kant.mazzo.li} and presents itself as in
-figure \ref{fig:kant-web}.
+\subsubsection{Quotation and definitional equality}
+\label{sec:kant-irr}
-\begin{figure}
- \centering{
- \includegraphics[scale=1.0]{kant-web.png}
+Now we can give an account of definitional equality, by explaining how
+to perform quotation (as defined in Section \ref{sec:eta-expand})
+towards the goal described in Section \ref{sec:ott-quot}.
+
+We want to:
+\begin{itemize}
+\item Perform $\eta$-expansion on functions and records.
+
+\item As a consequence of the previous point, identify all records with
+no projections as equal, since they will have only one element.
+
+\item Identify all members of types with no constructors (and thus no
+ elements) as equal.
+
+\item Identify all equivalent proofs as equal---with `equivalent proof'
+we mean those proving the same propositions.
+
+\item Advance coercions working across definitionally equal types.
+\end{itemize}
+Towards these goals and following the intuition between bidirectional
+type checking we define two mutually recursive functions, one quoting
+canonical terms against their types (since we need the type to type check
+canonical terms), one quoting neutral terms while recovering their
+types.
+\begin{mydef}[Quotation for \mykant]
+The full procedure for quotation is shown in Figure
+\ref{fig:kant-quot}.
+\end{mydef}
+We $\boxed{\text{box}}$ the neutral proofs and
+neutral members of empty types, following the notation in
+\cite{Altenkirch2007}, and we make use of $\mydefeq_{\mybox}$ which
+compares terms syntactically up to $\alpha$-renaming, but also up to
+equivalent proofs: we consider all boxed content as equal.
+
+Our quotation will work on normalised terms, so that all defined values
+will have been replaced. Moreover, we match on data type eliminators
+and all their arguments, so that $\mynat.\myfun{elim} \myappsp \mytmm
+\myappsp \myse{P} \myappsp \vec{\mytmn}$ will stand for
+$\mynat.\myfun{elim}$ applied to the scrutinised $\mynat$, the
+predicate, and the two cases. This measure can be easily implemented by
+checking the head of applications and `consuming' the needed terms.
+Thus, we gain proof irrelevance, and not only for a more useful
+definitional equality, but also for example to eliminate all
+propositional content when compiling.
+
+\begin{figure}[t]
+ \mydesc{canonical quotation:}{\mycanquot(\myctx, \mytmsyn : \mytmsyn) \mymetagoes \mytmsyn}{
+ \small
+ $
+ \begin{array}{@{}l@{}l}
+ \mycanquot(\myctx,\ \mytmt : \mytyc{D} \myappsp \vec{A} &) \mymetaguard \mymeta{empty}(\myctx, \mytyc{D}) \mymetagoes \boxed{\mytmt} \\
+ \mycanquot(\myctx,\ \mytmt : \mytyc{D} \myappsp \vec{A} &) \mymetaguard \mymeta{record}(\myctx, \mytyc{D}) \mymetagoes
+ \mytyc{D}.\mydc{constr} \myappsp \cdots \myappsp \mycanquot(\myctx, \mytyc{D}.\myfun{f}_n : (\myctx(\mytyc{D}.\myfun{f}_n))(\vec{A};\mytmt)) \\
+ \mycanquot(\myctx,\ \mytyc{D}.\mydc{c} \myappsp \vec{t} : \mytyc{D} \myappsp \vec{A} &) \mymetagoes \cdots \\
+ \mycanquot(\myctx,\ \myse{f} : \myfora{\myb{x}}{\mytya}{\mytyb} &) \mymetagoes \myabs{\myb{x}}{\mycanquot(\myctx; \myb{x} : \mytya, \myapp{\myse{f}}{\myb{x}} : \mytyb)} \\
+ \mycanquot(\myctx,\ \myse{p} : \myprdec{\myse{P}} &) \mymetagoes \boxed{\myse{p}}
+ \\
+ \mycanquot(\myctx,\ \mytmt : \mytya &) \mymetagoes \mytmt'\ \text{\textbf{where}}\ \mytmt' : \myarg = \myneuquot(\myctx, \mytmt)
+ \end{array}
+ $
}
- \caption{The \mykant\ web prompt.}
+
+ \mynegder
+
+ \mydesc{neutral quotation:}{\myneuquot(\myctx, \mytmsyn) \mymetagoes \mytmsyn : \mytmsyn}{
+ \small
+ $
+ \begin{array}{@{}l@{}l}
+ \myneuquot(\myctx,\ \myb{x} &) \mymetagoes \myb{x} : \myctx(\myb{x}) \\
+ \myneuquot(\myctx,\ \mytyp &) \mymetagoes \mytyp : \mytyp \\
+ \myneuquot(\myctx,\ \myfora{\myb{x}}{\mytya}{\mytyb} & ) \mymetagoes
+ \myfora{\myb{x}}{\myneuquot(\myctx, \mytya)}{\myneuquot(\myctx; \myb{x} : \mytya, \mytyb)} : \mytyp \\
+ \myneuquot(\myctx,\ \mytyc{D} \myappsp \vec{A} &) \mymetagoes \mytyc{D} \myappsp \cdots \mycanquot(\myctx, \mymeta{head}((\myctx(\mytyc{D}))(\mytya_1 \cdots \mytya_{n-1}))) : \mytyp \\
+ \myneuquot(\myctx,\ \myprdec{\myjm{\mytmm}{\mytya}{\mytmn}{\mytyb}} &) \mymetagoes \\
+ \multicolumn{2}{l}{\myind{2}\myprdec{\myjm{\mycanquot(\myctx, \mytmm : \mytya)}{\mytya'}{\mycanquot(\myctx, \mytmn : \mytyb)}{\mytyb'}} : \mytyp} \\
+ \multicolumn{2}{@{}l}{\myind{2}\text{\textbf{where}}\ \mytya' : \myarg = \myneuquot(\myctx, \mytya)} \\
+ \multicolumn{2}{@{}l}{\myind{2}\phantom{\text{\textbf{where}}}\ \mytyb' : \myarg = \myneuquot(\myctx, \mytyb)} \\
+ \myneuquot(\myctx,\ \mytyc{D}.\myfun{f} \myappsp \mytmt &) \mymetaguard \mymeta{record}(\myctx, \mytyc{D}) \mymetagoes \mytyc{D}.\myfun{f} \myappsp \mytmt' : (\myctx(\mytyc{D}.\myfun{f}))(\vec{A};\mytmt) \\
+ \multicolumn{2}{@{}l}{\myind{2}\text{\textbf{where}}\ \mytmt' : \mytyc{D} \myappsp \vec{A} = \myneuquot(\myctx, \mytmt)} \\
+ \myneuquot(\myctx,\ \mytyc{D}.\myfun{elim} \myappsp \mytmt \myappsp \myse{P} &) \mymetaguard \mymeta{empty}(\myctx, \mytyc{D}) \mymetagoes \mytyc{D}.\myfun{elim} \myappsp \boxed{\mytmt} \myappsp \myneuquot(\myctx, \myse{P}) : \myse{P} \myappsp \mytmt \\
+ \myneuquot(\myctx,\ \mytyc{D}.\myfun{elim} \myappsp \mytmm \myappsp \myse{P} \myappsp \vec{\mytmn} &) \mymetagoes \mytyc{D}.\myfun{elim} \myappsp \mytmm' \myappsp \myneuquot(\myctx, \myse{P}) \cdots : \myse{P} \myappsp \mytmm\\
+ \multicolumn{2}{@{}l}{\myind{2}\text{\textbf{where}}\ \mytmm' : \mytyc{D} \myappsp \vec{A} = \myneuquot(\myctx, \mytmm)} \\
+ \myneuquot(\myctx,\ \myapp{\myse{f}}{\mytmt} &) \mymetagoes \myapp{\myse{f'}}{\mycanquot(\myctx, \mytmt : \mytya)} : \mysub{\mytyb}{\myb{x}}{\mytmt} \\
+ \multicolumn{2}{@{}l}{\myind{2}\text{\textbf{where}}\ \myse{f'} : \myfora{\myb{x}}{\mytya}{\mytyb} = \myneuquot(\myctx, \myse{f})} \\
+ \myneuquot(\myctx,\ \mycoee{\mytya}{\mytyb}{\myse{Q}}{\mytmt} &) \mymetaguard \myneuquot(\myctx, \mytya) \mydefeq_{\mybox} \myneuquot(\myctx, \mytyb) \mymetagoes \myneuquot(\myctx, \mytmt) \\
+\myneuquot(\myctx,\ \mycoee{\mytya}{\mytyb}{\myse{Q}}{\mytmt} &) \mymetagoes
+ \mycoee{\myneuquot(\myctx, \mytya)}{\myneuquot(\myctx, \mytyb)}{\boxed{\myse{Q}}}{\myneuquot(\myctx, \mytmt)}
+ \end{array}
+ $
+ }
+ \caption{Quotation in \mykant. Along the already used
+ $\mymeta{record}$ meta-operation on the context we make use of
+ $\mymeta{empty}$, which checks if a certain type constructor has
+ zero data constructor. The `data constructor' cases for non-record,
+ non-empty, data types are omitted for brevity.}
+ \label{fig:kant-quot}
+\end{figure}
+
+\subsubsection{Why $\myprop$?}
+
+It is worth to ask if $\myprop$ is needed at all. It is perfectly
+possible to have the type checker identify propositional types
+automatically, and in fact in some sense we already do during equality
+reduction and quotation. However, this has the considerable
+disadvantage that we can never identify abstracted
+variables\footnote{And in general neutral terms, although we currently
+ do not have neutral propositions apart from equalities on neutral
+ terms.} of type $\mytyp$ as $\myprop$, thus forbidding the user to
+talk about $\myprop$ explicitly.
