1005 lines
40 KiB
TeX
1005 lines
40 KiB
TeX
\documentclass[10pt, sigplan, 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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\usepackage{hyperref}
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\usepackage{url}
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\usepackage{stmaryrd}
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\usepackage{minted}
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\usemintedstyle{tango}
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\DeclareUnicodeCharacter{2200}{$\forall$}
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\DeclareUnicodeCharacter{03C3}{$\sigma$}
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\DeclareUnicodeCharacter{03C4}{$\tau$}
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\DeclareUnicodeCharacter{03BB}{$\lambda$}
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\DeclareUnicodeCharacter{21A6}{$\mapsto$}
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\DeclareUnicodeCharacter{039B}{$\Lambda$}
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\DeclareUnicodeCharacter{03B1}{$\alpha$}
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\DeclareUnicodeCharacter{03B2}{$\beta$}
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\DeclareUnicodeCharacter{03B3}{$\gamma$}
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\DeclareUnicodeCharacter{03B4}{$\delta$}
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\DeclareUnicodeCharacter{0393}{$\Gamma$}
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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 work explores the idea of \emph{representational polymorphism}
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to treat the coexistence of multiple equivalent representational forms for a single abstract concept.
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interchangeability
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%Our goal is a type system to support the seamless integration of
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%which may arise by consequence of external interfaces or internal optimization.
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For the study of its formalism, we extend the \emph{polymorphic lambda-calculus} by a new type-constructor,
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called the \emph{ladder-type} in order to encode a \emph{represented-as} relationship into our type-terms.
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Based on this extended type-structure, we first define a subtyping relation to capture
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a notion of structural embedding of higher-level types into lower-level types
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which is then relaxed into \emph{semantic subtyping},
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where for a certain expected type, an equivalent representation implementing the same abstract type
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is accepted as well. In that case, a coercion is inserted implicitly to transform the underlying datastructure
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while keeping all semantical properties of the type intact.
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We specify our typing-rules accordingly, give an algorithm that manifests all implicit coercions in a program
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and prove its \emph{soundness}.
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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{Introduction}
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While certain representational forms might be fixed already at the boundaries of an application,
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internally, some other representations might be desired for reasons of simplicity and efficiency.
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Further, differing complexity-profiles of certain representations might even have the potential to complement
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each other and coexist in a single application.
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Often however, implementations become heavily dependent on concrete data formats
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and require technical knowledge of the low-level data structures.
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Making use of multiple such representations additionally requires careful transformation of data.
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\todo{serialization}
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\todo{memory layout optimizations}
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\todo{difference to traditional coercions (static cast)}
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\todo{relation with inheritance based subtyping: bottom-up vs top-down inheritance vs ladder-types}
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\todo{related work: type specific languages}
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In order to facilitate programming at "high-level", we introduce a type-system that is able to disambiguate
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this multiplicity of representations and facilitate implicit coercions between them.
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We claim this to aid in (1) forgetting details about representational details during program composition
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and (2) keeping the system flexible enough to introduce representational optimizations at a later stage without
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compromising semantic correctness.
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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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Coq definitions of the abstract syntax can be found in \hyperref[coq:terms]{\texttt{terms.v}}.
