appendix.tex

% !TEX root = omar-thesis.tex
\ificfp\else \part*{Appendix} \fi

\appendix
\chapter{Conventions}
\section{Typographic Conventions}\label{appendix:typographic-conventions}
We adopt \emph{PFPL}'s typographic conventions for operational forms throughout the paper \cite{pfpl}. In particular, the names of operators and indexed families of operators are written in $\texttt{typewriter font}$, indexed families of operators specify indices within $[\text{braces}]$, and term arguments are grouped arbitrarily (roughly, by sort) using \texttt{\{}curly braces\texttt{\}} and \texttt{(}rounded braces\texttt{)}. We write $p.e$ for expressions binding the variables that appear in the pattern $p$.

We write $\mapschema{\tau}{i}{\labelset}$ for a sequence of arguments $\tau_i$, one for each $i\in \labelset$, and similarly for arguments of other valences. Operations  that are parameterized by label sets, e.g. $\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$, are identified up to mutual reordering of the label set and the corresponding argument sequence. Similarly, we write $\mapschema{J}{i}{\labelset}$ for the finite set of derivations $J_i$ for each $i \in \labelset$.

We write $\seqschemaX{r}$ for sequences of $n \geq 0$ rule arguments, and similarly for other finite sequences. 

Empty finite sets and finite functions are written $\emptyset$, or omitted entirely within judgements, and non-empty finite sets and finite functions are written as comma-separated sequences identified up to exchange and contraction.


\ificfp
\chapter{\texorpdfstring{A Calculus of Simple TLMs}{A Calculus of Simple TLMs}}
\else
\chapter{\texorpdfstring{$\miniVerseUE$ and $\miniVersePat$}{miniVerseSE and miniVerseS}}
\fi
\label{appendix:miniVerseSES}

\ificfp
This section defines $\miniVersePat$, the calculus of simple expression and pattern TLMs. For some readers, it might be useful to snip out pattern matching to get a language strictly of expression TLMs. To support that, one can omit the segments typeset in gray backgrounds below to recover $\miniVerseUE$, a calculus of simple expression TLMs. We have included the necessary eliminators below (they are technically redundant with pattern matching, but don't hurt things so they're left in white.)
\else

This section defines $\miniVersePat$, the language of Chapter \ref{chap:uptsms}. The language of Chapter \ref{chap:uetsms}, $\miniVerseUE$, can be recovered by omitting the segments typeset in a gray backgrounds below.
\fi

\clearpage

\section{Expanded Language (XL)}\label{appendix:SES-XL}
\subsection{Syntax}
% \begin{figure}[h!]
\[\begin{array}{lllllll}
\textbf{Sort} & & 
& \textbf{Operational Form} 
% & \textbf{Stylized Form} 
& \textbf{Description}\\
\mathsf{Typ} & \tau & ::= & t 
%& t 
& \text{variable}\\
&&& \aparr{\tau}{\tau} 
%& \parr{\tau}{\tau} 
& \text{partial function}\\
&&& \aall{t}{\tau} 
%& \forallt{t}{\tau} 
& \text{polymorphic}\\
&&& \arec{t}{\tau} 
%& \rect{t}{\tau} 
& \text{recursive}\\
&&& \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}} 
%& \prodt{\mapschema{\tau}{i}{\labelset}} 
& \text{labeled product}\\
&&& \asum{\labelset}{\mapschema{\tau}{i}{\labelset}} 
%& \sumt{\mapschema{\tau}{i}{\labelset}} 
& \text{labeled sum}\\
\mathsf{Exp} & e & ::= & x 
%& x 
& \text{variable}\\
&&& \aelam{\tau}{x}{e} 
%& \lam{x}{\tau}{e} 
& \text{abstraction}\\
&&& \aeap{e}{e} 
%& \ap{e}{e} 
& \text{application}\\
&&& \aetlam{t}{e} 
%& \Lam{t}{e} 
& \text{type abstraction}\\
&&& \aetap{e}{\tau} 
%& \App{e}{\tau} 
& \text{type application}\\
&&& \aefold{e} 
%& \fold{e} : \tau 
& \text{fold}\\
&&& \aeunfold{e} 
%& \unfold{e} 
& \text{unfold}\\
&&& \aetpl{\labelset}{\mapschema{e}{i}{\labelset}} 
%& \tpl{\mapschema{e}{i}{\labelset}} 
& \text{labeled tuple}\\
&&& \aepr{\ell}{e} 
%& \prj{e}{\ell} 
& \text{projection}\\
&&& \aein{\ell}{e} 
%& \inj{\ell}{e} 
& \text{injection}\\
&&& \aecase{\labelset}{e}{\mapschemab{x}{e}{i}{\labelset}} 
%& \caseof{e}{\mapschemab{x}{e}{i}{\labelset}} 
& \text{case analysis}\\
\LCC \color{light-gray} & \color{light-gray} & \color{light-gray} 
% & \color{light-gray} 
& \color{light-gray} & \color{light-gray} \\
&&
& \aematchwith{n}{e}{\seqschemaX{r}}
% & \matchwith{e}{\seqschemaX{r}} 
& \text{match}\\
\mathsf{Rule} & r & ::= 
& \aematchrule{p}{e} 
%& \matchrule{p}{e} 
& \text{rule}\\
\mathsf{Pat} & p & ::= 
& x  
%& x 
& \text{variable pattern}\\
&&& \aewildp 
%& \wildp 
& \text{wildcard pattern}\\
&&& \aefoldp{p} 
%& \foldp{p} 
& \text{fold pattern}\\
&&& \aetplp{\labelset}{\mapschema{p}{i}{\labelset}} 
%& \tplp{\mapschema{p}{i}{\labelset}} 
& \text{labeled tuple pattern}\\
&&& \aeinjp{\ell}{p} 
%& \injp{\ell}{p} 
& \text{injection pattern} \ECC
\end{array}\]
% \caption{Syntax of the $\miniVersePat$ expanded language (XL).}
% \end{figure}

\subsection{Statics}
\emph{Type formation contexts}, $\Delta$, are finite sets of hypotheses of the form $\Dhyp{t}$. We write $\Delta, \Dhyp{t}$ when $\Dhyp{t} \notin \Delta$ for $\Delta$ extended with the hypothesis $\Dhyp{t}$. %Finite sets are written as finite sequences identified up to exchange.% We write $\Dcons{\Delta}{\Delta'}$ for the union of $\Delta$ and $\Delta'$.

\emph{Typing contexts}, $\Gamma$, are finite functions that map each variable $x \in \domof{\Gamma}$, where $\domof{\Gamma}$ is a finite set of variables, to the hypothesis $\Ghyp{x}{\tau}$, for some $\tau$. We write $\Gamma, \Ghyp{x}{\tau}$, when $x \notin \domof{\Gamma}$, for the extension of $\Gamma$ with a mapping from $x$ to $\Ghyp{x}{\tau}$, and $\Gcons{\Gamma}{\Gamma'}$ when $\domof{\Gamma} \cap \domof{\Gamma'} = \emptyset$ for the typing context mapping each $x \in \domof{\Gamma} \cup \domof{\Gamma'}$ to $x : \tau$ if $x : \tau \in \Gamma$ or $x : \tau \in \Gamma'$. We write $\isctxU{\Delta}{\Gamma}$ if every type in $\Gamma$ is well-formed relative to $\Delta$.
\begin{definition}[Typing Context Formation] \label{def:isctxU}
$\isctxU{\Delta}{\Gamma}$ iff for each hypothesis $x : \tau \in \Gamma$, we have $\istypeU{\Delta}{\tau}$.
\end{definition}

\noindent\fbox{\strut$\istypeU{\Delta}{\tau}$}~~$\tau$ is a well-formed type
\begin{subequations}\label{rules:istypeU}
\begin{equation}\label{rule:istypeU-var}
\inferrule{ }{\istypeU{\Delta, \Dhyp{t}}{t}}
\end{equation}
\begin{equation}\label{rule:istypeU-parr}
\inferrule{
  \istypeU{\Delta}{\tau_1}\\
  \istypeU{\Delta}{\tau_2}
}{\istypeU{\Delta}{\aparr{\tau_1}{\tau_2}}}
\end{equation}
\begin{equation}\label{rule:istypeU-all}
  \inferrule{
    \istypeU{\Delta, \Dhyp{t}}{\tau}
  }{
    \istypeU{\Delta}{\aall{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:istypeU-rec}
  \inferrule{
    \istypeU{\Delta, \Dhyp{t}}{\tau}
  }{
    \istypeU{\Delta}{\arec{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:istypeU-prod}
  \inferrule{
    \{\istypeU{\Delta}{\tau_i}\}_{i \in \labelset}
  }{
    \istypeU{\Delta}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}
\begin{equation}\label{rule:istypeU-sum}
  \inferrule{
    \{\istypeU{\Delta}{\tau_i}\}_{i \in \labelset}
  }{
    \istypeU{\Delta}{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}
\end{subequations}

\noindent\fbox{\strut$\hastypeU{\Delta}{\Gamma}{e}{\tau}$}~~$e$ is assigned type $\tau$
\begin{subequations}\label{rules:hastypeU}\label{rules:hastypeUP}
\begin{equation}\label{rule:hastypeU-var}
  \inferrule{ }{
    \hastypeU{\Delta}{\Gamma, \Ghyp{x}{\tau}}{x}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:hastypeU-lam}
  \inferrule{
    \istypeU{\Delta}{\tau}\\
    \hastypeU{\Delta}{\Gamma, \Ghyp{x}{\tau}}{e}{\tau'}
  }{
    \hastypeU{\Delta}{\Gamma}{\aelam{\tau}{x}{e}}{\aparr{\tau}{\tau'}}
  }
\end{equation}
\begin{equation}\label{rule:hastypeU-ap}
  \inferrule{
    \hastypeU{\Delta}{\Gamma}{e_1}{\aparr{\tau}{\tau'}}\\
    \hastypeU{\Delta}{\Gamma}{e_2}{\tau}
  }{
    \hastypeU{\Delta}{\Gamma}{\aeap{e_1}{e_2}}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:hastypeU-tlam}
  \inferrule{
    \hastypeU{\Delta, \Dhyp{t}}{\Gamma}{e}{\tau}
  }{
    \hastypeU{\Delta}{\Gamma}{\aetlam{t}{e}}{\aall{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:hastypeU-tap}
  \inferrule{
    \hastypeU{\Delta}{\Gamma}{e}{\aall{t}{\tau}}\\
    \istypeU{\Delta}{\tau'}
  }{
    \hastypeU{\Delta}{\Gamma}{\aetap{e}{\tau'}}{[\tau'/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:hastypeU-fold}
  \inferrule{\
    % \istypeU{\Delta, \Dhyp{t}}{\tau}\\
    \hastypeU{\Delta}{\Gamma}{e}{[\arec{t}{\tau}/t]\tau}
  }{
    \hastypeU{\Delta}{\Gamma}{\aefold{e}}{\arec{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:hastypeU-unfold}
  \inferrule{
    \hastypeU{\Delta}{\Gamma}{e}{\arec{t}{\tau}}
  }{
    \hastypeU{\Delta}{\Gamma}{\aeunfold{e}}{[\arec{t}{\tau}/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:hastypeU-tpl}
  \inferrule{
    \{\hastypeU{\Delta}{\Gamma}{e_i}{\tau_i}\}_{i \in \labelset}
  }{
    \hastypeU{\Delta}{\Gamma}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}
\begin{equation}\label{rule:hastypeU-pr}
  \inferrule{
    \hastypeU{\Delta}{\Gamma}{e}{\aprod{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \ell \hookrightarrow \tau}}
  }{
    \hastypeU{\Delta}{\Gamma}{\aepr{\ell}{e}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:hastypeU-in}
  \inferrule{
    % \{\istypeU{\Delta}{\tau_i}\}_{i \in \labelset}\\
    % \istypeU{\Delta}{\tau}\\
    \hastypeU{\Delta}{\Gamma}{e}{\tau}
  }{
    \hastypeU{\Delta}{\Gamma}{\aein{\ell}{e}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \ell \hookrightarrow \tau}}
  }
\end{equation}
\begin{equation}\label{rule:hastypeU-case}
  \inferrule{
    \hastypeU{\Delta}{\Gamma}{e}{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}\\
    % \istypeU{\Delta}{\tau}\\
    \{\hastypeU{\Delta}{\Gamma, x_i : \tau_i}{e_i}{\tau}\}_{i \in \labelset}
  }{
    \hastypeU{\Delta}{\Gamma}{\aecase{\labelset}{e}{\mapschemab{x}{e}{i}{\labelset}}}{\tau}
  }
\end{equation}
\begin{grayparbox}
\begin{equation}\label{rule:hastypeUP-match}
\graybox{\inferrule{
  \hastypeU{\Delta}{\Gamma}{e}{\tau}\\
  % \istypeU{\Delta}{\tau'}\\
  \{\ruleType{\Delta}{\Gamma}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}\\
}{\hastypeU{\Delta}{\Gamma}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}}}
\end{equation}
\end{grayparbox}
\end{subequations}

\vspace{-5px}\begin{grayparbox}
\vspace{5px}\noindent\fcolorbox{black}{light-gray}{\strut$\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$}~~$r$ takes values of type $\tau$ to values of type $\tau'$
\begin{equation}\label{rule:ruleType}
\graybox{\inferrule{
  \patType{\pctx'}{p}{\tau}\\
  \hastypeU{\Delta}{\Gcons{\Gamma}{\pctx'}}{e}{\tau'}
}{\ruleType{\Delta}{\Gamma}{\aematchrule{p}{e}}{\tau}{\tau'}}}
\end{equation}
Rule (\ref{rule:ruleType}) is defined mutually inductively with Rules (\ref{rules:hastypeUP}).

\noindent\fcolorbox{black}{light-gray}{\strut$\patType{\Gamma}{p}{\tau}$}~~$p$ matches values of type $\tau$ and generates hypotheses $\pctx$
\begin{subequations}\label{rules:patType}
\begin{equation}\label{rule:patType-var}
\graybox{\inferrule{ }{\patType{\Ghyp{x}{\tau}}{x}{\tau}}}
\end{equation}
\begin{equation}\label{rule:patType-wild}
\graybox{\inferrule{ }{\patType{\emptyset}{\aewildp}{\tau}}}
\end{equation}
\begin{equation}\label{rule:patType-fold}
\graybox{\inferrule{
  \patType{\pctx}{p}{[\arec{t}{\tau}/t]\tau}
}{
  \patType{\pctx}{\aefoldp{p}}{\arec{t}{\tau}}
}}
\end{equation}
\begin{equation}\label{rule:patType-tpl}
\graybox{\inferrule{
  \{\patType{\pctx_i}{p_i}{\tau_i}\}_{i \in \labelset}
}{
  \patType{\Gconsi{i \in \labelset}{\pctx_i}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
}}
\end{equation}
\begin{equation}\label{rule:patType-inj}
\graybox{\inferrule{
  \patType{\pctx}{p}{\tau}
}{
  \patType{\pctx}{\aeinjp{\ell}{p}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
}}
\end{equation}
\end{subequations}
\end{grayparbox}

\subsubsection{Metatheory}
The rules above are syntax-directed, so we assume an inversion lemma for each rule as needed without stating it separately or proving it explicitly. The following standard lemmas also hold.

The Weakening Lemma establishes that extending the context with unnecessary hypotheses preserves well-formedness and typing.
\begin{lemma}[Weakening]\label{lemma:weakening-UP}\label{lemma:weakening-U} ~
\begin{enumerate} 
\item If $\istypeU{\Delta}{\tau}$ then $\istypeU{\Delta, \Dhyp{t}}{\tau}$.
%\item If $\isctxU{\Delta}{\Gamma}$ then $\isctxU{\Delta, \Dhyp{t}}{\Gamma}$.
\item \begin{enumerate}
  \item If $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ then $\hastypeU{\Delta, \Dhyp{t}}{\Gamma}{e}{\tau}$.
  \item \graytxtbox{If $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$ then $\ruleType{\Delta, \Dhyp{t}}{\Gamma}{r}{\tau}{\tau'}$.}
  \end{enumerate}
\item \begin{enumerate}
  \item If $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ and $\istypeU{\Delta}{\tau''}$ then $\hastypeU{\Delta}{\Gamma, \Ghyp{x}{\tau''}}{e}{\tau}$.
  \item \graytxtbox{If $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$ and $\istypeU{\Delta}{\tau''}$ then $\ruleType{\Delta}{\Gamma, \Ghyp{x}{\tau''}}{r}{\tau}{\tau'}$.}
  \end{enumerate}
\item \graytxtbox{If $\patType{\pctx}{p}{\tau}$ then $\patTypeD{\Delta, \Dhyp{t}}{\pctx}{p}{\tau}$.}
\end{enumerate}
\end{lemma}
\begin{proof-sketch} ~
\begin{enumerate}
\item By rule induction over Rules (\ref{rules:istypeU}).
%\item By rule induction over Rules (\ref{rules:isctxU}).
\item By \graytxtbox{mutual} rule induction over Rules (\ref{rules:hastypeUP}) \graytxtbox{and Rule (\ref{rule:ruleType})}, and part 1.
\item By \graytxtbox{mutual} rule induction over Rules (\ref{rules:hastypeUP}) \graytxtbox{and Rule (\ref{rule:ruleType})}, and part 1.
\item \graytxtbox{By rule induction over Rules (\ref{rules:patType}).}
\end{enumerate}
\end{proof-sketch}

\begin{grayparbox}
The {pattern typing judgement} is \emph{linear} in the pattern typing context, i.e. it does \emph{not} obey weakening of the pattern typing context. This is to ensure that the pattern typing context captures exactly those hypotheses generated by a pattern, and no others.
\end{grayparbox}

The Substitution Lemma establishes that substitution of a well-formed type for a type variable, or an expanded expression of the appropriate type for an expanded expression variable, preserves well-formedness and typing.
\begin{lemma}[Substitution]\label{lemma:substitution-UP} ~
\begin{enumerate}
\item If $\istypeU{\Delta, \Dhyp{t}}{\tau}$ and $\istypeU{\Delta}{\tau'}$ then $\istypeU{\Delta}{[\tau'/t]\tau}$.
%\item If $\isctxU{\Delta, \Dhyp{t}}{\Gamma}$ and $\istypeU{\Delta}{\tau'}$ then $\isctxU{\Delta}{[\tau'/t]\Gamma}$.
\item \begin{enumerate}
  \item If $\hastypeU{\Delta, \Dhyp{t}}{\Gamma}{e}{\tau}$ and $\istypeU{\Delta}{\tau'}$ then $\hastypeU{\Delta}{[\tau'/t]\Gamma}{[\tau'/t]e}{[\tau'/t]\tau}$.
  \item \begin{grayparbox} 
  {If} $\ruleType{\Delta, \Dhyp{t}}{\Gamma}{r}{\tau}{\tau''}$ and $\istypeU{\Delta}{\tau'}$ then $\ruleType{\Delta}{[\tau'/t]\Gamma}{[\tau'/t]r}{[\tau'/t]\tau}{[\tau'/t]\tau''}$.
  \end{grayparbox}
  \end{enumerate}
\item \begin{enumerate}
  \item If $\hastypeU{\Delta}{\Gamma, \Ghyp{x}{\tau'}}{e}{\tau}$ and $\hastypeU{\Delta}{\Gamma}{e'}{\tau'}$ then $\hastypeU{\Delta}{\Gamma}{[e'/x]e}{\tau}$.
  \item \graytxtbox{
  If $\ruleType{\Delta}{\Gamma, \Ghyp{x}{\tau'}}{r}{\tau}{\tau''}$ and $\hastypeU{\Delta}{\Gamma}{e'}{\tau''}$ then $\ruleType{\Delta}{\Gamma}{[e'/x]r}{\tau}{\tau''}$.}
  \end{enumerate}
\end{enumerate}\end{lemma}
\begin{proof-sketch} ~
\begin{enumerate}
\item By rule induction over Rules (\ref{rules:istypeU}).
\item By \graytxtbox{mutual} rule induction over Rules (\ref{rules:hastypeUP}) \graytxtbox{and Rule (\ref{rule:ruleType})}.
\item By \graytxtbox{mutual} rule induction over Rules (\ref{rules:hastypeUP}) \graytxtbox{and Rule (\ref{rule:ruleType})}.
\end{enumerate}
\end{proof-sketch}

The Decomposition Lemma is the converse of the Substitution Lemma.
\begin{lemma}[Decomposition]\label{lemma:decomposition-UP} ~
\begin{enumerate}
\item If $\istypeU{\Delta}{[\tau'/t]\tau}$ and $\istypeU{\Delta}{\tau'}$ then $\istypeU{\Delta, \Dhyp{t}}{\tau}$.
%\item If $\isctxU{\Delta}{[\tau'/t]\Gamma}$ and $\istypeU{\Delta}{\tau'}$ then $\isctxU{\Delta, \Dhyp{t}}{\Gamma}$.
\item \begin{enumerate}
  \item If $\hastypeU{\Delta}{[\tau'/t]\Gamma}{[\tau'/t]e}{[\tau'/t]\tau}$ and $\istypeU{\Delta}{\tau'}$ then $\hastypeU{\Delta, \Dhyp{t}}{\Gamma}{e}{\tau}$.
  \item \begin{grayparbox}
  If $\ruleType{\Delta}{[\tau'/t]\Gamma}{[\tau'/t]r}{[\tau'/t]\tau}{[\tau'/t]\tau''}$ and $\istypeU{\Delta}{\tau'}$ then $\ruleType{\Delta, \Dhyp{t}}{\Gamma}{r}{\tau}{\tau''}$.
  \end{grayparbox}
  \end{enumerate}
\item \begin{enumerate}
  \item If $\hastypeU{\Delta}{\Gamma}{[e'/x]e}{\tau}$ and $\hastypeU{\Delta}{\Gamma}{e'}{\tau'}$ then $\hastypeU{\Delta}{\Gamma, \Ghyp{x}{\tau'}}{e}{\tau}$.
  \item \graytxtbox{If $\ruleType{\Delta}{\Gamma}{[e'/x]r}{\tau}{\tau''}$ and $\hastypeU{\Delta}{\Gamma}{e'}{\tau'}$ then $\ruleType{\Delta}{\Gamma, \Ghyp{x}{\tau'}}{r}{\tau}{\tau''}$.}
  \end{enumerate}
\end{enumerate}\end{lemma}
\begin{proof-sketch} ~
\begin{enumerate}
\item By rule induction over Rules (\ref{rules:istypeU}) and case analysis over the definition of substitution. In all cases, the derivation of $\istypeU{\Delta}{[\tau'/t]\tau}$ does not depend on the form of $\tau'$.
%\item Context formation of $[\tau'/t]\Gamma$ does not depend on the structure of $\tau'$.
\item By \graytxtbox{mutual} rule induction over Rules (\ref{rules:hastypeUP}) \graytxtbox{and Rule (\ref{rule:ruleType})} and case analysis over the definition of substitution. In all cases, the derivation of $\hastypeU{\Delta}{[\tau'/t]\Gamma}{[\tau'/t]e}{[\tau'/t]\tau}$ \graytxtbox{or $\ruleType{\Delta}{[\tau'/t]\Gamma}{[\tau'/t]r}{[\tau'/t]\tau}{[\tau'/t]\tau''}$} does not depend on the form of $\tau'$.
\item By \graytxtbox{mutual} rule induction over Rules (\ref{rules:hastypeUP}) \graytxtbox{and Rule (\ref{rule:ruleType})} and case analysis over the definition of substitution. In all cases, the derivation of $\hastypeU{\Delta}{\Gamma}{[e'/x]e}{\tau}$ \graytxtbox{or $\ruleType{\Delta}{\Gamma}{[e'/x]r}{\tau}{\tau''}$} does not depend on the form of $e'$.
\end{enumerate}
\end{proof-sketch}

\begin{grayparbox}
% The Pattern Regularity Lemma establishes that the hypotheses generated by checking a pattern against a well-formed type involve only well-formed types.
\begin{lemma}[Pattern Regularity]\label{lemma:pattern-regularity-UP} 
If $\patType{\pctx}{p}{\tau}$ and $\istypeU{\Delta}{\tau}$ then $\isctxU{\Delta}{\pctx}$ and $\mathsf{patvars}({p}) = \domof{\pctx}$.
\end{lemma}
\begin{proof} By rule induction over Rules (\ref{rules:patType}).
\begin{byCases}
\item[\text{(\ref{rule:patType-var})}] ~
\begin{pfsteps*}
  \item $p=x$ \BY{assumption}
  \item $\pctx=x : \tau$ \BY{assumption}
  \item $\istypeU{\Delta}{\tau}$ \BY{assumption}\pflabel{istypeU}
  \item $\isctxU{\Delta}{\Ghyp{x}{\tau}}$ \BY{Definition \ref{def:isctxU} on \pfref{istypeU}}
  \item $\fvof{p} = \domof{\Gamma} = \{x\}$ \BY{definition}
 \end{pfsteps*}
 \resetpfcounter
\item[\text{(\ref{rule:patType-wild})}] ~
\begin{pfsteps}
\item p = \aewildp \BY{assumption}
\item \pctx=\emptyset \BY{assumption}
\item \isctxU{\Delta}{\emptyset} \BY{Definition \ref{def:isctxU}}
\item \mathsf{patvars}({p}) = \domof{\Gamma} = \emptyset \BY{definition}
\end{pfsteps}
\resetpfcounter

\item[\text{(\ref{rule:patType-tpl})}] ~
\begin{pfsteps*}
  \item $p=\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}$ \BY{assumption}
  \item $\tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$ \BY{assumption}
  \item $\pctx=\cup_{i \in \labelset} \pctx_i$ \BY{assumption}
  \item $\{\patType{\pctx_i}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{assumption}\pflabel{patType}
  \item $\istypeU{\Delta}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}$ \BY{assumption} \pflabel{istypeU}
  \item $\{\istypeU{\Delta}{\tau_i}\}_{i \in \labelset}$ \BY{Inversion of Rule (\ref{rule:istypeU-prod}) on \pfref{istypeU}}\pflabel{istypeU-each}
  \item $\{\isctxU{\Delta}{\pctx_i}\}_{i \in \labelset}$ \BY{IH over \pfref{patType} and \pfref{istypeU-each}} \pflabel{biggy}
  \item $\{\mathsf{patvars}({p_i}) = \domof{\pctx_i}\}_{i \in \labelset}$ \BY{IH over \pfref{patType} and \pfref{istypeU-each}} \pflabel{biggy2}
  \item $\isctxU{\Delta}{\cup_{i \in \labelset} \pctx_i}$ \BY{Definition \ref{def:isctxU} over \pfref{biggy}, then Definition \ref{def:isctxU} iteratively}
  \item $\mathsf{patvars}({p}) = \domof{\Gamma} = \emptyset$ \BY{definition and \pfref{biggy2}}
\end{pfsteps*}
\resetpfcounter

\item[\text{(\ref{rule:patType-inj})}] ~
\begin{pfsteps*}
  \item $p=\aeinjp{\ell}{p'}$ \BY{assumption}
  \item $\tau=\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}$ \BY{assumption}
  \item $\istypeU{\Delta}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}}$ \BY{assumption} \pflabel{istype}
  \item $\patType{\pctx}{p'}{\tau'}$ \BY{assumption} \pflabel{patType}
  \item $\istypeU{\Delta}{\tau'}$ \BY{Inversion of Rule (\ref{rule:istypeU-sum}) on \pfref{istype}} \pflabel{istypeTwo} 
  \item $\isctxU{\Delta}{\pctx}$ \BY{IH on \pfref{patType} and \pfref{istypeTwo}}
  \item $\mathsf{patvars}({p'}) = \domof{\pctx}$
  \BY{IH on \pfref{patType} and \pfref{istypeTwo}} \pflabel{fv1}
  \item $\mathsf{patvars}({p}) = \domof{\pctx}$ \BY{definition and \pfref{fv1}}
\end{pfsteps*}
\resetpfcounter
\end{byCases}
\end{proof}
\end{grayparbox}

% Finally, the Regularity Lemma establishes that the type assigned to an expression under a well-formed typing context is well-formed. 
% \begin{lemma}[Regularity]\label{lemma:regularity-UP} ~
% \begin{enumerate}
% \item If $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ and $\isctxU{\Delta}{\Gamma}$ then $\istypeU{\Delta}{\tau}$.
% \item \graytxtbox{If $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$ and $\isctxU{\Delta}{\Gamma}$ then $\istypeU{\Delta}{\tau'}$.}
% \end{enumerate}
% \end{lemma}
% \begin{proof-sketch} By \graytxtbox{mutual} rule induction over Rules (\ref{rules:hastypeUP}) \graytxtbox{and Rule (\ref{rule:ruleType})}, and Lemma \ref{lemma:substitution-UP} \graytxtbox{and Lemma \ref{lemma:pattern-regularity-UP}}.
% \end{proof-sketch}

\subsection{Structural Dynamics}\vspace{-4px}
The \emph{structural dynamics} is specified as a transition system, and is organized around judgements of the following form:
\vspace{-4px}\[\begin{array}{ll}
\textbf{Judgement Form} & \textbf{Description}\\
\stepsU{e}{e'} & \text{$e$ transitions to $e'$}\\
\isvalU{e} & \text{$e$ is a value}\\
\LCC \color{light-gray} & \color{light-gray} \\
\matchfail{e} & \text{$e$ raises match failure} \ECC
\end{array}\]\vspace{-4px}
We also define auxiliary judgements for \emph{iterated transition}, $\multistepU{e}{e'}$, and \emph{evaluation}, $\evalU{e}{e'}$.


\begin{definition}[Iterated Transition]\label{defn:iterated-transition-UP} Iterated transition, $\multistepU{e}{e'}$, is the reflexive, transitive closure of the transition judgement, $\stepsU{e}{e'}$.\end{definition}


\begin{definition}[Evaluation]\label{defn:evaluation-UP}  $\evalU{e}{e'}$ iff $\multistepU{e}{e'}$ and $\isvalU{e'}$. \end{definition}

Our subsequent developments do not make mention of particular rules in the dynamics, nor do they make mention of other judgements, not listed above,  that are used only for defining the dynamics of the match operator, so we do not produce these details here. Instead, it suffices to state the following conditions.

\begin{condition}[Canonical Forms]\label{condition:canonical-forms-UP} If $\hastypeUC{e}{\tau}$ and $\isvalU{e}$ then:
\begin{enumerate}
\item If $\tau=\aparr{\tau_1}{\tau_2}$ then $e=\aelam{\tau_1}{x}{e'}$ and $\hastypeUCO{\Ghyp{x}{\tau_1}}{e'}{\tau_2}$.
\item If $\tau=\aall{t}{\tau'}$ then $e=\aetlam{t}{e'}$ and $\hastypeUCO{\Dhyp{t}}{e'}{\tau'}$.
\item If $\tau=\arec{t}{\tau'}$ then $e=\aefold{e'}$ and $\hastypeUC{e'}{[\abop{rec}{t.\tau'}/t]\tau'}$ and $\isvalU{e'}$. 
\item If $\tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$ then $e=\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}$ and $\hastypeUC{e_i}{\tau_i}$ and $\isvalU{e_i}$ for each $i \in \labelset$.
\item If $\tau=\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}$ then for some label set $L'$ and label $\ell$ and type $\tau'$, we have that $\labelset=\labelset', \ell$ and $\tau=\asum{\labelset', \ell}{\mapschema{\tau}{i}{\labelset'}; \mapitem{\ell}{\tau'}}$ and $e=\aein{\ell}{e'}$ and $\hastypeUC{e'}{\tau'}$ and $\isvalU{e'}$.
\end{enumerate}\end{condition}


\begin{condition}[Preservation]\label{condition:preservation-UP} If $\hastypeUC{e}{\tau}$ and $\stepsU{e}{e'}$ then $\hastypeUC{e'}{\tau}$. \end{condition}

\begin{condition}[Progress]\label{condition:progress-UP} If $\hastypeUC{e}{\tau}$ then either $\isvalU{e}$ \graytxtbox{or $\matchfail{e}$} or there exists an $e'$ such that $\stepsU{e}{e'}$. \end{condition}

\section{Unexpanded Language (UL)}\label{appendix:SES-uexps}
\subsection{Syntax}\label{appendix:SES-syntax}\label{appendix:SES-shared-forms}
\subsubsection{Stylized Syntax}
\[\begin{array}{lllllll}
\textbf{Sort} & &  
%&\textbf{Operational Form} 
& \textbf{Stylized Form} & \textbf{Description}\\
\mathsf{UTyp} & \utau & ::= 
% &\ut 
& \ut & \text{identifier}\\
&& 
%& \auparr{\utau}{\utau} 
& \parr{\utau}{\utau} & \text{partial function}\\
&&
%& \auall{\ut}{\utau} 
& \forallt{\ut}{\utau} & \text{polymorphic}\\
&&
%& \aurec{\ut}{\utau} 
& \rect{\ut}{\utau} & \text{recursive}\\
&&
%& \auprod{\labelset}{\mapschema{\utau}{i}{\labelset}} 
& \prodt{\mapschema{\utau}{i}{\labelset}} & \text{labeled product}\\
&&
%& \ausum{\labelset}{\mapschema{\utau}{i}{\labelset}} 
& \sumt{\mapschema{\utau}{i}{\labelset}} & \text{labeled sum}\\
\mathsf{UExp} & \ue & ::= 
%& \ux 
& \ux & \text{identifier}\\
&&
%
& \asc{\ue}{\utau} & \text{ascription}\\
&&
%
& \letsyn{\ux}{\ue}{\ue} & \text{value binding}\\
&&
%& \aulam{\utau}{\ux}{\ue} 
& \lam{\ux}{\utau}{\ue} & \text{abstraction}\\
&&
%& \auap{\ue}{\ue} 
& \ap{\ue}{\ue} & \text{application}\\
&&
%& \autlam{\ut}{\ue} 
& \Lam{\ut}{\ue} & \text{type abstraction}\\
&&
%& \autap{\ue}{\utau} 
& \App{\ue}{\utau} & \text{type application}\\
&&
%& \aufold{\ut}{\utau}{\ue} 
& \fold{\ue} & \text{fold}\\
&&
%& \auunfold{\ue} 
& \unfold{\ue} & \text{unfold}\\
&&
%& \autpl{\labelset}{\mapschema{\ue}{i}{\labelset}} 
& \tpl{\mapschema{\ue}{i}{\labelset}} & \text{labeled tuple}\\
&&
%& \aupr{\ell}{\ue} 
& \prj{\ue}{\ell} & \text{projection}\\
&&
%& \auin{\labelset}{\ell}{\mapschema{\utau}{i}{\labelset}}{\ue} 
& \inj{\ell}{\ue} & \text{injection}\\
&&
%& \aucase{\labelset}{\utau}{\ue}{\mapschemab{\ux}{\ue}{i}{\labelset}} 
& \caseof{\ue}{\mapschemab{\ux}{\ue}{i}{\labelset}} & \text{case analysis}\\
&&
%& \audefuetsm{\utau}{e}{\tsmv}{\ue} 
& \uesyntax{\tsmv}{\utau}{e}{\ue} & \text{seTLM definition}\\ 
&&
%& \autsmap{b}{\tsmv} 
& \utsmap{\tsmv}{b} & \text{seTLM application}\\%\ECC
\LCC  \color{light-gray} & \color{light-gray} & \color{light-gray}
& \color{light-gray} 
& \color{light-gray} & \color{light-gray} \\
&&
%& \aumatchwith{n}{\utau}{\ue}{\seqschemaX{\urv}} 
& \matchwith{\ue}{\seqschemaX{\urv}} & \text{match}\\
&&
%& \audefuptsm{\utau}{e}{\tsmv}{\ue} 
& \usyntaxup{\tsmv}{\utau}{e}{\ue}
& \text{spTLM definition}\\
\mathsf{URule} & \urv & ::= 
%& \aumatchrule{\upv}{\ue} 
& \matchrule{\upv}{\ue} & \text{match rule}\\
\mathsf{UPat} & \upv & ::= 
%& \ux 
& \ux & \text{identifier pattern}\\
&&
%& \auwildp 
& \wildp & \text{wildcard pattern}\\
&&
%& \aufoldp{\upv} 
& \foldp{\upv} & \text{fold pattern}\\
&&
%& \autplp{\labelset}{\mapschema{\upv}{i}{\labelset}} 
& \tplp{\mapschema{\upv}{i}{\labelset}} & \text{labeled tuple pattern}\\
&&
%& \auinjp{\ell}{\upv} 
& \injp{\ell}{\upv} & \text{injection pattern}\\
% \LCC &&& \color{light-gray} & \color{light-gray} & \color{light-gray}\\
&&
%& \auapuptsm{b}{\tsmv} 
& \utsmap{\tsmv}{b} & \text{spTLM application}\ECC
\end{array}\]

\clearpage

\paragraph{Body Lengths}\label{appendix:SES-body-lengths}
We write $\sizeof{b}$ for the length of $b$. The metafunction $\sizeof{\ue}$ computes the sum of the lengths of expression literal bodies in $\ue$:
\[
\begin{array}{ll}
\sizeof{\ux} & = 0\\
\sizeof{\asc{\ue}{\utau}} & = \sizeof{\ue}\\
\sizeof{\letsyn{\ux}{\ue_1}{\ue_2}} & = \sizeof{\ue_1} + \sizeof{\ue_2}\\
\sizeof{\lam{\ux}{\utau}{\ue}} &= \sizeof{\ue}\\
\sizeof{\ap{\ue_1}{\ue_2}} & = \sizeof{\ue_1} + \sizeof{\ue_2}\\
\sizeof{\Lam{\ut}{\ue}} & = \sizeof{\ue}\\
\sizeof{\App{\ue}{\utau}} & = \sizeof{\ue}\\
\sizeof{\fold{\ue}} & = \sizeof{\ue}\\
\sizeof{\unfold{\ue}} & = \sizeof{\ue}\\
%\end{align*}
%\begin{align*}
\sizeof{\tpl{\mapschema{\ue}{i}{\labelset}}} & = \sum_{i \in \labelset} \sizeof{\ue_i}\\
\sizeof{\prj{\ell}{\ue}} & = \sizeof{\ue}\\
\sizeof{\inj{\ell}{\ue}} & = \sizeof{\ue}\\
\sizeof{\caseof{\ue}{\mapschemab{\ux}{\ue}{i}{\labelset}}} & = \sizeof{\ue} + \sum_{i \in \labelset} \sizeof{\ue_i}\\
\sizeof{\uesyntax{\tsmv}{\utau}{\eparse}{\ue}} & = \sizeof{\ue}\\
\sizeof{\utsmap{\tsmv}{b}} & = \sizeof{b}\\
\LCC \color{light-gray} & \color{light-gray}\\
\sizeof{\matchwith{\ue}{\seqschemaX{\urv}}} & = \sizeof{\ue} + \sum_{1 \leq i \leq n} \sizeof{r_i}\\
\sizeof{\usyntaxup{\tsmv}{\utau}{\eparse}{\ue}} & = \sizeof{\ue}\ECC
\end{array}
\]
\vspace{-3px}\begin{grayparbox}\vspace{3px}and $\sizeof{\urv}$ computes the sum of the lengths of expression literal bodies in $\urv$:
\begin{align*}
\sizeof{\matchrule{\upv}{\ue}} & = \sizeof{\ue}
\end{align*}
Similarly, the metafunction $\sizeof{\upv}$ computes the sum of the lengths of the pattern literal bodies in $\upv$:
\begin{align*}
\sizeof{\ux} & = 0\\
\sizeof{\foldp{\upv}} & = \sizeof{\upv}\\
\sizeof{\tplp{\mapschema{\upv}{i}{\labelset}}} & = \sum_{i \in \labelset} \sizeof{\upv_i}\\
\sizeof{\injp{\ell}{\upv}} & = \sizeof{\upv}\\
\sizeof{\utsmap{\tsmv}{b}} & = \sizeof{b}
\end{align*}
\end{grayparbox}

\paragraph{Common Unexpanded Forms} Each expanded form maps onto an unexpanded form. We refer to these as the \emph{common forms}. In particular:
\begin{itemize}
\item Each type variable, $t$, maps onto a unique {type identifier}, written $\sigilof{t}$.
\item Each type, $\tau$, maps onto an unexpanded type, $\Uof{\tau}$, as follows: 
  \begin{align*}
  \Uof{t} &= \sigilof{t}\\
  \Uof{\aparr{\tau_1}{\tau_2}} & = \parr{\Uof{\tau_1}}{\Uof{\tau_2}}\\
  \Uof{\aall{t}{\tau}} & = \forallt{\sigilof{t}}{\Uof{\tau}}\\
  \Uof{\arec{t}{\tau}} & = \rect{\sigilof{t}}{\Uof{\tau}}\\
  \Uof{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}} & = \prodt{\mapschemax{\Uofv}{\tau}{i}{\labelset}}\\
  \Uof{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}} & = \sumt{\mapschemax{\Uofv}{\tau}{i}{\labelset}}
  \end{align*}
\item Each expression variable, $x$, maps onto a unique expression identifier, written $\sigilof{x}$.
\item Each expanded expression, $e$, maps onto an unexpanded expression, $\Uof{e}$, as follows:
\[\arraycolsep=1pt\begin{array}{rl}
\Uof{x} & = \sigilof{x}\\
\Uof{\aelam{\tau}{x}{e}} & = \lam{\sigilof{x}}{\Uof{\tau}}{\Uof{e}}\\
\Uof{\aeap{e_1}{e_2}} & = \ap{\Uof{e_1}}{\Uof{e_2}}\\
\Uof{\aetlam{t}{e}} & = \Lam{\sigilof{t}}{\Uof{e}}\\
\Uof{\aetap{e}{\tau}} & = \App{\Uof{e}}{\Uof{\tau}}\\
\Uof{\aefold{e}} & = \fold{\Uof e}\\
\Uof{\aeunfold{e}} & = \unfold{\Uof{e}}\\
\Uof{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}} & = \tpl{\mapschemax{\Uofv}{e}{i}{\labelset}}\\
\Uof{\aepr{\ell}{e}} & = \prj{\Uof{e}}{\ell}\\
\Uof{\aein{\ell}{e}} &= \inj{\ell}{\Uof{e}}\\
\LCC \color{light-gray} & \color{light-gray} \\
\Uof{\aematchwith{n}{e}{\seqschemaX{r}}} & = \matchwith{\Uof{e}}{\seqschemaXx{\Uofv}{r}}\ECC
\end{array}\]
\end{itemize}
\begin{grayparbox}
\begin{itemize}
\item Each expanded rule, $r$, maps onto an unexpanded rule, $\Uof{r}$, as follows:
\[\arraycolsep=1pt\begin{array}{rl}
\LCC \color{light-gray} & \color{light-gray} \\
\Uof{\aematchrule{p}{e}} & = \aumatchrule{\Uof{p}}{\Uof{e}}\ECC
\end{array}\]
\item Each expanded pattern, $p$, maps onto the unexpanded pattern, $\Uof{p}$, as follows:
\[\arraycolsep=1pt\begin{array}{rl}
\LCC \color{light-gray} & \color{light-gray} \\
\Uof{x} & = \sigilof{x}\\
\Uof{\aewildp} &= \auwildp\\
\Uof{\aefoldp{p}} &= \aufoldp{\Uof{p}}\\
\Uof{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}} & = \autplp{\labelset}{\mapschemax{\Uofv}{p}{i}{\labelset}}\\
\Uof{\aeinjp{\ell}{p}} & = \auinjp{\ell}{\Uof{p}}\ECC
\end{array}\]
\end{itemize}
\end{grayparbox}
\vspace{-10px}
\subsubsection{Textual Syntax}\vspace{-3px} In addition to the stylized syntax, there is also a context-free textual syntax for the UL. For our purposes, we need only posit the existence of partial metafunctions $\parseUTypF{b}$ and $\parseUExpF{b}$\graytxtbox{~and $\parseUPatF{b}$}. 

\begin{condition}[Textual Representability]\label{condition:textual-representability-SES} ~
\begin{enumerate}
\item For each $\utau$, there exists $b$ such that $\parseUTyp{b}{\utau}$. 
\item For each $\ue$, there exists $b$ such that $\parseUExp{b}{\ue}$.
% \item For each $\urv$, there exists $b$ such that $\parseURule{b}{\urv}$.
\item \graytxtbox{For each $\upv$, there exists $b$ such that $\parseUPat{b}{\upv}$.}
\end{enumerate}
\end{condition}

We also impose the following technical condition\graytxtbox{s}.

\begin{condition}[Expression Parsing Monotonicity]\label{condition:body-parsing} If $\parseUExp{b}{\ue}$ then $\sizeof{\ue} < \sizeof{b}$.\end{condition}

\begin{grayparbox}\begin{condition}[Pattern Parsing Monotonicity]\label{condition:pattern-parsing} If $\parseUPat{b}{\upv}$ then $\sizeof{\upv} < \sizeof{b}$.\end{condition}\end{grayparbox}

\subsection{Type Expansion}
\emph{Unexpanded type formation contexts}, $\uDelta$, are of the form $\uDD{\uD}{\Delta}$, i.e. they consist of a \emph{type identifier expansion context}, $\uD$, paired with a type formation context, $\Delta$. 

A \emph{type identifier expansion context}, $\uD$, is a finite function that maps each type identifier $\ut \in \domof{\uD}$ to the hypothesis $\vExpands{\ut}{t}$, for some type variable $t$. We write $\ctxUpdate{\uD}{\ut}{t}$ for the type identifier expansion context that maps $\ut$ to $\vExpands{\ut}{t}$ and defers to $\uD$ for all other type identifiers (i.e. the previous mapping is \emph{updated}.) 

We define $\uDelta, \uDhyp{\ut}{t}$ when $\uDelta=\uDD{\uD}{\Delta}$ as an abbreviation of  \[\uDD{\ctxUpdate{\uD}{\ut}{t}}{\Delta, \Dhyp{t}}\]%type identifier expansion context is always extended/updated together with 

\begin{definition}[Unexpanded Type Formation Context Formation] $\uDOK{\uDD{\uD}{\Delta}}$ iff for each $\uDhyp{\ut}{t} \in \uD$ we have $\Dhyp{t} \in \Delta$. \end{definition}

\vspace{10px}\noindent\fbox{\strut$\expandsTU{\uDelta}{\utau}{\tau}$}~~$\utau$ has well-formed expansion $\tau$
\begin{subequations}\label{rules:expandsTU}
\begin{equation}\label{rule:expandsTU-var}
\inferrule{ }{\expandsTU{\uDelta, \uDhyp{\ut}{t}}{\ut}{t}}
\end{equation}
\begin{equation}\label{rule:expandsTU-parr}
\inferrule{
  \expandsTU{\uDelta}{\utau_1}{\tau_1}\\
  \expandsTU{\uDelta}{\utau_2}{\tau_2}
}{\expandsTU{\uDelta}{\auparr{\utau_1}{\utau_2}}{\aparr{\tau_1}{\tau_2}}}
\end{equation}
\begin{equation}\label{rule:expandsTU-all}
  \inferrule{
    \expandsTU{\uDelta, \uDhyp{\ut}{t}}{\utau}{\tau}
  }{
    \expandsTU{\uDelta}{\auall{\ut}{\utau}}{\aall{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:expandsTU-rec}
  \inferrule{
    \expandsTU{\uDelta, \uDhyp{\ut}{t}}{\utau}{\tau}
  }{
    \expandsTU{\uDelta}{\aurec{\ut}{\utau}}{\arec{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:expandsTU-prod}
  \inferrule{
    \{\expandsTU{\uDelta}{\utau_i}{\tau_i}\}_{i \in \labelset}
  }{
    \expandsTU{\uDelta}{\auprod{\labelset}{\mapschema{\utau}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}
\begin{equation}\label{rule:expandsTU-sum}
  \inferrule{
    \{\expandsTU{\uDelta}{\utau_i}{\tau_i}\}_{i \in \labelset}
  }{
    \expandsTU{\uDelta}{\ausum{\labelset}{\mapschema{\utau}{i}{\labelset}}}{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}
\end{subequations}
% \emph{Unexpanded type formation contexts}, $\uDelta$, are of the form $\uDD{\uD}{\Delta}$, where $\uD$ is a \emph{type identifier expansion context}, and $\Delta$ is a type formation context. A type identifier expansion context, $\uD$, is a finite function that maps each type identifier $\ut \in \domof{\uD}$ to the hypothesis $\vExpands{\ut}{t}$, for some type variable $t$. We write $\ctxUpdate{\uD}{\ut}{t}$ for the type identifier expansion context that maps $\ut$ to $\vExpands{\ut}{t}$ and defers to $\uD$ for all other type identifiers (i.e. the previous mapping, if it exists, is updated). 
% We define $\uDelta, \uDhyp{\ut}{t}$ when $\uDelta=\uDD{\uD}{\Delta}$ as an abbreviation of  \[\uDD{\ctxUpdate{\uD}{\ut}{t}}{\Delta, \Dhyp{t}}\]%type identifier expansion context is always extended/updated together with 
% %We write $\uDeltaOK{\uDelta}$ when $\uDelta=\uDD{\uD}{\Delta}$ and each type variable in $\uD$ also appears in $\Delta$.
% %\begin{definition}\label{def:uDeltaOK} $\uDeltaOK{\uDD{\uD}{\Delta}}$ iff for each $\vExpands{\ut}{t} \in \uD$, we have $\Dhyp{t} \in \Delta$.\end{definition}

\subsection{Typed Expression Expansion}\label{appendix:typed-expression-expansion-S}
\subsubsection{Contexts}
\emph{Unexpanded typing contexts}, $\uGamma$, are, similarly, of the form $\uGG{\uG}{\Gamma}$, where $\uG$ is an \emph{expression identifier expansion context}, and $\Gamma$ is a typing context. An expression identifier expansion context, $\uG$, is a finite function that maps each expression identifier $\ux \in \domof{\uG}$ to the hypothesis $\vExpands{\ux}{x}$, for some expression variable, $x$. We write $\ctxUpdate{\uG}{\ux}{x}$ for the expression identifier expansion context that maps $\ux$ to $\vExpands{\ux}{x}$ and defers to $\uG$ for all other expression identifiers (i.e. the previous mapping is updated.) 
%We write $\uGammaOK{\uGamma}$ when $\uGamma=\uGG{\uG}{\Gamma}$ and each expression variable in $\uG$ is assigned a type by $\Gamma$.
%\noindent 

We define $\uGamma, \uGhyp{\ux}{x}{\tau}$ when $\uGamma = \uGG{\uG}{\Gamma}$ as an abbreviation of \[\uGG{\ctxUpdate{\uG}{\ux}{x}}{\Gamma, \Ghyp{x}{\tau}}\]

\begin{definition}[Unexpanded Typing Context Formation] $\uGammaOK{\uGG{\uG}{\Gamma}}$ iff $\isctxU{\Delta}{\Gamma}$ and for each $\vExpands{\ux}{x} \in \uG$, we have $x \in \domof{\Gamma}$.\end{definition}


\subsubsection{Body Encoding and Decoding}
An assumed type abbreviated $\tBody$ classifies encodings of literal bodies, $b$. The mapping from literal bodies to values of type $\tBody$ is defined by the \emph{body encoding judgement} $\encodeBody{b}{\ebody}$. An inverse mapping is defined   by the \emph{body decoding judgement} $\decodeBody{\ebody}{b}$.
\[\begin{array}{ll}
\textbf{Judgement Form} & \textbf{Description}\\
\encodeBody{b}{e} & \text{$b$ has encoding $e$}\\
\decodeBody{e}{b} & \text{$e$ has decoding $b$}
\end{array}\]
The following condition establishes an isomorphism between literal bodies and values of type $\tBody$ mediated by the judgements above.
\begin{condition}[Body Isomorphism]\label{condition:body-isomorphism} ~
\begin{enumerate}
\item For every literal body $b$, we have that $\encodeBody{b}{\ebody}$ for some $\ebody$ such that $\hastypeUC{\ebody}{\tBody}$ and $\isvalU{\ebody}$.
\item If $\hastypeUC{\ebody}{\tBody}$ and $\isvalU{\ebody}$ then $\decodeBody{\ebody}{b}$ for some $b$.
\item If $\encodeBody{b}{\ebody}$ then $\decodeBody{\ebody}{b}$.
\item If $\hastypeUC{\ebody}{\tBody}$ and $\isvalU{\ebody}$ and $\decodeBody{\ebody}{b}$ then $\encodeBody{b}{\ebody}$. 
\item If $\encodeBody{b}{\ebody}$ and $\encodeBody{b}{\ebody'}$ then $\ebody = \ebody'$.
\item If $\hastypeUC{\ebody}{\tBody}$ and $\isvalU{\ebody}$ and $\decodeBody{\ebody}{b}$ and $\decodeBody{\ebody}{b'}$ then $b=b'$.
\end{enumerate}
\end{condition}
We also assume a partial metafunction, $\bsubseq{b}{m}{n}$, which extracts a subsequence of $b$ starting at position $m$ and ending at position $n$, inclusive, where $m$ and $n$ are natural numbers. The following condition is technically necessary.
\begin{condition}[Body Subsequencing]\label{condition:body-subsequences} If $\bsubseq{b}{m}{n}=b'$ then $\sizeof{b'} \leq \sizeof{b}$. \end{condition}

\subsubsection{Parse Results}
 The type abbreviated $\tParseResultExp$, and an auxiliary abbreviation used below, is defined as follows:
\begin{align*}
L_\mathtt{SE} & \defeq \lbltxt{ParseError}, \lbltxt{SuccessE}\\
\tParseResultExp & \defeq \asum{L_\mathtt{SE}}{
  \mapitem{\lbltxt{ParseError}}{\prodt{}}, 
  \mapitem{\lbltxt{SuccessE}}{\tCEExp}
}\\
\end{align*} %[\mapitem{\lbltxt{ParseError}}{\prodt{}}, \mapitem{\lbltxt{SuccessE}}{\tCEExp}]

\begin{grayparbox}
 The type abbreviated $\tParseResultPat$, and an auxiliary abbreviation used below, is defined as follows:
\begin{align*}
L_\mathtt{SP} & \defeq \lbltxt{ParseError}, \lbltxt{SuccessP}\\
\tParseResultExp & \defeq \asum{L_\mathtt{SP}}{
  \mapitem{\lbltxt{ParseError}}{\prodt{}}, 
  \mapitem{\lbltxt{SuccessP}}{\tCEPat}
}\\
\end{align*} %[\mapitem{\lbltxt{ParseError}}{\prodt{}}, \mapitem{\lbltxt{SuccessE}}{\tCEExp}]
\end{grayparbox}

\subsubsection{seTLM Contexts}

\emph{seTLM contexts}, $\uPsi$, are of the form $\uAS{\uA}{\Psi}$, where $\uA$ is a \emph{TLM identifier expansion context} and $\Psi$ is a \emph{seTLM definition context}. 

A \emph{TLM identifier expansion context}, $\uA$, is a finite function mapping each TLM identifier $\tsmv \in \domof{\uA}$ to the \emph{TLM identifier expansion}, $\vExpands{\tsmv}{a}$, for some \emph{TLM name}, $a$. We write $\ctxUpdate{\uA}{\tsmv}{a}$ for the TLM identifier expansion context that maps $\tsmv$ to $\vExpands{\tsmv}{a}$, and defers to $\uA$ for all other TLM identifiers (i.e. the previous mapping is \emph{updated}.)

An \emph{seTLM definition context}, $\Psi$, is a finite function mapping each TLM name $a \in \domof{\Psi}$ to an \emph{expanded seTLM definition}, $\xuetsmbnd{a}{\tau}{\eparse}$, where $\tau$ is the seTLM's type annotation, and $\eparse$ is its parse function. We write $\Psi, \xuetsmbnd{a}{\tau}{\eparse}$ when $a \notin \domof{\Psi}$ for the extension of $\Psi$ that maps $a$ to $\xuetsmbnd{a}{\tau}{\eparse}$. We write $\uetsmenv{\Delta}{\Psi}$  when all the type annotations in $\Psi$ are well-formed assuming $\Delta$, and the parse functions in $\Psi$ are closed and of the appropriate type.

\begin{definition}[seTLM Definition Context Formation]\label{def:seTLM-def-ctx-formation} $\uetsmenv{\Delta}{\Psi}$ iff for each $\xuetsmbnd{a}{\tau}{\eparse} \in \Psi$, we have $\istypeU{\Delta}{\tau}$ and $\hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultExp}}$.\end{definition}

\begin{definition}[seTLM Context Formation] $\uetsmctx{\Delta}{\uAS{\uA}{\Psi}}$ iff $\uetsmenv{\Delta}{\Psi}$ and for each $\vExpands{\tsmv}{a} \in \uA$ we have $a \in \domof{\Psi}$.
\end{definition}

We define $\uPsi, \uShyp{\tsmv}{a}{\tau}{\eparse}$, when $\uPsi=\uAS{\uA}{\Phi}$, as an abbreviation of \[\uAS{\ctxUpdate{\uA}{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}\]
%\vspace{10px}

\begin{grayparbox}\vspace{-15px}\subsubsection{spTLM Contexts}
\emph{spTLM contexts}, $\uPhi$, are of the form $\uAS{\uA}{\Phi}$, where $\uA$ is a {TLM identifier expansion context}, defined above, and $\Phi$ is a \emph{spTLM definition context}. 

