\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}}
\newcommand{\exprterminal}[1]{\textcolor{Sepia}{#1}}

\newcommand{\seltype}[0]{ {\textsc{\footnotesize{Typ}}} }
\newcommand{\selexpr}[0]{ {\textsc{\footnotesize{Exp}}} }
\newcommand{\selval}[0]{ {\textsc{\footnotesize{Val}}} }

\newcommand{\exprvars}[0]{ V }
\newcommand{\typevars}[0]{ \Upsilon }
\newcommand{\typenames}[0]{ \Sigma }

\newcommand{\typenonterm}[1]{ \nonterm{ T_\seltype \footnotesize{\textsf{$(\typenames, #1)$}}}}
\newcommand{\exprnonterm}[2]{ \nonterm{ T_\selexpr \footnotesize{\textsf{$(\typenames, #1, #2)$}}}}
\newcommand{\valnonterm}[2]{ \nonterm{ T_\selval \footnotesize{\textsf{$(\typenames, #1, #2)$}}}}


\newcommand{\todo}[1]{ {\textcolor{red}{\textbf{TODO:} #1}} }

\title{A functional core calculus with ladder-types}
\author{Michael Sippel}
\email{michael.sippel@mailbox.tu-dresden.de}

\makeatletter
\let\@authorsaddresses\@empty
\makeatother
\begin{document}


\begin{abstract}
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}.
Using ladder-types, multi-layered embeddings of higher-level data-types into lower-level data-types can be described by a type-level structure.
By facilitating automatic transformations between semantically compatible datatypes, ladder-typing opens up a new paradigm of abstraction.
We formally define the syntax \& semantics of this calculus and prove its \emph{type soundness}.
Further we show how the Boehm-Berarducci encoding can be used to implement algebraic datatypes on the basis of the introduced core calculus.
\end{abstract}

\maketitle
\tableofcontents


%\newpage
\section{Core Calculus}
\subsection{Syntax}

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}.
In addition to \emph{function types} \(\tau_1 \rightarrow \tau_2\), \emph{universal types} \(\forall\alpha.\tau\) and type specialization \(<\tau_1 \tau_2>\),
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 }{
	\metavariable{\sigma}
}{Base Type}

\otherform{
	\metavariable{\alpha}
}{Type Variable}

\otherform{
	\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \nonterm{T}
}{Universal Type}

\otherform{
	\typeterminal{<} \nonterm{T} \quad \nonterm{T} \typeterminal{>}
}{Specialized Type}

\otherform{
	\nonterm{T} \quad \typeterminal{\rightarrow} \quad \nonterm{T}
}{Function Type}

\otherform{
	\nonterm{T} \quad \typeterminal{\rightarrow_\text{morph}} \quad \nonterm{T}
}{Morphism Type}

\otherform{
	\nonterm{T} \quad \typeterminal{\sim} \quad \nonterm{T}
}{Ladder Type}

$$\\$$

\firstcase{ E
% T_\selexpr
}
	{ \metavariable{x}
} {Variable}

\otherform{
	$$ \exprterminal{\Lambda} \metavariable{\alpha}
	\quad \exprterminal{\mapsto} \quad $$
	\nonterm{ E }
}{Type Abstraction}

\otherform{
	$$ \exprterminal{\lambda} \metavariable{x} $$
	\exprterminal{:} \nonterm{ T }
    \quad $$\exprterminal{\mapsto}$$ \quad
	\nonterm{ E }
}{Value Abstraction}

\otherform{
	$$ \exprterminal{\lambda} \metavariable{x} $$
	\exprterminal{:} \nonterm{ T }
    \quad $$\exprterminal{\mapsto_\text{morph}}$$ \quad
	\nonterm{ E }
}{Value Morphism}

\otherform{
	\exprterminal{let} \quad \metavariable{x} \quad \exprterminal{=} \quad 
	\nonterm{ E }
	\quad \exprterminal{in} \quad
	\nonterm{ E }
}{Variable Binding}

