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paper/main.tex
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paper/main.tex
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\documentclass[10pt, nonacm]{acmart}
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\usepackage[utf8]{inputenc}
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\usepackage{formal-grammar}
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\usepackage[dvipsnames]{xcolor}
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\usepackage{mathpartir}
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\newcommand{\metavariable}[1]{\textcolor{teal}{#1}}
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\newcommand{\typeterminal}[1]{\textcolor{brown}{#1}}
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\newcommand{\exprterminal}[1]{\textcolor{Sepia}{#1}}
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\newcommand{\seltype}[0]{ {\textsc{\footnotesize{Typ}}} }
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\newcommand{\selexpr}[0]{ {\textsc{\footnotesize{Exp}}} }
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\newcommand{\selval}[0]{ {\textsc{\footnotesize{Val}}} }
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\newcommand{\exprvars}[0]{ V }
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\newcommand{\typevars}[0]{ \Upsilon }
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\newcommand{\typenames}[0]{ \Sigma }
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\newcommand{\typenonterm}[1]{ \nonterm{ T_\seltype \footnotesize{\textsf{$(\typenames, #1)$}}}}
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\newcommand{\exprnonterm}[2]{ \nonterm{ T_\selexpr \footnotesize{\textsf{$(\typenames, #1, #2)$}}}}
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\newcommand{\valnonterm}[2]{ \nonterm{ T_\selval \footnotesize{\textsf{$(\typenames, #1, #2)$}}}}
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\newcommand{\todo}[1]{ {\textcolor{red}{\textbf{TODO:} #1}} }
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\title{A functional core calculus with ladder-types}
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\author{Michael Sippel}
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\email{michael.sippel@mailbox.tu-dresden.de}
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\makeatletter
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\let\@authorsaddresses\@empty
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\makeatother
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\begin{document}
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\begin{abstract}
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This paper presents a minimal core calculus extending the \(\lambda\)-calculus by a polymorphic type-system similar to SystemF, but in addition it introduces a new type-constructor called the \emph{ladder-type}.
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Using ladder-types, multi-layered embeddings of higher-level data-types into lower-level data-types can be described by a type-level structure.
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By facilitating automatic transformations between semantically compatible datatypes, ladder-typing opens up a new paradigm of abstraction.
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We formally define the syntax \& semantics of this calculus and prove its \emph{type soundness}.
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Further we show how the Boehm-Berarducci encoding can be used to implement algebraic datatypes on the basis of the introduced core calculus.
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\end{abstract}
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\maketitle
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\tableofcontents
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%\newpage
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\section{Core Calculus}
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\subsection{Syntax}
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The calculus is made up of a \emph{type-level} language, and an \emph{expression-level} language. The formal grammar for both is given in \ref{gr:core}.
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In addition to \emph{function types} \(\tau_1 \rightarrow \tau_2\), \emph{universal types} \(\forall\alpha.\tau\) and type specialization \(<\tau_1 \tau_2>\),
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which are already known from SystemF,
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types can be of the form \(\tau_1 \sim \tau_2\) to denote a \emph{ladder type} to formalizes the notion of a type \(\tau_1\) being represented in terms of type \(\tau_2\).
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Similar to SystemF, expressions can be \emph{variables}, \emph{type-abstractions}, \emph{}
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\begin{figure}[h]
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\label{gr:core}
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\begin{grammar}
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\firstcase{ T_\seltype \textsf{$(\typenames, \typevars)$} }{
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\metavariable{\sigma}
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}{Type Literal \quad \textsf{where $ \metavariable{\sigma} \in \typenames $}}
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\otherform{
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\metavariable{\alpha}
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}{Type Variable \quad \textsf{where $ \metavariable{\alpha} \in \typevars $}}
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\otherform{
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$$\typeterminal{\forall}$$ \metavariable{\alpha} \typeterminal{.} \quad \typenonterm{\typevars \cup \{\metavariable{\alpha}\}}
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}{Universal Type}
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\otherform{
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\typeterminal{<} \typenonterm{\typevars} \quad \typenonterm{\typevars} \typeterminal{>}
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}{Specialization}
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\otherform{
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\typenonterm{\typevars}
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\quad $$\typeterminal{\rightarrow}$$ \quad
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\typenonterm{\typevars}
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}{Function Type}
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\otherform{
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\typenonterm{\typevars}
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\quad $$\typeterminal{\rightarrow_{morph}}$$ \quad
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\typenonterm{\typevars}
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}{Morphism Type}
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\otherform{
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\typenonterm{\typevars}
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\quad $$\typeterminal{\sim}$$ \quad
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\typenonterm{\typevars}