+
+This is a considerable impediment, for example when implementing
+\emph{quotient types}. With quotients, we let the user specify an
+equivalence class over a certain type, and then exploit this in various
+way---crucially, we need to be sure that the equivalence given is
+propositional, a fact which prevented the use of quotients in dependent
+type theories \citep{Jacobs1994}.
+
+\section{\mykant : the practice}
+\label{sec:kant-practice}
+
+\epigraph{\emph{It's alive!}}{Henry Frankenstein}
+
+The codebase consists of around 2500 lines of Haskell,\footnote{The full
+ source code is available under the GPL3 license at
+ \url{https://github.com/bitonic/kant}. `Kant' was a previous
+ incarnation of the software, and the name remained.} as reported by
+the \texttt{cloc} utility.
+
+We implement the type theory as described in Section
+\ref{sec:kant-theory}. The author learnt the hard way the
+implementation challenges for such a project, and ran out of time while
+implementing observational equality. While the constructs and typing
+rules are present, the machinery to make it happen (equality reduction,
+coercions, quotation, etc.) is not present yet.
+
+This considered, everything else presented in Section
+\ref{sec:kant-theory} is implemented and working well---and in fact all
+the examples presented in this thesis, apart from the ones that are
+equality related, have been encoded in \mykant\ in the Appendix.
+Moreover, given the detailed plan in the previous section, finishing off
+should not prove too much work.
+
+The interaction with the user takes place in a loop living in and
+updating a context of \mykant\ declarations, which presents itself as in
+Figure \ref{fig:kant-web}. Files with lists of declarations can also be
+loaded. The REPL is a available both as a command-line application and in
+a web interface, which is available at \url{bertus.mazzo.li}.
+
+A REPL cycle starts with the user inputting a \mykant\
+declaration or another REPL command, which then goes through various
+stages that can end up in a context update, or in failures of various
+kind. The process is described diagrammatically in figure
+\ref{fig:kant-process}.
+
+\begin{figure}[b!]
+{\small\begin{Verbatim}[frame=leftline,xleftmargin=3cm]
+B E R T U S
+Version 0.0, made in London, year 2013.
+>>> :h
+<decl> Declare value/data type/record
+:t <term> Typecheck
+:e <term> Normalise
+:p <term> Pretty print
+:l <file> Load file
+:r <file> Reload file (erases previous environment)
+:i <name> Info about an identifier
+:q Quit
+>>> :l data/samples/good/common.ka
+OK
+>>> :e plus three two
+suc (suc (suc (suc (suc zero))))
+>>> :t plus three two
+Type: Nat
+\end{Verbatim}
+}
+
+ \caption{A sample run of the \mykant\ prompt.}
\label{fig:kant-web}
\end{figure}
-The interaction with the user takes place in a loop living in and updating a
-context \mykant\ declarations. The user inputs a new declaration that goes
-through various stages starts with the user inputing a \mykant\ declaration or
-another REPL command, which then goes through various stages that can end up
-in a context update, or in failures of various kind. The process is described
-diagrammatically in figure \ref{fig:kant-process}:
\begin{description}
+
\item[Parse] In this phase the text input gets converted to a sugared
- version of the core language.
+ version of the core language. For example, we accept multiple
+ arguments in arrow types and abstractions, and we represent variables
+ with names, while as we will see in Section \ref{sec:term-repr} the
+ final term types uses a \emph{nameless} representation.
\item[Desugar] The sugared declaration is converted to a core term.
+ Most notably we go from names to nameless.
+
+\item[ConDestr] Short for `Constructors/Destructors', converts
+ applications of data destructors and constructors to a special form,
+ to perform bidirectional type checking.
\item[Reference] Occurrences of $\mytyp$ get decorated by a unique reference,
which is necessary to implement the type hierarchy check.
-\item[Elaborate] Convert the declaration to some context item, which might be
- a value declaration (type and body) or a data type declaration (constructors
- and destructors). This phase works in tandem with \textbf{Typechecking},
+\item[Elaborate/Typecheck/Evaluate] \textbf{Elaboration} converts the
+ declaration to some context items, which might be a value declaration
+ (type and body) or a data type declaration (constructors and
+ destructors). This phase works in tandem with \textbf{Type checking},
which in turns needs to \textbf{Evaluate} terms.
-\item[Distill] and report the result. `Distilling' refers to the process of
- converting a core term back to a sugared version that the user can
- visualise. This can be necessary both to display errors including terms or
- to display result of evaluations or type checking that the user has
- requested.
+\item[Distill] and report the result. `Distilling' refers to the
+ process of converting a core term back to a sugared version that we
+ can show to the user. This can be necessary both to display errors
+ including terms or to display result of evaluations or type checking
+ that the user has requested. Among the other things in this stage we
+ go from nameless back to names by recycling the names that the user
+ used originally, as to fabricate a term which is as close as possible
+ to what it originated from.
-\item[Pretty print] Format the terms in a nice way, and display the result to
+\item[Pretty print] Format the terms in a nice way, and display them to
the user.
\end{description}
\begin{figure}
- \centering{\small
+ \centering{\mysmall
\tikzstyle{block} = [rectangle, draw, text width=5em, text centered, rounded
corners, minimum height=2.5em, node distance=0.7cm]
\node [cloud] (user) {User};
\node [block, below left=1cm and 0.1cm of user] (parse) {Parse};
\node [block, below=of parse] (desugar) {Desugar};
- \node [block, below=of desugar] (reference) {Reference};
+ \node [block, below=of desugar] (condestr) {ConDestr};
+ \node [block, below=of condestr] (reference) {Reference};
\node [block, below=of reference] (elaborate) {Elaborate};
\node [block, left=of elaborate] (tycheck) {Typecheck};
\node [block, left=of tycheck] (evaluate) {Evaluate};
\node [decision, right=of elaborate] (error) {Error?};
- \node [block, right=of parse] (distill) {Distill};
- \node [block, right=of desugar] (update) {Update context};
+ \node [block, right=of parse] (pretty) {Pretty print};
+ \node [block, below=of pretty] (distill) {Distill};
+ \node [block, below=of distill] (update) {Update context};
\path [line] (user) -- (parse);
\path [line] (parse) -- (desugar);
- \path [line] (desugar) -- (reference);
+ \path [line] (desugar) -- (condestr);
+ \path [line] (condestr) -- (reference);
\path [line] (reference) -- (elaborate);
\path [line] (elaborate) edge[bend right] (tycheck);
\path [line] (tycheck) edge[bend right] (elaborate);
\path [line] (error) edge[out=0,in=0] node [near start] {yes} (distill);
\path [line] (error) -- node [near start] {no} (update);
\path [line] (update) -- (distill);
- \path [line] (distill) -- (user);
+ \path [line] (pretty) -- (user);
+ \path [line] (distill) -- (pretty);
\path [line] (tycheck) edge[bend right] (evaluate);
\path [line] (evaluate) edge[bend right] (tycheck);
\end{tikzpicture}
\label{fig:kant-process}
\end{figure}
-\subsection{Parsing and \texttt{Sugar}}
+Here we will review only a sampling of the more interesting
+implementation challenges present when implementing an interactive
+theorem prover.
+
+\subsection{Syntax}
+
+The syntax of \mykant\ is presented in Figure \ref{fig:syntax}.
+Examples showing the usage of most of the constructs---excluding the
+OTT-related ones---are present in Appendices \ref{app:kant-itt},
+\ref{app:kant-examples}, and \ref{app:hurkens}; plus a tutorial in
+Section \ref{sec:type-holes}. The syntax has grown organically with the
+needs of the language, and thus is not very sophisticated. The grammar
+is specified in and processed by the \texttt{happy} parser generator for
+Haskell.\footnote{Available at \url{http://www.haskell.org/happy}.}
+Tokenisation is performed by a simple hand written lexer.