\otherform{
	\nonterm{ E }
	\quad
	\nonterm{ T }
}{Type Application}

\otherform{
	\nonterm{ E }
	\quad
    \nonterm{ E }
}{Value Application}

\otherform{
    \nonterm{ E }
	\quad
	\exprterminal{as}
	\quad
    \nonterm{ T }
}{Up-Cast}

\otherform{
    \nonterm{ E }
	\quad
	\exprterminal{to}
	\quad
    \nonterm{ T }
}{Transformation}
\otherform{\exprterminal{(} \quad \nonterm{E} \quad \exprterminal{)}}{Parenthesis}


$$\\$$

\firstcase{V}{
    \exprterminal{\epsilon}
}{Empty Value}

\otherform{
	\exprterminal{\Lambda} \metavariable{\alpha} \quad
	\exprterminal{\mapsto} \quad
	\nonterm{ V }
}{Type-Abstraction Value}

\otherform{
	\exprterminal{\lambda} \metavariable{x}
	\exprterminal{:}
	\nonterm{ T } \quad
	\exprterminal{\mapsto} \quad
	\nonterm{ E }
}{Abstraction Value}

\otherform{
	\nonterm{ V } \quad
	\exprterminal{as} \quad
	\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 (\(\typenames\)), typevariables (\(\typevars\)), and expression variables (\(\exprvars\)).
By default, assume \(\metavariable{\sigma} \in \typenames\), \(\metavariable{\alpha} \in \typevars\) and \(\metavariable{x} \in \exprvars\)
$$\\$$}

\end{figure}

\begin{example}
\label{ex:terms}
Let \(\Sigma = \{ \text{Digit}, \text{Char}, \text{Seq}, \text{UTF-8}, \mathbb{N}, PosInt \} \cup \{ 0, 1, 2, ... \}\).
The following terms are valid types over \(\Sigma\):

\begin{enumerate}
\item \typeterminal{<Seq Char>}\\ %\( \in \typenonterm{\emptyset}\)\\
"sequence of characters"
\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}\\ %\( \in \typenonterm{\{Item\}}\)\\
"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 }\\ %\( \in \typenonterm{\emptyset}\)\\
"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{
	\(\forall\) 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>\\
}\\ %\(\in \typenonterm{\emptyset} \)\\
"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}


\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, 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\\
\metavariable{\xi} \quad \text{if }  \metavariable{\xi} \in \{\metavariable{\alpha_1}, \metavariable{\alpha_2}, \dots\} \text{ and } \metavariable{\xi} \notin \text{Dom}(\psi_t)\\
\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\\
\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)\\
\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\\
\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{>}\\
(\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}\\
(\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}\\
(\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}\\
\end{cases}\]

\[\overline{\psi_t} \metavariable{e} = \begin{cases}
\metavariable{e} \quad \text{ if } \metavariable{e} \text{ is a variable}
\exprterminal{\Lambda \metavariable{\alpha} \mapsto} \overline{\psi_t}\metavariable{e'}  \quad \text{ if \metavariable{e} is of the form } \exprterminal{\Lambda \metavariable{\alpha} \mapsto \metavariable{e'}}\\
\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto} \overline{\psi_t}\metavariable{e'}  \quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto \metavariable{e'}}\\
\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto_\text{morph}} \overline{\psi_t}\metavariable{e'}  \quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto_\text{morph} \metavariable{e'}}\\
\end{cases}\]


\end{definition}





\begin{definition}[Substitution in Expressions]
\todo{complete}
Given an expression-variable assignment \(\psi_e = \{ \metavariable{x_1} \mapsto \metavariable{t_1}, \quad \metavariable{x_2} \mapsto \metavariable{t_2}, \quad \dots \}\),
the thereby induced, lexically scoped substitution \(\overline{\psi_e}\) replaces all \emph{free} occurences of the expression variables \(\metavariable{x_i}\)
in an expression \(
	e \in
	\exprnonterm
	{\typevars}
	{\{\metavariable{x_1}, \quad \metavariable{x_2}, \quad \dots\}}
\). 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.