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}{Ladder Type}
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\otherform{
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$$\typeterminal{(}$$ \quad
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\typenonterm{\typevars}
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\quad $$\typeterminal{)}$$
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}{Parenthesis}
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$$\\$$
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\firstcase{ T_\selexpr \textsc{$(\typenames, \typevars, \exprvars)$} }
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{ \metavariable{x}
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} {Variable \quad \textsf{where $\metavariable{x} \in \exprvars$} }
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\otherform{
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$$ \exprterminal{\Lambda} \metavariable{\alpha}
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\quad \exprterminal{\mapsto} \quad $$
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\exprnonterm{\typevars \cup \{\metavariable{\alpha}\}}{\exprvars}
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}{Type Abstraction}
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\otherform{
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$$ \exprterminal{\lambda} \metavariable{x} $$
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\exprterminal{:} \typenonterm{\typevars}
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\quad $$\exprterminal{\mapsto}$$ \quad
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\exprnonterm{\typevars}{\exprvars \cup \{\metavariable{x}\}}
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}{Value Abstraction}
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\otherform{
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$$ \exprterminal{\lambda} \metavariable{x} $$
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\exprterminal{:} \typenonterm{\typevars}
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\quad $$\exprterminal{\mapsto_{morph}}$$ \quad
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\exprnonterm{\typevars}{\exprvars \cup \{\metavariable{x}\}}
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}{Value Morphism}
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\otherform{
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\exprterminal{let} \quad \metavariable{x} \quad \exprterminal{=} \quad
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\exprnonterm{\typevars}{\exprvars}
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\quad \exprterminal{in} \quad
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\exprnonterm{\typevars}{\exprvars \cup \{\metavariable{x}\}}
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}{Variable Binding}
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\otherform{
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\exprnonterm{\typevars}{\exprvars}
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\quad
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\typenonterm{\typevars}
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}{Type Application}
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\otherform{
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\exprnonterm{\typevars}{\exprvars}
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\quad
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\exprnonterm{\typevars}{\exprvars}
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}{Value Application}
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\otherform{
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\exprnonterm{\typevars}{\exprvars}
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\quad
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\exprterminal{as}
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\quad
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\typenonterm{\typevars}
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}{Type Cast}
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\otherform{
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\exprterminal{(} \quad
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\exprnonterm{\typevars}{\exprvars}
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\quad \exprterminal{)}
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}{Parenthesis}
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$$\\$$
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\firstcase{ T_\textsc{Val} \textsc{$(\typenames, \typevars, \exprvars)$} }{
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\exprterminal{\epsilon}
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}{Empty Value}
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\otherform{
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\metavariable{x} \quad \valnonterm{\typevars}{\exprvars}
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}{Value Conactenation}
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%\otherform{
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% \exprterminal{\Lambda} \metavariable{\alpha} \quad
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% \exprterminal{\mapsto} \quad
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% \valnonterm{ \typevars \cup \{ \metavariable{\alpha} \} }
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%}{Type-Function Value}
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\otherform{
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\exprterminal{\lambda} \metavariable{x} \quad
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\exprterminal{:} \quad
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\typenonterm{\emptyset} \quad
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\exprterminal{\mapsto} \quad
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\exprnonterm{\typevars}{\{\metavariable{x}\}}
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}{Function Value}
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\otherform{
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\valnonterm{ \typevars } \quad
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\exprterminal{as} \quad
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\typenonterm{ \typevars }
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}{Value}
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\end{grammar}
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\caption{Syntax of the core calculus with colors for \metavariable{metavariables}, \typeterminal{type-level terminal symbols}, \exprterminal{expression-level terminal symbols}
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where $\typenames, \typevars, \exprvars$ are mutually disjoint, countable sets of symbols to denote atomic type identifiers (\(\typenames\)), free typevariables (\(\typevars\)), and free expression variables (\(\exprvars\)).
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$$\\$$}
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\end{figure}
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\begin{example}
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\label{ex:terms}
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Let \(\Sigma = \{ \text{Digit}, \text{Char}, \text{Seq}, \text{UTF-8}, \mathbb{N}, PosInt \} \cup \{ 0, 1, 2, ... \}\).