An \emph{spTLM definition context}, $\Phi$, is a finite function mapping each TLM name $a \in \domof{\Phi}$ to an \emph{expanded seTLM definition}, $\xuptsmbnd{a}{\tau}{\eparse}$, where $\tau$ is the spTLM's type annotation, and $\eparse$ is its parse function. We write $\Phi, \xuptsmbnd{a}{\tau}{\eparse}$ when $a \notin \domof{\Phi}$ for the extension of $\Phi$ that maps $a$ to $\xuptsmbnd{a}{\tau}{\eparse}$. We write $\uptsmenv{\Delta}{\Phi}$  when all the type annotations in $\Phi$ are well-formed assuming $\Delta$, and the parse functions in $\Phi$ are closed and of the appropriate type.

\begin{definition}[spTLM Definition Context Formation]\label{def:spTLM-def-ctx-formation} $\uptsmenv{\Delta}{\Phi}$ iff for each $\xuptsmbnd{a}{\tau}{\eparse} \in \Phi$, we have $\istypeU{\Delta}{\tau}$ and $\hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPat}}$.\end{definition}

\begin{definition}[spTLM Context Formation] $\uptsmctx{\Delta}{\uAS{\uA}{\Phi}}$ iff $\uptsmenv{\Delta}{\Phi}$ and for each $\vExpands{\tsmv}{a} \in \uA$ we have $a \in \domof{\Phi}$.
\end{definition}

We define $\uPhi, \uPhyp{\tsmv}{a}{\tau}{\eparse}$, when $\uPhi=\uAS{\uA}{\Phi}$, as an abbreviation of \[\uAS{\ctxUpdate{\uA}{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}\]
\end{grayparbox}

\subsubsection{Typed Expression Expansion}\label{appendix:typed-expression-expansion-SES}
\vspace{8px}\noindent\fbox{\strut$\expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}$}~~$\ue$ has expansion $e$ of type $\tau$
\begin{subequations}\label{rules:expandsU}
\begin{equation}\label{rule:expandsU-var}
  \inferrule{ }{
    \expandsSG{\uDelta}{\uGamma, \uGhyp{\ux}{x}{\tau}}{\uPsi}{\uPhi}{\ux}{x}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-asc}
  \inferrule{
    \expandsTU{\uDelta}{\utau}{\tau}\\
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}
  }{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\asc{\ue}{\utau}}{e}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-letsyn}
  \inferrule{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue_1}{e_1}{\tau_1}\\
    \expandsSG{\uDelta}{\uGamma, \uGhyp{\ux}{x}{\tau_1}}{\uPsi}{\uPhi}{\ue_2}{e_2}{\tau_2}
  }{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\letsyn{\ux}{\ue_1}{\ue_2}}{
      \aeap{\aelam{\tau_1}{x}{e_2}}{e_1}
    }{\tau_2}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-lam}
  \inferrule{
    \expandsTU{\uDelta}{\utau}{\tau}\\
    \expandsSG{\uDelta}{\uGamma, \uGhyp{\ux}{x}{\tau}}{\uPsi}{\uPhi}{\ue}{e}{\tau'}
  }{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\lam{\ux}{\utau}{\ue}}{\aelam{\tau}{x}{e}}{\aparr{\tau}{\tau'}}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-ap}
  \inferrule{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue_1}{e_1}{\aparr{\tau}{\tau'}}\\
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue_2}{e_2}{\tau}
  }{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ap{\ue_1}{\ue_2}}{\aeap{e_1}{e_2}}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-tlam}
  \inferrule{
    \expandsSG{\uDelta, \uDhyp{\ut}{t}}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}
  }{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\Lam{\ut}{\ue}}{\aetlam{t}{e}}{\aall{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-tap}
  \inferrule{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\aall{t}{\tau}}\\
    \expandsTU{\uDelta}{\utau'}{\tau'}
  }{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\App{\ue}{\utau'}}{\aetap{e}{\tau'}}{[\tau'/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-fold}
  \inferrule{
    % \istypeU{\Delta, \Dhyp{t}}{\tau}\\
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{[\arec{t}{\tau}/t]\tau}
  }{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\fold{\ue}}{\aefold{e}}{\arec{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-unfold}
  \inferrule{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\arec{t}{\tau}}
  }{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\unfold{\ue}}{\aeunfold{e}}{[\arec{t}{\tau}/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-tpl}
  \inferrule{
    \{\expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue_i}{e_i}{\tau_i}\}_{i \in \labelset}
  }{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\tpl{\mapschema{\ue}{i}{\labelset}}}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-pr}
  \inferrule{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\aprod{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \ell \hookrightarrow \tau}}
  }{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\prj{\ue}{\ell}}{\aepr{\ell}{e}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-in}
  \inferrule{
   % \{\istypeU{\Delta}{\tau_i}\}_{i \in \labelset}\\
    % \istypeU{\Delta}{\tau'}\\
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau'}
  }{
    % \left(\shortstack{
    %   $\uDelta~\uGamma~{\vdash_{\uPhi}}{\setlength{\fboxsep}{0px}\colorbox{light-gray}{$_{\mathstrut; \uPsi}$}}~ \inj{\ell}{\ue}$\\
    %   $\leadsto$\\
    %   $\aein{\ell}{e} : \asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \ell \hookrightarrow \tau}$\vspace{-1.2em}}\right)
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\inj{\ell}{\ue}}{\aein{\ell}{e}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \ell \hookrightarrow \tau'}}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-case}
  \inferrule{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}\\
    % \istypeU{\Delta}{\tau}\\
    \{\expandsSG{\uDelta}{\uGamma, \uGhyp{\ux_i}{x_i}{\tau_i}}{\uPsi}{\uPhi}{\ue_i}{e_i}{\tau}\}_{i \in \labelset}
  }{
    \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\caseof{\ue}{\mapschemab{\ux}{\ue}{i}{\labelset}}}{\aecase{\labelset}{e}{\mapschemab{x}{e}{i}{\labelset}}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:expandsU-syntax}
\inferrule{
  \expandsTU{\uDelta}{\utau}{\tau}\\
  \hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultExp}}\\\\
  \evalU{\eparse}{\eparse'}\\
  \expandsU{\uDelta}{\uGamma}{\uPsi, \uShyp{\tsmv}{a}{\tau}{\eparse'}}{\ue}{e}{\tau'}
}{
  \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\uesyntax{\tsmv}{\utau}{\eparse}{\ue}}{e}{\tau'}
}
\end{equation}
\begin{equation}\label{rule:expandsU-tsmap}
\inferrule{
  \uPsi = \uPsi', \uShyp{\tsmv}{a}{\tau}{\eparse}\\\\
  \encodeBody{b}{\ebody}\\
 \evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessE}}{\ecand}}\\
   \decodeCondE{\ecand}{\ce}\\\\
    \segOK{\segof{\ce}}{b}\\
  \cvalidE{\emptyset}{\emptyset}{\esceneSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}
}{
  \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\utsmap{\tsmv}{b}}{e}{\tau}
}
\end{equation}
\begin{grayparbox}
\begin{equation}\label{rule:expandsU-match}
\inferrule{
  \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}\\
  % \istypeU{\Delta}{\tau'}\\
  \{\ruleExpands{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\urv_i}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}\\
}{
  \expandsSG
    {\uDelta}{\uGamma}{\uPsi}{\uPhi}
    {\matchwith
      {\ue}
      {\seqschemaX{\urv}}
    }{\aematchwith
      {n}
      {e}
      {\seqschemaX{r}}
    }{\tau'}
}
\end{equation}
\begin{equation}\label{rule:expandsU-defuptsm}
\graybox{\inferrule{
  \expandsTU{\uDelta}{\utau}{\tau}\\
  \hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPat}}\\\\
  \evalU{\eparse}{\eparse'}\\
  \expandsUP{\uDelta}{\uGamma}{\uPsi}{\uPhi, \uPhyp{\tsmv}{a}{\tau}{\eparse'}}{\ue}{e}{\tau'}
}{
  \expandsUPX{\usyntaxup{\tsmv}{\utau}{\eparse}{\ue}}{e}{\tau'}
}}
\end{equation}
\end{grayparbox}
\end{subequations}

% \begin{subequations}\label{rules:expandsU}
% Rules (\ref*{rule:expandsU-var}) through (\ref*{rule:expandsU-case}) handle unexpanded expressions of common form. The first five of these rules are defined below:
% %Each of these rules is based on the corresponding typing rule, i.e. Rules (\ref{rule:hastypeU-var}) through (\ref{rule:hastypeU-case}), respectively. For example, the following typed expansion rules are based on the typing rules (\ref{rule:hastypeU-var}), (\ref{rule:hastypeU-lam}) and (\ref{rule:hastypeU-ap}), respectively:% for unexpanded expressions of variable, function and application form, respectively: 
% \begin{equation}\label{rule:expandsU-var}
%   \inferrule{ }{\expandsU{\uDelta}{\uGamma, \uGhyp{\ux}{x}{\tau}}{\uPsi}{\ux}{x}{\tau}}
% \end{equation}
% \begin{equation}\label{rule:expandsU-lam}
%   \inferrule{
%     \expandsTU{\uDelta}{\utau}{\tau}\\
%     \expandsU{\uDelta}{\uGamma, \uGhyp{\ux}{x}{\tau}}{\uPsi}{\ue}{e}{\tau'}
%   }{\expandsUX{\aulam{\utau}{\ux}{\ue}}{\aelam{\tau}{x}{e}}{\aparr{\tau}{\tau'}}}
% \end{equation}
% \begin{equation}\label{rule:expandsU-ap}
%   \inferrule{
%     \expandsUX{\ue_1}{e_1}{\aparr{\tau}{\tau'}}\\
%     \expandsUX{\ue_2}{e_2}{\tau}
%   }{
%     \expandsUX{\auap{\ue_1}{\ue_2}}{\aeap{e_1}{e_2}}{\tau'}
%   }
% \end{equation}
% \begin{equation}\label{rule:expandsU-tlam}
%   \inferrule{
%     \expandsU{\uDelta, \uDhyp{\ut}{t}}{\uGamma}{\uPsi}{\ue}{e}{\tau}
%   }{
%     \expandsUX{\autlam{\ut}{\ue}}{\aetlam{t}{e}}{\aall{t}{\tau}}
%   }
% \end{equation}
% \begin{equation}\label{rule:expandsU-tap}
%   \inferrule{
%     \expandsUX{\ue}{e}{\aall{t}{\tau}}\\
%     \expandsTU{\uDelta}{\utau'}{\tau'}
%   }{
%     \expandsUX{\autap{\ue}{\utau'}}{\aetap{e}{\tau'}}{[\tau'/t]\tau}
%   }
% \end{equation}
% Observe that, in each of these rules, the unexpanded and expanded expression forms in the conclusion correspond, and the premises correspond to those of the typing rule for the expanded expression form, i.e. Rules (\ref{rule:hastypeU-var}) through (\ref{rule:hastypeU-tap}), respectively. In particular, each type expansion premise in each rule above corresponds to a  type formation premise in the corresponding typing rule, and each typed expression expansion premise in each rule above corresponds to a typing premise in the corresponding typing rule. The type assigned in the conclusion of each rule above is identical to the type assigned in the conclusion of the corresponding typing rule. The ueTLM context, $\uPsi$, passes opaquely through these rules (we will define ueTLM contexts below). Rules (\ref{rules:expandsTU}) were similarly generated by mechanically transforming Rules (\ref{rules:istypeU}).

% We can express this scheme more precisely with the following rule transformation. For each rule in Rules (\ref{rules:istypeU}) and Rules (\ref{rules:hastypeU}),
% \begin{mathpar}
% \refstepcounter{equation}
% % \label{rule:expandsU-tlam}
% % \refstepcounter{equation}
% % \label{rule:expandsU-tap}
% % \refstepcounter{equation}
% \label{rule:expandsU-fold}
% \refstepcounter{equation}
% \label{rule:expandsU-unfold}
% \refstepcounter{equation}
% \label{rule:expandsU-tpl}
% \refstepcounter{equation}
% \label{rule:expandsU-pr}
% \refstepcounter{equation}
% \label{rule:expandsU-in}
% \refstepcounter{equation}
% \label{rule:expandsU-case}
% \inferrule{J_1\\ \cdots \\ J_k}{J}
% \end{mathpar}
% the corresponding typed expansion rule is 
% \begin{mathpar}
% \inferrule{
%   \Uof{J_1} \\
%   \cdots\\
%   \Uof{J_k}
% }{
%   \Uof{J}
% }
% \end{mathpar}
% where
% \[\begin{split}
% \Uof{\istypeU{\Delta}{\tau}} & = \expandsTU{\Uof{\Delta}}{\Uof{\tau}}{\tau} \\
% \Uof{\hastypeU{\Gamma}{\Delta}{e}{\tau}} & = \expandsU{\Uof{\Gamma}}{\Uof{\Delta}}{\uPsi}{\Uof{e}}{e}{\tau}\\
% \Uof{\{J_i\}_{i \in \labelset}} & = \{\Uof{J_i}\}_{i \in \labelset}
% \end{split}\]
% and where:
% \begin{itemize}
% \item $\Uof{\tau}$ is defined as follows:
%   \begin{itemize}
%   \item When $\tau$ is of definite form, $\Uof{\tau}$ is defined as in Sec. \ref{sec:syntax-U}.
%   \item When $\tau$ is of indefinite form, $\Uof{\tau}$ is a uniquely corresponding metavariable of sort $\mathsf{UTyp}$ also of indefinite form. For example, in Rule (\ref{rule:istypeU-parr}), $\tau_1$ and $\tau_2$ are of indefinite form, i.e. they match arbitrary types. The rule transformation simply ``hats'' them, i.e. $\Uof{\tau_1}=\utau_1$ and $\Uof{\tau_2}=\utau_2$.
%   \end{itemize}
% \item $\Uof{e}$ is defined as follows
% \begin{itemize}
% \item When $e$ is of definite form, $\Uof{e}$ is defined as in Sec. \ref{sec:syntax-U}. 
% \item When $e$ is of indefinite form, $\Uof{e}$ is a uniquely corresponding metavariable of sort $\mathsf{UExp}$ also of indefinite form. For example, $\Uof{e_1}=\ue_1$ and $\Uof{e_2}=\ue_2$.
% \end{itemize}
% \item $\Uof{\Delta}$ is defined as follows:
%   \begin{itemize} 
%   \item When $\Delta$ is of definite form, $\Uof{\Delta}$ is defined as above.
%   \item When $\Delta$ is of indefinite form, $\Uof{\Delta}$ is a uniquely corresponding metavariable ranging over unexpanded type formation contexts. For example, $\Uof{\Delta} = \uDelta$.
%   \end{itemize}
% \item $\Uof{\Gamma}$ is defined as follows:
%   \begin{itemize}
%   \item When $\Gamma$ is of definite form, $\Uof{\Gamma}$ produces the corresponding unexpanded typing context as follows:
% \begin{align*}
% \Uof{\emptyset} & = \uGG{\emptyset}{\emptyset}\\
% \Uof{\Gamma, \Ghyp{x}{\tau}} & = \Uof{\Gamma}, \uGhyp{\sigilof{x}}{x}{\tau}
% \end{align*}
%   \item When $\Gamma$ is of indefinite form, $\Uof{\Gamma}$ is a uniquely corresponding metavariable ranging over unexpanded typing contexts. For example, $\Uof{\Gamma} = \uGamma$.
% \end{itemize}
% \end{itemize}

% It is instructive to use this rule transformation to generate Rules (\ref{rules:expandsTU}) and Rules (\ref{rule:expandsU-var}) through (\ref{rule:expandsU-tap}) above. We omit the remaining rules, i.e. Rules (\ref*{rule:expandsU-fold}) through (\ref*{rule:expandsU-case}). By instead defining these rules solely by the rule transformation just described, we avoid having to write down a number of rules that are of limited marginal interest. Moreover, this demonstrates the general technique for generating typed expansion rules for unexpanded types and expressions of common form, so our exposition is somewhat ``robust'' to changes to the inner core. 
\vspace{-5px}\begin{grayparbox}
\vspace{8px}
\noindent\fcolorbox{black}{light-gray}{\strut$\ruleExpands{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\urv}{r}{\tau}{\tau'}$}~~$\urv$ has expansion $r$ taking values of type $\tau$ to values of type $\tau'$
\begin{equation}\label{rule:ruleExpands}
\graybox{\inferrule{
  \patExpands{\uAS{\uG'}{\pctx'}}{\uPhi}{\upv}{p}{\tau}\\
  \expandsUP{\uDelta}{\uGG{\uGcons{\uG}{\uG'}}{\Gcons{\Gamma}{\pctx'}}}{\uPsi}{\uPhi}{\ue}{e}{\tau'} 
}{
  \ruleExpands{\uDelta}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\aumatchrule{\upv}{\ue}}{\aematchrule{p}{e}}{\tau}{\tau'}
}}
\end{equation}

% Rule (\ref{rule:ruleExpands}) is defined mutually with Rules (\ref{rules:expandsU}).

\subsubsection{Typed Pattern Expansion}
% \vspace{8px}
\noindent\fcolorbox{black}{light-gray}{\strut$\patExpands{\uGamma}{\uPhi}{\upv}{p}{\tau}$}~~$\upv$ has expansion $p$ matching against $\tau$ generating hypotheses $\uGamma$
\begin{subequations}\label{rules:patExpands}
\begin{equation}\label{rule:patExpands-var}
\graybox{\inferrule{ }{
  \patExpands{\uGG{\vExpands{\ux}{x}}{\Ghyp{x}{\tau}}}{\uPhi}{\ux}{x}{\tau}
}}
\end{equation}
\begin{equation}\label{rule:patExpands-wild}
\graybox{\inferrule{ }{
  \patExpands{\uGG{\emptyset}{\emptyset}}{\uPhi}{\wildp}{\aewildp}{\tau}
}}
\end{equation}
\begin{equation}\label{rule:patExpands-fold}
\graybox{\inferrule{ 
  \patExpands{\upctx}{\uPhi}{\upv}{p}{[\arec{t}{\tau}/t]\tau}
}{
  \patExpands{\upctx}{\uPhi}{\foldp{\upv}}{\aefoldp{p}}{\arec{t}{\tau}}
}}
\end{equation}
\begin{equation}\label{rule:patExpands-tpl}
\graybox{
  \inferrule{
    \tau = \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}\\\\
    \{\patExpands{{\upctx_i}}{\uPhi}{\upv_i}{p_i}{\tau_i}\}_{i \in \labelset}
  }{
    % \left(\shortstack{
    %   $\Delta \vdash_{\uPhi} \tplp{\mapschema{\upv}{i}{\labelset}}$\\
    %   $\leadsto$\\
    %   $\aetplp{\labelset}{\mapschema{p}{i}{\labelset}} : \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$\vspace{-1.2em}}\right)
    \patExpands{\GIconsi{i \in \labelset}{\upctx_i}}{\uPhi}{\tplp{\mapschema{\upv}{i}{\labelset}}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\tau}
  }
}
% \graybox{\inferrule{
%   \{\patExpands{{\upctx_i}}{\uPhi}{\upv_i}{p_i}{\tau_i}\}_{i \in \labelset}\\
% }{
%   % \patExpands{\Gconsi{i \in \labelset}{\pctx_i}}{\Phi}{
%   %   \autplp{\labelset}{\mapschema{\upv}{i}{\labelset}}
%   % }{
%   %   \aetplp{\labelset}{\mapschema{p}{i}{\labelset}}
%   % }{
%   %   \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}
%   % } %{\autplp{\labelset}{\mapschema{\upv}{i}{\labelset}}}{\aetplp{\labelset}{\mapschema}{p}{i}{\labelset}}{...}
%   \left(\shortstack{$\Delta \vdash_{\uPhi} \autplp{\labelset}{\mapschema{\upv}{i}{\labelset}}$\\$\leadsto$\\$\aetplp{\labelset}{\mapschema{p}{i}{\labelset}} : \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}} \dashV \Gconsi{i \in \labelset}{\upctx_i}$\vspace{-1.2em}}\right)
% }}
\end{equation}
\begin{equation}\label{rule:patExpands-in}
\graybox{\inferrule{
  \patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}
}{
  \patExpands{\upctx}{\uPhi}{\injp{\ell}{\upv}}{\aeinjp{\ell}{p}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
}}
\end{equation}
\begin{equation}\label{rule:patExpands-apuptsm}
\graybox{\inferrule{
  \uPhi = \uPhi', \uPhyp{\tsmv}{a}{\tau}{\eparse}\\\\
  \encodeBody{b}{\ebody}\\
  \evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessP}}{\ecand}}\\
  \decodeCEPat{\ecand}{\cpv}\\\\
    \segOK{\segof{\cpv}}{b}\\
  \cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}
}{
  \patExpands{\upctx}{\uPhi}{\utsmap{\tsmv}{b}}{p}{\tau}
}}
\end{equation}

\end{subequations}

In Rule (\ref{rule:patExpands-tpl}), $\upctx_i$ is shorthand for $\uGG{\uG_i}{\pctx_i}$ and $\GIconsi{i \in \labelset}{\upctx_i}$ is shorthand for \[\uGG{\GIconsi{i \in \labelset}{\uG_i}}{\Gconsi{i \in \labelset}{\pctx_i}}\] 
\end{grayparbox}


% \end{subequations}
% \clearpage
\section{Proto-Expansion Validation}\label{appendix:proto-expansions-SES}
\subsection{Syntax of Proto-Expansions}
$\arraycolsep=2pt\begin{array}{lllllll}
\textbf{Sort} & & & \textbf{Operational Form} & \textbf{Stylized Form} & \textbf{Description}\\
\mathsf{PrTyp} & \ctau & ::= & t & t & \text{variable}\\
&&& \aceparr{\ctau}{\ctau} & \parr{\ctau}{\ctau} & \text{partial function}\\
&&& \aceall{t}{\ctau} & \forallt{t}{\ctau} & \text{polymorphic}\\
&&& \acerec{t}{\ctau} & \rect{t}{\ctau} & \text{recursive}\\
&&& \aceprod{\labelset}{\mapschema{\ctau}{i}{\labelset}} & \prodt{\mapschema{\ctau}{i}{\labelset}} & \text{labeled product}\\
&&& \acesum{\labelset}{\mapschema{\ctau}{i}{\labelset}} & \sumt{\mapschema{\ctau}{i}{\labelset}} & \text{labeled sum}\\
% \LCC &&& \color{light-gray} & \color{light-gray} & \color{light-gray}\\
&&& \acesplicedt{m}{n} & \splicedt{m}{n} & \text{spliced type ref.}\\%\ECC
\mathsf{PrExp} & \ce & ::= & x & x & \text{variable}\\
&&& \aceasc{\ctau}{\ce} & \asc{\ce}{\ctau} & \text{ascription}\\
&&& \aceletsyn{x}{\ce}{\ce} & \letsyn{x}{\ce}{\ce} & \text{value binding}\\
&&& \acelam{\ctau}{x}{\ce} & \lam{x}{\ctau}{\ce} & \text{abstraction}\\
&&& \aceap{\ce}{\ce} & \ap{\ce}{\ce} & \text{application}\\
&&& \acetlam{t}{\ce} & \Lam{t}{\ce} & \text{type abstraction}\\
&&& \acetap{\ce}{\ctau} & \App{\ce}{\ctau} & \text{type application}\\
&&& \acefold{\ce} & \fold{\ce} & \text{fold}\\
&&& \aceunfold{\ce} & \unfold{\ce} & \text{unfold}\\
&&& \acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}} & \tpl{\mapschema{\ce}{i}{\labelset}} & \text{labeled tuple}\\
&&& \acepr{\ell}{\ce} & \prj{\ce}{\ell} & \text{projection}\\
&&& \acein{\ell}{\ce} & \inj{\ell}{\ce} & \text{injection}\\
&&& \acecase{\labelset}{\ce}{\mapschemab{x}{\ce}{i}{\labelset}} & \caseof{\ce}{\mapschemab{x}{\ce}{i}{\labelset}} & \text{case analysis}\\
&&& \acesplicede{m}{n}{\ctau} & \splicede{m}{n}{\ctau} & \text{spliced expr. ref.}\\
\LCC \color{light-gray} &\color{light-gray} & \color{light-gray}& \color{light-gray} & \color{light-gray} & \color{light-gray}\\
&&& \acematchwith{n}{\ce}{\seqschemaX{\crv}} & \matchwith{\ce}{\seqschemaX{\crv}} & \text{match}\\
\mathsf{PrRule} & \crv & ::= & \acematchrule{p}{\ce} & \matchrule{p}{\ce} & \text{rule}\\
\mathsf{PrPat} & \cpv & ::= & \acewildp & \wildp & \text{wildcard pattern}\\
&&& \acefoldp{\cpv} & \foldp{\cpv} & \text{fold pattern}\\
&&& \acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}} & \tplp{\mapschema{\cpv}{i}{\labelset}} & \text{labeled tuple pattern}\\
&&& \aceinjp{\ell}{\cpv} & \injp{\ell}{\cpv} & \text{injection pattern}\\
% \LCC &&& \color{Yellow} & \color{Yellow} & \color{Yellow}\\
&&& \acesplicedp{m}{n}{\ctau} & \splicedp{m}{n}{\ctau} & \text{spliced pattern ref.}\ECC
\end{array}$

\subsubsection{Common Proto-Expansion Terms} Each expanded term\graytxtbox{, except variable patterns,} maps onto a proto-expansion term. We refer to these as the \emph{common proto-expansion terms}. In particular:
\begin{itemize}
  \item Each type, $\tau$, maps onto a proto-type, $\Cof{\tau}$, as follows:
  \[\arraycolsep=1pt\begin{array}{rl}
  \Cof{t} & = t\\
  \Cof{\aparr{\tau_1}{\tau_2}} & = \aceparr{\Cof{\tau_1}}{\Cof{\tau_2}}\\
  \Cof{\aall{t}{\tau}} & = \aceall{t}{\Cof{\tau}}\\
  \Cof{\arec{t}{\tau}} & = \acerec{t}{\Cof{\tau}}\\
  \Cof{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}} & = \aceprod{\labelset}{\mapschemax{\Cofv}{\tau}{i}{\labelset}}\\
  \Cof{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}} & = \acesum{\labelset}{\mapschemax{\Cofv}{\tau}{i}{\labelset}}
  \end{array}\]
  \item Each expanded expression, $e$, maps onto a proto-expression, $\Cof{e}$, as follows:
  \[\arraycolsep=1pt\begin{array}{rl}
  \Cof{x} & = x\\
  \Cof{\aelam{\tau}{x}{e}} & = \acelam{\Cof{\tau}}{x}{\Cof{e}}\\
  \Cof{\aeap{e_1}{e_2}} & = \aceap{\Cof{e_1}}{\Cof{e_2}}\\
  \Cof{\aetlam{t}{e}} & = \acetlam{t}{\Cof{e}}\\
  \Cof{\aetap{e}{\tau}} & = \acetap{\Cof{e}}{\Cof{\tau}}\\
  \Cof{\aefold{e}} & = \acefold{\Cof e}\\
  \Cof{\aeunfold{e}} & = \aceunfold{\Cof{e}}\\
  \Cof{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}} & = \acetpl{\labelset}{\mapschemax{\Cofv}{e}{i}{\labelset}}\\
  \Cof{\aein{\ell}{e}} &= \acein{\ell}{\Cof{e}}\\
  \LCC \color{light-gray} & \color{light-gray} \\
  \Cof{\aematchwith{n}{e}{\seqschemaX{r}}} & = \acematchwith{n}{\Cof{e}}{\seqschemaXx{\Cofv}{r}} \ECC
  \end{array}\]
  \end{itemize}
  \begin{grayparbox}
  \begin{itemize}
  \item Each expanded rule, $r$, maps onto the proto-rule, $\Cof{r}$, as follows:
  \begin{align*}
  \Cof{\aematchrule{p}{e}} & = \acematchrule{p}{\Cof{e}}
  \end{align*}
  Notice that proto-rules bind expanded patterns, not proto-patterns. This is because proto-rules appear in proto-expressions, which are generated by seTLMs. It would not be sensible for an seTLM to splice a pattern out of a literal body.
  \item Each expanded pattern, $p$, except for the variable patterns, maps onto a proto-pattern, $\Cof{p}$, as follows:
  \begin{align*}
  \Cof{\aewildp} & = \acewildp\\
  \Cof{\aefoldp{p}} & = \acefoldp{\Cof{p}}\\
  \Cof{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}} & = \acetplp{\labelset}{\mapschemax{\Cofv}{p}{i}{\labelset}}\\
  \Cof{\aeinjp{\ell}{p}} & = \aceinjp{\ell}{\Cof{p}}
  \end{align*}
\end{itemize}
\end{grayparbox}

\subsubsection{Proto-Expression Encoding and Decoding}
The type abbreviated $\tCEExp$ classifies encodings of \emph{proto-expressions}. The mapping from proto-expressions to values of type $\tCEExp$ is defined by the \emph{proto-expression encoding judgement}, $\encodeCondE{\ce}{e}$. An inverse mapping is defined by the \emph{proto-expression decoding judgement}, $\decodeCondE{e}{\ce}$.

\[\begin{array}{ll}
\textbf{Judgement Form} & \textbf{Description}\\
\encodeCondE{\ce}{e} & \text{$\ce$ has encoding $e$}\\
\decodeCondE{e}{\ce} & \text{$e$ has decoding $\ce$}
\end{array}\]

Rather than picking a particular definition of $\tCEExp$ and defining the judgements above inductively against it, we only state the following condition, which establishes an isomorphism between values of type $\tCEExp$ and proto-expressions.

\begin{condition}[Proto-Expression Isomorphism]\label{condition:proto-expression-isomorphism} ~
\begin{enumerate}
\item For every $\ce$, we have $\encodeCondE{\ce}{\ecand}$ for some $\ecand$ such that $\hastypeUC{\ecand}{\tCEExp}$ and $\isvalU{\ecand}$.
\item If $\hastypeUC{\ecand}{\tCEExp}$ and $\isvalU{\ecand}$ then $\decodeCondE{\ecand}{\ce}$ for some $\ce$.
\item If $\encodeCondE{\ce}{\ecand}$ then $\decodeCondE{\ecand}{\ce}$.
\item If $\hastypeUC{\ecand}{\tCEExp}$ and $\isvalU{\ecand}$ and $\decodeCondE{\ecand}{\ce}$ then $\encodeCondE{\ce}{\ecand}$.
\item If $\encodeCondE{\ce}{\ecand}$ and $\encodeCondE{\ce}{\ecand'}$ then $\ecand=\ecand'$.
\item If $\hastypeUC{\ecand}{\tCEExp}$ and $\isvalU{\ecand}$ and $\decodeCondE{\ecand}{\ce}$ and $\decodeCondE{\ecand}{\ce'}$ then $\ce=\ce'$.
\end{enumerate}
\end{condition}\vspace{10px}

\begin{grayparbox}\vspace{-16px}
\subsubsection{Proto-Pattern Encoding and Decoding}
The type abbreviated $\tCEPat$ classifies encodings of \emph{proto-patterns}. The mapping from proto-patterns to values of type $\tCEPat$ is defined by the \emph{proto-pattern encoding judgement}, $\encodeCEPat{\cpv}{p}$. An inverse mapping is defined by the \emph{proto-expression decoding judgement}, $\decodeCEPat{p}{\cpv}$.

\[\begin{array}{ll}
\textbf{Judgement Form} & \textbf{Description}\\
\encodeCEPat{\cpv}{p} & \text{$\cpv$ has encoding $p$}\\
\decodeCEPat{p}{\cpv} & \text{$p$ has decoding $\cpv$}
\end{array}\]

Again, rather than picking a particular definition of $\tCEPat$ and defining the judgements above inductively against it, we only state the following condition, which establishes an isomorphism between values of type $\tCEPat$ and proto-patterns.

\begin{condition}[Proto-Pattern Isomorphism]\label{condition:proto-pattern-isomorphism} ~
\begin{enumerate}
\item For every $\cpv$, we have $\encodeCEPat{\cpv}{\ecand}$ for some $\ecand$ such that $\hastypeUC{\ecand}{\tCEPat}$ and $\isvalU{\ecand}$.
\item If $\hastypeUC{\ecand}{\tCEPat}$ and $\isvalU{\ecand}$ then $\decodeCEPat{\ecand}{\cpv}$ for some $\cpv$.
\item If $\encodeCEPat{\cpv}{\ecand}$ then $\decodeCEPat{\ecand}{\cpv}$.
\item If $\hastypeUC{\ecand}{\tCEPat}$ and $\isvalU{\ecand}$ and $\decodeCEPat{\ecand}{\cpv}$ then $\encodeCEPat{\cpv}{\ecand}$.
\item If $\encodeCEPat{\cpv}{\ecand}$ and $\encodeCEPat{\cpv}{\ecand'}$ then $\ecand=\ecand'$.
\item If $\hastypeUC{\ecand}{\tCEPat}$ and $\isvalU{\ecand}$ and $\decodeCEPat{\ecand}{\cpv}$ and $\decodeCEPat{\ecand}{\cpv'}$ then $\cpv=\cpv'$.
\end{enumerate}
\end{condition}
\end{grayparbox}

\subsubsection{Segmentations}\label{appendix:segmentations-U}
The \emph{segmentation}, $\psi$, of a proto-type, $\segof{\ctau}$ or proto-expression, $\segof{\ce}$, is the finite set of references to spliced types and expressions that it mentions.
\[
\begin{array}{lll}
\segof{t} & = & \emptyset\\
\segof{\aceparr{\ctau_1}{\ctau_2}} & = & \segof{\ctau_1} \cup \segof{\ctau_2}\\
\segof{\aceall{t}{\ctau}} & = & \segof{\ctau}\\
\segof{\acerec{t}{\ctau}} & = & \segof{\ctau}\\
\segof{\aceprod{\labelset}{\mapschema{\ctau}{i}{\labelset}}} & = & \bigcup_{i \in \labelset} \segof{\ctau_i}\\
\segof{\acesum{\labelset}{\mapschema{\ctau}{i}{\labelset}}} & = & \bigcup_{i \in \labelset} \segof{\ctau_i}\\
\segof{\acesplicedt{m}{n}} & = & \{ \acesplicedt{m}{n} \}\\
~\\
\segof{x} & = & \emptyset\\
\segof{\aceasc{\ctau}{\ce}} & = & \segof{\ctau} \cup \segof{\ce}\\
\segof{\aceletsyn{x}{\ce_1}{\ce_2}} & = & \segof{\ce_1} \cup \segof{\ce_2}\\
\segof{\acelam{\ctau}{x}{\ce}} & = & \segof{\ctau} \cup \segof{\ce}\\
\segof{\aceap{\ce_1}{\ce_2}} & = & \segof{\ce_1} \cup \segof{\ce_2}\\
\segof{\acetlam{t}{\ce}} & = & \segof{\ce}\\
\segof{\acetap{\ce}{\ctau}} & = & \segof{\ce} \cup \segof{\ctau}\\
\segof{\acefold{\ce}} & = & \segof{\ce}\\
\segof{\aceunfold{\ce}} & = & \segof{\ce}\\
\segof{\acetpl{\labelset}{\mapschemab{x}{\ce}{i}{\labelset}}} & = & \bigcup_{i \in \labelset} \segof{\ce_i}\\
\segof{\acepr{\ell}{\ce}} & = & \segof{\ce}\\
\segof{\acein{\ell}{\ce}} & = & \segof{\ce}\\
\segof{\acecase{\labelset}{\ce}{\mapschemab{x}{\ce}{i}{\labelset}}} & = & \segof{\ce} \cup \bigcup_{i \in \labelset} \segof{\ce_i}\\
\segof{\acesplicede{m}{n}{\ctau}} & = & \{ \acesplicede{m}{n}{\ctau} \} \cup \segof{\ctau}\\
\LCC \color{light-gray} & \color{light-gray} & \color{light-gray}\\
\segof{\acematchwith{n}{\ce}{\seqschemaX{\crv}}} & = & \segof{\ce} \cup \bigcup_{1 \leq i \leq n} \segof{\crv_i}\\
~\\
\segof{\acematchrule{p}{\ce}} & = & \segof{\ce}\ECC
\end{array}
\]

\begin{grayparbox}
The splice summary of a proto-pattern, $\segof{\cpv}$, is the finite set of references to spliced types and patterns that it mentions.
\[
\begin{array}{lll}
\segof{\acewildp} & = & \emptyset\\
\segof{\acefoldp{\cpv}} & = & \segof{\cpv}\\
\segof{\acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}}} & = & \bigcup_{i \in \labelset} \segof{\cpv_i}\\
\segof{\aceinjp{\ell}{\cpv}} & = & \segof{\cpv}\\
\segof{\acesplicedp{m}{n}{\ctau}} & = & \{ \acesplicedp{m}{n}{\ctau} \} \cup \segof{\ctau}
\end{array}\]
\end{grayparbox}

The predicate $\segOK{\psi}{b}$ checks that each segment in $\psi$, has positive length and is within bounds of $b$, and that the segments in $\psi$ do not overlap and operate at consistent sorts or types.

\begin{definition}[Segmentation Validity] $\segOK{\psi}{b}$ iff 
\begin{enumerate}
  \item For each $\acesplicedt{m}{n} \in \psi$, all of the following hold:
    \begin{enumerate}
      \item $0 \leq m < n \leq \sizeof{b}$
      \item For each $\acesplicedt{m'}{n'} \in \psi$, either 
        \begin{enumerate}
          \item $m=m'$ and $n=n'$; or 
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicede{m'}{n'}{\ctau} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
    \end{enumerate}
    \begin{grayparbox}
    \begin{enumerate}
      \item[(d)] For each $\acesplicedp{m'}{n'}{\ctau} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or
          \item $m' > n$
        \end{enumerate}
    \end{enumerate}
    \end{grayparbox}
  \item For each $\acesplicede{m}{n}{\ctau} \in \psi$, all of the following hold:
    \begin{enumerate}
      \item $0 \leq m < n \leq \sizeof{b}$
      \item For each $\acesplicedt{m'}{n'} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicede{m'}{n'}{\ctau'} \in \psi$, either 
        \begin{enumerate}
          \item $m=m'$ and $n=n'$ and $\ctau=\ctau'$; or 
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
    \end{enumerate}
\end{enumerate}
\begin{grayparbox}
\begin{enumerate}
\item[3.] For each $\acesplicedp{m}{n}{\ctau} \in \psi$, all of the following hold:
    \begin{enumerate}
      \item $0 \leq m < n \leq \sizeof{b}$
      \item For each $\acesplicedt{m'}{n'} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicede{m'}{n'}{\ctau'} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicedp{m'}{n'}{\ctau'} \in \psi$, either 
        \begin{enumerate}
          \item $m=m'$ and $n=n'$ and $\ctau=\ctau'$; or 
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
    \end{enumerate}
\end{enumerate}
\end{grayparbox}
\end{definition}

\subsection{Proto-Type Validation}\label{appendix:proto-type-validation-SES}
%Each of these rules is defined based on the corresponding type formation rule, i.e. Rules (\ref{rule:istypeU-var}) through (\ref{rule:istypeU-sum}), respectively. For example, the following candidate expansion type validation rules are based on type formation rules (\ref{rule:istypeU-var}), (\ref{rule:istypeU-parr}) and (\ref{rule:istypeU-all}), respectively: 
\emph{Type splicing scenes}, $\tscenev$, are of the form $\tsceneUP{\uDelta}{b}$.

\vspace{10px}\noindent\fbox{\strut$\cvalidT{\Delta}{\tscenev}{\ctau}{\tau}$}~~$\ctau$ has well-formed expansion $\tau$
\begin{subequations}\label{rules:cvalidT-U}
\begin{equation}\label{rule:cvalidT-U-tvar}
\inferrule{ }{
  \cvalidT{\Delta, \Dhyp{t}}{\tscenev}{t}{t}
}
\end{equation}
\begin{equation}\label{rule:cvalidT-U-parr}
  \inferrule{
    \cvalidT{\Delta}{\tscenev}{\ctau_1}{\tau_1}\\
    \cvalidT{\Delta}{\tscenev}{\ctau_2}{\tau_2}
  }{
    \cvalidT{\Delta}{\tscenev}{\aceparr{\ctau_1}{\ctau_2}}{\aparr{\tau_1}{\tau_2}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidT-U-all}
  \inferrule {
    \cvalidT{\Delta, \Dhyp{t}}{\tscenev}{\ctau}{\tau}
  }{
    \cvalidT{\Delta}{\tscenev}{\aceall{t}{\ctau}}{\aall{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidT-U-rec}
  \inferrule{
    \cvalidT{\Delta, \Dhyp{t}}{\tscenev}{\ctau}{\tau}
  }{
    \cvalidT{\Delta}{\tscenev}{\acerec{t}{\ctau}}{\arec{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidT-U-prod}
  \inferrule{
    \{\cvalidT{\Delta}{\tscenev}{\ctau_i}{\tau_i}\}_{i \in \labelset}
  }{
    \cvalidT{\Delta}{\tscenev}{\aceprod{\labelset}{\mapschema{\ctau}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidT-U-sum}
  \inferrule{
    \{\cvalidT{\Delta}{\tscenev}{\ctau_i}{\tau_i}\}_{i \in \labelset}
  }{
    \cvalidT{\Delta}{\tscenev}{\acesum{\labelset}{\mapschema{\ctau}{i}{\labelset}}}{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidT-U-splicedt}
  \inferrule{
    \parseUTyp{\bsubseq{b}{m}{n}}{\utau}\\
    \expandsTU{\uDD{\uD}{\Delta_\text{app}}}{\utau}{\tau}\\
    \Delta \cap \Delta_\text{app} = \emptyset
  }{
    \cvalidT{\Delta}{\tsceneU{\uDD{\uD}{\Delta_\text{app}}}{b}}{\acesplicedt{m}{n}}{\tau}
  }
\end{equation}
\end{subequations}


%Rule (\ref*{rule:cvalidT-U-splicedt}) governs this form:
%\chapter{Dependent Labeled Product Kinds}
% \begin{landscape}
% \begin{equation}\label{rule:iskind-dlprod}
% \inferrule{
% 	\{\iskind{\Omega}{\Delta \cup \{u_{i, j} :: \kappa_j\}_{1 \leq j < i}}{\Gamma}{\kappa_i}\}_{1 \leq i \leq n}
% }{
% 	\iskindX{\akdprodstd}
% }
% \end{equation}

% \begin{equation}\label{rule:kequal-dlprod}
% \inferrule{
% 	\{\kequal{\Omega}{\Delta \cup \{u_{i, j} :: \kappa_j\}_{1 \leq j < i}}{\Gamma}{\kappa_i}{\kappa'_i}\}_{1 \leq i \leq n}
% }{
% 	\kequalX{\akdprodstd}{\akdprod{n}{\seqschemaX{\ell}}{\seqschemaijb{u}{\kappa'}{i}{1}{n}{j}{1}{i}}}
% }
% \end{equation}
% \begin{equation}\label{rule:ksub-dlprod}
% \inferrule{
% 	\{\ksub{\Omega}{\Delta \cup \{u_{i,j} :: \kappa_j\}_{1 \leq j < i}}{\Gamma}{\kappa_i}{\kappa'_i}\}_{1 \leq i \leq n}
% }{
% 	\ksubX{\akdprodstd}{\akdprod{n}{\seqschemaX{\ell}}{\seqschemaijb{u}{\kappa'}{i}{1}{n}{j}{1}{i}}}
% }
% \end{equation}
% \begin{equation}\label{rule:haskind-dtpl}
% \inferrule{
% 	\{\haskind{\Omega}{\Delta \cup \{u_{i, j} :: \aksing{c_j}\}_{1 \leq j < i}}{\Gamma}{c_i}{\kappa_i}\}_{1 \leq i \leq n}
% }{
% 	\haskindX{\adtplX}{\akdprodstd}
% }
% \end{equation}
% \begin{equation}\label{rule:haskind-prj}
% \inferrule{
% 	\haskindX{c}{
% 		\akdprod{
% 			n' + 1 + n''
% 		}{
% 			\seqschema{\ell'}{i}{1}{n'}, \ell, \seqschema{\ell''}{i}{1}{n''}
% 		}{
% 			\seqschemaijb{u'}{\kappa'}{i}{1}{n'}{j}{1}{i};
% 			\{u_{j}\}_{1 \leq j \leq n'}.\kappa;
% 			\seqschemaijb{u''}{\kappa''}{i}{1}{n''}{j}{1}{i}
% 		}
% 	}
% }{
% 	\haskindX{\adprj{\ell}{c}}{[\{\adprj{\ell'_j}{c}/u_{j}\}_{1 \leq j \leq n'}]\kappa}
% }
% \end{equation}
% \begin{equation}\label{rule:cequal-dtpl}
% \inferrule{
% 	c=\adtplX\\
% 	c'=\adtpl{n}{\seqschemaX{\ell}}{\seqschemaijb{u}{c'}{i}{1}{n}{j}{1}{i}}\\\\
% 	\{\cequal{\Omega}{\Delta \cup \{u_{i, j} :: \aksing{\kappa_j}\}_{1 \leq j < i}}{\Gamma}{c_i}{c'_i}{\kappa_i}\}_{1 \leq i \leq n}
% }{
% 	\cequalX{c}{c'}{\akdprodstd}
% }
% \end{equation}
% \begin{equation}\label{rule:cequal-prj-1}
% \inferrule{
%   \cequalX{c}{c'}{
% 		\akdprod{
% 			n' + 1 + n''
% 		}{
% 			\seqschema{\ell'}{i}{1}{n'}, \ell, \seqschema{\ell''}{i}{1}{n''}
% 		}{
% 			\seqschemaijb{u'}{\kappa'}{i}{1}{n'}{j}{1}{i};
% 			\{u_{j}\}_{1 \leq j \leq n'}.\kappa;
% 			\seqschemaijb{u''}{\kappa''}{i}{1}{n''}{j}{1}{i}
% 		}
% 	}	
% }{
% 	\cequalX{\adprj{\ell}{c}}{\adprj{\ell}{c'}}{\kappa}
% }
% \end{equation}
% \begin{equation}\label{rule:cequal-prj-2}
% \inferrule{
% 	c = \adtpl{
% 				n' + 1 + n''
% 			}{
% 				\seqschema{\ell'}{i}{1}{n'}, \ell, \seqschema{\ell''}{i}{1}{n''}
% 			}{
% 				\seqschemaijb{u'}{c'}{i}{1}{n'}{j}{1}{i};
% 				\{u_j\}_{1 \leq j \leq n}.c_\ell; 
% 				\seqschemaijb{u''}{c''}{i}{1}{n''}{j}{1}{i}
% 			}\\
% 	\haskindX{c}{
% 		\akdprod{
% 			n' + 1 + n''
% 		}{
% 			\seqschema{\ell'}{i}{1}{n'}, \ell, \seqschema{\ell''}{i}{1}{n''}
% 		}{
% 			\seqschemaijb{u'}{\kappa'}{i}{1}{n'}{j}{1}{i};
% 			\{u_{j}\}_{1 \leq j \leq n'}.\kappa;
% 			\seqschemaijb{u''}{\kappa''}{i}{1}{n''}{j}{1}{i}
% 		}
% 	}
% }{
% 	\cequalX{
% 		\adprj{\ell}{
% 			c
% 		}
% 	}{[\{\adprj{\ell'_j}{c}/u_j\}_{1 \leq j \leq n'}]c_\ell}{[\{\adprj{\ell'_j}{c}/u_j\}_{1 \leq j \leq n'}]\kappa}
% }
% \end{equation}

% \end{landscape}

\subsection{Proto-Expression Validation}

\emph{Expression splicing scenes}, $\escenev$, are of the form $\esceneSGB{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}$. We write $\tsfrom{\escenev}$ for the type splicing scene constructed by dropping unnecessary contexts from $\escenev$:
\[\tsfrom{\esceneSGB{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}} = \tsceneUP{\uDelta}{b}\]

\vspace{10px}\noindent\fbox{\strut$\cvalidE{\Delta}{\Gamma}{\escenev}{\ce}{e}{\tau}$}~~$\ce$ has expansion $e$ of type $\tau$
\begin{subequations}\label{rules:cvalidE-U}
\begin{equation}\label{rule:cvalidE-U-var}
\inferrule{ }{
  \cvalidE{\Delta}{\Gamma, \Ghyp{x}{\tau}}{\escenev}{x}{x}{\tau}
}
\end{equation}
\begin{equation}\label{rule:cvalidE-U-asc}
\inferrule{
  \cvalidT{\Delta}{\tsfrom{\escenev}}{\ctau}{\tau}\\
  \cvalidE{\Delta}{\Gamma}{\escenev}{\ce}{e}{\tau}
}{
  \cvalidE{\Delta}{\Gamma}{\escenev}{\aceasc{\ctau}{\ce}}{e}{\tau}
}
\end{equation}
\begin{equation}\label{rule:cvalidE-U-letsyn}
  \inferrule{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\ce_1}{e_1}{\tau_1}\\
    \cvalidE{\Delta}{\Gamma, x : \tau_1}{\ce_2}{e_2}{\tau_2}
  }{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\aceletsyn{x}{\ce_1}{\ce_2}}{
      \aeap{\aelam{\tau_1}{x}{e_2}}{e_1}
    }{\tau_2}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-U-lam}
\inferrule{
  \cvalidT{\Delta}{\tsfrom{\escenev}}{\ctau}{\tau}\\
  \cvalidE{\Delta}{\Gamma, \Ghyp{x}{\tau}}{\escenev}{\ce}{e}{\tau'}
}{
  \cvalidE{\Delta}{\Gamma}{\escenev}{\acelam{\ctau}{x}{\ce}}{\aelam{\tau}{x}{e}}{\aparr{\tau}{\tau'}}
}
\end{equation}
\begin{equation}\label{rule:cvalidE-U-ap}
  \inferrule{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\ce_1}{e_1}{\aparr{\tau}{\tau'}}\\
    \cvalidE{\Delta}{\Gamma}{\escenev}{\ce_2}{e_2}{\tau}
  }{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\aceap{\ce_1}{\ce_2}}{\aeap{e_1}{e_2}}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-U-tlam}
  \inferrule{
    \cvalidE{\Delta, \Dhyp{t}}{\Gamma}{\escenev}{\ce}{e}{\tau}
  }{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\acetlam{t}{\ce}}{\aetlam{t}{e}}{\aall{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-U-tap}
  \inferrule{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\ce}{e}{\aall{t}{\tau}}\\
    \cvalidT{\Delta}{\tsfrom{\escenev}}{\ctau'}{\tau'}
  }{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\acetap{\ce}{\ctau'}}{\aetap{e}{\tau'}}{[\tau'/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-U-fold}
  \inferrule{\
    % \cvalidT{\Delta, \Dhyp{t}}{\tsfrom{\escenev}}{\ctau}{\tau}\\
    \cvalidE{\Delta}{\Gamma}{\escenev}{\ce}{e}{[\arec{t}{\tau}/t]\tau}
  }{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\acefold{\ce}}{\aefold{e}}{\arec{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-U-unfold}
  \inferrule{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\ce}{e}{\arec{t}{\tau}}
  }{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\aceunfold{\ce}}{\aeunfold{e}}{[\arec{t}{\tau}/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-U-tpl}
  \inferrule{
    \tau = \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}\\\\
    \{\cvalidE{\Delta}{\Gamma}{\escenev}{\ce_i}{e_i}{\tau_i}\}_{i \in \labelset}
  }{
  % \left(\shortstack{$\Delta~\Gamma \vdash^{\escenev} \acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}}$\\$\leadsto$\\$\aetpl{\labelset}{\mapschema{e}{i}{\labelset}} : \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$\vspace{-1.2em}}\right)
    \cvalidE{\Delta}{\Gamma}{\escenev}{\acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}}}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-U-pr}
  \inferrule{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\ce}{e}{\aprod{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \ell \hookrightarrow \tau}}
  }{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\acepr{\ell}{\ce}}{\aepr{\ell}{e}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-U-in}
  \inferrule{
    % \{\cvalidT{\Delta}{\tsfrom{\escenev}}{\ctau_i}{\tau_i}\}_{i \in \labelset}\\
    % \cvalidT{\Delta}{\tsfrom{\escenev}}{\ctau}{\tau}\\
    \cvalidE{\Delta}{\Gamma}{\escenev}{\ce}{e}{\tau'}
  }{
    % \left(\shortstack{
    %   $\Delta~\Gamma \vdash^{\escenev} \acein{\ell}{\ce}$\\
    %   $\leadsto$\\
    %   $\aein{\ell}{e} : \asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \ell \hookrightarrow \tau}$\vspace{-1.2em}
    % }\right)
    \cvalidE{\Delta}{\Gamma}{\escenev}{\acein{\ell}{\ce}}{\aein{\ell}{e}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \ell \hookrightarrow \tau'}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-U-case}
  \inferrule{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\ce}{e}{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}\\
    % \cvalidT{\Delta}{\tsfrom{\escenev}}{\ctau}{\tau}\\
    \{\cvalidE{\Delta}{\Gamma, x_i : \tau_i}{\escenev}{\ce_i}{e_i}{\tau}\}_{i \in \labelset}
  }{
    \cvalidE{\Delta}{\Gamma}{\escenev}{\acecase{\labelset}{\ce}{\mapschemab{x}{\ce}{i}{\labelset}}}{\aecase{\labelset}{e}{\mapschemab{x}{e}{i}{\labelset}}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-U-splicede}
\inferrule{
  \cvalidT{\emptyset}{\tsfrom{\escenev}}{\ctau}{\tau}\\
  \escenev=\esceneSGB{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}\\
  \parseUExp{\bsubseq{b}{m}{n}}{\ue}\\
  \expandsSG{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{\ue}{e}{\tau}\\\\
  \Delta \cap \Delta_\text{app} = \emptyset\\
  \domof{\Gamma} \cap \domof{\Gamma_\text{app}} = \emptyset
}{
  \cvalidE{\Delta}{\Gamma}{\escenev}{\acesplicede{m}{n}{\ctau}}{e}{\tau}
}
\end{equation}
\begin{grayparbox}
\begin{equation}\label{rule:cvalidE-U-match}
\graybox{\inferrule{
  % \escenev = \esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGamma}{\uPsi}{\uPhi}{b}\\\\
  % \istypeU{\Delta \cup \Delta_\text{app}}{\tau'}\\
  \cvalidE{\Delta}{\Gamma}{\escenev}{\ce}{e}{\tau}\\
  % \cvalidT{\Delta}{\tsfrom{\escenev}}{\ctau'}{\tau'}\\\\
  \{\cvalidR{\Delta}{\Gamma}{\escenev}{\crv_i}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}\\
}{\cvalidE{\Delta}{\Gamma}{\escenev}{\acematchwith{n}{\ce}{\seqschemaX{\crv}}}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}}}
\end{equation}
\end{grayparbox}
\end{subequations}
% \clearpage
\vspace{-5px}
\begin{grayparbox}
\vspace{15px}
\noindent\fbox{\strut$\cvalidR{\Delta}{\Gamma}{\escenev}{\crv}{r}{\tau}{\tau'}$}~~$\crv$ has expansion $r$ taking values of type $\tau$ to values of type $\tau'$
\begin{equation}\label{rule:cvalidR-UP}
\inferrule{
  % \escenev = \esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGamma}{\uPsi}{\uPhi}{b}\\\\
  \patTypeD{\Delta \cup \Delta_\text{app}}{\pctx'}{p}{\tau}\\
  \cvalidE{\Delta}{\Gcons{\Gamma}{\pctx'}}{\escenev}{\ce}{e}{\tau'}
}{
  \cvalidR{\Delta}{\Gamma}{\escenev}{\acematchrule{p}{\ce}}{\aematchrule{p}{e}}{\tau}{\tau'}
}
\end{equation}
\end{grayparbox}
\vspace{-5px}\begin{grayparbox}
\subsection{Proto-Pattern Validation}\label{appendix:proto-pattern-validation-P}
\emph{Pattern splicing scenes}, $\pscenev$, are of the form $\pscene{\uDelta}{\uPhi}{b}$.

\vspace{10px}\noindent\fbox{\strut$\cvalidP{\upctx}{\pscenev}{\cpv}{p}{\tau}$}~~$\cpv$ has expansion $p$ matching against $\tau$ generating hypotheses $\upctx$
% \begin{grayparbox}
\begin{subequations}\label{rules:cvalidP-UP}
\begin{equation}\label{rule:cvalidP-UP-wild}
\inferrule{ }{
  \cvalidP{\uGG{\emptyset}{\emptyset}}{\pscenev}{\acewildp}{\aewildp}{\tau}
}
\end{equation}
\begin{equation}\label{rule:cvalidP-UP-fold}
\inferrule{
  \cvalidP{\upctx}{\pscenev}{\cpv}{p}{[\arec{t}{\tau}/t]\tau}
}{
  \cvalidP{\upctx}{\pscenev}{\acefoldp{\cpv}}{\aefoldp{p}}{\arec{t}{\tau}}
}
\end{equation}
\begin{equation}\label{rule:cvalidP-UP-tpl}
\inferrule{
  \tau = \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}\\\\
  \{\cvalidP{\upctx_i}{\pscenev}{\cpv_i}{p_i}{\tau_i}\}_{i \in \labelset}
}{
% \left(\shortstack{$\vdash^{\pscenev} \acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}}$\\$\leadsto$\\$\aetplp{\labelset}{\mapschema{p}{i}{\labelset}} : \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}~\dashVx^{\,\Gconsi{i \in \labelset}{\upctx_i}}$\vspace{-1.2em}}\right)
  \cvalidP{\GIconsi{i \in \labelset}{\upctx_i}}{\pscenev}{\acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\tau}
}
\end{equation}
\begin{equation}\label{rule:cvalidP-UP-in}
\inferrule{
  \cvalidP{\upctx}{\pscenev}{\cpv}{p}{\tau}
}{
  \cvalidP{\upctx}{\pscenev}{\aceinjp{\ell}{\cpv}}{\aeinjp{\ell}{p}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
}
\end{equation}
\begin{equation}\label{rule:cvalidP-UP-spliced}
\inferrule{
  \cvalidT{\emptyset}{\tsceneUP{\uDelta}{b}}{\ctau}{\tau}\\
  \parseUPat{\bsubseq{b}{m}{n}}{\upv}\\
  \patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}
}{
  \cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\acesplicedp{m}{n}{\ctau}}{p}{\tau}
}
\end{equation}
\end{subequations}
\end{grayparbox}
% Observe that, in each of these rules, the proto-expression form and the expanded expression form in the conclusion correspond, and the premises correspond to those of the corresponding typing rule, i.e. Rules (\ref{rule:hastypeU-var}) through (\ref{rule:hastypeU-ap}), respectively. The expression splicing scene, $\escenev$, passes opaquely through these rules.