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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 }{
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\metavariable{\sigma}
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}{Base Type}
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\otherform{
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\metavariable{\alpha}
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}{Type Variable}
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\otherform{
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\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \nonterm{T}
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}{Universal Type}
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\otherform{
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\typeterminal{<} \nonterm{T} \quad \nonterm{T} \typeterminal{>}
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}{Specialized Type}
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\otherform{
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\nonterm{T} \quad \typeterminal{\rightarrow} \quad \nonterm{T}
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}{Function Type}
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\otherform{
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\nonterm{T} \quad \typeterminal{\rightarrow_\text{morph}} \quad \nonterm{T}
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}{Morphism Type}
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\otherform{
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\nonterm{T} \quad \typeterminal{\sim} \quad \nonterm{T}
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}{Ladder Type}
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$$\\$$
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\firstcase{ E
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% T_\selexpr
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}
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{ \metavariable{x}
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} {Variable}
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\otherform{
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\exprterminal{let} \quad \metavariable{x} \quad \exprterminal{=} \quad
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\nonterm{ E }
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\quad \exprterminal{in} \quad
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\nonterm{ E }
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}{Variable Binding}
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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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\nonterm{ E }
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}{Type Abstraction}
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\otherform{
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$$ \exprterminal{\lambda} \metavariable{x} $$
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\exprterminal{:} \nonterm{ T }
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\quad $$\exprterminal{\mapsto}$$ \quad
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\nonterm{ E }
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}{Value Abstraction}
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\otherform{
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$$ \exprterminal{\lambda} \metavariable{x} $$
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\exprterminal{:} \nonterm{ T }
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\quad $$\exprterminal{\mapsto_\text{morph}}$$ \quad
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\nonterm{ E }
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}{Value Morphism}
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\otherform{
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\nonterm{ E }
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\quad
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\nonterm{ T }
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}{Type Application}
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\otherform{
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\nonterm{ E }
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\quad
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\nonterm{ E }
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}{Value Application}
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\otherform{
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\nonterm{ E }
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\quad
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\exprterminal{as}
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\quad
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\nonterm{ T }
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}{Ascription}
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%\otherform{
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% \nonterm{ E }
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% \quad
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% \exprterminal{to}
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% \quad
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% \nonterm{ T }
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%}{Transformation}
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%\otherform{\exprterminal{(} \quad \nonterm{E} \quad \exprterminal{)}}{Parenthesis}
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$$\\$$
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\firstcase{V}{
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\nonterm{ V } \quad
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\exprterminal{as} \quad
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\nonterm{ T }
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}{Ascribed Value}
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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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\nonterm{ V }
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}{Type-Abstraction Value}
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\otherform{
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\exprterminal{\lambda} \metavariable{x}
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\exprterminal{:}
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\nonterm{ T } \quad
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\exprterminal{\mapsto} \quad
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\nonterm{ E }
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}{Abstraction 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 (\(\typenames\)), typevariables (\(\typevars\)), and expression variables (\(\exprvars\)).
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By default, assume \(\metavariable{\sigma} \in \typenames\), \(\metavariable{\alpha} \in \typevars\) and \(\metavariable{x} \in \exprvars\).
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For simplicity, we write \(\metavariable{e} \in \nonterm{E}\) to say that the term \metavariable{e} is contained in the language generated by the nonterminal \(\nonterm{E}\).
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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 substitution \(\overline{\psi_t}\) replaces all \emph{free} occurences of the variables \(\metavariable{\alpha_i}\) in a type-term \(\metavariable{\xi} \in \nonterm{T}\) recursively with the type-term given by \(\psi_t(\metavariable{\alpha_i})\)
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, while occurences of bound variables are left untouched.
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Further, we assume that for all \(\tau_i\), all variable names are disjunct with the free variables of the term to which the substitution is applied.