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The following terms are valid types over \(\Sigma\):
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\begin{enumerate}
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\item \typeterminal{<Seq Char>} \( \in \typenonterm{\emptyset}\)\\
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"sequence of characters"
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\item \typeterminal{<Seq <Digit 10> \(\sim\) Char>} \( \in \typenonterm{\emptyset}\)\\
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"sequence of decimal digits, where each digit is represented as character"
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\item \typeterminal{<Seq Item> \(\rightarrow \mathbb{N} \rightarrow\) Item} \( \in \typenonterm{\{Item\}}\)\\
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"function that maps a sequence of items and a natural number to an item"\\
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Note: this type contains the free variable \typeterminal{Item}
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\item \typeterminal{<Seq Char> \(\sim\) UTF-8 \(\rightarrow \mathbb{N} \rightarrow\) Char } \( \in \typenonterm{\emptyset}\)\\
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"function that takes a sequence of chars, represented as UTF-8 string, and a natural number to return a character"
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\item \typeterminal{\(\forall\) Radix . <Seq <Digit Radix> \(\sim\) Char>} \(\in \typenonterm{\emptyset} \)\\
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"given the parameter \typeterminal{Radix}, a sequence of digits where each digit is represented as character"\\
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Note: this type-term is \emph{ground} (i.e. \(\typevars = \emptyset\)), since \typeterminal{Radix} is bound by \(\typeterminal{\forall}\)
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\item \typeterminal{
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\(\forall\) SrcRadix.\\
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\(\forall\) DstRadix.\\
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\(\mathbb{N} \sim\) <PosInt SrcRadix> \(\sim\) <Seq <Digit SrcRadix> \(\sim\) Char>\\
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\(\rightarrow_{morph}\)\\
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\(\mathbb{N} \sim\) <PosInt DstRadix> \(\sim\) <Seq <Digit DstRadix> \(\sim\) Char>\\
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} \(\in \typenonterm{\emptyset} \)\\
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"morphism function that maps the \typeterminal{PosInt} representation of \(\typeterminal{\mathbb{N}}\) with radix \typeterminal{SrcRadix} to the \typeterminal{PosInt} representation of radix \typeterminal{DstRadix}"
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\end{enumerate}
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\end{example}
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\begin{definition}[Substitution in Types]
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Given a type-variable assignment \(\psi_t = \{ \metavariable{\alpha_1} \mapsto \metavariable{\tau_1}, \quad \metavariable{\alpha_2} \mapsto \metavariable{\tau_2}, \quad \dots \}\),
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the thereby induced, lexically scoped substitution \(\overline{\psi_t}\) replaces all \emph{free} occurences of the variables \(\metavariable{\alpha_i}\) in a type-term \(\metavariable{\xi} \in \typenonterm{\{\metavariable{\alpha_1}, \quad \metavariable{\alpha_2}, \quad \dots\}}\) recursively with the type-term given by \(\psi_t(\metavariable{\alpha_i})\).
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Lexical scoping is implemented by simply not substituting any bound occourences of variables \(\metavariable{\alpha_i}\). This allows to skip \(\alpha\)-conversion as done in classical \(\lambda\)-calculus.
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\[\overline{\psi_t} \metavariable{\xi} = \begin{cases}
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\metavariable{\xi} \quad \text{if } \metavariable{\xi} \in \typenames\\
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\metavariable{\xi} \quad \text{if } \metavariable{\xi} \in \{\metavariable{\alpha_1}, \metavariable{\alpha_2}, \dots\} \text{ and } \metavariable{\xi} \notin \text{Dom}(\psi_t)\\
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\metavariable{\tau} \quad \text{if } \metavariable{\xi} \in \{\metavariable{\alpha_1}, \metavariable{\alpha_2}, \dots\} \text{ and } (\metavariable{\xi} \mapsto \metavariable{\tau}) \in \psi_t\\
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\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \overline{\psi_t}\metavariable{\xi'} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \text{ and } \metavariable{\alpha} \notin \text{Dom}(\psi_t)\\
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\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \overline{\psi_t \setminus \{\metavariable{\alpha} \mapsto \metavariable{\tau}\}}\metavariable{\xi'} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \text{ and } \{\metavariable{\alpha} \mapsto \metavariable{\tau}\} \in \psi_t\\
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\typeterminal{<} (\overline{\psi_t} \metavariable{\xi_1}) \quad (\overline{\psi_t} \metavariable{\xi_2}) \typeterminal{>} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{<} \metavariable{\xi_1} \quad \metavariable{\xi_2} \typeterminal{>}\\
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(\overline{\psi_t} \metavariable{\xi_1}) \typeterminal{\rightarrow} (\overline{\psi_t} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\rightarrow} \metavariable{\xi_2}\\
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(\overline{\psi_t} \metavariable{\xi_1}) \typeterminal{\rightarrow_{morph}} (\overline{\psi_t} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\rightarrow_{morph}} \metavariable{\xi_2}\\
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(\overline{\psi_t} \metavariable{\xi_1}) \typeterminal{\sim} (\overline{\psi_t} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\sim} \metavariable{\xi_2}\\
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\end{cases}\]
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\[\overline{\psi_t} \metavariable{e} = \begin{cases}
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\metavariable{e} \quad \text{ if } \metavariable{e} \text{ is a variable}
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\exprterminal{\Lambda \metavariable{\alpha} \mapsto} \overline{\psi_t}\metavariable{e'} \quad \text{ if \metavariable{e} is of the form } \exprterminal{\Lambda \metavariable{\alpha} \mapsto \metavariable{e'}}\\
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\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto} \overline{\psi_t}\metavariable{e'} \quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto \metavariable{e'}}\\
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\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto_\text{morph}} \overline{\psi_t}\metavariable{e'} \quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto_\text{morph} \metavariable{e'}}\\
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\end{cases}\]
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\end{definition}
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\begin{definition}[Substitution in Expressions]
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\todo{complete}
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Given an expression-variable assignment \(\psi_e = \{ \metavariable{x_1} \mapsto \metavariable{t_1}, \quad \metavariable{x_2} \mapsto \metavariable{t_2}, \quad \dots \}\),
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the thereby induced, lexically scoped substitution \(\overline{\psi_e}\) replaces all \emph{free} occurences of the expression variables \(\metavariable{x_i}\)
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in an expression \(
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e \in
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\exprnonterm
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{\typevars}
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{\{\metavariable{x_1}, \quad \metavariable{x_2}, \quad \dots\}}
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\). Lexical scoping is implemented by simply not substituting any bound occourences of variables \(\metavariable{\alpha_i}\). This allows to skip \(\alpha\)-conversion as done in classical \(\lambda\)-calculus.
