-\documentclass[report]{article}
+\documentclass[11pt, fleqn]{article}
\usepackage{etex}
+\usepackage[sc,slantedGreek]{mathpazo}
+% \linespread{1.05}
+% \usepackage{times}
+
+\oddsidemargin 0in
+\evensidemargin 0in
+\textheight 9.5in
+\textwidth 6.2in
+\topmargin -7mm
+%% \parindent 10pt
+
+\headheight 0pt
+\headsep 0pt
+
+
%% Narrow margins
% \usepackage{fullpage}
%% Subfigure
\usepackage{subcaption}
+\usepackage{verbatim}
+\usepackage{fancyvrb}
+
+\RecustomVerbatimEnvironment
+ {Verbatim}{Verbatim}
+ {xleftmargin=9mm}
+
%% diagrams
\usepackage{tikz}
\usetikzlibrary{shapes,arrows,positioning}
\DeclareUnicodeCharacter{8799}{\ensuremath{\stackrel{?}{=}}}
\DeclareUnicodeCharacter{9655}{\ensuremath{\rhd}}
+\renewenvironment{code}%
+{\noindent\ignorespaces\advance\leftskip\mathindent\AgdaCodeStyle\pboxed\small}%
+{\endpboxed\par\noindent%
+\ignorespacesafterend\small}
+
%% -----------------------------------------------------------------------------
%% Commands
+\newcommand{\mysmall}{}
\newcommand{\mysyn}{\AgdaKeyword}
\newcommand{\mytyc}{\AgdaDatatype}
-% TODO have this with math mode so I can have subscripts
\newcommand{\mydc}{\AgdaInductiveConstructor}
\newcommand{\myfld}{\AgdaField}
\newcommand{\myfun}{\AgdaFunction}
\newcommand{\myb}[1]{\AgdaBound{$#1$}}
\newcommand{\myfield}{\AgdaField}
\newcommand{\myind}{\AgdaIndent}
-\newcommand{\mykant}{\textsc{Kant}}
+\newcommand{\mykant}{\textsc{Bertus}}
\newcommand{\mysynel}[1]{#1}
\newcommand{\myse}{\mysynel}
\newcommand{\mytmsyn}{\mysynel{term}}
\newcommand{\myappsp}{\hspace{0.07cm}}
\newcommand{\myapp}[2]{#1 \myappsp #2}
\newcommand{\mysynsep}{\ \ |\ \ }
+\newcommand{\myITE}[3]{\myfun{If}\, #1\, \myfun{Then}\, #2\, \myfun{Else}\, #3}
+\newcommand{\mycumul}{\preceq}
\FrameSep0.2cm
\newcommand{\mydesc}[3]{
\noindent
\mbox{
\parbox{\textwidth}{
- {\small
- \vspace{0.3cm}
+ {\mysmall
+ \vspace{0.2cm}
\hfill \textbf{#1} $#2$
-
\framebox[\textwidth]{
\parbox{\textwidth}{
\vspace{0.1cm}
\centering{
#3
}
- \vspace{0.1cm}
+ \vspace{0.2cm}
}
}
+ \vspace{0.2cm}
}
}
}
}
-% TODO is \mathbin the correct thing for arrow and times?
-
\newcommand{\mytmt}{\mysynel{t}}
\newcommand{\mytmm}{\mysynel{m}}
\newcommand{\mytmn}{\mysynel{n}}
\newcommand{\mysub}[3]{#1[#2 / #3]}
\newcommand{\mytysyn}{\mysynel{type}}
\newcommand{\mybasetys}{K}
-% TODO change this name
\newcommand{\mybasety}[1]{B_{#1}}
\newcommand{\mytya}{\myse{A}}
\newcommand{\mytyb}{\myse{B}}
\newcommand{\myvalid}[1]{#1 \vdash \underline{\mathrm{valid}}}
\newcommand{\myjudd}[3]{#1 \vdash #2 : #3}
\newcommand{\myjud}[2]{\myjudd{\myctx}{#1}{#2}}
-% TODO \mathbin or \mathrel here?
\newcommand{\myabss}[3]{\mydc{$\lambda$} #1 {:} #2 \mathrel{\mydc{$\mapsto$}} #3}
\newcommand{\mytt}{\mydc{$\langle\rangle$}}
\newcommand{\myunit}{\mytyc{Unit}}
\newcommand{\myfst}{\myfld{fst}}
\newcommand{\mysnd}{\myfld{snd}}
\newcommand{\myconst}{\myse{c}}
-\newcommand{\myemptyctx}{\cdot}
+\newcommand{\myemptyctx}{\varepsilon}
\newcommand{\myhole}{\AgdaHole}
\newcommand{\myfix}[3]{\mysyn{fix} \myappsp #1 {:} #2 \mapsto #3}
\newcommand{\mysum}{\mathbin{\textcolor{AgdaDatatype}{+}}}
\newcommand{\myempty}{\mytyc{Empty}}
\newcommand{\mycase}[2]{\mathopen{\myfun{[}}#1\mathpunct{\myfun{,}} #2 \mathclose{\myfun{]}}}
\newcommand{\myabsurd}[1]{\myfun{absurd}_{#1}}
-\newcommand{\myarg}{-}
-\newcommand{\myderivsp}{\vspace{0.3cm}}
+\newcommand{\myarg}{\_}
+\newcommand{\myderivsp}{}
+\newcommand{\myderivspp}{\vspace{0.3cm}}
\newcommand{\mytyp}{\mytyc{Type}}
\newcommand{\myneg}{\myfun{$\neg$}}
\newcommand{\myar}{\,}
\newcommand{\mycons}{\mathbin{\mydc{∷}}}
\newcommand{\myfoldr}{\myfun{foldr}}
\newcommand{\myw}[3]{\myapp{\myapp{\mytyc{W}}{(#1 {:} #2)}}{#3}}
-\newcommand{\mynode}[2]{\mathbin{\mydc{$\lhd$}_{#1.#2}}}
+\newcommand{\mynodee}{\mathbin{\mydc{$\lhd$}}}
+\newcommand{\mynode}[2]{\mynodee_{#1.#2}}
\newcommand{\myrec}[4]{\myfun{rec}\,#1 / {#2.#3}\,\myfun{with}\,#4}
\newcommand{\mylub}{\sqcup}
\newcommand{\mydefeq}{\cong}
\newcommand{\myrefl}{\mydc{refl}}
\newcommand{\mypeq}[1]{\mathrel{\mytyc{=}_{#1}}}
-\newcommand{\myjeqq}{\myfun{=-elim}}
+\newcommand{\myjeqq}{\myfun{$=$-elim}}
\newcommand{\myjeq}[3]{\myapp{\myapp{\myapp{\myjeqq}{#1}}{#2}}{#3}}
\newcommand{\mysubst}{\myfun{subst}}
\newcommand{\myprsyn}{\myse{prop}}
-\newcommand{\myprdec}[1]{\mathopen{\mytyc{$\llbracket$}} #1 \mathopen{\mytyc{$\rrbracket$}}}
+\newcommand{\myprdec}[1]{\mathopen{\mytyc{$\llbracket$}} #1 \mathclose{\mytyc{$\rrbracket$}}}
\newcommand{\myand}{\mathrel{\mytyc{$\wedge$}}}
+\newcommand{\mybigand}{\mathrel{\mytyc{$\bigwedge$}}}
\newcommand{\myprfora}[3]{\forall #1 {:} #2. #3}
\newcommand{\myimpl}{\mathrel{\mytyc{$\Rightarrow$}}}
\newcommand{\mybot}{\mytyc{$\bot$}}
\newcommand{\myann}[2]{#1 : #2}
\newcommand{\mydeclsyn}{\myse{decl}}
\newcommand{\myval}[3]{#1 : #2 \mapsto #3}
-\newcommand{\mypost}[2]{\mysyn{postulate}\ #1 : #2}
-\newcommand{\myadt}[4]{\mysyn{data}\ #1 : #2\ \mysyn{where}\ #3\{ #4 \}}
-\newcommand{\myreco}[4]{\mysyn{record}\ #1 : #2\ \mysyn{where}\ #3\ \{ #4 \}}
-% TODO change vdash
+\newcommand{\mypost}[2]{\mysyn{abstract}\ #1 : #2}
+\newcommand{\myadt}[4]{\mysyn{data}\ #1 #2\ \mysyn{where}\ #3\{ #4 \}}
+\newcommand{\myreco}[4]{\mysyn{record}\ #1 #2\ \mysyn{where}\ \{ #4 \}}
\newcommand{\myelabt}{\vdash}
\newcommand{\myelabf}{\rhd}
\newcommand{\myelab}[2]{\myctx \myelabt #1 \myelabf #2}
\newcommand{\mymeta}{\textsc}
\newcommand{\myhyps}{\mymeta{hyps}}
\newcommand{\mycc}{;}
-\newcommand{\myemptytele}{\cdot}
+\newcommand{\myemptytele}{\varepsilon}
\newcommand{\mymetagoes}{\Longrightarrow}
% \newcommand{\mytesctx}{\
\newcommand{\mytelesyn}{\myse{telescope}}
\newcommand{\myrecs}{\mymeta{recs}}
\newcommand{\myle}{\mathrel{\lcfun{$\le$}}}
+\newcommand{\mylet}{\mysyn{let}}
+\newcommand{\myhead}{\mymeta{head}}
+\newcommand{\mytake}{\mymeta{take}}
+\newcommand{\myix}{\mymeta{ix}}
+\newcommand{\myapply}{\mymeta{apply}}
+\newcommand{\mydataty}{\mymeta{datatype}}
+\newcommand{\myisreco}{\mymeta{record}}
+\newcommand{\mydcsep}{\ |\ }
+\newcommand{\mytree}{\mytyc{Tree}}
+\newcommand{\myproj}[1]{\myfun{$\pi_{#1}$}}
+\newcommand{\mysigma}{\mytyc{$\Sigma$}}
+\newcommand{\mynegder}{\vspace{-0.3cm}}
+\newcommand{\myquot}{\Uparrow}
+\newcommand{\mynquot}{\, \Downarrow}
+
+\renewcommand{\[}{\begin{equation*}}
+\renewcommand{\]}{\end{equation*}}
+\newcommand{\mymacol}[2]{\text{\textcolor{#1}{$#2$}}}
%% -----------------------------------------------------------------------------
\author{Francesco Mazzoli \href{mailto:fm2209@ic.ac.uk}{\nolinkurl{<fm2209@ic.ac.uk>}}}
\date{June 2013}
+ \iffalse
+ \begin{code}
+ module thesis where
+ \end{code}
+ \fi
+
\begin{document}
-\iffalse
-\begin{code}
-module thesis where
-\end{code}
-\fi
+\begin{titlepage}
+ \centering
+
+ \maketitle
+ \thispagestyle{empty}
+
+ \begin{minipage}{0.4\textwidth}
+ \begin{flushleft} \large
+ \emph{Supervisor:}\\
+ Dr. Steffen \textsc{van Bakel}
+ \end{flushleft}
+\end{minipage}
+\begin{minipage}{0.4\textwidth}
+ \begin{flushright} \large
+ \emph{Co-marker:} \\
+ Dr. Philippa \textsc{Gardner}
+ \end{flushright}
+\end{minipage}
-\maketitle
+\end{titlepage}
\begin{abstract}
The marriage between programming and logic has been a very fertile one. In
particular, since the simply typed lambda calculus (STLC), a number of type
systems have been devised with increasing expressive power.
- Section \ref{sec:types} will give a very brief overview of STLC, and then
- illustrate how it can be interpreted as a natural deduction system. Section
- \ref{sec:itt} will introduce Inutitionistic Type Theory (ITT), which expands
- on this concept, employing a more expressive logic. The exposition is quite
- dense since there is a lot of material to cover; for a more complete treatment
- of the material the reader can refer to \citep{Thompson1991, Pierce2002}.
- Section \ref{sec:equality} will explain why equality has always been a tricky
- business in these theories, and talk about the various attempts that have been
- made to make the situation better. One interesting development has recently
- emerged: Observational Type theory.
-
- Section \ref{sec:practical} will describe common extensions found in the
- systems currently in use. Finally, section \ref{sec:kant} will describe a
- system developed by the author that implements a core calculus based on the
- principles described.
+ Among this systems, Inutitionistic Type Theory (ITT) has been a very
+ 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.
+\end{abstract}
+
+\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 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
+ Observational Type Theory as a topic of study: this thesis would not
+ exist without him. Before them, Tony Fields 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,
+ who answered many of my doubts through these months. I also owe him
+ the colours.
\end{abstract}
\clearpage
\label{sec:types}
\subsection{The untyped $\lambda$-calculus}
+\label{sec:untyped}
Along with Turing's machines, the earliest attempts to formalise computation
lead to the $\lambda$-calculus \citep{Church1936}. This early programming
$
\begin{array}{l}
\myapp{(\myabs{\myb{x}}{\mytmm})}{\mytmn} \myred \mysub{\mytmm}{\myb{x}}{\mytmn}\text{, where} \\
- \myind{1}
+ \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{1} \text{with $\myb{x} \neq \myb{y}$ and $\myb{z}$ not free in $\myapp{\mytmm}{\mytmn}$}}
+ \multicolumn{3}{l}{\myind{2} \text{with $\myb{x} \neq \myb{y}$ and $\myb{z}$ not free in $\myapp{\mytmm}{\mytmn}$}}
\end{array}
\end{array}
$
\[
(\myapp{\omega}{\omega}) \myred (\myapp{\omega}{\omega}) \myred \cdots \text{, with $\omega = \myabs{x}{\myapp{x}{x}}$}
\]
-
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
correspond to some `primitive' types; and $\myarr$, the type former for
`arrow' types, the types of functions. The language is explicitly
typed: when we bring a variable into scope with an abstraction, we
-explicitly declare its type. Reduction is unchanged from the untyped
+declare its type. Reduction is unchanged from the untyped
$\lambda$-calculus.
\begin{figure}[t]
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
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:
\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{$\myjud{\mytmm}{\mytya}$}
is no obvious computational behaviour for laws like the excluded middle.
Theories of this kind are called \emph{intuitionistic}, or \emph{constructive},
and all the systems analysed will have this characteristic since they build on
-the foundation of the STLC\footnote{There is research to give computational
- behaviour to classical logic, but I will not touch those subjects.}.
+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:
-\[
-(\myfix{\myb{x}}{\mytya}{\myb{x}}) : \mytya
-\]
+\[(\myfix{\myb{x}}{\mytya}{\myb{x}}) : \mytya\]
\subsection{Inductive data}
\label{sec:ind-data}
\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}
The STLC can be made more expressive in various ways. \cite{Barendregt1991}
succinctly expressed geometrically how we can add expressivity:
-
$$
\xymatrix@!0@=1.5cm{
& \lambda\omega \ar@{-}[rr]\ar@{-}'[d][dd]
\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: \[\displaystyle(\myabss{\myb{A} \myar \myb{R}}{\mytyp}{(\myb{A}
- \myarr \myb{R}) \myarr \myb{R}}) : \mytyp \myarr \mytyp \myarr \mytyp\]
+ continuation passing style:
+ \[\displaystyle(\myabss{\myb{A} \myar \myb{R}}{\mytyp}{(\myb{A}
+ \myarr \myb{R}) \myarr \myb{R}}) : \mytyp \myarr \mytyp \myarr \mytyp
+ \]
\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:
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
+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}.