% We can express this scheme more precisely with the following rule transformation. For each rule in Rules (\ref{rules:hastypeU}),
% \begin{mathpar}\refstepcounter{equation}
% \label{rule:cvalidE-U-tlam}
% \refstepcounter{equation}
% \label{rule:cvalidE-U-tap}
% \refstepcounter{equation}
% \label{rule:cvalidE-U-fold}
% \refstepcounter{equation}
% \label{rule:cvalidE-U-unfold}
% \refstepcounter{equation}
% \label{rule:cvalidE-U-tpl}
% \refstepcounter{equation}
% \label{rule:cvalidE-U-pr}
% \refstepcounter{equation}
% \label{rule:cvalidE-U-in}
% \refstepcounter{equation}
% \label{rule:cvalidE-U-case}
%   \inferrule{
%     J_1\\
%     \cdots\\
%     J_k
%   }{
%     J
%   }
% \end{mathpar}
% the corresponding proto-expression validation rule is 
% \begin{mathpar}
%   \inferrule{
%     \Cof{J_1}\\
%     \cdots\\
%     \Cof{J_k}
%   }{
%     \Cof{J}
%   }
% \end{mathpar}
% where 
% \[\begin{split}
%   \Cof{\istypeU{\Delta}{\tau}} & = \cvalidT{\Delta}{\tsfrom{\escenev}}{\Cof{\tau}}{\tau}\\
%   \Cof{\hastypeU{\Delta}{\Gamma}{e}{\tau}} & = \cvalidE{\Delta}{\Gamma}{\escenev}{\Cof{e}}{e}{\tau}\\
%   \Cof{\{J_i\}_{i \in \labelset}} & = \{\Cof{J_i}\}_{i \in \labelset}
% \end{split}\]
% and where:
% \begin{itemize}
% \item $\Cof{\tau}$ is defined as follows:
%   \begin{itemize}
%   \item When $\tau$ is of definite form, $\Cof{\tau}$ is defined as in Sec. \ref{sec:ce-syntax-U}.
%   \item When $\tau$ is of indefinite form, $\Cof{\tau}$ is a uniquely corresponding metavariable of sort $\mathsf{CETyp}$ also of indefinite form. For example, $\Cof{\tau_1}=\ctau_1$ and $\Cof{\tau_2}=\ctau_2$.
%   \end{itemize}
% \item $\Cof{e}$ is defined as follows
%   \begin{itemize}
%   \item When $e$ is of definite form, $\Cof{e}$ is defined as in Sec. \ref{sec:ce-syntax-U}. 
%   \item When $e$ is of indefinite form, $\Cof{e}$ is a uniquely corresponding metavariable of sort $\mathsf{CEExp}$ also of indefinite form. For example, $\Cof{e_1}=\ce_1$ and $\Cof{e_2}=\ce_2$.
%   \end{itemize}
% \end{itemize}

% It is instructive to use this rule transformation to generate Rules (\ref{rule:cvalidE-U-var}) through (\ref{rule:cvalidE-U-ap}) above. We omit the remaining rules for common forms, i.e. Rules (\ref*{rule:cvalidE-U-tlam}) through (\ref*{rule:cvalidE-U-case}).

\section{Metatheory}\label{appendix:metatheory-SES}
\subsection{Type Expansion}
% The Type Expansion Lemma establishes that the expansion of an unexpanded type is a well-formed type.

\begin{lemma}[Type Expansion]\label{lemma:type-expansion-U} If $\expandsTU{\uDD{\uD}{\Delta}}{\utau}{\tau}$ then $\istypeU{\Delta}{\tau}$.\end{lemma}
\begin{proof} By rule induction over Rules (\ref{rules:expandsTU}). In each case, we apply the IH to or over each premise, then apply the corresponding type formation rule in Rules (\ref{rules:istypeU}). \end{proof}

\begin{lemma}[Proto-Type Validation]\label{lemma:candidate-expansion-type-validation}
If $\cvalidT{\Delta}{\tsceneU{\uDD{\uD}{\Delta_\text{app}}}{b}}{\ctau}{\tau}$ and $\Delta \cap \Delta_\text{app}=\emptyset$ then $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\tau}$.
\end{lemma}
\begin{proof} By rule induction over Rules (\ref{rules:cvalidT-U}).
\begin{byCases}
\item[\text{(\ref{rule:cvalidT-U-tvar})}] ~
\begin{pfsteps*}
   \item $\Delta=\Delta', \Dhyp{t}$ \BY{assumption}
   \item $\ctau=t$ \BY{assumption}
   \item $\tau=t$ \BY{assumption}
   \item $\istypeU{\Delta', \Dhyp{t}}{t}$ \BY{Rule (\ref{rule:istypeU-var})} \pflabel{istype}
   \item $\istypeU{\Dcons{\Delta', \Dhyp{t}}{\Delta_\text{app}}}{t}$ \BY{Lemma \ref{lemma:weakening-U} over $\Delta_\text{app}$ to \pfref{istype}}
 \end{pfsteps*} 
\resetpfcounter

\item[\text{(\ref{rule:cvalidT-U-parr})}] ~
\begin{pfsteps*}
  \item $\ctau=\aceparr{\ctau_1}{\ctau_2}$ \BY{assumption}
  \item $\tau=\aparr{\tau_1}{\tau_2}$ \BY{assumption}
  \item $\cvalidT{\Delta}{\tsceneU{\uDD{\uD}{\Delta_\text{app}}}{b}}{\ctau_1}{\tau_1}$ \BY{assumption} \pflabel{cvalid-ctau1}
  \item $\cvalidT{\Delta}{\tsceneU{\uDD{\uD}{\Delta_\text{app}}}{b}}{\ctau_2}{\tau_2}$ \BY{assumption} \pflabel{cvalid-ctau2}
  \item $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\tau_1}$ \BY{IH on \pfref{cvalid-ctau1}} \pflabel{istype1}
  \item $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\tau_2}$ \BY{IH on \pfref{cvalid-ctau2}} \pflabel{istype2}
  \item $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\aparr{\tau_1}{\tau_2}}$ \BY{Rule (\ref{rule:istypeU-parr}) on \pfref{istype1} and \pfref{istype2}}
\end{pfsteps*}
\resetpfcounter

\item[\text{(\ref{rule:cvalidT-U-all})}] ~
\begin{pfsteps*}
  \item $\ctau=\aceall{t}{\ctau'}$ \BY{assumption}
  \item $\tau=\aall{t}{\tau'}$ \BY{assumption}
  \item $\cvalidT{\Delta, \Dhyp{t}}{\tsceneU{\uDD{\uD}{\Delta_\text{app}}}{b}}{\ctau'}{\tau'}$ \BY{assumption} \label{cvalidT}
  \item $\istypeU{\Dcons{\Delta, \Dhyp{t}}{\Delta_\text{app}}}{\tau'}$ \BY{IH on \pfref{cvalidT}} \pflabel{istypeU1}
  \item $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}, \Dhyp{t}}{\tau'}$ \BY{exchange over $\Delta_\text{app}$ on \pfref{istypeU1}} \pflabel{istypeU2}
  \item $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\aall{t}{\tau'}}$ \BY{Rule (\ref{rule:istypeU-all}) on \pfref{istypeU2}}
\end{pfsteps*}
\resetpfcounter

% \item[{\text{(\ref{rule:cvalidT-U-rec})}}~\textbf{through}~{\text{(\ref{rule:cvalidT-U-sum})}}] These cases follow analagously, i.e. we apply the IH to or over all proto-type validation premises, apply exchange as needed, and then apply the corresponding type formation rule.
% \\
\item[\text{(\ref{rule:cvalidT-U-rec})}] ~
\begin{pfsteps*}
  \item $\ctau=\acerec{t}{\ctau'}$ \BY{assumption}
  \item $\tau=\arec{t}{\tau'}$ \BY{assumption}
  \item $\cvalidT{\Delta, \Dhyp{t}}{\tsceneU{\Delta_\text{app}}{b}}{\ctau'}{\tau'}$ \BY{assumption} \label{cvalidT}
  \item $\istypeU{\Dcons{\Delta, \Dhyp{t}}{\Delta_\text{app}}}{\tau'}$ \BY{IH on \pfref{cvalidT}} \pflabel{istypeU1}
  \item $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}, \Dhyp{t}}{\tau'}$ \BY{exchange over $\Delta_\text{app}$ on \pfref{istypeU1}} \pflabel{istypeU2}
  \item $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\arec{t}{\tau'}}$ \BY{Rule (\ref{rule:istypeU-rec}) on \pfref{istypeU2}}
\end{pfsteps*}
\resetpfcounter

\item[\text{(\ref{rule:cvalidT-U-prod})}] ~
\begin{pfsteps*}
\item $\ctau=\aceprod{\labelset}{\mapschema{\ctau}{i}{\labelset}}$ \BY{assumption}  
\item $\tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$ \BY{assumption}
\item $\{\cvalidT{\Delta}{\tsceneU{\Delta_\text{app}}{b}}{\ctau_i}{\tau_i}\}_{i \in \labelset}$ \BY{assumption} \pflabel{cvalidT-ass}
\item $\{\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\tau_i}\}_{i \in \labelset}$ \BY{IH over \pfref{cvalidT-ass}} \pflabel{istype}
\item $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}$ \BY{Rule (\ref{rule:istypeU-prod}) on \pfref{istype}}
\end{pfsteps*}
\resetpfcounter 

\item[\text{(\ref{rule:cvalidT-U-sum})}] ~
\begin{pfsteps*}
\item $\ctau=\acesum{\labelset}{\mapschema{\ctau}{i}{\labelset}}$ \BY{assumption}  
\item $\tau=\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}$ \BY{assumption}
\item $\{\cvalidT{\Delta}{\tsceneU{\Delta_\text{app}}{b}}{\ctau_i}{\tau_i}\}_{i \in \labelset}$ \BY{assumption} \pflabel{cvalidT-ass}
\item $\{\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\tau_i}\}_{i \in \labelset}$ \BY{IH over \pfref{cvalidT-ass}} \pflabel{istype}
\item $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}$ \BY{Rule (\ref{rule:istypeU-sum}) on \pfref{istype}}
\end{pfsteps*}
\resetpfcounter

\item[\text{(\ref{rule:cvalidT-U-splicedt})}] ~
\begin{pfsteps*}
\item $\ctau=\acesplicedt{m}{n}$ \BY{assumption}
\item $\parseUTyp{\bsubseq{b}{m}{n}}{\utau}$ \BY{assumption}
\item $\expandsTU{\uDD{\uD}{\Delta_\text{app}}}{\utau}{\tau}$ \BY{assumption} \label{expandsTU}
\item $\Delta \cap \Delta_\text{app} = \emptyset$ \BY{assumption}
\item $\istypeU{\Delta_\text{app}}{\tau}$ \BY{Lemma \ref{lemma:type-expansion-U} on \pfref{expandsTU}}\pflabel{istype}
\item $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\tau}$ \BY{Lemma \ref{lemma:weakening-U} over $\Delta$ on \pfref{istype} and exchange over $\Delta$}
\end{pfsteps*}
\resetpfcounter
\end{byCases}
\end{proof}

\vspace{15px}
\begin{grayparbox}\vspace{-15px}
\subsection{Typed Pattern Expansion}\label{appendix:SES-typed-pattern-expansion}
\begin{theorem}[Typed Pattern Expansion]\label{thm:typed-pattern-expansion} ~
\begin{enumerate}
  \item If $\pExpandsSP{\uDD{\uD}{\Delta}}{\uAS{\uA}{\Phi}}{\upv}{p}{\tau}{\uGG{\uG}{\pctx}}$ then $\patType{\pctx}{p}{\tau}$.
  \item If $\cvalidP{\uGG{\uG}{\pctx}}{\pscene{\uDD{\uD}{\Delta}}{\uAP{\uA}{\Phi}}{b}}{\cpv}{p}{\tau}$ then $\patType{\pctx}{p}{\tau}$.
\end{enumerate}
\end{theorem}
\begin{proof}
  By mutual rule induction over Rules (\ref{rules:patExpands}) and Rules (\ref{rules:cvalidP-UP}).
  \begin{enumerate}
  \item We induct on the premise. In the following, let $\uDelta=\uDD{\uD}{\Delta}$ and $\upctx=\uGG{\uG}{\pctx}$ and $\uPhi=\uAP{\uA}{\Phi}$.
  \begin{byCases}
    \item[\text{(\ref{rule:patExpands-var})}] ~
      \begin{pfsteps*}
        \item $\upv=\ux$ \BY{assumption}
        \item $p=x$ \BY{assumption}
        \item $\pctx=\Ghyp{x}{\tau}$ \BY{assumption}
        \item $\patType{\Ghyp{x}{\tau}}{x}{\tau}$ \BY{Rule (\ref{rule:patType-var})}
      \end{pfsteps*}
      \resetpfcounter
    \item[\text{(\ref{rule:patExpands-wild})}] ~
      \begin{pfsteps*}
        \item $p=\aewildp$ \BY{assumption}
        \item $\pctx = \emptyset$ \BY{assumption}
        \item $\patType{\emptyset}{\aewildp}{\tau}$ \BY{Rule (\ref{rule:patType-wild})}
      \end{pfsteps*}
      \resetpfcounter
    \item[\text{(\ref{rule:patExpands-fold})}] ~
      \begin{pfsteps*}
        \item $\upv=\foldp{\upv'}$ \BY{assumption}
        \item $p=\aefoldp{p'}$ \BY{assumption}
        \item $\tau=\arec{t}{\tau'}$ \BY{assumption}
        %\item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
        \item $\patExpands{\upctx}{\uPhi}{\upv'}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{assumption} \pflabel{patExpands}
        \item $\patType{\pctx}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{IH, part 1 on \pfref{patExpands}} \pflabel{patType}
        \item $\patType{\pctx}{\aefoldp{p'}}{\arec{t}{\tau'}}$ \BY{Rule (\ref{rule:patType-fold}) on \pfref{patType}}
      \end{pfsteps*}
      \resetpfcounter
    \item[\text{(\ref{rule:patExpands-tpl})}] ~
      \begin{pfsteps*}
        \item $\upv=\tplp{\mapschema{\upv}{i}{\labelset}}$ \BY{assumption}
        \item $p=\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}$ \BY{assumption}
        \item $\tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$ \BY{assumption}
        \item $\{\patExpands{\uGG{\uG_i}{\pctx_i}}{\uPhi}{\upv_i}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{assumption} \pflabel{patExpands}
        \item $\pctx = \Gconsi{i \in \labelset}{\pctx_i}$ \BY{assumption}
        %\item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
        \item $\{\patType{\pctx_i}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{IH, part 1 over \pfref{patExpands}}\pflabel{patType}
        \item $\patType{\Gconsi{i \in \labelset}{\pctx_i}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}$ \BY{Rule (\ref{rule:patType-tpl}) on \pfref{patType}}
      \end{pfsteps*}
      \resetpfcounter
    \item[\text{(\ref{rule:patExpands-in})}] ~
      \begin{pfsteps*}
        \item $\upv=\injp{\ell}{\upv'}$ \BY{assumption}
        \item $p=\aeinjp{\ell}{p'}$ \BY{assumption}
        \item $\tau=\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}$ \BY{assumption}
        \item $\patExpands{\upctx}{\uPhi}{\upv'}{p'}{\tau'}$ \BY{assumption} \pflabel{patExpands}
%        \item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
        \item $\patType{\pctx}{p'}{\tau'}$ \BY{IH, part 1 on \pfref{patExpands}} \pflabel{patType}
        \item $\patType{\pctx}{\aeinjp{\ell}{p'}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}}$ \BY{Rule (\ref{rule:patType-inj}) on \pfref{patType}}
      \end{pfsteps*}
      \resetpfcounter
    \item[\text{(\ref{rule:patExpands-apuptsm})}] ~
      \begin{pfsteps*}
        \item $\upv=\utsmap{\tsmv}{b}$ \BY{assumption}
        \item $\uA=\uA', \vExpands{\tsmv}{a}$ \BY{assumption}
        \item $\Phi=\Phi', \xuptsmbnd{a}{\tau}{\eparse}$ \BY{assumption}
        \item $\encodeBody{b}{\ebody}$ \BY{assumption}
        \item $\evalU{\eparse(\ebody)}{\aein{\mathtt{SuccessP}}{\ecand}}$ \BY{assumption}
        \item $\decodeCEPat{\ecand}{\cpv}$ \BY{assumption}
        \item $\cvalidP{\uGG{\uG}{\pctx}}{\pscene{\uDelta}{\uAP{\uA}{\Phi}}{b}}{\cpv}{p}{\tau}$ \BY{assumption} \pflabel{cvalidP}
%        \item $\uptsmenv{\Delta}{\Phi', \xuptsmbnd{a}{\tau}{\eparse}}$ \BY{assumption} \pflabel{env}
        \item $\patType{\pctx}{p}{\tau}$ \BY{IH, part 2 on \pfref{cvalidP}}
      \end{pfsteps*}
      \resetpfcounter
  \end{byCases}

  \item We induct on the premise. In the following, let $\upctx=\uGG{\uG}{\pctx}$ and $\uDelta = \uDD{\uD}{\Delta}$ and $\uPhi=\uAP{\uA}{\Phi}$.
  \begin{byCases}
    \item[\text{(\ref{rule:cvalidP-UP-wild})}] ~
      \begin{pfsteps*}
        \item $p=\aewildp$ \BY{assumption}
        \item $\pctx=\emptyset$ \BY{assumption}
        \item $\patType{\emptyset}{\aewildp}{\tau}$ \BY{Rule (\ref{rule:patType-wild})}
      \end{pfsteps*}
      \resetpfcounter
    \item[\text{(\ref{rule:cvalidP-UP-fold})}] ~
      \begin{pfsteps*}
        \item $\cpv=\acefoldp{\cpv'}$ \BY{assumption}
        \item $p=\aefoldp{p'}$ \BY{assumption}
        \item $\tau=\arec{t}{\tau'}$ \BY{assumption}
        % \item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
        \item $\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv'}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{assumption} \pflabel{cvalidP}
        \item $\patType{\pctx}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{IH, part 2 on \pfref{cvalidP}} \pflabel{patType}
        \item $\patType{\pctx}{\aefoldp{p'}}{\arec{t}{\tau'}}$ \BY{Rule (\ref{rule:patType-fold}) on \pfref{patType}}
      \end{pfsteps*}
      \resetpfcounter
    \item[\text{(\ref{rule:cvalidP-UP-tpl})}] ~
      \begin{pfsteps*}
        \item $\cpv=\acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}}$ \BY{assumption}
        \item $p=\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}$ \BY{assumption}
        \item $\tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$ \BY{assumption}
        \item $\{\cvalidP{\uGG{\uG_i}{\pctx_i}}{\pscene{\uDelta}{\uPhi}{b}}{\cpv_i}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{assumption} \pflabel{cvalidP}
        \item $\pctx = \Gconsi{i \in \labelset}{\pctx_i}$ \BY{assumption}
        %\item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
        \item $\{\patType{\pctx_i}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{IH, part 2 over \pfref{cvalidP}}\pflabel{patType}
        \item $\patType{\Gconsi{i \in \labelset}{\pctx_i}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}$ \BY{Rule (\ref{rule:patType-tpl}) on \pfref{patType}}
      \end{pfsteps*}
      \resetpfcounter
    \item[\text{(\ref{rule:cvalidP-UP-in})}] ~
      \begin{pfsteps*}
        \item $\cpv=\aceinjp{\ell}{\cpv'}$ \BY{assumption}
        \item $p=\aeinjp{\ell}{p'}$ \BY{assumption}
        \item $\tau=\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}$ \BY{assumption}
        \item $\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv'}{p'}{\tau'}$ \BY{assumption} \pflabel{cvalidP}
%        \item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
        \item $\patType{\pctx}{p'}{\tau'}$ \BY{IH, part 2 on \pfref{cvalidP}} \pflabel{patType}
        \item $\patType{\pctx}{\aeinjp{\ell}{p'}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}}$ \BY{Rule (\ref{rule:patType-inj}) on \pfref{patType}}
      \end{pfsteps*}
      \resetpfcounter
    \item[\text{(\ref{rule:cvalidP-UP-spliced})}] ~
      \begin{pfsteps*}
        \item $\cpv=\acesplicedp{m}{n}{\ctau}$ \BY{assumption}
        \item $\cvalidT{\emptyset}{\tsceneUP{\uDelta}{b}}{\ctau}{\tau}$ \BY{assumption}
        \item $\parseUExp{\bsubseq{b}{m}{n}}{\upv}$ \BY{assumption}
        \item $\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}$ \BY{assumption} \pflabel{patExpands}
        \item $\patType{\pctx}{p}{\tau}$ \BY{IH, part 1 on \pfref{patExpands}}
      \end{pfsteps*}
      \resetpfcounter
  \end{byCases}
  \end{enumerate}
The mutual induction can be shown to be well-founded by showing that the following numeric metric on the judgements that we induct on is decreasing:
\begin{align*}
\sizeof{\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}} & = \sizeof{\upv}\\
\sizeof{{\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}}} & = \sizeof{b}
\end{align*}
where $\sizeof{b}$ is the length of $b$ and $\sizeof{\upv}$ is the sum of the lengths of the literal bodies in $\upv$, as defined in Sec. \ref{appendix:SES-syntax}.

The only case in the proof of part 1 that invokes part 2 is Case (\ref{rule:patExpands-apuptsm}). There, we have that the metric remains stable: \begin{align*}
 & \sizeof{\patExpands{\upctx}{\uPhi}{\utsmap{\tsmv}{b}}{p}{\tau}}\\
=& \sizeof{{\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}}}\\
=&\sizeof{b}\end{align*}

The only case in the proof of part 2 that invokes part 1 is Case (\ref{rule:cvalidP-UP-spliced}). There, we have that $\parseUPat{\bsubseq{b}{m}{n}}{\upv}$ and the IH is applied to the judgement $\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}$. Because the metric is stable when passing from part 1 to part 2, we must have that it is strictly decreasing in the other direction:
\[\sizeof{\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}} < \sizeof{{\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\acesplicedp{m}{n}{\ctau}}{p}{\tau}}}\]
i.e. by the definitions above, 
\[\sizeof{\upv} < \sizeof{b}\]

This is established by appeal to Condition \ref{condition:body-subsequences}, which states that subsequences of $b$ are no longer than $b$, and the Condition \ref{condition:pattern-parsing}, which states that an unexpanded pattern constructed by parsing a textual sequence $b$ is strictly smaller, as measured by the metric defined above, than the length of $b$, because some characters must necessarily be used to apply the pattern TLM and delimit each literal body. Combining Conditions \ref{condition:body-subsequences} and \ref{condition:pattern-parsing}, we have that $\sizeof{\upv} < \sizeof{b}$ as needed.
\end{proof}

\end{grayparbox}
\subsection{Typed Expression Expansion}\label{appendix:SES-typed-expression-expansion-metatheory}
\begin{theorem}[Typed Expansion (Strong)]\label{thm:typed-expansion-full-U} ~
\begin{enumerate}
  \item \begin{enumerate}
    \item If $\expandsSG{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\ue}{e}{\tau}$ then $\hastypeU{\Delta}{\Gamma}{e}{\tau}$.
    \item \graytxtbox{If $\ruleExpands{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\urv}{r}{\tau}{\tau'}$  then $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$.}
  \end{enumerate}
  \item \begin{enumerate}
    \item If $\cvalidE{\Delta}{\Gamma}{\esceneSG{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}$ and $\Delta \cap \Delta_\text{app}=\emptyset$ and $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ then $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e}{\tau}$. 
    \item \begin{grayparbox}If $\cvalidR{\Delta}{\Gamma}{\esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}}{\crv}{r}{\tau}{\tau'}$ and $\Delta \cap \Delta_\text{app}=\emptyset$ and $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ then $\ruleType{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{r}{\tau}{\tau'}$.\end{grayparbox}
  \end{enumerate}
\end{enumerate}
\end{theorem}
\begin{proof}
By mutual rule induction over Rules (\ref{rules:expandsU}), \graytxtbox{Rule (\ref{rule:ruleExpands}),} Rules (\ref{rules:cvalidE-U}) \graytxtbox{and Rule (\ref{rule:cvalidR-UP})}.

\begin{enumerate}
\item In the following, let $\uDelta=\uDD{\uD}{\Delta}$ and $\uGamma=\uGG{\uG}{\Gamma}$.
  \begin{enumerate}
  \item 
  \begin{byCases} \item[\text{(\ref{rule:expandsU-var})}] ~
\begin{pfsteps}
  \item \ue=\ux \BY{assumption}
  \item e=x \BY{assumption}
  \item \Gamma=\Gamma', \Ghyp{x}{\tau} \BY{assumption}
  \item \hastypeU{\Delta}{\Gamma', \Ghyp{x}{\tau}}{x}{\tau} \BY{Rule (\ref{rule:hastypeU-var})}
\end{pfsteps}
\resetpfcounter

\item[\text{(\ref{rule:expandsU-asc})}] ~
\begin{pfsteps}
  \item \ue=\asc{\ue'}{\utau} \BY{assumption}
  \item \expandsTU{\uDelta}{\utau}{\tau} \BY{assumption} \pflabel{expandsTU}
  \item \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue'}{e}{\tau} \BY{assumption} \pflabel{expandsSG}
  \item \hastypeU{\Delta}{\Gamma}{e}{\tau} \BY{IH, part 1(a) on \pfref{expandsSG}}
\end{pfsteps}
\resetpfcounter

\item[\text{(\ref{rule:expandsU-letsyn})}] ~
\begin{pfsteps}
  \item \ue=\letsyn{\ux}{\ue_1}{\ue_2} \BY{assumption}
  \item e =   \aeap{\aelam{\tau_1}{x}{e_2}}{e_1} \BY{assumption}
  \item     \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue_1}{e_1}{\tau_1} \BY{assumption} \pflabel{expandsSG1}
  \item     \expandsSG{\uDelta}{\uGamma, \uGhyp{\ux}{x}{\tau_1}}{\uPsi}{\uPhi}{\ue_2}{e_2}{\tau} \BY{assumption} \pflabel{expandsSG2}
  \item \hastypeU{\Delta}{\Gamma}{e_1}{\tau_1} \BY{IH, part 1(a) on \pfref{expandsSG1}} \pflabel{hastype1}
  \item \hastypeU{\Delta}{\Gamma, x : \tau}{e_2}{\tau} \BY{IH, part 1(a) on \pfref{expandsSG2}} \pflabel{hastype2}
  \item \hastypeU{\Delta}{\Gamma}{\aelam{\tau_1}{x}{e_2}}{\aparr{\tau_1}{\tau}} \BY{Rule (\ref{rule:hastypeU-lam}) on \pfref{hastype2}} \pflabel{hastype3}
  \item \hastypeU{\Delta}{\Gamma}{\aeap{\aelam{\tau_1}{x}{e_2}}{e_1}}{\tau} \BY{Rule (\ref{rule:hastypeU-ap}) on \pfref{hastype3} and \pfref{hastype1}}
\end{pfsteps}
\resetpfcounter

\item[\text{(\ref{rule:expandsU-lam})}] ~
\begin{pfsteps}
  \item \ue=\lam{\ux}{\utau_1}{\ue'} \BY{assumption}
  \item e=\aelam{\tau_1}{x}{e'} \BY{assumption}
  \item \tau=\aparr{\tau_1}{\tau_2} \BY{assumption}
  \item \expandsTU{\uDelta}{\utau_1}{\tau_1} \BY{assumption} \pflabel{istype}
  \item \expandsSG{\uDelta}{\uGamma, \uGhyp{\ux}{x}{\tau_1}}{\uPsi}{\uPhi}{\ue'}{e'}{\tau_2} \BY{assumption} \pflabel{expandsU}
%  \item \uetsmenv{\Delta}{\Psi} \BY{assumption} \pflabel{uetsmenv}
  \item \istypeU{\Delta}{\tau_1} \BY{Lemma \ref{lemma:type-expansion-U} on \pfref{istype}} \pflabel{istype2}
  \item \hastypeU{\Delta}{\Gamma, \Ghyp{x}{\tau_1}}{e'}{\tau_2} \BY{IH, part 1(a) on \pfref{expandsU}} \pflabel{hastypeU}
  \item \hastypeU{\Delta}{\Gamma}{\aelam{\tau_1}{x}{e'}}{\aparr{\tau_1}{\tau_2}} \BY{Rule (\ref{rule:hastypeU-lam}) on \pfref{istype2} and \pfref{hastypeU}}
\end{pfsteps}
\resetpfcounter

\item[\text{(\ref{rule:expandsU-ap})}] ~
\begin{pfsteps}
  \item \ue=\ap{\ue_1}{\ue_2} \BY{assumption}
  \item e=\aeap{e_1}{e_2} \BY{assumption}
  \item \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue_1}{e_1}{\aparr{\tau_2}{\tau}} \BY{assumption}\pflabel{expandsU1}
  \item \expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue_2}{e_2}{\tau_2} \BY{assumption}\pflabel{expandsU2}
%  \item \uetsmenv{\Delta}{\Psi} \BY{assumption} \pflabel{uetsmenv}
  \item \hastypeU{\Delta}{\Gamma}{e_1}{\aparr{\tau_2}{\tau}} \BY{IH, part 1(a) on \pfref{expandsU1}}\pflabel{hastypeU1}
  \item \hastypeU{\Delta}{\Gamma}{e_2}{\tau_2} \BY{IH, part 1(a) on \pfref{expandsU2}}\pflabel{hastypeU2}
  \item \hastypeU{\Delta}{\Gamma}{\aeap{e_1}{e_2}}{\tau} \BY{Rule (\ref{rule:hastypeU-ap}) on \pfref{hastypeU1} and \pfref{hastypeU2}}
\end{pfsteps}
\resetpfcounter

\item[\text{(\ref{rule:expandsU-tlam})}~\textbf{through}~\text{(\ref{rule:expandsU-case})}] These cases follow analagously, i.e. we apply Lemma \ref{lemma:type-expansion-U} to or over the type expansion premises and the IH part 1(a) to or over the typed expression expansion premises and then apply the corresponding typing rule in Rules (\ref{rule:hastypeU-tlam}) through (\ref{rule:hastypeU-case}).
\\
\item[\text{(\ref{rule:expandsU-syntax})}] ~ 
\begin{pfsteps}
  \item \ue=\uesyntax{\tsmv}{\utau'}{\eparse}{\ue'} \BY{assumption}
  \item \expandsTU{\uDelta}{\utau'}{\tau'} \BY{assumption} \pflabel{expandsTU}
 \item \hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultExp}} \BY{assumption}\pflabel{eparse}
  \item \expandsSG{\uDelta}{\uGamma}{\uPsi, \uShyp{\tsmv}{a}{\tau'}{\eparse}}{\uPhi}{\ue'}{e}{\tau} \BY{assumption}\pflabel{expandsU}
%  \item \uetsmenv{\Delta}{\Psi} \BY{assumption}\pflabel{uetsmenv1}
 \item \istypeU{\Delta}{\tau'} \BY{Lemma \ref{lemma:type-expansion-U} to \pfref{expandsTU}} \pflabel{istype}
%  \item \uetsmenv{\Delta}{\Psi, \xuetsmbnd{\tsmv}{\tau'}{\eparse}} \BY{Definition \ref{def:seTLM-def-ctx-formation} on \pfref{uetsmenv1}, \pfref{istype} and \pfref{eparse}}\pflabel{uetsmenv3}
  \item \hastypeU{\Delta}{\Gamma}{e}{\tau} \BY{IH, part 1(a) on \pfref{expandsU}}
\end{pfsteps}
\resetpfcounter 

\item[\text{(\ref{rule:expandsU-tsmap})}] ~ 
\begin{pfsteps}
  \item \ue=\utsmap{\tsmv}{b} \BY{assumption}
  \item \uA = \uA', \vExpands{\tsmv}{a} \BY{assumption}
  \item \Psi=\Psi', \xuetsmbnd{a}{\tau}{\eparse} \BY{assumption}
  \item \encodeBody{b}{\ebody} \BY{assumption}
  \item \evalU{\eparse(\ebody)}{\aein{\mathtt{SuccessE}}{\ecand}} \BY{assumption}
  \item \decodeCondE{\ecand}{\ce} \BY{assumption}
  \item \cvalidE{\emptyset}{\emptyset}{\esceneSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau} \BY{assumption}\pflabel{cvalidE}
%  \item \uetsmenv{\Delta}{\Psi} \BY{assumption} \pflabel{uetsmenv}
  \item \emptyset \cap \Delta = \emptyset \BY{finite set intersection} \pflabel{delta-cap}
  \item {\emptyset} \cap \domof{\Gamma} = \emptyset \BY{finite set intersection} \pflabel{gamma-cap}
  \item \hastypeU{\emptyset \cup \Delta}{\emptyset \cup \Gamma}{e}{\tau} \BY{IH, part 2(a) on \pfref{cvalidE}, \pfref{delta-cap}, and \pfref{gamma-cap}} \pflabel{penultimate}
  \item \hastypeU{\Delta}{\Gamma}{e}{\tau} \BY{finite set and finite function identity over \pfref{penultimate}}
\end{pfsteps}
\resetpfcounter
\end{byCases}
\end{enumerate}
\begin{grayparbox}
\begin{enumerate}
\item[\hphantom{(a)}] \begin{byCases}
    \item[\text{(\ref{rule:expandsU-match})}] ~
      \begin{pfsteps*}
        \item $\ue=\matchwith{\ue'}{\seqschemaX{\urv}}$ \BY{assumption}
        \item $e=\aematchwith{n}{e'}{\seqschemaX{r}}$ \BY{assumption}
        \item $\expandsUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue'}{e'}{\tau'}$ \BY{assumption} \pflabel{expandsUP}
        % \item $\istypeU{\Delta}{\tau}$ \BY{assumption}\pflabel{istype}
        % \item $\expandsTU{\uDelta}{\utau}{\tau}$ \BY{assumption} \pflabel{expandsTU}
        \item $\{\ruleExpands{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\urv_i}{r_i}{\tau'}{\tau}\}_{1 \leq i \leq n}$ \BY{assumption}\pflabel{ruleExpands}
        \item $\hastypeU{\Delta}{\Gamma}{e'}{\tau'}$ \BY{IH, part 1(a) on \pfref{expandsUP}}\pflabel{hasType}
        \item $\{\ruleType{\Delta}{\Gamma}{r_i}{\tau'}{\tau}\}_{1 \leq i \leq n}$ \BY{IH, part 1(b) over \pfref{ruleExpands}}\pflabel{ruleType}
        \item $\hastypeU{\Delta}{\Gamma}{\aematchwith{n}{e'}{\seqschemaX{r}}}{\tau}$ \BY{Rule (\ref{rule:hastypeUP-match}) on \pfref{hasType} and \pfref{ruleType}}
      \end{pfsteps*}
      \resetpfcounter

    \item[\text{(\ref{rule:expandsU-defuptsm})}] ~
      \begin{pfsteps}
          \item \ue=\usyntaxup{\tsmv}{\utau'}{\eparse}{\ue'} \BY{assumption}
          \item \expandsTU{\uDelta}{\utau'}{\tau'} \BY{assumption} \pflabel{expandsTU}
         \item \hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultExp}} \BY{assumption}\pflabel{eparse}
          \item \expandsUP{\uDelta}{\uGamma}{\uPsi}{\uPhi, \uPhyp{\tsmv}{a}{\tau'}{\eparse}}{\ue'}{e}{\tau} \BY{assumption}\pflabel{expandsU}
        %  \item \uetsmenv{\Delta}{\Psi} \BY{assumption}\pflabel{uetsmenv1}
         \item \istypeU{\Delta}{\tau'} \BY{Lemma \ref{lemma:type-expansion-U} to \pfref{expandsTU}} \pflabel{istype}
        %  \item \uetsmenv{\Delta}{\Psi, \xuetsmbnd{\tsmv}{\tau'}{\eparse}} \BY{Definition \ref{def:seTLM-def-ctx-formation} on \pfref{uetsmenv1}, \pfref{istype} and \pfref{eparse}}\pflabel{uetsmenv3}
          \item \hastypeU{\Delta}{\Gamma}{e}{\tau} \BY{IH, part 1(a) on \pfref{expandsU}}
        \end{pfsteps}
        \resetpfcounter 
  \end{byCases}
  \end{enumerate}
  \end{grayparbox}
  \vspace{-4px}\begin{grayparbox}\vspace{4px}
  \begin{enumerate}
  \item[(b)] \begin{byCases}
    \item[\text{(\ref{rule:ruleExpands})}] ~
      \begin{pfsteps*}
        \item $\urv=\matchrule{\upv}{\ue}$ \BY{assumption}
        \item $r=\aematchrule{p}{e}$ \BY{assumption}
        \item $\patExpands{\uGG{\uA'}{\pctx}}{\uPhi}{\upv}{p}{\tau}$ \BY{assumption} \pflabel{patExpands}
        \item $\expandsUP{\uDelta}{\uGG{{\uA}\uplus{\uA'}}{\Gcons{\Gamma}{\pctx}}}{\uPsi}{\uPhi}{\ue}{e}{\tau'}$ \BY{assumption} \pflabel{expandsUP}
        \item $\patType{\pctx}{p}{\tau}$ \BY{Theorem \ref{thm:typed-pattern-expansion}, part 1 on \pfref{patExpands}}\pflabel{patType}
        \item $\hastypeU{\Delta}{\Gcons{\Gamma}{\pctx}}{e}{\tau'}$ \BY{IH, part 1(a) on \pfref{expandsUP}} \pflabel{hasType}
        \item $\ruleType{\Delta}{\Gamma}{\aematchrule{p}{e}}{\tau}{\tau'}$ \BY{Rule (\ref{rule:ruleType}) on \pfref{patType} and \pfref{hasType}}
      \end{pfsteps*}
      \resetpfcounter
  \end{byCases}
  \end{enumerate}
  \end{grayparbox}

\item In the following, let $\uDelta=\uDD{\uD}{\Delta_\text{app}}$ and $\uGamma=\uGG{\uG}{\Gamma_\text{app}}$. \begin{enumerate}
  \item 
  \begin{byCases}
    \item[\text{(\ref{rule:cvalidE-U-var})}] ~
\begin{pfsteps*}
  \item $\ce=x$ \BY{assumption}
  \item $e=x$ \BY{assumption}
  \item $\Gamma=\Gamma', \Ghyp{x}{\tau}$ \BY{assumption}
  \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gamma', \Ghyp{x}{\tau}}{x}{\tau}$ \BY{Rule (\ref{rule:hastypeU-var})} \pflabel{hastypeU}
  \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma', \Ghyp{x}{\tau}}{\Gamma_\text{app}}}{x}{\tau}$ \BY{Lemma \ref{lemma:weakening-U} over $\Gamma_\text{app}$ to \pfref{hastypeU}}
\end{pfsteps*}
\resetpfcounter

\item[\text{(\ref{rule:cvalidE-U-lam})}] ~
\begin{pfsteps*}
  \item $\ce=\acelam{\ctau_1}{x}{\ce'}$ \BY{assumption}
  \item $e=\aelam{\tau_1}{x}{e'}$ \BY{assumption}
  \item $\tau=\aparr{\tau_1}{\tau_2}$ \BY{assumption}
  \item $\cvalidT{\Delta}{\tsceneU{\uDelta_\text{app}}{b}}{\ctau_1}{\tau_1}$ \BY{assumption} \pflabel{cvalidT}
  \item $\cvalidE{\Delta}{\Gamma, \Ghyp{x}{\tau_1}}{\esceneSG{\uDelta_\text{app}}{\uGamma_\text{app}}{\uPsi}{\uPhi}{b}}{\ce'}{e'}{\tau_2}$ \BY{assumption} \pflabel{cvalidE}
%  \item $\uetsmenv{\Delta_\text{app}}{\Psi}$ \BY{assumption} \pflabel{uetsmenv}
  \item $\Delta \cap \Delta_\text{app}=\emptyset$ \BY{assumption} \pflabel{delta-disjoint}
  \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ \BY{assumption} \pflabel{gamma-disjoint}
  \item $x \notin \domof{\Gamma_\text{app}}$ \BY{identification convention} \pflabel{x-fresh}
  \item $\domof{\Gamma, x : \tau_1} \cap \domof{\Gamma_\text{app}}=\emptyset$ \BY{\pfref{gamma-disjoint} and \pfref{x-fresh}} \pflabel{gamma-disjoint2}
  \item $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\tau_1}$ \BY{Lemma \ref{lemma:candidate-expansion-type-validation} on \pfref{cvalidT} and \pfref{delta-disjoint}} \pflabel{istype}
  \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma, \Ghyp{x}{\tau_1}}{\Gamma_\text{app}}}{e'}{\tau_2}$ \BY{IH, part 2(a) on \pfref{cvalidE}, \pfref{delta-disjoint} and \pfref{gamma-disjoint2}} \pflabel{hastype1}
  \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}, \Ghyp{x}{\tau_1}}{e'}{\tau_2}$ \BY{exchange over $\Gamma_\text{app}$ on \pfref{hastype1}} \pflabel{hastype2}
  \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{\aelam{\tau_1}{x}{e'}}{\aparr{\tau_1}{\tau_2}}$ \BY{Rule (\ref{rule:hastypeU-lam}) on \pfref{istype} and \pfref{hastype2}}
\end{pfsteps*}
\resetpfcounter

\item[\text{(\ref{rule:cvalidE-U-ap})}] ~
\begin{pfsteps*}
  \item $\ce=\aceap{\ce_1}{\ce_2}$ \BY{assumption}
  \item $e=\aeap{e_1}{e_2}$ \BY{assumption}
  \item $\cvalidE{\Delta}{\Gamma}{\esceneSG{\uDelta_\text{app}}{\uGamma_\text{app}}{\uPsi}{\uPhi}{b}}{\ce_1}{e_1}{\aparr{\tau_2}{\tau}}$ \BY{assumption} \pflabel{cvalidE1}
  \item $\cvalidE{\Delta}{\Gamma}{\esceneSG{\uDelta_\text{app}}{\uGamma_\text{app}}{\uPsi}{\uPhi}{b}}{\ce_2}{e_2}{\tau_2}$ \BY{assumption} \pflabel{cvalidE2}
%  \item $\uetsmenv{\Delta_\text{app}}{\Psi}$ \BY{assumption} \pflabel{uetsmenv}
  \item $\Delta \cap \Delta_\text{app}=\emptyset$ \BY{assumption} \pflabel{delta-disjoint}
  \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ \BY{assumption} \pflabel{gamma-disjoint}
  \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e_1}{\aparr{\tau_2}{\tau}}$ \BY{IH, part 2(a) on \pfref{cvalidE1}, \pfref{delta-disjoint} and \pfref{gamma-disjoint}} \pflabel{hastypeU1}
  \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e_2}{\tau_2}$ \BY{IH, part 2(a) on \pfref{cvalidE2}, \pfref{delta-disjoint} and \pfref{gamma-disjoint}} \pflabel{hastypeU2}
  \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{\aeap{e_1}{e_2}}{\tau}$ \BY{Rule (\ref{rule:hastypeU-ap}) on \pfref{hastypeU1} and \pfref{hastypeU2}}
\end{pfsteps*}
\resetpfcounter

\item[\text{(\ref{rule:cvalidE-U-tlam})}] ~
\begin{pfsteps}
  \item \ce=\acetlam{t}{\ce'} \BY{assumption}
  \item e = \aetlam{t}{e'} \BY{assumption}
  \item \tau = \aall{t}{\tau'}\BY{assumption}
  \item \cvalidE{\Delta, \Dhyp{t}}{\Gamma}{\esceneSG{\uDelta_\text{app}}{\uGamma_\text{app}}{\uPsi}{\uPhi}{b}}{\ce'}{e'}{\tau'} \BY{assumption} \pflabel{cvalidE}
%  \item \uetsmenv{\Delta_\text{app}}{\Psi} \BY{assumption} \pflabel{uetsmenv}
  \item \Delta \cap \Delta_\text{app}=\emptyset \BY{assumption} \pflabel{delta-disjoint}
  \item \domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset \BY{assumption} \pflabel{gamma-disjoint}
  \item \Dhyp{t} \notin \Delta_\text{app} \BY{identification convention}\pflabel{t-fresh}
  \item \Delta, \Dhyp{t} \cap \Delta_\text{app} = \emptyset \BY{\pfref{delta-disjoint} and \pfref{t-fresh}}\pflabel{delta-disjoint2}
  \item \hastypeU{\Dcons{\Delta, \Dhyp{t}}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e'}{\tau'} \BY{IH, part 2(a) on \pfref{cvalidE}, \pfref{delta-disjoint2} and \pfref{gamma-disjoint}}\pflabel{hastype1}
  \item \hastypeU{\Dcons{\Delta}{\Delta_\text{app}, \Dhyp{t}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e'}{\tau'} \BY{exchange over $\Delta_\text{app}$ on \pfref{hastype1}}\pflabel{hastype2}
  \item \hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{\aetlam{t}{e'}}{\aall{t}{\tau'}} \BY{Rule (\ref{rule:hastypeU-tlam}) on \pfref{hastype2}}
\end{pfsteps}
\resetpfcounter

\item[{\text{(\ref{rule:cvalidE-U-tap})}}~\textbf{through}~{\text{(\ref{rule:cvalidE-U-case})}}] These cases follow analagously, i.e. we apply the IH, part 2(a) to all proto-expression validation judgements, Lemma \ref{lemma:candidate-expansion-type-validation} to all proto-type validation judgements, the identification convention to ensure that extended contexts remain disjoint, weakening and exchange as needed, and the corresponding typing rule in Rules (\ref{rule:hastypeU-tap}) through (\ref{rule:hastypeU-case}).
\\

\item[\text{(\ref{rule:cvalidE-U-splicede})}] ~
\begin{pfsteps*}
  \item $\ce=\acesplicede{m}{n}{\ctau}$ \BY{assumption}
  \item $\escenev=\esceneU{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{b}$ \BY{assumption}
  \item   $\cvalidT{\emptyset}{\tsfrom{\escenev}}{\ctau}{\tau}$ \BY{assumption}
  \item $\parseUExp{\bsubseq{b}{m}{n}}{\ue}$ \BY{assumption}
  \item $\expandsU{\uDelta_\text{app}}{\uGamma_\text{app}}{\uPsi}{\ue}{e}{\tau}$ \BY{assumption} \pflabel{expands}
%  \item $\uetsmenv{\Delta_\text{app}}{\Psi}$ \BY{assumption} \pflabel{uetsmenv}
  \item $\Delta \cap \Delta_\text{app}=\emptyset$ \BY{assumption} \pflabel{delta-disjoint}
  \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ \BY{assumption} \pflabel{gamma-disjoint}
  \item $\hastypeU{\Delta_\text{app}}{\Gamma_\text{app}}{e}{\tau}$ \BY{IH, part 1 on \pfref{expands}} \pflabel{hastype}
  \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e}{\tau}$ \BY{Lemma \ref{lemma:weakening-U} over $\Delta$ and $\Gamma$ and exchange on \pfref{hastype}}
\end{pfsteps*}
\resetpfcounter
\end{byCases}
\end{enumerate}
\begin{grayparbox}
\begin{enumerate}
\item[\hphantom{(a)}] \begin{byCases}
    \item[\text{(\ref{rule:cvalidE-U-match})}] ~
      \begin{pfsteps*}
        \item $\ce=\acematchwith{n}{\ce'}{\seqschemaX{\crv}}$ \BY{assumption}
        \item $e=\aematchwith{n}{e'}{\seqschemaX{r}}$ \BY{assumption}
        \item $\cvalidE{\Delta}{\Gamma}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce'}{e'}{\tau'}$ \BY{assumption} \pflabel{cvalidE}        
        % \item $\istypeU{\Delta \cup \Delta_\text{app}}{\tau}$ \BY{assumption} \pflabel{istype}
        % \item $\cvalidT{\Delta}{\tsceneUP{\uDelta}{b}}{\ctau}{\tau}$ \BY{assumption} \pflabel{cvalidT}
        \item $\{\cvalidR{\Delta}{\Gamma}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\crv_i}{r_i}{\tau'}{\tau}\}_{1 \leq i \leq n}$ \BY{assumption} \pflabel{cvalidR}
        \item $\Delta \cap \Delta_\text{app} = \emptyset$ \BY{assumption} \pflabel{delta-disjoint}
        \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}} = \emptyset$ \BY{assumption} \pflabel{gamma-disjoint}
        \item $\hastypeU{\Delta \cup \Delta_\text{app}}{\Gamma \cup \Gamma_\text{app}}{e'}{\tau'}$ \BY{IH, part 2(a) on \pfref{cvalidE}, \pfref{delta-disjoint} and \pfref{gamma-disjoint}} \pflabel{hastype}
        \item $\ruleType{\Delta \cup \Delta_\text{app}}{\Gamma \cup \Gamma_\text{app}}{r}{\tau'}{\tau}$ \BY{IH, part 2(b) on \pfref{cvalidR}, \pfref{delta-disjoint} and \pfref{gamma-disjoint}} \pflabel{ruleType}
        \item $\hastypeU{\Delta \cup \Delta_\text{app}}{\Gamma \cup \Gamma_\text{app}}{\aematchwith{n}{e'}{\seqschemaX{r}}}{\tau}$ \BY{Rule (\ref{rule:hastypeUP-match}) on \pfref{hastype} and \pfref{ruleType}}
      \end{pfsteps*}
      \resetpfcounter
  \end{byCases}
    \end{enumerate}
  \end{grayparbox}\vspace{-3px}
  \begin{grayparbox}\vspace{3px}
  \begin{enumerate}
  \item[(b)] There is only one case. 
    \begin{byCases}
     \item[\text{(\ref{rule:cvalidR-UP})}] ~
      \begin{pfsteps*}
        \item $\crv=\acematchrule{p}{\ce}$ \BY{assumption}
        \item $r=\aematchrule{p}{e}$ \BY{assumption}
        \item $\patType{\pctx'}{p}{\tau}$ \BY{assumption} \pflabel{patType}
        \item $\cvalidE{\Delta}{\Gcons{\Gamma}{\pctx'}}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau'}$ \BY{assumption} \pflabel{cvalidE}
        \item $\Delta \cap \Delta_\text{app} = \emptyset$ \BY{assumption}\pflabel{delta-disjoint}
        \item $\domof{\Gamma} \cap \domof{\pctx'} = \emptyset$ \BY{identification convention}\pflabel{gamma-disjoint1}
        \item $\domof{\Gamma_\text{app}} \cap \domof{\pctx'} = \emptyset$ \BY{identification convention}\pflabel{gamma-disjoint2}
        \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}} = \emptyset$ \BY{assumption}\pflabel{gamma-disjoint3}
        \item $\domof{\Gcons{\Gamma}{\pctx'}} \cap \domof{\Gamma_\text{app}} = \emptyset$ \BY{standard finite set definitions and identities on \pfref{gamma-disjoint1}, \pfref{gamma-disjoint2} and \pfref{gamma-disjoint3}}\pflabel{gamma-disjoint4}
        \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gcons{\Gamma}{\pctx'}}{\Gamma_\text{app}}}{e}{\tau'}$ \BY{IH, part 2(a) on \pfref{cvalidE}, \pfref{delta-disjoint} and \pfref{gamma-disjoint4}}\pflabel{hastype}
        \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gcons{\Gamma}{\Gamma_\text{app}}}{\pctx'}}{e}{\tau'}$ \BY{exchange of $\pctx'$ and $\Gamma_\text{app}$ on \pfref{hastype}}\pflabel{hastype2}
        \item $\ruleType{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{\aematchrule{p}{e}}{\tau}{\tau'}$ \BY{Rule (\ref{rule:ruleType}) on \pfref{patType} and \pfref{hastype2}}
      \end{pfsteps*}
      \resetpfcounter
   \end{byCases} 
\end{enumerate}
\end{grayparbox}
\end{enumerate}
\vspace{10px}

The mutual induction can be shown to be well-founded by showing that the following numeric metric on the judgements that we induct on is decreasing:
\begin{align*}
\sizeof{\expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}} & = \sizeof{\ue}\\
\sizeof{\cvalidE{\Delta}{\Gamma}{\esceneSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}} & = \sizeof{b}
\end{align*}
where $\sizeof{b}$ is the length of $b$ and $\sizeof{\ue}$ is the sum of the lengths of the seTLM literal bodies in $\ue$, as defined in Sec. \ref{appendix:SES-syntax}.

The only case in the proof of part 1 that invokes part 2 is Case (\ref{rule:expandsU-tsmap}). There, we have that the metric remains stable: \begin{align*}
 & \sizeof{\expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\utsmap{\tsmv}{b}}{e}{\tau}}\\
=& \sizeof{\cvalidE{\emptyset}{\emptyset}{\esceneSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}}\\
=&\sizeof{b}\end{align*}

The only case in the proof of part 2 that invokes part 1 is Case (\ref{rule:cvalidE-U-splicede}). There, we have that $\parseUExp{\bsubseq{b}{m}{n}}{\ue}$ and the IH is applied to the judgement $\expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}$. Because the metric is stable when passing from part 1 to part 2, we must have that it is strictly decreasing in the other direction:
\[\sizeof{\expandsSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}} < \sizeof{\cvalidE{\Delta}{\Gamma}{\esceneSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\acesplicede{m}{n}{\ctau}}{e}{\tau}}\]
i.e. by the definitions above, 
\[\sizeof{\ue} < \sizeof{b}\]

This is established by appeal to Condition \ref{condition:body-subsequences}, which states that subsequences of $b$ are no longer than $b$, and Condition \ref{condition:body-parsing}, which states that an unexpanded expression constructed by parsing a textual sequence $b$ is strictly smaller, as measured by the metric defined above, than the length of $b$, because some characters must necessarily be used to apply a TLM and delimit each literal body. 
Combining these conditions, we have that $\sizeof{\ue} < \sizeof{b}$ as needed.
\end{proof}

\begin{theorem}[Typed Expression Expansion]\label{thm:typed-expansion-short-U} If $\expandsSG{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}\hspace{-3px}}{\uPsi}{\uPhi}{\ue}{e}{\tau}$ then $\hastypeU{\Delta}{\Gamma}{e}{\tau}$.
\end{theorem}
\begin{proof} This theorem follows immediately from Theorem \ref{thm:typed-expansion-full-U}, part 1(a). \end{proof}

% % \subsection{Expressibility}
% The following lemma establishes that each type can be expressed as a well-formed proto-type, under the same type formation context and any type splicing scene.
% \begin{lemma}[Proto-Expansion Type Expressibility]\label{lemma:proto-type-expressibility-U} If $\istypeU{\Delta}{\tau}$ then $\cvalidT{\Delta}{\tscenev}{\Cof{\tau}}{\tau}$. \end{lemma}
% \begin{proof}
% By rule induction over Rules (\ref{rules:istypeU}). In each case, we apply the IH on or over each premise, then apply the corresponding proto-type validation rule in Rules (\ref{rules:cvalidT-U}).
% \end{proof}

% The Type Expressibility Lemma establishes that every well-formed type, $\tau$, can be expressed as a well-formed unexpanded type, $\Uof{\tau}$. This requires defining the metafunction $\Uof{\Delta}$ which maps $\Delta$ onto an unexpanded type formation context as follows:
% \begin{align*}
% \Uof{\emptyset} &= \uDD{\emptyset}{\emptyset}\\
% \Uof{\Delta, \Dhyp{t}} &= \Uof{\Delta}, \uDhyp{\sigilof{t}}{t}
% \end{align*}
% \begin{lemma}[Type Expressibility]\label{lemma:type-expressibility} If $\istypeU{\Delta}{\tau}$ then $\expandsTU{\Uof{\Delta}}{\Uof{\tau}}{\tau}$.\end{lemma}
% \begin{proof} By rule induction over Rules (\ref{rules:istypeU}) using the definitions of $\Uof{\tau}$ and $\Uof{\Delta}$ above. In each case, we apply the IH to or over each premise, then apply the corresponding type expansion rule in Rules (\ref{rules:expandsTU}).\end{proof}


% The following lemma establishes that each well-typed expanded expression, $e$, can be expressed as a valid proto-expression, $\Cof{e}$, that is assigned the same type under any expression splicing scene.
% \begin{theorem}[Proto-Expansion Expression Expressibility]\label{theorem:proto-expressions-expressibility-U} If $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ then $\cvalidE{\Delta}{\Gamma}{\escenev}{\Cof{e}}{e}{\tau}$.\end{theorem}
% \begin{proof} By rule induction over Rules (\ref{rules:hastypeU}). The rule transformation above guarantees that this lemma holds by construction. In particular, in each case, we apply Lemma \ref{lemma:proto-type-expressibility-U} to or over each type formation premise, the IH to or over each typing premise, then apply the corresponding proto-expression validation rule in Rules (\ref{rule:cvalidE-U-var}) through (\ref{rule:cvalidE-U-case}).
% \end{proof}

% The following lemma establishes that each well-typed expanded expression, $e$, can be expressed as a valid ce-expression, $\Cof{e}$, that is assigned the same type under any expression splicing scene.
% \begin{theorem}[Candidate Expansion Expression Expressibility]\label{lemma:ce-expressions-expressibility-UP} Both of the following hold:
% \begin{enumerate}
% \item If $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ then $\cvalidE{\Delta}{\Gamma}{\escenev}{\Cof{e}}{e}{\tau}$.
% \item If $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$ then $\cvalidR{\Delta}{\Gamma}{\escenev}{\Cof{r}}{r}{\tau}{\tau'}$.
% \end{enumerate}
% \end{theorem}
% \begin{proof} By mutual rule induction over Rules (\ref{rules:hastypeUP}) and Rule (\ref{rule:ruleType}). 