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Coq definition is at \hyperref[coq:subst-type]{subst.v:\ref{coq:subst-type}}.
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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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\\
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\exprterminal{\text{let } \metavariable{x} = }\overline{\psi_t}\metavariable{a} \exprterminal{\text{ in }} \overline{\psi_t}\metavariable{e'}
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\quad \text{ if \metavariable{e} is of the form }
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\exprterminal{\text{let } \metavariable{x} = \metavariable{a} \text{ in } \metavariable{e'}}
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\\
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\exprterminal{\Lambda \metavariable{\alpha} \mapsto} \overline{\psi_t}\metavariable{e'}
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\quad \text{ if \metavariable{e} is of the form } \exprterminal{\Lambda \metavariable{\alpha} \mapsto \metavariable{e'}}
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\\
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\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto} \overline{\psi_t}\metavariable{e'}
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\quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto \metavariable{e'}}
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\\
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\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto_\text{morph}} \overline{\psi_t}\metavariable{e'}
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\quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto_\text{morph} \metavariable{e'}}
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\\
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\overline{\psi_t} \metavariable{e'} \overline{\psi_t}\metavariable{\tau}
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\quad \text{ if \metavariable{e} is of the form }
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\exprterminal{( \metavariable{e'} \metavariable{\tau} )}
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\\
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\overline{\psi_t} \metavariable{e_1} \overline{\psi_t} \metavariable{e_2}
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\quad \text{ if \metavariable{e} is of the form }
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\exprterminal{(\metavariable{e_1} \metavariable{e_2})}
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\\
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\overline{\psi_t} \metavariable{e'} \exprterminal{\text{ as }} \overline{\psi_t}\metavariable{\tau}
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\quad \text{ if \metavariable{e} is of the form }
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\exprterminal{ \metavariable{e'} \text{ as } \metavariable{\tau} }
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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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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 substitution \(\overline{\psi_e}\) replaces all \emph{free} occurences of the expression variables \(\metavariable{x_i}\)
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in an expression \(e \in \nonterm{E} \) with the \(\psi_e(\metavariable{x_i})\)
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\[\overline{\psi_e} \metavariable{e} = \begin{cases}
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\metavariable{e} \quad \text{if } \metavariable{e} \in \exprvars \text{ and } \metavariable{e} \notin \text{Dom}(\psi_e)\\
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\metavariable{t} \quad \text{if } \metavariable{e} \in \exprvars \text{ and } (\metavariable{e}\mapsto\metavariable{t}) \in \psi_e\\
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\exprterminal{\text{let } \metavariable{x} \text{ = }} \overline{\psi_e}\metavariable{a} \exprterminal{\text{ in }} \overline{\psi_e}\metavariable{e'}
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\quad \text{if }
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\\
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\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto } \overline{\psi_e} \metavariable{e'}
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\quad \text{if \metavariable{e} is of the form }
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\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto \metavariable{e'}}
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\text{ and } \metavariable{x} \notin \text{Dom}(\psi_e)
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\\
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\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto } \overline{\psi_e \setminus \{\metavariable{x} \mapsto \metavariable{t}\}} \metavariable{e'}
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\quad \text{if \metavariable{e} is of the form }
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\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto \metavariable{e'}}
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\text{ and } (\metavariable{x}\mapsto\metavariable{t}) \in \psi_e
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\\
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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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%We want distributivity of ladders over type-specialization as well as over function/morphism types.
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\begin{definition}[Distributivity in Types]
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\begin{mathpar}
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\typeterminal{< \metavariable{\sigma}\sim\metavariable{\sigma'} \quad \metavariable{\tau} >}
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\rightarrow_\text{distribute}
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\typeterminal{< \metavariable{\sigma} \quad \metavariable{\tau} > \sim < \metavariable{\sigma'} \quad \metavariable{\tau} > }
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\typeterminal{< \metavariable{\sigma} \quad \metavariable{\tau}\sim\metavariable{\tau'} >}
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\rightarrow_\text{distribute}
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\typeterminal{< \metavariable{\sigma} \quad \metavariable{\tau} > \sim < \metavariable{\sigma} \quad \metavariable{\tau'} > }
|
|
|
|
\typeterminal{ \metavariable{\sigma}\sim\metavariable{\sigma'} \rightarrow \metavariable{\tau} }
|
|
\rightarrow_\text{distribute}
|
|
\typeterminal{ (\metavariable{\sigma} \rightarrow \metavariable{\tau} ) \sim ( \metavariable{\sigma'} \rightarrow \metavariable{\tau} ) }
|
|
|
|
\typeterminal{ \metavariable{\sigma} \rightarrow \metavariable{\tau}\sim\metavariable{\tau'} }
|
|
\rightarrow_\text{distribute}
|
|
\typeterminal{ (\metavariable{\sigma} \rightarrow \metavariable{\tau} ) \sim ( \metavariable{\sigma} \rightarrow \metavariable{\tau'} ) }
|
|
|
|
\typeterminal{ \metavariable{\sigma}\sim\metavariable{\sigma'} \rightarrow_\text{morph} \metavariable{\tau} }
|
|
\rightarrow_\text{distribute}
|
|
\typeterminal{ (\metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau} ) \sim ( \metavariable{\sigma'} \rightarrow_\text{morph} \metavariable{\tau} ) }
|
|
|
|
\typeterminal{ \metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau}\sim\metavariable{\tau'} }
|
|
\rightarrow_\text{distribute}
|
|
\typeterminal{ (\metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau} ) \sim ( \metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau'} ) }
|
|
|
|
\end{mathpar}
|
|
|
|
|
|
Let \(\rightarrow_\text{condense}\) be the inverse to \(\rightarrow_\text{distribute}\).
|
|
|
|
See \hyperref[coq:type-dist]{equiv.v:\ref{coq:type-dist}}.