+
+\begin{figure}[p]
+ \centering
+ $
+ \begin{array}{@{\ \ }l@{\ }c@{\ }l}
+ \multicolumn{3}{@{}l}{\text{A name, in regexp notation.}} \\
+ \mysee{name} & ::= & \texttt{[a-zA-Z] [a-zA-Z0-9'\_-]*} \\
+ \multicolumn{3}{@{}l}{\text{A binder might or might not (\texttt{\_}) bind a name.}} \\
+ \mysee{binder} & ::= & \mytermi{\_} \mysynsep \mysee{name} \\
+ \multicolumn{3}{@{}l}{\text{A series of typed bindings.}} \\
+ \mysee{telescope}\, \ \ \ & ::= & (\mytermi{[}\ \mysee{binder}\ \mytermi{:}\ \mysee{term}\ \mytermi{]}){*} \\
+ \multicolumn{3}{@{}l}{\text{Terms, including propositions.}} \\
+ \multicolumn{3}{@{}l}{
+ \begin{array}{@{\ \ }l@{\ }c@{\ }l@{\ \ \ \ \ }l}
+ \mysee{term} & ::= & \mysee{name} & \text{A variable.} \\
+ & | & \mytermi{*} & \text{\mytyc{Type}.} \\
+ & | & \mytermi{\{|}\ \mysee{term}{*}\ \mytermi{|\}} & \text{Type holes.} \\
+ & | & \mytermi{Prop} & \text{\mytyc{Prop}.} \\
+ & | & \mytermi{Top} \mysynsep \mytermi{Bot} & \text{$\mytop$ and $\mybot$.} \\
+ & | & \mysee{term}\ \mytermi{/\textbackslash}\ \mysee{term} & \text{Conjuctions.} \\
+ & | & \mytermi{[|}\ \mysee{term}\ \mytermi{|]} & \text{Proposition decoding.} \\
+ & | & \mytermi{coe}\ \mysee{term}\ \mysee{term}\ \mysee{term}\ \mysee{term} & \text{Coercion.} \\
+ & | & \mytermi{coh}\ \mysee{term}\ \mysee{term}\ \mysee{term}\ \mysee{term} & \text{Coherence.} \\
+ & | & \mytermi{(}\ \mysee{term}\ \mytermi{:}\ \mysee{term}\ \mytermi{)}\ \mytermi{=}\ \mytermi{(}\ \mysee{term}\ \mytermi{:}\ \mysee{term}\ \mytermi{)} & \text{Heterogeneous equality.} \\
+ & | & \mytermi{(}\ \mysee{compound}\ \mytermi{)} & \text{Parenthesised term.} \\
+ \mysee{compound} & ::= & \mytermi{\textbackslash}\ \mysee{binder}{*}\ \mytermi{=>}\ \mysee{term} & \text{Untyped abstraction.} \\
+ & | & \mytermi{\textbackslash}\ \mysee{telescope}\ \mytermi{:}\ \mysee{term}\ \mytermi{=>}\ \mysee{term} & \text{Typed abstraction.} \\
+ & | & \mytermi{forall}\ \mysee{telescope}\ \mysee{term} & \text{Universal quantification.} \\
+ & | & \mysee{arr} \\
+ \mysee{arr} & ::= & \mysee{telescope}\ \mytermi{->}\ \mysee{arr} & \text{Dependent function.} \\
+ & | & \mysee{term}\ \mytermi{->}\ \mysee{arr} & \text{Non-dependent function.} \\
+ & | & \mysee{term}{+} & \text {Application.}
+ \end{array}
+ } \\
+ \multicolumn{3}{@{}l}{\text{Typed names.}} \\
+ \mysee{typed} & ::= & \mysee{name}\ \mytermi{:}\ \mysee{term} \\
+ \multicolumn{3}{@{}l}{\text{Declarations.}} \\
+ \mysee{decl}& ::= & \mysee{value} \mysynsep \mysee{abstract} \mysynsep \mysee{data} \mysynsep \mysee{record} \\
+ \multicolumn{3}{@{}l}{\text{Defined values. The telescope specifies named arguments.}} \\
+ \mysee{value} & ::= & \mysee{name}\ \mysee{telescope}\ \mytermi{:}\ \mysee{term}\ \mytermi{=>}\ \mysee{term} \\
+ \multicolumn{3}{@{}l}{\text{Abstracted variables.}} \\
+ \mysee{abstract} & ::= & \mytermi{postulate}\ \mysee{typed} \\
+ \multicolumn{3}{@{}l}{\text{Data types, and their constructors.}} \\
+ \mysee{data} & ::= & \mytermi{data}\ \mysee{name}\ \mysee{telescope}\ \mytermi{->}\ \mytermi{*}\ \mytermi{=>}\ \mytermi{\{}\ \mysee{constrs}\ \mytermi{\}} \\
+ \mysee{constrs} & ::= & \mysee{typed} \\
+ & | & \mysee{typed}\ \mytermi{|}\ \mysee{constrs} \\
+ \multicolumn{3}{@{}l}{\text{Records, and their projections. The $\mysee{name}$ before the projections is the constructor name.}} \\
+ \mysee{record} & ::= & \mytermi{record}\ \mysee{name}\ \mysee{telescope}\ \mytermi{->}\ \mytermi{*}\ \mytermi{=>}\ \mysee{name}\ \mytermi{\{}\ \mysee{projs}\ \mytermi{\}} \\
+ \mysee{projs} & ::= & \mysee{typed} \\
+ & | & \mysee{typed}\ \mytermi{,}\ \mysee{projs}
+ \end{array}
+ $
+
+ \caption{\mykant' syntax. The non-terminals are marked with
+ $\langle\text{angle brackets}\rangle$ for greater clarity. The
+ syntax in the implementation is actually more liberal, for example
+ giving the possibility of using arrow types directly in
+ constructor/projection declarations.\\
+ Additionally, we give the user the possibility of using Unicode
+ characters instead of their ASCII counterparts, e.g. \texttt{→} in
+ place of \texttt{->}, \texttt{λ} in place of
+ \texttt{\textbackslash}, etc.}
+ \label{fig:syntax}
+\end{figure}
-\subsection{Term representation and context}
+\subsection{Term representation}
\label{sec:term-repr}
-\subsection{Type checking}
+\subsubsection{Naming and substituting}
+
+Perhaps surprisingly, one of the most difficult challenges in
+implementing a theory of the kind presented is choosing a good data type
+for terms, and specifically handling substitutions in a sane way.
+
+There are two broad schools of thought when it comes to naming
+variables, and thus substituting:
+\begin{description}
+\item[Nameful] Bound variables are represented by some enumerable data
+ type, just as we have described up to now, starting from Section
+ \ref{sec:untyped}. The problem is that avoiding name capturing is a
+ nightmare, both in the sense that it is not performant---given that we
+ need to rename rename substitute each time we `enter' a binder---but
+ most importantly given the fact that in even slightly more complicated
+ systems it is very hard to get right, even for experts.
+
+ One of the sore spots of explicit names is comparing terms up to
+ $\alpha$-renaming, which again generates a huge amounts of
+ substitutions and requires special care.
+
+\item[Nameless] We can capture the relationship between variables and
+ their binders, by getting rid of names altogether, and representing
+ bound variables with an index referring to the `binding' structure, a
+ notion introduced by \cite{de1972lambda}. Usually $0$ represents the
+ variable bound by the innermost binding structure, $1$ the
+ second-innermost, and so on. For instance with simple abstractions we
+ might have
+ \[
+ \begin{array}{@{}l}
+ \mymacol{red}{\lambda}\, (\mymacol{blue}{\lambda}\, \mymacol{blue}{0}\, (\mymacol{AgdaInductiveConstructor}{\lambda\, 0}))\, (\mymacol{AgdaFunction}{\lambda}\, \mymacol{red}{1}\, \mymacol{AgdaFunction}{0}) : ((\mytya \myarr \mytya) \myarr \mytyb) \myarr \mytyb\text{, which corresponds to} \\
+ \myabs{\myb{f}}{(\myabs{\myb{g}}{\myapp{\myb{g}}{(\myabs{\myb{x}}{\myb{x}})}}) \myappsp (\myabs{\myb{x}}{\myapp{\myb{f}}{\myb{x}}})} : ((\mytya \myarr \mytya) \myarr \mytyb) \myarr \mytyb
+ \end{array}
+ \]
+
+ While this technique is obviously terrible in terms of human
+ usability,\footnote{With some people going as far as defining it akin
+ to an inverse Turing test.} it is much more convenient as an
+ internal representation to deal with terms mechanically---at least in
+ simple cases. $\alpha$-renaming ceases to be an issue, and
+ term comparison is purely syntactical.
+
+ Nonetheless, more complex constructs such as pattern matching put
+ some strain on the indices and many systems end up using explicit
+ names anyway.
+
+\end{description}
+
+In the past decade or so advancements in the Haskell's type system and
+in general the spread new programming practices have made the nameless
+option much more amenable. \mykant\ thus takes the nameless path
+through the use of Edward Kmett's excellent \texttt{bound}
+library.\footnote{Available at
+ \url{http://hackage.haskell.org/package/bound}.} We describe the
+advantages of \texttt{bound}'s approach, but also its pitfalls in the
+previously relatively unknown territory of dependent
+types---\texttt{bound} being created mostly to handle more simply typed
+systems.
+
+ \texttt{bound} builds on the work of \cite{Bird1999}, who suggested to
+ parametrising the term type over the type of the variables, and `nest'
+ the type each time we enter a scope. If we wanted to define a term
+ for the untyped $\lambda$-calculus, we might have
+\begin{Verbatim}
+-- A type with no members.
+data Empty
+
+data Var v = Bound | Free v
+
+data Tm v
+ = V v -- Bound variable
+ | App (Tm v) (Tm v) -- Term application
+ | Lam (Tm (Var v)) -- Abstraction
+\end{Verbatim}
+Closed terms would be of type \texttt{Tm Empty}, so that there would be
+no occurrences of \texttt{V}. However, inside an abstraction, we can
+have \texttt{V Bound}, representing the bound variable, and inside a
+second abstraction we can have \texttt{V Bound} or \texttt{V (Free
+Bound)}. Thus the term
+\[\myabs{\myb{x}}{\myabs{\myb{y}}{\myb{x}}}\]
+can be represented as
+\begin{Verbatim}
+-- The top level term is of type `Tm Empty'.
+-- The inner term `Lam (Free Bound)' is of type `Tm (Var Empty)'.
+-- The second inner term `V (Free Bound)' is of type `Tm (Var (Var
+-- Empty))'.
+Lam (Lam (V (Free Bound)))
+\end{Verbatim}
+This allows us to reflect the `nestedness' of a type at the type level,
+and since we usually work with functions polymorphic on the parameter
+\texttt{v} it's very hard to make mistakes by putting terms of the wrong
+nestedness where they do not belong.
+
+Even more interestingly, the substitution operation is perfectly
+captured by the \verb|>>=| (bind) operator of the \texttt{Monad}
+type class:
+\begin{Verbatim}
+class Monad m where
+ return :: m a
+ (>>=) :: m a -> (a -> m b) -> m b
+
+instance Monad Tm where
+ -- `return'ing turns a variable into a `Tm'
+ return = V
+
+ -- `t >>= f' takes a term `t' and a mapping from variables to terms
+ -- `f' and applies `f' to all the variables in `t', replacing them
+ -- with the mapped terms.
+ V v >>= f = f v
+ App m n >>= f = App (m >>= f) (n >>= f)
+
+ -- `Lam' is the tricky case: we modify the function to work with bound
+ -- variables, so that if it encounters `Bound' it leaves it untouched
+ -- (since the mapping refers to the outer scope); if it encounters a
+ -- free variable it asks `f' for the term and then updates all the
+ -- variables to make them refer to the outer scope they were meant to
+ -- be in.