% For part 1, we induct on the assumption. 
% \begin{byCases}
% \item[\text{(\ref{rule:hastypeUP-var}) through (\ref{rule:hastypeUP-in})}] In each of these cases, we apply Lemma \ref{lemma:ce-type-expressibility-U} to or over each type formation premise, the IH (part 1) to or over each typing premise, then apply the corresponding ce-expression validation rule in Rules (\ref{rule:cvalidE-UP-var}) through (\ref{rule:cvalidE-UP-in}).
% \item[\text{(\ref{rule:hastypeUP-match})}] ~
%   \begin{pfsteps}
%   \item e = \aematchwith{n}{e'}{\seqschemaX{r}} \BY{assumption}
%   \item \Cof{e} = \acematchwith{n}{\Cof{\tau}}{\Cof{e'}}{\seqschemaXx{\Cofv}{r}} \BY{definition of $\Cof{e}$}
%   \item \hastypeU{\Delta}{\Gamma}{e'}{\tau'} \BY{assumption} \pflabel{hasType}
%   \item \istypeU{\Delta}{\tau} \BY{assumption} \pflabel{isType}
%   \item \{\ruleType{\Delta}{\Gamma}{r_i}{\tau'}{\tau}\}_{1 \leq i \leq n} \BY{assumption} \pflabel{ruleType}
%   \item \cvalidE{\Delta}{\Gamma}{\escenev}{\Cof{e'}}{e'}{\tau'} \BY{IH, part 1 on \pfref{hasType}} \pflabel{cvalidE}
%   \item \cvalidT{\Delta}{\tsfrom{\escenev}}{\Cof{\tau}}{\tau} \BY{Lemma \ref{lemma:candidate-expansion-type-validation} on \pfref{isType}} \pflabel{cvalidT}
%   \item \{\cvalidR{\Delta}{\Gamma}{\escenev}{\Cof{r_i}}{r_i}{\tau'}{\tau}\}_{1 \leq i \leq n} \BY{IH, part 2 over \pfref{ruleType}} \pflabel{cvalidR}
%   \item \cvalidE{\Delta}{\Gamma}{\escenev}{\acematchwith{n}{\Cof{\tau}}{\Cof{e'}}{\seqschemaXx{\Cofv}{r}}}{\aematchwith{n}{e'}{\seqschemaX{r}}}{\tau} \BY{Rule (\ref{rule:cvalidE-UP-match}) on \pfref{cvalidE}, \pfref{cvalidT} and \pfref{cvalidR}}
%   \end{pfsteps}
% \end{byCases}
% \resetpfcounter

% For part 2, we induct on the assumption. There is only one case.
% \begin{byCases}
% \item[\text{(\ref{rule:ruleType})}] ~
%   \begin{pfsteps}
%     \item r = \aematchrule{p}{e} \BY{assumption}
%     \item \Cof{r} = \acematchrule{p}{\Cof{e}} \BY{definition of $\Cof{r}$}
%     \item \patType{\pctx}{p}{\tau} \BY{assumption} \pflabel{patType}
%     \item \hastypeU{\Delta}{\Gcons{\Gamma}{\pctx}}{e}{\tau'} \BY{assumption} \pflabel{hasType}
%     \item \cvalidE{\Delta}{\Gcons{\Gamma}{\pctx}}{\escenev}{\Cof{e}}{e}{\tau'} \BY{IH, part 1 on \pfref{hasType}} \pflabel{cvalidE}
%     \item \cvalidR{\Delta}{\Gamma}{\escenev}{\acematchrule{p}{\Cof{e}}}{\aematchrule{p}{e}}{\tau}{\tau'} \BY{Rule (\ref{rule:cvalidR-UP}) on \pfref{patType} and \pfref{cvalidE}}
%   \end{pfsteps}
%   \resetpfcounter
% \end{byCases}
% \end{proof}

% The following lemma establishes that every well-typed expanded pattern that generates no hypotheses can be expressed as a ce-pattern.
% \begin{lemma}[Candidate Expansion Pattern Expressibility]\label{lemma:ce-pattern-expressibility-U} If $\patType{\emptyset}{p}{\tau}$ then $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\Cof{p}}{p}{\tau}$.\end{lemma}
% \begin{proof} By rule induction over Rules (\ref{rules:patType}).
% \begin{byCases}
% \item[\text{(\ref{rule:patType-var})}] This case does not apply.
% \item[\text{(\ref{rule:patType-wild})}] ~
%   \begin{pfsteps*}
%     \item $p=\aewildp$ \BY{assumption}
%     \item $\Cof{p}=\acewildp$ \BY{definition of $\Cof{p}$}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\acewildp}{\aewildp}{\tau}$ \BY{Rule (\ref{rule:cvalidP-UP-wild})}
%   \end{pfsteps*}
%   \resetpfcounter
% \item[\text{(\ref{rule:patType-fold})}] ~
%   \begin{pfsteps*}
%     \item $p=\aefoldp{p'}$ \BY{assumption}
%     \item $\Cof{p}=\acefoldp{\Cof{p'}}$ \BY{definition of $\Cof{p}$}
%     \item $\tau=\arec{t}{\tau'}$ \BY{assumption}
%     \item $\patType{\emptyset}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{assumption} \pflabel{patType}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\Cof{p'}}{p}{[\arec{t}{\tau'}/t]\tau'}$ \BY{IH on \pfref{patType}} \pflabel{cvalidP}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\acefoldp{\Cof{p'}}}{\aefoldp{p'}}{\arec{t}{\tau'}}$ \BY{Rule (\ref{rule:cvalidP-UP-fold}) on \pfref{cvalidP}}
%   \end{pfsteps*}
%   \resetpfcounter
% \item[\text{(\ref{rule:patType-tpl})}] ~
%   \begin{pfsteps*}
%     \item $p=\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}$ \BY{assumption}
%     \item $\Cof{p}=\acetpl{\labelset}{\mapschemax{\Cofv}{p}{i}{\labelset}}$ \BY{definition of $\Cof{p}$}
%     \item $\tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$ \BY{assumption}
%     \item $\{\patType{\emptyset}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{assumption} \pflabel{patType}
%     \item $\{\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\Cof{p_i}}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{IH over \pfref{patType}} \pflabel{cvalidP}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\acetpl{\labelset}{\mapschemax{\Cofv}{p}{i}{\labelset}}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}$ \BY{Rule (\ref{rule:cvalidP-UP-tpl}) on \pfref{cvalidP}}
%   \end{pfsteps*}
%   \resetpfcounter
% \item[\text{(\ref{rule:patType-inj})}] ~
%   \begin{pfsteps*}
%     \item $p=\aeinjp{\ell}{p'}$ \BY{assumption}
%     \item $\Cof{p}=\aceinjp{\ell}{\Cof{p'}}$ \BY{definition of $\Cof{p}$}
%     \item $\tau=\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}$ \BY{assumption}
%     \item $\patType{\emptyset}{p'}{\tau'}$ \BY{assumption}\pflabel{patType}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\Cof{p'}}{p'}{\tau'}$ \BY{IH on \pfref{patType}}\pflabel{cvalidP}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\aceinjp{\ell}{\Cof{p'}}}{\aeinjp{\ell}{p'}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}}$ \BY{Rule (\ref{rule:cvalidP-UP-in}) on \pfref{cvalidP}}
%   \end{pfsteps*}
%   \resetpfcounter
% \end{byCases}
% \end{proof}

% \subsubsection{Expressibility}
% The following lemma establishes that each well-typed expanded pattern can be expressed as an unexpanded pattern matching values of the same type and generating the same hypotheses and corresponding identifier updates. The metafunction $\Uof{\pctx}$ maps $\pctx$ to an unexpanded typing context as follows:
% \begin{align*}
% \Uof{\emptyset} & = \uGG{\emptyset}{\emptyset}\\
% \Uof{\pctx, x : \tau} & = \Uof{\pctx}, \uGhyp{\sigilof{x}}{x}{\tau}\\
% \Uof{\Gconsi{i \in \labelset}{\pctx_i}} & = \Gconsi{i \in \labelset}{\Uof{\pctx_i}}
% \end{align*}
% \begin{lemma}[Pattern Expressibility]\label{lemma:pattern-expressibility} If $\patType{\pctx}{p}{\tau}$ then $\patExpands{\Uof{\pctx}}{\uPhi}{\Uof{p}}{p}{\tau}$.\end{lemma}
% \begin{proof} By rule induction over Rules (\ref{rules:patType}), using the definitions of $\Uof{\pctx}$ and $\Uof{p}$ given above. In each case, we can apply the IH to or over each premise, then apply the corresponding rule in Rules (\ref{rules:patExpands}).\end{proof}

% We can now establish the Expressibility Theorem -- that each well-typed expanded expression, $e$, can be expressed as an unexpanded expression, $\ue$, and assigned the same type under the corresponding contexts.

% \begin{theorem}[Expressibility] Both of the following hold:
% \begin{enumerate}
% \item If $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ then $\expandsUP{\Uof{\Delta}}{\Uof{\Gamma}}{\uPsi}{\uPhi}{\Uof{e}}{e}{\tau}$.
% \item If $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$ then $\ruleExpands{\Uof{\Delta}}{\Uof{\Gamma}}{\uPsi}{\uPhi}{\Uof{r}}{r}{\tau}{\tau'}$.
% \end{enumerate}
% \end{theorem}
% \begin{proof} By mutual rule induction over Rules (\ref{rules:hastypeUP}) and Rule (\ref{rule:ruleType}). 

% For part 1, we induct on the assumption. The rule transformation defined above guarantees that this part holds by its construction. In particular, in each case, we can apply Lemma \ref{lemma:type-expressibility} to or over each type formation premise, the IH (part 1) to or over each typing premise, the IH (part 2) over each rule typing premise, then apply the corresponding rule in Rules (\ref{rules:expandsUP}).

% For part 2, we induct on the assumption. There is only one case:
% \begin{byCases}
% \item[(\ref{rule:ruleType})] ~
% \begin{pfsteps*}
% \item $r = \aematchrule{p}{e}$ \BY{assumption}
% \item $\patType{\pctx}{p}{\tau}$ \BY{assumption} \pflabel{patType}
% \item $\hastypeU{\Delta}{\Gamma \cup \pctx}{e}{\tau'}$ \BY{assumption} \pflabel{hasType}
% \item $\Uof{\Gamma}=\uGG{\uG}{\Gamma}$, for some $\uG$ \BY{definition of $\Uof{\Gamma}$}
% \item $\Uof{\pctx} =\uGG{\uG'}{\pctx}$, for some $\uG'$ \BY{definition of $\Uof{\pctx}$}
% \item $\Uof{\Gamma \cup \pctx} = \uGG{\uG \uplus \uG'}{\Gamma \cup \pctx}$ \BY{definition of $\Uof{\pctx}$}
% \item $\Uof{r} = \aumatchrule{\Uof{p}}{\Uof{e}}$ \BY{definition of $\Uof{r}$}
% \item $\patExpands{\uGG{\uG'}{\pctx}}{\uPhi}{\Uof{p}}{p}{\tau}$ \BY{Lemma \ref{lemma:pattern-expressibility} on \pfref{patType}} \pflabel{patExpands}
% \item $\expandsUP{\uDelta}{\uGG{\uGcons{\uG}{\uG'}}{\Gcons{\Gamma}{\pctx}}}{\uPsi}{\uPhi}{\Uof{e}}{e}{\tau'}$ \BY{IH, part 1 on \pfref{hasType}} \pflabel{expandsUP}
% \item $\ruleExpands{\Uof{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\aumatchrule{\Uof{p}}{\Uof{e}}}{\aematchrule{p}{e}}{\tau}{\tau'}$ \BY{Rule (\ref{rule:ruleExpands}) on \pfref{patExpands} and \pfref{expandsUP}}
% \end{pfsteps*}
% \resetpfcounter
% \end{byCases}
% \end{proof}

\subsection{Abstract Reasoning Principles}\label{appendix:SES-reasoning-principles}
\begin{lemma}[Proto-Type Expansion Decomposition] 
\label{thm:proto-type-expansion-decomposition-SES}
If $\cvalidT{\Delta}{\tsceneU{\uDD{\uD}{\Delta_\text{app}}}{b}}{\ctau}{\tau}$ where $\segof{\ctau} = \sseq{\acesplicedt{m_i}{n_i}}{n}$ then all of the following hold:
\begin{enumerate}
\item $\sseq{\expandsTU{\uDD{\uD}{\Delta_\text{app}}}{
  \parseUTypF{\bsubseq{b}{m_i}{n_i}}
}{\tau_i}}{n}$
% \item $\sseq{\istypeU{\Delta_\text{app}}{\tau_i}}{n}$
\item $\tau = [\sseq{\tau_i/t_i}{n}]\tau'$ for some $\tau'$ and fresh $\sseq{t_i}{n}$ (i.e.  $\sseq{t_i \notin \domof{\Delta}}{n}$ and $\sseq{t_i \notin \domof{\Delta_\text{app}}}{n}$)
\item $\mathsf{fv}(\tau') \subset \domof{\Delta} \cup \sseq{t_i}{n}$
\end{enumerate}
\end{lemma}
\begin{proof}
By rule induction over Rules (\ref{rules:cvalidT-U}). In the following, let $\uDelta = \uDD{\uD}{\Delta_\text{app}}$ and $\tscenev=\tsceneU{\uDelta}{b}$.
\begin{byCases}
  \item[\text{(\ref{rule:cvalidT-U-tvar})}] 
    \begin{pfsteps}
    \item \ctau = t \BY{assumption}
    \item \tau = t \BY{assumption}
    \item \Delta = \Delta', \Dhyp{t} \BY{assumption} \pflabel{Delta}
    \item \segof{\ctau} = \emptyset \BY{definition}
    \item \fvof{t} = \{ t \} \BY{definition} \pflabel{fv}
    \item \{ t \} \subset \domof{\Delta} \cup \emptyset \BY{definition} \pflabel{subset}
    \end{pfsteps}
    The conclusions hold as follows:
    \begin{enumerate}
    \item This conclusion holds trivially because $n=0$.
    % \item This conclusion holds trivially because $n=0$.
    \item Choose $\tau'=t$ and $\emptyset$.
    \item \pfref{subset}
    \end{enumerate}
    \resetpfcounter
  \item[\text{(\ref{rule:cvalidT-U-parr})}] 
    \begin{pfsteps}
    \item \ctau = \aceparr{\ctau_1}{\ctau_2} \BY{assumption}
    \item \tau = \aparr{\tau'_1}{\tau'_2} \BY{assumption}
    \item \cvalidT{\Delta}{\tscenev}{\ctau_1}{\tau'_1} \BY{assumption} \pflabel{cvalidT1}
    \item \cvalidT{\Delta}{\tscenev}{\ctau_2}{\tau'_2} \BY{assumption} \pflabel{cvalidT2}
    \item \segof{\ctau} = \segof{\ctau_1} \cup \segof{\ctau_2} \BY{definition} \pflabel{summaryOf}
    \item \segof{\ctau_1} = \sseqS{\acesplicedt{m_{i}}{n_{i}}}{0}{n'} \BY{definition} \pflabel{summaryOf1}
    \item \segof{\ctau_2} = \sseqS{\acesplicedt{m_{i}}{n_{i}}}{n'}{n} \BY{definition} \pflabel{summaryOf2}
    \item \sseq{\expandsTU{\uDD{\uD}{\Delta_\text{app}}}{
  \parseUTypF{\bsubseq{b}{m_i}{n_i}}
}{\tau_i}}{n'} 
    \BY{IH on \pfref{cvalidT1} and \pfref{summaryOf1}}
    \pflabel{expands1}
    % \item \sseq{\istypeU{\Delta_\text{app}}{\tau_i}}{n}
    \item \tau'_1 = [\sseq{\tau_i/t_i}{n'}]\tau''_1 \text{~for some $\tau''_1$ and fresh $\sseq{t_i}{n'}$} \BY{IH on \pfref{cvalidT1} and \pfref{summaryOf1}
        \pflabel{decompose1}}
    % \item \istypeU{\Delta \cup \sseq{\Dhyp{t_i}}{n'}}{\tau''_1}
    %     \BY{IH on \pfref{cvalidT1} and \pfref{summaryOf1}}
    %     \pflabel{istype1}
    \item \fvof{\tau''_1} \subset \domof{\Delta} \cup \sseq{t_i}{n'} 
        \BY{IH on \pfref{cvalidT1} and \pfref{summaryOf1}
        \pflabel{fv1}}
    \item \sseqS{\expandsTU{\uDD{\uD}{\Delta_\text{app}}}{
  \parseUTypF{\bsubseq{b}{m_i}{n_i}}
}{\tau_i}}{n'}{n} 
    \BY{IH on \pfref{cvalidT2} and \pfref{summaryOf2}}
    \pflabel{expands2}
    % \item \sseq{\istypeU{\Delta_\text{app}}{\tau_i}}{n}
    \item \tau'_2 = [\sseqS{\tau_i/t_i}{n'}{n}]\tau''_2 \text{~for some $\tau''_2$ and fresh $\sseqS{t_i}{n'}{n}$}
        \BY{IH on \pfref{cvalidT2} and \pfref{summaryOf2}}
        \pflabel{decompose2}
    % \item \istypeU{\Delta \cup \sseqS{\Dhyp{t_i}}{n'}{n}}{\tau''_2}
    %     \BY{IH on \pfref{cvalidT2} and \pfref{summaryOf2}}
    %     \pflabel{istype2}
    \item \fvof{\tau''_2} \subset \domof{\Delta} \cup \sseqS{t_i}{n'}{n}
        \BY{IH on \pfref{cvalidT2} and \pfref{summaryOf2}}
        \pflabel{fv2}
    \item \sseq{t_i}{n'} \cap \sseqS{t_i}{n'}{n} = \emptyset \BY{identification convention} \pflabel{fvid}
    \item \fvof{\tau''_1} \subset \domof{\Delta} \cup \sseq{t_i}{n} \BY{\pfref{fv1} and \pfref{fvid}} \pflabel{fvf1}
\item \fvof{\tau''_2} \subset \domof{\Delta} \cup \sseq{t_i}{n} \BY{\pfref{fv2} and \pfref{fvid}}     \pflabel{fvf2}
    \item \tau_1' = [\sseq{\tau_i/t_i}{n}]\tau_1'' \BY{substitution properties and \pfref{decompose1} and \pfref{fvid}} \pflabel{fdecompose1}
    \item \tau_2' = [\sseq{\tau_i/t_i}{n}]\tau_2'' \BY{substitution properties and \pfref{decompose2} and \pfref{fvid}} \pflabel{fdecompose2}
    \item \aparr{\tau_1'}{\tau_2'} = [\sseq{\tau_i/t_i}{n}]\aparr{\tau_1''}{\tau_2''} \BY{substitution and \pfref{fdecompose1} and \pfref{fdecompose2}}
    \pflabel{parr}
    % \item \istypeU{\Delta \cup \sseq{\Dhyp{t_i}}{n}}{\tau''_1} \BY{Weakening and \pfref{istype1}} \pflabel{istype3}
    % \item \istypeU{\Delta \cup \sseq{\Dhyp{t_i}}{n}}{\tau''_2} \BY{Weakening and \pfref{istype2}} \pflabel{istype4}
    % \item \istypeU{\Delta \cup \sseq{\Dhyp{t_i}}{n}}{\aparr{\tau''_1}{\tau''_2}} \BY{Rule (\ref{rule:istypeU-parr}) on \pfref{istype3} and \pfref{istype4}}
    % \pflabel{finalistype}
    \item \fvof{\aparr{\tau''_1}{\tau''_2}} = \fvof{\tau''_1} \cup \fvof{\tau''_2} \BY{definition} \pflabel{fvparr}
    \item \fvof{\aparr{\tau''_1}{\tau''_2}} \subset \domof{\Delta} \cup \sseq{t_i}{n} \BY{\pfref{fvparr} and \pfref{fvf1} and \pfref{fvf2}} \pflabel{finalfv}
    \end{pfsteps}
    The conclusions hold as follows:
    \begin{enumerate}
      \item \pfref{expands1} $\cup$ \pfref{expands2}
      \item Choosing $\sseq{t_i}{n}$ and $\aparr{\tau''_1}{\tau''_2}$, by \pfref{parr}
      \item \pfref{finalfv}
    \end{enumerate}
    \resetpfcounter
  \item[\text{(\ref{rule:cvalidT-U-all}) \textbf{through} (\ref{rule:cvalidT-U-sum})}] These cases follow by analagous inductive argument.
  \item[\text{(\ref{rule:cvalidT-U-splicedt})}] ~
  \begin{pfsteps}
  \item \ctau = \acesplicedt{m}{n} \BY{assumption}
  \item \segof{\acesplicedt{m}{n}} = \{ \acesplicedt{m}{n} \} \BY{definition}
  \item \parseUTyp{\bsubseq{b}{m}{n}}{\utau} \BY{assumption} \pflabel{parseUTyp}
  \item \expandsTU{\uDD{\uD}{\Delta_\text{app}}}{\utau}{\tau} \BY{assumption} \pflabel{expandsTU}
  \item t \notin \domof{\Delta} \BY{identification convention} \pflabel{notin}
  \item t \notin \domof{\Delta_\text{app}} \BY{identification} \pflabel{notin2}
  \item \tau = [\tau/t]\tau \BY{definition} \pflabel{subst}
  \item \fvof{t} \subset \Delta \cup \{ t \} \BY{definition} \pflabel{fv}
  \end{pfsteps}
  The conclusions hold as follows:
  \begin{enumerate}
    \item \pfref{parseUTyp} and \pfref{expandsTU}
    \item Choosing $\{t\}$ and $t$, by \pfref{notin}, \pfref{notin2} and \pfref{subst}
    \item \pfref{fv}
  \end{enumerate}
  \resetpfcounter
\end{byCases}
\end{proof}

\begin{lemma}[Proto-Expression \graytxtbox{and Proto-Rule} Expansion Decomposition] 
\label{thm:proto-expression-expansion-decomposition} ~
\begin{enumerate}
\item If $\cvalidE
  {\Delta}{\Gamma}
  {\esceneSG
    {\uDD{\uD}{\Delta_\text{app}}}
    {\uGG{\uG}{\Gamma_\text{app}}}
    {\uPsi}{\uPhi}{b}
  }{\ce}{e}{\tau}$ where $\segof{\ce} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicede{m_i}{n_i}{\ctau_i}}{\nexp}$ then all of the following hold:
  \begin{enumerate}
    \item $\sseq{
          \expandsTU{\uDD{\uD}{\Delta_\text{app}}}
          {
            \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
          }{\tau'_i}
        }{\nty}$
    \item $\sseq{
      \cvalidT{\emptyset}{
        \tsceneUP
          {\uDD
            {\uD}{\Delta_\text{app}}
          }{b}
      }{
        \ctau_i
      }{\tau_i}
    }{\nexp}$
    \item $\sseq{
      \expandsSG
        {\uDD{\uD}{\Delta_\text{app}}}
        {\uGG{\uG}{\Gamma_\text{app}}}
        {\uPsi}
        {\uPhi}
        {\parseUExpF{\bsubseq{b}{m_i}{n_i}}}
        {e_i}
        {\tau_i}
    }{\nexp}$
    \item $e = [\sseq{\tau'_i/t_i}{\nty}, \sseq{e_i/x_i}{\nexp}]e'$ for some $e'$ and $\sseq{t_i}{\nty}$ and $\sseq{x_i}{\nexp}$ such that $\sseq{t_i}{\nty}$ fresh (i.e.  $\sseq{t_i \notin \domof{\Delta}}{\nty}$ and $\sseq{t_i \notin \domof{\Delta_\text{app}}}{\nty}$) and $\sseq{x_i}{\nexp}$ fresh (i.e.  $\sseq{x_i \notin \domof{\Gamma}}{\nexp}$ and $\sseq{x_i \notin \domof{\Gamma_\text{app}}}{\nty}$)
    \item $\mathsf{fv}(e') \subset \domof{\Delta} \cup \domof{\Gamma} \cup \sseq{t_i}{\nty} \cup \sseq{x_i}{\nexp}$
    % \item $\hastypeU
    %   {\Delta \cup \sseq{\Dhyp{t_i}}{\nty}}
    %   {\Gamma \cup \sseq{x_i : \tau_i}{\nexp}}
    %   {e'}{\tau}$
  \end{enumerate}
\end{enumerate}

\begin{grayparbox}
\begin{enumerate}
\item[2.] If $\cvalidR{\Delta}{\Gamma}{\esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}}{\crv}{r}{\tau}{\tau'}$ and \[\segof{\crv} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicede{m_i}{n_i}{\ctau_i}}{\nexp}\] then all of the following hold:
  \begin{enumerate}
    \item $\sseq{
          \expandsTU{\uDD{\uD}{\Delta_\text{app}}}
          {
            \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
          }{\tau'_i}
        }{\nty}$
    \item $\sseq{
      \cvalidT{\emptyset}{
        \tsceneUP
          {\uDD
            {\uD}{\Delta_\text{app}}
          }{b}
      }{
        \ctau_i
      }{\tau_i}
    }{\nexp}$
    \item $\sseq{
      \expandsSG
        {\uDD{\uD}{\Delta_\text{app}}}
        {\uGG{\uG}{\Gamma_\text{app}}}
        {\uPsi}
        {\uPhi}
        {\parseUExpF{\bsubseq{b}{m_i}{n_i}}}
        {e_i}
        {\tau_i}
    }{\nexp}$
    \item $r = [\sseq{\tau'_i/t_i}{\nty}, \sseq{e_i/x_i}{\nexp}]r'$ for some $e'$ and fresh $\sseq{t_i}{\nty}$ and fresh $\sseq{x_i}{\nexp}$ 
    \item $\mathsf{fv}(r') \subset \domof{\Delta} \cup \domof{\Gamma} \cup \sseq{t_i}{\nty} \cup \sseq{x_i}{\nexp}$
    % \item $\ruleType
    %   {\Delta \cup \sseq{\Dhyp{t_i}}{\nty}}
    %   {\Gamma \cup \sseq{x_i : \tau_i}{\nexp}}
    %   {r'}{\tau}{\tau'}$
  \end{enumerate}
\end{enumerate}
\end{grayparbox}
\end{lemma}
\begin{proof} By rule induction over Rules (\ref{rules:cvalidE-U}) and Rule (\ref{rule:cvalidR-UP}). In the following, let $\uDelta=\uDD{\uD}{\Delta_\text{app}}$ and $\uGamma=\uGG{\uG}{\Gamma_\text{app}}$ and $\escenev=\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}$.
\begin{enumerate}
\item \begin{byCases}
  \item[\text{(\ref{rule:cvalidE-U-var})}] \begin{pfsteps}
    \item \ce=x \BY{assumption}
    \item e=x \BY{assumption}
    \item \Gamma = \Gamma', x : \tau \BY{assumption}
    \item \segof{x}=\{ \} \BY{definition}
    \item \fvof{x} = \{ x \} \BY{definition}
    \item \fvof{x} \subset \domof{\Gamma} \BY{definition} \pflabel{fv}
    \item \fvof{x} \subset \domof{\Gamma} \cup \domof{\Delta} \BY{\pfref{fv} and definition of subset} \pflabel{fv2}
  \end{pfsteps}
  The conclusions hold as follows:
  \begin{enumerate}
    \item This conclusion holds trivially because $\nty=0$.
    \item This conclusion holds trivially because $\nexp=0$.
    \item This conclusion holds trivially because $\nexp=0$.
    \item Choose $x$, $\emptyset$ and $\emptyset$. 
    \item \pfref{fv2}
  \end{enumerate}
  \resetpfcounter
  \item[\text{(\ref{rule:cvalidE-U-asc}) \textbf{through} (\ref{rule:cvalidE-U-case})}] These cases follow by straightforward inductive argument.
  \item[\text{(\ref{rule:cvalidE-U-splicede})}] \begin{pfsteps}
    \item \ce=\acesplicede{m}{n}{\ctau} \BY{assumption}
    \item \segof{\acesplicede{m}{n}{\ctau}} = \segof{\ctau} \cup \{ \acesplicede{m}{n}{\ctau} \} \BY{definition}
    \item \segof{\ctau} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \BY{definition} \pflabel{ctau}
    \item \cvalidT{\emptyset}{\tsfrom{\escenev}}{\ctau}{\tau} \BY{assumption} \pflabel{cvalidT}
    \item \parseUExp{\bsubseq{b}{m}{n}}{\ue} \BY{assumption} \pflabel{parseUExp}
    \item \expandsSG{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{\ue}{e}{\tau}
    \BY{assumption} \pflabel{expandsSG}
    \item \sseq{\expandsTU{\uDD{\uD}{\Delta_\text{app}}}{
  \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
}{\tau'_i}}{\nty}
% \item $\sseq{\istypeU{\Delta_\text{app}}{\tau_i}}{n}$
   \BY{Lemma \ref{thm:proto-type-expansion-decomposition-SES} on \pfref{cvalidT} and \pfref{ctau}} \pflabel{parseUTyp}
   \item x \notin \domof{\Gamma} \BY{identification convention} \pflabel{notin1}
   \item x \notin \domof{\Gamma_\text{app}} \BY{identification convention} \pflabel{notin2}
   \item x \notin \domof{\Delta} \BY{identificaiton convention} \pflabel{notin3}
   \item x \notin \domof{\Delta_\text{app}} \BY{identification convention} \pflabel{notin4}
   \item e = [\sseq{\tau'_i/t_i}{\nty}, e/x]x \BY{definition} \pflabel{subst}
   \item \fvof{x} = \{ x \} \BY{definition}
   \item \fvof{x} \subset \domof{\Delta} \cup \domof{\Gamma} \cup \sseq{t_i}{\nty} \cup \{x\} \BY{definition} \pflabel{subset}
  \end{pfsteps}
  The conclusions hold as follows:
  \begin{enumerate}
    \item \pfref{parseUTyp}
    \item $\{$\pfref{cvalidT}$\}$
    \item $\{$\pfref{expandsSG}$\}$
    \item Choosing $x$, $\sseq{t_i}{\nty}$ and $\{x\}$, by \pfref{notin1}, \pfref{notin2}, \pfref{notin3}, \pfref{notin4} and \pfref{subst}.
    \item \pfref{subset}
  \end{enumerate}
  \resetpfcounter
\end{byCases}
\begin{grayparbox}
\begin{byCases}
  \item[\text{(\ref{rule:cvalidE-U-match})}] \begin{pfsteps*}
    \item $\ce = \acematchwith{\nrules}{\ce'}{\seqschemaX{\crv}}$ \BY{assumption}
    \item $e = \aematchwith{\nrules}{\tau}{e'}{\seqschemaX{r}}$ \BY{assumption}
    % \item $\istypeU{\Delta \cup \Delta_\text{app}}{\tau'}$ \BY{assumption} \pflabel{istype}
    \item $\cvalidE{\Delta}{\Gamma}{\escenev}{\ce}{e}{\tau'}$ \BY{assumption} \pflabel{cvalidE}
    \item $\{\cvalidR{\Delta}{\Gamma}{\escenev}{\crv_j}{r_j}{\tau'}{\tau}\}_{1 \leq j \leq \nrules}$
    \BY{assumption} \pflabel{cvalidR}
    \item $\segof{\acematchwith{\nrules}{\ce'}{\seqschemaX{\crv}}} = \segof{\ce} \cup \bigcup_{0 \leq i < \nrules} \segof{\crv_i}$ \BY{definition} 
    \item $\segof{\ce'} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty'} \cup \sseq{\acesplicede{m_i}{n_i}{\ctau_i}}{\nexp'}$ \BY{definition} \pflabel{summaryOfscrut}
    \item $\{\segof{\crv_j} = \sseq{\acesplicedt{m'_{i,j}}{n'_{i,j}}}{\ntyj} \cup \sseq{\acesplicede{m_{i,j}}{n_{i,j}}{\ctau_{i,j}}}{\nexpj}\}_{0 \leq j < \nrules}$ \BY{definition} \pflabel{summaryOfr}
    \item $\sseq{
              \expandsTU{\uDD{\uD}{\Delta_\text{app}}}
              {
                \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
              }{\tau'_i}
            }{\nty'}$
        \BY{IH, part 1 on \pfref{cvalidE} and \pfref{summaryOfscrut}} \pflabel{c1scrut}
    \item $\sseq{
          \cvalidT{\emptyset}{
            \tsceneUP
              {\uDD
                {\uD}{\Delta_\text{app}}
              }{b}
          }{
            \ctau_i
          }{\tau_i}
        }{\nexp'}$
            \BY{IH, part 1 on \pfref{cvalidE} and \pfref{summaryOfscrut}} \pflabel{c2scrut}
    \item $\sseq{
          \expandsSG
            {\uDD{\uD}{\Delta_\text{app}}}
            {\uGG{\uG}{\Gamma_\text{app}}}
            {\uPsi}
            {\uPhi}
            {\parseUExpF{\bsubseq{b}{m_i}{n_i}}}
            {e_i}
            {\tau_i}
        }{\nexp'}$
            \BY{IH, part 1 on \pfref{cvalidE} and \pfref{summaryOfscrut}} \pflabel{c3scrut}
    \item $e' = [\sseq{\tau'_i/t_i}{\nty'}, \sseq{e_i/x_i}{\nexp'}]e''$ for some $e''$ and fresh $\sseq{t_i}{\nty'}$ and fresh $\sseq{x_i}{\nexp'}$ 
        \BY{IH, part 1 on \pfref{cvalidE} and \pfref{summaryOfscrut}} \pflabel{c4scrut}
    \item $\mathsf{fv}(e'') \subset \domof{\Delta} \cup \domof{\Gamma} \cup \sseq{t_i}{\nty'} \cup \sseq{x_i}{\nexp'}$ \BY{IH, part 1 on \pfref{cvalidE} and \pfref{summaryOfscrut}} \pflabel{c5scrut}
    % \item $\hastypeU
    %       {\Delta \cup \sseq{\Dhyp{t_i}}{\nty'}}
    %       {\Gamma \cup \sseq{x_i : \tau_i}{\nexp'}}
    %       {e''}{\tau'}$
    %           \BY{IH, part 1 on \pfref{cvalidE} and \pfref{summaryOfscrut}} \pflabel{c5scrut}
    \item $\{\sseq{
              \expandsTU{\uDD{\uD}{\Delta_\text{app}}}
              {
                \parseUTypF{\bsubseq{b}{m'_{i,j}}{n'_{i,j}}}
              }{\tau'_{i,j}}
            }{\ntyj}\}_{0 \leq j < \nrules}$
        \BY{IH, part 2 over \pfref{cvalidR} and \pfref{summaryOfr}} \pflabel{c1r}
    \item $\{\sseq{
          \cvalidT{\emptyset}{
            \tsceneUP
              {\uDD
                {\uD}{\Delta_\text{app}}
              }{b}
          }{
            \ctau_{i,j}
          }{\tau_{i,j}}
        }{\nexpj}\}_{0 \leq j < \nrules}$
            \BY{IH, part 2 over \pfref{cvalidR} and \pfref{summaryOfr}} \pflabel{c2r}
    \item $\{\sseq{
          \expandsSG
            {\uDD{\uD}{\Delta_\text{app}}}
            {\uGG{\uG}{\Gamma_\text{app}}}
            {\uPsi}
            {\uPhi}
            {\parseUExpF{\bsubseq{b}{m_{i,j}}{n_{i,j}}}}
            {e_{i,j}}
            {\tau_{i,j}}
        }{\nexpj}\}_{0 \leq j < \nrules}$
            \BY{IH, part 2 over \pfref{cvalidR} and \pfref{summaryOfr}} \pflabel{c3r}
    \item $\{r_j = [\sseq{\tau'_{i,j}/t_{i,j}}{\ntyj}, \sseq{e_{i,j}/x_{i,j}}{\nexpj}]r_j'\}_{0 \leq j < \nrules}$ for some  $\{r_j'\}_{0 \leq j < \nrules}$  and fresh $\{\sseq{t_{i,j}}{\ntyj}\}_{0 \leq j < \nrules}$ and fresh $\{\sseq{x_{i,j}}{\nexpj}\}_{0 \leq j < \nrules}$ 
        \BY{IH, part 2 over \pfref{cvalidR} and \pfref{summaryOfr}} \pflabel{c4r}
        \item $\{\mathsf{fv}(r_j') \subset \domof{\Delta} \cup \domof{\Gamma} \cup \sseq{t_{i,j}}{\ntyj} \cup \sseq{x_{i,j}}{\nexpj}\}_{0 \leq j < \nrules}$ \BY{IH, part 2 over \pfref{cvalidR} and \pfref{summaryOfr}} \pflabel{c5r}
    % \item $\{\ruleType
    %       {\Delta \cup \sseq{\Dhyp{t_{i,j}}}{\ntyj}}
    %       {\Gamma \cup \sseq{x_{i,j} : \tau_{i,j}}{\nexpj}}
    %       {r_j'}{\tau'}{\tau}\}_{0 \leq j < \nrules}$
    %           \BY{IH, part 2 over \pfref{cvalidR} and \pfref{summaryOfr}} \pflabel{c5r}
    \item $(\cup_{0 \leq j < \nrules} \sseq{t_{i,j}}{\ntyj}) \cap \sseq{t_i}{\nty'} = \emptyset$ \BY{identification convention}\pflabel{tfresh}
    \item $(\cup_{0 \leq j < \nrules} \sseq{x_{i,j}}{\nexpj}) \cap \sseq{x_i}{\nexp'} = \emptyset$ \BY{identification convention} \pflabel{xfresh}
    \item $e' = [\sseq{\tau'_i/t_i}{\nty'} \cup_{0 \leq j < \nrules} \sseq{\tau_{i,j}/t_{i,j}}{\ntyj}, \sseq{e_i/x_i}{\nexp} \cup_{0 \leq j < \nrules} \sseq{\tau_{i,j}/t_{i,j}}{\ntyj} ]e''$ \BY{substitution properties and \pfref{c4scrut} and \pfref{c5scrut} and \pfref{tfresh} and \pfref{xfresh}} \pflabel{c4fscrut}
    \item $\{r_j = [\sseq{\tau'_i/t_i}{\nty'} \cup_{0 \leq j < \nrules} \sseq{\tau_{i,j}/t_{i,j}}{\ntyj}, \sseq{e_i/x_i}{\nexp} \cup_{0 \leq j < \nrules} \sseq{\tau_{i,j}/t_{i,j}}{\ntyj} ]r_j'\}_{0 \leq j < \nrules}$ \BY{substitution properties and \pfref{c4r} and \pfref{c5r} and \pfref{tfresh} and \pfref{xfresh}} \pflabel{c4fr}
    \item $e = [\sseq{\tau'_i/t_i}{\nty'} \cup_{0 \leq j < \nrules} \sseq{\tau_{i,j}/t_{i,j}}{\ntyj} , \sseq{e_i/x_i}{\nexp'} \cup_{0 \leq j < \nrules} \sseq{e_{i,j}/x_{i,j}}{\nexpj}]\aematchwith{\nrules}{e''}{\seqschemaX{r'}}$ \BY{\pfref{c4fscrut} and \pfref{c4fr} and definition of substitution} \pflabel{c4e}
    \item $\mathsf{fv}(e'') \subset \domof{\Delta} \cup \domof{\Gamma} \cup \sseq{t_i}{\nty'} \cup_{0 \leq j < \nrules} \sseq{t_{i,j}}{\ntyj} \cup \sseq{x_i}{\nexp'} \cup_{0 \leq j < \nrules} \sseq{x_{i,j}}{\nexpj}$ \BY{\pfref{c5scrut} and \pfref{tfresh} and \pfref{xfresh}} \pflabel{c5fscrut}
    \item $\{\mathsf{fv}(r'_j) \subset \domof{\Delta} \cup \domof{\Gamma} \cup \sseq{t_i}{\nty'} \cup_{0 \leq j < \nrules} \sseq{t_{i,j}}{\ntyj} \cup \sseq{x_i}{\nexp'} \cup_{0 \leq j < \nrules} \sseq{x_{i,j}}{\nexpj}\}_{0 \leq j < \nrules}$ \BY{\pfref{c5r} and \pfref{tfresh} and \pfref{xfresh}} \pflabel{c5fr}
    \item $\mathsf{fv}(\aematchwith{\nrules}{e''}{\seqschemaX{r'}}) \subset \domof{\Delta} \cup \domof{\Gamma} \cup \sseq{t_i}{\nty'} \cup_{0 \leq j < \nrules} \sseq{t_{i,j}}{\ntyj} \cup \sseq{x_i}{\nexp'} \cup_{0 \leq j < \nrules} \sseq{x_{i,j}}{\nexpj}$ \BY{\pfref{c5fscrut} and \pfref{c5fr}} \pflabel{c5ff}
    % \item $\hastypeU
    %       {\Delta \cup \sseq{\Dhyp{t_i}}{\nty'} \cup \{ \sseq{t_{i,j}}{\ntyj} \}_{0 \leq j < \nrules}}
    %       {\Gamma \cup \sseq{x_i : \tau_i}{\nexp'} \cup \{ \sseq{x_{i,j}}{\nexpj} \}_{0 \leq j < \nrules}}
    %       {e''}{\tau'}$ \BY{Weakening on \pfref{c5scrut}} \pflabel{c5scrutf}
    % \item $\{\ruleType
    %       {\Delta \cup \sseq{\Dhyp{t_i}}{\nty'} \cup \{ \sseq{t_{i,j}}{\ntyj} \}_{0 \leq j < \nrules}}
    %       {\Gamma \cup \sseq{x_{i,j} : \tau_{i,j}}{\nexpj} \cup \{ \sseq{x_{i,j}}{\nexpj} \}_{0 \leq j < \nrules}}
    %       {r_j'}{\tau'}{\tau}\}_{0 \leq j < \nrules}$ \BY{Weakening over \pfref{c5r}} \pflabel{c5rf}
    % \item $\hastypeU
    %           {\Delta \cup \sseq{\Dhyp{t_i}}{\nty'} \cup \{ \sseq{t_{i,j}}{\ntyj} \}_{0 \leq j < \nrules}}
    %           {\Gamma \cup \sseq{x_i : \tau_i}{\nexp'} \cup \{ \sseq{x_{i,j}}{\nexpj} \}_{0 \leq j < \nrules}}
    %           {\aematchwith{\nrules}{e''}{\seqschemaX{r'}}}{\tau}$ \BY{Rule (\ref{rule:hastypeUP-match}) on \pfref{c5scrutf} and \pfref{c5rf}} \pflabel{c5ff}
  \end{pfsteps*} 

  The conclusions hold as follows:
  \begin{enumerate}
    \item $\text{\pfref{c1scrut}} \cup \bigcup_{0 \leq j < \nrules} \text{\pfref{c1r}}_j$
    \item $\text{\pfref{c2scrut}} \cup \bigcup_{0 \leq j < \nrules} \text{\pfref{c2r}}_j$
    \item $\text{\pfref{c3scrut}} \cup \bigcup_{0 \leq j < \nrules} \text{\pfref{c3r}}_j$
    \item Choose:
      \begin{enumerate}
              \item $\aematchwith{\nrules}{e''}{\seqschemaX{r'}}$
      \item $\sseq{t_i}{\nty'} \cup \{ \sseq{t_{i,j}}{\ntyj} \}_{0 \leq j < \nrules}$; and
      \item $\sseq{x_i}{\nexp'} \cup \{ \sseq{x_{i,j}}{\nexpj} \}_{0 \leq j < \nrules}$; and 
      \end{enumerate}
      We have $e = [\sseq{\tau'_i/t_i}{\nty'} \cup \{ \sseq{\tau_{i,j}/t_{i,j}}{\ntyj} \}_{0 \leq j < \nrules}, \sseq{e_i/x_i}{\nexp'} \cup \{ \sseq{e_{i,j}/x_{i,j}}{\nexpj} \}_{0 \leq j < \nrules}]\aematchwith{\nrules}{e''}{\seqschemaX{r'}}$ by \pfref{c4e}.
    \item \pfref{c5ff}
  \end{enumerate}
  \resetpfcounter
\end{byCases}
\end{grayparbox}
\end{enumerate}
\vspace{-2px}\begin{grayparbox}\vspace{2px}
\begin{enumerate}
\item[2.] By rule induction over the rule typing assumption. There is only one case. In the following, let $\uDelta=\uDD{\uD}{\Delta_\text{app}}$ and $\uGamma=\uGG{\uG}{\Gamma_\text{app}}$ and $\escenev=\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}$.
  \begin{byCases}
    \item[\text{(\ref{rule:cvalidR-UP})}] ~
    \begin{pfsteps*}
      \item $\crv = \acematchrule{p}{\ce}$ \BY{assumption}
      \item $r=\aematchrule{p}{e}$ \BY{assumption}
      \item $\patType{\pctx'}{p}{\tau}$ \BY{assumption} \pflabel{patType}
      \item $\cvalidE{\Delta}{\Gcons{\Gamma}{\pctx'}}{\escenev}{\ce}{e}{\tau'}$ \BY{assumption} \pflabel{cvalid}
      \item $\segof{\crv} = \segof{\ce}$ \BY{definition}
      \item $\segof{\ce} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicede{m_i}{n_i}{\ctau_i}}{\nexp}$ \BY{definition} \pflabel{summaryOfce}
      \item $\sseq{
            \expandsTU{\uDD{\uD}{\Delta_\text{app}}}
            {
              \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
            }{\tau'_i}
          }{\nty}$
          \BY{IH, part 1 on \pfref{cvalid} and \pfref{summaryOfce}}
          \pflabel{c1e}
      \item $\sseq{
        \cvalidT{\emptyset}{
          \tsceneUP
            {\uDD
              {\uD}{\Delta_\text{app}}
            }{b}
        }{
          \ctau_i
        }{\tau_i}
      }{\nexp}$
                \BY{IH, part 1 on \pfref{cvalid} and \pfref{summaryOfce}}
          \pflabel{c2e}
      \item $\sseq{
        \expandsSG
          {\uDD{\uD}{\Delta_\text{app}}}
          {\uGG{\uG}{\Gamma_\text{app}}}
          {\uPsi}
          {\uPhi}
          {\parseUExpF{\bsubseq{b}{m_i}{n_i}}}
          {e_i}
          {\tau_i}
      }{\nexp}$
          \BY{IH, part 1 on \pfref{cvalid} and \pfref{summaryOfce}}
          \pflabel{c3e}      
      \item $e = [\sseq{\tau'_i/t_i}{\nty}, \sseq{e_i/x_i}{\nexp}]e'$ for some $e'$ and fresh $\sseq{t_i}{\nty}$ and fresh $\sseq{x_i}{\nexp}$
          \BY{IH, part 1 on \pfref{cvalid} and \pfref{summaryOfce}}
          \pflabel{c4e}
      % \item $\hastypeU
      %   {\Delta \cup \sseq{\Dhyp{t_i}}{\nty}}
      %   {\Gamma \cup \pctx' \cup \sseq{x_i : \tau_i}{\nexp}}
      %   {e'}{\tau}$
      %     \BY{IH, part 1 on \pfref{cvalid} and \pfref{summaryOfce}}
      \item  $\mathsf{fv}(e') \subset \domof{\Delta} \cup \domof{\Gamma} \cup \domof{\Gamma'} \cup \sseq{t_i}{\nty} \cup \sseq{x_i}{\nexp}$ \BY{IH, part 1 on \pfref{cvalid} and \pfref{summaryOfce}}
          \pflabel{c5e}
      \item $r=[\sseq{\tau'_i/t_i}{\nty}, \sseq{e_i/x_i}{\nexp}]\aematchrule{p}{e'}$ \BY{substitution properties and \pfref{c4e}} \pflabel{c4r}
      \item $\fvof{p} = \domof{\Gamma'}$ \BY{Lemma \ref{lemma:pattern-regularity-UP} on \pfref{patType}}\pflabel{fvofp}
         \item  $\mathsf{fv}(\aematchrule{p}{e'}) \subset \domof{\Delta} \cup \domof{\Gamma} \cup \sseq{t_i}{\nty} \cup \sseq{x_i}{\nexp}$ \BY{definition of $\fvof{r}$ and \pfref{c5e} and \pfref{fvofp}}
          \pflabel{c5r}
      % \item $\ruleType
      %         {\Delta \cup \sseq{\Dhyp{t_i}}{\nty}}
      %         {\Gamma \cup \sseq{x_i : \tau_i}{\nexp}}
      %         {\aematchrule{p}{e'}}
      %         {\tau}{\tau'}$ \BY{Rule (\ref{rule:ruleType}) on \pfref{patType} and \pfref{c5e}} \pflabel{c5r}
    \end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
      \item \pfref{c1e}
      \item \pfref{c2e}
      \item \pfref{c3e}
      \item Choosing $\aematchrule{p}{e'}$ and $\sseq{t_i}{\nty}$ and $\sseq{x_i}{\nexp}$, by \pfref{c4r}
      \item \pfref{c5r}
    \end{enumerate}
    \resetpfcounter
  \end{byCases}
\end{enumerate}
\end{grayparbox}
\end{proof}
% \begin{equation}\label{rule:cvalidR-UP}
% \inferrule{
%   \patType{\pctx}{p}{\tau}\\
%   \cvalidE{\Delta}{\Gcons{\Gamma}{\pctx}}{\escenev}{\ce}{e}{\tau'}
% }{
%   \cvalidR{\Delta}{\Gamma}{\escenev}{\acematchrule{p}{\ce}}{\aematchrule{p}{e}}{\tau}{\tau'}
% }
% \end{equation}

\begin{theorem}[seTLM Abstract Reasoning Principles]
\label{thm:tsc-SES}
If $\expandsSG{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\utsmap{\tsmv}{b}}{e}{\tau}$ then:
\begin{enumerate}
\item (\textbf{Typing 1}) $\uPsi = \uPsi', \uShyp{\tsmv}{a}{\tau}{\eparse}$ and $\hastypeU{\Delta}{\Gamma}{e}{\tau}$
\item $\encodeBody{b}{\ebody}$
\item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessE}}{\ecand}}$
\item $\decodeCondE{\ecand}{\ce}$
\item (\textbf{Segmentation}) $\segOK{\segof{\ce}}{b}$
\item $\segof{\ce} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicede{m_i}{n_i}{\ctau_i}}{\nexp}$
\item \textbf{(Typing 2)} $\sseq{
      \expandsTU{\uDD{\uD}{\Delta}}
      {
        \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
      }{\tau'_i}
    }{\nty}$ and $\sseq{\istypeU{\Delta}{\tau'_i}}{\nty}$
\item \textbf{(Typing 3)} $\sseq{
  \cvalidT{\emptyset}{
    \tsceneUP
      {\uDD
        {\uD}{\Delta}
      }{b}
  }{
    \ctau_i
  }{\tau_i}
}{\nexp}$ and $\sseq{\istypeU{\Delta}{\tau_i}}{\nexp}$
\item \textbf{(Typing 4)} $\sseq{
  \expandsSG
    {\uDD{\uD}{\Delta}}
    {\uGG{\uG}{\Gamma}}
    {\uPsi}
    {\uPhi}
    {\parseUExpF{\bsubseq{b}{m_i}{n_i}}}
    {e_i}
    {\tau_i}
}{\nexp}$ and $\sseq{\hastypeU{\Delta}{\Gamma}{e_i}{\tau_i}}{\nexp}$
\item (\textbf{Capture Avoidance}) $e = [\sseq{\tau'_i/t_i}{\nty}, \sseq{e_i/x_i}{\nexp}]e'$ for some $\sseq{t_i}{\nty}$ and $\sseq{x_i}{\nexp}$ and $e'$
\item (\textbf{Context Independence}) $\mathsf{fv}(e') \subset \sseq{t_i}{\nty} \cup \sseq{x_i}{\nexp}$
  % $\hastypeU
  % {\sseq{\Dhyp{t_i}}{\nty}}
  % {\sseq{x_i : \tau_i}{\nexp}}
  % {e'}{\tau}$
\end{enumerate}
\end{theorem}
\begin{proof} By rule induction over Rules (\ref{rules:expandsU}). There is only one rule that applies. In the following, let $\uDelta = \uDD{\uD}{\Delta}$ and $\uGamma = \uGG{\uG}{\Gamma}$.
\begin{byCases}
\item[\text{(\ref{rule:expandsU-tsmap})}] ~
\begin{pfsteps*}
  \item $\uPsi = \uPsi', \uShyp{\tsmv}{a}{\tau}{\eparse}$ \BY{assumption} \pflabel{uPsidef}
  \item $\expandsSG{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\utsmap{\tsmv}{b}}{e}{\tau}$ \BY{assumption} \pflabel{expandsSG}
  \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{Theorem \ref{thm:typed-expansion-short-U} on \pfref{expandsSG}} \pflabel{hastype}
  \item $\encodeBody{b}{\ebody}$ \BY{assumption} \pflabel{encodeBody}
  \item $\evalU{\eparse(\ebody)}{\aein{\mathtt{SuccessE}}{\ecand}}$ \BY{assumption} \pflabel{evalU}
  \item $\decodeCondE{\ecand}{\ce}$ \BY{assumption} \pflabel{decodeCondE}
  \item $\segOK{\segof{\ce}}{b}$ \BY{assumption} \pflabel{segOK}
  \item $\cvalidE{\emptyset}{\emptyset}{\esceneSG{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}$ \BY{assumption}\pflabel{cvalidE}
  \item $\segof{\ce} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicede{m_i}{n_i}{\ctau_i}}{\nexp}$ \BY{definition} \pflabel{summaryOf}
  \item $\sseq{
      \expandsTU{\uDD{\uD}{\Delta}}
      {
        \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
      }{\tau'_i}
    }{\nty}$ \BY{Lemma \ref{thm:proto-expression-expansion-decomposition} on \pfref{cvalidE} and \pfref{summaryOf}} \pflabel{c1}
\item $\sseq{\istypeU{\Delta}{\tau'_i}}{\nty}$ \BY{Lemma \ref{lemma:type-expansion-U}, part 1 over \pfref{c1}} \pflabel{t1}
\item $\sseq{
  \cvalidT{\emptyset}{
    \tsceneUP
      {\uDD
        {\uD}{\Delta}
      }{b}
  }{
    \ctau_i
  }{\tau_i}
}{\nexp}$ \BY{Lemma \ref{thm:proto-expression-expansion-decomposition} on \pfref{cvalidE} and \pfref{summaryOf}} \pflabel{c2}
\item $\emptyset \cap \Delta = \emptyset$ \BY{definition} \pflabel{emptyintersect}
\item $\sseq{\istypeU{\Delta}{\tau_i}}{\nexp}$ \BY{Lemma \ref{lemma:type-expansion-U}, part 2 over \pfref{c2} and \pfref{emptyintersect}} \pflabel{t2}
\item $\sseq{
  \expandsSG
    {\uDD{\uD}{\Delta}}
    {\uGG{\uG}{\Gamma}}
    {\uPsi}
    {\uPhi}
    {\parseUExpF{\bsubseq{b}{m_i}{n_i}}}
    {e_i}
    {\tau_i}
}{\nexp}$ \BY{Lemma \ref{thm:proto-expression-expansion-decomposition} on \pfref{cvalidE} and \pfref{summaryOf}} \pflabel{c3}
\item $\sseq{\hastypeU{\Delta}{\Gamma}{e_i}{\tau_i}}{\nexp}$ \BY{Theorem \ref{thm:typed-expansion-short-U} over \pfref{c3}} \pflabel{t3}
\item $e = [\sseq{\tau'_i/t_i}{\nty}, \sseq{e_i/x_i}{\nexp}]e'$ for some $e'$ and fresh $\sseq{t_i}{\nty}$ and fresh $\sseq{x_i}{\nexp}$ \BY{Lemma \ref{thm:proto-expression-expansion-decomposition} on \pfref{cvalidE} and \pfref{summaryOf}} \pflabel{c4}
\item $\fvof{e'} \subset \sseq{t_i}{\nty} \cup \sseq{x_i}{\nexp}$ \BY{Lemma \ref{thm:proto-expression-expansion-decomposition} on \pfref{cvalidE} and \pfref{summaryOf}} \pflabel{c5}
% \item $\hastypeU
%   {\sseq{\Dhyp{t_i}}{\nty}}
%   {\sseq{x_i : \tau_i}{\nexp}}
%   {e'}{\tau}$ \BY{Lemma \ref{thm:proto-expression-expansion-decomposition} on \pfref{cvalidE} and \pfref{summaryOf}} \pflabel{c5}
\end{pfsteps*}
The conclusions hold as follows:
\begin{enumerate}
  \item \pfref{uPsidef} and \pfref{hastype}
  \item \pfref{encodeBody}
  \item \pfref{evalU}
  \item \pfref{decodeCondE}
  \item \pfref{segOK}
  \item \pfref{summaryOf}
  \item \pfref{c1} and \pfref{t1}
  \item \pfref{c2} and \pfref{t2}
  \item \pfref{c3} and \pfref{t3}
  \item \pfref{c4}
  \item \pfref{c5}
\end{enumerate}
\resetpfcounter
\end{byCases}
\end{proof}

% The following theorem establishes a prohibition on \textbf{Shadowing} as discussed in Sec. \ref{sec:uetsms-validation}.