|
|
\end{definition}
|
|
|
|
\begin{definition}[Alpha Conversion in Types]
|
|
\begin{mathpar}
|
|
\typeterminal{\forall \metavariable{\alpha} . \metavariable{\tau}}
|
|
\rightarrow_{\alpha}
|
|
\typeterminal{\forall \metavariable{\alpha'} . } \{ \metavariable{\alpha} \mapsto \metavariable{\alpha'} \} \metavariable{\tau}
|
|
|
|
\end{mathpar}
|
|
\end{definition}
|
|
|
|
\begin{definition}[Equivalence Relation]
|
|
Transitive closure over \(\rightarrow_\text{distribute}\), \(\rightarrow_\text{condense}\) and \(\rightarrow_\alpha\).
|
|
|
|
|
|
\begin{mathpar}
|
|
|
|
\inferrule[E-Refl]{
|
|
\metavariable{\tau} \in \nonterm{T}
|
|
}{
|
|
\metavariable{\tau} \equiv \metavariable{\tau}
|
|
}\and
|
|
\inferrule[E-Trans]{
|
|
\metavariable{\tau_1} \equiv \metavariable{\tau_2}\\
|
|
\metavariable{\tau_2} \equiv \metavariable{\tau_3}
|
|
}{
|
|
\metavariable{\tau_1} \equiv \metavariable{\tau_3}
|
|
}
|
|
|
|
\inferrule[E-Rename]{
|
|
\metavariable{\tau_1} \rightarrow_\alpha \metavariable{\tau_2}
|
|
}{
|
|
\metavariable{\tau_1} \equiv \metavariable{\tau_2}
|
|
}
|
|
|
|
\inferrule[E-Distribute]{
|
|
\metavariable{\tau_1} \rightarrow_\text{distribute} \metavariable{\tau_2}
|
|
}{
|
|
\metavariable{\tau_1} \equiv \metavariable{\tau_2}
|
|
}\and
|
|
\inferrule[E-Condense]{
|
|
\metavariable{\tau_1} \rightarrow_\text{condense} \metavariable{\tau_2}
|
|
}{
|
|
\metavariable{\tau_1} \equiv \metavariable{\tau_2}
|
|
}
|
|
|
|
\end{mathpar}
|
|
|
|
See \hyperref[coq:type-equiv]{equiv.v:\ref{coq:type-equiv}}.
|
|
\end{definition}
|
|
|
|
|
|
\begin{lemma}[Symmetry of \(\equiv\)]
|
|
|
|
\begin{mathpar}
|
|
\inferrule[E-Symm]{
|
|
\metavariable{\tau_1} \equiv \metavariable{\tau_2}
|
|
}{
|
|
\metavariable{\tau_2} \equiv \metavariable{\tau_1}
|
|
}
|
|
\end{mathpar}
|
|
|
|
\begin{proof}
|
|
\(\rightarrow_{distribute}\) is the inverse of \(\rightarrow_{condense}\) and \(\rightarrow_{\alpha}\) is symmetric by itself.
|
|
\end{proof}
|
|
|
|
\end{lemma}
|
|
|
|
\subsubsection{Normal Forms}
|
|
|
|
\begin{definition}[Ladder Normal Form]
|
|
LNF is reached by exhaustive application of \(\rightarrow_\text{distribute}\).
|
|
\end{definition}
|
|
|
|
\begin{definition}[Parameter Normal Form]
|
|
PNF reached by exhaustive application of \(\rightarrow_\text{condense}\).
|
|
\end{definition}
|
|
|
|
\subsubsection{Subtype Relations}
|
|
|
|
We define two relations: first the syntatic subtype relation \(\leq\) and second the semantic subtype relation \(\precsim\).