A related development is the polymorphic $\lambda$-calculus, and specifically
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.
-
\subsection{A simple type theory}
\label{sec:core-tt}
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. An Agda rendition of the
-presented theory and all the examples is reproduced in appendix
-\ref{app:agda-itt}.
+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}.
\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
\[
\myabss{\myb{x}}{\mytya}{\myb{x}} \mydefeq \myabss{\myb{y}}{\mytya}{\myb{y}}
\]
-Types that are definitionally equal can be used interchangeably. Here the
-`conversion' rule is not syntax directed, however we will see how it is
-possible to employ $\myred$ to decide term equality in a systematic
-way. % 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.
-
-Moreover, we specify a \emph{type hierarchy} to talk about `large' types:
-$\mytyp_0$ will be the type of types inhabited by data: $\mybool$, $\mynat$,
-$\mylist$, etc. $\mytyp_1$ will be the type of $\mytyp_0$, and so on---for
-example we have $\mytrue : \mybool : \mytyp_0 : \mytyp_1 : \cdots$. Each type
-`level' is often called a universe in the literature. While it is possible,
-to simplify things by having only one universe $\mytyp$ with $\mytyp :
-\mytyp$, this plan is inconsistent for much the same reason that impredicative
-na\"{\i}ve set theory is \citep{Hurkens1995}. Moreover, various techniques
-can be employed to lift the burden of explicitly handling universes.
-% TODO add sectioon about universes
+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.
+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.
+
+Moreover, we specify a \emph{type hierarchy} to talk about `large'
+types: $\mytyp_0$ will be the type of types inhabited by data:
+$\mybool$, $\mynat$, $\mylist$, etc. $\mytyp_1$ will be the type of
+$\mytyp_0$, and so on---for example we have $\mytrue : \mybool :
+\mytyp_0 : \mytyp_1 : \cdots$. Each type `level' is often called a
+universe in the literature. While it is possible to simplify things by
+having only one universe $\mytyp$ with $\mytyp : \mytyp$, this plan is
+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}.
\subsubsection{Contexts}
\UnaryInfC{$\myjud{\myfalse}{\mybool}$}
\DisplayProof
\end{tabular}
- \myderivsp
+ \myderivspp
\AxiomC{$\myjud{\mytmt}{\mybool}$}
\AxiomC{$\myjudd{\myctx : \mybool}{\mytya}{\mytyp_l}$}
\DisplayProof
}
-With booleans we get the first taste of `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
-\begin{verbatim}
+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
+\begin{Verbatim}
if null xs then head xs else 0
-\end{verbatim}
+\end{Verbatim}
are a necessary, well typed, danger.
However, in a more expressive system, we can do better: the branches' type can
\BinaryInfC{$\myjud{\myfora{\myb{x}}{\mytya}{\mytyb}}{\mytyp_{l_1 \mylub l_2}}$}
\DisplayProof
- \myderivsp
+ \myderivspp
\begin{tabular}{cc}
\AxiomC{$\myjudd{\myctx; \myb{x} : \mytya}{\mytmt}{\mytyb}$}
language, using dependent functions we would be able to
write:
\[
-\begin{array}{c@{\ }c@{\ }l@{\ }}
-\myfun{length} & : & (\myb{A} {:} \mytyp_0) \myarr \myapp{\mylist}{\myb{A}} \myarr \mynat \\
-\myarg \myfun{$>$} \myarg & : & \mynat \myarr \mynat \myarr \mytyp_0 \\
-\myfun{head} & : & (\myb{A} {:} \mytyp_0) \myarr (\myb{l} {:} \myapp{\mylist}{\myb{A}})
- \myarr \myapp{\myapp{\myfun{length}}{\myb{A}}}{\myb{l}} \mathrel{\myfun{>}} 0 \myarr
+\begin{array}{l}
+\myfun{length} : (\myb{A} {:} \mytyp_0) \myarr \myapp{\mylist}{\myb{A}} \myarr \mynat \\
+\myarg \myfun{$>$} \myarg : \mynat \myarr \mynat \myarr \mytyp_0 \\
+\myfun{head} : (\myb{A} {:} \mytyp_0) \myarr (\myb{l} {:} \myapp{\mylist}{\myb{A}})
+ \myarr \myapp{\myapp{\myfun{length}}{\myb{A}}}{\myb{l}} \mathrel{\myfun{$>$}} 0 \myarr
\myb{A}
\end{array}
\]
-\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-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.
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
type-level binders, $\myprod$ and $\mytyc{W}$.
\subsubsection{$\myprod$, or dependent product}
+\label{sec:disju}
\mydesc{typing:}{\myjud{\mytmsyn}{\mytmsyn}}{
\AxiomC{$\myjud{\mytya}{\mytyp_{l_1}}$}
\BinaryInfC{$\myjud{\myexi{\myb{x}}{\mytya}{\mytyb}}{\mytyp_{l_1 \mylub l_2}}$}
\DisplayProof
- \myderivsp
+ \myderivspp
\begin{tabular}{cc}
\AxiomC{$\myjud{\mytmm}{\mytya}$}
a witness. This means, for instance, that $\neg \forall \neg$ is not
equivalent to $\exists$.
+Another perhaps more `dependent' application of products, paired with
+$\mybool$, is to offer choice between different types. For example we
+can easily recover disjunctions:
+\[
+\begin{array}{l}
+ \myarg\myfun{$\vee$}\myarg : \mytyp_0 \myarr \mytyp_0 \myarr \mytyp_0 \\
+ \myb{A} \mathrel{\myfun{$\vee$}} \myb{B} \mapsto \myexi{\myb{x}}{\mybool}{\myite{\myb{x}}{\myb{A}}{\myb{B}}} \\ \ \\
+ \myfun{case} : (\myb{A}\ \myb{B}\ \myb{C} {:} \mytyp_0) \myarr (\myb{A} \myarr \myb{C}) \myarr (\myb{B} \myarr \myb{C}) \myarr \myb{A} \mathrel{\myfun{$\vee$}} \myb{B} \myarr \myb{C} \\
+ \myfun{case} \myappsp \myb{A} \myappsp \myb{B} \myappsp \myb{C} \myappsp \myb{f} \myappsp \myb{g} \myappsp \myb{x} \mapsto \\
+ \myind{2} \myapp{(\myitee{\myapp{\myfst}{\myb{x}}}{\myb{b}}{(\myite{\myb{b}}{\myb{A}}{\myb{B}})}{\myb{f}}{\myb{g}})}{(\myapp{\mysnd}{\myb{x}})}
+\end{array}
+\]
+
\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{$\myjudd{\myctx; \myb{w} : \myw{\myb{x}}{\myse{S}}{\myse{T}}}{\myse{P}}{\mytyp_l}$}
\DisplayProof
}
+Finally, the well-order type, or in short $\mytyc{W}$-type, which will
+let us represent inductive data in a general (but clumsy) way. We can
+form `nodes' of the shape $\mytmt \mynode{\myb{x}}{\mytyb} \myse{f} :
+\myw{\myb{x}}{\mytya}{\mytyb}$ that contain data ($\mytmt$) of type and
+one `child' 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 an a function dealing with the inductive
+case---we will gain more intuition about inductive data in ITT in
+Section \ref{sec:user-type}.
+
+For example, if we want to form natural numbers, we can take
+\[
+ \begin{array}{@{}l}
+ \mytyc{Tr} : \mybool \myarr \mytyp_0 \\
+ \mytyc{Tr} \myappsp \myb{b} \mapsto \myfun{if}\, \myb{b}\, \myunit\, \myfun{else}\, \myempty \\
+ \ \\
+ \mynat : \mytyp_0 \\
+ \mynat \mapsto \myw{\myb{b}}{\mybool}{(\mytyc{Tr}\myappsp\myb{b})}
+ \end{array}
+\]
+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 $\myempty$:
+\[
+ \begin{array}{@{}l}
+ \mydc{zero} : \mynat \\
+ \mydc{zero} \mapsto \myfalse \mynodee (\myabs{\myb{z}}{\myabsurd{\mynat} \myappsp \myb{x}}) \\
+ \ \\
+ \mydc{suc} : \mynat \myarr \mynat \\
+ \mydc{suc}\myappsp \myb{x} \mapsto \mytrue \mynodee (\myabs{\myarg}{\myb{x}})
+ \end{array}
+\]
+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 $\mynat$---which gives a taste of the
+`clumsiness' of $\mytyc{W}$-types, which while useful as a reasoning
+tool are useless to the user modelling data types.
+
\section{The struggle for equality}
\label{sec:equality}
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-code}.
+formalised in Agda in appendix \ref{app:agda-itt}.
\subsection{Propositional equality}
$
\myjeq{\myse{P}}{(\myapp{\myrefl}{\mytmm})}{\mytmn} \myred \mytmn
$
- \vspace{0.9cm}
+ \vspace{1.1cm}
}
\end{minipage}
-
+\mynegder
\mydesc{typing:}{\myjud{\mytmsyn}{\mytmsyn}}{
\AxiomC{$\myjud{\mytya}{\mytyp_l}$}
\AxiomC{$\myjud{\mytmm}{\mytya}$}
\TrinaryInfC{$\myjud{\mytmm \mypeq{\mytya} \mytmn}{\mytyp_l}$}
\DisplayProof
- \myderivsp
+ \myderivspp
\begin{tabular}{cc}
- \AxiomC{\phantom{$\myctx P \mytyp_l$}}
- \noLine
- \UnaryInfC{$\myjud{\mytmm}{\mytya}\hspace{1.1cm}\mytmm \mydefeq \mytmn$}
+ \AxiomC{$\begin{array}{c}\ \\\myjud{\mytmm}{\mytya}\hspace{1.1cm}\mytmm \mydefeq \mytmn\end{array}$}
\UnaryInfC{$\myjud{\myapp{\myrefl}{\mytmm}}{\mytmm \mypeq{\mytya} \mytmn}$}
\DisplayProof
&
- \AxiomC{$\myjud{\myse{P}}{\myfora{\myb{x}\ \myb{y}}{\mytya}{\myfora{q}{\myb{x} \mypeq{\mytya} \myb{y}}{\mytyp_l}}}$}
- \noLine
- \UnaryInfC{$\myjud{\myse{q}}{\mytmm \mypeq{\mytya} \mytmn}\hspace{1.1cm}\myjud{\myse{p}}{\myapp{\myapp{\myapp{\myse{P}}{\mytmm}}{\mytmm}}{(\myapp{\myrefl}{\mytmm})}}$}
+ \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})}}
+ \end{array}
+ $}
\UnaryInfC{$\myjud{\myjeq{\myse{P}}{\myse{q}}{\myse{p}}}{\myapp{\myapp{\myapp{\myse{P}}{\mytmm}}{\mytmn}}{q}}$}
\DisplayProof
\end{tabular}
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---in the typing rule we display the term as `the same', meaning
-`the same up to $\mydefeq$'. % TODO maybe mention this earlier
+equal terms.
Finally, we have one eliminator for $\mypeq{}$, $\myjeqq$. $\myjeq{\myse{P}}{\myse{q}}{\myse{p}}$ takes
\begin{itemize}
$\mytya$) 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
+\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
replaces equal for equal in predicates:
\[
\begin{array}{l}
-(\myabs{\myb{A}\ \myb{P}\ \myb{x}\ \myb{y}\ \myb{q}\ \myb{p}}{
- \myjeq{(\myabs{\myb{x}\ \myb{y}\ \myb{q}}{\myapp{\myb{P}}{\myb{y}}})}{\myb{q}}{\myb{p}}}) : \\
-\myind{1} \myfora{\myb{A}}{\mytyp}{\myfora{\myb{P}}{\myb{A} \myarr \mytyp}{\myfora{\myb{x}\ \myb{y}}{\myb{A}}{\myb{x} \mypeq{\myb{A}} \myb{y} \myarr \myapp{\myb{P}}{\myb{x}} \myarr \myapp{\myb{P}}{\myb{y}}}}}
+\myfun{subst} : \myfora{\myb{A}}{\mytyp}{\myfora{\myb{P}}{\myb{A} \myarr \mytyp}{\myfora{\myb{x}\ \myb{y}}{\myb{A}}{\myb{x} \mypeq{\myb{A}} \myb{y} \myarr \myapp{\myb{P}}{\myb{x}} \myarr \myapp{\myb{P}}{\myb{y}}}}} \\
+\myfun{subst}\myappsp \myb{A}\myappsp\myb{P}\myappsp\myb{x}\myappsp\myb{y}\myappsp\myb{q}\myappsp\myb{p} \mapsto
+ \myjeq{(\myabs{\myb{x}\ \myb{y}\ \myb{q}}{\myapp{\myb{P}}{\myb{y}}})}{\myb{p}}{\myb{q}}
\end{array}
\]
-This rule is often called $\myfun{subst}$---here we will invoke it without
-specifying the type ($\myb{A}$) and the sides of the equality
-($\myb{x},\myb{y}$).
-
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
+equality, such as symmetry, transitivity, congruence laws, etc.
\subsection{Common extensions}
-eta law
+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 $\eta$-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 based on their types, a process also 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
+by 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}.
+
+To justify this process in our type system we will add a congruence law
+for abstractions and a similar law for products, plus the fact that all
+elements of $\myunit$ or $\myempty$ are equal.
+
+\mydesc{definitional equality:}{\myjud{\mytmm \mydefeq \mytmn}{\mytmsyn}}{
+ \begin{tabular}{cc}
+ \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{$\myjud{\mypair{\myapp{\myfst}{\mytmm}}{\myapp{\mysnd}{\mytmm}} \mydefeq \mypair{\myapp{\myfst}{\mytmn}}{\myapp{\mysnd}{\mytmn}}}{\myexi{\myb{x}}{\mytya}{\mytyb}}$}
+ \UnaryInfC{$\myjud{\mytmm \mydefeq \mytmn}{\myexi{\myb{x}}{\mytya}{\mytyb}}$}
+ \DisplayProof
+ \end{tabular}
+
+ \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}
+}
+
+\subsubsection{Uniqueness of identity proofs}
-congruence
+Another common but controversial addition to propositional equality is
+the $\myfun{K}$ axiom, which essentially states that all equality proofs
+are by reflexivity.
-UIP
+\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{\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{t} \myappsp \myse{p} \myappsp \myse{q}}{\myse{P} \myappsp \mytmt \myappsp \myse{q}}$}
+ \DisplayProof
+}
+
+\cite{Hofmann1994} showed that $\myfun{K}$ is not derivable from the
+$\myjeqq$ axiom that we presented, and \cite{McBride2004} showed that it is
+needed to implement `dependent pattern matching', as first proposed by
+\cite{Coquand1992}. 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 `Univalent Foundations', which includes 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}.}
\subsection{Limitations}
}
\]
To see why this is the case, consider the functions
-\[\myabs{\myb{x}}{0 \mathrel{\myfun{+}} \myb{x}}$ and $\myabs{\myb{x}}{\myb{x} \mathrel{\myfun{+}} 0}\]
-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
\[
-\myfora{\myb{x}}{\mynat}{(0 \mathrel{\myfun{+}} \myb{x}) \mypeq{\mynat} (\myb{x} \mathrel{\myfun{+}} 0)}
+\myfora{\myb{x}}{\mynat}{(0 \mathrel{\myfun{$+$}} \myb{x}) \mypeq{\mynat} (\myb{x} \mathrel{\myfun{$+$}} 0)}
\]
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.
