ladder-calculus/paper/main.tex

\documentclass[10pt, sigplan, nonacm]{acmart}

\usepackage[utf8]{inputenc}
\usepackage{formal-grammar}
\usepackage[dvipsnames]{xcolor}
\usepackage{mathpartir}
\usepackage{hyperref}
\usepackage{url}
\usepackage{stmaryrd}
\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}{$\Gamma$}
\DeclareUnicodeCharacter{211D}{$\mathbb{R}$}

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

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

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

\otherform{
    \nonterm{ E }
	\quad
	\exprterminal{as}
	\quad
    \nonterm{ T }
}{Ascription}

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


$$\\$$

\firstcase{V}{
	\nonterm{ V } \quad
	\exprterminal{as} \quad
	\nonterm{ T }
}{Ascribed 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}

\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 \(\metavariable{x} \in \exprvars\).
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 \(\nonterm{E}\).
$$\\$$}

\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 \(\metavariable{\xi} \in \nonterm{T}\) recursively with the type-term given by \(\psi_t(\metavariable{\alpha_i})\)
, while occurences of bound variables are left untouched.
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.
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{\text{let } \metavariable{x} = }\overline{\psi_t}\metavariable{a} \exprterminal{\text{ in }} \overline{\psi_t}\metavariable{e'}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{\text{let } \metavariable{x} = \metavariable{a} \text{ in } \metavariable{e'}}
\\

\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'}}
\\

\overline{\psi_t} \metavariable{e'} \overline{\psi_t}\metavariable{\tau}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{( \metavariable{e'} \metavariable{\tau} )}
\\

\overline{\psi_t} \metavariable{e_1} \overline{\psi_t} \metavariable{e_2}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{(\metavariable{e_1} \metavariable{e_2})}
\\

\overline{\psi_t} \metavariable{e'} \exprterminal{\text{ as }} \overline{\psi_t}\metavariable{\tau}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{ \metavariable{e'} \text{ as } \metavariable{\tau} }

\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 substitution \(\overline{\psi_e}\) replaces all \emph{free} occurences of the expression variables \(\metavariable{x_i}\)
in an expression \(e \in \nonterm{E} \) with the \(\psi_e(\metavariable{x_i})\)

\[\overline{\psi_e} \metavariable{e} = \begin{cases}

\metavariable{e} \quad \text{if } \metavariable{e} \in \exprvars \text{ and } \metavariable{e} \notin \text{Dom}(\psi_e)\\
\metavariable{t} \quad \text{if } \metavariable{e} \in \exprvars \text{ and } (\metavariable{e}\mapsto\metavariable{t}) \in \psi_e\\

\exprterminal{\text{let } \metavariable{x} \text{ = }} \overline{\psi_e}\metavariable{a} \exprterminal{\text{ in }} \overline{\psi_e}\metavariable{e'}
\quad \text{if }
\\

\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto }  \overline{\psi_e} \metavariable{e'}
\quad \text{if \metavariable{e} is of the form }
\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto \metavariable{e'}}
\text{ and } \metavariable{x} \notin \text{Dom}(\psi_e)
\\

\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto }  \overline{\psi_e \setminus \{\metavariable{x} \mapsto \metavariable{t}\}} \metavariable{e'}
\quad \text{if \metavariable{e} is of the form }
\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto \metavariable{e'}}
\text{ and } (\metavariable{x}\mapsto\metavariable{t}) \in \psi_e
\\


\end{cases}\]

\end{definition}

\subsection{Typing}

\subsubsection{Equivalence of Type Terms}

%We want distributivity of ladders over type-specialization as well as over function/morphism types.
\begin{definition}[Distributivity in Types]

\begin{mathpar}
\typeterminal{< \metavariable{\sigma}\sim\metavariable{\sigma'} \quad \metavariable{\tau} >}
\rightarrow_\text{distribute}
\typeterminal{< \metavariable{\sigma} \quad \metavariable{\tau} > \sim < \metavariable{\sigma'} \quad \metavariable{\tau} > }

\typeterminal{< \metavariable{\sigma} \quad \metavariable{\tau}\sim\metavariable{\tau'} >}
\rightarrow_\text{distribute}
\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[T-Eq-Refl]{
	\metavariable{\tau} \in \nonterm{T}
}{
	\metavariable{\tau} \equiv \metavariable{\tau}
}\and
\inferrule[T-Eq-Trans]{
	\metavariable{\tau_1} \equiv \metavariable{\tau_2}\\
	\metavariable{\tau_2} \equiv \metavariable{\tau_3}
}{
	\metavariable{\tau_1} \equiv \metavariable{\tau_3}
}

\inferrule[T-Eq-Alpha]{
	\metavariable{\tau_1} \rightarrow_\alpha \metavariable{\tau_2}
}{
	\metavariable{\tau_1} \equiv \metavariable{\tau_2}
}