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\[\overline{\psi} \metavariable{e} = \begin{cases}
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\metavariable{e} \quad \text{if } \metavariable{\xi} \in \typenames\\
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\metavariable{\tau} \quad \text{if } (\metavariable{\xi} \mapsto \metavariable{\tau}) \in \psi\\
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\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \overline{\psi}\metavariable{\xi'} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \text{ and } \metavariable{\alpha} \notin \text{Dom}(\psi)\\
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\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \text{ and } \metavariable{\alpha} \in \text{Dom}(\psi)\\
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\typeterminal{<} (\overline{\psi} \metavariable{\xi_1}) \quad (\overline{\psi} \metavariable{\xi_2}) \typeterminal{>} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{<} \metavariable{\xi_1} \quad \metavariable{\xi_2} \typeterminal{>}\\
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(\overline{\psi} \metavariable{\xi_1}) \typeterminal{\rightarrow} (\overline{\psi} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\rightarrow} \metavariable{\xi_2}\\
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(\overline{\psi} \metavariable{\xi_1}) \typeterminal{\rightarrow_{morph}} (\overline{\psi} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\rightarrow_{morph}} \metavariable{\xi_2}\\
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(\overline{\psi} \metavariable{\xi_1}) \typeterminal{\sim} (\overline{\psi} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\sim} \metavariable{\xi_2}\\
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\end{cases}\]
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\end{definition}
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\subsection{Typing}
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\subsubsection{Equivalence of Type Terms}
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\begin{definition}[Distributivity]
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\todo{}
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\end{definition}
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\begin{definition}[Equivalence Relation]
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\todo{}
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\end{definition}
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\subsubsection{Normal Forms}
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\begin{definition}[Ladder Normal Form]
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\todo{}
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\end{definition}
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\subsubsection{Subtyping}
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\begin{definition}[Syntactic Subtyping]
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\todo{}
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\end{definition}
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\begin{definition}[Semantic Subtyping]
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\todo{}
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\end{definition}
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\subsubsection{Inference of Expression Types}
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The type-context \(\Gamma = \{ \metavariable{x_1} : \metavariable{\tau_1} , \quad \metavariable{x_2} : \metavariable{\tau_2} , \quad \ldots \}\) is a finite mapping from variables \(\metavariable{x_i} \in \exprvars\) to ground types \(\metavariable{\tau_i} \in \typenonterm{\emptyset}\).
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Using the inference rules given in \ref{def:typerules}, further typing-judgements
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of the form
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\begin{itemize}
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\item \(\metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)" and
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\item \(\metavariable{e} :\approx \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is compatible with type \(\metavariable{\tau}\)"
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\end{itemize}
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can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \exprnonterm{\emptyset}{\exprvars}\) and \(\metavariable{\tau} \in \typenonterm{\emptyset}\)
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\begin{definition}[Syntactic Well-Typedness]
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An expression \(\metavariable{e} \in \exprnonterm{\emptyset}{\emptyset}\) is \textbf{syntactically well-typed} if there exists a type \(\metavariable{\tau} \in \typenonterm{\emptyset}\),
|
||||
such that \( \emptyset \vdash \metavariable{e} : \metavariable{\tau} \) by \ref{def:typerules}.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Semantic Well-Typedness]
|
||||
An expression \(\metavariable{e} \in \exprnonterm{\emptyset}{\emptyset}\) is \textbf{semantically well-typed} if there exists a type \(\metavariable{\tau} \in \typenonterm{\emptyset}\),
|
||||
such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) by \ref{def:typerules}.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Inference Rules for the Typing Relation.]