+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.
This is quite an annoyance that often makes reasoning awkward to execute. It
also extends to other fields, for example proving bisimulation between
\item The rule is 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.
- Consider for example
- \[
- \myabss{\myb{q}}{\mytya \mypeq{\mytyp} (\mytya \myarr \mytya)}{\myhole{?}}
- \]
- Using the assumed proof in tandem with equality reflection we could easily
- write a classic Y combinator, sending the compiler into a loop.
+ 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
\[\myb{A}, \myb{B} : \mytyp; \myb{f}, \myb{g} : \myb{A} \myarr \myb{B}; \myb{q} : \myfora{\myb{x}}{\myb{A}}{\myapp{\myb{f}}{\myb{x}} \mypeq{} \myapp{\myb{g}}{\myb{x}}}\]
We can then derive
\begin{prooftree}
+ \mysmall
\AxiomC{$\hspace{1.1cm}\myjudd{\myctx; \myb{x} : \myb{A}}{\myapp{\myb{q}}{\myb{x}}}{\myapp{\myb{f}}{\myb{x}} \mypeq{} \myapp{\myb{g}}{\myb{x}}}\hspace{1.1cm}$}
\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{$\hspace{1.4cm}\myjud{\myb{f} \mydefeq \myb{g}}{\myb{A} \myarr \myb{B}}\hspace{1.4cm}$}
+ \UnaryInfC{$\hspace{1.45cm}\myjud{\myb{f} \mydefeq \myb{g}}{\myb{A} \myarr \myb{B}}\hspace{1.45cm}$}
\RightLabel{$\myrefl$}
\UnaryInfC{$\myjud{\myapp{\myrefl}{\myb{f}}}{\myb{f} \mypeq{} \myb{g}}$}
\end{prooftree}
Now, the question is: do we need to give up well-behavedness of our theory to
gain extensionality?
-\subsection{Observational equality}
+\subsection{Some alternatives}
+
+% TODO finish
+% TODO add `extentional axioms' (Hoffman), setoid models (Thorsten)
+
+\section{Observational equality}
\label{sec:ott}
-% TODO should we explain this in detail?
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 that OTT gains
- \emph{all} the equality proofs of ETT, but no proof 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 value-level equality based on similar principles. A
-brief overview is given below,
+ Theory} (OTT), promises to keep the well behavedness of ITT while
+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
+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
+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}
\mydesc{syntax}{ }{
- $\mytyp_l$ is replaced by $\mytyp$. \\
+ $\mytyp_l$ is replaced by $\mytyp$. \\\ \\
$
\begin{array}{r@{\ }c@{\ }l}
- \mytmsyn & ::= & \cdots \\
- & | & \myprdec{\myprsyn} \mysynsep
- \mycoee{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn} \mysynsep
- \mycohh{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn} \\
+ \mytmsyn & ::= & \cdots \mysynsep \myprdec{\myprsyn} \mysynsep
+ \myITE{\mytmsyn}{\mytmsyn}{\mytmsyn} \\
\myprsyn & ::= & \mybot \mysynsep \mytop \mysynsep \myprsyn \myand \myprsyn
- \mysynsep \myprfora{\myb{x}}{\mytmsyn}{\myprsyn} \\\
- & | & \mytmsyn \myeq \mytmsyn \mysynsep
- \myjm{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn}
+ \mysynsep \myprfora{\myb{x}}{\mytmsyn}{\myprsyn}
\end{array}
$
}
\mydesc{typing:}{\myjud{\mytmsyn}{\mytmsyn}}{
-
- There is only $\mytyp$, which corresponds to $\mytyp_0$. \\ Thus all
- the type-formers take $\mytyp$ arguments and form a $\mytyp$. \\ \ \\
-
- % TODO insert large eliminator
-
\begin{tabular}{cc}
\AxiomC{$\myjud{\myse{P}}{\myprop}$}
\UnaryInfC{$\myjud{\myprdec{\myse{P}}}{\mytyp}$}
\DisplayProof
&
- \AxiomC{$\myjud{\myse{P}}{\myprdec{\mytya \myeq \mytyb}}$}
- \AxiomC{$\myjud{\mytmt}{\mytya}$}
- \BinaryInfC{$\myjud{\mycoee{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\mytyb}$}
+ \AxiomC{$\myjud{\mytmt}{\mybool}$}
+ \AxiomC{$\myjud{\mytya}{\mytyp}$}
+ \AxiomC{$\myjud{\mytyb}{\mytyp}$}
+ \TrinaryInfC{$\myjud{\myITE{\mytmt}{\mytya}{\mytyb}}{\mytyp}$}
\DisplayProof
\end{tabular}
-
- \myderivsp
-
- \AxiomC{$\myjud{\myse{P}}{\myprdec{\mytya \myeq \mytyb}}$}
- \AxiomC{$\myjud{\mytmt}{\mytya}$}
- \BinaryInfC{$\myjud{\mycohh{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\myprdec{\myjm{\mytmt}{\mytya}{\mycoee{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\mytyb}}}$}
- \DisplayProof
}
\mydesc{propositions:}{\myjud{\myprsyn}{\myprop}}{
- \begin{tabular}{cc}
+ \begin{tabular}{ccc}
\AxiomC{\phantom{$\myjud{\myse{P}}{\myprop}$}}
\UnaryInfC{$\myjud{\mytop}{\myprop}$}
\noLine
\noLine
\UnaryInfC{\phantom{$\myjud{\mybot}{\myprop}$}}
\DisplayProof
- \end{tabular}
-
- \myderivsp
-
- \begin{tabular}{cc}
+ &
\AxiomC{$\myjud{\myse{A}}{\mytyp}$}
\AxiomC{$\myjudd{\myctx; \myb{x} : \mytya}{\myse{P}}{\myprop}$}
\BinaryInfC{$\myjud{\myprfora{\myb{x}}{\mytya}{\myse{P}}}{\myprop}$}
- \DisplayProof
- &
- \AxiomC{$\myjud{\myse{A}}{\mytyp}$}
- \AxiomC{$\myjud{\myse{B}}{\mytyp}$}
- \BinaryInfC{$\myjud{\mytya \myeq \mytyb}{\myprop}$}
+ \noLine
+ \UnaryInfC{\phantom{$\myjud{\mybot}{\myprop}$}}
\DisplayProof
\end{tabular}
-
- \myderivsp
-
- \AxiomC{$\myjud{\myse{A}}{\mytyp}$}
- \AxiomC{$\myjud{\mytmm}{\myse{A}}$}
- \AxiomC{$\myjud{\myse{B}}{\mytyp}$}
- \AxiomC{$\myjud{\mytmn}{\myse{B}}$}
- \QuaternaryInfC{$\myjud{\myjm{\mytmm}{\myse{A}}{\mytmn}{\myse{B}}}{\myprop}$}
- \DisplayProof
}
+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
+types thanks to the hierarchy (which is gone), we need to reintroduce an
+ad-hoc conditional for types, where the reduction rule is the obvious
+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 $\myprdec{\myarg}$: \\
\mydesc{proposition decoding:}{\myprdec{\mytmsyn} \myred \mytmsyn}{
\begin{tabular}{cc}
$
\myfora{\myb{x}}{\mytya}{\myprdec{\myse{P}}}
\end{array}
$
+ \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.
+
+ 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.
+
+\subsection{Equality proofs}
+
+\mydesc{syntax}{ }{
+ $
+ \begin{array}{r@{\ }c@{\ }l}
+ \mytmsyn & ::= & \cdots \mysynsep
+ \mycoee{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn} \mysynsep
+ \mycohh{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn} \\
+ \myprsyn & ::= & \cdots \mysynsep \mytmsyn \myeq \mytmsyn \mysynsep
+ \myjm{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn}
+ \end{array}
+ $
+}
+
+\mydesc{typing:}{\myjud{\mytmsyn}{\mytmsyn}}{
+ \begin{tabular}{cc}
+ \AxiomC{$\myjud{\myse{P}}{\myprdec{\mytya \myeq \mytyb}}$}
+ \AxiomC{$\myjud{\mytmt}{\mytya}$}
+ \BinaryInfC{$\myjud{\mycoee{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\mytyb}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myjud{\myse{P}}{\myprdec{\mytya \myeq \mytyb}}$}
+ \AxiomC{$\myjud{\mytmt}{\mytya}$}
+ \BinaryInfC{$\myjud{\mycohh{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\myprdec{\myjm{\mytmt}{\mytya}{\mycoee{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\mytyb}}}$}
+ \DisplayProof
+
+ \end{tabular}
+}
+
+\mydesc{propositions:}{\myjud{\myprsyn}{\myprop}}{
+ \begin{tabular}{cc}
+ \AxiomC{$
+ \begin{array}{l}
+ \ \\
+ \myjud{\myse{A}}{\mytyp} \hspace{1cm} \myjud{\myse{B}}{\mytyp}
+ \end{array}
+ $}
+ \UnaryInfC{$\myjud{\mytya \myeq \mytyb}{\myprop}$}
+ \DisplayProof
+ &
+ \AxiomC{$
+ \begin{array}{c}
+ \myjud{\myse{A}}{\mytyp} \hspace{1cm} \myjud{\mytmm}{\myse{A}} \\
+ \myjud{\myse{B}}{\mytyp} \hspace{1cm} \myjud{\mytmn}{\myse{B}}
+ \end{array}
+ $}
+ \UnaryInfC{$\myjud{\myjm{\mytmm}{\myse{A}}{\mytmn}{\myse{B}}}{\myprop}$}
+ \DisplayProof
+
\end{tabular}
}
+
+While isolating a propositional universe as presented can be a useful
+exercises on its own, what we are really after is a useful notion of
+equality. In OTT we want to maintain the notion that things judged to
+be equal are still always repleaceable for one another with no
+additional changes. Note that this is not the same as saying that they
+are definitionally equal, since as we saw extensionally equal functions,
+while satisfying the above requirement, are not definitionally equal.
+
+Towards this goal we introduce two equality constructs in
+$\myprop$---the fact that they are in $\myprop$ indicates that they
+indeed have no computational content. The first construct, $\myarg
+\myeq \myarg$, relates types, the second,
+$\myjm{\myarg}{\myarg}{\myarg}{\myarg}$, relates values. The
+value-level equality is different from our old propositional equality:
+instead of ranging over only one type, we might form equalities between
+values of different types---the usefulness of this construct will be
+clear soon. In the literature this equality is known as `heterogeneous'
+or `John Major', since
+
+\begin{quote}
+ John Major's `classless society' widened people's aspirations to
+ equality, but also the gap between rich and poor. After all, aspiring
+ to be equal to others than oneself is the politics of envy. In much
+ the same way, forms equations between members of any type, but they
+ cannot be treated as equals (ie substituted) unless they are of the
+ same type. Just as before, each thing is only equal to
+ itself. \citep{McBride1999}.
+\end{quote}
+
+Correspondingly, at the term level, $\myfun{coe}$ (`coerce') lets us
+transport values between equal types; and $\myfun{coh}$ (`coherence')
+guarantees that $\myfun{coe}$ respects the value-level equality, or in
+other words that it really has no computational component: if we
+transport $\mytmm : \mytya$ to $\mytmn : \mytyb$, $\mytmm$ and $\mytmn$
+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.
+
+\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 $\myfun{coe}$.
+
+\begin{figure}[t]
+
\mydesc{equality reduction:}{\myprsyn \myred \myprsyn}{
$
\begin{array}{c@{\ }c@{\ }c@{\ }l}
\myempty & \myeq & \myempty & \myred \mytop \\
\myunit & \myeq & \myunit & \myred \mytop \\
\mybool & \myeq & \mybool & \myred \mytop \\
- \myexi{\myb{x_1}}{\mytya_1}{\mytyb_1} & \myeq & \myexi{\myb{x_2}}{\mytya_2}{\mytyb_2} & \myred \\
+ \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
- \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}
+ \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 \\
\myw{\myb{x_1}}{\mytya_1}{\mytyb_1} & \myeq & \myw{\myb{x_2}}{\mytya_2}{\mytyb_2} & \myred \cdots \\
- \mytya & \myeq & \mytyb & \myred \mybot\ \text{for other canonical types.}
+ \mytya & \myeq & \mytyb & \myred \mybot\ \text{if $\mytya$ and $\mytyb$ are canonical.}
\end{array}
$
}
-
+\myderivsp
\mydesc{reduction}{\mytmsyn \myred \mytmsyn}{
$
- \begin{array}{l@{\ }l@{\ }l@{\ }l@{\ }l@{\ }c@{\ }l@{\ }}
+ \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}) &
(\myexi{\myb{x_2}}{\mytya_2}{\mytyb_2}) & \myse{Q} &
- \mytmt_1 & \myred &
- foo \\
+ \mytmt_1 & \myred & \\
+ \multicolumn{7}{l}{
+ \myind{2}\begin{array}[t]{l@{\ }l@{\ }c@{\ }l}
+ \mysyn{let} & \myb{\mytmm_1} & \mapsto & \myapp{\myfst}{\mytmt_1} : \mytya_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{\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}} \\
+
\mycoe & (\myfora{\myb{x_1}}{\mytya_1}{\mytyb_1}) &
(\myfora{\myb{x_2}}{\mytya_2}{\mytyb_2}) & \myse{Q} &
- \mytmt_1 & \myred &
+ \mytmt & \myred &
\cdots \\
\mycoe & (\myw{\myb{x_1}}{\mytya_1}{\mytyb_1}) &
(\myw{\myb{x_2}}{\mytya_2}{\mytyb_2}) & \myse{Q} &
- \mytmt_1 & \myred &
+ \mytmt & \myred &
\cdots \\
- \mycoe & \mytya & \mytyb & \myse{Q} & \mytmt & \myred & \myapp{\myabsurd{\mytyb}}{\myse{Q}}
+ \mycoe & \mytya & \mytyb & \myse{Q} & \mytmt & \myred & \myapp{\myabsurd{\mytyb}}{\myse{Q}}\ \text{if $\mytya$ and $\mytyb$ are canonical.}
\end{array}
$
}
+\caption{Reducing type equalities, and using them when
+ $\myfun{coe}$rcing.}
+\label{fig:eqred}
+\end{figure}
-The original presentation of OTT employs the theory presented above. It is
-close to the one presented in section \ref{sec:itt}, with the additions
-presented above, and the change that only one the `first' universe, the type
-of small types ($\mytyp_0$), is present.
-
-The propositional universe is meant to be where equality proofs live in. The
-equality proofs are respectively between types ($\mytya = \mytyb$), and
-between values
-
-
-
-However, only one universe is present ($\mytyp_0$), and a \emph{propositional}
-universe is isolated, intended to be the universe where equality proofs live
-in. Propositions (as long as our system is consistent) are inhabited only by
-one element, and thus can all be treated as definitionally equal.
-
-
-
-% \section{Augmenting ITT}
-% \label{sec:practical}
-
-% \subsection{A more liberal hierarchy}
+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 $\myunit$, we can do better: we return its only member
+without matching on the term. 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.