\[\overline{\psi} \metavariable{e} = \begin{cases}
\metavariable{e} \quad \text{if } \metavariable{\xi} \in \typenames\\
\metavariable{\tau} \quad \text{if } (\metavariable{\xi} \mapsto \metavariable{\tau}) \in \psi\\
\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \overline{\psi}\metavariable{\xi'} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{\forall}  \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \text{ and } \metavariable{\alpha} \notin \text{Dom}(\psi)\\
\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{\forall}  \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \text{ and } \metavariable{\alpha} \in \text{Dom}(\psi)\\
\typeterminal{<} (\overline{\psi} \metavariable{\xi_1}) \quad (\overline{\psi} \metavariable{\xi_2}) \typeterminal{>} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{<} \metavariable{\xi_1} \quad \metavariable{\xi_2} \typeterminal{>}\\
(\overline{\psi} \metavariable{\xi_1}) \typeterminal{\rightarrow} (\overline{\psi} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\rightarrow} \metavariable{\xi_2}\\
(\overline{\psi} \metavariable{\xi_1}) \typeterminal{\rightarrow_{morph}} (\overline{\psi} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\rightarrow_{morph}} \metavariable{\xi_2}\\
(\overline{\psi} \metavariable{\xi_1}) \typeterminal{\sim} (\overline{\psi} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\sim} \metavariable{\xi_2}\\
\end{cases}\]

\end{definition}

\subsection{Typing}
\subsubsection{Equivalence of Type Terms}

\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}

\begin{definition}[Ladder Normal Form]
\todo{}
\end{definition}

\subsubsection{Subtyping}

\begin{definition}[Syntactic Subtyping]
\todo{}
\end{definition}

\begin{definition}[Semantic Subtyping]
\todo{}
\end{definition}

\subsubsection{Inference of Expression Types}

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 \(\metavariable{\tau_i} \in \typenonterm{\emptyset}\).
Using the inference rules given in \ref{def:typerules}, further typing-judgements
of the form
\begin{itemize}
\item \(\metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)" and
\item \(\metavariable{e} :\approx \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is compatible with type \(\metavariable{\tau}\)"
\end{itemize}
can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \exprnonterm{\emptyset}{\exprvars}\) and \(\metavariable{\tau} \in \typenonterm{\emptyset}\)

\begin{definition}[Syntactic Well-Typedness]
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 \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-ValueAbs]{
		\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}
	}

	\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 \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}
	}\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}

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]
\label{lemma:preserve-delta-normalform}
Assume \metavariable{e} is in \(\delta\)-normalform and \(\metavariable{e} \rightarrow_\beta \metavariable{e'}\). Then \(\metavariable{e'}\) is in \(\delta\)-normalform as well.
\begin{proof}
\todo{}
\end{proof}
\end{lemma}

\begin{lemma}[\(\delta\)-normalform eliminates compatibility]
\label{lemma:eliminate-compat}
Assume \(\emptyset \vdash \metavariable{e} :\approx \metavariable{\tau}\) and \(\metavariable{e} \rightarrow_{\delta}^* \metavariable{e'}\) such that \(\metavariable{e'}\) is in \(\delta\)-normalform.
Then \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) 

\begin{proof}
\end{proof}

\end{lemma}

\subsection{Soundness}

\begin{lemma}[\(\beta\)-Preservation]
\label{lemma:beta-preservation}
Assume the expression \(\metavariable{e}\) is \textbf{syntactically 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}[\(\delta\)-Preservation]
\label{lemma:delta-preservation}

\begin{proof}
\todo{}
\end{proof}
\end{lemma}

\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 \(\metavariable{e} \rightarrow_{eval} \metavariable{e'}\) 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 \(\metavariable{e} \rightarrow_{eval} \metavariable{e'}\)

\begin{proof}
\todo{}
\end{proof}
\end{lemma}

\begin{theorem}[Soundness]
If \(\emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\), then it never occurs that \(\metavariable{e} \rightarrow_{eval}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.

\begin{proof}
By \ref{lemma:}
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{}

\input{appendix}

\end{document}