% \begin{theorem}[Shadowing Prohibition]
% \label{thm:shadowing-prohibition-SES} ~
% \begin{enumerate}
% \item If $\cvalidT{\Delta}{\tsceneU{\uDD{\uD}{\Delta_\text{app}}}{b}}{\acesplicedt{m}{n}}{\tau}$ then:\begin{enumerate}
% \item $\parseUTyp{\bsubseq{b}{m}{n}}{\utau}$
% \item $\expandsTU{\uDD{\uD}{\Delta_\text{app}}}{\utau}{\tau}$
% \item $\Delta \cap \Delta_\text{app} = \emptyset$
% \end{enumerate}
% \item If $\cvalidE{\Delta}{\Gamma}{\escenev}{\acesplicede{m}{n}{\ctau}}{e}{\tau}$ then:
% \begin{enumerate}
% \item $\cvalidT{\emptyset}{\tsfrom{\escenev}}{\ctau}{\tau}$
% \item $  \escenev=\esceneU{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{b}$
% \item $\parseUExp{\bsubseq{b}{m}{n}}{\ue}$
% \item $\expandsU{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\ue}{e}{\tau}$
% \item $\Delta \cap \Delta_\text{app} = \emptyset$
% \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}} = \emptyset$
% \end{enumerate}
% \end{enumerate}
% \end{theorem}
% \begin{proof} ~
% \begin{enumerate}
% \item By rule induction over Rules (\ref{rules:cvalidT-U}). The only rule that applies is Rule (\ref{rule:cvalidT-U-splicedt}). The conclusions are the premises of this rule.
% \item By rule induction over Rules (\ref{rules:cvalidE-U}). The only rule that applies is Rule (\ref{rule:cvalidE-U-splicede}). The conclusions are the premises of this rule.
% \end{enumerate}
% \end{proof}

\begin{grayparbox}
\begin{lemma}[Proto-Pattern Expansion Decomposition]
\label{lemma:proto-pattern-expansion-decomposition-S}
If $\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}$ where  
\[ 
\segof{\cpv} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicedp{m_i}{n_i}{\ctau_i}}{\npat}
\]
then all of the following hold:
\begin{enumerate}
    \item $\sseq{
          \expandsTU{\uDelta}
          {
            \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
          }{\tau'_i}
        }{\nty}$
    \item $\sseq{
      \cvalidT{\emptyset}{
        \tsceneUP
          {\uDelta}{b}
      }{
        \ctau_i
      }{\tau_i}
    }{\npat}$
    \item $\sseq{
      \patExpands
        {\upctx_i}
        {\uPhi}
        {\parseUPatF{\bsubseq{b}{m_i}{n_i}}}
        {p_i}
        {\tau_i}
    }{\npat}$
  \item $\upctx = \biguplus_{0 \leq i < \npat} \upctx_i$
\end{enumerate}
\end{lemma}
\begin{proof} By rule induction over Rules (\ref{rules:cvalidP-UP}). In the following, let $\pscenev=\pscene{\uDelta}{\uPhi}{b}$.
\begin{byCases}
  \item[\text{(\ref{rule:cvalidP-UP-wild})}] ~
    \begin{pfsteps*}
      \item $\cpv=\acewildp$ \BY{assumption}
      \item $e = \aewildp$ \BY{assumption}
      \item $\upctx = \uGG{\emptyset}{\emptyset}$ \BY{assumption}
      \item $\segof{\acewildp} = \emptyset$ \BY{definition}
    \end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
      \item This conclusion holds trivially because $\nty=0$.
      \item This conclusion holds trivially because $\npat=0$.
      \item This conclusion holds trivially because $\npat=0$.
      \item This conclusion holds trivially because $\upctx=\emptyset$ and $\npat=0$.
    \end{enumerate}
    \resetpfcounter
  \item[\text{(\ref{rule:cvalidP-UP-fold})}] ~
    \begin{pfsteps*}
      \item $\cpv=\acefoldp{\cpv'}$ \BY{assumption}
      \item $p=\aefoldp{p'}$ \BY{assumption}
      \item $\tau=\arec{t}{\tau'}$ \BY{assumption}
      \item $\cvalidP{\upctx}{\pscenev}{\cpv}{p}{[\arec{t}{\tau'}/t]\tau'}$ \BY{assumption} \pflabel{cvalidP}
      \item $\segof{\acefoldp{\cpv'}} = \segof{\cpv'}$ \BY{definition} \pflabel{summaryOf}
      \item $\segof{\cpv'} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicedp{m_i}{n_i}{\ctau_i}}{\npat}$ \BY{definition} \pflabel{summaryOf2}
      \item $\sseq{
            \expandsTU{\uDelta}
            {
              \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
            }{\tau'_i}
          }{\nty}$ \BY{IH on \pfref{cvalidP} and \pfref{summaryOf2}} \pflabel{c1}
      \item $\sseq{
        \cvalidT{\emptyset}{
          \tsceneUP
            {\uDelta}{b}
        }{
          \ctau_i
        }{\tau_i}
      }{\npat}$ \BY{IH on \pfref{cvalidP} and \pfref{summaryOf2}} \pflabel{c2}
      \item $\sseq{
        \patExpands
          {\upctx_i}
          {\uPhi}
          {\parseUPatF{\bsubseq{b}{m_i}{n_i}}}
          {p_i}
          {\tau_i}
      }{\npat}$ \BY{IH on \pfref{cvalidP} and \pfref{summaryOf2}} \pflabel{c3}
    \item $\upctx = \biguplus_{0 \leq i < \npat} \upctx_i$ \BY{IH on \pfref{cvalidP} and \pfref{summaryOf2}}\pflabel{c4}\end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
    \item \pfref{c1}
    \item \pfref{c2}
    \item \pfref{c3}
    \item \pfref{c4}
    \end{enumerate}
    \resetpfcounter
  \item[\text{(\ref{rule:cvalidP-UP-tpl})}] ~
    \begin{pfsteps*}
      \item $\cpv=\acetplp{\labelset}{\mapschema{\cpv}{j}{\labelset}}$ \BY{assumption}
      \item $p=\aetplp{\labelset}{\mapschema{p}{j}{\labelset}}$ \BY{assumption}
      \item $\tau = \aprod{\labelset}{\mapschema{\tau}{j}{\labelset}}$ \BY{assumption}
      \item $\upctx=\biguplus_{j \in \labelset}{\upctx_j}$ \BY{assumption} \pflabel{Gconsi}
      \item $\{\cvalidP{\upctx_j}{\pscenev}{\cpv_j}{p_j}{\tau_j}\}_{j \in \labelset}$ \BY{assumption} \pflabel{cvalidP}
      \item $\segof{\acetplp{\labelset}{\mapschema{\cpv}{j}{\labelset}}} = \bigcup_{j \in \labelset} \segof{\cpv_j}$ \BY{definition}
      \item $\{ \segof{\cpv_j} = \sseq{\acesplicedt{m'_{i,j}}{n'_{i,j}}}{\ntyj} \cup \sseq{\acesplicedp{m_{i,j}}{n_{i,j}}{\ctau_{i,j}}}{\npatj} \}_{j \in \labelset}$ \BY{definition}\pflabel{summaryOf2}
      \item $\npat = \Sigma_{j \in \labelset} \npatj$ \BY{definition}
      \item $\{\sseq{
            \expandsTU{\uDelta}
            {
              \parseUTypF{\bsubseq{b}{m'_{i,j}}{n'_{i,j}}}
            }{\tau'_{i,j}}
          }{\ntyj}\}_{j \in \labelset}$ \BY{IH over \pfref{cvalidP} and \pfref{summaryOf2}} \pflabel{c1}
      \item $\{\sseq{
        \cvalidT{\emptyset}{
          \tsceneUP
            {\uDelta}{b}
        }{
          \ctau_{i,j}
        }{\tau_{i,j}}
      }{\npatj}\}_{j \in \labelset}$ \BY{IH over \pfref{cvalidP} and \pfref{summaryOf2}} \pflabel{c2}
      \item $\{\sseq{
        \patExpands
          {\upctx_{i,j}}
          {\uPhi}
          {\parseUPatF{\bsubseq{b}{m_{i,j}}{n_{i,j}}}}
          {p_{i,j}}
          {\tau_{i,j}}
      }{\npatj}\}_{j \in \labelset}$ \BY{IH over \pfref{cvalidP} and \pfref{summaryOf2}} \pflabel{c3}
    \item $\{\upctx_j = \biguplus_{0 \leq i < \npatj} \upctx_{i,j}\}_{j \in \labelset}$ \BY{IH over \pfref{cvalidP} and \pfref{summaryOf2}}\pflabel{c4}
    \item $\biguplus_{j \in \labelset} \upctx_{j} = \biguplus_{j \in \labelset} \biguplus_{i \in \npatj} \upctx_{i,j}$ \BY{definition and \pfref{c4}}\pflabel{c4x}
    \end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
      \item $\bigcup_{j \in \labelset} \bigcup_{i \in \ntyj} \text{\pfref{c1}}_{i,j}$
      \item $\bigcup_{j \in \labelset} \bigcup_{i \in \npatj} \text{\pfref{c2}}_{i,j}$
      \item $\bigcup_{j \in \labelset} \bigcup_{i \in \npatj} \text{\pfref{c3}}_{i,j}$
      \item \pfref{c4x}
    \end{enumerate}
    \resetpfcounter
  \item[\text{(\ref{rule:cvalidP-UP-in})}]
    \begin{pfsteps*}
      \item $\cpv=\aceinjp{\ell}{\cpv'}$ \BY{assumption}
      \item $p=\aeinjp{\ell}{p'}$ \BY{assumption}
      \item $\tau=\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}$ \BY{assumption}
      \item $\cvalidP{\upctx}{\pscenev}{\cpv}{p}{\tau'}$ \BY{assumption} \pflabel{cvalidP}
      \item $\segof{\aceinjp{\ell}{\cpv'}} = \segof{\cpv'}$ \BY{definition} \pflabel{summaryOf}
      \item $\segof{\cpv'} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicedp{m_i}{n_i}{\ctau_i}}{\npat}$ \BY{definition} \pflabel{summaryOf2}
      \item $\sseq{
            \expandsTU{\uDelta}
            {
              \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
            }{\tau'_i}
          }{\nty}$ \BY{IH on \pfref{cvalidP} and \pfref{summaryOf2}} \pflabel{c1}
      \item $\sseq{
        \cvalidT{\emptyset}{
          \tsceneUP
            {\uDelta}{b}
        }{
          \ctau_i
        }{\tau_i}
      }{\npat}$ \BY{IH on \pfref{cvalidP} and \pfref{summaryOf2}} \pflabel{c2}
      \item $\sseq{
        \patExpands
          {\upctx_i}
          {\uPhi}
          {\parseUPatF{\bsubseq{b}{m_i}{n_i}}}
          {p_i}
          {\tau_i}
      }{\npat}$ \BY{IH on \pfref{cvalidP} and \pfref{summaryOf2}} \pflabel{c3}
    \item $\upctx = \biguplus_{0 \leq i < \npat} \upctx_i$ \BY{IH on \pfref{cvalidP} and \pfref{summaryOf2}}\pflabel{c4}\end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
    \item \pfref{c1}
    \item \pfref{c2}
    \item \pfref{c3}
    \item \pfref{c4}
    \end{enumerate}
    \resetpfcounter
  \item[\text{(\ref{rule:cvalidP-UP-spliced})}] ~
    \begin{pfsteps*}
      \item $\cpv=\acesplicedp{m}{n}{\ctau}$ \BY{assumption}
      \item $\cvalidT{\emptyset}{\tsceneUP{\uDelta}{b}}{\ctau}{\tau}$ \BY{assumption} \pflabel{cvalidT}
      \item $\parseUPat{\bsubseq{b}{m}{n}}{\upv}$ \BY{assumption} \pflabel{parseUPat}
      \item $\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}$ \BY{assumption} \pflabel{patExpands}
      \item $\segof{\acesplicedp{m}{n}{\ctau}} = \segof{\ctau} \cup \{ \acesplicedp{m}{n}{\ctau} \}$ \BY{definition} \pflabel{summaryOf}
      \item $\segof{\ctau} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty}$ \BY{definition} \pflabel{summaryOf2}
      \item $\sseq{\expandsTU{\uDD{\uD}{\Delta_\text{app}}}{
  \parseUTypF{\bsubseq{b}{m_i}{n_i}}
}{\tau_i}}{n}$ \BY{Lemma \ref{thm:proto-type-expansion-decomposition-SES} on \pfref{cvalidT} and \pfref{summaryOf2}} \pflabel{expandsTU}
% \item $\sseq{\istypeU{\Delta_\text{app}}{\tau_i}}{n}$
    \end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
      \item \pfref{expandsTU}
      \item \pfref{cvalidT}
      \item \pfref{parseUPat} and \pfref{patExpands}
      \item This conclusion holds by \pfref{patExpands} because $\npat=1$.
    \end{enumerate}
    \resetpfcounter
\end{byCases}
\end{proof}

\begin{theorem}[spTLM Abstract Reasoning Principles]
\label{thm:spTLM-Typing-Segmentation}
If $\patExpands{\upctx}{\uPhi}{\utsmap{\tsmv}{b}}{p}{\tau}$ where $\uDelta=\uDD{\uD}{\Delta}$ and $\uGamma=\uGG{\uG}{\Gamma}$ then all of the following hold:
\begin{enumerate}
        \item (\textbf{Typing 1}) $\uPhi=\uPhi', \uPhyp{\tsmv}{a}{\tau}{\eparse}$ and $\patType{\pctx}{p}{\tau}$
        \item $\encodeBody{b}{\ebody}$
        \item $\evalU{\eparse(\ebody)}{\aein{\mathtt{SuccessP}}{\ecand}}$
        \item $\decodeCEPat{\ecand}{\cpv}$
        \item (\textbf{Segmentation}) $\segOK{\segof{\cpv}}{b}$
        \item $\segof{\cpv} = \sseq{\acesplicedt{n'_i}{m'_i}}{\nty} \cup \sseq{\acesplicedp{m_i}{n_i}{\ctau_i}}{\npat}$
        \item (\textbf{Typing 2}) $\sseq{
              \expandsTU{\uDelta}
              {
                \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
              }{\tau'_i}
            }{\nty}$ and $\sseq{\istypeU{\Delta}{\tau'_i}}{\nty}$
        \item (\textbf{Typing 3}) $\sseq{
          \cvalidT{\emptyset}{
            \tsceneUP
              {\uDelta}{b}
          }{
            \ctau_i
          }{\tau_i}
        }{\npat}$ and $\sseq{\istypeU{\Delta}{\tau_i}}{\npat}$
        \item (\textbf{Typing 4}) $\sseq{
          \patExpands
            {\uGG{\uG_i}{\pctx_i}}
            {\uPhi}
            {\parseUPatF{\bsubseq{b}{m_i}{n_i}}}
            {p_i}
            {\tau_i}
        }{\npat}$  and $\sseq{\patType{\pctx_i}{p_i}{\tau_i}}{\npat}$
      \item (\textbf{Visibility}) $\uG = \biguplus_{0 \leq i < \npat} \uG_i$ and $\Gamma = \bigcup_{0 \leq i < \npat} \pctx_i$
\end{enumerate}
\end{theorem}
\begin{proof} By rule induction over Rules (\ref{rules:patExpands}). There is only one rule that applies.
\begin{byCases}
  \item[\text{(\ref{rule:patExpands-apuptsm})}] 
    \begin{pfsteps*}
      \item $\patExpands{\upctx}{\uPhi}{\utsmap{\tsmv}{b}}{p}{\tau}$ \BY{assumption} \pflabel{patExpands}
      \item $\uPhi = \uPhi', \uPhyp{\tsmv}{a}{\tau}{\eparse}$ \BY{assumption} \pflabel{uPhidef}
      \item $\patType{\pctx}{p}{\tau}$ \BY{Theorem \ref{thm:typed-pattern-expansion} on \pfref{patExpands}} \pflabel{patType}
      \item $\encodeBody{b}{\ebody}$ \BY{assumption} \pflabel{encodeBody}
      \item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessP}}{\ecand}}$ \BY{assumption} \pflabel{evalU}
      \item $\decodeCEPat{\ecand}{\cpv}$ \BY{assumption} \pflabel{decodeCEPat}
      \item $\segOK{\segof{\cpv}}{b}$ \BY{assumption} \pflabel{segOK}
      \item $\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}$ \BY{assumption} \pflabel{cvalidP}
      \item $\segof{\cpv} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicedp{m_i}{n_i}}{\npat}$ \BY{definition}\pflabel{summaryOf}
      \item $\sseq{
            \expandsTU{\uDelta}
            {
              \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
            }{\tau'_i}
          }{\nty}$ \BY{Lemma \ref{lemma:proto-pattern-expansion-decomposition-S} on \pfref{cvalidP} and \pfref{summaryOf}}\pflabel{c1}
      \item $\sseq{\istypeU{\Delta}{\tau'_i}}{\nty}$ \BY{Lemma \ref{lemma:type-expansion-U}, part 1 over \pfref{c1}}\pflabel{t1}
      \item $\sseq{
        \cvalidT{\emptyset}{
          \tsceneUP
            {\uDelta}{b}
        }{
          \ctau_i
        }{\tau_i}
      }{\npat}$  \BY{Lemma \ref{lemma:proto-pattern-expansion-decomposition-S} on \pfref{cvalidP} and \pfref{summaryOf}}\pflabel{c2}
      \item $\sseq{\istypeU{\Delta}{\tau_i}}{\npat}$ \BY{Lemma \ref{lemma:type-expansion-U}, part 2 over \pfref{c2}}\pflabel{t2}
      \item $\sseq{
        \patExpands
          {\upctx_i}
          {\uPhi}
          {\parseUPatF{\bsubseq{b}{m_i}{n_i}}}
          {p_i}
          {\tau_i}
      }{\npat}$ \BY{Lemma \ref{lemma:proto-pattern-expansion-decomposition-S} on \pfref{cvalidP} and \pfref{summaryOf}}\pflabel{c3}
    \item $\sseq{\patType{\pctx_i}{p_i}{\tau_i}}{\npat}$ \BY{Theorem \ref{thm:typed-pattern-expansion} over \pfref{c3}}\pflabel{t3}
    \item $\uG = \biguplus_{0 \leq i < \npat} \uG_i$ and $\Gamma = \bigcup_{0 \leq i < \npat} \pctx_i$ \BY{Lemma \ref{lemma:proto-pattern-expansion-decomposition-S} on \pfref{cvalidP} and \pfref{summaryOf}}\pflabel{c4}

    \end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
      \item \pfref{uPhidef} and \pfref{patType}
      \item \pfref{encodeBody}
      \item \pfref{evalU}
      \item \pfref{decodeCEPat}
      \item \pfref{segOK}
      \item \pfref{summaryOf}
      \item \pfref{c1} and \pfref{t1}
      \item \pfref{c2} and \pfref{t2}
      \item \pfref{c3} and \pfref{t3} 
      \item \pfref{c4}
    \end{enumerate}
    \resetpfcounter
\end{byCases}
\end{proof}
\end{grayparbox}

% \inferrule{
%   \uPhi = \uPhi', \uPhyp{\tsmv}{a}{\tau}{\eparse}\\\\
%   \encodeBody{b}{\ebody}\\
%   \evalU{\ap{\eparse}{\ebody}}{{\lbltxt{SuccessP}}\cdot{\ecand}}\\
%   \decodeCEPat{\ecand}{\cpv}\\\\
%     \segOK{\segof{\cpv}}{b}\\
%   \cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}
% }{
%   \patExpands{\upctx}{\uPhi}{\utsmap{\tsmv}{b}}{p}{\tau}
% }}

\ificfp
\chapter{\texorpdfstring{A Calculus of Parametric TLMs}{A Calculus of Parametric TLMs}}
\else
\chapter{\texorpdfstring{$\miniVerseParam$}{miniVerseP}}
\fi
\label{appendix:miniVerseParam}

\clearpage
\section{Expanded Language (XL)}
\subsection{Syntax}
\subsubsection{Signatures and Module Expressions}
\[\begin{array}{lllllll}
\textbf{Sort} & & & \textbf{Operational Form} 
%& \textbf{Stylized Form} 
& \textbf{Description}\\
\mathsf{Sig} & \sigma & ::= & \asignature{\kappa}{u}{\tau} 
%& \signature{u}{\kappa}{\tau} 
& \text{signature}\\
\mathsf{Mod} & M & ::= & X 
%& X 
& \text{module variable}\\
&&& \astruct{c}{e} 
%& \struct{c}{e} 
& \text{structure}\\
&&& \aseal{\sigma}{M} 
%& \seal{M}{\sigma} 
& \text{seal}\\
&&& \amlet{\sigma}{M}{X}{M} %& \mlet{X}{M}{M}{\sigma} 
& \text{definition}
\end{array}\]

\subsubsection{Kinds and Constructions}
\[\begin{array}{lrlllll}
\textbf{Sort} & & & \textbf{Operational Form} 
%& \textbf{Stylized Form} 
& \textbf{Description}\\
\mathsf{Kind} & \kappa & ::= & k & \text{kind variable}\\
&&& \akdarr{\kappa}{u}{\kappa} 
%& \kdarr{u}{\kappa}{\kappa} 
& \text{dependent function}\\
&&& \akunit 
%& \kunit 
& \text{nullary product}\\
&&& \akdbprod{\kappa}{u}{\kappa} 
%& \kdbprod{u}{\kappa}{\kappa} 
& \text{dependent product}\\
%&&& \akdprodstd & \kdprodstd & \text{labeled dependent product}\\
&&& \akty 
%& \kty
& \text{type}\\
&&& \aksing{\tau} 
%& \ksing{\tau} 
& \text{singleton}\\
\mathsf{Con} & c, \tau & ::= & u 
%& u 
& \text{construction variable}\\
&&& t 
%& t 
& \text{type variable}
\\
&&& \acabs{u}{c} 
%& \cabs{u}{c} 
& \text{abstraction}\\
&&& \acapp{c}{c} 
%& \capp{c}{c} 
& \text{application}\\
&&& \actriv 
%& \ctriv 
& \text{trivial}\\
&&& \acpair{c}{c}
% & \cpair{c}{c} 
& \text{pair}\\
&&& \acprl{c} 
%& \cprl{c} 
& \text{left projection}\\
&&& \acprr{c} 
%& \cprr{c} 
& \text{right projection}\\
%&&& \adtplX & \dtplX & \text{labeled dependent tuple}\\
%&&& \adprj{\ell}{c} & \prj{c}{\ell} & \text{projection}\\
&&& \aparr{\tau}{\tau} 
%& \parr{\tau}{\tau} 
& \text{partial function}\\
&&& \aallu{\kappa}{u}{\tau} 
%& \forallu{u}{\kappa}{\tau} 
& \text{polymorphic}\\
&&& \arec{t}{\tau} 
%& \rect{t}{\tau} 
& \text{recursive}\\
&&& \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}} 
%& \prodt{\mapschema{\tau}{i}{\labelset}} 
& \text{labeled product}\\
&&& \asum{\labelset}{\mapschema{\tau}{i}{\labelset}} 
%& \sumt{\mapschema{\tau}{i}{\labelset}} 
& \text{labeled sum}\\
&&& \amcon{M} 
%& \mcon{M} 
& \text{construction component}
\end{array}\]
\clearpage

\subsubsection{Expressions, Rules and Patterns}
\[\begin{array}{lllllll}
\textbf{Sort} & & & \textbf{Operational Form} 
%& \textbf{Stylized Form} 
& \textbf{Description}\\
\mathsf{Exp} & e & ::= & x 
%& x 
& \text{variable}\\
&&& \aelam{\tau}{x}{e} 
%& \lam{x}{\tau}{e} 
& \text{abstraction}\\
&&& \aeap{e}{e} 
%& \ap{e}{e} 
& \text{application}\\
&&& \aeclam{\kappa}{u}{e} %& \clam{u}{\kappa}{e} 
& \text{construction abstraction}\\
&&& \aecap{e}{\kappa} %& \cAp{e}{\kappa} 
& \text{construction application}\\
&&& \aefold{e} %& \fold{e} 
& \text{fold}\\
&&& \aeunfold{e} %& \unfold{e} 
& \text{unfold}\\
&&& \aetpl{\labelset}{\mapschema{e}{i}{\labelset}} 
%& \tpl{\mapschema{e}{i}{\labelset}} 
& \text{labeled tuple}\\
&&& \aepr{\ell}{e} 
%& \prj{e}{\ell} 
& \text{projection}\\
&&& \aein{\ell}{e} 
%& \inj{\ell}{e} 
& \text{injection}\\
&&& \aematchwith{n}{e}{\seqschemaX{r}} 
%& \matchwith{e}{\seqschemaX{r}} 
& \text{match}\\
&&& \amval{M} 
%& \mval{M} 
& \text{value component}\\
\mathsf{Rule} & r & ::= & \aematchrule{p}{e} 
%& \matchrule{p}{e} 
& \text{rule}\\
\mathsf{Pat} & p & ::= & x 
%& x 
& \text{variable pattern}\\
&&& \aewildp 
%& \wildp 
& \text{wildcard pattern}\\
&&& \aefoldp{p} 
%& \foldp{p} 
& \text{fold pattern}\\
&&& \aetplp{\labelset}{\mapschema{p}{i}{\labelset}} 
%& \tplp{\mapschema{p}{i}{\labelset}} 
& \text{labeled tuple pattern}\\
&&& \aeinjp{\ell}{p} 
%& \injp{\ell}{p} 
& \text{injection pattern}
\end{array}\]

\subsection{Statics}\label{appendix:P-statics}
\subsubsection{Unified Contexts}
A \emph{unified context}, $\Omega$, is an ordered finite function. 
We write
\begin{itemize}
\item $\Omega, X : \sigma$ when $X \notin \domof{\Omega}$ for the extension of $\Omega$ with a mapping from $X$ to the hypothesis $X : \sigma$.
\item $\Omega, x : \tau$ when $x \notin \domof{\Omega}$ for the extension of $\Omega$ with a mapping from $x$ to the hypothesis $x : \tau$.
\item $\Omega, u :: \kappa$ when $u \notin \domof{\Omega}$ for the extension of $\Omega$ with a mapping from $u$ to the hypothesis $u :: \kappa$.
\end{itemize}

% \begin{definition}[Unified Context Formation] $\isctxU{\Omega}$ iff:
% \begin{enumerate}
% \item $\Omega = \emptyset$; or
% \item $\Omega = \Omega', X : \sigma$ and $\issig{\Omega'}{\sigma}$; or 
% \item $\Omega = \Omega', u :: \kappa$ and $\iskind{\Omega'}{\kappa}$; or 
% \item $\Omega = \Omega', x : \tau$ and $\isTypeP{\Omega'}{\tau}$.
% \end{enumerate}
% \end{definition}

\subsubsection{Signatures and Structures}
\noindent\fbox{$\strut\issigX{\sigma}$}~~$\sigma$ is a signature
\begin{equation}\label{rule:issig}
\inferrule{
  \iskindX{\kappa}\\
  \haskind{\Omega, u :: \kappa}{\tau}{\akty}
}{
  \issigX{\asignature{\kappa}{u}{\tau}}
}
\end{equation}

\noindent\fbox{$\strut\sigequalX{\sigma}{\sigma'}$}~~$\sigma$ and $\sigma'$ are definitionally equal
\begin{equation}\label{rule:sigequal}
\inferrule{
  \kequalX{\kappa}{\kappa'}\\
  \cequal{\Omega, u :: \kappa}{\tau}{\tau'}{\akty}
}{
  \sigequalX{\asignature{\kappa}{u}{\tau}}{\asignature{\kappa'}{u}{\tau'}}
}
\end{equation}

\noindent\fbox{$\strut\sigsubX{\sigma}{\sigma'}$}~~$\sigma$ is a subsignature of $\sigma'$
\begin{equation}\label{rule:sigsub}
\inferrule{
  \ksubX{\kappa}{\kappa'}\\
  \issubtypeP{\Omega, u :: \kappa}{\tau}{\tau'}
}{
  \sigsubX{\asignature{\kappa}{u}{\tau}}{\asignature{\kappa'}{u}{\tau'}}
}
\end{equation}

\noindent\fbox{$\strut\hassigX{M}{\sigma}$}~~$M$ matches $\sigma$
\begin{subequations}\label{rules:hassig}
\begin{equation}\label{rule:hassig-subsume}
\inferrule{
  \hassigX{M}{\sigma}\\
  \sigsubX{\sigma}{\sigma'}
}{
  \hassigX{M}{\sigma'}
}
\end{equation}
\begin{equation}\label{rule:hassig-var}
\inferrule{ }{
  \hassig{\Omega, X : \sigma}{X}{\sigma}
}
\end{equation}
\begin{equation}\label{rule:hassig-struct}
\inferrule{
  \haskindX{c}{\kappa}\\
  \hastypeP{\Omega}{e}{[c/u]\tau}
}{
  \hassigX{\astruct{c}{e}}{\asignature{\kappa}{u}{\tau}}
}
\end{equation}
\begin{equation}\label{rule:hassig-seal}
\inferrule{
  \issigX{\sigma}\\
  \hassigX{M}{\sigma}
}{
  \hassigX{\aseal{\sigma}{M}}{\sigma}
}
\end{equation}
\begin{equation}\label{rule:hassig-let}
\inferrule{
  \hassigX{M}{\sigma}\\
  \issigX{\sigma'}\\
  \hassig{\Omega, X : \sigma}{M'}{\sigma'}  
}{
  \hassigX{\amlet{\sigma'}{M}{X}{M'}}{\sigma'}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\ismvalX{M}$}~~$M$ is, or stands for, a module value
\begin{subequations}\label{rules:ismval}
\begin{equation}\label{rule:ismval-struct}
\inferrule{ }{
  \ismvalX{\astruct{c}{e}}
}
\end{equation}
\begin{equation}\label{rule:ismval-var}
\inferrule{ }{
  \ismval{\Omega, X : \sigma}{X}
}
\end{equation}
\end{subequations}

\subsubsection{Kinds and Constructions}
\noindent\fbox{$\strut\iskindX{\kappa}$}~~$\kappa$ is a kind
\begin{subequations}\label{rules:iskind}
\begin{equation}\label{rule:iskind-darr}
\inferrule{
  \iskindX{\kappa_1}\\
  \iskind{\Omega, u :: \kappa_1}{\kappa_2}
}{
  \iskindX{\akdarr{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:iskind-unit}
\inferrule{ }{
  \iskindX{\akunit}
}
\end{equation}
\begin{equation}\label{rule:iskind-dprod}
\inferrule{
  \iskindX{\kappa_1}\\
  \iskind{\Omega, u :: \kappa_1}{\kappa_2}
}{
  \iskindX{\akdbprod{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:iskind-ty}
\inferrule{ }{
  \iskindX{\akty}
}
\end{equation}
\begin{equation}\label{rule:iskind-sing}
\inferrule{
  \haskindX{\tau}{\akty}
}{
  \iskindX{\aksing{\tau}}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\kequalX{\kappa}{\kappa'}$}~~$\kappa$ and $\kappa'$ are definitionally equal
\begin{subequations}\label{rules:kequal}
\begin{equation}\label{rule:kequal-refl}
\inferrule{
  \iskindX{\kappa}
}{
  \kequalX{\kappa}{\kappa}
}
\end{equation}
\begin{equation}\label{rule:kequal-sym}
\inferrule{
  \kequalX{\kappa}{\kappa'}
}{
  \kequalX{\kappa'}{\kappa}
}
\end{equation}
\begin{equation}\label{rule:kequal-trans}
\inferrule{
  \kequalX{\kappa}{\kappa'}\\
  \kequalX{\kappa'}{\kappa''}
}{
  \kequalX{\kappa}{\kappa''}
}
\end{equation}
\begin{equation}\label{rule:kequal-darr}
\inferrule{
  \kequalX{\kappa_1}{\kappa_1'}\\
  \kequal{\Omega, u :: \kappa_1}{\kappa_2}{\kappa_2'}
}{
  \kequalX{\akdarr{\kappa_1}{u}{\kappa_2}}{\akdarr{\kappa_1'}{u}{\kappa_2'}}
}
\end{equation}
\begin{equation}\label{rule:kequal-dprod}
\inferrule{
  \kequalX{\kappa_1}{\kappa'_1}\\
  \kequal{\Omega, u :: \kappa_1}{\kappa_2}{\kappa'_2}
}{
  \kequalX{\akdbprod{\kappa_1}{u}{\kappa_2}}{\akdbprod{\kappa'_1}{u}{\kappa'_2}}  
}
\end{equation}
\begin{equation}\label{rule:kequal-sing}
\inferrule{
  \cequalX{c}{c'}{\akty}
}{
  \kequalX{\aksing{c}}{\aksing{c'}}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\ksubX{\kappa}{\kappa'}$}~~$\kappa$ is a subkind of $\kappa'$
\begin{subequations}\label{rules:ksub}
\begin{equation}\label{rule:ksub-equal}
\inferrule{
  \kequalX{\kappa}{\kappa'}
}{
  \ksubX{\kappa}{\kappa'}
}
\end{equation}
\begin{equation}\label{rule:ksub-trans}
\inferrule{
  \ksubX{\kappa}{\kappa'}\\
  \ksubX{\kappa'}{\kappa''}
}{
  \ksubX{\kappa}{\kappa''}
}
\end{equation}
\begin{equation}\label{rule:ksub-darr}
\inferrule{
  \ksubX{\kappa'_1}{\kappa_1}\\
  \ksub{\Omega, u :: \kappa'_1}{\kappa_2}{\kappa'_2}  
}{
  \ksubX{\akdarr{\kappa_1}{u}{\kappa_2}}{\akdarr{\kappa'_1}{u}{\kappa'_2}}
}
\end{equation}
\begin{equation}\label{rule:ksub-dprod}
\inferrule{
  \ksubX{\kappa_1}{\kappa'_1}\\
  \ksub{\Omega, u :: \kappa_1}{\kappa_2}{\kappa'_2}
}{
  \ksubX{\akdbprod{\kappa_1}{u}{\kappa_2}}{\akdbprod{\kappa'_1}{u}{\kappa'_2}}
}
\end{equation}
\begin{equation}\label{rule:ksub-sing}
\inferrule{
  \haskindX{\tau}{\akty}
}{
  \ksubX{\aksing{\tau}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:ksub-sing-2}
\inferrule{
  \issubtypePX{\tau}{\tau'}
}{
  \ksubX{\aksing{\tau}}{\aksing{\tau'}}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\haskindX{c}{\kappa}$}~~$c$ has kind $\kappa$
\begin{subequations}\label{rules:haskind}
\begin{equation}\label{rule:haskind-subsume}
\inferrule{
  \haskindX{c}{\kappa_1}\\
  \ksubX{\kappa_1}{\kappa_2}
}{
  \haskindX{c}{\kappa_2}
}
\end{equation}
\begin{equation}\label{rule:haskind-var}
\inferrule{ }{\haskind{\Omega, u :: \kappa}{u}{\kappa}}
\end{equation}
\begin{equation}\label{rule:haskind-abs}
\inferrule{
  \haskind{\Omega, u :: \kappa_1}{c_2}{\kappa_2}
}{
  \haskindX{\acabs{u}{c_2}}{\akdarr{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:haskind-app}
\inferrule{
  \haskindX{c_1}{\akdarr{\kappa_2}{u}{\kappa}}\\
  \haskindX{c_2}{\kappa_2}
}{
  \haskindX{\acapp{c_1}{c_2}}{[c_1/u]\kappa}
}
\end{equation}
\begin{equation}\label{rule:haskind-unit}
\inferrule{ }{
  \haskindX{\actriv}{\akunit}
}
\end{equation}
\begin{equation}\label{rule:haskind-pair}
\inferrule{
  \haskindX{c_1}{\kappa_1}\\
  \haskindX{c_2}{[c_1/u]\kappa_2}
}{
  \haskindX{\acpair{c_1}{c_2}}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:haskind-prl}
\inferrule{
  \haskindX{c}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}{
  \haskindX{\acprl{c}}{\kappa_1}
}
\end{equation}
\begin{equation}\label{rule:haskind-prr}
\inferrule{
  \haskindX{c}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}{
  \haskindX{\acprr{c}}{[\acprl{c}/u]\kappa_2}
}
\end{equation}
\begin{equation}\label{rule:haskind-parr}
\inferrule{
  \haskindX{\tau_1}{\akty}\\
  \haskindX{\tau_2}{\akty}
}{
  \haskindX{\aparr{\tau_1}{\tau_2}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:haskind-all}
\inferrule{
  \iskindX{\kappa}\\
  \haskind{\Omega, u :: \kappa}{\tau}{\akty}
}{
  \haskindX{\aallu{\kappa}{u}{\tau}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:haskind-rec}
\inferrule{
  \haskind{\Omega, t :: \akty}{\tau}{\akty}
}{
  \haskindX{\arec{t}{\tau}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:haskind-prod}
\inferrule{
  \{\haskindX{\tau_i}{\akty}\}_{1 \leq i \leq n}
}{
  \haskindX{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:haskind-sum}
\inferrule{
  \{\haskindX{\tau_i}{\akty}\}_{1 \leq i \leq n}
}{
  \haskindX{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:haskind-sing}
\inferrule{
  \haskindX{c}{\akty}
}{
  \haskindX{c}{\aksing{c}}
}
\end{equation}
\begin{equation}\label{rule:haskind-stat}
\inferrule{
  \ismvalX{M}\\
  \hassigX{M}{\asignature{\kappa}{u}{\tau}}
}{
  \haskindX{\amcon{M}}{\kappa}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\cequalX{c}{c'}{\kappa}$}~~$c$ and $c'$ are definitionally equal as constructions of kind $\kappa$
\begin{subequations}\label{rules:cequal}
\begin{equation}\label{rule:cequal-refl}
\inferrule{
  \haskindX{c}{\kappa}
}{
  \cequalX{c}{c}{\kappa}
}
\end{equation}
\begin{equation}\label{rule:cequal-sym}
\inferrule{
  \cequalX{c}{c'}{\kappa}
}{
  \cequalX{c'}{c}{\kappa}
}
\end{equation}
\begin{equation}\label{rule:cequal-trans}
\inferrule{
  \cequalX{c}{c'}{\kappa}\\
  \cequalX{c'}{c''}{\kappa}
}{
  \cequalX{c}{c''}{\kappa}
}
\end{equation}
\begin{equation}\label{rule:cequal-lam}
\inferrule{
  \cequal{\Omega, u :: \kappa_1}{c}{c'}{\kappa_2}
}{
  \cequalX{\acabs{u}{c}}{\acabs{u}{c'}}{\akdarr{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:cequal-app-1}
\inferrule{
  \cequalX{c_1}{c_1'}{\akdarr{\kappa_2}{u}{\kappa}}\\
  \cequalX{c_2}{c_2'}{\kappa_2}
}{
  \cequalX{\acapp{c_1}{c_2}}{\acapp{c'_1}{c'_2}}{\kappa}
}
\end{equation}
\begin{equation}\label{rule:cequal-app-2}
\inferrule{
  \haskindX{\acabs{u}{c}}{\akdarr{\kappa_2}{u}{\kappa}}\\
  \haskindX{c_2}{\kappa_2}
}{
  \cequalX{\acapp{\acabs{u}{c}}{c_2}}{[c_2/u]c}{[c_2/u]\kappa}
}
\end{equation}
\begin{equation}\label{rule:cequal-pair}
\inferrule{
  \cequalX{c_1}{c'_1}{\kappa_1}\\
  \cequalX{c_2}{c'_2}{[c_1/u]\kappa_2}
}{
  \cequalX{\acpair{c_1}{c_2}}{\acpair{c'_1}{c'_2}}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:cequal-prl-1}
\inferrule{
  \cequalX{c}{c'}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}{
  \cequalX{\acprl{c}}{\acprl{c'}}{\kappa_1}
}
\end{equation}
\begin{equation}\label{rule:cequal-prl-2}
\inferrule{
  \haskindX{c_1}{\kappa_1}\\
  \haskindX{c_2}{\kappa_2}
}{
  \cequalX{\acprl{\acpair{c_1}{c_2}}}{c_1}{\kappa_1}
}
\end{equation}
\begin{equation}\label{rule:cequal-prr-1}
\inferrule{
  \cequalX{c}{c'}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}{
  \cequalX{\acprr{c}}{\acprr{c'}}{[\acprl{c}/u]\kappa_2}
}
\end{equation}
\begin{equation}\label{rule:cequal-prr-2}
\inferrule{
  \haskindX{c_1}{\kappa_1}\\
  \haskindX{c_2}{\kappa_2}
}{
  \cequalX{\acprr{\acpair{c_1}{c_2}}}{c_2}{\kappa_2}
}
\end{equation}
\begin{equation}\label{rule:cequal-parr}
\inferrule{
  \cequalX{\tau_1}{\tau'_1}{\akty}\\
  \cequalX{\tau_2}{\tau'_2}{\akty}
}{
  \cequalX{\aparr{\tau_1}{\tau_2}}{\aparr{\tau'_1}{\tau'_2}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cequal-all}
\inferrule{
  \kequalX{\kappa}{\kappa'}\\
  \cequal{\Omega, u :: \kappa}{\tau}{\tau'}{\akty}
}{
  \cequalX{\aallu{\kappa}{u}{\tau}}{\aallu{\kappa'}{u}{\tau'}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cequal-rec}
\inferrule{
  \cequal{\Omega, t :: \akty}{\tau}{\tau'}{\akty}
}{
  \cequalX{\arec{t}{\tau}}{\arec{t}{\tau'}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cequal-prod}
\inferrule{
  \{\cequalX{\tau_i}{\tau'_i}{\akty}\}_{1 \leq i \leq n}
}{
  \cequalX{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau'}{i}{\labelset}}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cequal-sum}
\inferrule{
  \{\cequalX{\tau_i}{\tau'_i}{\akty}\}_{1 \leq i \leq n}
}{
  \cequalX{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\asum{\labelset}{\mapschema{\tau'}{i}{\labelset}}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cequal-sing}
\inferrule{
  \haskindX{c}{\aksing{c'}}
}{
  \cequalX{c}{c'}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cequal-stat}
\inferrule{
  % \ismvalX{\astruct{c}{e}}\\
  \hassigX{\astruct{c}{e}}{\asignature{\kappa}{u}{\tau}}
}{
  \cequalX{\amcon{\astruct{c}{e}}}{c}{\kappa}
}
\end{equation}
\end{subequations}
\subsubsection{Expressions, Rules and Patterns}
% \noindent\fbox{$\strut\istypeP{\Omega}{\tau}$}~~$\tau$ is a type

% \vspace{6px}\noindent Types, $\tau$, classify expressions. The constructions of kind $\akty$ coincide with the types of $\miniVerseParam$.
% \begin{equation}\label{rule:istypeP}
% \inferrule{
%   \haskindX{\tau}{\akty}
% }{
%   \istypeP{\Omega}{\tau}
% }
% \end{equation}

% \noindent\fbox{$\strut\tequalPX{\tau}{\tau'}$}~~$\tau$ and $\tau'$ are definitionally equal types

% \vspace{6px}\noindent Type equality then coincides with construction equality at kind $\akty$.
% \begin{equation}\label{rule:tequalP}
% \inferrule{
%   \cequalX{\tau}{\tau}{\akty}
% }{
%   \tequalPX{\tau}{\tau'}
% }
% \end{equation}


\noindent\fbox{$\strut\issubtypePX{\tau}{\tau'}$}~~$\tau$ is a subtype of $\tau'$

\begin{subequations}\label{rules:issubtypeP}  
\begin{equation}\label{rule:issubtypeP-equal}
\inferrule{
  \cequalX{\tau_1}{\tau_2}{\akty}
}{
  \issubtypePX{\tau_1}{\tau_2}
}
\end{equation}
\begin{equation}\label{rule:issubtypeP-trans}
\inferrule{
  \issubtypePX{\tau}{\tau'}\\
  \issubtypePX{\tau'}{\tau''}
}{
  \issubtypePX{\tau}{\tau''}
}
\end{equation}
\begin{equation}\label{rule:issubtypeP-parr}
\inferrule{
  \issubtypePX{\tau_1'}{\tau_1}\\
  \issubtypePX{\tau_2}{\tau_2'}
}{
  \issubtypePX{\aparr{\tau_1}{\tau_2}}{\aparr{\tau_1'}{\tau_2'}}
}
\end{equation}
\begin{equation}\label{rule:issubtypeP-all}
\inferrule{
  \ksubX{\kappa'}{\kappa}\\
  \issubtypeP{\Omega, u :: \kappa'}{\tau}{\tau'}
}{
  \issubtypePX{\aallu{\kappa}{u}{\tau}}{\aallu{\kappa'}{u}{\tau'}}
}
\end{equation}
\begin{equation}\label{rule:issubtypeP-prod}
\inferrule{
  \{\issubtypePX{\tau_i}{\tau'_i}\}_{i \in \labelset}
}{
  \issubtypePX{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau'}{i}{\labelset}}}
}
\end{equation}
\begin{equation}\label{rule:issubtypeP-sum}
\inferrule{
  \{\issubtypePX{\tau_i}{\tau'_i}\}_{i \in \labelset}
}{
  \issubtypePX{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\asum{\labelset}{\mapschema{\tau'}{i}{\labelset}}}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\hastypeP{\Omega}{e}{\tau}$}~~$e$ has type $\tau$
\begin{subequations}\label{rules:hastypeP}
\begin{equation}\label{rule:hastypeP-subsume}
\inferrule{
  \hastypeP{\Omega}{e}{\tau}\\
  \issubtypePX{\tau}{\tau'}
}{
  \hastypeP{\Omega}{e}{\tau'}
}
\end{equation}
\begin{equation}\label{rule:hastypeP-var}
  \inferrule{ }{
    \hastypeP{\Omega, \Ghyp{x}{\tau}}{x}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:hastypeP-lam}
  \inferrule{
    \haskind{\Omega}{\tau}{\akty}\\
    \hastypeP{\Omega, \Ghyp{x}{\tau}}{e}{\tau'}
  }{
    \hastypeP{\Omega}{\aelam{\tau}{x}{e}}{\aparr{\tau}{\tau'}}
  }
\end{equation}
\begin{equation}\label{rule:hastypeP-ap}
  \inferrule{
    \hastypeP{\Omega}{e_1}{\aparr{\tau}{\tau'}}\\
    \hastypeP{\Omega}{e_2}{\tau}
  }{
    \hastypeP{\Omega}{\aeap{e_1}{e_2}}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:hastypeP-clam}
  \inferrule{
    \iskindX{\kappa}\\
    \hastypeP{\Omega, u :: \kappa}{e}{\tau}
  }{
    \hastypeP{\Omega}{\aeclam{\kappa}{u}{e}}{\aallu{\kappa}{u}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:hastypeP-cap}
  \inferrule{
    \hastypeP{\Omega}{e}{\aallu{\kappa}{u}{\tau}}\\
    \haskindX{c}{\kappa}
  }{
    \hastypeP{\Omega}{\aecap{e}{c}}{[c/u]\tau}
  }
\end{equation}
\begin{equation}\label{rule:hastypeP-fold}
  \inferrule{\
    % \haskind{\Omega, t :: \akty}{\tau}{\akty}\\
    \hastypeP{\Omega}{e}{[\arec{t}{\tau}/t]\tau}
  }{
    \hastypeP{\Omega}{\aefold{e}}{\arec{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:hastypeP-unfold}
  \inferrule{
    \hastypeP{\Omega}{e}{\arec{t}{\tau}}
  }{
    \hastypeP{\Omega}{\aeunfold{e}}{[\arec{t}{\tau}/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:hastypeP-tpl}
  \inferrule{
    \{\hastypeP{\Omega}{e_i}{\tau_i}\}_{i \in \labelset}
  }{
    \hastypeP{\Omega}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}
\begin{equation}\label{rule:hastypeP-pr}
  \inferrule{
    \hastypeP{\Omega}{e}{\aprod{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \ell \hookrightarrow \tau}}
  }{
    \hastypeP{\Omega}{\aepr{\ell}{e}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:hastypeP-in}
  \inferrule{
    % \{\haskind{\Omega}{\tau_i}{\akty}\}_{i \in \labelset}\\
    % \haskind{\Omega}{\tau}{\akty}\\
    \hastypeP{\Omega}{e}{\tau}
  }{
    \hastypeP{\Omega}{\aein{\ell}{e}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \ell \hookrightarrow \tau}}
  }
\end{equation}
\begin{equation}\label{rule:hastypeP-match}
\inferrule{
  \hastypeP{\Omega}{e}{\tau}\\
  % \haskind{\Omega}{\tau'}{\akty}\\
  \{\ruleTypeP{\Omega}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}\\
}{\hastypeP{\Omega}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}}
\end{equation}
\begin{equation}\label{rule:hastypeP-dyn}
\inferrule{
  \ismvalX{M}\\
  \hassigX{M}{\asignature{\kappa}{u}{\tau}}
}{
  \hastypeP{\Omega}{\amval{M}}{[\amcon{M}/u]\tau}
}
\end{equation}
\end{subequations}
\noindent\fbox{$\strut\ruleTypeP{\Omega}{r}{\tau}{\tau'}$}~~$r$ takes values of type $\tau$ to values of type $\tau'$
\begin{equation}\label{rule:ruleTypeP}
\inferrule{
  \patTypeP{\Omega'}{p}{\tau}\\
  \hastypeP{\Gcons{\Omega}{\Omega'}}{e}{\tau'}
}{
  \ruleTypeP{\Omega}{\aematchrule{p}{e}}{\tau}{\tau'}
}
\end{equation}

\noindent\fbox{$\strut\patTypeP{\Omega'}{p}{\tau}$}~~$p$ matches values of type $\tau$ generating hypotheses $\Omega'$

\begin{subequations}\label{rules:patTypeP}
\begin{equation}\label{rule:patTypeP-subsume}
\inferrule{
  \patTypeP{\Omega'}{p}{\tau}\\
  \issubtypePX{\tau}{\tau'}
}{
  \patTypeP{\Omega'}{p}{\tau'}
}
\end{equation}
\begin{equation}\label{rule:patTypeP-var}
\inferrule{ }{\patTypeP{\Ghyp{x}{\tau}}{x}{\tau}}
\end{equation}
\begin{equation}\label{rule:patTypeP-wild}
\inferrule{ }{\patTypeP{\emptyset}{\aewildp}{\tau}}
\end{equation}
\begin{equation}\label{rule:patTypeP-fold}
\inferrule{
  \patTypeP{\Omega'}{p}{[\arec{t}{\tau}/t]\tau}
}{
  \patTypeP{\Omega'}{\aefoldp{p}}{\arec{t}{\tau}}
}
\end{equation}
\begin{equation}\label{rule:patTypeP-tpl}
\inferrule{
  \{\patTypeP{\Omega_i}{p_i}{\tau_i}\}_{i \in \labelset}
}{
  \patTypeP{\Gconsi{i \in \labelset}{\Omega_i}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
}
\end{equation}
\begin{equation}\label{rule:patTypeP-inj}
\inferrule{
  \patTypeP{\Omega'}{p}{\tau}
}{
  \patTypeP{\Omega'}{\aeinjp{\ell}{p}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
}
\end{equation}
\end{subequations}

\subsubsection{Metatheory}
The rules above are syntax-directed, so we assume an inversion lemma for each rule as needed without stating it separately or proving it explicitly. The following standard lemmas also hold, for all basic judgements $J$ above.

\begin{lemma}[Weakening]\label{lemma:weakening-P}  If $\Omega \vdash J$ then $\Omega \cup \Omega' \vdash J$.
% \begin{enumerate}
% \item \begin{enumerate}
%   \item If $\issigX{\sigma}$ then $\issig{\Omega \cup \Omega'}{\sigma}$.
%   \item If $\sigequal{\Omega}{\sigma}{\sigma'}$ then $\sigequal{\Omega \cup \Omega'}{\sigma}{\sigma'}$.
%   \item If $\sigsub{\Omega}{\sigma}{\sigma'}$ then $\sigsub{\Omega \cup \Omega'}{\sigma}{\sigma'}$.
%   \item If $\hassigX{M}{\sigma}$ then $\hassig{\Omega \cup \Omega'}{M}{\sigma}$.
%   \item If $\ismvalX{M}$ then $\ismval{\Omega \cup \Omega'}{M}$.
%   \end{enumerate}
% \item \begin{enumerate}
% \item If $\iskindX{\kappa}$ then $\iskind{\Omega \cup \Omega'}{\kappa}$.
% \item If $\kequalX{\kappa}{\kappa'}$ then $\kequal{\Omega \cup \Omega'}{\kappa}{\kappa'}$.
% \item If $\ksubX{\kappa}{\kappa'}$ then $\ksub{\Omega \cup \Omega'}{\kappa}{\kappa'}$.
% \item If $\haskindX{c}{\kappa}$ then $\haskind{\Omega \cup \Omega'}{c}{\kappa}$.
% \item If $\cequalX{c}{c'}{\kappa}$ then $\cequal{\Omega \cup \Omega'}{c}{c'}{\kappa}$.
% \end{enumerate}
% \item \begin{enumerate}
% \item If $\istypeP{\Omega}{\tau}$ then $\istypeP{\Omega \cup \Omega'}{\tau}$.
% \item If $\tequalPX{\tau}{\tau'}$ then $\tequalP{\Omega \cup \Omega'}{\tau}{\tau'}$.
% \item If $\issubtypePX{\tau}{\tau'}$ then $\issubtypeP{\Omega \cup \Omega'}{\tau}{\tau'}$.
% \item If $\hastypeP{\Omega}{e}{\tau}$ then $\hastypeP{\Omega \cup \Omega'}{e}{\tau}$.
% \item If $\ruleTypeP{\Omega}{r}{\tau}{\tau'}$ then $\ruleTypeP{\Omega \cup \Omega'}{r}{\tau}{\tau'}$.
% \item If $\patTypePC{\Omega}{\Omega''}{p}{\tau}$ then $\patTypePC{\Omega \cup \Omega'}{\Omega''}{p}{\tau}$.
% \end{enumerate}
% \end{enumerate}
\end{lemma}
\begin{proof-sketch} By straightforward mutual rule induction.
\end{proof-sketch}

\begin{definition} 
\label{def:substitution-p}
A \emph{substitution}, $\omega$, is a finite function that maps:
\begin{itemize}
\item each $X \in \domof{\omega}$ to a module expression subtitution, $M/X$; 
\item each $u \in \domof{\omega}$ to a construction substitution, $c/u$; and 
\item each $x \in \domof{\omega}$ to an expression substitution, $e/x$.
\end{itemize}

We write $\hastypeP{\Omega}{\omega}{\Omega'}$ iff $\domof{\omega}=\domof{\Omega'}$ and:
\begin{itemize}
\item for each $M/X \in \omega$, we have $X : \sigma \in \Omega'$ and $\hassigX{M}{\sigma}$ and $\ismvalX{M}$; and
\item for each $c/u \in \omega$, we have $u :: \kappa \in \Omega'$ and $\haskindX{c}{\kappa}$; and 
\item for each $e/x \in \omega$, we have $x : \tau \in \Omega'$ and $\hastypeP{\Omega}{e}{\tau}$.
\end{itemize}

We simultaneously apply a substitution by placing it in prefix position. For example, $[\omega]e$ applies the substitutions $\omega$ simultaneously to $e$.
\end{definition}

\begin{lemma}[Substitution]\label{lemma:substitution-P} If $\Omega \cup \Omega' \cup \Omega'' \vdash J$ and $\hastypeP{\Omega}{\omega}{\Omega'}$ then $\Omega \cup [\omega]\Omega'' \vdash [\omega]J$.
\end{lemma}
\begin{proof-sketch} By straightforward rule induction. 
\end{proof-sketch}

\begin{lemma}[Decomposition]\label{lemma:decomposition-P} 
If $\Omega \cup [\omega]\Omega'' \vdash [\omega]J$ and $\hastypeP{\Omega}{\omega}{\Omega'}$ then $\Omega \cup \Omega' \cup \Omega'' \vdash J$.
\end{lemma}
\begin{proof-sketch} By straightforward rule induction.
\end{proof-sketch}

\begin{lemma}[Pattern Binding]\label{lemma:pattern-binding}
If $\patTypeP{\Omega'}{p}{\tau}$ then $\domof{\Omega'} = \mathsf{patvars}(p)$.
\end{lemma}
\begin{proof-sketch} By straightforward rule induction over Rules (\ref{rules:patTypeP}). \end{proof-sketch}
% \begin{lemma}[Regularity]\label{lemma:regularity-P} ~
% \begin{enumerate}
% \item ...
% \end{enumerate}
% \end{lemma}

\subsection{Structural Dynamics}
The structural dynamics of modules is defined as a transition system, and is organized around judgements of the following form:

\vspace{10px}
$\begin{array}{ll}
\textbf{Judgement Form} & \textbf{Description}\\
\stepsU{M}{M'} & \text{$M$ transitions to $M'$}\\
\isvalP{M} & \text{$M$ is a module value}\\
\matchfail{M} & \text{$M$ raises match failure}
\end{array}$
\vspace{10px}

The structural dynamics of expressions is also defined as a transition system, and is organized around judgements of the following form:

\vspace{10px}
$\begin{array}{ll}
\textbf{Judgement Form} & \textbf{Description}\\
\stepsU{e}{e'} & \text{$e$ transitions to $e'$}\\
\isvalP{e} & \text{$e$ is a value}\\
\matchfail{e} & \text{$e$ raises match failure}
\end{array}$
\vspace{10px}

We also define auxiliary judgements for \emph{iterated transition}, $\multistepU{e}{e'}$, and \emph{evaluation}, $\evalU{e}{e'}$ of expressions.

\begin{definition}[Iterated Transition]\label{defn:iterated-transition-P} Iterated transition, $\multistepU{e}{e'}$, is the reflexive, transitive closure of the transition judgement, $\stepsU{e}{e'}$.\end{definition}

\begin{definition}[Evaluation]\label{defn:evaluation-P} $\evalU{e}{e'}$ iff $\multistepU{e}{e'}$ and $\isvalU{e'}$. \end{definition}

Similarly, we lift these definitions to the level of module expressions as well.

\begin{definition}[Iterated Module Transition]\label{defn:iterated-transition-modules-P} Iterated transition, $\multistepU{M}{M'}$, is the reflexive, transitive closure of the transition judgement, $\stepsU{M}{M'}$.\end{definition}

\begin{definition}[Module Evaluation]\label{defn:evaluation-modules-P} $\evalU{M}{M'}$ iff $\multistepU{M}{M'}$ and $\isvalU{M'}$. \end{definition}



As in $\miniVersePat$, our subsequent developments do not make mention of particular rules in the dynamics, nor do they make mention of other judgements, not listed above, that are used only for defining the dynamics of the match operator, so we do not produce these details here. Instead, it suffices to state the following conditions.

