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@ -37,7 +37,7 @@ Include Terms.
Module Equiv.
(** Define all rewrite steps *)
(** Define all rewrite steps $\label{coq:type-dist}$ *)
Reserved Notation "S '-->distribute-ladder' T" (at level 40).
Inductive type_distribute_ladder : type_term -> type_term -> Prop :=
@ -111,7 +111,7 @@ Proof.
Qed.
(** Define the equivalence relation as reflexive, transitive hull. *)
(** Define the equivalence relation as reflexive, transitive hull. $\label{coq:type-equiv}$ *)
Reserved Notation " S '===' T " (at level 40).
Inductive type_eq : type_term -> type_term -> Prop :=
@ -163,7 +163,7 @@ Proof.
apply IHtype_eq1.
Qed.
(** "flat" types do not contain ladders *)
(** "flat" types do not contain ladders $\label{coq:type-flat}$ *)
Inductive type_is_flat : type_term -> Prop :=
| FlatUnit : (type_is_flat type_unit)
| FlatVar : forall x, (type_is_flat (type_var x))

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@ -4,8 +4,7 @@ Require Import terms.
Include Terms.
Module Subst.
(* scoped variable substitution in type terms *)
(* scoped variable substitution in type terms $\label{coq:subst-type}$ *)
Fixpoint type_subst (v:string) (n:type_term) (t0:type_term) :=
match t0 with
| type_var name => if (eqb v name) then n else t0

30
paper/appendix.tex Normal file
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@ -0,0 +1,30 @@
\onecolumn
\appendix
\section{Coq Listings}
This document lists the Coq-Sourcecode from commit
\input{|"git log -n 1 --oneline | cut -d' ' -f1"}.
\subsection{terms.v}
\label{coq:terms}
\inputminted[fontsize=\footnotesize, linenos, mathescape]{coq}{../coq/terms.v}
\subsection{subst.v}
\label{coq:subst}
\inputminted[fontsize=\footnotesize, linenos, mathescape]{coq}{../coq/subst.v}
\subsection{equiv.v}
\label{coq:equiv}
\inputminted[fontsize=\footnotesize, linenos, mathescape]{coq}{../coq/equiv.v}
\subsection{typing.v}
\label{coq:typing}
\inputminted[fontsize=\footnotesize, linenos, mathescape]{coq}{../coq/typing.v}
\subsection{smallstep.v}
\label{coq:smallstep}
\inputminted[fontsize=\footnotesize, linenos, mathescape]{coq}{../coq/smallstep.v}

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@ -1,10 +1,26 @@
\documentclass[10pt, nonacm]{acmart}
\documentclass[10pt, sigplan, nonacm]{acmart}
\usepackage[utf8]{inputenc}
\usepackage{formal-grammar}
\usepackage[dvipsnames]{xcolor}
\usepackage{mathpartir}
\usepackage{hyperref}
\usepackage{url}
\usepackage{minted}
\usemintedstyle{tango}
\DeclareUnicodeCharacter{2200}{$\forall$}
\DeclareUnicodeCharacter{03C3}{$\sigma$}
\DeclareUnicodeCharacter{03C4}{$\tau$}
\DeclareUnicodeCharacter{03BB}{$\lambda$}
\DeclareUnicodeCharacter{21A6}{$\mapsto$}
\DeclareUnicodeCharacter{039B}{$\Lambda$}
\DeclareUnicodeCharacter{03B1}{$\alpha$}
\DeclareUnicodeCharacter{03B2}{$\beta$}
\DeclareUnicodeCharacter{03B3}{$\gamma$}
\DeclareUnicodeCharacter{03B4}{$\delta$}
\DeclareUnicodeCharacter{0393}{$\Gamma$}
\newcommand{\metavariable}[1]{\textcolor{teal}{#1}}
\newcommand{\typeterminal}[1]{\textcolor{brown}{#1}}
@ -57,151 +73,139 @@ which are already known from SystemF,
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 \(\tau_2\).