|
||||
\label{def:typerules}
|
||||
|
||||
As usual, each rule is composed of premises (above the horizontal line) and a conclusion (below the line):
|
||||
\begin{mathpar}
|
||||
|
||||
\inferrule[T-Variable]{
|
||||
\metavariable{x} \in \exprvars\\
|
||||
\metavariable{\tau} \in \typenonterm{\emptyset}\\
|
||||
\metavariable{x}:\metavariable{\tau} \in \Gamma\\
|
||||
}{
|
||||
\Gamma \vdash \metavariable{x}:\metavariable{\tau}
|
||||
}\and
|
||||
|
||||
|
||||
\inferrule[T-LetBinding]{
|
||||
\Gamma \vdash \metavariable{e} : \metavariable{\sigma} \\
|
||||
\Gamma , \metavariable{x}:\metavariable{\sigma} \vdash \metavariable{t} : \metavariable{\tau}
|
||||
}{
|
||||
\Gamma \vdash (\exprterminal{\text{let }}\metavariable{x}\exprterminal{\text{ = }}\metavariable{e}\exprterminal{\text{ in }} \metavariable{t}) : \metavariable{\tau}
|
||||
}
|
||||
|
||||
|
||||
\inferrule[T-TypeAbs]{
|
||||
\metavariable{\tau} \in \typenonterm{\typevars \cup \{\metavariable{\alpha}\}} \\
|
||||
\metavariable{e} \in \exprnonterm{\typevars \cup \{ \metavariable{\alpha} \}}{\exprvars} \\
|
||||
\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
|
||||
}{
|
||||
\Gamma \vdash (\exprterminal{\Lambda} \metavariable{\alpha} \exprterminal{\mapsto} \metavariable{e}) : \typeterminal{\forall}\metavariable{\alpha}\typeterminal{.}\metavariable{\tau}
|
||||
}
|
||||
|
||||
\inferrule[T-TypeApp]{
|
||||
\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
|
||||
\metavariable{\tau} \in \typenonterm{\typevars \cup \metavariable{\alpha}} \\
|
||||
\metavariable{\sigma} \in \typenonterm{\typevars}
|
||||
}{
|
||||
\Gamma \vdash ( \metavariable{e} \quad \metavariable{\sigma} ) : \{\metavariable{\alpha} \mapsto \metavariable{\sigma}\} \metavariable{\tau}
|
||||
}
|
||||
|
||||
|
||||
\inferrule[T-ValueAbs]{
|
||||
\metavariable{\sigma}, \metavariable{\tau} \in \typenonterm{\typevars} \\
|
||||
\metavariable{e} \in \exprnonterm{\typevars}{\exprvars \cup \{ \metavariable{x} \} } \\
|
||||
\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
|
||||
}{
|
||||
\Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\sigma} \exprterminal{\mapsto} \metavariable{e}) : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau}
|
||||
}
|
||||
|
||||
\inferrule[T-ValueApp]{
|
||||
\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau} \\
|
||||
\Gamma \vdash \metavariable{a} : \metavariable{\sigma} \\
|
||||
}{
|
||||
\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
|
||||
}\and
|
||||
|
||||
|
||||
\inferrule[T-Compatible]{
|
||||
\Gamma \vdash \metavariable{e} : \metavariable{\tau}
|
||||
}{
|
||||
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau}
|
||||
}\and
|
||||
|
||||
\inferrule[T-MorphAbs]{
|
||||
\metavariable{\sigma}, \metavariable{\tau} \in \typenonterm{\typevars} \\
|
||||
\metavariable{e} \in \exprnonterm{\typevars}{\exprvars \cup \{ \metavariable{x} \} } \\
|
||||
\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
|
||||
}{
|
||||
\Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\tau} \exprterminal{\mapsto_{morph}} \metavariable{e}) : \metavariable{\sigma}\typeterminal{\rightarrow_{morph}}\metavariable{\tau}
|
||||
}\and
|
||||
|
||||
\inferrule[T-MorphApp]{
|
||||
\Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
|
||||
\exists \metavariable{h} . \Gamma \vdash \metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_{morph}} \metavariable{\tau'}\\
|
||||
\metavariable{\tau} \precsim \metavariable{\tau'}
|
||||
}{
|
||||
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
|
||||
}\and
|
||||
|
||||
\inferrule[T-Ascension]{
|
||||
\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
|
||||
\metavariable{\tau'} \leq \metavariable{\tau}
|
||||
}{
|
||||
\Gamma \vdash (\metavariable{e} \exprterminal{\text{ as }} \metavariable{\tau'}) : \metavariable{\tau'}
|
||||
}\and
|
||||
|
||||
\inferrule[T-Descension]{
|
||||
\Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
|
||||
\metavariable{\tau} \leq \metavariable{\tau'}
|
||||
}{
|
||||
\Gamma \vdash \metavariable{e} : \metavariable{\tau'}
|
||||
}\and
|
||||
|
||||
\end{mathpar}
|
||||
\end{definition}
|
||||
|
||||
|
||||
\subsection{Evaluation Semantics}
|
||||
|
||||
Evaluation of an expression \(\metavariable{e} \in \exprnonterm{\emptyset}{\emptyset}\) is defined by exhaustive application of the rewrite rules \(\rightarrow_\beta\) and \(\rightarrow_\delta\),
|
||||
which are given in \ref{def:evalrules}.