+
+$\myprod$ 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 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}
+
+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 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:
-% \subsection{Type inference}
+\mydesc{typing:}{\myjud{\mytmsyn}{\mytmsyn}}{
+ \AxiomC{$\myjud{\mytmt}{\mytya}$}
+ \UnaryInfC{$\myjud{\myapp{\myrefl}{\mytmt}}{\myprdec{\myjm{\mytmt}{\mytya}{\mytmt}{\mytya}}}$}
+ \DisplayProof
-% \subsubsection{Bidirectional type checking}
+ \myderivspp
-% \subsubsection{Pattern unification}
+ \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
+}
-% \subsection{Pattern matching and explicit fixpoints}
+$\myrefl$ is the equivalent of the reflexivity rule in propositional
+equality, and $\mytyc{R}$ asserts that if we have a we have a $\mytyp$
+abstracting over a value we can substitute equal for equal---this lets
+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.
-% \subsection{Induction-recursion}
+\subsubsection{Value-level equality}
-% \subsection{Coinduction}
+\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 & : & \myunit&) & \myeq & (&\mytmt_2 & : & \myunit&) & \myred \mytop \\
+ (&\mytrue & : & \mybool&) & \myeq & (&\mytrue & : & \mybool&) & \myred \mytop \\
+ (&\myfalse & : & \mybool&) & \myeq & (&\myfalse & : & \mybool&) & \myred \mytop \\
+ (&\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
+ \myjm{\myapp{\mysnd}{\mytmt_1}}{\mysub{\mytyb_1}{\myb{x_1}}{\myapp{\myfst}{\mytmt_1}}}{\myapp{\mysnd}{\mytmt_2}}{\mysub{\mytyb_2}{\myb{x_2}}{\myapp{\myfst}{\mytmt_2}}}
+ } \\
+ (&\myse{f}_1 & : & \myfora{\mytya_1}{\myb{x_1}}{\mytyb_1}&) & \myeq & (&\myse{f}_2 & : & \myfora{\mytya_2}{\myb{x_2}}{\mytyb_2}&) & \myred \\
+ & \multicolumn{11}{@{}l}{
+ \myind{2} \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{\myse{f}_1}{\myb{x_1}}}{\mytyb_1[\myb{x_1}]}{\myapp{\myse{f}_2}{\myb{x_2}}}{\mytyb_2[\myb{x_2}]}
+ }}
+ } \\
+ (&\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 \mybot\ \text{if $\mytya_1$ and $\mytya_2$ are canonical.}
+ \end{array}
+ $
+}
-% \subsection{Dealing with partiality}
+As with type-level equality, we want value-level equality to reduce
+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.
+
+\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.
-% \subsection{Type holes}
+This process of identifying every proof as equivalent and removing
+$\myfun{coe}$rcions is known as \emph{quotation}.
\section{\mykant : the theory}
\label{sec:kant-theory}
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.
+We will first present the features of the system, and then describe the
+implementation we have developed in Section \ref{sec:kant-practice}.
-The features currently implemented in \mykant\ are:
+The defining features of \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}.
+ Section \ref{sec:itt}.
\item[Implicit, cumulative universe hierarchy] The user does not need to
specify universe level explicitly, and universes are \emph{cumulative}.
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:
+\item[Bidirectional type checking] While no `fancy' inference via
+ unification is present, we take advantage of a type synthesis system
+ in the style of \cite{Pierce2000}, extending the concept for user
+ defined data types.
-\begin{description}
-\item[Observational equality] As described in section \ref{sec:ott} but
+\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 analyse the features one by one, along with motivations and
+tradeoffs for the design decisions made.
\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 runs through the semantics of the
+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 having top-level canonical
+terms. The types are the same as those found in the normal STLC.
-% TODO do this --- is it even necessary
-
-% \subsubsection{Declarations and contexts}
+\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 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.
+
+\mydesc{typing:}{\myctx \vdash \mytmsyn \Leftrightarrow \mytmsyn}{
+ \begin{tabular}{ccc}
+ \AxiomC{$\myctx(x) = A$}
+ \UnaryInfC{$\myinf{\myb{x}}{A}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myjudd{\myctx;\myb{x} : A}{\mytmt}{\mytyb}$}
+ \UnaryInfC{$\mychk{\myabs{x}{\mytmt}}{\mytyb}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myjud{\mytmm}{\mytya \myarr \mytyb}$}
+ \AxiomC{$\myjud{\mytmn}{\mytya}$}
+ \BinaryInfC{$\myjud{\myapp{\mytmm}{\mytmn}}{\mytyb}$}
+ \DisplayProof
+ \end{tabular}
-% A \mykant declaration can be one of 4 kinds:
+ \myderivspp
-% \begin{description}
-% \item[Value] A declared variable, together with a type and a body.
-% \item[Postulate] 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. We will explain the need for records
-% later.
-% \end{description}
+ \begin{tabular}{cc}
+ \AxiomC{$\mychk{\mytmt}{\mytya}$}
+ \UnaryInfC{$\myinf{\myann{\mytmt}{\mytya}}{\mytya}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myinf{\mytmt}{\mytya}$}
+ \UnaryInfC{$\mychk{\mytmt}{\mytya}$}
+ \DisplayProof
+ \end{tabular}
+}
-% The syntax of
+For example, if we wanted to type function composition (in this case for
+naturals), we would have to annotate the term:
+\[
+ \myfun{comp} \mapsto (\myabs{\myb{f}\, \myb{g}\, \myb{x}}{\myb{f}\myappsp(\myb{g}\myappsp\myb{x})}) : (\mynat \myarr \mynat) \myarr (\mynat \myarr \mynat) \myarr \mynat \myarr \mynat
+\]
+But we would not have to annotate functions passed to it, since the type would be propagated to the arguments:
+\[
+ \myfun{comp}\myappsp (\myabs{\myb{x}}{\myb{x} \mathrel{\myfun{$+$}} 3}) \myappsp (\myabs{\myb{x}}{\myb{x} \mathrel{\myfun{$*$}} 4}) \myappsp 42
+\]
\subsection{Base terms and types}
-Let us begin by describing the primitives available without the user defining
-any data types, and without equality. The syntax given here is the one of the
-core (`desugared') terms, and the way we handle 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 rules too much, and just treat types as
-impredicative for the time being.
+Let us begin by describing the primitives available without the user
+defining any data types, and without equality. The way we handle
+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
+rules too much, and just treat types as impredicative for the time
+being.
\mydesc{syntax}{ }{
$
\begin{array}{r@{\ }c@{\ }l}
\mytmsyn & ::= & \mynamesyn \mysynsep \mytyp \\
- & | & \myfora{\myb{x}}{\mytmsyn}{\mytmsyn} \mysynsep
- \myabss{\myb{x}}{\mytmsyn}{\mytmsyn} \mysynsep
- (\myapp{\mytmsyn}{\mytmsyn}) \mysynsep
- (\myann{\mytmsyn}{\mytmsyn}) \\
+ & | & \myfora{\myb{x}}{\mytmsyn}{\mytmsyn} \mysynsep
+ \myabs{\myb{x}}{\mytmsyn} \mysynsep
+ (\myapp{\mytmsyn}{\mytmsyn}) \mysynsep
+ (\myann{\mytmsyn}{\mytmsyn}) \\
\mynamesyn & ::= & \myb{x} \mysynsep \myfun{f}
\end{array}
$
}
-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. We also want to make
-sure not to have duplicate top names, so we enforce that.
+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.
-% \mytyc{D} \mysynsep \mytyc{D}.\mydc{c}
-% \mysynsep \mytyc{D}.\myfun{f} \mysynsep
+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:
\mydesc{context validity:}{\myvalid{\myctx}}{
\begin{tabular}{ccc}
\end{tabular}
}
-Now we can present the reduction rules, which are unsurprising. We have the
-usual functional application ($\beta$-reduction), but also a rule to replace
-names with their bodies, if in the context ($\delta$-reduction), and one to
-discard type annotations. For this reason the new reduction rules are
-dependent on the context:
+Now we can present the reduction rules, which are unsurprising. We have
+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:
\mydesc{reduction:}{\myctx \vdash \mytmsyn \myred \mytmsyn}{
\begin{tabular}{ccc}
\end{tabular}
}
-We want to define a \emph{weak head normal form} (WHNF) for our terms, to give
-a syntax directed presentation of type rules with no `conversion' rule. We
-will consider all \emph{canonical} forms (abstractions and data constructors)
-to be in weak head normal form... % TODO finish
-
-We can now give types to our terms. Using our definition of WHNF, I will use
-$\mytmm \mynf \mytmn$ to indicate that $\mytmm$'s normal form is $\mytmn$.
-This way, we can avoid the non syntax-directed conversion rule, giving a more
-algorithmic presentation of type checking.
+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}
- \AxiomC{$\myb{x} : A \in \myctx$ or $\myb{x} \mapsto \mytmt : A \in \myctx$}
- \UnaryInfC{$\myinf{\myb{x}}{A}$}
+ \begin{tabular}{cccc}
+ \AxiomC{$\myse{name} : A \in \myctx$}
+ \UnaryInfC{$\myinf{\myse{name}}{A}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myfun{f} \mapsto \mytmt : A \in \myctx$}
+ \UnaryInfC{$\myinf{\myfun{f}}{A}$}
\DisplayProof
&
\AxiomC{$\mychk{\mytmt}{\mytya}$}
\UnaryInfC{$\myinf{\myann{\mytmt}{\mytya}}{\mytya}$}
\DisplayProof
+ &
+ \AxiomC{$\myinf{\mytmt}{\mytya}$}
+ \AxiomC{$\myctx \vdash \mytya \mydefeq \mytyb$}
+ \BinaryInfC{$\mychk{\mytmt}{\mytyb}$}
+ \DisplayProof
\end{tabular}
- \myderivsp
+ \myderivspp
- \AxiomC{$\myinf{\mytmm}{\mytya}$}
- \AxiomC{$\myctx \vdash \mytya \mynf \myfora{\myb{x}}{\mytyb}{\myse{C}}$}
- \AxiomC{$\mychk{\mytmn}{\mytyb}$}
- \TrinaryInfC{$\myinf{\myapp{\mytmm}{\mytmn}}{\mysub{\myse{C}}{\myb{x}}{\mytmn}}$}
- \DisplayProof
+ \begin{tabular}{ccc}
+ \AxiomC{$\myinf{\mytmm}{\myfora{\myb{x}}{\mytya}{\mytyb}}$}
+ \AxiomC{$\mychk{\mytmn}{\mytya}$}
+ \BinaryInfC{$\myinf{\myapp{\mytmm}{\mytmn}}{\mysub{\mytyb}{\myb{x}}{\mytmn}}$}
+ \DisplayProof
- \myderivsp
+ &
- \AxiomC{$\myctx \vdash \mytya \mynf \myfora{\myb{x}}{\mytyb}{\myse{C}}$}
- \AxiomC{$\mychkk{\myctx; \myb{x}: \mytyb}{\mytmt}{\myse{C}}$}
- \BinaryInfC{$\mychk{\myabs{\myb{x}}{\mytmt}}{\mytya}$}
- \DisplayProof
+ \AxiomC{$\mychkk{\myctx; \myb{x}: \mytya}{\mytmt}{\mytyb}$}
+ \UnaryInfC{$\mychk{\myabs{\myb{x}}{\mytmt}}{\myfora{\myb{x}}{\mytyb}{\mytyb}}$}
+ \DisplayProof
+ \end{tabular}
}
\subsection{Elaboration}
+As we mentioned, $\mykant$\ allows the user to define not only values
+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},
+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.
+
+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
+\[
+ (\myb{x} {:} \mynat); (\myb{p} : \myapp{\myfun{even}}{\myb{x}}) \myarr \mynat =
+ (\myb{x} {:} \mynat) \myarr (\myb{p} : \myapp{\myfun{even}}{\myb{x}}) \myarr \mynat
+\]
+
+We make use of various operations to manipulate telescopes:
+\begin{itemize}
+\item $\myhead(\mytele)$ refers to the first type appearing in
+ $\mytele$: $\myhead((\myb{x} {:} \mynat); (\myb{p} :
+ \myapp{\myfun{even}}{\myb{x}})) = \mynat$. Similarly,
+ $\myix_i(\mytele)$ refers to the $i^{th}$ type in a telescope
+ (1-indexed).
+\item $\mytake_i(\mytele)$ refers to the telescope created by taking the
+ first $i$ elements of $\mytele$: $\mytake_1((\myb{x} {:} \mynat); (\myb{p} :
+ \myapp{\myfun{even}}{\myb{x}})) = (\myb{x} {:} \mynat)$.
+\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, I'll use $\mytmsyn^n$
+to indicate a term vector composed of $n$ elements, or
+$\mytmsyn^{\mytele}$ for one composed by as many elements as the
+telescope.
+
+\subsubsection{Declarations syntax}
+
\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)
+ \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}
$
}
-\mydesc{typing:}{\myctx \vdash \mytmsyn \Leftrightarrow \mytmsyn}{
-
-}
+In \mykant\ we have four kind of declarations:
-\subsubsection{Values and postulated variables}
+\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.
+\end{description}
-As mentioned, in \mykant\ we can defined top level variables, with or without
-a body. We call the variables
+Elaborating defined variables consists of type checking 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:
\mydesc{context elaboration:}{\myelab{\mydeclsyn}{\myctx}}{
\begin{tabular}{cc}
}
\subsubsection{User defined types}
+\label{sec:user-type}
-\mydesc{syntax}{ }{
- $
- \begin{array}{l}
- \mynamesyn ::= \cdots \mysynsep \mytyc{D} \mysynsep \mytyc{D}.\mydc{c} \mysynsep \mytyc{D}.\myfun{f}
- \end{array}
- $
-}
-
-\mydesc{typing:}{ }{
- \AxiomC{$\mytyc{D} : \mytele \myarr \mytyp \in \myctx$}
- \AxiomC{$\mytyc{D}.\mydc{c} : \mytele \mycc \mytele' \myarr
- \myapp{\mytyc{D}}{\mytelee} \in \myctx$}
- \BinaryInfC{$\mychk{\myapp{\mytyc{D}.\mydc{c}}{\vec{t}}}{\myapp{\mytyc{D}}{\vec{A}}}$}
- \DisplayProof
- % TODO
-
- \myderivsp
-
- \AxiomC{$\mytyc{D} : \mytele \myarr \mytyp \in \myctx$}
- \AxiomC{$\mytyc{D}.\myfun{f} : \mytele \myarr
- \myapp{\mytyc{D}}{\mytelee} \myarr \myse{F}$}
- \AxiomC{$\myjud{\mytmt}{\myapp{\mytyc{D}}{\vec{A}}}$}
- \TrinaryInfC{$\myinf{\myapp{\mytyc{D}.\myfun{f}}{\mytmt}}{TODO}$}
- \DisplayProof
-}
-
-\subsubsection{Data types}
+Elaborating user defined types is the real effort. First, let's explain
+what we can defined, with some examples.