Similar to SystemF, expressions can be \emph{variables}, \emph{type-abstractions}, \emph{}
Coq definitions of the abstract syntax can be found in \hyperref[coq:terms]{\texttt{terms.v}}.
\begin{figure}[h]
\label{gr:core}
\begin{grammar}
\firstcase{ T_\seltype \textsf{$(\typenames, \typevars)$} }{
\firstcase{ T }{
\metavariable{\sigma}
}{Type Literal \quad \textsf{where $ \metavariable{\sigma} \in \typenames $}}
}{Base Type}
\otherform{
\metavariable{\alpha}
}{Type Variable \quad \textsf{where $ \metavariable{\alpha} \in \typevars $}}
\metavariable{\alpha}
}{Type Variable}
\otherform{
$$\typeterminal{\forall}$$ \metavariable{\alpha} \typeterminal{.} \quad \typenonterm{\typevars \cup \{\metavariable{\alpha}\}}
\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \nonterm{T}
}{Universal Type}
\otherform{
\typeterminal{<} \typenonterm{\typevars} \quad \typenonterm{\typevars} \typeterminal{>}
}{Specialization}
\typeterminal{<} \nonterm{T} \quad \nonterm{T} \typeterminal{>}
}{Specialized Type}
\otherform{
\typenonterm{\typevars}
\quad $$\typeterminal{\rightarrow}$$ \quad
\typenonterm{\typevars}
\nonterm{T} \quad \typeterminal{\rightarrow} \quad \nonterm{T}
}{Function Type}
\otherform{
\typenonterm{\typevars}
\quad $$\typeterminal{\rightarrow_{morph}}$$ \quad
\typenonterm{\typevars}
\nonterm{T} \quad \typeterminal{\rightarrow_\text{morph}} \quad \nonterm{T}
}{Morphism Type}
\otherform{
\typenonterm{\typevars}
\quad $$\typeterminal{\sim}$$ \quad
\typenonterm{\typevars}
\nonterm{T} \quad \typeterminal{\sim} \quad \nonterm{T}
}{Ladder Type}
\otherform{
$$\typeterminal{(}$$ \quad
\typenonterm{\typevars}
\quad $$\typeterminal{)}$$
}{Parenthesis}
$$\\$$
\firstcase{ T_\selexpr \textsc{$(\typenames, \typevars, \exprvars)$} }
\firstcase{ E
% T_\selexpr
}
{ \metavariable{x}
} {Variable \quad \textsf{where $\metavariable{x} \in \exprvars$} }
} {Variable}
\otherform{
$$ \exprterminal{\Lambda} \metavariable{\alpha}
\quad \exprterminal{\mapsto} \quad $$
\exprnonterm{\typevars \cup \{\metavariable{\alpha}\}}{\exprvars}
\nonterm{ E }
}{Type Abstraction}
\otherform{
$$ \exprterminal{\lambda} \metavariable{x} $$
\exprterminal{:} \typenonterm{\typevars}
\exprterminal{:} \nonterm{ T }
\quad $$\exprterminal{\mapsto}$$ \quad
\exprnonterm{\typevars}{\exprvars \cup \{\metavariable{x}\}}
\nonterm{ E }
}{Value Abstraction}
\otherform{
$$ \exprterminal{\lambda} \metavariable{x} $$
\exprterminal{:} \typenonterm{\typevars}
\quad $$\exprterminal{\mapsto_{morph}}$$ \quad
\exprnonterm{\typevars}{\exprvars \cup \{\metavariable{x}\}}
\exprterminal{:} \nonterm{ T }
\quad $$\exprterminal{\mapsto_\text{morph}}$$ \quad
\nonterm{ E }
}{Value Morphism}
\otherform{
\exprterminal{let} \quad \metavariable{x} \quad \exprterminal{=} \quad
\exprnonterm{\typevars}{\exprvars}
\nonterm{ E }
\quad \exprterminal{in} \quad
\exprnonterm{\typevars}{\exprvars \cup \{\metavariable{x}\}}
\nonterm{ E }
}{Variable Binding}
\otherform{
\exprnonterm{\typevars}{\exprvars}
\nonterm{ E }
\quad
\typenonterm{\typevars}
\nonterm{ T }
}{Type Application}
\otherform{
\exprnonterm{\typevars}{\exprvars}
\nonterm{ E }
\quad
\exprnonterm{\typevars}{\exprvars}
\nonterm{ E }
}{Value Application}
\otherform{
\exprnonterm{\typevars}{\exprvars}
\nonterm{ E }
\quad
\exprterminal{as}
\quad
\typenonterm{\typevars}
}{Type Cast}
\nonterm{ T }
}{Up-Cast}
\otherform{
\exprterminal{(} \quad
\exprnonterm{\typevars}{\exprvars}
\quad \exprterminal{)}
}{Parenthesis}
\nonterm{ E }
\quad
\exprterminal{to}
\quad
\nonterm{ T }
}{Transformation}
\otherform{\exprterminal{(} \quad \nonterm{E} \quad \exprterminal{)}}{Parenthesis}
$$\\$$
\firstcase{ T_\textsc{Val} \textsc{$(\typenames, \typevars, \exprvars)$} }{
\firstcase{V}{
\exprterminal{\epsilon}
}{Empty Value}
\otherform{
\metavariable{x} \quad \valnonterm{\typevars}{\exprvars}
}{Value Conactenation}
\otherform{
\exprterminal{\Lambda} \metavariable{\alpha} \quad
\exprterminal{\mapsto} \quad
\valnonterm{ \typevars \cup \{ \metavariable{\alpha} \} }
\{Type-Function Value}
\nonterm{ V }
}{Type-Abstraction Value}
\otherform{
\exprterminal{\lambda} \metavariable{x} \quad
\exprterminal{:} \quad