|
||||
|
||||
\begin{definition}[Inference Rules for Evaluation]
|
||||
\label{def:evalrules}
|
||||
|
||||
\begin{mathpar}
|
||||
|
||||
\inferrule[E-App1]{
|
||||
\metavariable{e_1} \rightarrow_\beta \metavariable{e_1'}
|
||||
}{
|
||||
\metavariable{e_1} \metavariable{e_2}
|
||||
\rightarrow_\beta
|
||||
\metavariable{e_1'} \metavariable{e_2}
|
||||
}\and
|
||||
|
||||
\inferrule[E-App2]{
|
||||
\metavariable{e_2} \rightarrow_\beta \metavariable{e_2'}
|
||||
}{
|
||||
\metavariable{e_1} \metavariable{e_2}
|
||||
\rightarrow_\beta
|
||||
\metavariable{e_1} \metavariable{e_2'}
|
||||
}
|
||||
|
||||
\inferrule[E-TypApp]{
|
||||
\metavariable{\tau} \in \typenonterm{\emptyset}\\
|
||||
\metavariable{e} \rightarrow_\beta \metavariable{e'}
|
||||
}{
|
||||
\metavariable{e}
|
||||
\metavariable{\tau}
|
||||
\rightarrow_\beta
|
||||
\metavariable{e'}
|
||||
\metavariable{\tau}
|
||||
}
|
||||
|
||||
\newline
|
||||
|
||||
\inferrule[E-TypAppLam]{
|
||||
}{
|
||||
\exprterminal{(\Lambda} \metavariable{\alpha}
|
||||
\exprterminal{\mapsto} \metavariable{e}
|
||||
\exprterminal{)}
|
||||
\metavariable{\tau}
|
||||
\rightarrow_\beta
|
||||
\{ \metavariable{\alpha} \mapsto \metavariable{\tau} \} \metavariable{e}
|
||||
}\and
|
||||
|
||||
\inferrule[E-AppLam]{
|
||||
}{
|
||||
\exprterminal{(\lambda} \metavariable{x}
|
||||
\exprterminal{:} \metavariable{\sigma}
|
||||
\exprterminal{\mapsto} \metavariable{e}
|
||||
\exprterminal{)}
|
||||
\metavariable{a}
|
||||
\rightarrow_\beta
|
||||
\{ \metavariable{x} \mapsto \metavariable{a} \} \metavariable{e}
|
||||
}\and
|
||||
|
||||
\inferrule[E-AppLet]{
|
||||
}{
|
||||
\exprterminal{\text{let }}\metavariable{x}
|
||||
\exprterminal{\text{ = }}\metavariable{a}
|
||||
\exprterminal{\text{ in }}\metavariable{e}
|
||||
\rightarrow_\beta
|
||||
\{ \metavariable{x} \mapsto \metavariable{a} \} \metavariable{e}
|
||||
}
|
||||
|
||||
|
||||
\inferrule[E-ImplicitCast]{
|
||||
\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau} \\
|
||||
\Gamma \vdash \metavariable{h} : \metavariable{\sigma'} \typeterminal{\rightarrow_{morph}} \metavariable{\sigma} \\
|
||||
\Gamma \vdash \metavariable{a} : \metavariable{\sigma'}
|
||||
}{
|
||||
\exprterminal{(} \metavariable{f} \quad \metavariable{a} \exprterminal{)}
|
||||
\rightarrow_\delta
|
||||
\exprterminal{(} \metavariable{f} \quad \exprterminal{(} \metavariable{h} \quad \metavariable{a} \exprterminal{))}
|
||||
}
|
||||
|
||||
\end{mathpar}
|
||||
\end{definition}
|
||||
|
||||
|
||||
\begin{lemma}[\(\beta\)-reduction preserves \(\delta\)-normalform]
|
||||
Assume \metavariable{e} is in \(\delta\)-normalform and \(\metavariable{e} \rightarrow \metavariable{e'}\). Then \(\metavariable{e'}\) is in \(\delta\)-normalform as well.
|
||||
\begin{proof}
|
||||
\todo{}
|
||||
\end{proof}
|
||||
\end{lemma}
|
||||
|
||||
\subsection{Proof of Syntactic Type Soundness}
|
||||
|
||||
\begin{lemma}[Progress]
|
||||
\label{lemma:progress}
|
||||
|
||||
If \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\), then either \(\metavariable{e}\) is a value or there exists some \(\metavariable{e'}\) such that \(\metavariable{e} \rightarrow_{eval} \metavariable{e'}\)
|
||||
|
||||
\begin{proof}
|
||||
\todo{}
|
||||
\end{proof}
|
||||
|
||||
\end{lemma}
|
||||
|
||||
|
||||
\begin{lemma}[Preservation]
|
||||
\label{lemma:preservation}
|
||||
|
||||
\begin{proof}
|
||||
\todo{}
|
||||
\end{proof}
|
||||
|
||||
\end{lemma}
|
||||
|
||||
|
||||
|
||||
|
||||
\begin{theorem}[Type Soundness]
|
||||
If \(\emptyset \vdash \metavariable{e}:\metavariable{\tau}\), then it never occurs that \(\metavariable{e} \rightarrow_{eval}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.
|
||||
|
||||
\begin{proof}
|
||||
Follows from \ref{lemma:progress} and \ref{lemma:preservation}.
|
||||
\end{proof}
|
||||
\end{theorem}
|
||||
|
||||
|
||||
\section{Boehm-Berarducci Encoding}
|
||||
|
||||
|
||||
\newcommand{\typvar}[1]{\textcolor{Emerald}{#1}}
|
||||
\newcommand{\expvar}[1]{\textcolor{MidnightBlue}{#1}}
|
||||
|
||||
|
||||
\subsection{Booleans}
|
||||
|
||||
\begin{definition}
|
||||
Let \(\Sigma = \{ \text{Bool}, \text{BB-Bool} \}\).