|
|
|
|
\begin{definition}[Syntactic Subtype (\(\tau_1\leq\tau_2\))]
|
|
\begin{mathpar}
|
|
\inferrule[S-Refl]{
|
|
\metavariable{\tau} \equiv \metavariable{\tau'}
|
|
}{
|
|
\metavariable{\tau} \leq \metavariable{\tau'}
|
|
}
|
|
|
|
\inferrule[S-Syntactic]{
|
|
\metavariable{\sigma} \leq \metavariable{\tau}
|
|
}{
|
|
\metavariable{\sigma'} \typeterminal{\sim} \metavariable{\sigma} \leq \metavariable{\tau}
|
|
}
|
|
\end{mathpar}
|
|
\end{definition}
|
|
|
|
\begin{definition}[Semantic Subtype (\(\tau_1\precsim\tau_2\))]
|
|
\begin{mathpar}
|
|
|
|
\inferrule[S-Refl]{
|
|
\metavariable{\tau} \equiv \metavariable{\tau'}
|
|
}{
|
|
\metavariable{\tau} \precsim \metavariable{\tau'}
|
|
}
|
|
|
|
\inferrule[S-Syntactic]{
|
|
\metavariable{\sigma} \precsim \metavariable{\tau}
|
|
}{
|
|
\metavariable{\sigma'} \typeterminal{\sim} \metavariable{\sigma} \precsim \metavariable{\tau}
|
|
}
|
|
|
|
\inferrule[S-Semantic]{
|
|
\metavariable{\sigma} \equiv \metavariable{\tau}
|
|
}{
|
|
\metavariable{\sigma} \typeterminal{\sim} \metavariable{\sigma'} \precsim \metavariable{\tau} \typeterminal{\sim} \metavariable{\tau'}
|
|
}
|
|
|
|
|
|
\end{mathpar}
|
|
\end{definition}
|
|
|
|
|
|
\begin{example}[Syntactic \& Semantic Subtypes]$\\$
|
|
|
|
\begin{enumerate}
|
|
\item \typeterminal{ <Digit 10> \(\sim\) Char \( \quad \leq \quad \) Char }\\
|
|
.. is a \emph{syntactic subtype}, because the representation of \typeterminal{<Digit 10>} is \emph{embedded} into \typeterminal{Char}.\\
|
|
|
|
\item \typeterminal{ <Digit 10> \(\sim\) Char \( \quad \precsim \quad \) <Digit 10> \(\sim\) machine.UInt64}\\
|
|
.. is a \emph{semantic subtype}, because the \typeterminal{Char} based representation can be transformed into a representation based on \typeterminal{machine.UInt64},
|
|
while preserving its semantics.
|
|
\end{enumerate}
|
|
\end{example}
|
|
|
|
\subsubsection{Inference of Expression Types}
|
|
|
|
As usual, the typing-context \(\Gamma = \{ \metavariable{x_1} : \metavariable{\tau_1} , \quad \metavariable{x_2} : \metavariable{\tau_2} , \quad \ldots \}\)
|
|
is a finite mapping which assigns variables \(\metavariable{x_i} \in \exprvars\) to types \(\metavariable{\tau_i} \in \nonterm{T}\).
|
|
Using the inference rules given in \ref{def:typerules}, further typing-judgements
|
|
of the form
|
|
\begin{itemize}
|
|
\item \(\Gamma \vdash \metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)" and
|
|
\item \(\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is compatible with type \(\metavariable{\tau}\)"
|
|
\end{itemize}
|
|
can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \nonterm{E}\) and \(\metavariable{\tau} \in \nonterm{T}\).
|
|
|
|
|
|
|
|
\begin{definition}[Syntactic Well-Typedness]
|
|
An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{syntactically well-typed} if there exists a type \(\metavariable{\tau} \in \nonterm{T}\),
|
|
such that \( \emptyset \vdash \metavariable{e} : \metavariable{\tau} \) by \ref{def:typerules}.
|
|
\end{definition}
|
|
|
|
\begin{definition}[Semantic Well-Typedness]
|
|
An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{semantically well-typed} if there exists a type \(\metavariable{\tau} \in \nonterm{T}\),
|
|
such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) by \ref{def:typerules} and \ref{def:semtyperules}.