\typenonterm{\emptyset} \quad
\exprterminal{\lambda} \metavariable{x}
\exprterminal{:}
\nonterm{ T } \quad
\exprterminal{\mapsto} \quad
\exprnonterm{\typevars}{\{\metavariable{x}\}}
}{Function Value}
\nonterm{ E }
}{Abstraction Value}
\otherform{
\valnonterm{ \typevars } \quad
\nonterm{ V } \quad
\exprterminal{as} \quad
\typenonterm{ \typevars }
}{Value}
\nonterm{ T }
}{Cast Value}
\end{grammar}
\caption{Syntax of the core calculus with colors for \metavariable{metavariables}, \typeterminal{type-level terminal symbols}, \exprterminal{expression-level terminal symbols}
where $\typenames, \typevars, \exprvars$ are mutually disjoint, countable sets of symbols to denote atomic type identifiers (\(\typenames\)), free typevariables (\(\typevars\)), and free expression variables (\(\exprvars\)).
where $\typenames, \typevars, \exprvars$ are mutually disjoint, countable sets of symbols to denote atomic type identifiers (\(\typenames\)), typevariables (\(\typevars\)), and expression variables (\(\exprvars\)).
By default, assume \(\metavariable{\sigma} \in \typenames\), \(\metavariable{\alpha} \in \typevars\) and \(\metavariable{x} \in \exprvars\)
$$\\$$}
\end{figure}
@ -212,25 +216,25 @@ Let \(\Sigma = \{ \text{Digit}, \text{Char}, \text{Seq}, \text{UTF-8}, \mathbb{N
The following terms are valid types over \(\Sigma\):
\begin{enumerate}
\item \typeterminal{<Seq Char>} \( \in \typenonterm{\emptyset}\)\\
\item \typeterminal{<Seq Char>}\\ %\( \in \typenonterm{\emptyset}\)\\
"sequence of characters"
\item \typeterminal{<Seq <Digit 10> \(\sim\) Char>} \( \in \typenonterm{\emptyset}\)\\
\item \typeterminal{<Seq <Digit 10> \(\sim\) Char>}\\ %\( \in \typenonterm{\emptyset}\)\\
"sequence of decimal digits, where each digit is represented as character"
\item \typeterminal{<Seq Item> \(\rightarrow \mathbb{N} \rightarrow\) Item} \( \in \typenonterm{\{Item\}}\)\\
\item \typeterminal{<Seq Item> \(\rightarrow \mathbb{N} \rightarrow\) Item}\\ %\( \in \typenonterm{\{Item\}}\)\\
"function that maps a sequence of items and a natural number to an item"\\
Note: this type contains the free variable \typeterminal{Item}
\item \typeterminal{<Seq Char> \(\sim\) UTF-8 \(\rightarrow \mathbb{N} \rightarrow\) Char } \( \in \typenonterm{\emptyset}\)\\
%Note: this type contains the free variable \typeterminal{Item}
\item \typeterminal{<Seq Char> \(\sim\) UTF-8 \(\rightarrow \mathbb{N} \rightarrow\) Char }\\ %\( \in \typenonterm{\emptyset}\)\\
"function that takes a sequence of chars, represented as UTF-8 string, and a natural number to return a character"
\item \typeterminal{\(\forall\) Radix . <Seq <Digit Radix> \(\sim\) Char>} \(\in \typenonterm{\emptyset} \)\\
"given the parameter \typeterminal{Radix}, a sequence of digits where each digit is represented as character"\\
Note: this type-term is \emph{ground} (i.e. \(\typevars = \emptyset\)), since \typeterminal{Radix} is bound by \(\typeterminal{\forall}\)
\item \typeterminal{\(\forall\) Radix . <Seq <Digit Radix> \(\sim\) Char>}\\ %\(\in \typenonterm{\emptyset} \)\\
"given the parameter \typeterminal{Radix}, a sequence of digits where each digit is represented as character"
%Note: this type-term is \emph{ground} (i.e. \(\typevars = \emptyset\)), since \typeterminal{Radix} is bound by \(\typeterminal{\forall}\)
\item \typeterminal{
\(\forall\) SrcRadix.\\
\(\forall\) DstRadix.\\
\(\mathbb{N} \sim\) <PosInt SrcRadix> \(\sim\) <Seq <Digit SrcRadix> \(\sim\) Char>\\
\(\rightarrow_{morph}\)\\
\(\mathbb{N} \sim\) <PosInt DstRadix> \(\sim\) <Seq <Digit DstRadix> \(\sim\) Char>\\
} \(\in \typenonterm{\emptyset} \)\\
}\\ %\(\in \typenonterm{\emptyset} \)\\
"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}"
\end{enumerate}
\end{example}
@ -238,8 +242,10 @@ Note: this type-term is \emph{ground} (i.e. \(\typevars = \emptyset\)), since \t
\begin{definition}[Substitution in Types]
Given a type-variable assignment \(\psi_t = \{ \metavariable{\alpha_1} \mapsto \metavariable{\tau_1}, \quad \metavariable{\alpha_2} \mapsto \metavariable{\tau_2}, \quad \dots \}\),
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 \(\psi_t(\metavariable{\alpha_i})\).