|
||||
|
||||
\begin{itemize}
|
||||
\item let {\textcolor{magenta}{bb-true}} = \exprterminal{\((\Lambda\typvar{\alpha}\mapsto\lambda\expvar{t}:\typvar{\alpha}\mapsto\lambda\expvar{f}:\typvar{\alpha}\mapsto\expvar{t})\) as} \typeterminal{Bool \(\sim\) BB-Bool}
|
||||
\item let {\textcolor{magenta}{bb-false}} = \exprterminal{\((\Lambda\typvar{\alpha}\mapsto\lambda\expvar{t}:\typvar{\alpha}\mapsto\lambda\expvar{f}:\typvar{\alpha}\mapsto\expvar{f})\) as} \typeterminal{Bool \(\sim\) BB-Bool}
|
||||
\end{itemize}
|
||||
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}
|
||||
{\textcolor{magenta}{bb-true}} and {\textcolor{magenta}{bb-false}} are \emph{values}.
|
||||
|
||||
\begin{proof}
|
||||
\todo{}
|
||||
\end{proof}
|
||||
\end{lemma}
|
||||
|
||||
\subsection{Conditionals}
|
||||
|
||||
\begin{itemize}
|
||||
\item let {\textcolor{magenta}{bb-if}} = \exprterminal{
|
||||
\(\Lambda\typvar{\alpha}\)\\
|
||||
\(\mapsto\lambda\) \expvar{cond} : \typeterminal{ Bool \(\sim\) BB-Bool \(\sim \forall\typvar{\beta}.(\typvar{\beta}\rightarrow\typvar{\beta}\rightarrow\typvar{\beta})\)}\\
|
||||
\(\mapsto\lambda\) \expvar{then} : \(\typvar{\alpha}\)\\
|
||||
\(\mapsto\lambda\) \expvar{else} : \(\typvar{\alpha}\)\\
|
||||
\(\mapsto\) \expvar{cond} \typvar{\(\alpha\)} \expvar{then} \expvar{else}
|
||||
}
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Natural Numbers}
|
||||
\begin{definition}
|
||||
|
||||
Let \(\Sigma = \{ \mathbb{N}, \text{BB-Nat}\}\).
|
||||
|
||||
\begin{itemize}
|
||||
\item let {\textcolor{magenta}{bb-zero}} = \(\exprterminal{( \Lambda \typvar{\alpha} \mapsto \lambda \expvar{s} : \typeterminal{\typvar{\alpha} \rightarrow \typvar{\alpha}} \mapsto \lambda \expvar{z} : \typvar{\alpha} \mapsto \expvar{z} ) \text{ as } }\)
|
||||
\typeterminal{\(\mathbb{N} \sim \text{BB-Nat} \sim \forall \typvar{\alpha} . (\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha} \)}
|
||||
|
||||
|
||||
\item let {\textcolor{magenta}{bb-one}} = \(\exprterminal{(\Lambda \typvar{\alpha} \mapsto \lambda \expvar{s} : \typeterminal{\typvar{\alpha} \rightarrow \typvar{\alpha}} \mapsto \lambda \expvar{z} : \typvar{\alpha} \mapsto \expvar{s} \expvar{z} ) \text{ as }}\)
|
||||
\typeterminal{\(\mathbb{N} \sim \text{BB-Nat} \sim \forall \typvar{\alpha} . (\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha} \)}
|
||||
|
||||
|
||||
\item let {\textcolor{magenta}{bb-two}} = \(\exprterminal{(\Lambda \typvar{\alpha} \mapsto \lambda \expvar{s} : \typeterminal{\typvar{\alpha} \rightarrow \typvar{\alpha}} \mapsto \lambda \expvar{z} : \typvar{\alpha} \mapsto \expvar{s} (\expvar{s} \expvar{z}) ) \text{ as }}\)
|
||||
\typeterminal{\(\mathbb{N} \sim \text{BB-Nat} \sim \forall \typvar{\alpha} . (\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha} \)}
|
||||
|
||||
\end{itemize}
|
||||
|
||||
\end{definition}
|
||||
|
||||
|
||||
\begin{definition}
|
||||
The successor function is given by:
|
||||
|
||||
let {\textcolor{magenta}{bb-succ}} =
|
||||
\newline\(\exprterminal{\quad \lambda \expvar{n} :
|
||||
\typeterminal{ \mathbb{N}
|
||||
\sim \text{BB-Nat}
|
||||
\sim \forall \typvar{\alpha} . ((\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha})}\\
|
||||
}\)
|
||||
\newline
|
||||
\( \exprterminal{\quad \quad \mapsto ( \Lambda \typvar{\alpha} \mapsto \lambda \expvar{s} : \typeterminal{(\typvar{\alpha} \rightarrow \typvar{\alpha})} \mapsto \lambda \expvar{z} : \typvar{\alpha} }\)
|
||||
\( \exprterminal{\mapsto \expvar{s} ( \expvar{n} \typvar{\alpha} \expvar{s} \expvar{z} )) \text{ as } }\)
|
||||
\typeterminal{
|
||||
\(\mathbb{N} \sim \text{BB-Nat} \sim \forall \typvar{\alpha} . (\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha}\)
|
||||
}
|
||||
\end{definition}
|
||||
|
||||
\begin{example}
|
||||
\(\emptyset \vdash ( \textcolor{magenta}{\text{bb-succ} \quad \text{bb-one}} ) : \typeterminal{ \mathbb{N} \sim \text{BB-Nat} \sim \forall \typvar{\alpha} . (\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha}) } \)
|
||||
\end{example}
|
||||
|
||||
\begin{example}
|
||||
\textcolor{magenta}{bb-succ bb-one} \(\rightarrow_\beta^{*}\) \textcolor{magenta}{bb-two}.