The Preservation condition ensures that evaluation preserves typing.
\begin{condition}[Preservation]\label{condition:preservation-P} ~
\begin{enumerate}
\item If $\hassig{}{M}{\sigma}$ and $\stepsU{M}{M'}$ then $\hassig{}{M}{\sigma}$.
\item If $\hastypeUC{e}{\tau}$ and $\stepsU{e}{e'}$ then $\hastypeUC{e'}{\tau}$.
\end{enumerate}
\end{condition}

The Progress condition ensures that evaluation of a well-typed expanded expression cannot ``get stuck''. We must consider the possibility of match failure in this condition.
\begin{condition}[Progress]\label{condition:progress-P} ~
\begin{enumerate}
\item If $\hassig{}{M}{\sigma}$ then either $\isvalU{M}$ or $\matchfail{M}$ or there exists an $M'$ such that $\stepsU{M}{M'}$.
\item If $\hastypeUC{e}{\tau}$ then either $\isvalU{e}$ or $\matchfail{e}$ or there exists an $e'$ such that $\stepsU{e}{e'}$.
\end{enumerate}
\end{condition}

\section{Unexpanded Language (UL)}
\subsection{Syntax}
\subsubsection{Stylized Syntax -- Unexpanded Signatures and Modules}
\[\begin{array}{lllllll}
\textbf{Sort} & & 
%& \textbf{Operational Form} 
& \textbf{Stylized Form} & \textbf{Description}\\
\mathsf{USig} & \usigma & ::= 
%& \ausignature{\ukappa}{\uu}{\utau} 
& \signature{\uu}{\ukappa}{\utau} & \text{signature}\\
\mathsf{UMod} & \uM & ::= 
%& \uX 
& \uX & \text{module identifier}\\
&&
%& \austruct{\uc}{\ue} 
& \struct{\uc}{\ue} & \text{structure}\\
&&
%& \auseal{\usigma}{\uM} 
& \seal{\uM}{\usigma} & \text{seal}\\
&&
%& \aumlet{\usigma}{\uM}{\uX}{\uM} 
& \mlet{\uX}{\uM}{\uM}{\usigma} & \text{definition}\\
% \LCC &&
%& \color{light-gray} 
% & \color{Yellow} & \color{Yellow}\\
&&
%& \aumdefpetsm{\urho}{e}{\tsmv}{\uM} 
& \defpetsm{\tsmv}{\urho}{e}{\uM} & \text{peTLM definition}\\
%&&&                                    & \texttt{expressions}~\{e\}~\texttt{in}~\uM\\
&&
%& \aumletpetsm{\uepsilon}{\tsmv}{\uM} 
& \uletpetsm{\tsmv}{\uepsilon}{\uM} & \text{peTLM binding}\\
% &&&                                  & \texttt{expressions}~\texttt{in}~\uM\\
% &&& ... & ... & \text{peTLM designation}\\
&&
%& \audefpptsm{\urho}{e}{\tsmv}{\uM} 
& \defpptsm{\tsmv}{\urho}{e}{\uM} & \text{ppTLM definition}\\
% &&&                                    & \texttt{patterns}~\{e\}~\texttt{in}~\uM\\
&&
%& \auletpptsm{\uepsilon}{\tsmv}{\uM} 
& \uletpptsm{\tsmv}{\uepsilon}{\uM} & \text{ppTLM binding}%\ECC%
% &&& & \texttt{patterns}~\texttt{in}~\uM\\
% &&& ... & ... & \text{ppTLM designation}\ECC
\end{array}\]%\vspace{-15px}
% \caption[Syntax of unexpanded module expressions and signatures in $\miniVerseParam$]{Syntax of unexpanded module expressions and signatures in $\miniVerseParam$.}\vspace{-5px}
% \label{fig:P-unexpanded-modules-signatures}
% \end{figure}
% \begin{figure}[p] \vspace{-10px}

\subsubsection{Stylized Syntax -- Unexpanded Kinds and Constructions}
\[\begin{array}{lrlllll}
\textbf{Sort} & & 
%& \textbf{Operational Form} 
& \textbf{Stylized Form} & \textbf{Description}\\
\mathsf{UKind} & \ukappa & ::= 
%& \aukdarr{\ukappa}{\uu}{\ukappa} 
& \kdarr{\uu}{\ukappa}{\ukappa} & \text{dependent function}\\
&&
%& \aukunit 
& \kunit & \text{nullary product}\\
&&
%& \aukdbprod{\ukappa}{\uu}{\ukappa} 
& \kdbprod{\uu}{\ukappa}{\ukappa} & \text{dependent product}\\
%&&& \akdprodstd & \kdprodstd & \text{labeled dependent product}\\
&&
%& \aukty 
& \kty & \text{type}\\
&&
%& \auksing{\utau} 
& \ksing{\utau} & \text{singleton}\\
\mathsf{UCon} & \uc, \utau & ::= 
%& \uu 
& \uu & \text{construction identifier}\\
&&
%& \ut 
& \ut & \\
&&
%& \aucasc{\ukappa}{\uc} 
& \casc{\uc}{\ukappa} & \text{ascription}\\
&&
%& \aucabs{\uu}{\uc} 
& \cabs{\uu}{\uc} & \text{abstraction}\\
&&
%& \aucapp{c}{c} 
& \capp{c}{c} & \text{application}\\
&&
%& \auctriv 
& \ctriv & \text{trivial}\\
&&
%& \aucpair{\uc}{\uc} 
& \cpair{\uc}{\uc} & \text{pair}\\
&&
%& \aucprl{\uc} 
& \cprl{\uc} & \text{left projection}\\
&&
%& \aucprr{\uc} 
& \cprr{\uc} & \text{right projection}\\
%&&& \adtplX & \dtplX & \text{labeled dependent tuple}\\
%&&& \adprj{\ell}{c} & \prj{c}{\ell} & \text{projection}\\
&&
%& \auparr{\utau}{\utau} 
& \parr{\utau}{\utau} & \text{partial function}\\
&&
%& \auallu{\ukappa}{\uu}{\utau} 
& \forallu{\uu}{\ukappa}{\utau} & \text{polymorphic}\\
&&
%& \aurec{\ut}{\utau} 
& \rect{\ut}{\utau} & \text{recursive}\\
&&
%& \auprod{\labelset}{\mapschema{\utau}{i}{\labelset}} 
& \prodt{\mapschema{\utau}{i}{\labelset}} & \text{labeled product}\\
&&
%& \ausum{\labelset}{\mapschema{\utau}{i}{\labelset}} 
& \sumt{\mapschema{\utau}{i}{\labelset}} & \text{labeled sum}\\
&&
%& \aumcon{\uX} 
& \mcon{\uX} & \text{construction component}
\end{array}\]%\vspace{-15px}
% \caption[Syntax of unexpanded kinds and constructions in $\miniVerseParam$]{Syntax of unexpanded kinds and constructions in $\miniVerseParam$.}\vspace{-10px}
% \label{fig:P-unexpanded-kinds-constructions}
% \end{figure}
\clearpage

\subsubsection{Stylized Syntax -- Unexpanded Expressions, Rules and Patterns}
% \clearpage
% \begin{figure}[p]
\[\begin{array}{lllllll}
\textbf{Sort} & & 
%& \textbf{Operational Form} 
& \textbf{Stylized Form} & \textbf{Description}\\
\mathsf{UExp} & \ue & ::= 
%& \ux 
& \ux & \text{identifier}\\
&&
% & \auasc{\utau}{\ue} 
& \asc{\ue}{\utau} & \text{ascription}\\
&&
% & \auletsyn{\ux}{\ue}{\ue} 
& \letsyn{\ux}{\ue}{\ue} & \text{value binding}\\
% &&
%& \auanalam{\ux}{\ue} 
% & \analam{\ux}{\ue} & \text{abstraction (unannotated)}\\
&&
%& \aulam{\utau}{\ux}{\ue} 
& \lam{\ux}{\utau}{\ue} & \text{abstraction}\\
&&
%& \auap{\ue}{\ue} 
& \ap{\ue}{\ue} & \text{application}\\
&&
%& \auclam{\ukappa}{\uu}{\ue} 
& \clam{\uu}{\ukappa}{\ue} & \text{construction abstraction}\\
&&
%& \aucap{\ue}{\uc} 
& \cAp{\ue}{\uc} & \text{construction application}\\
&&
%& \auanafold{\ue} 
& \fold{\ue} & \text{fold}\\
&&
%& \auunfold{\ue} 
& \unfold{\ue} & \text{unfold}\\
&&
%& \autpl{\labelset}{\mapschema{\ue}{i}{\labelset}} 
& \tpl{\mapschema{\ue}{i}{\labelset}} & \text{labeled tuple}\\
&&
%& \aupr{\ell}{\ue} 
& \prj{\ue}{\ell} & \text{projection}\\
&&
%& \auanain{\ell}{\ue} 
& \inj{\ell}{\ue} & \text{injection}\\
&&
%& \aumatchwithb{n}{\ue}{\seqschemaX{\urv}} 
& \matchwith{\ue}{\seqschemaX{\urv}} & \text{match}\\
&&
%& \aumval{\uX} 
& \mval{\uX} & \text{value component}\\
% \LCC &&
% %& \color{Yellow} 
% & \color{Yellow} & \color{Yellow} \\
% &&& \audefpetsm{\urho}{e}{\tsmv}{\ue} & \texttt{syntax}~\tsmv~\texttt{at}~\urho~\texttt{for} & \text{peTLM definition}\\
% &&&                                    & \texttt{expressions}~\{e\}~\texttt{in}~\ue\\
% &&& \auletpetsm{\uepsilon}{\tsmv}{\ue} & \texttt{let}~\texttt{syntax}~\tsmv=\uepsilon~\texttt{for} & \text{peTLM binding}\\
% &&&                                  & \texttt{expressions}~\texttt{in}~\ue\\
% &&& ... & ... & \text{peTLM designation}\\
&&
%& \auappetsm{b}{\uepsilon} 
& \utsmap{\uepsilon}{b} & \text{peTLM application}\\%\ECC\\%\ECC
% &&& \auelit{b} & {\lit{b}}  & \text{peTLM unadorned literal}\\
% &&& \audefpptsm{\urho}{e}{\tsmv}{\ue} & \texttt{syntax}~\tsmv~\texttt{at}~\urho~\texttt{for} & \text{ppTLM definition}\\
% &&&                                    & \texttt{patterns}~\{e\}~\texttt{in}~\ue\\
% &&& \auletpptsm{\uepsilon}{\tsmv}{\ue} & \texttt{let}~\texttt{syntax}~\tsmv=\uepsilon~\texttt{for} & \text{ppTLM binding}\\
% &&& & \texttt{patterns}~\texttt{in}~\ue\\
% &&& ... & ... & \text{ppTLM designation}\\\ECC
\mathsf{URule} & \urv & ::= 
%& \aumatchrule{\upv}{\ue} 
& \matchrule{\upv}{\ue} & \text{match rule}\\
\mathsf{UPat} & \upv & ::= 
%& \ux 
& \ux & \text{identifier pattern}\\
&&
%& \auwildp 
& \wildp & \text{wildcard pattern}\\
&&
%& \aufoldp{\upv} 
& \foldp{\upv} & \text{fold pattern}\\
&&
%& \autplp{\labelset}{\mapschema{\upv}{i}{\labelset}} 
& \tplp{\mapschema{\upv}{i}{\labelset}} & \text{labeled tuple pattern}\\
&&
%& \auinjp{\ell}{\upv} 
& \injp{\ell}{\upv} 
& \text{injection pattern}\\
% \LCC &&
%& \color{light-gray} 
% & \color{Yellow} & \color{Yellow}\\
&&
%& \auappptsm{b}{\uepsilon} 
& \utsmap{\uepsilon}{b} & \text{ppTLM application}%\ECC
% &&& \auplit{b} & \lit{b} & \text{ppTLM unadorned literal}\ECC
\end{array}\]
% \caption[Syntax of unexpanded expressions, rules and patterns in $\miniVerseParam$]{Syntax of unexpanded expressions, rules and patterns in $\miniVerseParam$.}
% \label{fig:P-unexpanded-terms}
% \end{figure}

\subsubsection{Stylized Syntax -- Unexpanded TLM Types and Expressions}
% \begin{figure}[p]
\[\begin{array}{lllllll}
\textbf{Sort} & & 
%& \textbf{Operational Form} 
& \textbf{Stylized Form} 
& \textbf{Description}\\
% \LCC \color{Yellow}&\color{Yellow}& \color{Yellow}
%& \color{light-gray} 
% & \color{Yellow} & \color{Yellow}\\
\mathsf{UMType} & \urho & ::= 
%& \autype{\utau} 
& \utau & \text{type annotation}\\
% &&
%& \aualltypes{\ut}{\urho} 
% & \alltypes{\ut}{\urho} & \text{type parameterization}\\
&&
%& \auallmods{\usigma}{\uX}{\urho} 
& \allmods{\uX}{\usigma}{\urho} & \text{module parameterization}\\
\mathsf{UMExp} & \uepsilon & ::= 
%& \abindref{\tsmv} 
& \tsmv & \text{TLM identifier reference}\\
% &&
%& \auabstype{\ut}{\uepsilon} 
% & \abstype{\ut}{\uepsilon} & \text{type abstraction}\\
&&
%& \auabsmod{\usigma}{\uX}{\uepsilon} 
& \absmod{\uX}{\usigma}{\uepsilon} & \text{module abstraction}\\
% &&
%& \auaptype{\utau}{\uepsilon} 
% & \aptype{\uepsilon}{\utau} & \text{type application}\\
&&
%& \auapmod{\uM}{\uepsilon} 
& \apmod{\uepsilon}{\uX} & \text{module application}%\ECC
\end{array}
\]
% \caption{Syntax of unexpanded TLM types and expressions.}
% \label{fig:P-macro-expressions-types-u}
% \end{figure}

\subsubsection{Stylized Syntax -- TLM Types and Expressions}

% \clearpage
% \begin{figure}[p]
\[\begin{array}{lllllll}
\textbf{Sort} & & & \textbf{Operational Form} 
%& \textbf{Stylized Form} 
& \textbf{Description}\\
% \LCC \color{Yellow}&\color{Yellow}& \color{Yellow}
%& \color{light-gray} 
% & \color{Yellow} & \color{Yellow}\\
\mathsf{MType} & \rho & ::= & \aetype{\tau} 
%& \tau 
& \text{type annotation}\\
% &&& \aealltypes{t}{\rho} 
%& \alltypes{t}{\rho} 
% & \text{type parameterization}\\
&&& \aeallmods{\sigma}{X}{\rho} 
%& \allmods{X}{\sigma}{\rho} 
& \text{module parameterization}\\
\mathsf{MExp} & \epsilon & ::= & \adefref{a} 
%& a 
& \text{TLM definition reference}\\
% &&& \aeabstype{t}{\epsilon} 
%& \abstype{t}{\epsilon} 
% & \text{type abstraction}\\
&&& \aeabsmod{\sigma}{X}{\epsilon} 
%& \absmod{X}{\sigma}{\epsilon} 
& \text{module abstraction}\\
% &&& \aeaptype{\tau}{\epsilon} 
%& \aptype{\epsilon}{\tau} 
% & \text{type application}\\
&&& \aeapmod{M}{\epsilon} 
%& \aptype{\epsilon}{M} 
& \text{module application}%\ECC
\end{array}\]
% \caption[Syntax of TLM types and expressions in $\miniVerseParam$]{Syntax of TLM types and expressions.}
% \label{fig:P-macro-expressions-types}
% \end{figure}

\subsubsection{Body Lengths}
We write $\sizeof{b}$ for the length of $b$. 
The metafunction $\sizeof{\uM}$ computes the sum of the lengths of expression literal bodies in $\uM$:
\[
\begin{array}{ll}
\sizeof{\uX} & = 0\\
\sizeof{\struct{\uc}{\ue}} & = \sizeof{\ue}\\
\sizeof{\seal{\uM}{\usigma}} & = \sizeof{\uM}\\
\sizeof{\mlet{\uX}{\uM}{\uM'}{\usigma}} & = \sizeof{\uM} + \sizeof{\uM'}\\
\sizeof{\defpetsm{\tsmv}{\urho}{e}{\uM}} & = \sizeof{\uM}\\
\sizeof{\uletpetsm{\tsmv}{\uepsilon}{\uM}} & = \sizeof{\uM}\\
\sizeof{\defpptsm{\tsmv}{\urho}{e}{\uM}} & = \sizeof{\uM}\\
\sizeof{\uletpptsm{\tsmv}{\uepsilon}{\uM}} & = \sizeof{\uM}
\end{array}
\]
and $\sizeof{\ue}$ computes the sum of the lengths of expression literal bodies in $\ue$:
\[
\begin{array}{ll}
\sizeof{\ux} & = 0\\
\sizeof{\lam{\ux}{\utau}{\ue}} &= \sizeof{\ue}\\
\sizeof{\ap{\ue_1}{\ue_2}} & = \sizeof{\ue_1} + \sizeof{\ue_2}\\
\sizeof{\clam{\uu}{\ukappa}{\ue}} & = \sizeof{\ue}\\
\sizeof{\cAp{\ue}{\uc}} & = \sizeof{\ue}\\
\sizeof{\fold{\ue}} & = \sizeof{\ue}\\
\sizeof{\unfold{\ue}} & = \sizeof{\ue}\\
%\end{align*}
%\begin{align*}
\sizeof{\tpl{\mapschema{\ue}{i}{\labelset}}} & = \sum_{i \in \labelset} \sizeof{\ue_i}\\
\sizeof{\prj{\ell}{\ue}} & = \sizeof{\ue}\\
\sizeof{\inj{\ell}{\ue}} & = \sizeof{\ue}\\
\sizeof{\matchwith{\ue}{\seqschemaX{\urv}}} & = \sizeof{\ue} + \sum_{1 \leq i \leq n} \sizeof{r_i}\\
\sizeof{\mval{\uX}} & = 0\\
% \sizeof{\caseof{\ue}{\mapschemab{\ux}{\ue}{i}{\labelset}}} & = \sizeof{\ue} + \sum_{i \in \labelset} \sizeof{\ue_i}\\
% \sizeof{\uesyntax{\tsmv}{\utau}{\eparse}{\ue}} & = \sizeof{\ue}\\
\sizeof{\utsmap{\uepsilon}{b}} & = \sizeof{b}
\end{array}
\]
and $\sizeof{\urv}$ computes the sum of the lengths of expression literal bodies in $\urv$:
\begin{align*}
\sizeof{\matchrule{\upv}{\ue}} & = \sizeof{\ue}
\end{align*}
% and $\sizeof{\uepsilon}$ computes the sum of the lengths of expression literal bodies in $\uepsilon$:
% \begin{align*}
% \sizeof{\tsmv} & = 0\\
% \sizeof{\abstype{\ut}{\uepsilon}} & = \sizeof{\uepsilon}\\
% \sizeof{\absmod{\uX}{\usigma}{\uepsilon}} & = \sizeof{\uepsilon}\\
% \sizeof{\aptype{\uepsilon}{\utau}} & = 0\\
% \sizeof{\apmod{\uepsilon}{\uM}} & = \sizeof{\uM}
% \end{align*}

Similarly, the metafunction $\sizeof{\upv}$ computes the sum of the lengths of the pattern literal bodies in $\upv$:
\begin{align*}
\sizeof{\ux} & = 0\\
\sizeof{\foldp{\upv}} & = \sizeof{\upv}\\
\sizeof{\tplp{\mapschema{\upv}{i}{\labelset}}} & = \sum_{i \in \labelset} \sizeof{\upv_i}\\
\sizeof{\injp{\ell}{\upv}} & = \sizeof{\upv}\\
\sizeof{\utsmap{\uepsilon}{b}} & = \sizeof{b}
\end{align*}

\subsubsection{Common Unexpanded Forms}\label{appendix:P-shared-forms}
Each expanded form, with a few minor exceptions noted below, maps onto an unexpanded form. We refer to these as the \emph{common forms}. In particular:
\begin{itemize}
\item Each module variable, $X$, maps onto a unique module identifier, written $\sigilof{X}$.
\item Each signature, $\sigma$, maps onto an unexpanded signature, $\Uof{\sigma}$, as follows:
\begin{align*}
\Uof{\asignature{\kappa}{u}{c}} & = \signature{\sigilof{u}}{\Uof{\kappa}}{\Uof{c}}
\end{align*}
\item Each module expression, $M$, maps onto an unexpanded module expression, $\uM$, as follows:
\begin{align*}
\Uof{X} & = \sigilof{X}\\
\Uof{\astruct{\uc}{\ue}} & = \struct{\Uof{\uc}}{\Uof{\ue}}\\
\Uof{\aseal{\sigma}{M}} & = \seal{\Uof{M}}{\Uof{\sigma}}\\
\Uof{\amlet{\sigma}{M}{X}{M'}} & = \mlet{\sigilof{X}}{\Uof{M}}{\Uof{M'}}{\Uof{\sigma}}
\end{align*}
\item Each construction variable, $u$, maps onto a unique {type identifier}, written $\sigilof{u}$.
\item Each kind, $\kappa$, maps onto an unexpanded kind, $\Uof{\kappa}$, as follows:
\begin{align*}
\Uof{\akdarr{\kappa}{u}{\kappa'}} & = \kdarr{\sigilof{u}}{\Uof{\kappa}}{\Uof{\kappa'}}\\
\Uof{\akunit} & = \kunit\\
\Uof{\akdbprod{\kappa}{u}{\kappa'}} & = \kdbprod{\sigilof{u}}{\Uof{\kappa}}{\Uof{\kappa'}}\\
\Uof{\akty} & = \kty\\
\Uof{\aksing{\tau}} & = \ksing{\Uof{\tau}}
\end{align*}
\item Each construction, $c$, except for constructions of the form $\amcon{M}$ where $M$ is not a module variable, maps onto an unexpanded type, $\Uof{c}$, as follows: 
  \begin{align*}
  \Uof{u} &= \sigilof{u}\\
  \Uof{\acabs{u}{c}} & = \cabs{\sigilof{u}}{\Uof{c}}\\
  \Uof{\acapp{c}{c'}} & = \capp{\Uof{c}}{\Uof{c'}}\\
  \Uof{\actriv} & = \ctriv\\
  \Uof{\acpair{c}{c'}} & = \cpair{\Uof{c}}{\Uof{c'}}\\
  \Uof{\acprl{c}} & = \cprl{\Uof{c}}\\
  \Uof{\acprr{c}} & = \cprr{\Uof{c}}\\
  \Uof{\aparr{\tau_1}{\tau_2}} & = \parr{\Uof{\tau_1}}{\Uof{\tau_2}}\\
  \Uof{\aallu{\kappa}{u}{\tau}} & = \forallu{\sigilof{u}}{\Uof{\kappa}}{\Uof{\tau}}\\
  \Uof{\arec{t}{\tau}} & = \rect{\sigilof{t}}{\Uof{\tau}}\\
  \Uof{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}} & = \prodt{\mapschemax{\Uofv}{\tau}{i}{\labelset}}\\
  \Uof{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}} & = \sumt{\mapschemax{\Uofv}{\tau}{i}{\labelset}}\\
  \Uof{\amcon{X}} & = \mcon{\sigilof{X}}
  \end{align*}
\item Each expression variable, $x$, maps onto a unique expression identifier, written $\sigilof{x}$.
\item Each expanded expression, $e$, except expressions of the form $\amval{M}$ where $M$ is not a module variable, maps onto an unexpanded expression, $\Uof{e}$, as follows:
\begin{align*}
\Uof{x} & = \sigilof{x}\\
\Uof{\aelam{\tau}{x}{e}} & = \lam{\sigilof{x}}{\Uof{\tau}}{\Uof{e}}\\
\Uof{\aeap{e_1}{e_2}} & = \ap{\Uof{e_1}}{\Uof{e_2}}\\
\Uof{\aeclam{\kappa}{u}{e}} & = \clam{\sigilof{u}}{\Uof{\kappa}}{\Uof{e}}\\
\Uof{\aecap{e}{c}} & = \cAp{\Uof{e}}{\Uof{c}}\\
\Uof{\aefold{e}} & = \fold{\Uof e}\\
\Uof{\aeunfold{e}} & = \unfold{\Uof{e}}\\
\Uof{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}} & = \tpl{\mapschemax{\Uofv}{e}{i}{\labelset}}\\
\Uof{\aepr{\ell}{e}} & = \prj{\Uof{e}}{\ell}\\
\Uof{\aein{\ell}{e}} &= \inj{\ell}{\Uof{e}}\\
\Uof{\aematchwith{n}{e}{\seqschemaX{r}}} & = \matchwith{\Uof{e}}{\seqschemaXx{\Uofv}{r}}\\
\Uof{\amval{X}} & = \mval{\sigilof{X}}
\end{align*}
\end{itemize}
\begin{itemize}
\item Each expanded rule, $r$, maps onto an unexpanded rule, $\Uof{r}$, as follows:
\begin{align*}
\Uof{\aematchrule{p}{e}} & = \aumatchrule{\Uof{p}}{\Uof{e}}
\end{align*}
\item Each expanded pattern, $p$, maps onto an unexpanded pattern, $\Uof{p}$, as follows:
\begin{align*}
\Uof{x} & = \sigilof{x}\\
\Uof{\aewildp} &= \auwildp\\
\Uof{\aefoldp{p}} &= \aufoldp{\Uof{p}}\\
\Uof{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}} & = \autplp{\labelset}{\mapschemax{\Uofv}{p}{i}{\labelset}}\\
\Uof{\aeinjp{\ell}{p}} & = \auinjp{\ell}{\Uof{p}}
\end{align*}
\end{itemize}

\subsubsection{Textual Syntax}
There is also a context-free textual syntax for the UL. We need only posit the existence of partial metafunctions that satisfy the following condition. 
\begin{condition}[Textual Representability]\label{condition:textual-representability-P} ~
\begin{enumerate}
% \item For each $\usigma$, there exists $b$ such that $\parseUSig{b}{\usigma}$.
% \item For each $\uM$, there exists $b$ such that $\parseUMod{b}{\uM}$.
\item For each $\ukappa$, there exists $b$ such that $\parseUKind{b}{\ukappa}$.
\item For each $\uc$, there exists $b$ such that $\parseUCon{b}{\uc}$.
\item For each $\ue$, there exists $b$ such that $\parseUExp{b}{\ue}$.
\item For each $\upv$, there exists $b$ such that $\parseUPat{b}{\upv}$.
\end{enumerate}
\end{condition}

\begin{condition}[Expression Parsing Monotonicity]\label{condition:body-parsing-P} If $\parseUExp{b}{\ue}$ then $\sizeof{\ue} < \sizeof{b}$.\end{condition}

\begin{condition}[Pattern Parsing Monotonicity]\label{condition:pattern-parsing-P} If $\parseUPat{b}{\upv}$ then $\sizeof{\upv} < \sizeof{b}$.\end{condition}

\subsection{Typed Expansion}\label{appendix:typed-expansion-P}
\subsubsection{Unexpanded Unified Contexts}\label{appendix:u-unified-ctxs}
A \emph{unexpanded unified context}, $\uOmega$, takes the form $\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}$, where $\uMctx$ is a \emph{module identifier expansion context}, $\uD$ is a \emph{construction identifier expansion context}, $\uG$ is an \emph{expression identifier expansion context}, and $\Omega$ is a unified context.

% \subsubsection{Identifier Expansion Contexts}
A module identifier expansion context, $\uMctx$, is a finite function that maps each module identifier $\uX \in \domof{\uMctx}$ to the module identifier expansion $\vExpands{\uX}{X}$. We write $\uOmega, \uMhyp{\uX}{X}{\sigma}$ when $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}$ as an abbreviation of \[\uOmegaEx{\uD}{\uG}{\uMctx \uplus \vExpands{\uX}{X}}{\Omega, X : \sigma}\]

A construction identifier expansion context, $\uD$, is a finite function that maps each construction identifier $\uu \in \domof{\uD}$ to the construction identifier expansion $\vExpands{\uu}{u}$. We write $\uOmega, \uKhyp{\uu}{u}{\kappa}$ when $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}$ as an abbreviation of \[\uOmegaEx{\uD \uplus \vExpands{\uu}{u}}{\uG}{\uMctx}{\Omega, u :: \kappa}\]

An expression identifier expansion context, $\uG$, is a finite function that maps each expression identifier $\ux \in \domof{\uG}$ to the expression identifier expansion $\vExpands{\ux}{x}$. We write $\uOmega, \uGhyp{\ux}{x}{\tau}$ when $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}$ as an abbreviation of \[\uOmegaEx{\uD}{\uG \uplus \vExpands{\ux}{x}}{\uMctx}{\Omega, x : \tau}\]

\subsubsection{Body Encoding and Decoding}
An assumed type abbreviated $\tBody$ classifies encodings of literal bodies, $b$. The mapping from literal bodies to values of type $\tBody$ is defined by the \emph{body encoding judgement} $\encodeBody{b}{\ebody}$. An inverse mapping is defined   by the \emph{body decoding judgement} $\decodeBody{\ebody}{b}$.
\[\begin{array}{ll}
\textbf{Judgement Form} & \textbf{Description}\\
\encodeBody{b}{e} & \text{$b$ has encoding $e$}\\
\decodeBody{e}{b} & \text{$e$ has decoding $b$}
\end{array}\]
The following condition establishes an isomorphism between literal bodies and values of type $\tBody$ mediated by the judgements above.
\begin{condition}[Body Isomorphism]\label{condition:body-isomorphism-P} ~
\begin{enumerate}
\item For every literal body $b$, we have that $\encodeBody{b}{\ebody}$ for some $\ebody$ such that $\hastypeUC{\ebody}{\tBody}$ and $\isvalU{\ebody}$.
\item If $\hastypeUC{\ebody}{\tBody}$ and $\isvalU{\ebody}$ then $\decodeBody{\ebody}{b}$ for some $b$.
\item If $\encodeBody{b}{\ebody}$ then $\decodeBody{\ebody}{b}$.
\item If $\hastypeUC{\ebody}{\tBody}$ and $\isvalU{\ebody}$ and $\decodeBody{\ebody}{b}$ then $\encodeBody{b}{\ebody}$. 
\item If $\encodeBody{b}{\ebody}$ and $\encodeBody{b}{\ebody'}$ then $\ebody = \ebody'$.
\item If $\hastypeUC{\ebody}{\tBody}$ and $\isvalU{\ebody}$ and $\decodeBody{\ebody}{b}$ and $\decodeBody{\ebody}{b'}$ then $b=b'$.
\end{enumerate}
\end{condition}
We also assume a partial metafunction, $\bsubseq{b}{m}{n}$, which extracts a subsequence of $b$ starting at position $m$ and ending at position $n$, inclusive, where $m$ and $n$ are natural numbers. The following condition is technically necessary.
\begin{condition}[Body Subsequencing]\label{condition:body-subsequences-P} If $\bsubseq{b}{m}{n}=b'$ then $\sizeof{b'} \leq \sizeof{b}$. \end{condition}

\subsubsection{Parse Results}
 The type function abbreviated $\tParseResultF$, and auxiliary abbreviations used below, is defined as follows:
\begin{align*}
L_\mathtt{P} & \defeq \lbltxt{ParseError}, \lbltxt{Success}\\
\tParseResultF & \defeq \acabs{t}{\asum{L_\mathtt{P}}{
  \mapitem{\lbltxt{ParseError}}{\prodt{}}, 
  \mapitem{\lbltxt{Success}}{t}
}}\\
\tParseResult{\tau} & \defeq \acapp{\tParseResultF}{\tau}\\
% \lbltxt{SuccessE}\cdot e & \defeq \aein{L_\mathtt{P}}{\mathtt{Success}}{\mapitem{\mathtt{ParseError}}{\tpl{}}, \mapitem{\mathtt{Success}}{\tPProtoExpr}}{e}\\
% \lbltxt{SuccessP}\cdot e & \defeq \aein{L_\mathtt{P}}{\mathtt{Success}}{\mapitem{\mathtt{ParseError}}{\tpl{}}, \mapitem{\mathtt{Success}}{\tCEPat}}{e}
\end{align*} %[\mapitem{\lbltxt{ParseError}}{\prodt{}}, \mapitem{\lbltxt{SuccessE}}{\tCEExp}]

\subsubsection{TLM Contexts}
\emph{peTLM contexts}, $\uPsi$, are of the form $\uAS{\uA}{\Psi}$, where $\uA$ is a \emph{TLM identifier expansion context} and $\Psi$ is a \emph{peTLM definition context}.

\emph{ppTLM contexts}, $\uPhi$, are of the form $\uAS{\uA}{\Phi}$, where $\uA$ is a TLM identifier expansion context and $\Phi$ is a \emph{ppTLM definition context}.

A \emph{TLM identifier expansion context}, $\uA$, is a finite function mapping each TLM identifier $\tsmv \in \domof{\uA}$ to the \emph{TLM identifier expansion}, $\vExpands{\tsmv}{\epsilon}$, for some \emph{TLM expression}, $\epsilon$. We write $\ctxUpdate{\uA}{\tsmv}{\epsilon}$ for the TLM identifier expansion context that maps $\tsmv$ to $\vExpands{\tsmv}{\epsilon}$, and defers to $\uA$ for all other TLM identifiers (i.e. the previous mapping is \emph{updated}.)

A \emph{peTLM definition context}, $\Psi$, is a finite function mapping each TLM name $a \in \domof{\Psi}$ to an \emph{expanded peTLM definition}, $\petsmdefn{a}{\rho}{\eparse}$, where $\rho$ is the peTLM's type annotation, and $\eparse$ is its parse function. We write $\Psi, \petsmdefn{a}{\rho}{\eparse}$ when $a \notin \domof{\Psi}$ for the extension of $\Psi$ that maps $a$ to $\petsmdefn{a}{\rho}{\eparse}$. We write $\petsmenv{\Omega}{\Psi}$  when all the TLM type annotations in $\Psi$ are well-formed assuming $\Omega$, and the parse functions in $\Psi$ are closed and of the appropriate type.


\begin{definition}[peTLM Definition Context Formation]\label{def:peTLM-def-ctx-formation} $\petsmenv{\Omega}{\Psi}$ iff for each ${\petsmdefn{a}{\rho}{\eparse}} \in \Psi$, we have $\istsmty{\Omega}{\rho}$ and \[\hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPCEExp}}\]\end{definition}

\begin{definition}[peTLM Context Formation] $\petsmctx{\Omega}{\uAS{\uA}{\Psi}}$ iff $\petsmenv{\Omega}{\Psi}$ and for each $\vExpands{\tsmv}{\epsilon} \in \uA$ we have $\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}$ for some $\rho$.
\end{definition}

A \emph{ppTLM definition context}, $\Phi$, is a finite function mapping each TLM name $a \in \domof{\Phi}$ to an \emph{expanded ppTLM definition}, $\pptsmdefn{a}{\rho}{\eparse}$, where $\rho$ is the ppTLM's type annotation, and $\eparse$ is its parse function. We write $\Phi, \pptsmdefn{a}{\rho}{\eparse}$ when $a \notin \domof{\Phi}$ for the extension of $\Phi$ that maps $a$ to $\pptsmdefn{a}{\rho}{\eparse}$. We write $\pptsmenv{\Omega}{\Phi}$  when all the type annotations in $\Phi$ are well-formed assuming $\Omega$, and the parse functions in $\Phi$ are closed and of the appropriate type.

\begin{definition}[ppTLM Definition Context Formation]\label{def:ppTLM-def-ctx-formation} $\pptsmenv{\Omega}{\Phi}$ iff for each $\pptsmdefn{\tsmv}{\rho}{\eparse} \in \Phi$, we have $\istsmty{\Omega}{\rho}$ and \[\hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultCEPat}}\]\end{definition}

\begin{definition}[ppTLM Context Formation] $\pptsmctx{\Omega}{\uAS{\uA}{\Phi}}$ iff $\pptsmenv{\Omega}{\Phi}$ and for each $\vExpands{\tsmv}{\epsilon} \in \uA$ we have $\hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho}$ for some $\rho$.
\end{definition}

\subsubsection{Signature and Module Expansion}
\noindent\fbox{$\strut\sigExpandsPX{\usigma}{\sigma}$}~~$\usigma$ has well-formed expansion $\sigma$
\begin{equation}\label{rule:sigExpandsP}
\inferrule{
  \kExpandsX{\ukappa}{\kappa}\\
  \cExpands{\uOmega, \uKhyp{\uu}{u}{\kappa}}{\utau}{\tau}{\akty}
}{
  \sigExpandsPX{\signature{\uu}{\ukappa}{\utau}}{\asignature{\kappa}{u}{\tau}}
}
\end{equation}

\noindent\fbox{$\strut\mExpandsPX{\uM}{M}{\sigma}$}~~$\uM$ has expansion $M$ matching $\sigma$
\begin{subequations}\label{rules:mExpandsP}
\begin{equation}\label{rule:mExpandsP-subsumes}
\inferrule{
  \mExpandsPX{\uM}{M}{\sigma}\\
  \sigsub{\uOmega}{\sigma}{\sigma'}
}{
  \mExpandsPX{\uM}{M}{\sigma'}
}
\end{equation}
\begin{equation}\label{rule:mExpandsP-var}
\inferrule{ }{
  \mExpandsP{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\uPsi}{\uPhi}{\uX}{X}{\sigma}
}
\end{equation}
\begin{equation}\label{rule:mExpandsP-struct}
\inferrule{
  \cExpandsX{\uc}{c}{\kappa}\\
  \expandsPX{\ue}{e}{[c/u]\tau}
}{
  \mExpandsPX{\struct{\uc}{\ue}}{\astruct{c}{e}}{\asignature{\kappa}{u}{\tau}}
}
\end{equation}
\begin{equation}\label{rule:mExpandsP-seal}
\inferrule{
  \sigExpandsPX{\usigma}{\sigma}\\
  \mExpandsPX{\uM}{M}{\sigma}
}{
  \mExpandsPX{\seal{\uM}{\usigma}}{\aseal{\sigma}{M}}{\sigma} 
}
\end{equation}
\begin{equation}\label{rule:mExpandsP-mlet}
\inferrule{
  \mExpandsPX{\uM}{M}{\sigma}\\
  \sigExpandsPX{\usigma'}{\sigma'}\\\\
  \mExpandsP{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\uPsi}{\uPhi}{\uM'}{M'}{\sigma'}
}{
  \mExpandsPX{\mlet{\uX}{\uM}{\uM'}{\usigma'}}{\amlet{\sigma'}{M}{X}{M'}}{\sigma'}
}
\end{equation}
\begin{equation}\label{rule:mExpandsP-syntaxpe}
\inferrule{
  \tsmtyExpands{\uOmega}{\urho}{\rho}\\
  \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPCEExp}}\\\\
  \evalU{\eparse}{\eparse'}\\
  \mExpandsP{\uOmega}{\uAS{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Psi, \petsmdefn{a}{\rho}{\eparse'}}}{\uPhi}{\uM}{M}{\sigma}
}{
  \mExpandsP{\uOmega}{\uAS{\uA}{\Psi}}{\uPhi}{\defpetsm{\tsmv}{\urho}{\eparse}{\uM}}{M}{\sigma}
}
\end{equation}
\begin{equation}\label{rule:mExpandsP-letpetsm}
\inferrule{
  % \uOmega = \uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}\\
  \tsmexpExpandsExp{\uOmega}{\uAS{\uA}{\Psi}}{\uepsilon}{\epsilon}{\rho}\\
  % \tsmexpEvalsExp{\Omega}{\Psi}{\epsilon}{\epsilon_\text{normal}}\\\\
  \mExpandsP{\uOmega}{\uAS{\uA\uplus\mapitem{\tsmv}{\epsilon_\text{normal}}}{\Psi}}{\uPhi}{\uM}{M}{\sigma}
}{
  \mExpandsP{\uOmega}{\uAS{\uA}{\Psi}}{\uPhi}{\uletpetsm{\tsmv}{\uepsilon}{\uM}}{M}{\sigma}
}
\end{equation}
\begin{equation}\label{rule:mExpandsP-syntaxpp}
\inferrule{ 
  \tsmtyExpands{\uOmega}{\urho}{\rho}\\
  \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultCEPat }}\\\\
  \evalU{\eparse}{\eparse'}\\
  \mExpandsP{\uOmega}{\uPsi}{\uAS{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Phi, \pptsmdefn{a}{\rho}{\eparse'}}}{\uM}{M}{\sigma}
}{
  \mExpandsP{\uOmega}{\uPsi}{\uAS{\uA}{\Phi}}{\defpptsm{\tsmv}{\urho}{\eparse}{\uM}}{M}{\sigma}
}
\end{equation}
\begin{equation}\label{rule:mExpandsP-letpptsm}
\inferrule{
  \tsmexpExpandsPat{\uOmega}{\uAS{\uA}{\Phi}}{\uepsilon}{\epsilon}{\rho}\\
  \mExpandsP{\uOmega}{\uPsi}{\uAS{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Phi}}{\uM}{M}{\sigma}
}{
  \mExpandsP{\uOmega}{\uPsi}{\uAS{\uA}{\Phi}}{\uletpptsm{\tsmv}{\uepsilon}{\uM}}{M}{\sigma}
}
\end{equation}
\end{subequations}

\subsubsection{Kind and Construction Expansion}
\noindent\fbox{$\strut\kExpandsX{\ukappa}{\kappa}$}~~$\ukappa$ has well-formed expansion $\kappa$
\begin{subequations}\label{rules:kExpands}
\begin{equation}\label{rule:kExpands-darr}
\inferrule{
  \kExpandsX{\ukappa_1}{\kappa_1}\\
  \kExpands{\uOmega, \uKhyp{\uu}{u}{\kappa_1}}{\ukappa_2}{\kappa_2}
}{
  \kExpandsX{\kdarr{\uu}{\ukappa_1}{\ukappa_2}}{\akdarr{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:kExpands-unit}
\inferrule{ }{
  \kExpandsX{\kunit}{\akunit}
}
\end{equation}
\begin{equation}\label{rule:kExpands-dprod}
\inferrule{
  \kExpandsX{\ukappa_1}{\kappa_1}\\
  \kExpands{\uOmega, \uKhyp{\uu}{u}{\kappa_1}}{\ukappa_2}{\kappa_2}
}{
  \kExpandsX{\kdbprod{\uu}{\ukappa_1}{\ukappa_2}}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:kExpands-ty}
\inferrule{ }{
  \kExpandsX{\kty}{\akty}
}
\end{equation}
\begin{equation}\label{rule:kExpands-sing}
\inferrule{
  \cExpandsX{\utau}{\tau}{\akty}
}{
  \kExpandsX{\ksing{\utau}}{\aksing{\tau}}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\cExpandsX{\uc}{c}{\kappa}$}~~$\uc$ has expansion $c$ of kind $\kappa$
\begin{subequations}\label{rules:cExpands}
\begin{equation}\label{rule:cExpands-subsume}
\inferrule{
  \cExpandsX{\uc}{c}{\kappa_1}\\
  \ksubX{\kappa_1}{\kappa_2}
}{
  \cExpandsX{\uc}{c}{\kappa_2}
}
\end{equation}
\begin{equation}\label{rule:cExpands-var}
\inferrule{ }{\cExpands{\uOmega, \uKhyp{\uu}{u}{\kappa}}{\uu}{u}{\kappa}}
\end{equation}
\begin{equation}\label{rule:cExpands-abs}
\inferrule{
  \cExpands{\uOmega, \uKhyp{\uu}{u}{\kappa_1}}{\uc_2}{c_2}{\kappa_2}
}{
  \cExpandsX{\cabs{\uu}{\uc_2}}{\acabs{u}{c_2}}{\akdarr{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:cExpands-app}
\inferrule{
  \cExpandsX{\uc_1}{c_1}{\akdarr{\kappa_2}{u}{\kappa}}\\
  \cExpandsX{\uc_2}{c_2}{\kappa_2}
}{
  \cExpandsX{\capp{\uc_1}{\uc_2}}{\acapp{c_1}{c_2}}{[c_1/u]\kappa}
}
\end{equation}
\begin{equation}\label{rule:cExpands-unit}
\inferrule{ }{
  \cExpandsX{\ctriv}{\actriv}{\akunit}
}
\end{equation}
\begin{equation}\label{rule:cExpands-pair}
\inferrule{
  \cExpandsX{\uc_1}{c_1}{\kappa_1}\\
  \cExpandsX{\uc_2}{c_2}{[c_1/u]\kappa_2}
}{
  \cExpandsX{\cpair{\uc_1}{\uc_2}}{\acpair{c_1}{c_2}}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:cExpands-prl}
\inferrule{
  \cExpandsX{\uc}{c}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}{
  \cExpandsX{\cprl{\uc}}{\acprl{c}}{\kappa_1}
}
\end{equation}
\begin{equation}\label{rule:cExpands-prr}
\inferrule{
  \cExpandsX{\uc}{c}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}{
  \cExpandsX{\cprr{\uc}}{\acprr{c}}{[\acprl{c}/u]\kappa_2}
}
\end{equation}
\begin{equation}\label{rule:cExpands-parr}
\inferrule{
  \cExpandsX{\utau_1}{\tau_1}{\akty}\\
  \cExpandsX{\utau_2}{\tau_2}{\akty}
}{
  \cExpandsX{\parr{\utau_1}{\utau_2}}{\aparr{\tau_1}{\tau_2}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cExpands-all}
\inferrule{
  \kExpandsX{\ukappa}{\kappa}\\
  \cExpands{\uOmega, \uKhyp{\uu}{u}{\kappa}}{\utau}{\tau}{\akty}
}{
  \cExpandsX{\forallu{\uu}{\ukappa}{\utau}}{\aallu{\kappa}{u}{\tau}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cExpands-rec}
\inferrule{
  \cExpands{\uOmega, \uKhyp{\ut}{t}{\akty}}{\utau}{\tau}{\akty}
}{
  \cExpandsX{\rect{\ut}{\utau}}{\arec{t}{\tau}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cExpands-prod}
\inferrule{
  \{\cExpandsX{\utau_i}{\tau_i}{\akty}\}_{1 \leq i \leq n}
}{
  \cExpandsX{\prodt{\mapschema{\utau}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cExpands-sum}
\inferrule{
  \{\cExpandsX{\utau_i}{\tau_i}{\akty}\}_{1 \leq i \leq n}
}{
  \cExpandsX{\sumt{\mapschema{\utau}{i}{\labelset}}}{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cExpands-sing}
\inferrule{
  \cExpandsX{\uc}{c}{\akty}
}{
  \cExpandsX{\uc}{c}{\aksing{c}}
}
\end{equation}
\begin{equation}\label{rule:cExpands-stat}
\inferrule{ }{
  \cExpands{\uOmega, \uMhyp{\uX}{X}{\asignature{\kappa}{u}{\tau}}}{\mcon{\uX}}{\amcon{X}}{\kappa}
}
\end{equation}
\end{subequations}

\subsubsection{Type, Expression, Rule and Pattern Expansion}
% \noindent\fbox{$\strut\tExpandsPX{\utau}{\tau}$}~~$\utau$ has well-formed expansion $\tau$
% \begin{equation}\label{rule:tExpandsP}
% \inferrule{
%   \cExpandsX{\utau}{\tau}{\akty}
% }{
%   \tExpandsPX{\utau}{\tau}
% }
% \end{equation}

\noindent\fbox{$\strut\expandsPX{\ue}{e}{\tau}$}~~$\ue$ has expansion $e$ of type $\tau$
\begin{subequations}\label{rules:expandsP}
\begin{equation}\label{rule:expandsP-subsume}
  \inferrule{
    \expandsPX{\ue}{e}{\tau}\\
    \issubtypePX{\tau}{\tau'}
  }{
    \expandsPX{\ue}{e}{\tau'}
  }
\end{equation}

\begin{equation}\label{rule:expandsP-var}
  \inferrule{ }{ 
    \expandsP{\uOmega, \uGhyp{\ux}{x}{\tau}}{\uPsi}{\uPhi}{\ux}{x}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:expandsP-asc}
  \inferrule{
    \cExpands{\uOmega}{\utau}{\tau}{\akty}\\
    \expandsP{\uOmega}{\uPsi}{\uPhi}{\ue}{e}{\tau}
  }{
    \expandsP{\uOmega}{\uPsi}{\uPhi}{\asc{\ue}{\utau}}{e}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:expandsP-letsyn}
  \inferrule{
    \expandsP{\uOmega}{\uPsi}{\uPhi}{\ue_1}{e_1}{\tau_1}\\
    \expandsP{\uOmega, \uGhyp{\ux}{x}{\tau_1}}{\uPsi}{\uPhi}{\ue_2}{e_2}{\tau_2}
  }{
    \expandsP{\uOmega}{\uPsi}{\uPhi}{\letsyn{\ux}{\ue_1}{\ue_2}}{
      \aeap{\aelam{\tau_1}{x}{e_2}}{e_1}
    }{\tau_2}
  }
\end{equation}
%Functions with an argument type annotation can appear in synthetic position.
\begin{equation}\label{rule:expandsP-lam}
  \inferrule{
    \cExpands{\uOmega}{\utau_1}{\tau_1}{\akty}\\
    \expandsP{\uOmega, \uGhyp{\ux}{x}{\tau_1}}{\uPsi}{\uPhi}{\ue}{e}{\tau_2}
  }{
    \expandsPX{\lam{\ux}{\utau_1}{\ue}}{\aelam{\tau_1}{x}{e}}{\aparr{\tau_1}{\tau_2}}
  }
\end{equation}

%Function applications can appear in synthetic position. The argument is analyzed against the argument type synthesized by the function.
\begin{equation}\label{rule:expandsP-ap}
  \inferrule{
    \expandsPX{\ue_1}{e_1}{\aparr{\tau_2}{\tau}}\\
    \expandsPX{\ue_2}{e_2}{\tau_2}
  }{
    \expandsPX{\ap{\ue_1}{\ue_2}}{\aeap{e_1}{e_2}}{\tau}
  }
\end{equation}

%Type lambdas and type applications can appear in synthetic position.
\begin{equation}\label{rule:expandsP-tlam}
  \inferrule{
    \kExpandsX{\ukappa}{\kappa}\\
    \expandsP{\uOmega, \uKhyp{\uu}{u}{\kappa}}{\uPsi}{\uPhi}{\ue}{e}{\tau}
  }{
    \expandsPX{\clam{\uu}{\ukappa}{\ue}}{\aeclam{\kappa}{u}{e}}{\aallu{\kappa}{u}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:expandsP-tap}
  \inferrule{
    \expandsPX{\ue}{e}{\aallu{\kappa}{u}{\tau}}\\
    \ksynX{\uc}{c}{\kappa}
  }{
    \expandsPX{\cAp{\ue}{\uc}}{\aecap{e}{c}}{[c/t]\tau}
  }
\end{equation}
% Values of recursive types can be introduced only in analytic position.
\begin{equation}\label{rule:expandsP-fold}
  \inferrule{
    \expandsPX{\ue}{e}{[\arec{t}{\tau}/t]\tau}
  }{
    \expandsPX{\fold{\ue}}{\aefold{e}}{\arec{t}{\tau}}
  }
\end{equation}

%Unfoldings can appear in synthetic position.
\begin{equation}\label{rule:expandsP-unfold}
  \inferrule{
    \expandsPX{\ue}{e}{\arec{t}{\tau}}
  }{
    \expandsPX{\unfold{\ue}}{\aeunfold{e}}{[\arec{t}{\tau}/t]\tau}
  }
\end{equation}

%Labeled tuples can appear in synthetic position. Each of the field values are then in synthetic position. 
\begin{equation}\label{rule:expandsP-tpl}
  \inferrule{
    \{\expandsPX{\ue_i}{e_i}{\tau_i}\}_{i \in \labelset}
  }{
    \expandsPX{\tpl{\mapschema{\ue}{i}{\labelset}}}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}

%Fields can be projected out of a labeled tuple in synthetic position.
\begin{equation}\label{rule:expandsP-pr}
  \inferrule{
    \expandsPX{\ue}{e}{\aprod{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
  }{
    \expandsPX{\prj{\ue}{\ell}}{\aepr{\ell}{e}}{\tau}
  }
\end{equation}

% Values of labeled sum type can appear only in analytic position.
\begin{equation}\label{rule:expandsP-in}
  \inferrule{
    \expandsPX{\ue'}{e'}{\tau'}
  }{
    \expandsPX{\inj{\ell}{\ue}}{\aein{\ell}{e'}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}}
    % \uOmega \vdash_{\uPsi; \uPhi} \left(\shortstack{$\ue \leadsto $\\$\Leftarrow$\vspace{-1.2em}}\right)
    %\expandsPX{\auanain{\ell}{\ue}}{\aein{\ell}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
  }
\end{equation}

%Match expressions can appear in synthetic position.
\begin{equation}\label{rule:expandsP-match}
  \inferrule{
    % \uOmega = \uOmegaEx{\uD}{\uG}{\uM}{\Omega}\\\\
    \expandsPX{\ue}{e}{\tau}\\
    % \haskind{\Omega}{\tau'}{\akty}\\
    \{\rsynPX{\urv_i}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}
  }{
    \expandsPX{\matchwith{\ue}{\seqschemaX{\urv}}}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}
  }
\end{equation}

\begin{equation}\label{rule:expandsP-mval}
  \inferrule{ }{
    \expandsP{\uOmega, \uMhyp{\uX}{X}{\asignature{\kappa}{u}{\tau}}}{\uPsi}{\uPhi}{\mval{\uX}}{\amval{X}}{[\amcon{X}/u]\tau}
  }
\end{equation}

% ueTLMs can be defined and applied in synthetic position.
% \begin{equation}\label{rule:expandsP-defpetsm}
% \inferrule{
%   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
%   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPCEExp}}\\\\
%   \expandsP{\uOmega}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\uI}}{\uPhi}{\ue}{e}{\tau}
% }{
%   \expandsP{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\usyntaxueP{\tsmv}{\urho}{\eparse}{\ue}}{e}{\tau}
% }
% \end{equation}

% \begin{equation}\label{rule:expandsP-letpetsm}
% \inferrule{
%   \tsmexpExpandsExp{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
%   \expandsP{\uOmega}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Psi}{\uI}}{\uPhi}{\ue}{e}{\tau}
% }{
%   \expandsP{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\uletpetsm{\tsmv}{\uepsilon}{\ue}}{e}{\tau}
% }
% \end{equation}

\begin{equation}\label{rule:expandsP-apuetsm}
\inferrule{
  \uOmega = \uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
  \uPsi=\uAS{\uA}{\Psi}\\\\
  \tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\aetype{\tau_\text{final}}}\\
  \tsmexpEvalsExp{\Omega_\text{app}}{\Psi}{\epsilon}{\epsilon_\text{normal}}\\\\
  \tsmdefof{\epsilon_\text{normal}}=a\\
  \Psi = \Psi', \petsmdefn{a}{\rho}{\eparse}\\\\
  \encodeBody{b}{\ebody}\\
  \evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessE}}{e_\text{pproto}}}\\
  \decodePCEExp{e_\text{pproto}}{\pce}\\\\
  \prepce{\Omega_\text{app}}{\Psi}{\pce}{\ce}{\epsilon_\text{normal}}{\aetype{\tau_\text{proto}}}{\omega}{\Omega_\text{params}}\\\\
     \segOKP{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\segof{\ce}}{b}\\
  \cvalidEP{\Omega_\text{params}}{\esceneP{\omega : \OParams}{\uOmega}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau_\text{proto}}
}{
  \expandsP{\uOmega}{\uPsi}{\uPhi}{\utsmap{\uepsilon}{b}}{[\omega]e}{[\omega]\tau_\text{proto}}
}
\end{equation}

% These rules are nearly identical to Rules (\ref{rule:expandsUP-syntax}) and (\ref{rule:expandsUP-tsmap}), differing only in that the typed expansion premises have been replaced by corresponding synthetic typed expansion premises. The premises of these rules can be understood as described in Sections \ref{sec:U-uetsm-definition} and \ref{sec:U-uetsm-application}. The body encoding judgement and candidate expansion expression decoding judgements were characterized in Sec. \ref{sec:typed-expansion-UP}. We discuss candidate expansion validation in Sec. \ref{sec:ce-validation-B} below.