-\begin{figure}[t]
- \mydesc{syntax elaboration:}{\mydeclsyn \myelabf \mytmsyn ::= \cdots}{
- $
- \begin{array}{r@{\ }l}
- & \myadt{\mytyc{D}}{\mytele}{}{\cdots\ |\ \mydc{c}_n : \myvec{(\myb{x} {:} \mytya)} \ |\ \cdots } \\
- \myelabf &
-
+\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 ($\mydc{suc}$):
+ \[
+ \begin{array}{@{}l}
+ \myadt{\mynat}{ }{ }{
+ \mydc{zero} \mydcsep \mydc{suc} \myappsp \mynat
+ }
+ \end{array}
+ \]
+ This is very similar to what we would write in Haskell:
+ \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}
+ \mysmall
+ \begin{tabular}{ccc}
+ \AxiomC{\phantom{$\mychk{\mytmt}{\mynat}$}}
+ \UnaryInfC{$\myinf{\mynat}{\mytyp}$}
+ \DisplayProof
+ &
+ \AxiomC{\phantom{$\mychk{\mytmt}{\mynat}$}}
+ \UnaryInfC{$\myinf{\mytyc{\mynat}.\mydc{zero}}{\mynat}$}
+ \DisplayProof
+ &
+ \AxiomC{$\mychk{\mytmt}{\mynat}$}
+ \UnaryInfC{$\myinf{\mytyc{\mynat}.\mydc{suc} \myappsp \mytmt}{\mynat}$}
+ \DisplayProof
+ \end{tabular}
+ \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
+ 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
+ to each data constructor. When using data constructors in examples I
+ will omit the type constructor prefix for brevity.
+
+ 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}
+ \mysmall
+ \AxiomC{$\mychk{\mytmt}{\mynat}$}
+ \UnaryInfC{$
+ \myinf{\mytyc{\mynat}.\myfun{elim} \myappsp \mytmt}{
+ \begin{array}{@{}l}
+ \myfora{\myb{P}}{\mynat \myarr \mytyp}{ \\ \myapp{\myb{P}}{\mydc{zero}} \myarr (\myfora{\myb{x}}{\mynat}{\myapp{\myb{P}}{\myb{x}} \myarr \myapp{\myb{P}}{(\myapp{\mydc{suc}}{\myb{x}})}}) \myarr \\ \myapp{\myb{P}}{\mytmt}}
+ \end{array}
+ }$}
+ \end{prooftree}
+ $\mynat.\myfun{elim}$ corresponds to the induction principle for
+ natural numbers: if we have a predicate on numbers ($\myb{P}$), and we
+ know that predicate holds for the base case
+ ($\myapp{\myb{P}}{\mydc{zero}}$) and for each inductive step
+ ($\myfora{\myb{x}}{\mynat}{\myapp{\myb{P}}{\myb{x}} \myarr
+ \myapp{\myb{P}}{(\myapp{\mydc{suc}}{\myb{x}})}}$), then $\myb{P}$
+ holds for any number. As with the data constructors, we require the
+ eliminator to be applied to the `destructed' element.
+
+ 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
+ $\mydc{suc}$cessor:
+ \[
+ \begin{array}{@{}l@{}l}
+ \mytyc{\mynat}.\myfun{elim} \myappsp \mydc{zero} & \myappsp \myse{P} \myappsp \myse{pz} \myappsp \myse{ps} \myred \myse{pz} \\
+ \mytyc{\mynat}.\myfun{elim} \myappsp (\mydc{suc} \myappsp \mytmt) & \myappsp \myse{P} \myappsp \myse{pz} \myappsp \myse{ps} \myred \myse{ps} \myappsp \mytmt \myappsp (\mynat.\myfun{elim} \myappsp \mytmt \myappsp \myse{P} \myappsp \myse{pz} \myappsp \myse{ps})
+ \end{array}
+ \]
+ The Haskell equivalent would be
+ \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.
+
+\item[Binary trees] Now for a polymorphic data type: binary trees, since
+ lists are too similar to natural numbers to be interesting.
+ \[
+ \begin{array}{@{}l}
+ \myadt{\mytree}{\myappsp (\myb{A} {:} \mytyp)}{ }{
+ \mydc{leaf} \mydcsep \mydc{node} \myappsp (\myapp{\mytree}{\myb{A}}) \myappsp \myb{A} \myappsp (\myapp{\mytree}{\myb{A}})
+ }
+ \end{array}
+ \]
+ Now the purpose of constructors as syntax can be explained: what would
+ 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:
+ \[
+ \begin{array}{@{}l@{\ }l}
+ \mydc{leaf} & : \myfora{\myb{A}}{\mytyp}{\myapp{\mytree}{\myb{A}}} \\
+ \mydc{node} & : \myfora{\myb{A}}{\mytyp}{\myapp{\mytree}{\myb{A}} \myarr \myb{A} \myarr \myapp{\mytree}{\myb{A}} \myarr \myapp{\mytree}{\myb{A}}}
+ \end{array}
+ \]
+ 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
+ \[
+ \mydc{node} \myappsp \mynat \myappsp (\mydc{node} \myappsp \mynat \myappsp (\mydc{leaf} \myappsp \mynat) \myappsp (\myapp{\mydc{suc}}{\mydc{zero}}) \myappsp (\mydc{leaf} \myappsp \mynat)) \myappsp \mydc{zero} \myappsp (\mydc{leaf} \myappsp \mynat)
+ \]
+ 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:
+ \begin{center}
+ \mysmall
+ \begin{tabular}{cc}
+ \AxiomC{$\mychk{\mytya}{\mytyp}$}
+ \UnaryInfC{$\mychk{\mydc{leaf}}{\myapp{\mytree}{\mytya}}$}
+ \DisplayProof
+ &
+ \AxiomC{$\mychk{\mytmm}{\mytree \myappsp \mytya}$}
+ \AxiomC{$\mychk{\mytmt}{\mytya}$}
+ \AxiomC{$\mychk{\mytmm}{\mytree \myappsp \mytya}$}
+ \TrinaryInfC{$\mychk{\mydc{node} \myappsp \mytmm \myappsp \mytmt \myappsp \mytmn}{\mytree \myappsp \mytya}$}
+ \DisplayProof
+ \end{tabular}
+ \end{center}
+ Which enables us to write, much more concisely
+ \[
+ \mydc{node} \myappsp (\mydc{node} \myappsp \mydc{leaf} \myappsp (\myapp{\mydc{suc}}{\mydc{zero}}) \myappsp \mydc{leaf}) \myappsp \mydc{zero} \myappsp \mydc{leaf} : \myapp{\mytree}{\mynat}
+ \]
+ 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}
+ \small
+ \AxiomC{$\myinf{\mytmt}{\myapp{\mytree}{\mytya}}$}
+ \UnaryInfC{$
+ \myinf{\mytree.\myfun{elim} \myappsp \mytmt}{
+ \begin{array}{@{}l}
+ (\myb{P} {:} \myapp{\mytree}{\mytya} \myarr \mytyp) \myarr \\
+ \myapp{\myb{P}}{\mydc{leaf}} \myarr \\
+ ((\myb{l} {:} \myapp{\mytree}{\mytya}) (\myb{x} {:} \mytya) (\myb{r} {:} \myapp{\mytree}{\mytya}) \myarr \myapp{\myb{P}}{\myb{l}} \myarr
+ \myapp{\myb{P}}{\myb{r}} \myarr \myb{P} \myappsp (\mydc{node} \myappsp \myb{l} \myappsp \myb{x} \myappsp \myb{r})) \myarr \\
+ \myapp{\myb{P}}{\mytmt}
+ \end{array}
+ }
+ $}
+ \end{prooftree}
+ As expected, the eliminator embodies structural induction on trees.
+
+\item[Empty type] We have presented types that have at least one
+ constructors, but nothing prevents us from defining types with
+ \emph{no} constructors:
+ \[\myadt{\mytyc{Empty}}{ }{ }{ }\]
+ 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:
+ \begin{prooftree}
+ \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}
+
+\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
+ 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
+ $\mytmm \le \mytmn$:
+ \[
+ \begin{array}{@{}l}
+ \myfun{le} : \mynat \myarr \mynat \myarr \mytyp \\
+ \myfun{le} \myappsp \myb{n} \mapsto \\
+ \myind{2} \mynat.\myfun{elim} \\
+ \myind{2}\myind{2} \myb{n} \\
+ \myind{2}\myind{2} (\myabs{\myarg}{\mynat \myarr \mytyp}) \\
+ \myind{2}\myind{2} (\myabs{\myarg}{\myunit}) \\
+ \myind{2}\myind{2} (\myabs{\myb{n}\, \myb{f}\, \myb{m}}{
+ \mynat.\myfun{elim} \myappsp \myb{m} \myappsp (\myabs{\myarg}{\mytyp}) \myappsp \myempty \myappsp (\myabs{\myb{m'}\, \myarg}{\myapp{\myb{f}}{\myb{m'}}})
+ })
+ \end{array}
+ \]
+ We return $\myunit$ if the scrutinised is $\mydc{zero}$ (every
+ 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
+ function:
+ \[
+ \begin{array}{@{}l}
+ \myadt{\mytyc{Lift}}{ }{ }{\mydc{bot} \mydcsep \mydc{lift} \myappsp \mynat \mydcsep \mydc{top}}\\
+ \myfun{le'} : \mytyc{Lift} \myarr \mytyc{Lift} \myarr \mytyp\\
+ \myfun{le'} \myappsp \myb{l_1} \mapsto \\
+ \myind{2} \mytyc{Lift}.\myfun{elim} \\
+ \myind{2}\myind{2} \myb{l_1} \\
+ \myind{2}\myind{2} (\myabs{\myarg}{\mytyc{Lift} \myarr \mytyp}) \\
+ \myind{2}\myind{2} (\myabs{\myarg}{\myunit}) \\
+ \myind{2}\myind{2} (\myabs{\myb{n_1}\, \myb{n_2}}{
+ \mytyc{Lift}.\myfun{elim} \myappsp \myb{l_2} \myappsp (\myabs{\myarg}{\mytyp}) \myappsp \myempty \myappsp (\myabs{\myb{n_2}}{\myfun{le} \myappsp \myb{n_1} \myappsp \myb{n_2}}) \myappsp \myunit
+ }) \\
+ \myind{2}\myind{2} (\myabs{\myb{n_1}\, \myb{n_2}}{
+ \mytyc{Lift}.\myfun{elim} \myappsp \myb{l_2} \myappsp (\myabs{\myarg}{\mytyp}) \myappsp \myempty \myappsp (\myabs{\myarg}{\myempty}) \myappsp \myunit
+ })
+ \end{array}
+ \]
+ 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:
+ \[
+ \begin{array}{@{}l}
+ \myadt{\mytyc{OList}}{\myappsp (\myb{low}\ \myb{upp} {:} \mytyc{Lift})}{\\ \myind{2}}{
+ \mydc{nil} \myappsp (\myfun{le'} \myappsp \myb{low} \myappsp \myb{upp}) \mydcsep \mydc{cons} \myappsp (\myb{n} {:} \mynat) \myappsp (\mytyc{OList} \myappsp (\myfun{lift} \myappsp \myb{n}) \myappsp \myb{upp}) \myappsp (\myfun{le'} \myappsp \myb{low} \myappsp (\myfun{lift} \myappsp \myb{n})
+ }
+ \end{array}
+ \]
+ If we want we can then employ this structure to write and prove
+ correct various sorting algorithms.\footnote{See this presentation by
+ Conor McBride:
+ \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}.}
+
+\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
+ fields of the said constructor.
+
+ For example, we can recover dependent products:
+ \[
+ \begin{array}{@{}l}
+ \myreco{\mytyc{Prod}}{\myappsp (\myb{A} {:} \mytyp) \myappsp (\myb{B} {:} \myb{A} \myarr \mytyp)}{\\ \myind{2}}{\myfst : \myb{A}, \mysnd : \myapp{\myb{B}}{\myb{fst}}}
+ \end{array}
+ \]
+ 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.
+
+ 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}
+ \mysmall
+ \begin{tabular}{cc}
+ \AxiomC{$\mychk{\mytmm}{\mytya}$}
+ \AxiomC{$\mychk{\mytmn}{\myapp{\mytyb}{\mytmm}}$}
+ \BinaryInfC{$\mychk{\mytyc{Prod}.\mydc{constr} \myappsp \mytmm \myappsp \mytmn}{\mytyc{Prod} \myappsp \mytya \myappsp \mytyb}$}
+ \noLine
+ \UnaryInfC{\phantom{$\myinf{\myfun{snd} \myappsp \mytmt}{\mytyb \myappsp (\myfst \myappsp \mytmt)}$}}
+ \DisplayProof
+ &
+ \AxiomC{$\myinf{\mytmt}{\mytyc{Prod} \myappsp \mytya \myappsp \mytyb}$}
+ \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.
+\end{description}
+
+\begin{figure}[p]
+ \mydesc{syntax}{ }{
+ \footnotesize
+ $
+ \begin{array}{l}
+ \mynamesyn ::= \cdots \mysynsep \mytyc{D} \mysynsep \mytyc{D}.\mydc{c} \mysynsep \mytyc{D}.\myfun{f}
+ \end{array}
+ $
+ }
+
+ \mynegder
+
+ \mydesc{syntax elaboration:}{\mydeclsyn \myelabf \mytmsyn ::= \cdots}{
+ \footnotesize
+ $
+ \begin{array}{r@{\ }l}
+ & \myadt{\mytyc{D}}{\mytele}{}{\cdots\ |\ \mydc{c}_n : \mytele_n } \\
+ \myelabf &
+
\begin{array}{r@{\ }c@{\ }l}
- \mytmsyn & ::= & \cdots \mysynsep \myapp{\mytyc{D}}{\myvec{\mytmsyn}} \mysynsep
- \mytyc{D}.\mydc{c}_n \myappsp \myvec{\mytmsyn} \mysynsep \cdots \mysynsep \mytyc{D}.\myfun{elim} \myappsp \mytmsyn \\
+ \mytmsyn & ::= & \cdots \mysynsep \myapp{\mytyc{D}}{\mytmsyn^{\mytele}} \mysynsep \cdots \mysynsep
+ \mytyc{D}.\mydc{c}_n \myappsp \mytmsyn^{\mytele_n} \mysynsep \mytyc{D}.\myfun{elim} \myappsp \mytmsyn \\
\end{array}
\end{array}
$
}
+ \mynegder
+
\mydesc{context elaboration:}{\myelab{\mydeclsyn}{\myctx}}{
- \AxiomC{$\myinf{\mytele \myarr \mytyp}{\mytyp}$}
- \AxiomC{$\mytyc{D} \not\in \myctx$}
- \noLine
- \BinaryInfC{$\myinff{\myctx;\ \mytyc{D} : \mytele \myarr \mytyp}{\mytele \mycc \mytele_i \myarr \myapp{\mytyc{D}}{\mytelee}}{\mytyp}\ \ \ (1 \leq i \leq n)$}
- \noLine
- \UnaryInfC{For each $(\myb{x} {:} \mytya)$ in each $\mytele_i$, if $\mytyc{D} \in \mytya$, then $\mytya = \myapp{\mytyc{D}}{\vec{\mytmt}}$.}
+ \footnotesize
+
+ \AxiomC{$
+ \begin{array}{c}
+ \myinf{\mytele \myarr \mytyp}{\mytyp}\hspace{0.8cm}
+ \mytyc{D} \not\in \myctx \\
+ \myinff{\myctx;\ \mytyc{D} : \mytele \myarr \mytyp}{\mytele \mycc \mytele_i \myarr \myapp{\mytyc{D}}{\mytelee}}{\mytyp}\ \ \ (1 \leq i \leq n) \\
+ \text{For each $(\myb{x} {:} \mytya)$ in each $\mytele_i$, if $\mytyc{D} \in \mytya$, then $\mytya = \myapp{\mytyc{D}}{\vec{\mytmt}}$.}
+ \end{array}
+ $}
\UnaryInfC{$
\begin{array}{r@{\ }c@{\ }l}
- \myctx & \myelabt & \myadt{\mytyc{D}}{\mytele}{}{ \mydc{c} : \mytele_1 \ |\ \cdots \ |\ \mydc{c}_n : \mytele_n } \\
+ \myctx & \myelabt & \myadt{\mytyc{D}}{\mytele}{}{ \cdots \ |\ \mydc{c}_n : \mytele_n } \\
& & \vspace{-0.2cm} \\
- & \myelabf & \myctx;\ \mytyc{D} : \mytele \mycc \mytyp;\ \mytyc{D}.\mydc{c}_1 : \mytele \mycc \mytele_1 \myarr \myapp{\mytyc{D}}{\mytelee};\ \cdots;\ \mytyc{D}.\mydc{c}_n : \mytele \mycc \mytele_n \myarr \myapp{\mytyc{D}}{\mytelee}; \\
+ & \myelabf & \myctx;\ \mytyc{D} : \mytele \myarr \mytyp;\ \cdots;\ \mytyc{D}.\mydc{c}_n : \mytele \mycc \mytele_n \myarr \myapp{\mytyc{D}}{\mytelee}; \\
& &
\begin{array}{@{}r@{\ }l l}
\mytyc{D}.\myfun{elim} : & \mytele \myarr (\myb{x} {:} \myapp{\mytyc{D}}{\mytelee}) \myarr & \textbf{target} \\
& (\myb{P} {:} \myapp{\mytyc{D}}{\mytelee} \myarr \mytyp) \myarr & \textbf{motive} \\
& \left.