\inferrule[T-Eq-Distribute]{
	\metavariable{\tau_1} \rightarrow_\text{distribute} \metavariable{\tau_2}
}{
	\metavariable{\tau_1} \equiv \metavariable{\tau_2}
}\and
\inferrule[T-Eq-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[T-Eq-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 representation subtype relation \(\leq\) and second the transformation subtype relation \(\precsim\).

\begin{definition}[Representation Subtype (\(\tau_1\leq\tau_2\))]
\begin{mathpar}
\inferrule[TSubRepr-Refl]{	
	\metavariable{\tau} \equiv \metavariable{\tau'}
}{
	\metavariable{\tau} \leq \metavariable{\tau'}
}

\inferrule[TSubRepr-Trans]{	
	\metavariable{\sigma} \leq \metavariable{\tau}\\
	\metavariable{\tau} \leq \metavariable{\nu}
}{
	\metavariable{\sigma} \leq \metavariable{\nu}
}

\inferrule[TSubRepr-Ladder]{
	\metavariable{\sigma} \leq \metavariable{\tau}
}{
	\metavariable{\sigma'} \typeterminal{\sim} \metavariable{\sigma} \leq \metavariable{\tau}
}
\end{mathpar}
\end{definition}

\begin{definition}[Transformation Subtype (\(\tau_1\precsim\tau_2\))]
\begin{mathpar}
\inferrule[TSub-Refl]{	
	\metavariable{\tau} \equiv \metavariable{\tau'}
}{
	\metavariable{\tau} \precsim \metavariable{\tau'}
}\and
\inferrule[TSub-Trans]{	
	\metavariable{\sigma} \precsim \metavariable{\tau}\\
	\metavariable{\tau} \precsim \metavariable{\nu}
}{
	\metavariable{\sigma} \precsim \metavariable{\nu}
}

\\

\inferrule[TSub-Ladder]{
	\metavariable{\sigma} \precsim \metavariable{\tau}
}{
	\metavariable{\sigma'} \typeterminal{\sim} \metavariable{\sigma} \precsim \metavariable{\tau}
}\and
\inferrule[TSub-Morph]{
	\metavariable{\sigma} \equiv \metavariable{\tau}
}{
	\metavariable{\sigma} \typeterminal{\sim} \metavariable{\sigma'} \precsim \metavariable{\tau} \typeterminal{\sim} \metavariable{\tau'}
}
\end{mathpar}
\end{definition}


\begin{example}[Representation \& Transformation Subtypes]$\\$

\begin{enumerate}
\item \typeterminal{ <Digit 10> \(\sim\) Char \( \quad \leq \quad \) Char }\\
.. is a \emph{representation 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{transformation 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{Typing Context}

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:pathrules} \ref{def:typerules}, further typing-judgements
%of the form \(\Gamma \vdash \metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)"
%can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \nonterm{E}\) and \(\metavariable{\tau} \in \nonterm{T}\).

\subsubsection{Morphism Graph}
Every typing context \(\Gamma\) implies a \emph{Morphism Graph}, a directed graph whose vertices are types
and the edges represent a type-transformations, as defined by morphisms.
A type \(\metavariable{\tau}\) can be implicitly coerced into a type \(\metavariable{\tau'}\),
provided there is a path from \(\metavariable{\tau}\) to \(\metavariable{\tau'}\) in the \emph{Morphism-Graph} of \(\Gamma\),
written as \(\Gamma  \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\).

\begin{definition}[Morphism Paths]
\label{def:pathrules}
\begin{mathpar}

	\inferrule[M-Sub]{
		\metavariable{\tau} \leq \metavariable{\tau'}
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}

	\inferrule[M-Single]{
		(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}

	\inferrule[M-Chain]{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
		\Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
	}
	
	\inferrule[M-MapSeq]{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}{
		\Gamma \vdash
		\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
		\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
	}
\end{mathpar}

\end{definition}

\begin{example}[Morphism Graph]
Assume \(\Gamma := \{\\
	\exprterminal{\text{degrees-to-turns}} : \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R} \rightarrow_\text{morph} \text{Angle}\sim\text{Turns}\sim\mathbb{R}},\\
	\exprterminal{\text{turns-to-radians}} : \typeterminal{\text{Angle}\sim\text{Turns}\sim\mathbb{R} \rightarrow_\text{morph} \text{Angle}\sim\text{Radians}\sim\mathbb{R}},\\
\}\).
Then
\begin{itemize}
\item \(\Gamma \vdash \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R}} \leadsto \typeterminal{\mathbb{R}}\) (by \textsc{M-Sub})
\item \(\Gamma \vdash \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R}} \leadsto \typeterminal{\text{Angle}\sim\text{Radians}\sim\mathbb{R}}\) (by \textsc{M-Chain})
\item \(\Gamma \vdash \typeterminal{\langle\text{Seq }\text{Angle}\sim\text{Degrees}\sim\mathbb{R}\rangle} \leadsto \typeterminal{\langle\text{Seq }\text{Angle}\sim\text{Radians}\sim\mathbb{R}\rangle}\) (by \textsc{M-MapSeq})
\end{itemize}