+ Lam s >>= f = Lam (s >>= bump)
+ where bump Bound = return Bound
+ bump (Free v) = f v >>= V . Free
+\end{Verbatim}
+With this in mind, we can define functions which will not only work on
+\verb|Tm|, but on any \verb|Monad|!
+\begin{Verbatim}
+-- Replaces free variable `v' with `m' in `n'.
+subst :: (Eq v, Monad m) => v -> m v -> m v -> m v
+subst v m n = n >>= \v' -> if v == v' then m else return v'
+
+-- Replace the variable bound by `s' with term `t'.
+inst :: Monad m => m v -> m (Var v) -> m v
+inst t s = s >>= \v -> case v of
+ Bound -> t
+ Free v' -> return v'
+\end{Verbatim}
+The beauty of this technique is that with a few simple functions we have
+defined all the core operations in a general and `obviously correct'
+way, with the extra confidence of having the type checker looking our
+back. For what concerns term equality, we can just ask the Haskell
+compiler to derive the instance for the \verb|Eq| type class and since
+we are nameless that will be enough (modulo fancy quotation).
+
+Moreover, if we take the top level term type to be \verb|Tm String|, we
+get a representation of terms with top-level definitions; where closed
+terms contain only \verb|String| references to said definitions---see
+also \cite{McBride2004b}.
+
+What are then the pitfalls of this seemingly invincible technique? The
+most obvious impediment is the need for polymorphic recursion.
+Functions traversing terms parametrised by the variable type will have
+types such as
+\begin{Verbatim}
+-- Infer the type of a term, or return an error.
+infer :: Tm v -> Either Error (Tm v)
+\end{Verbatim}
+When traversing under a \verb|Scope| the parameter changes from \verb|v|
+to \verb|Var v|, and thus if we do not specify the type for our function explicitly
+inference will fail---type inference for polymorphic recursion being
+undecidable \citep{henglein1993type}. This causes some annoyance,
+especially in the presence of many local definitions that we would like
+to leave untyped.
+
+But the real issue is the fact that giving a type parametrised over a
+variable---say \verb|m v|---a \verb|Monad| instance means being able to
+only substitute variables for values of type \verb|m v|. This is a
+considerable inconvenience. Consider for instance the case of
+telescopes, which are a central tool to deal with contexts and other
+constructs. In Haskell we can give them a faithful representation
+with a data type along the lines of
+\begin{Verbatim}
+data Tele m v = Empty (m v) | Bind (m v) (Tele m (Var v))
+type TeleTm = Tele Tm
+\end{Verbatim}
+The problem here is that what we want to substitute for variables in
+\verb|Tele m v| is \verb|m v| (probably \verb|Tm v|), not \verb|Tele m v| itself! What we need is
+\begin{Verbatim}
+bindTele :: Monad m => Tele m a -> (a -> m b) -> Tele m b
+substTele :: (Eq v, Monad m) => v -> m v -> Tele m v -> Tele m v
+instTele :: Monad m => m v -> Tele m (Var v) -> Tele m v
+\end{Verbatim}
+Not what \verb|Monad| gives us. Solving this issue in an elegant way
+has been a major sink of time and source of headaches for the author,
+who analysed some of the alternatives---most notably the work by
+\cite{weirich2011binders}---but found it impossible to give up the
+simplicity of the model above.
+
+That said, our term type is still reasonably brief, as shown in full in
+Appendix \ref{app:termrep}. The fact that propositions cannot be
+factored out in another data type is an instance of the problem
+described above. However the real pain is during elaboration, where we
+are forced to treat everything as a type while we would much rather have
+telescopes. Future work would include writing a library that marries
+more flexibility with a nice interface similar to the one of
+\verb|bound|.
+
+We also make use of a `forgetful' data type (as provided by
+\verb|bound|) to store user-provided variables names along with the
+`nameless' representation, so that the names will not be considered when
+compared terms, but will be available when distilling so that we can
+recover variable names that are as close as possible to what the user
+originally used.
+
+\subsubsection{Evaluation}
+
+Another source of contention related to term representation is dealing
+with evaluation. Here \mykant\ does not make bold moves, and simply
+employs substitution. When type checking we match types by reducing
+them to their weak head normal form, as to avoid unnecessary evaluation.
+
+We treat data types eliminators and record projections in an uniform
+way, by elaborating declarations in a series of \emph{rewriting rules}:
+\begin{Verbatim}
+type Rewr =
+ forall v.
+ Tm v -> -- Term to which the destructor is applied
+ [Tm v] -> -- List of other arguments
+ -- The result of the rewriting, if the eliminator reduces.
+ Maybe [Tm v]
+\end{Verbatim}
+A rewriting rule is polymorphic in the variable type, guaranteeing that
+it just pattern matches on terms structure and rearranges them in some
+way, and making it possible to apply it at any level in the term. When
+reducing a series of applications we match the first term and check if
+it is a destructor, and if that's the case we apply the reduction rule
+and reduce further if it yields a new list of terms.
+
+This has the advantage of simplicity, at the expense of being quite poor
+in terms of performance and that we need to do quotation manually. An
+alternative that solves both of these is the already mentioned
+\emph{normalization by evaluation}, where we would compute by turning
+terms into Haskell values, and then reify back to terms to compare
+them---a useful tutorial on this technique is given by \cite{Loh2010}.
+
+However, quotation has its disadvantages. The most obvious one is that
+it is less simple: we need to set up some infrastructure to handle the
+quotation and reification, while with substitution we have a uniform
+representation through the process of type checking. The second is that
+performance advantages can be rendered less effective by the continuous
+quoting and reifying, although this can probably be mitigated with some
+heuristics.
+
+\subsubsection{Parametrise everything!}
+\label{sec:parame}
+
+Through the life of a REPL cycle we need to execute two broad
+`effectful' actions:
+\begin{itemize}
+\item Retrieve, add, and modify elements to an environment. The
+ environment will contain not only types, but also the rewriting rules
+ presented in the previous section, and a counter to generate fresh
+ references for the type hierarchy.
+
+\item Throw various kinds of errors when something goes wrong: parsing,
+ type checking, input/output error when reading files, and more.
+\end{itemize}
+Haskell taught us the value of monads in programming languages, and in
+\mykant\ we keep this lesson in mind. All of the plumbing required to do
+the two actions above is provided by a very general \emph{monad
+ transformer} that we use through the codebase, \texttt{KMonadT}:
+\begin{Verbatim}
+newtype KMonad f v m a = KMonad (StateT (f v) (ErrorT KError m) a)
+
+data KError
+ = OutOfBounds Id
+ | DuplicateName Id
+ | IOError IOError
+ | ...
+\end{Verbatim}
+Without delving into the details of what a monad transformer
+is,\footnote{See
+ \url{https://en.wikibooks.org/wiki/Haskell/Monad_transformers.}} this
+is what \texttt{KMonadT} works with and provides:
+\begin{itemize}
+\item The \verb|v| parameter represents the parametrised variable for
+ the term type that we spoke about at the beginning of this section.
+ More on this later.
+
+\item The \verb|f| parameter indicates what kind of environment we are
+ holding. Sometimes we want to traverse terms without carrying the
+ entire environment, for various reasons---\texttt{KMonatT} lets us do
+ that. Note that \verb|f| is itself parametrised over \verb|v|. The
+ inner \verb|StateT| monad transformer lets us retrieve and modify this
+ environment at any time.
+
+\item The \verb|m| is the `inner' monad that we can `plug in' to be able
+ to perform more effectful actions in \texttt{KMonatT}. For example if we
+ plug the \texttt{IO} monad in, we will be able to do input/output.
+
+\item The inner \verb|ErrorT| lets us throw errors at any time. The
+ error type is \verb|KError|, which describes all the possible errors
+ that a \mykant\ process can throw.
+
+\item Finally, the \verb|a| parameter represents the return type of the
+ computation we are executing.
+\end{itemize}
-\subsection{Type hierarchy}
+The clever trick in \texttt{KMonadT} is to have it to be parametrised
+over the same type as the term type. This way, we can easily carry the
+environment while traversing under binders. For example, if we only
+needed to carry types of bound variables in the environment, we can
+quickly set up the following infrastructure:
+\begin{Verbatim}
+data Tm v = ...
+
+-- A context is a mapping from variables to types.
+newtype Ctx v = Ctx (v -> Tm v)
+
+-- A context monad holds a context.
+type CtxMonad v m = KMonadT Ctx v m
+
+-- Enter into a scope binding a type to the variable, execute a
+-- computation there, and return exit the scope returning to the `current'
+-- context.
+nestM :: Monad m => Tm v -> CtxMonad (Var v) m a -> CtxMonad v m a
+nestM = ...
+\end{Verbatim}
+Again, the types guard our back guaranteeing that we add a type when we
+enter a scope, and we discharge it when we get out. The author
+originally started with a more traditional representation and often
+forgot to add the right variable at the right moment. Using this
+practices it is very difficult to do so---we achieve correctness through
+types.
+
+In the actual \mykant\ codebase, we have also abstracted the concept of
+`context' further, so that we can easily embed contexts into other
+structures and write generic operations on all context-like
+structures.\footnote{See the \texttt{Kant.Cursor} module for details.}
+
+\subsection{Turning a hierarchy into some graphs}
\label{sec:hier-impl}
-\subsection{Elaboration}
+In this section we will explain how to implement the typical ambiguity
+we have spoken about in \ref{sec:term-hierarchy} efficiently, a subject
+which is often dismissed in the literature. As mentioned, we have to
+verify a the consistency of a set of constraints each time we add a new
+one. The constraints range over some set of variables whose members we
+will denote with $x, y, z, \dots$. and are of two kinds:
+\begin{center}
+ \begin{tabular}{cc}
+ $x \le y$ & $x < y$
+ \end{tabular}
+\end{center}
+
+Predictably, $\le$ expresses a reflexive order, and $<$ expresses an
+irreflexive order, both working with the same notion of equality, where
+$x < y$ implies $x \le y$---they behave like $\le$ and $<$ do for natural
+numbers (or in our case, levels in a type hierarchy). We also need an
+equality constraint ($x = y$), which can be reduced to two constraints
+$x \le y$ and $y \le x$.