% To support ueTLM implicits, ueTLM contexts, $\uPsi$, are redefined to take the form $\uASI{\uA}{\Psi}{\uI}$. TLM naming contexts, $\uA$, and ueTLM definition contexts, $\Psi$, were defined in Sec. \ref{sec:typed-expansion-UP}. We write $\uPsi, \uShyp{\tsmv}{a}{\tau}{\eparse}$ when $\uPsi=\uASI{\uA}{\Psi}{\uI}$ as shorthand for \[\uASI{\ctxUpdate{\uA}{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}\]

% \emph{TLM designation contexts}, $\uI$, are finite functions that map each type $\tau \in \domof{\uI}$ to the \emph{TLM designation} $\designate{\tau}{a}$, for some symbol $a$. We write $\uI \uplus \designate{\tau}{a}$ for the TLM designation context that maps $\tau$ to $\designate{\tau}{a}$ and defers to $\uI$ for all other types (i.e. the previous designation, if any, is updated). 

% The TLM designation context in the ueTLM context is updated by expressions of ueTLM designation form. Such expressions can appear in synthetic position, where they are governed by the following rule:% We write $\uIOK{\Delta}{\uI}$ when each type in $\uI$ is well-formed assuming $\Delta$.
%\begin{definition}[TLM Designation Context Well-Formedness] $\uIOK{\Delta}{{\uI}$ iff for each $\designate{\tau}{a}$ we have $\istypeU{\Delta}{\tau}$.\end{definition}

% \todo{peTLM implicit designation}
% \begin{equation}\label{rule:expandsP-implicite}
%   \inferrule{
%     \esyn{\uDelta}{\uGamma}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\uPhi}{\ue}{e}{\tau'}
%   }{
%     \esyn{\uDelta}{\uGamma}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}}{\uPhi}{\implicite{\tsmv}{\ue}}{e}{\tau'}
%   }
% \end{equation}

% % Like ueTLMs, upTLMs can be defined in synthetic position.
% \begin{equation}\label{rule:expandsP-syntaxup}
% \inferrule{
%   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
%   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultCEPat}}\\\\
%   \expandsP{\uOmega}{\uPsi}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\uI}}{\ue}{e}{\tau}
% }{
%   \expandsP{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\usyntaxup{\tsmv}{\urho}{\eparse}{\ue}}{e}{\tau}
% }
% \end{equation}


% \begin{equation}\label{rule:expandsP-letpptsm}
% \inferrule{
%   \tsmexpExpandsPat{\uOmega}{\uASI{\uA}{\Phi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
%   \expandsP{\uOmega}{\uPsi}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Phi}{\uI}}{\ue}{e}{\tau}
% }{
%   \expandsP{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\uletpptsm{\tsmv}{\uepsilon}{\ue}}{e}{\tau}
% }
% \end{equation}

% % This rule is nearly identical to Rule (\ref{rule:expandsUP-defuptsm}), differing only in that the typed expansion premise has been replaced by the corresponding synthetic typed expansion premise. The premises can be understood as described in Section \ref{sec:uptsm-definition}.

% % To support upTLM implicits, upTLM contexts, $\uPhi$, are redefined to take the form $\uASI{\uA}{\Phi}{\uI}$. upTLM definition contexts, $\Phi$, were defined in Sec. \ref{sec:uptsm-definition}. We write $\uPhi, \uPhyp{\tsmv}{a}{\tau}{\eparse}$ when $\uPhi=\uASI{\uA}{\Phi}{\uI}$ as shorthand for \[\uASI{\ctxUpdate{\uA}{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI}\]

% % The TLM designation context in the upTLM context is updated by expressions of upTLM designation form. Such expressions can appear in synthetic position, where they are governed by the following rule:% We write $\uIOK{\Delta}{\uI}$ when each type in $\uI$ is well-formed assuming $\Delta$.
% %\begin{definition}[TLM Designation Context Well-Formedness] $\uIOK{\Delta}{{\uI}$ iff for each $\designate{\tau}{a}$ we have $\istypeU{\Delta}{\tau}$.\end{definition}
% \todo{ppTLM implicit designation}
% \begin{equation}\label{rule:expandsP-implicitp}
%   \inferrule{
%     \esyn{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\ue}{e}{\tau'}
%   }{
%     \esyn{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}}{\implicitp{\tsmv}{\ue}}{e}{\tau'}
%   }
% \end{equation}
\end{subequations}

% \begin{subequations}\label{rules:expandsP}
% Type analysis subsumes type synthesis, in that when a type can be synthesized for an unexpanded expression, that unexpanded expression can also be analyzed against that type, producing the same expansion. This is expressed by the following \emph{subsumption rule} for unexpanded expressions.

% Additional rules are needed for certain forms in order to propagate types for analysis into subexpressions, and for forms that can appear only in analytic position.



% Rule (\ref{rule:esyn-tpl}) governed labeled tuples in synthetic position. The following rule governs labeled tuples in analytic position.


% Rule (\ref{rule:esyn-match}) governed match expressions in synthetic position. The following rule governs match expressions in analytic position.

% Rule (\ref{rule:esyn-defuetsm}) governed ueTLM definitions in synthetic position. The following rule governs ueTLM definitions in analytic position.
% \begin{equation}\label{rule:expandsP-defpetsm}
% \inferrule{
%   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
%   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPCEExp}}\\\\
%   \expandsP{\uOmega}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\uI}}{\uPhi}{\ue}{e}{\tau}
% }{
%   \expandsP{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\usyntaxueP{\tsmv}{\urho}{\eparse}{\ue}}{e}{\tau}
% }
% \end{equation}

% \begin{equation}\label{rule:expandsP-letpetsm}
% \inferrule{
%   \tsmexpExpandsExp{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
%   \expandsP{\uOmega}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Psi}{\uI}}{\uPhi}{\ue}{e}{\tau}
% }{
%   \expandsP{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\uletpetsm{\tsmv}{\uepsilon}{\ue}}{e}{\tau}
% }
% \end{equation}

% \todo{peTLM implicit designation}
% Rule (\ref{rule:esyn-implicite}) governed ueTLM designations in synthetic position. The following rule governs ueTLM designations in analytic position.
% \begin{equation}\label{rule:expandsP-implicite}
%   \inferrule{
%     \eana{\uDelta}{\uGamma}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\uPhi}{\ue}{e}{\tau'}
%   }{
%     \eana{\uDelta}{\uGamma}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}}{\uPhi}{\implicite{\tsmv}{\ue}}{e}{\tau'}
%   }
% \end{equation}

% \todo{peTLM implicit application}
% % An expression of unadorned literal form can appear only in analytic position. The following rule extracts the TLM designated at the type that the expression is being analyzed against from the TLM designation context in the ueTLM context and applies it implicitly, i.e. the premises correspond to those of Rule (\ref{rule:esyn-apuetsm}).


% Rule (\ref{rule:esyn-defuptsm}) governed upTLM definitions in synthetic position. The following rule governs upTLM definitions in analytic position.
% \begin{equation}\label{rule:expandsP-syntaxup}
% \inferrule{
%   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
%   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultCEPat}}\\\\
%   \expandsP{\uOmega}{\uPsi}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\uI}}{\ue}{e}{\tau}
% }{
%   \expandsP{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\usyntaxup{\tsmv}{\urho}{\eparse}{\ue}}{e}{\tau}
% }
% \end{equation}


% \begin{equation}\label{rule:expandsP-letpptsm}
% \inferrule{
%   \tsmexpExpandsPat{\uOmega}{\uASI{\uA}{\Phi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
%   \expandsP{\uOmega}{\uPsi}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Phi}{\uI}}{\ue}{e}{\tau}
% }{
%   \expandsP{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\uletpptsm{\tsmv}{\uepsilon}{\ue}}{e}{\tau}
% }
% \end{equation}


% \todo{ppTLM implicit designation}
% % Rule (\ref{rule:esyn-implicitp}) governed upTLM designations in synthetic position. The following rule governs upTLM designations in analytic position.
% \begin{equation}\label{rule:expandsP-implicitp}
%   \inferrule{
%     \eana{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\ue}{e}{\tau'}
%   }{
%     \eana{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}}{\implicitp{\tsmv}{\ue}}{e}{\tau'}
%   }
% \end{equation}

% \end{subequations}

\noindent\fbox{$\strut\rExpandsSP{\uOmega}{\uPsi}{\uPhi}{\urv}{r}{\tau}{\tau'}$}~~$\urv$ has expansion $r$ taking values of type $\tau$ to values of type $\tau'$
\begin{equation}\label{rule:rExpandsP}
  \inferrule{
    \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}\\
    \patExpandsP{\uOmegaEx{\emptyset}{\uG'}{\emptyset}{\Omega'}}{\uPhi}{\upv}{p}{\tau}\\
    \expandsP{\uOmegaEx{\uD}{\uG \uplus \uG'}{\uMctx}{\Omega \cup \Omega'}}{\uPsi}{\uPhi}{\ue}{e}{\tau'}
  }{
    \rExpandsSP{\uOmega}{\uPsi}{\uPhi}{\matchrule{\upv}{\ue}}{\aematchrule{p}{e}}{\tau}{\tau'}
  }
\end{equation}


\noindent\fbox{$\strut\patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{\tau}$}~~$\upv$ has expansion $p$ matching against $\tau$ generating hypotheses $\uOmega'$
% The typed pattern expansion judgement is inductively defined by Rules (\ref{rules:patExpandsP}) as follows. %As in $\miniVersePat$, \emph{unexpanded pattern typing contexts}, $\upctx$, are defined identically to unexpanded typing contexts (i.e. we only use a distinct metavariable to emphasize their distinct roles in the judgements above). 

% The following rules are written identically to the typed pattern expansion rules for shared pattern forms in $\miniVersePat$, i.e. Rules (\ref{rule:patExpands-var}) through (\ref{rule:patExpands-in}).
\begin{subequations}\label{rules:patExpandsP}
\begin{equation}\label{rule:patExpandsP-subsume}
\inferrule{
  \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}\\\\
  \patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{\tau}\\
  \issubtypeP{\Omega}{\tau}{\tau'}
}{
  \patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{\tau'}
}
\end{equation}
\begin{equation}\label{rule:patExpandsP-var}
\inferrule{ }{
  \patExpandsP{\uOmegaEx{\emptyset}{\vExpands{\ux}{x}}{\emptyset}{\Ghyp{x}{\tau}}}{\uPhi}{\ux}{x}{\tau}
}
\end{equation}
\begin{equation}\label{rule:patExpandsP-wild}
\inferrule{ }{
  \patExpandsP{\uOmegaEx{\emptyset}{\emptyset}{\emptyset}{\emptyset}}{\uPhi}{\wildp}{\aewildp}{\tau}
}
\end{equation}
\begin{equation}\label{rule:patExpandsP-fold}
\inferrule{ 
  \patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{[\arec{t}{\tau}/t]\tau}
}{
  \patExpandsP{\uOmega'}{\uPhi}{\foldp{\upv}}{\aefoldp{p}}{\arec{t}{\tau}}
}
\end{equation}
\begin{equation}\label{rule:patExpandsP-tpl}
\inferrule{
  \tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}\\\\
  \{\patExpandsP{{\uOmega_i}}{\uPhi}{\upv_i}{p_i}{\tau_i}\}_{i \in \labelset}
}{
  %\patExpandsP{\Gconsi{i \in \labelset}{\upctx_i}}{A}{B}{C}
  \patExpandsP{\Gconsi{i \in \labelset}{\uOmega_i}}{\uPhi}{\tplp{\mapschema{\upv}{i}{\labelset}}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\tau}
  % \patExpands{\Gconsi{i \in \labelset}{\pctx_i}}{\Phi}{
  %   \autplp{\labelset}{\mapschema{\upv}{i}{\labelset}}
  % }{
  %   \aetplp{\labelset}{\mapschema{p}{i}{\labelset}}
  % }{
  %   \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}
  % } %{\autplp{\labelset}{\mapschema{\upv}{i}{\labelset}}}{\aetplp{\labelset}{\mapschema}{p}{i}{\labelset}}{...}
  %\left(\shortstack{$\Delta \vdash_{\uPhi} \autplp{\labelset}{\mapschema{\upv}{i}{\labelset}}$\\$\leadsto$\\$\aetplp{\labelset}{\mapschema{p}{i}{\labelset}} : \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}} \dashV \Gconsi{i \in \labelset}{\upctx_i}$\vspace{-1.2em}}\right)
}
\end{equation}
\begin{equation}\label{rule:patExpandsP-in}
\inferrule{
  \patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{\tau}
}{
  \patExpandsP{\uOmega'}{\uPhi}{\injp{\ell}{\upv}}{\aeinjp{\ell}{p}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
}
\end{equation}

\begin{equation}\label{rule:patExpandsP-apuptsm}
\inferrule{
  \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
  \uPhi=\uAS{\uA}{\Phi}\\\\
  \tsmexpExpandsPat{\uOmega}{\uPhi}{\uepsilon}{\epsilon}{\aetype{\tau_\text{final}}}\\
  \tsmexpEvalsPat{\Omega_\text{app}}{\Phi}{\epsilon}{\epsilon_\text{normal}}\\\\
  \tsmdefof{\epsilon_\text{normal}}=a\\
  \Phi = \Phi', \pptsmdefn{a}{\rho}{\eparse}\\\\
  \encodeBody{b}{\ebody}\\
  \evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessP}}{e_\text{pproto}}}\\
  \decodePCEPat{e_\text{pproto}}{\pcp}\\\\
  \prepcp{\Omega_\text{app}}{\Phi}{\pcp}{\cpv}{\epsilon_\text{normal}}{\aetype{\tau_\text{proto}}}{\omega}{\Omega_\text{params}}\\\\
     \segOKP{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\segof{\cpv}}{b}\\
  \cvalidPP{\uOmega'}{\psceneP{\omega : \Omega_\text{params}}{\uOmega}{\uPhi}{b}}{\cpv}{p}{\tau_\text{proto}}
}{
  \patExpandsP{\uOmega'}{\uPhi}{\utsmap{\uepsilon}{b}}{p}{[\omega]\tau_\text{proto}}
}
\end{equation}
\end{subequations}

\subsubsection{TLM Type and Expression Expansion}
\noindent\fbox{$\strut\tsmtyExpands{\uOmega}{\urho}{\rho}$}~~$\urho$ has well-formed expansion $\rho$
\begin{subequations}\label{rules:tsmtyExpands}
\begin{equation}\label{rule:tsmtyExpands-type}
\inferrule{
  \cExpandsX{\utau}{\tau}{\akty}
}{
  \tsmtyExpands{\uOmega}{{\utau}}{\aetype{\tau}}
}
\end{equation}
% \begin{equation}\label{rule:tsmtyExpands-alltypes}
% \inferrule{
%   \tsmtyExpands{\uOmega, \uKhyp{\ut}{t}{\akty}}{\urho}{\rho}
% }{
%   \tsmtyExpands{\uOmega}{\alltypes{\ut}{\urho}}{\aealltypes{t}{\rho}}
% }
% \end{equation}
\begin{equation}\label{rule:tsmtyExpands-allmods}
\inferrule{
  \sigExpandsPX{\usigma}{\sigma}\\
  \tsmtyExpands{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\urho}{\rho}
}{
  \tsmtyExpands{\uOmega}{\allmods{\uX}{\usigma}{\urho}}{\aeallmods{\sigma}{X}{\rho}}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\rho}$}~~$\uepsilon$ has peTLM expression expansion $\epsilon$ at $\rho$
\begin{subequations}\label{rules:tsmexpExpandsExp}
\begin{equation}\label{rule:tsmexpExpandsExp-bindref}
\inferrule{
  \hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}  
}{
  \tsmexpExpandsExp{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\uAS{\uA, \mapitem{\tsmv}{\epsilon}}{\Psi}}{{\tsmv}}{\epsilon}{\rho}
}
\end{equation}
% \begin{equation}\label{rule:tsmexpExpandsExp-abstype}
% \inferrule{
%   \tsmexpExpandsExp{\uOmega, \uKhyp{\ut}{t}{\akty}}{\uPsi}{\uepsilon}{\epsilon}{\rho}
% }{
%   \tsmexpExpandsExp{\uOmega}{\uPsi}{\abstype{\ut}{\uepsilon}}{\aeabstype{t}{\epsilon}}{\aealltypes{t}{\rho}}
% }
% \end{equation}
\begin{equation}\label{rule:tsmexpExpandsExp-absmod}
\inferrule{
  \sigExpandsPX{\usigma}{\sigma}\\
  \tsmexpExpandsExp{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\uPsi}{\uepsilon}{\epsilon}{\rho}
}{
  \tsmexpExpandsExp{\uOmega}{\uPsi}{\absmod{\uX}{\usigma}{\uepsilon}}{\aeabsmod{\sigma}{X}{\epsilon}}{\aeallmods{\sigma}{X}{\rho}}
}
\end{equation}
% \begin{equation}\label{rule:tsmexpExpandsExp-aptype}
% \inferrule{
%   \tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\aealltypes{t}{\rho}}\\
%   \cExpandsX{\utau}{\tau}{\akty}
% }{
%   \tsmexpExpandsExp{\uOmega}{\uPsi}{\aptype{\uepsilon}{\utau}}{\aeaptype{\tau}{\epsilon}}{[\tau/t]\rho} 
% }
% \end{equation}
\begin{equation}\label{rule:tsmexpExpandsExp-apmod}
\inferrule{
  \tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\aeallmods{\sigma}{X'}{\rho}}\\
  \mExpandsPX{\uX}{X}{\sigma}
}{
  \tsmexpExpandsExp{\uOmega}{\uPsi}{\apmod{\uepsilon}{\uX}}{\aeapmod{X}{\epsilon}}{[X/X']\rho}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\tsmexpExpandsPat{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\rho}$}~~$\uepsilon$ has ppTLM expression expansion $\epsilon$ at $\rho$
\begin{subequations}\label{rules:tsmexpExpandsPat}
\begin{equation}\label{rule:tsmexpExpandsPat-bindref}
\inferrule{
  \hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho}  
}{
  \tsmexpExpandsPat{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\uAS{\uA, \mapitem{\tsmv}{\epsilon}}{\Phi}}{{\tsmv}}{\epsilon}{\rho}
}
\end{equation}
% \begin{equation}\label{rule:tsmexpExpandsPat-abstype}
% \inferrule{
%   \tsmexpExpandsPat{\uOmega, \uKhyp{\ut}{t}{\akty}}{\uPhi}{\uepsilon}{\epsilon}{\rho}
% }{
%   \tsmexpExpandsPat{\uOmega}{\uPhi}{\abstype{\ut}{\uepsilon}}{\aeabstype{t}{\epsilon}}{\aealltypes{t}{\rho}}
% }
% \end{equation}
\begin{equation}\label{rule:tsmexpExpandsPat-absmod}
\inferrule{
  \sigExpandsPX{\usigma}{\sigma}\\
  \tsmexpExpandsPat{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\uPhi}{\uepsilon}{\epsilon}{\rho}
}{
  \tsmexpExpandsPat{\uOmega}{\uPhi}{\absmod{\uX}{\usigma}{\uepsilon}}{\aeabsmod{\sigma}{X}{\epsilon}}{\aeallmods{\sigma}{X}{\rho}}
}
\end{equation}
% \begin{equation}\label{rule:tsmexpExpandsPat-aptype}
% \inferrule{
%   \tsmexpExpandsPat{\uOmega}{\uPhi}{\uepsilon}{\epsilon}{\aealltypes{t}{\rho}}\\
%   \cExpandsX{\utau}{\tau}{\akty}
% }{
%   \tsmexpExpandsPat{\uOmega}{\uPhi}{\aptype{\uepsilon}{\utau}}{\aeaptype{\tau}{\epsilon}}{[\tau/t]\rho} 
% }
% \end{equation}
\begin{equation}\label{rule:tsmexpExpandsPat-apmod}
\inferrule{
  \tsmexpExpandsPat{\uOmega}{\uPhi}{\uepsilon}{\epsilon}{\aeallmods{\sigma}{X'}{\rho}}\\
  \mExpandsPX{\uX}{X}{\sigma}
}{
  \tsmexpExpandsPat{\uOmega}{\uPhi}{\apmod{\uepsilon}{\uX}}{\aeapmod{X}{\epsilon}}{[X/X']\rho}
}
\end{equation}
\end{subequations}

\subsubsection{Statics of the TLM Language}
\noindent\fbox{$\strut\istsmty{\Omega}{\rho}$}~~$\rho$ is a TLM type
\begin{subequations}\label{rules:istsmty}
\begin{equation}\label{rule:istsmty-type}
\inferrule{
  \haskindX{\tau}{\akty}
}{
  \istsmty{\Omega}{\aetype{\tau}}
}
\end{equation}
% \begin{equation}\label{rule:istsmty-alltypes}
% \inferrule{
%   \istsmty{\Omega, t :: \akty}{\rho}
% }{
%   \istsmty{\Omega}{\aealltypes{t}{\rho}}
% }
% \end{equation}
\begin{equation}\label{rule:istsmty-allmods}
\inferrule{
  \issig{\Omega}{\sigma}\\
  \istsmty{\Omega, X : \sigma}{\rho}
}{
  \istsmty{\Omega}{\aeallmods{\sigma}{X}{\rho}}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}$}~~$\epsilon$ is a peTLM expression at $\rho$
\begin{subequations}\label{rules:hastsmtypeExp}
\begin{equation}\label{rule:hastsmtypeExp-defref}
\inferrule{
  \istsmty{\Omega}{\rho}
}{
  \hastsmtypeExp{\Omega}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\adefref{a}}{\rho}
}
\end{equation}
% \begin{equation}\label{rule:hastsmtypeExp-abstype}
% \inferrule{
%   \hastsmtypeExp{\Omega, t :: \akty}{\Psi}{\epsilon}{\rho}
% }{
%   \hastsmtypeExp{\Omega}{\Psi}{\aeabstype{t}{\epsilon}}{\aealltypes{t}{\rho}}
% }
% \end{equation}
\begin{equation}\label{rule:hastsmtypeExp-absmod}
\inferrule{
  \issigX{\sigma}\\
  \hastsmtypeExp{\Omega, X : \sigma}{\Psi}{\epsilon}{\rho}
}{
  \hastsmtypeExp{\Omega}{\Psi}{\aeabsmod{\sigma}{X}{\epsilon}}{\aeallmods{\sigma}{X}{\rho}}
}
\end{equation}
% \begin{equation}\label{rule:hastsmtypeExp-aptype}
% \inferrule{
%   \hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\aealltypes{t}{\rho}}\\
%   \haskindX{\tau}{\akty}
% }{
%   \hastsmtypeExp{\Omega}{\Psi}{\aeaptype{\tau}{\epsilon}}{[\tau/t]\rho}
% }
% \end{equation}
\begin{equation}\label{rule:hastsmtypeExp-apmod}
\inferrule{
  \hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\aeallmods{\sigma}{X'}{\rho}}\\
  \hassig{\Omega}{X}{\sigma}
}{
  \hastsmtypeExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon}}{[X/X']\rho}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho}$}~~$\epsilon$ is a ppTLM expression at $\rho$
\begin{subequations}\label{rules:hastsmtypePat}
\begin{equation}\label{rule:hastsmtypePat-defref}
\inferrule{
  \istsmty{\Omega}{\rho}
}{
  \hastsmtypePat{\Omega}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\adefref{a}}{\rho}
}
\end{equation}
% \begin{equation}\label{rule:hastsmtypePat-abstype}
% \inferrule{
%   \hastsmtypePat{\Omega, t :: \akty}{\Phi}{\epsilon}{\rho}
% }{
%   \hastsmtypePat{\Omega}{\Phi}{\aeabstype{t}{\epsilon}}{\aealltypes{t}{\rho}}
% }
% \end{equation}
\begin{equation}\label{rule:hastsmtypePat-absmod}
\inferrule{
  \issigX{\sigma}\\
  \hastsmtypePat{\Omega, X : \sigma}{\Phi}{\epsilon}{\rho}
}{
  \hastsmtypePat{\Omega}{\Phi}{\aeabsmod{\sigma}{X}{\epsilon}}{\aeallmods{\sigma}{X}{\rho}}
}
\end{equation}
% \begin{equation}\label{rule:hastsmtypePat-aptype}
% \inferrule{
%   \hastsmtypePat{\Omega}{\Phi}{\epsilon}{\aealltypes{t}{\rho}}\\
%   \haskindX{\tau}{\akty}
% }{
%   \hastsmtypePat{\Omega}{\Phi}{\aeaptype{\tau}{\epsilon}}{[\tau/t]\rho}
% }
% \end{equation}
\begin{equation}\label{rule:hastsmtypePat-apmod}
\inferrule{
  \hastsmtypePat{\Omega}{\Phi}{\epsilon}{\aeallmods{\sigma}{X'}{\rho}}\\
  \hassig{\Omega}{X}{\sigma}
}{
  \hastsmtypePat{\Omega}{\Phi}{\aeapmod{X}{\epsilon}}{[X/X']\rho}
}
\end{equation}

\end{subequations}

The following metafunction extracts the TLM name from a TLM expression.
\begin{subequations}
\begin{align}
\tsmdefof{\adefref{a}} & = a \label{eqn:tsmdefof-adefref}\\
% \tsmdefof{\aeabstype{t}{\epsilon}} & = \tsmdefof{\epsilon} \label{eqn:tsmdefof-abstype}\\
\tsmdefof{\aeabsmod{\sigma}{X}{\epsilon}} & = \tsmdefof{\epsilon} \label{eqn:tsmdefof-absmod}\\
% \tsmdefof{\aeaptype{\tau}{\epsilon}} & = \tsmdefof{\epsilon} \label{eqn:tsmdefof-aptype}\\
\tsmdefof{\aeapmod{X}{\epsilon}} & = \tsmdefof{\epsilon} \label{eqn:tsmdefof-apmod}
\end{align}
\end{subequations}

\subsubsection{Dynamics of the TLM Language}

\noindent\fbox{$\strut\tsmexpStepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}$}~~peTLM expression $\epsilon$ transitions to $\epsilon'$
\begin{subequations}\label{rules:tsmexpStepsExp}
% \begin{equation}\label{rule:tsmexpStepsExp-aptype-1}
% \inferrule{
%   \tsmexpStepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}
% }{
%   \tsmexpStepsExp{\Omega}{\Psi}{\aeaptype{\tau}{\epsilon}}{\aeaptype{\tau}{\epsilon'}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpStepsExp-aptype-2}
% \inferrule{ }{
%   \tsmexpStepsExp{\Omega}{\Psi}{\aeaptype{\tau}{\aeabstype{t}{\epsilon}}}{[\tau/t]\epsilon}
% }
% \end{equation}
\begin{equation}\label{rule:tsmexpStepsExp-apmod-1}
\inferrule{
  \tsmexpStepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}
}{
  \tsmexpStepsExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon}}{\aeapmod{X}{\epsilon'}}
}
\end{equation}
\begin{equation}\label{rule:tsmexpStepsExp-apmod-2}
\inferrule{ }{
  \tsmexpStepsExp{\Omega}{\Psi}{\aeapmod{X}{\aeabsmod{\sigma}{X'}{\epsilon}}}{[X/X']\epsilon}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\tsmexpStepsPat{\Omega}{\Psi}{\epsilon}{\epsilon'}$}~~ppTLM expression $\epsilon$ transitions to $\epsilon'$
\begin{subequations}\label{rules:tsmexpStepsPat}
% \begin{equation}\label{rule:tsmexpStepsPat-aptype-1}
% \inferrule{
%   \tsmexpStepsPat{\Omega}{\Psi}{\epsilon}{\epsilon'}
% }{
%   \tsmexpStepsPat{\Omega}{\Psi}{\aeaptype{\tau}{\epsilon}}{\aeaptype{\tau}{\epsilon'}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpStepsPat-aptype-2}
% \inferrule{ }{
%   \tsmexpStepsPat{\Omega}{\Psi}{\aeaptype{\tau}{\aeabstype{t}{\epsilon}}}{[\tau/t]\epsilon}
% }
% \end{equation}
\begin{equation}\label{rule:tsmexpStepsPat-apmod-1}
\inferrule{
  \tsmexpStepsPat{\Omega}{\Psi}{\epsilon}{\epsilon'}
}{
  \tsmexpStepsPat{\Omega}{\Psi}{\aeapmod{X}{\epsilon}}{\aeapmod{X}{\epsilon'}}
}
\end{equation}
\begin{equation}\label{rule:tsmexpStepsPat-apmod-2}
\inferrule{ }{
  \tsmexpStepsPat{\Omega}{\Psi}{\aeapmod{X}{\aeabsmod{\sigma}{X'}{\epsilon}}}{[X/X']\epsilon}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}$}~~peTLM expression $\epsilon$ transitions in multiple steps to $\epsilon'$
\begin{subequations}\label{rules:tsmexpMultistepsExp}
\begin{equation}\label{rule:tsmexpMultistepsExp-refl}
\inferrule{ }{
  \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon}
}
\end{equation}
\begin{equation}\label{rule:tsmexpMultistepsExp-steps}
\inferrule{
  \tsmexpStepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}
}{
  \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}
}
\end{equation}
\begin{equation}\label{rule:tsmexpMultistepsExp-trans}
\inferrule{
  \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}\\
  \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon'}{\epsilon''}
}{
  \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon''}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\tsmexpMultistepsPat{\Omega}{\Psi}{\epsilon}{\epsilon'}$}~~ppTLM expression $\epsilon$ transitions in multiple steps to $\epsilon'$
\begin{subequations}\label{rules:tsmexpMultistepsPat}
\begin{equation}\label{rule:tsmexpMultistepsPat-refl}
\inferrule{ }{
  \tsmexpMultistepsPat{\Omega}{\Psi}{\epsilon}{\epsilon}
}
\end{equation}
\begin{equation}\label{rule:tsmexpMultistepsPat-steps}
\inferrule{
  \tsmexpStepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}
}{
  \tsmexpMultistepsPat{\Omega}{\Psi}{\epsilon}{\epsilon'}
}
\end{equation}
\begin{equation}\label{rule:tsmexpMultistepsPat-trans}
\inferrule{
  \tsmexpMultistepsPat{\Omega}{\Psi}{\epsilon}{\epsilon'}\\
  \tsmexpMultistepsPat{\Omega}{\Psi}{\epsilon'}{\epsilon''}
}{
  \tsmexpMultistepsPat{\Omega}{\Psi}{\epsilon}{\epsilon''}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\tsmexpEvalsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}$}~~peTLM expression $\epsilon$ normalizes to $\epsilon'$
\begin{equation}\label{rule:tsmexpEvalsExp}
\inferrule{
  \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}\\
  \tsmexpNormalExp{\Omega}{\Psi}{\epsilon'}
}{
  \tsmexpEvalsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}
}
\end{equation}


\noindent\fbox{$\strut\tsmexpEvalsPat{\Omega}{\Psi}{\epsilon}{\epsilon'}$}~~ppTLM expression $\epsilon$ normalizes to $\epsilon'$
\begin{equation}\label{rule:tsmexpEvalsPat}
\inferrule{
  \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}\\
  \tsmexpNormalExp{\Omega}{\Psi}{\epsilon'}
}{
  \tsmexpEvalsPat{\Omega}{\Psi}{\epsilon}{\epsilon'}
}
\end{equation}


\noindent\fbox{$\tsmexpNormalExp{\Omega}{\Psi}{\epsilon}$}~~$\epsilon$ is a normal peTLM expression
\begin{subequations}\label{rules:tsmexpNormalExp}
\begin{equation}\label{rule:tsmexpNormalExp-defref}
\inferrule{ }{
  \tsmexpNormalExp{\Omega}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\adefref{a}}
}
\end{equation}
% \begin{equation}\label{rule:tsmexpNormalExp-abstype}
% \inferrule{ }{
%   \tsmexpNormalExp{\Omega}{\Psi}{\aeabstype{t}{\epsilon}}
% }
% \end{equation}
\begin{equation}\label{rule:tsmexpNormalExp-absmod}
\inferrule{ }{
  \tsmexpNormalExp{\Omega}{\Psi}{\aeabsmod{\sigma}{X}{\epsilon}}
}
\end{equation}
% \begin{equation}\label{rule:tsmexpNormalExp-aptype}
% \inferrule{
%   \epsilon \neq \aeabstype{t}{\epsilon'}\\
%   \tsmexpNormalExp{\Omega}{\Psi}{\epsilon}
% }{
%   \tsmexpNormalExp{\Omega}{\Psi}{\aeaptype{\tau}{\epsilon}}
% }
% \end{equation}
\begin{equation}\label{rule:tsmexpNormalExp-apmod}
\inferrule{
  \epsilon \neq \aeabsmod{\sigma}{X'}{\epsilon'}\\
  \tsmexpNormalExp{\Omega}{\Psi}{\epsilon}
}{
  \tsmexpNormalExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon}}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\tsmexpNormalPat{\Omega}{\Psi}{\epsilon}$}~~$\epsilon$ is a normal ppTLM expression
\begin{subequations}\label{rules:tsmexpNormalPat}
\begin{equation}\label{rule:tsmexpNormalPat-defref}
\inferrule{ }{
  \tsmexpNormalPat{\Omega}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\adefref{a}}
}
\end{equation}
% \begin{equation}\label{rule:tsmexpNormalPat-abstype}
% \inferrule{ }{
%   \tsmexpNormalPat{\Omega}{\Psi}{\aeabstype{t}{\epsilon}}
% }
% \end{equation}
\begin{equation}\label{rule:tsmexpNormalPat-absmod}
\inferrule{ }{
  \tsmexpNormalPat{\Omega}{\Psi}{\aeabsmod{\sigma}{X}{\epsilon}}
}
\end{equation}
% \begin{equation}\label{rule:tsmexpNormalPat-aptype}
% \inferrule{
%   \epsilon \neq \aeabstype{t}{\epsilon'}\\
%   \tsmexpNormalPat{\Omega}{\Psi}{\epsilon}
% }{
%   \tsmexpNormalPat{\Omega}{\Psi}{\aeaptype{\tau}{\epsilon}}
% }
% \end{equation}
\begin{equation}\label{rule:tsmexpNormalPat-apmod}
\inferrule{
  \epsilon \neq \aeabsmod{\sigma}{X'}{\epsilon'}\\
  \tsmexpNormalPat{\Omega}{\Psi}{\epsilon}
}{
  \tsmexpNormalPat{\Omega}{\Psi}{\aeapmod{X}{\epsilon}}
}
\end{equation}
\end{subequations}


\section{Proto-Expansion Validation}\label{appendix:P-proto-expansion-validation}
\subsection{Syntax of Proto-Expansions}
\subsubsection{Syntax -- Parameterized Proto-Expressions}
\[\begin{array}{lllllll}
\textbf{Sort} & & & \textbf{Operational Form} & \textbf{Stylized Form} & \textbf{Description}\\
% \LCC \color{Yellow}&\color{Yellow}&\color{Yellow}& \color{Yellow} & \color{Yellow} & \color{Yellow}\\
\mathsf{PPrExpr} & \pce & ::= & \apceexp{\ce} & \pceexp{\ce} & \text{proto-expression}\\
% &&& \apcebindtype{t}{\pce} & \pcebindtype{t}{\pce} & \text{type binding}\\
&&& \apcebindmod{X}{\pce} & \pcebindmod{X}{\pce} & \text{module binding}%\ECC
\end{array}\]

\subsubsection{Syntax -- Parameterized Proto-Patterns}
\[\begin{array}{lllllll}
\textbf{Sort} & & & \textbf{Operational Form} & \textbf{Stylized Form} & \textbf{Description}\\
% \LCC \color{Yellow}&\color{Yellow}&\color{Yellow}& \color{Yellow} & \color{Yellow} & \color{Yellow}\\
\mathsf{PPrPat} & \pcp & ::= & \apcepat{\cpv} & {\cpv} & \text{proto-pattern}\\
% &&& \apcebindtype{t}{\pcp} & \pcebindtype{t}{\pcp} & \text{type binding}\\
&&& \apcebindmod{X}{\pcp} & \pcebindmod{X}{\pcp} & \text{module binding}%\ECC
\end{array}\]

\subsubsection{Syntax -- Proto-Kinds and Proto-Constructions}
\[\begin{array}{lrlllll}
\textbf{Sort} & & & \textbf{Operational Form} & \textbf{Stylized Form} & \textbf{Description}\\
\mathsf{PrKind} & \cekappa & ::= & \acekdarr{\cekappa}{u}{\cekappa} & \kdarr{u}{\cekappa}{\cekappa} & \text{dependent function}\\
&&& \acekunit & \kunit & \text{nullary product}\\
&&& \acekdbprod{\cekappa}{u}{\cekappa} & \kdbprod{u}{\cekappa}{\cekappa} & \text{dependent product}\\
%&&& \akdprodstd & \kdprodstd & \text{labeled dependent product}\\
&&& \acekty & \kty & \text{type}\\
&&& \aceksing{\ctau} & \ksing{\ctau} & \text{singleton}\\
% \LCC &&& \color{Yellow} & \color{Yellow} & \color{Yellow}\\
&&& \acesplicedk{m}{n} & \splicedk{m}{n} & \text{spliced kind}\\%\ECC\\
\mathsf{PrCon} & \cec, \ctau & ::= & u & u & \text{construction variable}\\
&&& t & t & \text{type variable}\\
% &&& \acecasc{\cekappa}{\cec} & \casc{\cec}{\cekappa} & \text{ascription}\\
&&& \acecabs{u}{\cec} & \cabs{u}{\cec} & \text{abstraction}\\
&&& \acecapp{\cec}{\cec} & \capp{\cec}{\cec} & \text{application}\\
&&& \acectriv & \ctriv & \text{trivial}\\
&&& \acecpair{\cec}{\cec} & \cpair{\cec}{\cec} & \text{pair}\\
&&& \acecprl{\cec} & \cprl{\cec} & \text{left projection}\\
&&& \acecprr{\cec} & \cprr{\cec} & \text{right projection}\\
%&&& \adtplX & \dtplX & \text{labeled dependent tuple}\\
%&&& \adprj{\ell}{c} & \prj{c}{\ell} & \text{projection}\\
&&& \aceparr{\ctau}{\ctau} & \parr{\ctau}{\ctau} & \text{partial function}\\
&&& \aceallu{\cekappa}{u}{\ctau} & \forallu{u}{\cekappa}{\ctau} & \text{polymorphic}\\
&&& \acerec{t}{\ctau} & \rect{t}{\ctau} & \text{recursive}\\
&&& \aceprod{\labelset}{\mapschema{\ctau}{i}{\labelset}} & \prodt{\mapschema{\ctau}{i}{\labelset}} & \text{labeled product}\\
&&& \acesum{\labelset}{\mapschema{\ctau}{i}{\labelset}} & \sumt{\mapschema{\ctau}{i}{\labelset}} & \text{labeled sum}\\
&&& \acemcon{X} & \mcon{X} & \text{construction component}\\
% \LCC &&& \color{Yellow} & \color{Yellow} & \color{Yellow}\\
&&& \acesplicedc{m}{n}{\cekappa} & \splicedc{m}{n}{\cekappa} & \text{spliced construction}%\ECC
\end{array}\]

\subsubsection{Syntax -- Proto-Expressions and Proto-Rules}
\[\arraycolsep=4pt\begin{array}{lllllll}
\textbf{Sort} & & & \textbf{Operational Form} & \textbf{Stylized Form} & \textbf{Description}\\
\mathsf{PrExp} & \ce & ::= & x & x & \text{variable}\\
&&& \aceasc{\ctau}{\ce} & \asc{\ce}{\ctau} & \text{ascription}\\
&&& \aceletsyn{x}{\ce}{\ce} & \letsyn{x}{\ce}{\ce} & \text{value binding}\\
% &&& \aceanalam{x}{\ce} & \analam{x}{\ce} & \text{abstraction (unannotated)}\\
&&& \acelam{\ctau}{x}{\ce} & \lam{x}{\ctau}{\ce} & \text{abstraction}\\
&&& \aceap{\ce}{\ce} & \ap{\ce}{\ce} & \text{application}\\
&&& \aceclam{\cekappa}{u}{\ce} & \clam{u}{\cekappa}{\ce} & \text{construction abstraction}\\
&&& \acecap{\ce}{\cec} & \cAp{\ce}{\cec} & \text{construction application}\\
&&& \acefold{\ce} & \fold{\ce} & \text{fold}\\
&&& \aceunfold{\ce} & \unfold{\ce} & \text{unfold}\\
&&& \acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}} & \tpl{\mapschema{\ce}{i}{\labelset}} & \text{labeled tuple}\\
&&& \acepr{\ell}{\ce} & \prj{\ce}{\ell} & \text{projection}\\
&&& \aceanain{\ell}{\ce} & \inj{\ell}{\ce} & \text{injection}\\
&&& \acematchwith{n}{\ce}{\seqschemaX{\crv}} & \matchwith{\ce}{\seqschemaX{\crv}} & \text{match}\\
&&& \acemval{X} & \mval{X} & \text{value component}\\
% \LCC &&& \color{Yellow} & \color{Yellow} & \color{Yellow}\\
&&& \acesplicede{m}{n}{\ctau} & \splicede{m}{n}{\ctau} & \text{spliced expression}\\%\ECC\\
\mathsf{PrRule} & \crv & ::= & \acematchrule{p}{\ce} & \matchrule{p}{\ce} & \text{rule}\end{array}\]

\subsubsection{Syntax -- Proto-Patterns}
\[\begin{array}{lllllll}
\mathsf{PrPat} & \cpv & ::= & \acewildp & \wildp & \text{wildcard pattern}\\
&&& \acefoldp{\cpv} & \foldp{\cpv} & \text{fold pattern}\\
&&& \acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}} & \tplp{\mapschema{\cpv}{i}{\labelset}} & \text{labeled tuple pattern}\\
&&& \aceinjp{\ell}{\cpv} & \injp{\ell}{\cpv} & \text{injection pattern}\\
% \LCC &&& \color{Yellow} & \color{Yellow} & \color{Yellow}\\
&&& \acesplicedp{m}{n}{\ctau} & \splicedp{m}{n}{\ctau} & \text{spliced pattern} %\ECC
\end{array}\]

\subsubsection{Common Proto-Expansion Terms}
Each expanded term, with a few exceptions noted below, maps onto a proto-expansion term. We refer to these as the \emph{common proto-expansion terms}. In particular:
\begin{itemize}
  \item Each kind, $\kappa$, maps onto a proto-kind, $\Cof{\kappa}$, as follows:
  \[\arraycolsep=1pt\begin{array}{rl}
  \Cof{\akdarr{\kappa_1}{u}{\kappa_2}} & = \acekdarr{\Cof{\kappa_1}}{u}{\Cof{\kappa_2}}\\
  \Cof{\akunit} & = \acekunit\\
  \Cof{\akdbprod{\kappa_1}{u}{\kappa_2}} & = \acekdbprod{\Cof{\kappa_1}}{u}{\Cof{\kappa_2}}\\
  \Cof{\akty} & = \acekty\\
  \Cof{\aksing{\tau}} & = \aceksing{\Cof{\tau}}
  \end{array}\]
  \item Each construction, $c$, maps onto a proto-construction, $\Cof{c}$, as follows:
  \[\arraycolsep=1pt\begin{array}{rl}
  \Cof{u} & = u\\
  \Cof{\acabs{u}{c}} & = \acecabs{u}{\Cof{c}}\\
  \Cof{\acapp{c_1}{c_2}} & = \acecapp{\Cof{c_1}}{\Cof{c_2}}\\
  \Cof{\actriv} & = \acectriv\\
  \Cof{\acpair{c_1}{c_2}} & = \acecpair{\Cof{c_1}}{\Cof{c_2}}\\
  \Cof{\acprl{c}} & = \acecprl{\Cof{c}}\\
  \Cof{\acprr{c}} & = \acecprr{\Cof{c}}\\
  \Cof{\aparr{\tau_1}{\tau_2}} & = \aceparr{\Cof{\tau_1}}{\Cof{\tau_2}}\\
  \Cof{\aall{t}{\tau}} & = \aceall{t}{\Cof{\tau}}\\
  \Cof{\arec{t}{\tau}} & = \acerec{t}{\Cof{\tau}}\\
  \Cof{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}} & = \aceprod{\labelset}{\mapschemax{\Cofv}{\tau}{i}{\labelset}}\\
  \Cof{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}} & = \acesum{\labelset}{\mapschemax{\Cofv}{\tau}{i}{\labelset}}\\
  \Cof{\amcon{X}} & = \acemcon{X}
  \end{array}\]
  \item Each expanded expression, $e$, except for the value projection of a module expression that is not of module variable form, maps onto a proto-expression, $\Cof{e}$, as follows:
  \[\arraycolsep=1pt\begin{array}{rl}
  \Cof{x} & = x\\
  \Cof{\aelam{\tau}{x}{e}} & = \acelam{\Cof{\tau}}{x}{\Cof{e}}\\
  \Cof{\aeap{e_1}{e_2}} & = \aceap{\Cof{e_1}}{\Cof{e_2}}\\
  \Cof{\aeclam{\kappa}{u}{e}} & = \aceclam{\Cof{\kappa}}{u}{\Cof{e}}\\
  \Cof{\aecap{e}{c}} & = \acecap{\Cof{e}}{\Cof{c}}\\
  \Cof{\aefold{e}} & = \acefold{\Cof e}\\
  \Cof{\aeunfold{e}} & = \aceunfold{\Cof{e}}\\
  \Cof{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}} & = \acetpl{\labelset}{\mapschemax{\Cofv}{e}{i}{\labelset}}\\
  \Cof{\aein{\ell}{e}} &= \acein{\ell}{\Cof{e}}\\
  \Cof{\aematchwith{n}{e}{\seqschemaX{r}}} & = \acematchwith{n}{\Cof{e}}{\seqschemaXx{\Cofv}{r}}\\
  \Cof{\amval{X}} & = \acemval{X}
  \end{array}\]
  \item Each expanded rule, $r$, maps onto the proto-rule, $\Cof{r}$, as follows:
  \begin{align*}
  \Cof{\aematchrule{p}{e}} & = \acematchrule{p}{\Cof{e}}
  \end{align*}
  Notice that proto-rules bind expanded patterns, not proto-patterns. This is because proto-rules appear in proto-expressions, which are generated by peTLMs. It would not be sensible for an peTLM to splice a pattern out of a literal body.
  \item Each expanded pattern, $p$, except for the variable patterns, maps onto a proto-pattern, $\Cof{p}$, as follows:
  \begin{align*}
  \Cof{\aewildp} & = \acewildp\\
  \Cof{\aefoldp{p}} & = \acefoldp{\Cof{p}}\\
  \Cof{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}} & = \acetplp{\labelset}{\mapschemax{\Cofv}{p}{i}{\labelset}}\\
  \Cof{\aeinjp{\ell}{p}} & = \aceinjp{\ell}{\Cof{p}}
  \end{align*}
\end{itemize}

\subsubsection{Parameterized Proto-Expression Encoding and Decoding}
The type abbreviated $\tPProtoExpr$ classifies encodings of \emph{parameterized proto-expressions}. The mapping from parameterized proto-expressions to values of type $\tPProtoExpr$ is defined by the \emph{parameterized proto-expression encoding judgement}, $\encodePCEExp{\pce}{e}$. An inverse mapping is defined by the \emph{parameterized proto-expression decoding judgement}, $\decodePCEExp{e}{\pce}$.

\[\begin{array}{ll}
\textbf{Judgement Form} & \textbf{Description}\\
\encodePCEExp{\pce}{e} & \text{$\pce$ has encoding $e$}\\
\decodePCEExp{e}{\pce} & \text{$e$ has decoding $\pce$}
\end{array}\]

Rather than picking a particular definition of $\tPProtoExpr$ and defining the judgements above inductively against it, we only state the following condition, which establishes an isomorphism between values of type $\tPProtoExpr$ and parameterized proto-expressions.

\begin{condition}[Parameterized Proto-Expression Isomorphism]\label{condition:parameterized-proto-expression-isomorphism} ~
\begin{enumerate}
\item For every $\pce$, we have $\encodePCEExp{\pce}{\ecand}$ for some $\ecand$ such that $\hastypeUC{\ecand}{\tPProtoExpr}$ and $\isvalU{\ecand}$.
\item If $\hastypeUC{\ecand}{\tPProtoExpr}$ and $\isvalU{\ecand}$ then $\decodePCEExp{\ecand}{\pce}$ for some $\pce$.
\item If $\encodePCEExp{\pce}{\ecand}$ then $\decodePCEExp{\ecand}{\pce}$.
\item If $\hastypeUC{\ecand}{\tPProtoExpr}$ and $\isvalU{\ecand}$ and $\decodePCEExp{\ecand}{\pce}$ then $\encodePCEExp{\pce}{\ecand}$.
\item If $\encodePCEExp{\pce}{\ecand}$ and $\encodePCEExp{\pce}{\ecand'}$ then $\ecand=\ecand'$.
\item If $\hastypeUC{\ecand}{\tPProtoExpr}$ and $\isvalU{\ecand}$ and $\decodePCEExp{\ecand}{\pce}$ and $\decodePCEExp{\ecand}{\pce'}$ then $\pce=\pce'$.
\end{enumerate}
\end{condition}

\subsubsection{Parameterized Proto-Pattern Encoding and Decoding}
The type abbreviated $\tPCEPat$ classifies encodings of \emph{parameterized proto-patterns}. The mapping from parameterized proto-patterns to values of type $\tPCEPat$ is defined by the \emph{parameterized proto-pattern encoding judgement}, $\encodePCEPat{\pcp}{p}$. An inverse mapping is defined by the \emph{parameterized proto-expression decoding judgement}, $\decodePCEPat{p}{\pcp}$.

\[\begin{array}{ll}
\textbf{Judgement Form} & \textbf{Description}\\
\encodePCEPat{\pcp}{p} & \text{$\pcp$ has encoding $p$}\\
\decodePCEPat{p}{\pcp} & \text{$p$ has decoding $\pcp$}
\end{array}\]

Again, rather than picking a particular definition of $\tPCEPat$ and defining the judgements above inductively against it, we only state the following condition, which establishes an isomorphism between values of type $\tPCEPat$ and parameterized proto-patterns.

\begin{condition}[Parameterized Proto-Pattern Isomorphism]\label{condition:proto-pattern-isomorphism-P} ~
\begin{enumerate}
\item For every $\pcp$, we have $\encodePCEPat{\pcp}{\ecand}$ for some $\ecand$ such that $\hastypeUC{\ecand}{\tPCEPat}$ and $\isvalU{\ecand}$.
\item If $\hastypeUC{\ecand}{\tPCEPat}$ and $\isvalU{\ecand}$ then $\decodePCEPat{\ecand}{\pcp}$ for some $\pcp$.
\item If $\encodePCEPat{\pcp}{\ecand}$ then $\decodePCEPat{\ecand}{\pcp}$.
\item If $\hastypeUC{\ecand}{\tPCEPat}$ and $\isvalU{\ecand}$ and $\decodePCEPat{\ecand}{\pcp}$ then $\encodePCEPat{\pcp}{\ecand}$.
\item If $\encodePCEPat{\pcp}{\ecand}$ and $\encodePCEPat{\pcp}{\ecand'}$ then $\ecand=\ecand'$.
\item If $\hastypeUC{\ecand}{\tPCEPat}$ and $\isvalU{\ecand}$ and $\decodePCEPat{\ecand}{\pcp}$ and $\decodePCEPat{\ecand}{\pcp'}$ then $\pcp=\pcp'$.
\end{enumerate}
\end{condition}

\subsubsection{Segmentations}\label{appendix:segmentations-P}
The \emph{segmentation}, $\psi$, of a proto-kind, $\segof{\cekappa}$, proto-construction, $\segof{\cec}$, proto-expression, $\segof{\ce}$, or proto-rule, $\segof{\crv}$, is the finite set of references to spliced kinds, constructions and expressions that it mentions.
\[
\begin{array}{lll}
\segof{\acekdarr{\cekappa_1}{u}{\cekappa_2}} & = & \segof{\cekappa_1} \cup \segof{\cekappa_2} \\
\segof{\acekunit} & = & \emptyset\\
\segof{\acekdbprod{\cekappa_1}{u}{\cekappa_2}} & = & \segof{\cekappa_1} \cup \cekappa_2\\
\segof{\acekty} & = & \emptyset\\
\segof{\aceksing{\ctau}} & = & \segof{\ctau}\\
\segof{\acesplicedk{m}{n}} & = & \{ \acesplicedk{m}{n} \}\\
~\\
\segof{u} & = & \emptyset\\
\segof{\acecabs{u}{\cec}} & = & \segof{\cec}\\
\segof{\acecapp{\cec_1}{\cec_2}} & = & \segof{\cec_1} \cup \segof{\cec_2}\\
\segof{\acectriv} & = & \emptyset\\
\segof{\acecpair{\cec_1}{\cec_2}} & = & \segof{\cec_1} \cup \segof{\cec_2}\\
\segof{\acecprl{\cec}} & = & \segof{\cec}\\
\segof{\acecprr{\cec}} & = & \segof{\cec}\\
\segof{\aceparr{\ctau_1}{\ctau_2}} & = & \segof{\ctau_1} \cup \segof{\ctau_2}\\
\segof{\aceallu{\cekappa}{u}{\ctau}} & = & \segof{\cekappa} \cup \segof{\ctau}\\
\segof{\acerec{t}{\ctau}} & = & \segof{\ctau}\\
\segof{\aceprod{\labelset}{\mapschema{\ctau}{i}{\labelset}}} & = & \bigcup_{i \in \labelset} \segof{\ctau_i}\\
\segof{\acesum{\labelset}{\mapschema{\ctau}{i}{\labelset}}} & = & \bigcup_{i \in \labelset} \segof{\ctau_i}\\
\segof{\acemcon{X}} & = & \emptyset\\
\segof{\acesplicedc{m}{n}{\cekappa}} & = & \{ \acesplicedc{m}{n}{\cekappa} \} \cup \segof{\cekappa}\\
~\\
\segof{x} & = & \emptyset\\
\segof{\aceasc{\ctau}{\ce}} & = & \segof{\ctau} \cup \segof{\ce}\\
\segof{\aceletsyn{x}{\ce_1}{\ce_2}} & = & \segof{\ce_1} \cup \segof{\ce_2}\\
\segof{\acelam{\ctau}{x}{\ce}} & = & \segof{\ctau} \cup \segof{\ce}\\
\segof{\aceap{\ce_1}{\ce_2}} & = & \segof{\ce_1} \cup \segof{\ce_2}\\
\segof{\aceclam{\cekappa}{u}{\ce}} & = & \segof{\cekappa} \cup \segof{\ce}\\
\segof{\acecap{\ce}{\cec}} & = & \segof{\ce} \cup \segof{\cec}\\
\segof{\acefold{\ce}} & = & \segof{\ce}\\
\segof{\aceunfold{\ce}} & = & \segof{\ce}\\
\segof{\acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}}} & = & \bigcup_{i \in \labelset} \segof{\ce_i}\\
\segof{\acepr{\ell}{\ce}} & = & \segof{\ce}\\
\segof{\aceanain{\ell}{\ce}} & = & \segof{\ce}\\
\segof{\acematchwith{n}{\ce}{\seqschemaX{\crv}}} & = & \segof{\ce} \cup \bigcup_{1 \leq i \leq n} \segof{\crv_i}\\
\segof{\acemval{X}} & = & \emptyset\\
\segof{\acesplicede{m}{n}{\ctau}} & = & \{\acesplicede{m}{n}{\ctau}\} \cup \segof{\ctau}\\
~\\
\segof{\acematchrule{p}{\ce}} & = & \segof{\ce}
\end{array}
\]

The segmentation of a proto-pattern, $\segof{\cpv}$, is the finite set of references to spliced patterns and types that it mentions.
\[
\begin{array}{lll}
\segof{\acewildp} & = & \emptyset\\
\segof{\acefoldp{\cpv}} & = & \segof{\cpv}\\
\segof{\acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}}} & = & \bigcup_{i \in \labelset} \segof{\cpv_i}\\
\segof{\aceinjp{\ell}{\cpv}} & = & \segof{\cpv}\\
\segof{\acesplicedp{m}{n}{\ctau}} & = & \{ \acesplicedp{m}{n}{\ctau} \} \cup \segof{\ctau}
\end{array}
\]

The predicate $\segOKP{\Omega}{\cscenev}{\psi}{b}$ checks that each segment in $\psi$, has non-negative length and is within bounds of $b$, and that the segments in $\psi$ do not overlap and operate at consistent sorts, kinds and types. The contexts are needed because kind and type equivalence are contextual.

\begin{definition}[Segmentation Validity] $\segOKP{\Omega}{\cscenev}{\psi}{b}$ where \[\cscenev=\csceneP{\omega : \OParams}{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}}{b}\] iff
\begin{enumerate}
  \item For each $\acesplicedk{m}{n} \in \psi$, all of the following hold:
    \begin{enumerate}
      \item $0 \leq m < n \leq \sizeof{b}$
      \item For each $\acesplicedk{m'}{n'} \in \psi$, either 
        \begin{enumerate}
          \item $m=m'$ and $n=n'$; or
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicedc{m'}{n'}{\cekappa} \in \psi$, either
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}        
      \item For each $\acesplicede{m'}{n'}{\ctau} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicedp{m'}{n'}{\ctau} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
    \end{enumerate}
  \item For each $\acesplicedc{m}{n}{\cekappa} \in \psi$, all of the following hold:
    \begin{enumerate}
      \item $0 \leq m < n \leq \sizeof{b}$
      \item For each $\acesplicedk{m'}{n'} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicedc{m'}{n'}{\cekappa'} \in \psi$, either
        \begin{enumerate}
          \item $m=m'$ and $n=n'$ and $\cvalidKX{\cekappa}{\kappa}$ and $\cvalidKX{\cekappa'}{\kappa'}$ and $\kequal{\Omega \cup \Omega_\text{app}}{\kappa}{\kappa'}$; or
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}        
      \item For each $\acesplicede{m'}{n'}{\ctau} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicedp{m'}{n'}{\ctau} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
    \end{enumerate}
  \item For each $\acesplicede{m}{n}{\ctau} \in \psi$, all of the following hold:
    \begin{enumerate}
      \item $0 \leq m < n \leq \sizeof{b}$
      \item For each $\acesplicedk{m'}{n'} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicedc{m'}{n'}{\cekappa'} \in \psi$, either
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}        
      \item For each $\acesplicede{m'}{n'}{\ctau'} \in \psi$, either 
        \begin{enumerate}
          \item $m=m'$ and $n=n'$ and $\cvalidCX{\ctau}{\tau}{\akty}$ and $\cvalidCX{\ctau'}{\tau'}{\akty}$ and $\cequal{\Omega \cup \Omega_\text{app}}{\tau}{\tau'}{\akty}$; or
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicedp{m'}{n'}{\ctau} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
    \end{enumerate}
  \item For each $\acesplicedp{m}{n}{\ctau} \in \psi$, all of the following hold:
    \begin{enumerate}
      \item $0 \leq m < n \leq \sizeof{b}$
      \item For each $\acesplicedk{m'}{n'} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicedc{m'}{n'}{\cekappa'} \in \psi$, either
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}        
      \item For each $\acesplicede{m'}{n'}{\ctau'} \in \psi$, either 
        \begin{enumerate}
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
      \item For each $\acesplicedp{m'}{n'}{\ctau} \in \psi$, either 
        \begin{enumerate}
          \item $m=m'$ and $n=n'$ and $\cvalidCX{\ctau}{\tau}{\akty}$ and $\cvalidCX{\ctau'}{\tau'}{\akty}$ and $\cequal{\Omega \cup \Omega_\text{app}}{\tau}{\tau'}{\akty}$; or
          \item $n' < m$; or 
          \item $m' > n$
        \end{enumerate}
    \end{enumerate}
\end{enumerate}
\end{definition}
% \subsubsection{Segmentations}
% A \emph{segmentation}, $\psi$, is a finite set of \emph{segments}. Segments consist of two natural numbers and a sort, i.e. segments are of the form $\segKind{m}{n}$ or $\segCon{m}{n}$ or $\segExp{m}{n}$ or $\segPat{m}{n}$.