|
|
\end{definition}
|
|
|
|
|
|
\begin{definition}[Syntactic Typing Relation]
|
|
|
|
\label{def:typerules}
|
|
\begin{mathpar}
|
|
|
|
\inferrule[T-Variable]{
|
|
% \metavariable{x} \in \exprvars\\
|
|
% \metavariable{\tau} \in \nonterm{T}\\
|
|
\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 \nonterm{T} \\
|
|
% \metavariable{e} \in \nonterm{E} \\
|
|
\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]{
|
|
% \metavariable{\tau} \in \nonterm{T} \\
|
|
\Gamma \vdash \metavariable{e} : \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\tau} \\
|
|
\metavariable{\sigma} \in \nonterm{T}
|
|
}{
|
|
\Gamma \vdash ( \metavariable{e} \quad \metavariable{\sigma} ) : \{\metavariable{\alpha} \mapsto \metavariable{\sigma}\} \metavariable{\tau}
|
|
}
|
|
|
|
|
|
\inferrule[T-Abs]{
|
|
% \metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
|
|
% \metavariable{e} \in \nonterm{E} \\
|
|
\Gamma,\metavariable{x}:\metavariable{\sigma} \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-App]{
|
|
\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-MorphAbs]{
|
|
% \metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
|
|
% \metavariable{e} \in \nonterm{E} \\
|
|
\Gamma,\metavariable{x}:\metavariable{\tau} \vdash \metavariable{e} : \metavariable{\tau'} \\
|
|
\metavariable{\tau} \precsim \metavariable{\tau'}
|
|
}{
|
|
\Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\tau} \exprterminal{\mapsto_{morph}} \metavariable{e}) : \metavariable{\tau}\typeterminal{\rightarrow_{morph}}\metavariable{\tau'}
|
|
}\and
|
|
|
|
\inferrule[T-MorphFun]{
|
|
\Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow_{morph}}\metavariable{\tau}
|
|
}{
|
|
\Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow}\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'}
|
|
}
|
|
|
|
\end{mathpar}
|
|
\end{definition}
|
|
|
|
\begin{definition}[Semantic Typing Relation]
|
|
\label{def:semtyperules}
|
|
\begin{mathpar}
|
|
\inferrule[T-NativeRepr]{
|
|
\Gamma\vdash \metavariable{e} : \metavariable{\tau}
|
|
}{
|
|
\Gamma\vdash \metavariable{e} :\approx \metavariable{\tau}
|
|
}
|
|
|
|
\inferrule[T-CoercedRepr]{
|
|
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau}\\
|
|
% \metavariable{\tau} \precsim \metavariable{\tau'}\\
|
|
%\exists \metavariable{h} \text{ s.t. }
|
|
\Gamma \vdash \metavariable{h}: \typeterminal{\metavariable{\tau}\rightarrow_\text{morph}\metavariable{\tau'}}
|
|
}{
|
|
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
|
|
}
|
|
|
|
|
|
\inferrule[T-CompatibleApp]{
|
|
\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \rightarrow \metavariable{\tau}\\
|
|
\Gamma \vdash \metavariable{a} :\approx \metavariable{\sigma}
|
|
}{
|
|
\Gamma \vdash \exprterminal{(\metavariable{f} \text{ } \metavariable{a})} : \metavariable{\tau}
|
|
}
|
|
\end{mathpar}
|
|
\end{definition}
|
|
|
|
\subsection{Coercion Semantics}
|
|
|
|
We define the translation function \(\llbracket . \rrbracket\) which completes a \emph{semantically well-typed} expression
|
|
by inserting all required coercions based on the typing derivation of the expression.
|
|
The result shall be a \emph{syntactically well-typed} expression.
|
|
|
|
We write \(C :: \sigma \precsim \tau\) to mean "C is a subtyping derivation tree whose conclusion is \(\sigma \precsim \tau\)".