+
+Given this specification, we have implemented a standalone Haskell
+module---that we plan to release as a library---to efficiently store and
+check the consistency of constraints. The problem predictably reduces
+to a graph algorithm, and for this reason we also implement a library
+for labelled graphs, since the existing Haskell graph libraries fell
+short in different areas.\footnote{We tried the \texttt{Data.Graph}
+ module in \url{http://hackage.haskell.org/package/containers}, and the
+ much more featureful \texttt{fgl} library
+ \url{http://hackage.haskell.org/package/fgl}.} The interfaces for
+these modules are shown in Appendix \ref{app:constraint}. The graph
+library is implemented as a modification of the code described by
+\cite{King1995}.
+
+We represent the set by building a graph where vertices are variables,
+and edges are constraints between them, labelled with the appropriate
+constraint: $x < y$ gives rise to a $<$-labelled edge from $x$ to $y$,
+and $x \le y$ to a $\le$-labelled edge from $x$ to $y$. As we add
+constraints, $\le$ constraints are replaced by $<$ constraints, so that
+if we started with an empty set and added
+\[
+ x < y,\ y \le z,\ z \le k,\ k < j,\ j \le y\
+\]
+it would generate the graph shown in Figure \ref{fig:graph-one-before},
+but adding $z < k$ would strengthen the edge from $z$ to $k$, as shown
+in \ref{fig:graph-one-after}.
+
+\begin{figure}[t]
+ \centering
+ \begin{subfigure}[b]{0.3\textwidth}
+ \begin{tikzpicture}[node distance=1.5cm]
+ % Place nodes
+ \node (x) {$x$};
+ \node [right of=x] (y) {$y$};
+ \node [right of=y] (z) {$z$};
+ \node [below of=z] (k) {$k$};
+ \node [left of=k] (j) {$j$};
+ %% Lines
+ \path[->]
+ (x) edge node [above] {$<$} (y)
+ (y) edge node [above] {$\le$} (z)
+ (z) edge node [right] {$\le$} (k)
+ (k) edge node [below] {$\le$} (j)
+ (j) edge node [left ] {$\le$} (y);
+ \end{tikzpicture}
+ \caption{Before $z < k$.}
+ \label{fig:graph-one-before}
+ \end{subfigure}%
+ ~
+ \begin{subfigure}[b]{0.3\textwidth}
+ \begin{tikzpicture}[node distance=1.5cm]
+ % Place nodes
+ \node (x) {$x$};
+ \node [right of=x] (y) {$y$};
+ \node [right of=y] (z) {$z$};
+ \node [below of=z] (k) {$k$};
+ \node [left of=k] (j) {$j$};
+ %% Lines
+ \path[->]
+ (x) edge node [above] {$<$} (y)
+ (y) edge node [above] {$\le$} (z)
+ (z) edge node [right] {$<$} (k)
+ (k) edge node [below] {$\le$} (j)
+ (j) edge node [left ] {$\le$} (y);
+ \end{tikzpicture}
+ \caption{After $z < k$.}
+ \label{fig:graph-one-after}
+ \end{subfigure}%
+ ~
+ \begin{subfigure}[b]{0.3\textwidth}
+ \begin{tikzpicture}[remember picture, node distance=1.5cm]
+ \begin{pgfonlayer}{foreground}
+ % Place nodes
+ \node (x) {$x$};
+ \node [right of=x] (y) {$y$};
+ \node [right of=y] (z) {$z$};
+ \node [below of=z] (k) {$k$};
+ \node [left of=k] (j) {$j$};
+ %% Lines
+ \path[->]
+ (x) edge node [above] {$<$} (y)
+ (y) edge node [above] {$\le$} (z)
+ (z) edge node [right] {$<$} (k)
+ (k) edge node [below] {$\le$} (j)
+ (j) edge node [left ] {$\le$} (y);
+ \end{pgfonlayer}{foreground}
+ \end{tikzpicture}
+ \begin{tikzpicture}[remember picture, overlay]
+ \begin{pgfonlayer}{background}
+ \fill [red, opacity=0.3, rounded corners]
+ (-2.7,2.6) rectangle (-0.2,0.05)
+ (-4.1,2.4) rectangle (-3.3,1.6);
+ \end{pgfonlayer}{background}
+ \end{tikzpicture}
+ \caption{SCCs.}
+ \label{fig:graph-one-scc}
+ \end{subfigure}%
+ \caption{Strong constraints overrule weak constraints.}
+ \label{fig:graph-one}
+\end{figure}
+
+\begin{mydef}[Strongly connected component]
+ A \emph{strongly connected component} in a graph with vertices $V$ is
+ a subset of $V$, say $V'$, such that for each $(v_1,v_2) \in V' \times
+ V$ there is a path from $v_1$ to $v_2$.
+\end{mydef}
+
+The SCCs in the graph for the constraints above is shown in Figure
+\ref{fig:graph-one-scc}. If we have a strongly connected component with
+a $<$ edge---say $x < y$---in it, we have an inconsistency, since there
+must also be a path from $y$ to $x$, and by transitivity it must either
+be the case that $y \le x$ or $y < x$, which are both at odds with $x <
+y$.
+
+Moreover, if we have a SCC with no $<$ edges, it means that all members
+of said SCC are equal, since for every $x \le y$ we have a path from $y$
+to $x$, which again by transitivity means that $y \le x$. Thus, we can
+\emph{condense} the SCC to a single vertex, by choosing a variable among
+the SCC as a representative for all the others. This can be done
+efficiently with disjoint set data structure, and is crucial to keep the
+graph compact, given the very large number of constraints generated when
+type checking.
+
+\subsection{(Web) REPL}
+
+Finally, we take a break from the types by giving a brief account of the
+design of our REPL, being a good example of modular design using various
+constructs dear to the Haskell programmer.
+
+Keeping in mind the \texttt{KMonadT} monad described in Section
+\ref{sec:parame}, the REPL is represented as a function in
+\texttt{KMonadT} consuming input and hopefully producing output. Then,
+frontends can very easily written by marshalling data in and out of the
+REPL:
+\begin{Verbatim}
+data Input
+ = ITyCheck String -- Type check a term
+ | IEval String -- Evaluate a term
+ | IDecl String -- Declare something
+ | ...
+
+data Output
+ = OTyCheck TmRefId [HoleCtx] -- Type checked term, with holes
+ | OPretty TmRefId -- Term to pretty print, after evaluation
+ -- Just holes, classically after loading a file
+ | OHoles [HoleCtx]
+ | ...
+
+-- KMonadT is parametrised over the type of the variables, which depends
+-- on how deep in the term structure we are. For the REPL, we only deal
+-- with top-level terms, and thus only `Id' variables---top level names.
+type REPL m = KMonadT Id m
+
+repl :: ReadFile m => Input -> REPL m Output
+repl = ...
+\end{Verbatim}
+The \texttt{ReadFile} monad embodies the only `extra' action that we
+need to have access too when running the REPL: reading files. We could
+simply use the \texttt{IO} monad, but this will not serve us well when
+implementing front end facing untrusted parties accessing the application
+running on our servers. In our case we expose the REPL as a web
+application, and we want the user to be able to load only from a
+pre-defined directory, not from the entire file system.
+
+For this reason we specify \texttt{ReadFile} to have just one function:
+\begin{Verbatim}
+class Monad m => ReadFile m where
+ readFile' :: FilePath -> m (Either IOError String)
+\end{Verbatim}
+While in the command-line application we will use the \texttt{IO} monad
+and have \texttt{readFile'} to work in the `obvious' way---by reading
+the file corresponding to the given file path---in the web prompt we
+will have it to accept only a file name, not a path, and read it from a
+pre-defined directory:
+\begin{Verbatim}
+-- The monad that will run the web REPL. The `ReaderT' holds the
+-- filepath to the directory where the files loadable by the user live.
+-- The underlying `IO' monad will be used to actually read the files.
+newtype DirRead a = DirRead (ReaderT FilePath IO a)
+
+instance ReadFile DirRead where
+ readFile' fp =
+ do -- We get the base directory in the `ReaderT' with `ask'
+ dir <- DirRead ask
+ -- Is the filepath provided an unqualified file name?
+ if snd (splitFileName fp) == fp
+ -- If yes, go ahead and read the file, by lifting
+ -- `readFile'' into the IO monad
+ then DirRead (lift (readFile' (dir </> fp)))
+ -- If not, return an error
+ else return (Left (strMsg ("Invalid file name `" ++ fp ++ "'")))
+\end{Verbatim}
+Once this light-weight infrastructure is in place, adding a web
+interface was an easy exercise. We use Jasper Van der Jeugt's
+\texttt{websockets} library\footnote{Available at
+ \url{http://hackage.haskell.org/package/websockets}.} to create a
+proxy that receives \texttt{JSON}\footnote{\texttt{JSON} is a popular data interchange
+ format, see \url{http://json.org} for more info.} messages with the
+user input, turns them into \texttt{Input} messages for the REPL, and
+then sends back a \texttt{JSON} message with the response. Moreover, each client
+is handled in a separate threads, so crashes of the REPL for a certain
+client will not bring the whole application down.
+
+On the frontend side, we had to write some JavaScript to accept input
+from a form, and to make the responses appear on the screen. The web
+prompt is publicly available at \url{http://bertus.mazzo.li}, a sample
+session is shown Figure \ref{fig:web-prompt-one}.
+
+\begin{figure}[t]
+ \includegraphics[width=\textwidth]{web-prompt.png}
+ \caption{A sample run of the web prompt.}
+ \label{fig:web-prompt-one}
+\end{figure}
+
+
\section{Evaluation}
+\label{sec:evaluation}
+
+Going back to our goals in Section \ref{sec:contributions}, we feel that
+this thesis fills a gap in the description of observational type theory.