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.
the thereby induced, substitution \(\overline{\psi_t}\) replaces all \emph{free} occurences of the variables \(\metavariable{\alpha_i}\) in a type-term \(\metavariable{\xi} \in \typenonterm{\{\metavariable{\alpha_1}, \quad \metavariable{\alpha_2}, \quad \dots\}}\) recursively with the type-term given by \(\psi_t(\metavariable{\alpha_i})\).
Occourences of bound variables \(\metavariable{\alpha_i}\) are
Coq definition is at \hyperref[coq:subst-type]{subst.v:\ref{coq:subst-type}}.
\[\overline{\psi_t} \metavariable{\xi} = \begin{cases}
\metavariable{\xi} \quad \text{if } \metavariable{\xi} \in \typenames\\
@ -296,10 +302,16 @@ in an expression \(
\begin{definition}[Distributivity]
\todo{}
See \hyperref[coq:type-dist]{equiv.v:\ref{coq:type-dist}}.
\end{definition}
\begin{definition}[Equivalence Relation]
\todo{}
See \hyperref[coq:type-equiv]{equiv.v:\ref{coq:type-equiv}}.
\end{definition}
\subsubsection{Normal Forms}
@ -347,7 +359,7 @@ As usual, each rule is composed of premises (above the horizontal line) and a co
\inferrule[T-Variable]{
\metavariable{x} \in \exprvars\\
\metavariable{\tau} \in \typenonterm{\emptyset}\\
\metavariable{\tau} \in \nonterm{T}\\
\metavariable{x}:\metavariable{\tau} \in \Gamma\\
}{
\Gamma \vdash \metavariable{x}:\metavariable{\tau}
@ -363,25 +375,25 @@ As usual, each rule is composed of premises (above the horizontal line) and a co
\inferrule[T-TypeAbs]{
\metavariable{\tau} \in \typenonterm{\typevars \cup \{\metavariable{\alpha}\}} \\
\metavariable{e} \in \exprnonterm{\typevars \cup \{ \metavariable{\alpha} \}}{\exprvars} \\
\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 \typenonterm{\typevars \cup \{\metavariable{\alpha}\}} \\
\metavariable{\tau} \in \nonterm{T} \\
\Gamma \vdash \metavariable{e} : \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\tau} \\
\metavariable{\sigma} \in \typenonterm{\typevars}
\metavariable{\sigma} \in \nonterm{T}
}{
\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} \} } \\
\metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
\metavariable{e} \in \nonterm{E} \\
\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}
@ -402,8 +414,8 @@ As usual, each rule is composed of premises (above the horizontal line) and a co
}\and
\inferrule[T-MorphAbs]{
\metavariable{\sigma}, \metavariable{\tau} \in \typenonterm{\typevars} \\
\metavariable{e} \in \exprnonterm{\typevars}{\exprvars \cup \{ \metavariable{x} \} } \\
\metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
\metavariable{e} \in \nonterm{E} \\
\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}
@ -435,7 +447,7 @@ As usual, each rule is composed of premises (above the horizontal line) and a co
\end{definition}
\subsection{Evaluation Semantics}
\subsection{Evaluation}
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}.
@ -537,7 +549,7 @@ Then \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\)
\end{lemma}
\subsection{Proof of Syntactic Type Soundness}
\subsection{Soundness}
\begin{lemma}[\(\beta\)-Preservation]
\label{lemma:beta-preservation}
@ -717,5 +729,7 @@ pairs
\subsection{Algebraic Datatypes}
\todo{}
\input{appendix}
\end{document}