\begin{array}{@{}l}
- (\mytele_1 \mycc \myhyps(\myb{P}, \mytele_1) \myarr \myapp{\myb{P}}{(\myapp{\mytyc{D}.\mydc{c}_1}{\mytelee_1})}) \myarr \\
\myind{3} \vdots \\
(\mytele_n \mycc \myhyps(\myb{P}, \mytele_n) \myarr \myapp{\myb{P}}{(\myapp{\mytyc{D}.\mydc{c}_n}{\mytelee_n})}) \myarr
\end{array} \right \}
& \textbf{methods} \\
& \myapp{\myb{P}}{\myb{x}} &
- \end{array} \\
- \\
- \multicolumn{3}{l}{
+ \end{array}
+ \end{array}
+ $}
+ \DisplayProof \\ \vspace{0.2cm}\ \\
+ $
\begin{array}{@{}l l@{\ } l@{} r c l}
\textbf{where} & \myhyps(\myb{P}, & \myemptytele &) & \mymetagoes & \myemptytele \\
& \myhyps(\myb{P}, & (\myb{r} {:} \myapp{\mytyc{D}}{\vec{\mytmt}}) \mycc \mytele &) & \mymetagoes & (\myb{r'} {:} \myapp{\myb{P}}{\myb{r}}) \mycc \myhyps(\myb{P}, \mytele) \\
& \myhyps(\myb{P}, & (\myb{x} {:} \mytya) \mycc \mytele & ) & \mymetagoes & \myhyps(\myb{P}, \mytele)
\end{array}
- }
- \end{array}
- $}
- \DisplayProof
+ $
+
}
+ \mynegder
+
\mydesc{reduction elaboration:}{\mydeclsyn \myelabf \myctx \vdash \mytmsyn \myred \mytmsyn}{
+ \footnotesize
+ $\myadt{\mytyc{D}}{\mytele}{}{ \cdots \ |\ \mydc{c}_n : \mytele_n } \ \ \myelabf$
\AxiomC{$\mytyc{D} : \mytele \myarr \mytyp \in \myctx$}
\AxiomC{$\mytyc{D}.\mydc{c}_i : \mytele;\mytele_i \myarr \myapp{\mytyc{D}}{\mytelee} \in \myctx$}
\BinaryInfC{$
- \begin{array}{c}
- \myctx \vdash \myapp{\myapp{\myapp{\mytyc{D}.\myfun{elim}}{(\myapp{\mytyc{D}.\mydc{c}_i}{\vec{\myse{t}}})}}{\myse{P}}}{\vec{\myse{m}}} \myred \myapp{\myapp{\myse{m}_i}{\vec{\mytmt}}}{\myrecs(\myse{P}, \vec{m}, \mytele_i)} \\ \\
+ \myctx \vdash \myapp{\myapp{\myapp{\mytyc{D}.\myfun{elim}}{(\myapp{\mytyc{D}.\mydc{c}_i}{\vec{\myse{t}}})}}{\myse{P}}}{\vec{\myse{m}}} \myred \myapp{\myapp{\myse{m}_i}{\vec{\mytmt}}}{\myrecs(\myse{P}, \vec{m}, \mytele_i)}
+ $}
+ \DisplayProof \\ \vspace{0.2cm}\ \\
+ $
\begin{array}{@{}l l@{\ } l@{} r c l}
\textbf{where} & \myrecs(\myse{P}, \vec{m}, & \myemptytele &) & \mymetagoes & \myemptytele \\
& \myrecs(\myse{P}, \vec{m}, & (\myb{r} {:} \myapp{\mytyc{D}}{\vec{A}}); \mytele & ) & \mymetagoes & (\mytyc{D}.\myfun{elim} \myappsp \myb{r} \myappsp \myse{P} \myappsp \vec{m}); \myrecs(\myse{P}, \vec{m}, \mytele) \\
& \myrecs(\myse{P}, \vec{m}, & (\myb{x} {:} \mytya); \mytele &) & \mymetagoes & \myrecs(\myse{P}, \vec{m}, \mytele)
- \end{array}
\end{array}
- $}
- \DisplayProof
+ $
}
- \caption{Elaborations for data types.}
- \label{fig:elab-adt}
-\end{figure}
-
-
-\subsubsection{Records}
+ \mynegder
-\begin{figure}[t]
-\mydesc{syntax elaboration:}{\myelab{\mydeclsyn}{\mytmsyn ::= \cdots}}{
+ \mydesc{syntax elaboration:}{\myelab{\mydeclsyn}{\mytmsyn ::= \cdots}}{
+ \footnotesize
$
\begin{array}{r@{\ }c@{\ }l}
- \myctx & \myelabt & \myadt{\mytyc{D}}{\mytele}{}{\cdots\ |\ \mydc{c}_n : \myvec{(\myb{x} {:} \mytya)} \ |\ \cdots } \\
+ \myctx & \myelabt & \myreco{\mytyc{D}}{\mytele}{}{ \cdots, \myfun{f}_n : \myse{F}_n } \\
& \myelabf &
\begin{array}{r@{\ }c@{\ }l}
- \mytmsyn & ::= & \cdots \mysynsep \myapp{\mytyc{D}}{\myvec{\mytmsyn}} \mysynsep
- \mytyc{D}.\mydc{c}_n \myappsp \myvec{\mytmsyn} \mysynsep \cdots \mysynsep \mytyc{D}.\myfun{elim} \myappsp \mytmsyn \\
+ \mytmsyn & ::= & \cdots \mysynsep \myapp{\mytyc{D}}{\mytmsyn^{\mytele}} \mysynsep \mytyc{D}.\mydc{constr} \myappsp \mytmsyn^{n} \mysynsep \cdots \mysynsep \mytyc{D}.\myfun{f}_n \myappsp \mytmsyn \\
\end{array}
\end{array}
$
}
+ \mynegder
\mydesc{context elaboration:}{\myelab{\mydeclsyn}{\myctx}}{
- \AxiomC{$\myinf{\mytele \myarr \mytyp}{\mytyp}$}
- \AxiomC{$\mytyc{D} \not\in \myctx$}
- \noLine
- \BinaryInfC{$\myinff{\myctx; \mytele; (\myb{f}_j : \myse{F}_j)_{j=1}^{i - 1}}{F_i}{\mytyp} \myind{3} (1 \le i \le n)$}
+ \footnotesize
+ \AxiomC{$
+ \begin{array}{c}
+ \myinf{\mytele \myarr \mytyp}{\mytyp}\hspace{0.8cm}
+ \mytyc{D} \not\in \myctx \\
+ \myinff{\myctx; \mytele; (\myb{f}_j : \myse{F}_j)_{j=1}^{i - 1}}{F_i}{\mytyp} \myind{3} (1 \le i \le n)
+ \end{array}
+ $}
\UnaryInfC{$
\begin{array}{r@{\ }c@{\ }l}
- \myctx & \myelabt & \myreco{\mytyc{D}}{\mytele}{}{ \myfun{f}_1 : \myse{F}_1, \cdots, \myfun{f}_n : \myse{F}_n } \\
+ \myctx & \myelabt & \myreco{\mytyc{D}}{\mytele}{}{ \cdots, \myfun{f}_n : \myse{F}_n } \\
& & \vspace{-0.2cm} \\
- & \myelabf & \myctx;\ \mytyc{D} : \mytele \myarr \mytyp;\\
- & & \mytyc{D}.\myfun{f}_1 : \mytele \myarr \myapp{\mytyc{D}}{\mytelee} \myarr \myse{F}_1;\ \cdots;\ \mytyc{D}.\myfun{f}_n : \mytele \myarr (\myb{x} {:} \myapp{\mytyc{D}}{\mytelee}) \myarr \mysub{\myse{F}_n}{\myb{f}_i}{\myapp{\myfun{f}_i}{\myb{x}}}_{i = 1}^{n-1}; \\
+ & \myelabf & \myctx;\ \mytyc{D} : \mytele \myarr \mytyp;\ \cdots;\ \mytyc{D}.\myfun{f}_n : \mytele \myarr (\myb{x} {:} \myapp{\mytyc{D}}{\mytelee}) \myarr \mysub{\myse{F}_n}{\myb{f}_i}{\myapp{\myfun{f}_i}{\myb{x}}}_{i = 1}^{n-1}; \\
& & \mytyc{D}.\mydc{constr} : \mytele \myarr \myse{F}_1 \myarr \cdots \myarr \myse{F}_n \myarr \myapp{\mytyc{D}}{\mytelee};
\end{array}
$}
\DisplayProof
}
+ \mynegder
+
\mydesc{reduction elaboration:}{\mydeclsyn \myelabf \myctx \vdash \mytmsyn \myred \mytmsyn}{
- \AxiomC{$\mytyc{D} \in \myctx$}
- \UnaryInfC{$\myctx \vdash \myapp{\mytyc{D}.\myfun{f}_i}{(\mytyc{D}.\mydc{constr} \myappsp \vec{t})} \myred t_i$}
- \DisplayProof
+ \footnotesize
+ $\myreco{\mytyc{D}}{\mytele}{}{ \cdots, \myfun{f}_n : \myse{F}_n } \ \ \myelabf$
+ \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
}
- \caption{Elaborations for records.}
- \label{fig:elab-adt}
+ \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.
+
+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 \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.
+
+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:}{\myctx \vdash \mytmsyn \Leftrightarrow \mytmsyn}{
+ \AxiomC{$
+ \begin{array}{c}
+ \mytyc{D} : \mytele \myarr \mytyp \in \myctx \hspace{1cm}
+ \mytyc{D}.\mydc{c} : \mytele \mycc \mytele' \myarr
+ \myapp{\mytyc{D}}{\mytelee} \in \myctx \\
+ \mytele'' = (\mytele;\mytele')\vec{A} \hspace{1cm}
+ \mychkk{\myctx; \mytake_{i-1}(\mytele'')}{t_i}{\myix_i( \mytele'')}\ \
+ (1 \le i \le \mytele'')
+ \end{array}
+ $}
+ \UnaryInfC{$\mychk{\myapp{\mytyc{D}.\mydc{c}}{\vec{t}}}{\myapp{\mytyc{D}}{\vec{A}}}$}
+ \DisplayProof
+
+ \myderivspp
-\subsection{Type hierarchy}
+ \AxiomC{$\mytyc{D} : \mytele \myarr \mytyp \in \myctx$}
+ \AxiomC{$\mytyc{D}.\myfun{f} : \mytele \mycc (\myb{x} {:}
+ \myapp{\mytyc{D}}{\mytelee}) \myarr \myse{F}$}
+ \AxiomC{$\myjud{\mytmt}{\myapp{\mytyc{D}}{\vec{A}}}$}
+ \TrinaryInfC{$\myinf{\myapp{\mytyc{D}.\myfun{f}}{\mytmt}}{(\mytele
+ \mycc (\myb{x} {:} \myapp{\mytyc{D}}{\mytelee}) \myarr
+ \myse{F})(\vec{A};\mytmt)}$}
+ \DisplayProof
+ }
+
+\subsubsection{Why user defined types? Why eliminators?}
+
+The `hardest' design choice when designing $\mykant$\ was to decide
+whether user defined types should be included. In the end, we can
+devise
+
+% TODO reference levitated theories, indexed containers
+
+foobar
+
+\subsection{Cumulative hierarchy and typical ambiguity}
\label{sec:term-hierarchy}
+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{sec:hurkens}.
+
+However, hierarchy as presented in section \label{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
+have a function that takes two $\mytyp_0$ and forms a new $\mytyp_0$.
+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!]
+
+ % TODO finish
+\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. Figure \ref{fig:cumulativity} gives a formal definition of
+cumulativity for types, abstractions, and data constructors.
+
+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
+$\myfun{$\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.
+
+We define a partial ordering on the levels, with both weak ($\le$) and
+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.
+
+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 $\myctx \vdash
+\myfora{\myb{x}}{\mytya}{\mytyb}$, 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}.
+
+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:
+\[
+\myarg\myfun{$\vee$}\myarg : (l_1\, l_2 : \mytyc{Level}) \myarr \mytyp_{l_1} \myarr \mytyp_{l_2} \myarr \mytyp_{l_1 \mylub l_2}
+\]
+Inference algorithms to automatically derive this kind of relationship
+are currently subject of research. We chose less flexible but more
+concise way, since it is easier to implement and better understood.
+
+% \begin{figure}[t]
+% % TODO do this
+% \caption{Constraints generated by the typical ambiguity engine. We
+% assume some global set of constraints with the ability of generating
+% fresh references.}
+% \label{fig:hierarchy}
+% \end{figure}
+
\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.