+In the design of \mykant\ we willingly patterned the core features
+against the ones present in Agda, with the hope that future implementors
+will be able to refer to this document without embarking on the same
+adventure themselves. We gave an original account of heterogeneous
+equality by showing that in a cumulative hierarchy we can keep
+equalities as small as we would be able too with a separate notion of
+type equality. As a side effect of developing \mykant, we also gave an
+original account of bidirectional type checking for user defined types,
+which get rid of many types while keeping the language very simple.
+
+Through the design of the theory of \mykant\ we have followed an
+approach where study and implementation were continuously interleaved,
+as a `reality check' for the ideas that we wished to implement. Given
+the great effort necessary to build a theorem prover capable of
+`real-world' proofs we have not attempted to compare \mykant's
+capabilities to those of Agda and Coq, the theorem provers that the
+author is most familiar with and in general two of the main players in
+the field. However we have ported a lot of simpler examples to check
+that the key features are working, some of which have been used in the
+previous sections and are reproduced in the appendices\footnote{The full
+list is available in the repository:
+\url{https://github.com/bitonic/kant/tree/master/data/samples/good}.}.
+A full example of interaction with \mykant\ is given in Section
+\ref{sec:type-holes}.
+
+The main culprits for the delays in the implementation are two issues
+that revealed themselves to be far less obvious than what the author
+predicted. The first, as we have already remarked in Section
+\ref{sec:term-repr}, is to have an adequate term representation that
+lets us express the right constructs in a safe way. There is still no
+widely accepted solution to this problem, which is approached in many
+different ways both in the literature and in the programming
+practice. The second aspect is the treatment of user defined data types.
+Again, the best techniques to implement them in a dependently typed
+setting still have not crystallised and implementors reinvent many
+wheels each time a new system is built. The author is still conflicted
+on whether having user defined types at all it is the right decision:
+while they are essential, the recent discovery of a paper by
+\cite{dagand2012elaborating} describing a way to efficiently encode
+user-defined data types to a set of core primitives---an option that
+seems very attractive.
+
+In general, implementing dependently typed languages is still a poorly
+understood practice, and almost every stage requires experimentation on
+behalf of the author. Another example is the treatment of the implicit
+hierarchy, where no resources are present describing the problem from an
+implementation perspective (we described our approach in Section
+\ref{sec:hier-impl}). Hopefully this state of things will change in the
+near future, and recent publications are promising in this direction,
+for example an unpublished paper by \cite{Brady2013} describing his
+implementation of the Idris programming language. Our ultimate goal is
+to be a part of this collective effort.
+
+\subsection{A type holes tutorial}
+\label{sec:type-holes}
+
+As a taster and showcase for the capabilities of \mykant, we present an
+interactive session with the \mykant\ REPL. While doing so, we present
+a feature that we still have not covered: type holes.
+
+Type holes are, in the author's opinion, one of the `killer' features of
+interactive theorem provers, and one that is begging to be exported to
+mainstream programming---although it is much more effective in a
+well-typed, functional setting. The idea is that when we are developing
+a proof or a program we can insert a hole to have the software tell us
+the type expected at that point. Furthermore, we can ask for the type
+of variables in context, to better understand our surroundings.
+
+In \mykant\ we use type holes by putting them where a term should go.
+We need to specify a name for the hole and then we can put as many terms
+as we like in it. \mykant\ will tell us which type it is expecting for
+the term where the hole is, and the type for each term that we have
+included. For example if we had:
+\begin{Verbatim}
+plus [m n : Nat] : Nat ⇒ (
+ {| h1 m n |}
+)
+\end{Verbatim}
+And we loaded the file in \mykant, we would get:
+\begin{Verbatim}[frame=leftline]
+>>> :l plus.ka
+Holes:
+ h1 : Nat
+ m : Nat
+ n : Nat
+\end{Verbatim}
+
+Suppose we wanted to define the `less or equal' ordering on natural
+numbers as described in Section \ref{sec:user-type}. We will
+incrementally build our functions in a file called \texttt{le.ka}.
+First we define the necessary types, all of which we know well by now:
+\begin{Verbatim}
+data Nat : ⋆ ⇒ { zero : Nat | suc : Nat → Nat }
+
+data Empty : ⋆ ⇒ { }
+absurd [A : ⋆] [p : Empty] : A ⇒ (
+ Empty-Elim p (λ _ ⇒ A)
+)
+
+record Unit : ⋆ ⇒ tt { }
+\end{Verbatim}
+Then fire up \mykant, and load the file:
+\begin{Verbatim}[frame=leftline]
+% ./bertus
+B E R T U S
+Version 0.0, made in London, year 2013.
+>>> :l le.ka
+OK
+\end{Verbatim}
+So far so good. Our definition will be defined by recursion on a
+natural number \texttt{n}, which will return a function that given
+another number \texttt{m} will return the trivial type \texttt{Unit} if
+$\texttt{n} \le \texttt{m}$, and the \texttt{Empty} type otherwise.
+However we are still not sure on how to define it, so we invoke
+$\texttt{Nat-Elim}$, the eliminator for natural numbers, and place holes
+instead of arguments. In the file we will write:
+\begin{Verbatim}
+le [n : Nat] : Nat → ⋆ ⇒ (
+ Nat-Elim n (λ _ ⇒ Nat → ⋆)
+ {|h1|}
+ {|h2|}
+)
+\end{Verbatim}
+And then we reload in \mykant:
+\begin{Verbatim}[frame=leftline]
+>>> :r le.ka
+Holes:
+ h1 : Nat → ⋆
+ h2 : Nat → (Nat → ⋆) → Nat → ⋆
+\end{Verbatim}
+Which tells us what types we need to satisfy in each hole. However, it
+is not that clear what does what in each hole, and thus it is useful to
+have a definition vacuous in its arguments just to clear things up. We
+will use \texttt{Le} aid in reading the goal, with \texttt{Le m n} as a
+reminder that we to return the type corresponding to $\texttt{m} ≤
+\texttt{n}$:
+\begin{Verbatim}
+Le [m n : Nat] : ⋆ ⇒ ⋆
+
+le [n : Nat] : [m : Nat] → Le n m ⇒ (
+ Nat-Elim n (λ n ⇒ [m : Nat] → Le n m)
+ {|h1|}
+ {|h2|}
+)
+\end{Verbatim}
+\begin{Verbatim}[frame=leftline]
+>>> :r le.ka
+Holes:
+ h1 : [m : Nat] → Le zero m
+ h2 : [x : Nat] → ([m : Nat] → Le x m) → [m : Nat] → Le (suc x) m
+\end{Verbatim}
+This is much better! \mykant, when printing terms, does not substitute
+top-level names for their bodies, since usually the resulting term is
+much clearer. As a nice side-effect, we can use tricks like this to
+find guidance.
+
+In this case in the first case we need to return, given any number
+\texttt{m}, $0 \le \texttt{m}$. The trivial type will do, since every
+number is less or equal than zero:
+\begin{Verbatim}
+le [n : Nat] : [m : Nat] → Le n m ⇒ (
+ Nat-Elim n (λ n ⇒ [m : Nat] → Le n m)
+ (λ _ ⇒ Unit)
+ {|h2|}
+)
+\end{Verbatim}
+\begin{Verbatim}[frame=leftline]
+>>> :r le.ka
+Holes:
+ h2 : [x : Nat] → ([m : Nat] → Le x m) → [m : Nat] → Le (suc x) m
+\end{Verbatim}
+Now for the important case. We are given our comparison function for a
+number, and we need to produce the function for the successor. Thus, we
+need to re-apply the induction principle on the other number, \texttt{m}:
+\begin{Verbatim}
+le [n : Nat] : [m : Nat] → Le n m ⇒ (
+ Nat-Elim n (λ n ⇒ [m : Nat] → Le n m)
+ (λ _ ⇒ Unit)
+ (λ n' f m ⇒ Nat-Elim m (λ m' ⇒ Le (suc n') m') {|h2|} {|h3|})
+)
+\end{Verbatim}
+\begin{Verbatim}[frame=leftline]
+>>> :r le.ka
+Holes:
+ h2 : ⋆
+ h3 : [x : Nat] → Le (suc n') x → Le (suc n') (suc x)
+\end{Verbatim}
+In the first hole we know that the second number is zero, and thus we
+can return empty. In the second case, we can use the recursive argument
+\texttt{f} on the two numbers:
+\begin{Verbatim}
+le [n : Nat] : [m : Nat] → Le n m ⇒ (
+ Nat-Elim n (λ n ⇒ [m : Nat] → Le n m)
+ (λ _ ⇒ Unit)
+ (λ n' f m ⇒
+ Nat-Elim m (λ m' ⇒ Le (suc n') m') Empty (λ f _ ⇒ f m'))
+)
+\end{Verbatim}
+We can verify that our function works as expected:
+\begin{Verbatim}[frame=leftline]
+>>> :e le zero zero
+Unit
+>>> :e le zero (suc zero)
+Unit
+>>> :e le (suc (suc zero)) (suc zero)
+Empty
+\end{Verbatim}
+The other functionality of type holes is examining types of things in
+context. Going back to the examples in Section \ref{sec:term-types}, we can
+implement the safe \texttt{head} function with our newly defined
+\texttt{le}:
+\begin{Verbatim}
+data List : [A : ⋆] → ⋆ ⇒
+ { nil : List A | cons : A → List A → List A }
+
+length [A : ⋆] [l : List A] : Nat ⇒ (
+ List-Elim l (λ _ ⇒ Nat) zero (λ _ _ n ⇒ suc n)
+)
+
+gt [n m : Nat] : ⋆ ⇒ (le (suc m) n)
+
+head [A : ⋆] [l : List A] : gt (length A l) zero → A ⇒ (
+ List-Elim l (λ l ⇒ gt (length A l) zero → A)
+ (λ p ⇒ {|h1 p|})
+ {|h2|}
+)
+\end{Verbatim}
+We define \texttt{List}s, a polymorphic \texttt{length} function, and
+express $<$ (\texttt{gt}) in terms of $\le$. Then, we set up the type
+for our \texttt{head} function. Given a list and a proof that its
+length is greater than zero, we return the first element. If we load
+this in \mykant, we get:
+\begin{Verbatim}[frame=leftline]
+>>> :r le.ka
+Holes:
+ h1 : A
+ p : Empty
+ h2 : [x : A] [x1 : List A] →
+ (gt (length A x1) zero → A) →
+ gt (length A (cons x x1)) zero → A
+\end{Verbatim}
+In the first case (the one for \texttt{nil}), we have a proof of
+\texttt{Empty}---surely we can use \texttt{absurd} to get rid of that
+case. In the second case we simply return the element in the
+\texttt{cons}:
+\begin{Verbatim}
+head [A : ⋆] [l : List A] : gt (length A l) zero → A ⇒ (
+ List-Elim l (λ l ⇒ gt (length A l) zero → A)
+ (λ p ⇒ absurd A p)
+ (λ x _ _ _ ⇒ x)
+)
+\end{Verbatim}
+Now, if we tried to get the head of an empty list, we face a problem:
+\begin{Verbatim}[frame=leftline]
+>>> :t head Nat nil
+Type: Empty → Nat
+\end{Verbatim}
+We would have to provide something of type \texttt{Empty}, which
+hopefully should be impossible. For non-empty lists, on the other hand,
+things run smoothly:
+\begin{Verbatim}[frame=leftline]
+>>> :t head Nat (cons zero nil)
+Type: Unit → Nat
+>>> :e head Nat (cons zero nil) tt
+zero
+\end{Verbatim}
+This should give a vague idea of why type holes are so useful and in
+more in general about the development process in \mykant. Most
+interactive theorem provers offer some kind of facility
+to... interactively develop proofs, usually much more powerful than the
+fairly bare tools present in \mykant. Agda in particular offers a
+celebrated interactive mode for the \texttt{Emacs} text editor.