\end{example}


\begin{definition}[Typing Relation]

\label{def:typerules}
\begin{mathpar}

	\inferrule[T-Variable]{
		\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]{
		\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
	}{
        \Gamma \vdash (\exprterminal{\Lambda} \metavariable{\alpha} \exprterminal{\mapsto} \metavariable{e}) : \typeterminal{\forall}\metavariable{\alpha}\typeterminal{.}\metavariable{\tau}
	}

	\inferrule[T-TypeApp]{
		\Gamma \vdash \metavariable{e} : \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]{
		\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-MorphAbs]{
		\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-App]{
		\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
		\Gamma \vdash \metavariable{a} : \metavariable{\sigma'}\\
		\Gamma \vdash \metavariable{\sigma'} \leadsto \metavariable{\sigma}
	}{
		\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \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}[Well-Typedness]
An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{well-typed} if there exist \(\Gamma\) and \(\metavariable{\tau}\)
such that \( \Gamma \vdash \metavariable{e} : \metavariable{\tau} \) by \ref{def:typerules}.
\end{definition}

\subsection{Coercion Semantics}
%We define the translation function \(\llbracket . \rrbracket\) which translates morphism-paths into
%expressions that define a transformation function, and also translates type-derivations into expressions with expanded type coercions.


%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 \(C :: \tau \leadsto \tau'\) to mean "C is a morphism-path derivation tree whose conclusion is \(\tau \leadsto \tau'\)".
Similarly, we write \(D :: \Gamma \vdash e : \tau\) to mean "D is a typing derivation whose conclusion is \(\Gamma \vdash e : \tau\)"

\begin{definition}[Morphism Translation]
%Translates a morphism-path derivation into an expression that defines a coercion function

\begin{mathpar}

\Big{\llbracket}
	\inferrule[M-Sub]{
		\metavariable{\tau} \leq \metavariable{\tau'}
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}
\Big{\rrbracket} = \exprterminal{\lambda x:\metavariable{\tau} \mapsto x}
\and

\Big{\llbracket}
	\inferrule[M-Single]{
		(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}
\Big{\rrbracket} = \metavariable{h}
\and


\Big{\llbracket}
	\inferrule[M-Chain]{
		C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
		C_2 :: \Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
	}{
		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
	}
\Big{\rrbracket} = \exprterminal{\lambda \text{x}:\metavariable{\tau} \mapsto}
		\Big{\llbracket} C_2 \Big{\rrbracket}
		\exprterminal{(}\Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{x})}
\and


\Big{\llbracket}
	\inferrule[M-MapSeq]{
		C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
	}{
		\Gamma \vdash
		\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
		\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
	}
\Big{\rrbracket} = \exprterminal{\lambda \text{xs}:\typeterminal{\langle\text{Seq }\metavariable{\tau}\rangle} \mapsto}
		\exprterminal{( \text{map}} \Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{xs})}

\end{mathpar}
\end{definition}

\begin{definition}[Expression Translation]
%Translates a type-derivation tree into a fully expanded expression

\begin{mathpar}
\Big{\llbracket} \inferrule[T-Variable]{
	\metavariable{x}:\metavariable{\tau} \in \Gamma
}{
	\Gamma \vdash \metavariable{x}:\metavariable{\tau}
}\Big{\rrbracket} = \metavariable{x}
\and

\Big{\llbracket} \inferrule[T-LetBinding]{
	D_1 ::\Gamma \vdash \metavariable{e} : \metavariable{\sigma} \\
	D_2 :: \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}
}\Big{\rrbracket} = \exprterminal{\text{let }\metavariable{x} = }
\Big{\llbracket} D_1 \Big{\rrbracket}
\exprterminal{\text{ in }}
\Big{\llbracket} D_2 \Big{\rrbracket}
\and


\Big{\llbracket} 
	\inferrule[T-TypeAbs]{
		D_1 :: \Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
	}{
        \Gamma \vdash (\exprterminal{\Lambda} \metavariable{\alpha} \exprterminal{\mapsto} \metavariable{e}) : \typeterminal{\forall}\metavariable{\alpha}\typeterminal{.}\metavariable{\tau}
	}
\Big{\rrbracket} = \exprterminal{\Lambda \metavariable{\alpha} \mapsto} \Big{\llbracket} D_1 \Big{\rrbracket}
\and