+
+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 broad design plan, although with some modifications. Most of
+the 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:
+\[
+\begin{array}{l}
+ \myadt{\mytyc{Empty}}{}{ }{ } \\
+ \myfun{absurd} : (\myb{A} {:} \mytyp) \myarr \mytyc{Empty} \myarr \myb{A} \mapsto \\
+ \myind{2} \myabs{\myb{A\ \myb{bot}}}{\mytyc{Empty}.\myfun{elim} \myappsp \myb{bot} \myappsp (\myabs{\_}{\myb{A}})} \\
+ \ \\
+
+ \myreco{\mytyc{Unit}}{}{}{ } \\ \ \\
+
+ \myreco{\mytyc{Prod}}{\myappsp (\myb{A}\ \myb{B} {:} \mytyp)}{ }{\myfun{fst} : \myb{A}, \myfun{snd} : \myb{B} }
+\end{array}
+\]
+When using $\mytyc{Prod}$, we shall use $\myprod$ to define `nested'
+products, and $\myproj{n}$ to project elements from them, so that
+\[
+\begin{array}{@{}l}
+\mytya \myprod \mytyb = \mytyc{Prod} \myappsp \mytya \myappsp (\mytyc{Prod} \myappsp \mytyb \myappsp \myunit) \\
+\mytya \myprod \mytyb \myprod \myse{C} = \mytyc{Prod} \myappsp \mytya \myappsp (\mytyc{Prod} \myappsp \mytyb \myappsp (\mytyc{Prod} \myappsp \mytyc \myappsp \myunit)) \\
+\myind{2} \vdots \\
+\myproj{1} : \mytyc{Prod} \myappsp \mytya \myappsp \mytyb \myarr \mytya \\
+\myproj{2} : \mytyc{Prod} \myappsp \mytya \myappsp (\mytyc{Prod} \myappsp \mytyb \myappsp \myse{C}) \myarr \mytyb \\
+\myind{2} \vdots
+\end{array}
+\]
+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:
+
+\mydesc{syntax}{ }{
+ $
+ \begin{array}{r@{\ }c@{\ }l}
+ \mytmsyn & ::= & \cdots \mysynsep \myprdec{\myprsyn} \\
+ \myprsyn & ::= & \mybot \mysynsep \mytop \mysynsep \myprsyn \myand \myprsyn \mysynsep \myprfora{\myb{x}}{\mytmsyn}{\myprsyn}
+ \end{array}
+ $
+}
+
+\mydesc{proposition decoding:}{\myprdec{\mytmsyn} \myred \mytmsyn}{
+ \begin{tabular}{cc}
+ $
+ \begin{array}{l@{\ }c@{\ }l}
+ \myprdec{\mybot} & \myred & \myempty \\
+ \myprdec{\mytop} & \myred & \myunit
+ \end{array}
+ $
+ &
+ $
+ \begin{array}{r@{ }c@{ }l@{\ }c@{\ }l}
+ \myprdec{&\myse{P} \myand \myse{Q} &} & \myred & \myprdec{\myse{P}} \myprod \myprdec{\myse{Q}} \\
+ \myprdec{&\myprfora{\myb{x}}{\mytya}{\myse{P}} &} & \myred &
+ \myfora{\myb{x}}{\mytya}{\myprdec{\myse{P}}}
+ \end{array}
+ $
+ \end{tabular}
+}
+
+Adopting the same convention as with $\mytyp$-level products, we will
+nest $\myand$ in the same way.
+
+\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,
+given what we already know about OTT, assuming the same propositional
+equalities.
+
+\begin{description}
+
+\item[Product types] Let's 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:
+ \[
+ \begin{array}{@{}l}
+ \myreco{\mysigma}{\myappsp (\myb{A} {:} \mytyp) \myappsp (\myb{B} {:} \myb{A} \myarr \mytyp)}{\\ \myind{2}}{\myfst : \myb{A}, \mysnd : \myapp{\myb{B}}{\myb{fst}}}
+ \end{array}
+ \]
+ Let's start with type-level equality. The result we want is
+ \[
+ \begin{array}{@{}l}
+ \mysigma \myappsp \mytya_1 \myappsp \mytyb_1 \myeq \mysigma \myappsp \mytya_2 \myappsp \mytyb_2 \myred \\
+ \myind{2} \mytya_1 \myeq \mytya_2 \myand \myprfora{\myb{x_1}}{\mytya_1}{\myprfora{\myb{x_2}}{\mytya_2}{\myjm{\myb{x_1}}{\mytya_1}{\myb{x_2}}{\mytya_2}} \myimpl \myapp{\mytyb_1}{\myb{x_1}} \myeq \myapp{\mytyb_2}{\myb{x_2}}}
+ \end{array}
+ \]
+ The difference here is that in the original presentation of OTT
+ the type binders are explicit, while here $\mytyb_1$ and $\mytyb_2$
+ 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.
+
+ If we take, just like we saw previously in OTT
+ \[
+ \begin{array}{@{}l}
+ \myjm{\myse{f}_1}{\myfora{\mytya_1}{\myb{x_1}}{\mytyb_1}}{\myse{f}_2}{\myfora{\mytya_2}{\myb{x_2}}{\mytyb_2}} \myred \\
+ \myind{2} \myprfora{\myb{x_1}}{\mytya_1}{\myprfora{\myb{x_2}}{\mytya_2}{
+ \myjm{\myb{x_1}}{\mytya_1}{\myb{x_2}}{\mytya_2} \myimpl
+ \myjm{\myapp{\myse{f}_1}{\myb{x_1}}}{\mytyb_1[\myb{x_1}]}{\myapp{\myse{f}_2}{\myb{x_2}}}{\mytyb_2[\myb{x_2}]}
+ }}
+ \end{array}
+ \]
+ Then we can simply take
+ \[
+ \begin{array}{@{}l}
+ \mysigma \myappsp \mytya_1 \myappsp \mytyb_1 \myeq \mysigma \myappsp \mytya_2 \myappsp \mytyb_2 \myred \\ \myind{2} \mytya_1 \myeq \mytya_2 \myand \myjm{\mytyb_1}{\mytya_1 \myarr \mytyp}{\mytyb_2}{\mytya_2 \myarr \mytyp}
+ \end{array}
+ \]
+ 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 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:
+ \[
+ \begin{array}{@{}l}
+ \myadt{\mylist}{\myappsp (\myb{A} {:} \mytyp)}{ }{\mydc{nil} \mydcsep \mydc{cons} \myappsp \myb{A} \myappsp (\myapp{\mylist}{\myb{A}})}
+ \end{array}
+ \]
+ Type equality is simple---we only need to compare the parameter:
+ \[
+ \mylist \myappsp \mytya_1 \myeq \mylist \myappsp \mytya_2 \myred \mytya_1 \myeq \mytya_2
+ \]
+ For coercions, we transport based on the constructor, recycling the
+ proof for the inductive occurrence:
+ \[
+ \begin{array}{@{}l@{\ }c@{\ }l}
+ \mycoe \myappsp (\mylist \myappsp \mytya_1) \myappsp (\mylist \myappsp \mytya_2) \myappsp \myse{Q} \myappsp \mydc{nil} & \myred & \mydc{nil} \\
+ \mycoe \myappsp (\mylist \myappsp \mytya_1) \myappsp (\mylist \myappsp \mytya_2) \myappsp \myse{Q} \myappsp (\mydc{cons} \myappsp \mytmm \myappsp \mytmn) & \myred & \\
+ \multicolumn{3}{l}{\myind{2} \mydc{cons} \myappsp (\mycoe \myappsp \mytya_1 \myappsp \mytya_2 \myappsp \myse{Q} \myappsp \mytmm) \myappsp (\mycoe \myappsp (\mylist \myappsp \mytya_1) \myappsp (\mylist \myappsp \mytya_2) \myappsp \myse{Q} \myappsp \mytmn)}
+ \end{array}
+ \]
+ Value equality is unsurprising---we match the constructors, and
+ return bottom for mismatches. However, we also need to equate the
+ parameter in $\mydc{nil}$:
+ \[
+ \begin{array}{r@{ }c@{\ }c@{\ }c@{}l@{\ }c@{\ }r@{}c@{\ }c@{\ }c@{}l@{\ }l}
+ (& \mydc{nil} & : & \myapp{\mylist}{\mytya_1} &) & \myeq & (& \mydc{nil} & : & \myapp{\mylist}{\mytya_2} &) \myred \mytya_1 \myeq \mytya_2 \\
+ (& \mydc{cons} \myappsp \mytmm_1 \myappsp \mytmn_1 & : & \myapp{\mylist}{\mytya_1} &) & \myeq & (& \mydc{cons} \myappsp \mytmm_2 \myappsp \mytmn_2 & : & \myapp{\mylist}{\mytya_2} &) \myred \\
+ & \multicolumn{11}{@{}l}{ \myind{2}
+ \myjm{\mytmm_1}{\mytya_1}{\mytmm_2}{\mytya_2} \myand \myjm{\mytmn_1}{\myapp{\mylist}{\mytya_1}}{\mytmn_2}{\myapp{\mylist}{\mytya_2}}
+ } \\
+ (& \mydc{nil} & : & \myapp{\mylist}{\mytya_1} &) & \myeq & (& \mydc{cons} \myappsp \mytmm_2 \myappsp \mytmn_2 & : & \myapp{\mylist}{\mytya_2} &) \myred \mybot \\
+ (& \mydc{cons} \myappsp \mytmm_1 \myappsp \mytmn_1 & : & \myapp{\mylist}{\mytya_1} &) & \myeq & (& \mydc{nil} & : & \myapp{\mylist}{\mytya_2} &) \myred \mybot
+ \end{array}
+ \]
+ % TODO quotation
+
+\item[Evil type]
+ Now for something useless but complicated.
+
+\end{description}
+
+\subsubsection{Only one equality}
+
+Given the examples above, a more `flexible' heterogeneous emerged, since
+of the fact that in $\mykant$ we re-gain the possibility of abstracting
+and in general handling sets in a way that was not possible in the
+original OTT presentation. Moreover, we found that the rules for value
+equality work very 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 very simple manner.
+
+In fact, we can drop a separate notion of type-equality, which will
+simply be served by $\myjm{\mytya}{\mytyp}{\mytyb}{\mytyp}$, from now on
+abbreviated as $\mytya \myeq \mytyb$. We shall still distinguish
+equalities relating types for hierarchical purposes. The full rules for
+equality reductions, along with the syntax for propositions, are given
+in figure \ref{fig:kant-eq-red}. 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.
+
+\begin{figure}[p]
\mydesc{syntax}{ }{
+ \small
$
\begin{array}{r@{\ }c@{\ }l}
- \mytmsyn & ::= & \mytmsyn \myeq \mytmsyn \mysynsep \myjm{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn} \\
- & | & \mycoee{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn} \mysynsep
- \mycohh{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn}
+ \myprsyn & ::= & \cdots \mysynsep \myjm{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn} \\
\end{array}
$
}
+ % \mytmsyn & ::= & \cdots \mysynsep \mycoee{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn} \mysynsep
+ % \mycohh{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn} \\
+ % \myprsyn & ::= & \cdots \mysynsep \myjm{\mytmsyn}{\mytmsyn}{\mytmsyn}{\mytmsyn} \\
+
+% \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{$\myinf{\mycoee{\mytya}{\mytyb}{\myse{P}}{\mytmt}}{\mytyb}$}
+% \DisplayProof
+% &
+% \AxiomC{$\myjud{\myse{P}}{\myprdec{\mytya \myeq \mytyb}}$}
+% \AxiomC{$\myjud{\mytmt}{\mytya}$}
+% \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}}{
+ \small
+ \begin{tabular}{cc}
+ \AxiomC{\phantom{$\myjud{\myse{P}}{\myprop}$}}
+ \UnaryInfC{$\myjud{\mytop}{\myprop}$}
+ \noLine
+ \UnaryInfC{$\myjud{\mybot}{\myprop}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myjud{\myse{P}}{\myprop}$}
+ \AxiomC{$\myjud{\myse{Q}}{\myprop}$}
+ \BinaryInfC{$\myjud{\myse{P} \myand \myse{Q}}{\myprop}$}
+ \noLine
+ \UnaryInfC{\phantom{$\myjud{\mybot}{\myprop}$}}
+ \DisplayProof
+ \end{tabular}
+
+ \myderivspp
+
+ \begin{tabular}{cc}
+ \AxiomC{$
+ \begin{array}{@{}c}
+ \phantom{\myjud{\myse{A}}{\mytyp} \hspace{0.8cm} \myjud{\mytmm}{\myse{A}}} \\
+ \myjud{\myse{A}}{\mytyp}\hspace{0.8cm}
+ \myjudd{\myctx; \myb{x} : \mytya}{\myse{P}}{\myprop}
+ \end{array}
+ $}
+ \UnaryInfC{$\myjud{\myprfora{\myb{x}}{\mytya}{\myse{P}}}{\myprop}$}
+ \DisplayProof
+ &
+ \AxiomC{$
+ \begin{array}{c}
+ \myjud{\myse{A}}{\mytyp} \hspace{0.8cm} \myjud{\mytmm}{\myse{A}} \\
+ \myjud{\myse{B}}{\mytyp} \hspace{0.8cm} \myjud{\mytmn}{\myse{B}}
+ \end{array}
+ $}
+ \UnaryInfC{$\myjud{\myjm{\mytmm}{\myse{A}}{\mytmn}{\myse{B}}}{\myprop}$}
+ \DisplayProof
+ \end{tabular}
+}
+
+\mynegder
+ % TODO equality for decodings
+\mydesc{equality reduction:}{\myctx \vdash \myprsyn \myred \myprsyn}{
+ \small
+ \begin{tabular}{cc}
+ \AxiomC{}
+ \UnaryInfC{$\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$}
+ \DisplayProof
+ \end{tabular}
+
+ \myderivspp
+
+ \AxiomC{}
+ \UnaryInfC{$
+ \begin{array}{@{}r@{\ }l}
+ \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[\myb{x_1}] \myeq \mytyb_2[\myb{x_2}]
+ }}
+ \end{array}
+ $}
+ \DisplayProof
+
+ \myderivspp
+
+ \AxiomC{}
+ \UnaryInfC{$
+ \begin{array}{@{}r@{\ }l}
+ \myctx \vdash &
+ \myjm{\myse{f}_1}{\myfora{\myb{x_1}}{\mytya_1}{\mytyb_1}}{\myse{f}_2}{\myfora{\myb{x_2}}{\mytya_2}{\mytyb_2}} \myred \\
+ & \myind{2} \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{\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}
+ $}
+ \DisplayProof
+
+
+ \myderivspp
+
+ \AxiomC{$\mytyc{D} : \mytele \myarr \mytyp \in \myctx$}
+ \UnaryInfC{$
+ \begin{array}{r@{\ }l}
+ \myctx \vdash &
+ \myjm{\mytyc{D} \myappsp \vec{A}}{\mytyp}{\mytyc{D} \myappsp \vec{B}}{\mytyp} \myred \\
+ & \myind{2} \mybigand_{i = 1}^n (\myjm{\mytya_n}{\myhead(\mytele(A_1 \cdots A_{i-1}))}{\mytyb_i}{\myhead(\mytele(B_1 \cdots B_{i-1}))})
+ \end{array}
+ $}
+ \DisplayProof
+
+ \myderivspp
+
+ \AxiomC{$
+ \begin{array}{@{}c}
+ \mydataty(\mytyc{D}, \myctx)\hspace{0.8cm}
+ \mytyc{D}.\mydc{c} : \mytele;\mytele' \myarr \mytyc{D} \myappsp \mytelee \in \myctx \hspace{0.8cm}
+ \mytele_A = (\mytele;\mytele')\vec{A}\hspace{0.8cm}
+ \mytele_B = (\mytele;\mytele')\vec{B}
+ \end{array}
+ $}
+ \UnaryInfC{$
+ \begin{array}{@{}l@{\ }l}
+ \myctx \vdash & \myjm{\mytyc{D}.\mydc{c} \myappsp \vec{\myse{l}}}{\mytyc{D} \myappsp \vec{A}}{\mytyc{D}.\mydc{c} \myappsp \vec{\myse{r}}}{\mytyc{D} \myappsp \vec{B}} \myred \\
+ & \myind{2} \mybigand_{i=1}^n(\myjm{\mytmm_i}{\myhead(\mytele_A (\mytya_i \cdots \mytya_{i-1}))}{\mytmn_i}{\myhead(\mytele_B (\mytyb_i \cdots \mytyb_{i-1}))})
+ \end{array}
+ $}
+ \DisplayProof
+
+ \myderivspp
+
+ \AxiomC{$\mydataty(\mytyc{D}, \myctx)$}
+ \UnaryInfC{$
+ \myctx \vdash \myjm{\mytyc{D}.\mydc{c} \myappsp \vec{\myse{l}}}{\mytyc{D} \myappsp \vec{A}}{\mytyc{D}.\mydc{c'} \myappsp \vec{\myse{r}}}{\mytyc{D} \myappsp \vec{B}} \myred \mybot
+ $}
+ \DisplayProof
+
+ \myderivspp
+
+ \AxiomC{$
+ \begin{array}{@{}c}
+ \myisreco(\mytyc{D}, \myctx)\hspace{0.8cm}
+ \mytyc{D}.\myfun{f}_i : \mytele; (\myb{x} {:} \myapp{\mytyc{D}}{\mytelee}) \myarr \myse{F}_i \in \myctx\\
+ \end{array}
+ $}
+ \UnaryInfC{$
+ \begin{array}{@{}l@{\ }l}
+ \myctx \vdash & \myjm{\myse{l}}{\mytyc{D} \myappsp \vec{A}}{\myse{r}}{\mytyc{D} \myappsp \vec{B}} \myred \\ & \myind{2} \mybigand_{i=1}^n(\myjm{\mytyc{D}.\myfun{f}_1 \myappsp \myse{l}}{(\mytele; (\myb{x} {:} \myapp{\mytyc{D}}{\mytelee}) \myarr \myse{F}_i)(\vec{\mytya};\myse{l})}{\mytyc{D}.\myfun{f}_i \myappsp \myse{r}}{(\mytele; (\myb{x} {:} \myapp{\mytyc{D}}{\mytelee}) \myarr \myse{F}_i)(\vec{\mytyb};\myse{r})})
+ \end{array}
+ $}
+ \DisplayProof
+
+ \myderivspp
+ \AxiomC{}
+ \UnaryInfC{$\myjm{\mytmm}{\mytya}{\mytmn}{\mytyb} \myred \mybot\ \text{if $\mytya$ and $\mytyb$ are canonical types.}$}
+ \DisplayProof
+}
+\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}
+
+% \begin{figure}[t]
+% \mydesc{reduction}{\mytmsyn \myred \mytmsyn}{
+
+% }
+% \caption{Coercions in \mykant.}
+% \label{fig:kant-coe}
+% \end{figure}
+
+% TODO finish
+
+\subsubsection{$\myprop$ and the hierarchy}
+
+Where is $\myprop$ 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 the same
+level as its reduction as to preserve subject reduction. In the example
+above, we'll 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 type $\mytyp_l$, where
+$\mytya : \mytyp_l$ and $\mytyb : \mytyp_l$. 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:
+$\myprdec{\myjm{\mytya}{\mytyp_l}{\mytyb}{\mytyp_l}} : \mytyp_{l + 1}$.