\section{Future work}
+\label{sec:future-work}
-\subsection{Coinduction}
-
-\subsection{Quotient types}
+The first move that the author plans to make is to work towards a simple
+but powerful term representation. A good plan seems to be to associate
+each type (terms, telescopes, etc.) with what we can substitute
+variables with, so that the term type will be associated with itself,
+while telescopes and propositions will be associated to terms. This can
+probably be accomplished elegantly with Haskell's \emph{type families}
+\citep{chakravarty2005associated}. After achieving a more solid
+machinery for terms, implementing observational equality fully should
+prove relatively easy.
-\subsection{Partiality}
+Beyond this steps, we can go in many directions to improve the
+system that we described---here we review the main ones.
-\subsection{Pattern matching}
+\begin{description}
+\item[Pattern matching and recursion] Eliminators are very clumsy,
+ and using them can be especially frustrating if we are used to writing
+ functions via explicit recursion. \cite{Gimenez1995} showed how to
+ reduce well-founded recursive definitions to primitive recursors.
+ Intuitively, defining a function through an eliminators corresponds to
+ pattern matching and recursively calling the function on the recursive
+ occurrences of the type we matched against.
+
+ Nested pattern matching can be justified by identifying a notion of
+ `structurally smaller', and allowing recursive calls on all smaller
+ arguments. Epigram goes all the way and actually implements recursion
+ exclusively by providing a convenient interface to the two constructs
+ above \citep{EpigramTut, McBride2004}.
+
+ However as we extend the flexibility in our recursion elaborating
+ definitions to eliminators becomes more and more laborious. For
+ example we might want mutually recursive definitions and definitions
+ that terminate relying on the structure of two arguments instead of
+ just one. For this reason both Agda and Coq (Agda putting more
+ effort) let the user write recursive definitions freely, and then
+ employ an external syntactic one the recursive calls to ensure that
+ the definitions are terminating.
+
+ Moreover, if we want to use dependently typed languages for
+ programming purposes, we will probably want to sidestep the
+ termination checker and write a possibly non-terminating function;
+ maybe because proving termination is particularly difficult. With
+ explicit recursion this amounts to turning off a check, if we have
+ only eliminators it is impossible.
+
+\item[More powerful data types] A popular improvement on basic data
+ types are inductive families \citep{Dybjer1991}, where the parameters
+ for the type constructors can change based on the data constructors,
+ which lets us express naturally types such as $\mytyc{Vec} : \mynat
+ \myarr \mytyp$, which given a number returns the type of lists of that
+ length, or $\mytyc{Fin} : \mynat \myarr \mytyp$, which given a number
+ $n$ gives the type of numbers less than $n$. This apparent omission
+ was motivated by the fact that inductive families can be represented
+ by adding equalities concerning the parameters of the type
+ constructors as arguments to the data constructor, in much the same
+ way that Generalised Abstract Data Types \citep{GHC} are handled in
+ Haskell. Interestingly the modified version of System F that lies at
+ the core of recent versions of GHC features coercions reminiscent of
+ those found in OTT, motivated precisely by the need to implement GADTs
+ in an elegant way \citep{Sulzmann2007}.
+
+ Another concept introduced by \cite{dybjer2000general} is
+ induction-recursion, where we define a data type in tandem with a
+ function on that type. This technique has proven extremely useful to
+ define embeddings of other calculi in an host language, by defining
+ the representation of the embedded language as a data type and at the
+ same time a function decoding from the representation to a type in the
+ host language. The decoding function is then used to define the data
+ type for the embedding itself, for example by reusing the host's
+ language functions to describe functions in the embedded language,
+ with decoded types as arguments.
+
+ It is also worth mentioning that in recent times there has been work
+ \citep{dagand2012elaborating, chapman2010gentle} to show how to define
+ a set of primitives that data types can be elaborated into. The big
+ advantage of the approach proposed is enabling a very powerful notion
+ of generic programming, by writing functions working on the
+ `primitive' types as to be workable by all the other `compatible'
+ elaborated user defined types. This has been a considerable problem
+ in the dependently type world, where we often define types which are
+ more `strongly typed' version of similar structures,\footnote{For
+ example the $\mytyc{OList}$ presented in Section \ref{sec:user-type}
+ being a `more typed' version of an ordinary list.} and then find
+ ourselves forced to redefine identical operations on both types.
+
+\item[Pattern matching and inductive families] The notion of inductive
+ family also yields a more interesting notion of pattern matching,
+ since matching on an argument influences the value of the parameters
+ of the type of said argument. This means that pattern matching
+ influences the context, which can be exploited to constraint the
+ possible data constructors for \emph{other} arguments
+ \citep{McBride2004}.
+
+\item[Type inference] While bidirectional type checking helps at a very
+ low cost of implementation and complexity, a much more powerful weapon
+ is found in \emph{pattern unification}, which allows Hindley-Milner
+ style inference for dependently typed languages. Unification for
+ higher order terms is undecidable and unification problems do not
+ always have a most general unifier \citep{huet1973undecidability}.
+ However \cite{miller1992unification} identified a decidable fragment
+ of higher order unification commonly known as pattern unification,
+ which is employed in most theorem provers to drastically reduce the
+ number of type annotations. \cite{gundrytutorial} provide a tutorial
+ on this practice.
+
+\item[Coinductive data types] When we specify inductive data types, we
+ do it by specifying its \emph{constructors}---functions with the type
+ we are defining as codomain. Then, we are offered way of compute by
+ recursively \emph{destructing} or \emph{eliminating} a member of the
+ defined data type.
+
+ Coinductive data types are the dual of this approach. We specify ways
+ to destruct data, and we are given a way to generate the defined type
+ by repeatedly `unfolding' starting from some seed data. For example,
+ we could defined infinite streams by specifying a $\myfun{head}$ and
+ $\myfun{tail}$ destructors---here using a syntax reminiscent of
+ \mykant\ records:
+ \[
+ \begin{array}{@{}l}
+ \mysyn{codata}\ \mytyc{Stream}\myappsp (\myb{A} {:} \mytyp)\ \mysyn{where} \\
+ \myind{2} \{ \myfun{head} : \myb{A}, \myfun{tail} : \mytyc{Stream} \myappsp \myb{A}\}
+ \end{array}
+ \]
+ which will hopefully give us something like
+ \[
+ \begin{array}{@{}l}
+ \myfun{head} : (\myb{A}{:}\mytyp) \myarr \mytyc{Stream} \myappsp \myb{A} \myarr \myb{A} \\
+ \myfun{tail} : (\myb{A}{:}\mytyp) \myarr \mytyc{Stream} \myappsp \myb{A} \myarr \mytyc{Stream} \myappsp \myb{A} \\
+ \mytyc{Stream}.\mydc{unfold} : (\myb{A}\, \myb{B} {:} \mytyp) \myarr (\myb{A} \myarr \myb{B} \myprod \myb{A}) \myarr \myb{A} \myarr \mytyc{Stream} \myappsp \myb{B}
+ \end{array}
+ \]
+ Where, in $\mydc{unfold}$, $\myb{B} \myprod \myb{A}$ represents the
+ fields of $\mytyc{Stream}$ but with the recursive occurrence replaced
+ by the `seed' type $\myb{A}$.
+
+ Beyond simple infinite types like $\mytyc{Stream}$, coinduction is
+ particularly useful to write non-terminating programs like servers or
+ software interacting with a user, while guaranteeing their liveliness.
+ Moreover it lets us model possibly non-terminating computations in an
+ elegant way \citep{Capretta2005}, enabling for example the study of
+ operational semantics for non-terminating languages
+ \citep{Danielsson2012}.