\Big{\llbracket}
	\inferrule[T-TypeApp]{
		D_1 :: \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}
	}
\Big{\rrbracket} =
\exprterminal{(}
\Big{\llbracket}
D_1
\Big{\rrbracket}
\metavariable{\sigma}
\exprterminal{)}
\and


\Big{\llbracket}
	\inferrule[T-Abs]{
		D_1 :: \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}
	}
\Big{\rrbracket} = 
\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\sigma}
\exprterminal{\mapsto} \Big{\llbracket}D_1\Big{\rrbracket}
\and


\Big{\llbracket}
	\inferrule[T-MorphAbs]{
		D_1 :: \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_\text{morph}}\metavariable{\tau'}
	}
\Big{\rrbracket} = 
\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\tau}
\exprterminal{\mapsto_\text{morph}} \Big{\llbracket}D_1\Big{\rrbracket}
\and



\Big{\llbracket}
	\inferrule[T-App]{
		D_1 :: \Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
		D_2 :: \Gamma \vdash \metavariable{a} : \metavariable{\sigma'}\\\\
		C :: \Gamma \vdash \metavariable{\sigma'} \leadsto \metavariable{\sigma}
	}{
		\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
	}
\Big{\rrbracket} =
	\Big{\llbracket}D_1\Big{\rrbracket}
	\exprterminal{(}
	\Big{\llbracket}C\Big{\rrbracket}
	\Big{\llbracket}D_2\Big{\rrbracket}
	\exprterminal{)}
\and

\Big{\llbracket}
	\inferrule[T-MorphFun]{
        D_1 :: \Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow_{morph}}\metavariable{\tau}
	}{
        \Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau}
	}
\Big{\rrbracket} = \Big{\llbracket} D_1 \Big{\rrbracket}
\and

\Big{\llbracket}
	\inferrule[T-Ascension]{
		D_1 :: \Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
		\metavariable{\tau'} \leq \metavariable{\tau}
	}{
		\Gamma \vdash (\metavariable{e} \exprterminal{\text{ as }} \metavariable{\tau'}) : \metavariable{\tau'}
	}
\Big{\rrbracket} =
	\Big{\llbracket}D_1\Big{\rrbracket} \exprterminal{\text{ as }} \metavariable{\tau'}
\and


\Big{\llbracket}
	\inferrule[T-Descension]{
		D_1 :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
		\metavariable{\tau} \leq \metavariable{\tau'}
	}{
		\Gamma \vdash \metavariable{e} : \metavariable{\tau'}
	}
\Big{\rrbracket} =
	\Big{\llbracket}
		D_1
	\Big{\rrbracket}


\end{mathpar}
\end{definition}




\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}
	}

	\inferrule[E-AppLamAscribe]{
	}{
		\exprterminal{(( \lambda \metavariable{x}:\metavariable{\sigma} \mapsto \metavariable{e} )}
		\exprterminal{\text{ as }}
		\typeterminal{\metavariable{\tau}}
		\exprterminal{)}
		\metavariable{a}
		\rightarrow_\beta
		\exprterminal{( \lambda \metavariable{x}:\metavariable{\sigma} \mapsto \metavariable{e} )}
		\metavariable{a}
	}
\end{mathpar}
\end{definition}


\subsection{Soundness}

\begin{lemma}[Preservation]
\label{lemma:preservation}
Assume the expression \(\metavariable{e}\) is well typed,
i.e. there is a type-derivation tree
\(D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\)
for some type \(\metavariable{\tau}\) and context \(\Gamma\).

Then forall \(\metavariable{e'}\) with \(\llbracket D \rrbracket \rightarrow_{\beta} \metavariable{e'}\)
it holds that \(\Gamma \vdash \metavariable{e'} : \metavariable{\tau}\) as well.

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

\begin{lemma}[Progress]
\label{lemma:progress}
Assume the expression \(\metavariable{e}\) is well typed,
i.e. there is a type-derivation tree
\(D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\)
for some type \(\metavariable{\tau}\) and context \(\Gamma\).

Then either \(\metavariable{e}\) is a value
or there exists some \(\metavariable{e'}\) such that \(\llbracket D \rrbracket \rightarrow_{\beta} \metavariable{e'}\)

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

\begin{theorem}[Soundness]
\label{theorem:semantic-soundness}
No well-typed expression is stuck.

Assume the typing derivation \(D :: \Gamma \vdash \metavariable{e}:\metavariable{\tau}\).
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}
\todo{}
%Assume the typing derivation \(D :: \Gamma \vdash \metavariable{e}:\approx\metavariable{\tau}\).
%By \ref{lemma:translation}, \(\Gamma \vdash \llbracket D \rrbracket : \metavariable{\tau}\)
%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}