|
|
Similarly, we write \(D :: \Gamma \vdash e : \tau\) to mean "D is a typing derivation whose conclusion is \(\Gamma \vdash e : \tau\)"
|
|
|
|
\begin{definition}[Translation]
|
|
\begin{mathpar}
|
|
|
|
\Big{\llbracket} \inferrule[T-SemanticSubtype]{
|
|
D_1 :: \Gamma \vdash \metavariable{h}:\metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}\\
|
|
D_2 :: \Gamma \vdash \metavariable{e}:\metavariable{\tau}\\
|
|
% C :: \metavariable{\tau} \precsim \metavariable{\tau'}
|
|
}{
|
|
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
|
|
}\Big{\rrbracket} = \exprterminal{(}
|
|
\llbracket D_1 \rrbracket \llbracket D_2 \rrbracket
|
|
%\metavariable{h} \llbracket D_2 \rrbracket
|
|
\exprterminal{)}
|
|
|
|
\Big{\llbracket} \inferrule[T-CoercedApp]{
|
|
D_1 :: \Gamma \vdash \metavariable{f}:\metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
|
|
D_2 :: \Gamma \vdash \metavariable{a}:\approx\metavariable{\sigma}
|
|
}{
|
|
\Gamma \vdash \exprterminal{(\metavariable{f} \text{ } \metavariable{a})} : \metavariable{\tau}
|
|
}\Big{\rrbracket} = \exprterminal{(}
|
|
%\exprterminal{(}\metavariable{f} \llbracket D_2 \rrbracket \exprterminal{)}
|
|
\llbracket D_1 \rrbracket \llbracket D_2 \rrbracket
|
|
\exprterminal{)}
|
|
|
|
\Big{\llbracket}
|
|
\inferrule[\emph{Otherwise}]{}{
|
|
D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}
|
|
}
|
|
\Big{\rrbracket} = \metavariable{e}
|
|
|
|
\end{mathpar}
|
|
\end{definition}
|
|
|
|
\begin{lemma}[Elimination of \(:\approx\)]
|
|
\label{lemma:translation}
|
|
|
|
For all \emph{semantically well-typed} expressions \metavariable{e} with the typing derivation \(D :: \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau}\),
|
|
the translation \(\llbracket D \rrbracket = \metavariable{e'}\), yields a \emph{syntactically well-typed} expression \metavariable{e'} with
|
|
\(\emptyset \vdash \metavariable{e'} : \metavariable{\tau} \)
|
|
|
|
\begin{proof}
|
|
\todo{}
|
|
\end{proof}
|
|
|
|
\end{lemma}
|
|
|
|
|
|
\subsection{Evaluation}
|
|
|
|
Evaluation of an expression \(\metavariable{e} \in \nonterm{E}\) is defined by exhaustive application of the rewrite rule \(\rightarrow_\beta\) as 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{v_1} \metavariable{e_2}
|
|
\rightarrow_\beta
|
|
\metavariable{v_1} \metavariable{e_2'}
|
|
}\and
|
|
|
|
\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}
|
|
}\and
|
|
\inferrule[E-Ascribe]{
|
|
}{
|
|
\metavariable{e}
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\exprterminal{\text{ as }}
|
|
\metavariable{\tau}
|
|
\rightarrow_\beta
|
|
\metavariable{e}
|
|
}
|
|
\end{mathpar}
|
|
\end{definition}
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|
|
|
|
|
\subsection{Soundness}
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|
|
|
\begin{lemma}[Preservation]
|
|
\label{lemma:preservation}
|
|
Assume the expression \(\metavariable{e}\) is well typed, i.e. \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\)
|
|
for some type \(\metavariable{\tau}\).
|
|
Then forall \(\metavariable{e'}\) with \(\metavariable{e} \rightarrow_{\beta} \metavariable{e'}\)
|
|
it holds that \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) as well.
|
|
|
|
\begin{proof}
|
|
\todo{}
|
|
\end{proof}
|
|
\end{lemma}
|
|
|
|
\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 \(\metavariable{e} \rightarrow_{\beta} \metavariable{e'}\)
|
|
|
|
\begin{proof}
|
|
\todo{}
|
|
\end{proof}
|
|
\end{lemma}
|
|
|
|
\begin{theorem}[Syntactic Type Soundness]
|
|
\label{theorem:syntactic-soundness}
|
|
No syntactically well-typed expression is stuck.
|
|
|
|
Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\metavariable{\tau}\).
|
|
Then it never occurs that \(\metavariable{e} \rightarrow_{\beta}^{*} \metavariable{e'}\) 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}
|
|
|
|
\begin{theorem}[Semantic Type Soundness]
|
|
\label{theorem:semantic-soundness}
|
|
No semantically well-typed expression is stuck.
|
|
|
|
Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\).
|
|
Then it never occurs that \(\llbracket D \rrbracket \rightarrow_{\beta}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.
|
|
|
|
\begin{proof}
|
|
Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\).
|
|
By \ref{lemma:translation}, \(\emptyset \vdash \llbracket D \rrbracket : \metavariable{\tau}\)
|
|
and thus it follows by \ref{theorem:syntactic-soundness} that \metavariable{e} is not stuck.
|
|
\end{proof}
|
|
\end{theorem}
|
|
|
|
|
|
|
|
\newpage
|
|
\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{}
|
|
|
|
\input{appendix}
|
|
|
|
\end{document}
|
|
|