+
+ \cite{cockett1992charity} pioneered this approach in their programming
+ language Charity, and coinduction has since been adopted in systems
+ such as Coq \citep{Gimenez1996} and Agda. However these
+ implementations are unsatisfactory, since Coq's break subject
+ reduction; and Agda, to avoid this problem, does not allow types to
+ depend on the unfolding of codata. \cite{mcbride2009let} has shown
+ how observational equality can help to resolve these issues, since we
+ can reason about the unfoldings in a better way, like we reason about
+ functions' extensional behaviour.
+\end{description}
-\subsection{Pattern unification}
+The author looks forward to the study and possibly the implementation of
+these ideas in the years to come.
-% TODO coinduction (obscoin, gimenez), pattern unification (miller,
-% gundry), partiality monad (NAD)
+\newpage{}
\appendix
\section{Notation and syntax}
+\label{app:notation}
Syntax, derivation rules, and reduction rules, are enclosed in frames describing
the type of relation being established and the syntactic elements appearing,
Typing derivations here.
}
-In the languages presented and Agda code samples I also highlight the syntax,
-following a uniform color and font convention:
+In the languages presented and Agda code samples we also highlight the syntax,
+following a uniform colour, capitalisation, and font style convention:
\begin{center}
\begin{tabular}{c | l}
\end{tabular}
\end{center}
-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{teletype} font. I assume that the reader is already familiar with
-Haskell, plenty of good introductions are available \citep{LYAH,ProgInHask}.
+When presenting grammars, we use a word in $\mysynel{math}$ font
+(e.g. $\mytmsyn$ or $\mytysyn$) to indicate indicate
+nonterminals. Additionally, we use quite flexibly a $\mysynel{math}$
+font to indicate a syntactic element in derivations or meta-operations.
+More specifically, terms are usually indicated by lowercase letters
+(often $\mytmt$, $\mytmm$, or $\mytmn$); and types by an uppercase
+letter (often $\mytya$, $\mytyb$, or $\mytycc$).
-When presenting grammars, I will use a word in $\mysynel{math}$ font
-(e.g. $\mytmsyn$ or $\mytysyn$) to indicate indicate nonterminals. Additionally,
-I will use quite flexibly a $\mysynel{math}$ font to indicate a syntactic
-element. More specifically, terms are usually indicated by lowercase letters
-(often $\mytmt$, $\mytmm$, or $\mytmn$); and types by an uppercase letter (often
-$\mytya$, $\mytyb$, or $\mytycc$).
-
-When presenting type derivations, I will often abbreviate and present multiple
+When presenting type derivations, we often abbreviate and present multiple
conclusions, each on a separate line:
\begin{prooftree}
\AxiomC{$\myjud{\mytmt}{\mytya \myprod \mytyb}$}
\noLine
\UnaryInfC{$\myjud{\myapp{\mysnd}{\mytmt}}{\mytyb}$}
\end{prooftree}
-
-I will often present `definition' in the described calculi and in
+We often present `definitions' in the described calculi and in
$\mykant$\ itself, like so:
-{\small\[
+\[
\begin{array}{@{}l}
\myfun{name} : \mytysyn \\
\myfun{name} \myappsp \myb{arg_1} \myappsp \myb{arg_2} \myappsp \cdots \mapsto \mytmsyn
\end{array}
-\]}
-To define operators, I use a mixfix notation similar
-to Agda, where $\myarg$s denote arguments, for example
-{\small\[
+\]
+To define operators, we use a mixfix notation similar
+to Agda, where $\myarg$s denote arguments:
+\[
\begin{array}{@{}l}
\myarg \mathrel{\myfun{$\wedge$}} \myarg : \mybool \myarr \mybool \myarr \mybool \\
\myb{b_1} \mathrel{\myfun{$\wedge$}} \myb{b_2} \mapsto \cdots
\end{array}
-\]}
+\]
+In explicitly typed systems, we omit type annotations when they
+are obvious, e.g. by not annotating the type of parameters of
+abstractions or of dependent pairs.\\
+We introduce multiple arguments in one go in arrow types:
+\[
+ (\myb{x}\, \myb{y} {:} \mytya) \myarr \cdots = (\myb{x} {:} \mytya) \myarr (\myb{y} {:} \mytya) \myarr \cdots
+\]
+and in abstractions:
+\[
+\myabs{\myb{x}\myappsp\myb{y}}{\cdots} = \myabs{\myb{x}}{\myabs{\myb{y}}{\cdots}}
+\]
+We also omit arrows to abbreviate types:
+\[
+(\myb{x} {:} \mytya)(\myb{y} {:} \mytyb) \myarr \cdots =
+(\myb{x} {:} \mytya) \myarr (\myb{y} {:} \mytyb) \myarr \cdots
+\]
+
+Meta operations names are displayed in $\mymeta{smallcaps}$ and
+written in a pattern matching style, also making use of boolean guards.
+For example, a meta operation operating on a context and terms might
+look like this:
+\[
+\begin{array}{@{}l}
+ \mymeta{quux}(\myctx, \myb{x}) \mymetaguard \myb{x} \in \myctx \mymetagoes \myctx(\myb{x}) \\
+ \mymeta{quux}(\myctx, \myb{x}) \mymetagoes \mymeta{outofbounds} \\
+ \myind{2} \vdots
+\end{array}
+\]
+
+From time to time we give examples in the Haskell programming
+language as defined by \cite{Haskell2010}, which we typeset in
+\texttt{teletype} font. I assume that the reader is already familiar
+with Haskell, plenty of good introductions are available
+\citep{LYAH,ProgInHask}.
+
+Examples of \mykant\ code will be typeset nicely with \LaTeX in Section
+\ref{sec:kant-theory}, to adjust with the rest of the presentation; and
+in \texttt{teletype} font in the rest of the document, including Section
+\ref{sec:kant-practice} and in the appendices. All the \mykant\ code
+shown is meant to be working and ready to be inputted in a \mykant\
+prompt or loaded from a file. Snippets of sessions in the \mykant\
+prompt will be displayed with a left border, to distinguish them from
+snippets of code:
+\begin{Verbatim}[frame=leftline]
+>>> :t ⋆
+Type: ⋆
+\end{Verbatim}
\section{Code}
then (λ _ f → (suc (f tt))) else (λ _ _ → y))
x
- List : (A : Set) → Set
- List A = W (A ∨ Unit) (λ s → Tr (fst s))
-
- [] : ∀ {A} → List A
- [] = (false , tt) ◁ absurd
-
- _∷_ : ∀ {A} → A → List A → List A
- x ∷ l = (true , x) ◁ (λ _ → l)
-
- _++_ : ∀ {A} → List A → List A → List A
- l₁ ++ l₂ = rec
- (λ _ → List _ → List _)
- (λ s f c l → {!!})
- l₁ l₂
-
module Equality where
open ITT
\end{code}
\subsubsection{\mykant}
+\label{app:kant-itt}
The following things are missing: $\mytyc{W}$-types, since our
positivity check is overly strict, and equality, since we haven't
}
\subsection{\mykant\ examples}
+\label{app:kant-examples}
{\small
\verbatiminput{examples.ka}
}
-\subsection{\mykant's hierachy}
+\subsection{\mykant' hierachy}
+\label{app:hurkens}
This rendition of the Hurken's paradox does not type check with the
hierachy enabled, type checks and loops without it. Adapted from an
\verbatiminput{hurkens.ka}
}
+\subsection{Term representation}
+\label{app:termrep}
+
+Data type for terms in \mykant.
+
+{\small\begin{verbatim}-- A top-level name.
+type Id = String
+-- A data/type constructor name.
+type ConId = String
+
+-- A term, parametrised over the variable (`v') and over the reference
+-- type used in the type hierarchy (`r').
+data Tm r v
+ = V v -- Variable.
+ | Ty r -- Type, with a hierarchy reference.
+ | Lam (TmScope r v) -- Abstraction.
+ | Arr (Tm r v) (TmScope r v) -- Dependent function.
+ | App (Tm r v) (Tm r v) -- Application.
+ | Ann (Tm r v) (Tm r v) -- Annotated term.
+ -- Data constructor, the first ConId is the type constructor and
+ -- the second is the data constructor.
+ | Con ADTRec ConId ConId [Tm r v]
+ -- Data destrutor, again first ConId being the type constructor
+ -- and the second the name of the eliminator.
+ | Destr ADTRec ConId Id (Tm r v)
+ -- A type hole.
+ | Hole HoleId [Tm r v]
+ -- Decoding of propositions.
+ | Dec (Tm r v)
+
+ -- Propositions.
+ | Prop r -- The type of proofs, with hierarchy reference.
+ | Top
+ | Bot
+ | And (Tm r v) (Tm r v)
+ | Forall (Tm r v) (TmScope r v)
+ -- Heterogeneous equality.
+ | Eq (Tm r v) (Tm r v) (Tm r v) (Tm r v)
+
+-- Either a data type, or a record.
+data ADTRec = ADT | Rec
+
+-- Either a coercion, or coherence.
+data Coeh = Coe | Coh\end{verbatim}
+}
+
+\subsection{Graph and constraints modules}
+\label{app:constraint}
+
+The modules are respectively named \texttt{Data.LGraph} (short for
+`labelled graph'), and \texttt{Data.Constraint}. The type class
+constraints on the type parameters are not shown for clarity, unless
+they are meaningful to the function. In practice we use the
+\texttt{Hashable} type class on the vertex to implement the graph
+efficiently with hash maps.
+
+\subsubsection{\texttt{Data.LGraph}}
+
+{\small
+\verbatiminput{graph.hs}
+}
+
+\subsubsection{\texttt{Data.Constraint}}
+
+{\small
+\verbatiminput{constraint.hs}
+}
+
+\newpage{}
+
\bibliographystyle{authordate1}
\bibliography{thesis}
projects / bitonic-mengthesis.git / blobdiff