+
+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}
+
+ \myderivspp
+
+ \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}
+
+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 $\myprop_0$ to be $\myprop_1$ too, which will be the case with
+cumulativity. This is not the most elegant of systems, but it buys us a
+cheap type level equality without having to replicate functionality with
+a dedicated construct.
+
+\subsubsection{Quotation and definitional equality}
+\label{sec:kant-irr}
+
+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 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 typecheck
+canonical terms), one quoting neutral terms while recovering their
+types.
+
+Our quotation will work on normalised terms, so that all defined values
+will have been replaced. Moreover, we match on datatype eliminators and
+all their arguments, so that $\mynat.\myfun{elim} \myappsp \vec{\mytmt}$
+will stand for $\mynat.\myfun{elim}$ applied to the scrutinised
+$\mynat$, plus its three arguments. This measure can be easily
+implemented by checking the head of applications and `consuming' the
+needed terms.
+
+% TODO finish this
+\begin{figure}[t]
+ \mydesc{canonical quotation}{\myctx \vdash \mytmsyn \myquot \mytmsyn \mapsto \mytmsyn}{
+
+ }
+
+ \mynegder
+
+ \mydesc{neutral quotation}{\myctx \vdash \mynquot \mytmsyn \mapsto \mytmsyn : \mytmsyn}{
+
+ }
+ \caption{Quotation in \mykant.}
+ \label{fig:kant-quot}
+\end{figure}
+
+% TODO finish
+
+\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
+ don't have neutral propositions.} 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}.
+
+% TODO finish
+
\section{\mykant : The practice}
\label{sec:kant-practice}
-The codebase consists of around 2500 lines of Haskell, as reported by the
-\texttt{cloc} utility. The high level design is heavily 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.
+The codebase consists of around 2500 lines of Haskell, as reported by
+the \texttt{cloc} utility. The high level design is inspired by the
+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.
-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}.
+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, the
+implementation of observational equality is not currently complete.
+However, given the detailed plan in the previous section, doing so
+should not prove to be too much work.
-\begin{figure}
- \centering{
- \includegraphics[scale=1.0]{kant-web.png}
- }
- \caption{The \mykant\ web prompt.}
+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}.
+
+\begin{figure}[t]
+{\small\begin{verbatim}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}
\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},
- which in turns needs to \textbf{Evaluate} terms.
+\item[Elaborate] 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
\end{description}
-The details of each phase will be described in section % TODO insert section
-
\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]
\label{fig:kant-process}
\end{figure}
-\subsection{Term representation}
+Here we will review only a sampling of the more interesting
+implementation challenges present when implementing an interactive
+theorem prover.
+
+\subsection{Term and context representation}
\label{sec:term-repr}
+\subsubsection{Naming and substituting}
+
+Perhaps surprisingly, one of the most fundamental 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 more slightly complicated
+ systems it is very hard to get right, even for experts.
+
+ One of the sore spots of explicit names is comparing terms up
+ $\alpha$-renaming, which again generates a huge amounts of
+ substitutions and requires special care. We can represent the
+ relationship between variables and their binders, by...
+
+\item[Nameless] ...getting rid of names altogether, and representing
+ bound variables with an index referring to the `binding' structure, a
+ notion introduced by \cite{de1972lambda}. Classically $0$ represents
+ the variable bound by the innermost binding structure, $1$ the
+ second-innermost, and so on. For instance with simple abstractions we
+ might have
+ \[\mymacol{red}{\lambda}\, (\mymacol{blue}{\lambda}\, \mymacol{blue}{0}\, (\mymacol{AgdaInductiveConstructor}{\lambda\, 0}))\, (\mymacol{AgdaFunction}{\lambda}\, \mymacol{red}{1}\, \mymacol{AgdaFunction}{0}) : ((\mytya \myarr \mytya) \myarr \mytyb) \myarr \mytyb \]
+
+ 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. Moreover, $\alpha$ renaming ceases to be an issue, and
+ term comparison is purely syntactical.
+
+ Nonetheless, more complex, more complex constructs such as pattern
+ matching put some strain on the indices and many systems end up using
+ explicit names anyway (Agda, Coq, \dots).
+
+\end{description}
+
+In the past decade or so advancements in the Haskell's type system and
+in general the spread new programming practices have enabled to make the
+second option much more amenable. \mykant\ thus takes the second path
+through the use of Edward Kmett's excellent \texttt{bound}
+library.\footnote{Available at
+\url{http://hackage.haskell.org/package/bound}.} We decribe its
+advantages but also 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 two core ideas. The first is the suggestion by
+\cite{Bird1999} consists of 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}
+data Void
+
+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 Void}, 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 $\myabs{\myb{x}}{\myabs{\myb{y}}{\myb{x}}}$ could be represented as
+\begin{Verbatim}
+Lam (Lam (Free Bound))
+\end{Verbatim}
+This allows us to reflect at the type level the `nestedness' of a type,
+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 don't belong.
+
+Even more interestingly, the substitution operation is perfectly
+captured by the \verb|>>=| (bind) operator of the \texttt{Monad}
+typeclass:
+\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 referred 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 = do v <- s
+ case v of
+ Bound -> t
+ Free v' -> return v'
+\end{Verbatim}
+The beauty of this technique is that in a few simple function 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.
+
+Moreover, if we take the top level term type to be \verb|Tm String|, we
+get for free 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:
+\begin{Verbatim}
+data Tele m v = End (m v) | Bind (m v) (Tele (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
+Figure \ref{fig:term}. The fact that propositions cannot be factored
+out in another data type is a consequence of the problems 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 a
+nice interface similar to the one of \verb|bound| with a more flexible
+interface.
+
+\begin{figure}[t]
+{\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}
+}
+ \caption{Data type for terms in \mykant.}
+ \label{fig:term}
+\end{figure}
+
+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{Context}
+
+% TODO finish
+
\subsection{Type hierarchy}
+\label{sec:hier-impl}
+
+As a breath of air amongst the type gas, we will explain how to
+implement the typical ambiguity we have spoken about in
+\ref{sec:term-hierarchy}. 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}
-\subsection{Elaboration}
+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 on 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. Its interface is shown in Figure
+\ref{fig:constraint}. The problem unpredictably reduces to a graph
+algorithm. If we
+
+\begin{figure}[t]
+{\small\begin{verbatim}module Data.Constraint where
+
+-- | Data type holding the set of constraints, parametrised over the
+-- type of the variables.
+data Constrs a
+
+-- | A representation of the constraints that we can add.
+data Constr a = a :<=: a | a :<: a | a :==: a
+
+-- | An empty set of constraints.
+empty :: Ord a => Constrs a
+
+-- | Adds one constraint to the set, returns the new set of constraints
+-- if consistent.
+addConstr :: Ord a => Constr a -> Constrs a -> Maybe (Constrs a)\end{verbatim}
+}
+
+ \caption{Interface for the module handling the constraints.}
+ \label{fig:constraint}
+\end{figure}
+
+
+\subsection{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.
\section{Evaluation}
\section{Future work}
+\subsection{Coinduction}
+
+\subsection{Quotient types}
+
+\subsection{Partiality}
+
+\subsection{Pattern matching}
+
+\subsection{Pattern unification}
+
+% TODO coinduction (obscoin, gimenez, jacobs), pattern unification (miller,
+% gundry), partiality monad (NAD)
+
\appendix
\section{Notation and syntax}
When presenting type derivations, I will often abbreviate and present multiple
conclusions, each on a separate line:
-
\begin{prooftree}
\AxiomC{$\myjud{\mytmt}{\mytya \myprod \mytyb}$}
\UnaryInfC{$\myjud{\myapp{\myfst}{\mytmt}}{\mytya}$}
\UnaryInfC{$\myjud{\myapp{\mysnd}{\mytmt}}{\mytyb}$}
\end{prooftree}
-\section{Agda rendition of ITT}
+I will often present `definitions' in the described calculi and in
+$\mykant$\ itself, like so:
+\[
+\begin{array}{@{}l}
+ \myfun{name} : \mytysyn \\
+ \myfun{name} \myappsp \myb{arg_1} \myappsp \myb{arg_2} \myappsp \cdots \mapsto \mytmsyn
+\end{array}
+\]
+To define operators, I use a mixfix notation similar
+to Agda, where $\myarg$s denote arguments:
+\[
+\begin{array}{@{}l}
+ \myarg \mathrel{\myfun{$\wedge$}} \myarg : \mybool \myarr \mybool \myarr \mybool \\
+ \myb{b_1} \mathrel{\myfun{$\wedge$}} \myb{b_2} \mapsto \cdots
+\end{array}
+\]
+
+In explicitly typed systems, I will also omit type annotations when they
+are obvious, e.g. by not annotating the type of parameters of
+abstractions or of dependent pairs.
+
+I will also 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:
+\[
+\myabs{\myb{x}\myappsp\myb{y}}{\cdots} = \myabs{\myb{x}}{\myabs{\myb{y}}{\cdots}}
+\]
+
+\section{Code}
+
+\subsection{ITT renditions}
+\label{app:itt-code}
+
+\subsubsection{Agda}
\label{app:agda-itt}
Note that in what follows rules for `base' types are
field
fst : A
snd : B fst
+ open _×_ public
data Bool : Set where
true false : Bool
- if_then_else_ : ∀ {a} {P : Bool → Set a} (x : Bool) → P true → P false → P x
- if true then x else _ = x
- if false then _ else x = x
+ if_/_then_else_ : ∀ {a} (x : Bool) (P : Bool → Set a) → P true → P false → P x
+ if true / _ then x else _ = x
+ if false / _ then _ else x = x
+
+ if_then_else_ : ∀ {a} (x : Bool) {P : Bool → Set a} → P true → P false → P x
+ if_then_else_ x {P} = if_/_then_else_ x P
data W {s p} (S : Set s) (P : S → Set p) : Set (s ⊔ p) where
_◁_ : (s : S) → (P s → W S P) → W S P
even (suc zero) = Empty
even (suc (suc n)) = even n
- 6-is-even : even 6
- 6-is-even = tt
+ 6-even : even 6
+ 6-even = tt
- 5-is-not-even : ¬ (even 5)
- 5-is-not-even = absurd
+ 5-not-even : ¬ (even 5)
+ 5-not-even = absurd
there-is-an-even-number : ℕ × even
- there-is-an-even-number = 6 , 6-is-even
+ there-is-an-even-number = 6 , 6-even
+
+ _∨_ : (A B : Set) → Set
+ A ∨ B = Bool × (λ b → if b then A else B)
+
+ left : ∀ {A B} → A → A ∨ B
+ left x = true , x
+
+ right : ∀ {A B} → B → A ∨ B
+ right x = false , x
+
+ [_,_] : {A B C : Set} → (A → C) → (B → C) → A ∨ B → C
+ [ f , g ] x =
+ (if (fst x) / (λ b → if b then _ else _ → _) then f else g) (snd x)
+
+module Examples-W where
+ open ITT
+ open Examples-×
+
+ Tr : Bool → Set
+ Tr b = if b then Unit else Empty
+
+ ℕ : Set
+ ℕ = W Bool Tr
+
+ zero : ℕ
+ zero = false ◁ absurd
+
+ suc : ℕ → ℕ
+ suc n = true ◁ (λ _ → n)
+
+ plus : ℕ → ℕ → ℕ
+ plus x y = rec
+ (λ _ → ℕ)
+ (λ b →
+ if b / (λ b → (Tr b → ℕ) → (Tr b → ℕ) → ℕ)
+ then (λ _ f → (suc (f tt))) else (λ _ _ → y))
+ x
module Equality where
open ITT
trans x y z p q = subst (λ z′ → x ≡ z′) q p
cong : ∀ {A B : Set} (x y : A) → x ≡ y → (f : A → B) → f x ≡ f y
- cong x y p f = subst (λ y′ → f x ≡ f y′) p (refl (f x))
+ cong x y p f = subst (λ z → f x ≡ f z) p (refl (f x))
\end{code}
-\nocite{*}
+\subsubsection{\mykant}
+
+The following things are missing: $\mytyc{W}$-types, since our
+positivity check is overly strict, and equality, since we haven't
+implemented that yet.
+
+{\small
+\verbatiminput{itt.ka}
+}
+
+\subsection{\mykant\ examples}
+
+{\small
+\verbatiminput{examples.ka}
+}
+
+\subsection{\mykant's hierachy}
+\label{sec: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
+Agda version, available at
+\url{http://code.haskell.org/Agda/test/succeed/Hurkens.agda}.
+
+{\small
+\verbatiminput{hurkens.ka}
+}
+
\bibliographystyle{authordate1}
\bibliography{thesis}
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