|
||||
\end{example}
|
||||
|
||||
\begin{example}
|
||||
\exprterminal{(}\textcolor{magenta}{bb-succ bb-false}\exprterminal{)} is stuck. \(\nexists\metavariable{v} . \) \exprterminal{(}\textcolor{magenta}{bb-succ bb-false}\exprterminal{)} \( \rightarrow_\text{eval} \metavariable{v}\) where \(\metavariable{v}\) is a \emph{value}.
|
||||
\end{example}
|
||||
|
||||
\begin{example}
|
||||
\exprterminal{(}\textcolor{magenta}{bb-succ bb-false}\exprterminal{)} is \emph{not} well-typed. \(\nexists\metavariable{\tau} . \emptyset \vdash \) \textcolor{magenta}{bb-succ bb-false} : \(\metavariable{\tau}\).
|
||||
\end{example}
|
||||
|
||||
\subsection{Pairs}
|
||||
|
||||
\begin{definition}
|
||||
pairs
|
||||
|
||||
\begin{itemize}
|
||||
\item let {\textcolor{magenta}{bb-pair}} = \(\exprterminal{\Lambda \typvar{\alpha} \mapsto \Lambda \typvar{\beta} \mapsto \lambda \expvar{a}:\typvar{\alpha} \mapsto \lambda \expvar{b} : \typvar{\beta}}\)\\
|
||||
\(\exprterminal{
|
||||
\mapsto (\Lambda\typvar{\gamma} \mapsto \lambda \expvar{s} : \typeterminal{\typvar{\alpha} \rightarrow \typvar{\beta} \rightarrow \typvar{\gamma}} \mapsto \expvar{s} \expvar{a} \expvar{b})}\)
|
||||
\exprterminal{ as } \typeterminal{\((\typvar{\alpha}\times\typvar{\beta})\) \(\sim\) <BB-Pair \(\typvar{\alpha}\text{ }\typvar{\beta}\)> \(\sim\) \(\forall \typvar{\gamma} . ((\typvar{\alpha} \rightarrow \typvar{\beta} \rightarrow \typvar{\gamma}) \rightarrow \typvar{\gamma})\) }\\
|
||||
|
||||
\item let {\textcolor{magenta}{bb-pair-first}} = \(\exprterminal{
|
||||
\Lambda \typvar{\alpha} \mapsto \Lambda \typvar{\beta}\\
|
||||
\mapsto \lambda \expvar{p} : \typeterminal{
|
||||
(\typvar{\alpha}\times\typvar{\beta}) \sim\text{ <BB-Pair \(\typvar{\alpha}\text{ }\typvar{\beta}\)> }\sim \forall \typvar{\gamma} . ((\typvar{\alpha} \rightarrow \typvar{\beta} \rightarrow \typvar{\gamma}) \rightarrow \typvar{\gamma} )
|
||||
}}\)\\
|
||||
\(\exprterminal{
|
||||
\mapsto \expvar{p} \typvar{\alpha} (\lambda \expvar{a}:\typvar{\alpha} \mapsto \lambda \expvar{b}:\typvar{\beta} \mapsto \expvar{a})
|
||||
}\)\\
|
||||
|
||||
\item let {\textcolor{magenta}{bb-pair-second}} = \(\exprterminal{
|
||||
\Lambda \typvar{\alpha} \mapsto \Lambda \typvar{\beta}
|
||||
\mapsto \lambda \expvar{p} : \typeterminal{
|
||||
(\typvar{\alpha}\times\typvar{\beta}) \sim\text{<BB-Pair \(\typvar{\alpha}\text{ }\typvar{\beta}\)> }\sim \forall \typvar{\gamma} . ((\typvar{\alpha} \rightarrow \typvar{\beta} \rightarrow \typvar{\gamma}) \rightarrow \typvar{\gamma} )
|
||||
}}\)\\
|
||||
\(\exprterminal{
|
||||
\mapsto \expvar{p} \typvar{\beta} (\lambda \expvar{a}:\typvar{\alpha} \mapsto \lambda \expvar{b}:\typvar{\beta} \mapsto \expvar{b})
|
||||
}\)\\
|
||||
|
||||
\end{itemize}
|
||||
\end{definition}
|
||||
|
||||
\subsection{Lists}
|
||||
|
||||
|
||||
\subsection{Algebraic Datatypes}
|
||||
\todo{}
|
||||
|
||||
\end{document}
|
||||
|
||||
Loading…
Reference in a new issue