% The metafunction $\segof{\ce}$ determines the segmentation of $\ce$ by generating one segment for each reference to a spliced term. More specifically:
% \begin{itemize}
% \item We define $\segof{\cekappa}$ as follows:
% \[\arraycolsep=1pt\begin{array}{rl}
%   \segof{\acekdarr{\cekappa_1}{u}{\cekappa_2}} & = \segof{\cekappa_1} \cup \segof{\cekappa_2}\\
%   \segof{\acekunit} & = \emptyset\\
%   \segof{\acekdbprod{\cekappa_1}{u}{\cekappa_2}} & = \segof{\cekappa_1} \cup \segof{\cekappa_2}\\
%   \segof{\acekty} & = \emptyset\\
%   \segof{\aceksing{\ctau}} & = \segof{\ctau}\\
%   \segof{\acesplicedk{m}{n}} & = \{ \segKind{m}{n} \}
% \end{array}\]
% \item We define $\segof{\cec}$ as follows:
% \[\arraycolsep=1pt\begin{array}{rl}
%   \segof{u} & = \emptyset\\
%   \segof{\acecabs{u}{\cec}} & = \segof{\cec}\\
%   \segof{\acecapp{\cec_1}{\cec_2}} & = \segof{\cec_1} \cup \segof{\cec_2}\\
%   \segof{\acectriv} & = \emptyset\\
%   \segof{\acecpair{\cec_1}{\cec_2}} & = \segof{\cec_1} \cup \segof{\cec_2}\\
%   \segof{\acecprl{\cec}} & = \segof{\cec}\\
%   \segof{\acecprr{\cec}} & = \segof{\cec}\\
%   \segof{\aceparr{\ctau_1}{\ctau_2}} & = \segof{\ctau_1} \cup \segof{\ctau_2}\\
%   \segof{\aceallu{u}{\cekappa}{\ctau}} &= \segof{\cekappa} \cup \segof{\ctau}\\
%   \segof{\acerec{t}{\ctau}} & = \segof{\ctau}\\
%   \segof{\aceprod{\labelset}{\mapschema{\ctau}{i}{\labelset}}} & = \cup_{i \in \labelset} \segof{\ctau_i}\\
%   \segof{\acesum{\labelset}{\mapschema{\ctau}{i}{\labelset}}} & = \cup_{i \in \labelset} \segof{\ctau_i}\\
%   \segof{\acemcon{X}} & = \emptyset\\
%   \segof{\acesplicedc{m}{n}} & = \{ \segCon{m}{n} \}
%   \end{array}\]
% \item We define $\segof{\ce}$ as follows:
% \[\arraycolsep=1pt\begin{array}{rl} 

% \segof{x} & = \emptyset\\
% \segof{\acelam{\ctau}{x}{\ce}} & = \segof{\ctau} \cup \segof{\ce} \\
% \segof{\aceclam{\cekappa}{u}{\ce}} & = \segof{\cekappa} \cup \segof{\ce}\\
% \segof{\acecap{\ce}{\cec}} & = \segof{\cec} \cup \segof{\ce}\\
% \segof{\acefold{\ce}} & = \segof{\ce}\\
% \segof{\aceunfold{\ce}} & = \segof{\ce}\\
% \segof{\acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}}} & = \cup_{i \in \labelset} \segof{\ce_i}\\
% \segof{\acepr{\ell}{\ce}} & = \segof{\ce}\\
% \segof{\acein{\ell}{\ce}} & = \segof{\ce}\\
% \segof{\acematchwith{n}{\ce}{\seqschemaX{\crv}}} & = \segof{\ce} \cup_{1 \leq i \leq n} \segof{\crv_i}\\
% \segof{\acemval{X}} & = \emptyset\\
% \segof{\acesplicede{m}{n}{\ctau}} & = \{ \segExp{m}{n} \} \cup \segof{\ctau}\\
% \end{array}\]
% \item We define $\segof{\crv}$ as follows:
% \[\arraycolsep=1pt\begin{array}{rl} 

% \segof{\acematchrule{p}{\ce}} & = \segof{\ce}
% \end{array}\]
% \end{itemize}

% The metafunction $\segof{\cpv}$ determines the segmentation of $\cpv$ by generating one segment for each reference to a spliced type or pattern:
% \[
% \arraycolsep=1pt\begin{array}{rl}

% \segof{\acewildp} & = \emptyset\\
% \segof{\acefoldp{\cpv}} & = \segof{\cpv}\\
% \segof{\acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}}} & = \cup_{i \in \labelset} \segof{\cpv_i}\\
% \segof{\aceinjp{\ell}{\cpv}} & = \segof{\cpv}\\
% \segof{\acesplicedp{m}{n}{\ctau}} & = \{ \segPat{m}{n} \} \cup \segof{\ctau}
% \end{array}
% \]

% The predicate $\segOK{\psi}{b}$ checks that each segment in $\psi$, has non-negative length and is within bounds of $b$, and that the segments in $\psi$ do not overlap.

\subsection{Deparameterization}
\begin{minipage}{0.42\textwidth}
\noindent\fbox{$\strut\prepce{\Omega_\text{app}}{\Psi}{\pce}{\ce}{\epsilon}{\rho}{\omega}{\Omega_\text{params}}$}\end{minipage}
\begin{minipage}{0.58\textwidth}
When applying peTLM $\epsilon$, $\pce$ has deparameterization $\ce$ leaving $\rho$ with parameter substitution $\omega$\end{minipage}
\begin{subequations}\label{rules:prepce}
\begin{equation}\label{rule:prepce-ceexp}
\inferrule{
  \istsmty{\Omega_\text{app}}{\rho}
}{
  \prepce{\Omega_\text{app}}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\apceexp{\ce}}{\ce}{\adefref{a}}{\rho}{\emptyset}{\emptyset}
}
\end{equation}
% \begin{equation}\label{rule:prepce-alltypes}
% \inferrule{
%   \prepce{\Omega_\text{app}}{\Psi}{\pce}{\ce}{\epsilon}{\aealltypes{t}{\rho}}{\omega}{\Omega}\\
%   t \notin \domof{\Omega_\text{app}}
% }{
%   \prepce{\Omega_\text{app}}{\Psi}{\apcebindtype{t}{\pce}}{\ce}{\aeaptype{\tau}{\epsilon}}{\rho}{\omega, \tau/t}{\Omega, t :: \akty}
% }
% \end{equation}
\begin{equation}\label{rule:prepce-allmods}
\inferrule{
  \prepce{\Omega_\text{app}}{\Psi}{\pce}{\ce}{\epsilon}{\aeallmods{\sigma}{X}{\rho}}{\omega}{\Omega}\\\\
  \hassig{\Omega_\text{app}}{X'}{\sigma}\\
  X \notin \domof{\Omega_\text{app}}
}{
  \prepce{\Omega_\text{app}}{\Psi}{\apcebindmod{X}{\pce}}{\ce}{\aeapmod{X'}{\epsilon}}{\rho}{(\omega, X'/X)}{(\Omega, X : \sigma)}
}
\end{equation}
\end{subequations}

\noindent\begin{minipage}{0.42\textwidth}
\fbox{$\strut\prepcp{\Omega_\text{app}}{\Phi}{\pcp}{\cpv}{\epsilon}{\rho}{\omega}{\Omega_\text{params}}$}\end{minipage}
\begin{minipage}{0.58\textwidth}
When applying ppTLM $\epsilon$, $\pcp$ has deparameterization $\cpv$ leaving $\rho$ with parameter substitution $\omega$\end{minipage}
\begin{subequations}\label{rules:prepcp}
\begin{equation}\label{rule:prepcp-cepat}
\inferrule{
  \istsmty{\Omega_\text{app}}{\rho}
}{
  \prepcp{\Omega_\text{app}}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\apcepat{\cpv}}{\cpv}{\adefref{a}}{\rho}{\emptyset}{\emptyset}
}
\end{equation}
% \begin{equation}\label{rule:prepcp-alltypes}
% \inferrule{
%   \prepcp{\Omega_\text{app}}{\Phi}{\pcp}{\cpv}{\epsilon}{\aealltypes{t}{\rho}}{\omega}{\Omega}\\
%   t \notin \domof{\Omega_\text{app}}
% }{
%   \prepcp{\Omega_\text{app}}{\Phi}{\apcebindtype{t}{\pcp}}{\cpv}{\aeaptype{\tau}{\epsilon}}{\rho}{\omega, \tau/t}{\Omega, t :: \akty}
% }
% \end{equation}
\begin{equation}\label{rule:prepcp-allmods}
\inferrule{
  \prepcp{\Omega_\text{app}}{\Phi}{\pcp}{\cpv}{\epsilon}{\aeallmods{\sigma}{X}{\rho}}{\omega}{\Omega}\\\\
  \hassig{\Omega_\text{app}}{X'}{\sigma}\\
  X \notin \domof{\Omega_\text{app}}
}{
  \prepcp{\Omega_\text{app}}{\Phi}{\apcebindmod{X}{\pcp}}{\cpv}{\aeapmod{X'}{\epsilon}}{\rho}{(\omega, X'/X)}{(\Omega, X : \sigma)}
}
\end{equation}
% \begin{equation}\label{rule:prepcp-ceexp}
% \inferrule{ }{
%   \prepcp{\Omega_\text{app}}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\adefref{a}}{\rho}{\emptyset}{\emptyset}
% }
% \end{equation}
% \begin{equation}\label{rule:prepcp-alltypes}
% \inferrule{
%   \prepcp{\Omega_\text{app}}{\Phi}{\epsilon}{\aealltypes{t}{\rho}}{\omega}{\Omega}\\
%   t \notin \domof{\Omega_\text{app}}
% }{
%   \prepcp{\Omega_\text{app}}{\Phi}{\aeaptype{\tau}{\epsilon}}{\rho}{\omega, \tau/t}{\Omega, t :: \akty}
% }
% \end{equation}
% \begin{equation}\label{rule:prepcp-allmods}
% \inferrule{
%   \prepcp{\Omega_\text{app}}{\Phi}{\epsilon}{\aeallmods{\sigma}{X}{\rho}}{\omega}{\Omega}\\
%   X \notin \domof{\Omega_\text{app}}
% }{
%   \prepcp{\Omega_\text{app}}{\Phi}{\aeapmod{X'}{\epsilon}}{\rho}{\omega, X'/X}{\Omega, X : \sigma}
% }
% \end{equation}
\end{subequations}

\subsection{Proto-Expansion Validation}
\subsubsection{Splicing Scenes}
\emph{Expression splicing scenes}, $\escenev$, are of the form $\esceneP{\omega : \Omega_\text{params}}{\uOmega}{\uPsi}{\uPhi}{b}$, \emph{construction splicing scenes}, $\cscenev$, are of the form $\csceneP{\omega : \Omega_\text{params}}{\uOmega}{b}$, and \emph{pattern splicing scenes}, $\pscenev$, are of the form $\psceneP{\omega : \Omega_\text{params}}{\uOmega}{\uPhi}{b}$. We write $\csfrom{\escenev}$ for the construction splicing scene constructed by dropping the TLM contexts from $\escenev$:
\[\csfrom{\esceneP{\omega : \OParams}{\uOmega}{\uPsi}{\uPhi}{b}} = \csceneP{\omega : \OParams}{\uOmega}{b}\]

\subsubsection{Proto-Kind and Proto-Construction Validation}
\noindent\fbox{$\strut\cvalidKX{\cekappa}{\kappa}$}~~$\cekappa$ has well-formed expansion $\kappa$
\begin{subequations}\label{rules:cvalidK}
\begin{equation}\label{rule:cvalidK-darr}
\inferrule{
  \cvalidKX{\cekappa_1}{\kappa_1}\\
  \cvalidK{\Omega, u :: \kappa_1}{\cscenev}{\cekappa_2}{\kappa_2}
}{
  \cvalidKX{\acekdarr{\cekappa_1}{u}{\cekappa_2}}{\akdarr{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:cvalidK-unit}
\inferrule{ }{
  \cvalidKX{\acekunit}{\akunit}
}
\end{equation}
\begin{equation}\label{rule:cvalidK-dprod}
\inferrule{
  \cvalidKX{\cekappa_1}{\kappa_1}\\
  \cvalidK{\Omega, u :: \kappa_1}{\cscenev}{\cekappa_2}{\kappa_2}
}{
  \cvalidKX{\acekdbprod{\cekappa_1}{u}{\cekappa_2}}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:cvalidK-ty}
\inferrule{ }{
  \cvalidKX{\acekty}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cvalidK-sing}
\inferrule{
  \cvalidCX{\ctau}{\tau}{\akty}
}{
  \cvalidKX{\aceksing{\ctau}}{\aksing{\tau}}
}
\end{equation}
\begin{equation}\label{rule:cvalidK-spliced}
\inferrule{
  \parseUKind{\bsubseq{b}{m}{n}}{\ukappa}\\
  \kExpands{\uOmega}{\ukappa}{\kappa}\\\\
  \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
  \domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset
}{
  \cvalidK{\Omega}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\acesplicedk{m}{n}}{\kappa}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\cvalidCX{\cec}{c}{\kappa}$}~~$\cec$ has expansion $c$ of kind $\kappa$
\begin{subequations}\label{rules:cvalidC}
\begin{equation}\label{rule:cvalidC-subsume}
\inferrule{
  \cvalidCX{\cec}{c}{\kappa_1}\\
  \ksubX{\kappa_1}{\kappa_2}
}{
  \cvalidCX{\cec}{c}{\kappa_2}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-var}
\inferrule{ }{\cvalidC{\Omega, {u} :: {\kappa}}{\cscenev}{u}{u}{\kappa}}
\end{equation}
\begin{equation}\label{rule:cvalidC-abs}
\inferrule{
  \cvalidC{\Omega, u :: \kappa_1}{\cscenev}{\cec_2}{c_2}{\kappa_2}
}{
  \cvalidCX{\acecabs{u}{\cec_2}}{\acabs{u}{c_2}}{\akdarr{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-app}
\inferrule{
  \cvalidCX{\cec_1}{c_1}{\akdarr{\kappa_2}{u}{\kappa}}\\
  \cvalidCX{\cec_2}{c_2}{\kappa_2}
}{
  \cvalidCX{\acecapp{\cec_1}{\cec_2}}{\acapp{c_1}{c_2}}{[c_1/u]\kappa}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-unit}
\inferrule{ }{
  \cvalidCX{\acectriv}{\actriv}{\akunit}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-pair}
\inferrule{
  \cvalidCX{\cec_1}{c_1}{\kappa_1}\\
  \cvalidCX{\cec_2}{c_2}{[c_1/u]\kappa_2}
}{
  \cvalidCX{\acecpair{\cec_1}{\cec_2}}{\acpair{c_1}{c_2}}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-prl}
\inferrule{
  \cvalidCX{\cec}{c}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}{
  \cvalidCX{\acecprl{\cec}}{\acprl{c}}{\kappa_1}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-prr}
\inferrule{
  \cvalidCX{\cec}{c}{\akdbprod{\kappa_1}{u}{\kappa_2}}
}{
  \cvalidCX{\acecprr{\cec}}{\acprr{c}}{[\acprl{c}/u]\kappa_2}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-parr}
\inferrule{
  \cvalidCX{\ctau_1}{\tau_1}{\akty}\\
  \cvalidCX{\ctau_2}{\tau_2}{\akty}
}{
  \cvalidCX{\aceparr{\ctau_1}{\ctau_2}}{\aparr{\tau_1}{\tau_2}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-all}
\inferrule{
  \cvalidKX{\cekappa}{\kappa}\\
  \cvalidC{\Omega, u :: \kappa}{\cscenev}{\ctau}{\tau}{\akty}
}{
  \cvalidCX{\aceallu{\cekappa}{u}{\ctau}}{\aallu{\kappa}{u}{\tau}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-rec}
\inferrule{
  \cvalidC{\Omega, t :: \akty}{\cscenev}{\ctau}{\tau}{\akty}
}{
  \cvalidCX{\acerec{t}{\ctau}}{\arec{t}{\tau}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-prod}
\inferrule{
  \{\cvalidCX{\ctau_i}{\tau_i}{\akty}\}_{1 \leq i \leq n}
}{
  \cvalidCX{\aceprod{\labelset}{\mapschema{\ctau}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-sum}
\inferrule{
  \{\cvalidCX{\ctau_i}{\tau_i}{\akty}\}_{1 \leq i \leq n}
}{
  \cvalidCX{\acesum{\labelset}{\mapschema{\ctau}{i}{\labelset}}}{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\akty}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-sing}
\inferrule{
  \cvalidCX{\cec}{c}{\akty}
}{
  \cvalidCX{\cec}{c}{\aksing{c}}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-stat}
\inferrule{ }{
  \cvalidC{\Omega, X : {\asignature{\kappa}{u}{\tau}}}{\cscenev}{\acemcon{X}}{\amcon{X}}{\kappa}
}
\end{equation}
\begin{equation}\label{rule:cvalidC-spliced}
\inferrule{
  \cscenev=\csceneP{\omega : \OParams}{\uOmega}{b}\\
  \cvalidK{\OParams}{\cscenev}{\cekappa}{\kappa}\\\\
  \parseUCon{\bsubseq{b}{m}{n}}{\uc}\\
  \cExpands{\uOmega}{\uc}{c}{[\omega]\kappa}\\\\
  \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
  \domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset
}{
  \cvalidC{\Omega}{\cscenev}{\acesplicedc{m}{n}{\cekappa}}{c}{\kappa}
}
\end{equation}
\end{subequations}
\subsubsection{Proto-Expression and Proto-Rule Validation}
% \begin{equation}
% \inferrule{
%   \ccanaX{\ctau}{\tau}{\akty}
% }{
%   \cvalidTP{\Omega}{\cscenev}{\ctau}{\tau}
% }
% \end{equation}
\noindent\fbox{$\strut\cvalidEPX{\ce}{e}{\tau}$}~~$\ce$ has expansion $e$ of type $\tau$
\begin{subequations}\label{rules:cvalidE-P}
\begin{equation}\label{rule:cvalidE-P-subsume}
  \inferrule{
    \cvalidEPX{\ce}{e}{\tau}\\
    \issubtypePX{\tau}{\tau'}
  }{
    \cvalidEPX{\ce}{e}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-var}
  \inferrule{ }{ 
    \cvalidEP{\Omega, \Ghyp{x}{\tau}}{\escenev}{x}{x}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-asc}
\inferrule{
  \cvalidC{\Omega}{\csfrom{\escenev}}{\ctau}{\tau}{\akty}\\
  \cvalidEP{\Omega}{\escenev}{\ce}{e}{\tau}
}{
  \cvalidEP{\Omega}{\escenev}{\aceasc{\ctau}{\ce}}{e}{\tau}
}
\end{equation}
\begin{equation}\label{rule:cvalidE-P-letsyn}
  \inferrule{
    \cvalidEP{\Omega}{\escenev}{\ce_1}{e_1}{\tau_1}\\
    \cvalidEP{\Omega, x : \tau_1}{\ce_2}{e_2}{\tau_2}
  }{
    \cvalidEP{\Omega}{\escenev}{\aceletsyn{x}{\ce_1}{\ce_2}}{
      \aeap{\aelam{\tau_1}{x}{e_2}}{e_1}
    }{\tau_2}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-lam}
  \inferrule{
    \cvalidC{\Omega}{\csfrom{\escenev}}{\ctau_1}{\tau_1}{\akty}\\
    \cvalidEP{\Omega, \Ghyp{x}{\tau_1}}{\escenev}{\ce}{e}{\tau_2}
  }{
    \cvalidEPX{\acelam{\ctau_1}{x}{\ce}}{\aelam{\tau_1}{x}{e}}{\aparr{\tau_1}{\tau_2}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-ap}
  \inferrule{
    \cvalidEPX{\ce_1}{e_1}{\aparr{\tau_2}{\tau}}\\
    \cvalidEPX{\ce_2}{e_2}{\tau_2}
  }{
    \cvalidEPX{\aceap{\ce_1}{\ce_2}}{\aeap{e_1}{e_2}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-clam}
  \inferrule{
    \cvalidK{\Omega}{\csfrom{\escenev}}{\cekappa}{\kappa}\\
    \cvalidEP{\Omega, u :: \kappa}{\escenev}{\ce}{e}{\tau}
  }{
    \csynX{\aceclam{\cekappa}{u}{\ce}}{\aeclam{\kappa}{u}{e}}{\aallu{\kappa}{u}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-cap}
  \inferrule{
    \cvalidEPX{\ce}{e}{\aallu{\kappa}{u}{\tau}}\\
    \cvalidC{\Omega}{\csfrom{\escenev}}{\cec}{c}{\kappa}
  }{
    \cvalidEPX{\acecap{\ce}{\cec}}{\aecap{e}{c}}{[c/u]\tau}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-fold}
  \inferrule{
    \cvalidEPX{\ce}{e}{[\arec{t}{\tau}/t]\tau}
  }{
    \cvalidEPX{\aceanafold{\ce}}{\aefold{e}}{\arec{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-unfold}
  \inferrule{
    \cvalidEPX{\ce}{e}{\arec{t}{\tau}}
  }{
    \cvalidEPX{\aceunfold{\ce}}{\aeunfold{e}}{[\arec{t}{\tau}/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-tpl}
  \inferrule{
    \tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}\\\\    
    \{\cvalidEPX{\ce_i}{e_i}{\tau_i}\}_{i \in \labelset}
  }{
    \cvalidEPX{\acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}}}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-pr}
  \inferrule{
    \cvalidEPX{\ce}{e}{\aprod{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
  }{
    \cvalidEPX{\acepr{\ell}{\ce}}{\aepr{\ell}{e}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-in}
  \inferrule{
    \asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}\\\\
    \cvalidEPX{\ce'}{e'}{\tau'}
  }{
    \cvalidEPX{\aceanain{\ell}{\ce'}}{\aein{\ell}{e'}}{\tau}
    %\left(\shortstack{$\Delta~\Gamma \vdash^{\escenev} $\\$\leadsto$\\$ \Leftarrow $\vspace{-1.2em}}\right)
    %\eanaX{\auanain{\ell}{\ue}}{\aein{\ell}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-match}
  \inferrule{
    % n > 0\\
    \cvalidEPX{\ce}{e}{\tau}\\
    \{\cvalidRP{\Omega}{\escenev}{\crv_i}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}
  }{
    \cvalidEPX{\acematchwithb{n}{\ce}{\seqschemaX{\crv}}}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:cvalidE-P-mval}
\inferrule{ }{
  \cvalidEP{\Omega, X : \asignature{\kappa}{u}{\tau}}{\escenev}{\acemval{X}}{\amval{X}}{[\amcon{X}/u]\tau}
}
\end{equation}
\begin{equation}\label{rule:cvalidE-P-splicede}
\inferrule{
  \escenev = \esceneP{\omega : \OParams}{\uOmega}{\uPsi}{\uPhi}{b}\\
  \cvalidC{\OParams}{\csfrom{\escenev}}{\ctau}{\tau}{\akty}\\\\
  \parseUExp{\bsubseq{b}{m}{n}}{\ue}\\
  \expandsP{\uOmega}{\uPsi}{\uPhi}{\ue}{e}{[\omega]\tau}\\\\
  \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
  \domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset
}{
  \cvalidEP{\Omega}{\escenev}{\acesplicede{m}{n}{\ctau}}{e}{\tau}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\cvalidRP{\Omega}{\escenev}{\crv}{r}{\tau}{\tau'}$}~~$\crv$ has expansion $r$ taking values of type $\tau$ to values of type $\tau'$
\begin{equation}\label{rule:cvalidR-P}
\inferrule{
  \patTypeP{\Omega'}{p}{\tau}\\
  \cvalidEP{\Gcons{\Omega}{\Omega'}}{\escenev}{\ce}{e}{\tau'}
}{
  \cvalidRP{\Omega}{\escenev}{\acematchrule{p}{\ce}}{\aematchrule{p}{e}}{\tau}{\tau'}
}
\end{equation}

\subsubsection{Proto-Pattern Validation}
\noindent\fbox{$\strut\cvalidPPE{\uOmega}{\pscenev}{\cpv}{p}{\tau}$}~~$\cpv$ has expansion $p$ matching against $\tau$ generating hypotheses $\uOmega$
\begin{subequations}\label{rules:cvalidPP}
\begin{equation}\label{rule:cvalidPP-wild}
\inferrule{ }{
  \cvalidPP{\uOmegaEx{\emptyset}{\emptyset}{\emptyset}{\emptyset}}{\pscenev}{\acewildp}{\aewildp}{\tau}
}
\end{equation}
\begin{equation}\label{rule:cvalidPP-fold}
\inferrule{
  \cvalidPP{\uOmega}{\pscenev}{\cpv}{p}{[\arec{t}{\tau}/t]\tau}
}{
  \cvalidPP{\uOmega}{\pscenev}{\acefoldp{\cpv}}{\aefoldp{p}}{\arec{t}{\tau}}
}
\end{equation}
\begin{equation}\label{rule:cvalidPP-tpl}
\inferrule{
  \cpv=\acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}}\\
  p=\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}\\\\
  \{\cvalidPP{\upctx_i}{\pscenev}{\cpv_i}{p_i}{\tau_i}\}_{i \in \labelset}
}{
  \cvalidPP{\GIconsi{i \in \labelset}{\uOmega_i}}{\pscenev}{\cpv}{p}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  %\cvalidPP{}{\cpv}{p}{}
%\left(\shortstack{$\vdash^{\pscenev} $\\$\leadsto$\\$ :~\dashVx^{\,\Gconsi{i \in \labelset}{\upctx_i}}$\vspace{-1.2em}}\right)
}
\end{equation}
\begin{equation}\label{rule:cvalidPP-in}
\inferrule{
  \cvalidPP{\uOmega}{\pscenev}{\cpv}{p}{\tau}
}{
  \cvalidPP{\uOmega}{\pscenev}{\aceinjp{\ell}{\cpv}}{\aeinjp{\ell}{p}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
}
\end{equation}
\begin{equation}\label{rule:cvalidPP-spliced}
\inferrule{
  \cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau}{\tau}{\akty}\\
  \parseUPat{\bsubseq{b}{m}{n}}{\upv}\\
  \patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{[\omega]\tau}
}{
  \cvalidPP{\uOmega'}{\psceneP{\omega : \Omega_\text{params}}{\uOmega}{\uPhi}{b}}{\acesplicedp{m}{n}{\ctau}}{p}{\tau}
}
\end{equation}
\end{subequations}


\section{Metatheory}\label{appendix:metatheory-P}
\subsection{TLM Expressions}
\begin{lemma}[peTLM Regularity]
\label{lemma:peTLM-regularity}
If $\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}$ then $\istsmty{\Omega}{\rho}$.
\end{lemma}
\begin{proof}
  By rule induction over Rules (\ref{rules:hastsmtypeExp}). 
  \begin{byCases}
    \item[\text{(\ref{rule:hastsmtypeExp-defref})}] ~
      \begin{pfsteps*}
        \item $\istsmty{\Omega}{\rho}$ \BY{assumption}
      \end{pfsteps*}
      \resetpfcounter
    \item[\text{(\ref{rule:hastsmtypeExp-absmod})}] ~
      \begin{pfsteps*}
        \item $\epsilon=\aeabsmod{\sigma}{X}{\epsilon'}$ \BY{assumption}
        \item $\rho=\aeallmods{\sigma}{X}{\rho'}$ \BY{assumption}
        \item $\hastsmtypeExp{\Omega, X : \sigma}{\Psi}{\epsilon'}{\rho'}$ \BY{assumption} \pflabel{hastsmtype}
        \item $\issigX{\sigma}$ \BY{assumption} \pflabel{issig}
        \item $\istsmty{\Omega, X : \sigma}{\rho'}$ \BY{IH on \pfref{hastsmtype}} \pflabel{istsmty}
        \item $\istsmty{\Omega}{\aeallmods{\sigma}{X}{\rho'}}$ \BY{Rule (\ref{rule:istsmty-allmods}) on \pfref{issig} and \pfref{istsmty}}
      \end{pfsteps*}
      \resetpfcounter
    \item[\text{(\ref{rule:hastsmtypeExp-apmod})}] ~
      \begin{pfsteps*}
        \item $\epsilon=\aeapmod{X}{\epsilon'}$ \BY{assumption}
        \item $\rho=[X/X']\rho'$ \BY{assumption}
        \item $\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\aeallmods{\sigma}{X'}{\rho'}}$ \BY{assumption} \pflabel{hastsmtype}
        \item $\hassig{\Omega}{X}{\sigma}$ \BY{assumption} \pflabel{hassig}
        \item $\istsmty{\Omega}{\aeallmods{\sigma}{X'}{\rho'}}$ \BY{IH on \pfref{hastsmtype}} \pflabel{istsmty}
        \item $\istsmty{\Omega, X' : \sigma}{\rho'}$ \BY{Inversion of Rule (\ref{rule:istsmty-allmods}) on \pfref{istsmty}} \pflabel{istsmtyprime}
        \item $\istsmty{\Omega}{[X'/X]\rho'}$ \BY{Substitution Lemma \ref{lemma:substitution-P} on \pfref{hassig} and \pfref{istsmtyprime}}
      \end{pfsteps*}
      \resetpfcounter
  \end{byCases}
\end{proof}

\begin{lemma}[ppTLM Regularity]
\label{lemma:ppTLM-regularity}
If $\hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho}$ then $\istsmty{\Omega}{\rho}$.
\end{lemma}
\begin{proof}
By rule induction over Rules (\ref{rules:hastsmtypePat}). The proof is nearly identical to the proof of Lemma \ref{lemma:peTLM-regularity}, differing only in that ppTLM contexts and the corresponding judgements are mentioned.
\end{proof}

\begin{lemma}[peTLM Unicity]
\label{lemma:peTLM-unicity}
If $\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}$ and $\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho'}$ then $\rho=\rho'$.
\end{lemma}
\begin{proof}
  By rule induction over Rules (\ref{rules:hastsmtypeExp}). The rules are syntax-directed, so the proof is by straightforward observations of syntactic contradictions.
\end{proof}

\begin{lemma}[ppTLM Unicity]
\label{lemma:ppTLM-unicity}
If $\hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho}$ and $\hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho'}$ then $\rho=\rho'$.
\end{lemma}
\begin{proof}
  By rule induction over Rules (\ref{rules:hastsmtypePat}). The rules are syntax-directed, so the proof is by straightforward observations of syntactic contradictions.
\end{proof}

\begin{theorem}[peTLM Preservation]
\label{thm:peTLM-preservation}
If $\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}$ and $\tsmexpStepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}$ then $\hastsmtypeExp{\Omega}{\Psi}{\epsilon'}{\rho}$.
\end{theorem}
\begin{proof}
By rule induction over Rules (\ref{rules:tsmexpStepsExp}).

\begin{byCases}
\item[\text{(\ref{rule:tsmexpStepsExp-apmod-1})}] By rule induction over Rules (\ref{rules:hastsmtypeExp}). There is only one rule that applies.
  \begin{byCases}
  \item[\text{(\ref{rule:hastsmtypeExp-apmod})}] ~
    \begin{pfsteps*}
      \item $\epsilon=\aeapmod{X}{\epsilon''}$ \BY{assumption}
      \item $\epsilon'=\aeapmod{X}{\epsilon'''}$ \BY{assumption}
      \item $\rho=[X/X']\rho'$ \BY{assumption}
      \item $\tsmexpStepsExp{\Omega}{\Psi}{\epsilon''}{\epsilon'''}$ \BY{assumption} \pflabel{steps}
      \item $\hastsmtypeExp{\Omega}{\Psi}{\epsilon''}{\aeallmods{\sigma}{X'}{\rho'}}$ \BY{assumption} \pflabel{tsmtype}
      \item $\hassig{\Omega}{X}{\sigma}$ \BY{assumption} \pflabel{hassig}
      \item $\hastsmtypeExp{\Omega}{\Psi}{\epsilon'''}{\aeallmods{\sigma}{X'}{\rho'}}$ \BY{IH on \pfref{tsmtype} and \pfref{steps}}\pflabel{IH}
      \item $\hastsmtypeExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon'''}}{[X/X']\rho}$ \BY{Rule (\ref{rule:hastsmtypeExp-apmod}) on \pfref{IH} and \pfref{hassig}}
    \end{pfsteps*}
    \resetpfcounter
  \end{byCases}
%   \begin{equation}\label{rule:tsmexpStepsExp-apmod-2}
% \inferrule{ }{
%   \tsmexpStepsExp{\Omega}{\Psi}{\aeapmod{X}{\aeabsmod{\sigma}{X'}{\epsilon}}}{[X/X']\epsilon}
% }
% \end{equation}
\item[\text{(\ref{rule:tsmexpStepsExp-apmod-2})}] By rule induction over Rules (\ref{rules:hastsmtypeExp}). There is only one rule that applies.
  \begin{byCases}
    \item[\text{(\ref{rule:hastsmtypeExp-apmod})}] ~
      \begin{pfsteps*}
        \item $\epsilon=\aeapmod{X}{\aeabsmod{\sigma}{X'}{\epsilon''}}$ \BY{assumption}
        \item $\epsilon'=[X/X']\epsilon''$ \BY{assumption}
        \item $\rho=[X/X']\rho'$ \BY{assumption}
        \item $\hastsmtypeExp{\Omega}{\Psi}{\aeabsmod{\sigma}{X'}{\epsilon''}}{\aeallmods{\sigma}{X'}{\rho'}}$ \BY{assumption} \pflabel{tsmtype}
        \item $\hassig{\Omega}{X}{\sigma}$ \BY{assumption} \pflabel{hassig}
        \item $\hastsmtypeExp{\Omega, X' : \sigma}{\Psi}{\epsilon''}{\rho'}$ \BY{Inversion Lemma for Rule (\ref{rule:hastsmtypeExp-absmod}) on \pfref{tsmtype}} \pflabel{tsmtype2}
        \item $\hastsmtypeExp{\Omega}{\Psi}{[X/X']\epsilon''}{[X/X']\rho'}$ \BY{Substitution Lemma \ref{lemma:substitution-P} on \pfref{tsmtype2}}
      \end{pfsteps*}
      \resetpfcounter
  \end{byCases}
\end{byCases}
\end{proof}

\begin{corollary}[peTLM Preservation (Multistep)]
\label{thm:peTLM-preservation-multistep}
If $\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}$ and $\tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}$ then $\hastsmtypeExp{\Omega}{\Psi}{\epsilon'}{\rho}$.
\end{corollary}
\begin{proof} The multistep relation is the reflexive, transitive closure of the single step relation, so the proof follows by applying Theorem \ref{thm:peTLM-preservation} over each step. \end{proof}

\begin{corollary}[peTLM Preservation (Evaluation)]
\label{thm:peTLM-preservation-evaluation}
If $\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}$ and $\tsmexpEvalsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}$ then $\hastsmtypeExp{\Omega}{\Psi}{\epsilon'}{\rho}$.
\end{corollary}
\begin{proof} The evaluation relation is the multistep relation with an additional requirement, so the proof follows directly from Corollary \ref{thm:peTLM-preservation-multistep}. \end{proof}

\begin{theorem}[ppTLM Preservation]
\label{thm:ppTLM-preservation}
If $\hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho}$ and $\tsmexpStepsPat{\Omega}{\Phi}{\epsilon}{\epsilon'}$ then $\hastsmtypePat{\Omega}{\Phi}{\epsilon'}{\rho}$.
\end{theorem}
\begin{proof} The proof is nearly identical to the proof of Theorem \ref{thm:peTLM-preservation}, differing only in that ppTLM contexts and the corresponding judgements are mentioned. \end{proof}

\begin{corollary}[ppTLM Preservation (Multistep)]
\label{thm:ppTLM-preservation-multistep}
If $\hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho}$ and $\tsmexpMultistepsPat{\Omega}{\Phi}{\epsilon}{\epsilon'}$ then $\hastsmtypePat{\Omega}{\Phi}{\epsilon'}{\rho}$.
\end{corollary}
\begin{proof} The multistep relation is the reflexive, transitive closure of the single step relation, so the proof follows by applying Theorem \ref{thm:ppTLM-preservation} over each step. \end{proof}

\begin{corollary}[ppTLM Preservation (Evaluation)]
\label{thm:ppTLM-preservation-evaluation}
If $\hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho}$ and $\tsmexpEvalsPat{\Omega}{\Phi}{\epsilon}{\epsilon'}$ then $\hastsmtypePat{\Omega}{\Phi}{\epsilon'}{\rho}$.
\end{corollary}
\begin{proof} The evaluation relation is the multistep relation with an additional requirement, so the proof follows directly from Corollary \ref{thm:ppTLM-preservation-multistep}. \end{proof}

\begin{theorem}[peTLM Progress]
\label{thm:peTLM-progress}
If $\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}$ then either $\tsmexpStepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}$ for some $\epsilon'$ or $\tsmexpNormalExp{\Omega}{\Psi}{\epsilon}$.
\end{theorem}
\begin{proof}
By rule induction over Rules (\ref{rules:hastsmtypeExp}).
\begin{byCases}
  \item[\text{(\ref{rule:hastsmtypeExp-defref})}] ~
    \begin{pfsteps*}
%     \begin{equation}\label{rule:hastsmtypeExp-defref}
% \inferrule{ }{
%   \hastsmtypeExp{\Omega}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\adefref{a}}{\rho}
% }
% \end{equation}
      \item $\epsilon=\adefref{a}$ \BY{assumption}
      \item $\Psi=\Psi', \petsmdefn{a}{\rho}{\eparse}$ \BY{assumption}
      \item $\tsmexpNormalExp{\Omega}{\Psi', \petsmdefn{a}{\rho}{\eparse}}{\adefref{a}}$ \BY{Rule (\ref{rule:tsmexpNormalExp-defref})}
    \end{pfsteps*}
    \resetpfcounter
  \item[\text{(\ref{rule:hastsmtypeExp-absmod})}] ~
    \begin{pfsteps*}
%       \begin{equation}\label{rule:hastsmtypeExp-absmod}
% \inferrule{
%   \issigX{\sigma}\\
%   \hastsmtypeExp{\Omega, X : \sigma}{\Psi}{\epsilon}{\rho}
% }{
%   \hastsmtypeExp{\Omega}{\Psi}{\aeabsmod{\sigma}{X}{\epsilon}}{\aeallmods{\sigma}{X}{\rho}}
% }
% \end{equation}
      \item $\epsilon=\aeabsmod{\sigma}{X}{\epsilon'}$ \BY{assumption}
      \item $\tsmexpNormalExp{\Omega}{\Psi}{\aeabsmod{\sigma}{X}{\epsilon'}}$ \BY{Rule (\ref{rule:tsmexpNormalExp-absmod})}
    \end{pfsteps*}
    \resetpfcounter
  \item[\text{(\ref{rule:hastsmtypeExp-apmod})}] ~
    \begin{pfsteps*}
%      \begin{equation}\label{rule:hastsmtypeExp-apmod}
% \inferrule{
%   \hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\aeallmods{\sigma}{X'}{\rho}}\\
%   \hassig{\Omega}{X}{\sigma}
% }{
%   \hastsmtypeExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon}}{[X/X']\rho}
% }
% \end{equation} 
      \item $\epsilon=\aeapmod{X}{\epsilon'}$ \BY{assumption}
      \item $\rho=[X/X']\rho'$ \BY{assumption}
      \item $\hastsmtypeExp{\Omega}{\Psi}{\epsilon'}{\aeallmods{\sigma}{X'}{\rho'}}$ \BY{assumption} \pflabel{tsmtype}
      \item $\tsmexpStepsExp{\Omega}{\Psi}{\epsilon'}{\epsilon''}$ for some $\epsilon''$ or $\tsmexpNormalExp{\Omega}{\Psi}{\epsilon'}$ \BY{IH on \pfref{tsmtype}} \pflabel{ih}
    \end{pfsteps*}
    Proceed by cases on \pfref{ih}.
    \begin{byCases}
      \item[\tsmexpStepsExp{\Omega}{\Psi}{\epsilon'}{\epsilon''}] 
        \begin{pfsteps*}
          \item $\tsmexpStepsExp{\Omega}{\Psi}{\epsilon'}{\epsilon''}$ \BY{assumption} \pflabel{steps}
          \item $\tsmexpStepsExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon'}}{\aeapmod{X}{\epsilon''}}$ \BY{Rule (\ref{rule:tsmexpStepsExp-apmod-1}) on \pfref{steps}}
        \end{pfsteps*}
      \item[\tsmexpNormalExp{\Omega}{\Psi}{\epsilon'}] Proceed by rule induction over Rules (\ref{rules:tsmexpNormalExp}).
        \begin{byCases}
          \item[\text{(\ref{rule:tsmexpNormalExp-defref})}] ~
            \begin{pfsteps*}
              \item $\epsilon' = \adefref{a}$ \BY{assumption}
              \item $\tsmexpNormalExp{\Omega}{\Psi}{\epsilon'}$ \BY{assumption} \pflabel{normal}
              \item $\tsmexpNormalExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon'}}$ \BY{Rule (\ref{rule:tsmexpNormalExp-apmod}) on \pfref{normal}}
            \end{pfsteps*}
          \item[\text{(\ref{rule:tsmexpNormalExp-absmod})}] ~
            \begin{pfsteps*}
              \item $\epsilon' = \aeabsmod{\sigma}{X}{\epsilon''}$ \BY{assumption}
              \item $\tsmexpStepsExp{\Omega}{\Psi}{\aeapmod{X}{\aeabsmod{\sigma}{X'}{\epsilon''}}}{[X/X']\epsilon''}$ \BY{Rule (\ref{rule:tsmexpStepsExp-apmod-2})}
            \end{pfsteps*}
          \item[\text{(\ref{rule:tsmexpNormalExp-apmod})}] ~
            \begin{pfsteps*}
              \item $\epsilon' = \aeapmod{X''}{\epsilon''}$ \BY{assumption}
              \item $\tsmexpNormalExp{\Omega}{\Psi}{\epsilon'}$ \BY{assumption} \pflabel{normal2}              
              \item $\tsmexpNormalExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon'}}$ \BY{Rule (\ref{rule:tsmexpNormalExp-apmod}) on \pfref{normal}}
            \end{pfsteps*}
        \end{byCases}
    \end{byCases}
    \resetpfcounter
\end{byCases}
\end{proof}

\begin{theorem}[ppTLM Progress]
\label{thm:ppTLM-progress}
If $\hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho}$ then either $\tsmexpStepsPat{\Omega}{\Phi}{\epsilon}{\epsilon'}$ for some $\epsilon'$ or $\tsmexpNormalPat{\Omega}{\Phi}{\epsilon}$.
\end{theorem}
\begin{proof} The proof is nearly identical to the proof of Theorem \ref{thm:peTLM-progress}, differing only in that ppTLM contexts and the corresponding judgements are mentioned. \end{proof}

\subsection{Typed Expansion}
\subsubsection{Kinds, Constructions and Signatures}
\begin{theorem}[Kind and Construction Expansion]
\label{thm:kind-and-constructor-expansion-P}
~
\begin{enumerate}
\item If $\kExpands{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\ukappa}{\kappa}$ then $\iskind{\Omega}{\kappa}$.
\item If $\cExpands{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\uc}{c}{\kappa}$ then $\haskind{\Omega}{c}{\kappa}$.
\end{enumerate}
\end{theorem}
\begin{proof}
By mutual rule induction over Rules (\ref{rules:kExpands}) and Rules (\ref{rules:cExpands}). In each case, we apply the IH to each premise and then apply the corresponding kind formation rule from Rules (\ref{rules:iskind}) or kinding rule from Rules (\ref{rules:haskind}).
\end{proof}

\begin{theorem}[Signature Expansion]
\label{thm:signature-expansion-P}
If $\sigExpandsP{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\usigma}{\sigma}$ then $\issig{\Omega}{\sigma}$.
\end{theorem}
\begin{proof} By rule induction over Rule (\ref{rule:sigExpandsP}). Apply Theorem \ref{thm:kind-and-constructor-expansion-P} to each premise, then apply Rule (\ref{rule:issig}). \end{proof}


\subsubsection{TLM Types and Expressions}
\begin{theorem}[TLM Type Expansion]
\label{thm:tsm-type-expansion-P}
If $\tsmtyExpands{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\urho}{\rho}$ then $\istsmty{\Omega}{\rho}$.
\end{theorem}
\begin{proof} By rule induction over Rules (\ref{rules:tsmtyExpands}).
\begin{byCases}
  \item[\text{(\ref{rule:tsmtyExpands-type})}] ~
    \begin{pfsteps*}
      \item $\urho = \utau$ \BY{assumption}
      \item $\rho = \aetype{\tau}$ \BY{assumption}
      \item $\cExpandsX{\utau}{\tau}{\akty}$ \BY{assumption} \pflabel{cexpands}
      \item $\haskindX{\tau}{\akty}$ \BY{Theorem \ref{thm:kind-and-constructor-expansion-P} on \pfref{cexpands}} \pflabel{haskind}
      \item $\istsmty{\Omega}{\aetype{\tau}}$ \BY{Rule (\ref{rule:istsmty-type}) on \pfref{haskind}}
    \end{pfsteps*}
    \resetpfcounter
  \item[\text{(\ref{rule:tsmtyExpands-allmods})}] ~
    \begin{pfsteps*}
      \item $\urho=\allmods{\uX}{\usigma}{\urho'}$ \BY{assumption}
      \item $\rho=\aeallmods{\sigma}{X}{\rho'}$ \BY{assumption}
      \item $\sigExpandsPX{\usigma}{\sigma}$ \BY{assumption} \pflabel{sigexpands}
      \item $\tsmtyExpands{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\urho'}{\rho'}$ \BY{assumption} \pflabel{tsmtyexpands}
      \item $\issig{\Omega}{\sigma}$ \BY{Theorem \ref{thm:signature-expansion-P} on \pfref{sigexpands}} \pflabel{issig}
      \item $\istsmty{\Omega, X : \sigma}{\rho'}$ \BY{IH on \pfref{tsmtyexpands}} \pflabel{istsmty} 
      \item $\istsmty{\Omega}{\aeallmods{\sigma}{X}{\rho'}}$ \BY{Rule (\ref{rule:istsmty-allmods}) on \pfref{issig} and \pfref{istsmty}}
    \end{pfsteps*}
    \resetpfcounter
\end{byCases}

\end{proof}

\begin{theorem}[peTLM Expression Expansion]
\label{thm:peTLM-expression-expansion}
If $\tsmexpExpandsExp{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\uAS{\uA}{\Psi}}{\uepsilon}{\epsilon}{\rho}$ then $\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}$.
\end{theorem}
\begin{proof}
By rule induction over Rules (\ref{rules:tsmexpExpandsExp}). In the following, let $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}$ and $\uPsi=\uAS{\uA}{\Psi}$.
\begin{byCases}
  \item[\text{(\ref{rule:tsmexpExpandsExp-bindref})}] ~
    \begin{pfsteps*}
      \item $\uepsilon = \tsmv$ \BY{assumption}
      \item $\uA = \uA', \mapitem{\tsmv}{\epsilon}$ \BY{assumption}
      \item $\hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}$ \BY{assumption} 
    \end{pfsteps*}
    \resetpfcounter
  \item[\text{(\ref{rule:tsmexpExpandsExp-absmod})}] ~
    \begin{pfsteps*}
      \item $\uepsilon=\absmod{\uX}{\usigma}{\uepsilon'}$ \BY{assumption}
      \item $\epsilon=\aeabsmod{\sigma}{X}{\epsilon'}$ \BY{assumption}
      \item $\rho=\aeallmods{\sigma}{X}{\rho'}$ \BY{assumption}
      \item $\sigExpandsPX{\usigma}{\sigma}$ \BY{assumption} \pflabel{sigexpands}
      \item $\tsmexpExpandsExp{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\uPsi}{\uepsilon'}{\epsilon'}{\rho'}$ \BY{assumption} \pflabel{tsmexpExpands}
      \item $\hastsmtypeExp{\Omega, X : \sigma}{\Psi}{\epsilon'}{\rho'}$ \BY{IH on \pfref{sigexpands} and \pfref{tsmexpExpands}} \pflabel{hastsmtype}
      \item $\issigX{\sigma}$ \BY{Theorem \ref{thm:signature-expansion-P} on \pfref{sigexpands}} \pflabel{issig}
      \item $\hastsmtypeExp{\Omega}{\Psi}{\aeabsmod{\sigma}{X}{\epsilon'}}{\aeallmods{\sigma}{X}{\rho'}}$ \BY{Rule (\ref{rule:hastsmtypeExp-absmod}) on \pfref{issig} and \pfref{hastsmtype}}
    \end{pfsteps*}
    \resetpfcounter
  \item[\text{(\ref{rule:tsmexpExpandsExp-apmod})}] ~
    \begin{pfsteps*}
      \item $\uepsilon=\apmod{\uepsilon'}{\uX}$ \BY{assumption}
      \item $\epsilon=\aeapmod{X}{\epsilon'}$ \BY{assumption}
      \item $\rho=[X/X']\rho'$ \BY{assumption}
      \item $\tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon'}{\epsilon'}{\aeallmods{\sigma}{X'}{\rho'}}$ \BY{assumption} \pflabel{tsmexpExpands}
      \item $\mExpandsPX{\uX}{X}{\sigma}$ \BY{assumption} \pflabel{mExpands}
      \item $\hastsmtypeExp{\Omega}{\Psi}{\epsilon'}{\aeallmods{\sigma}{X'}{\rho'}}$ \BY{IH on \pfref{tsmexpExpands}} \pflabel{hastsmtype}
      \item $\hassig{\Omega}{X}{\sigma}$ \BY{Theorem \ref{thm:module-expansion-P} on \pfref{mExpands}} \pflabel{hassig}
      \item $\hastsmtypeExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon'}}{[X/X']\rho'}$ \BY{Rule (\ref{rule:hastsmtypeExp-apmod}) on \pfref{hastsmtype} and \pfref{hassig}}
    \end{pfsteps*}
    \resetpfcounter
\end{byCases}

\end{proof}

\begin{theorem}[ppTLM Expression Expansion]
\label{thm:ppTLM-expression-expansion}
If $\tsmexpExpandsPat{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\uAS{\uA}{\Phi}}{\uepsilon}{\epsilon}{\rho}$ then $\hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho}$.
\end{theorem}
\begin{proof} The proof is nearly identical to the proof of Theorem \ref{thm:peTLM-expression-expansion}, differing only in that ppTLM contexts and the corresponding judgements are mentioned. \end{proof}

\subsubsection{Patterns}
\begin{lemma}[Proto-Pattern Deparameterization]
\label{lemma:pattern-deparameterization-P}
If $\prepcp{\Omega_\text{app}}{\Phi}{\pcp}{\cpv}{\epsilon}{\rho}{\omega}{\Omega_\text{params}}$ then $\domof{\Omega_\text{app}} \cap \domof{\Omega_\text{params}} = \emptyset$ and $\hastypeP{\Omega_\text{app}}{\omega}{\Omega_\text{params}}$ and $\hastsmtypePat{\Omega_\text{app}}{\Phi}{\epsilon}{[\omega]\rho}$. 
\end{lemma}
\begin{proof} By rule induction over Rules (\ref{rules:prepcp}).
\begin{byCases}
  \item[\text{(\ref{rule:prepcp-cepat})}] We have:
    \begin{pfsteps*}
      \item $\epsilon=\adefref{a}$ \BY{assumption}
      \item $\omega=\emptyset$ \BY{assumption}
      \item $\Phi=\Phi', \pptsmdefn{a}{\rho}{\eparse}$ \BY{assumption}
      \item $\Omega_\text{params}=\emptyset$ \BY{assumption}
      \item $\istsmty{\Omega_\text{app}}{\rho}$ \BY{assumption} \pflabel{istsmty}
      \item $\domof{\Omega_\text{app}} \cap \domof{\emptyset} = \emptyset$ \BY{definition}
      \item $\hastypeP{\Omega_\text{app}}{\emptyset}{\emptyset}$ \BY{definition}
      \item $[\emptyset]\rho=\rho$ \BY{definition}
      \item $\hastsmtypePat{\Omega_\text{app}}{\Phi', \pptsmdefn{a}{\rho}{\eparse}}{a}{\rho}$ \BY{Rule (\ref{rule:hastsmtypePat-defref}) on \pfref{istsmty}}
    \end{pfsteps*}
    \resetpfcounter
  \item[\text{(\ref{rule:prepcp-allmods})}] We have:
    \begin{pfsteps*}
      \item $\epsilon=\aeapmod{X}{\epsilon'}$ \BY{assumption}
      \item $\prepcp{\Omega_\text{app}}{\Phi}{\pcp}{\cpv}{\epsilon'}{\aeallmods{\sigma}{X'}{\rho}}{\omega'}{\Omega'}$ \BY{assumption} \pflabel{prepcp}
      \item $\hassig{\Omega_\text{app}}{X}{\sigma}$ \BY{assumption} \pflabel{hassig}
      \item $X' \notin \domof{\Omega_\text{app}}$ \BY{assumption} \pflabel{notin}
      \item $\omega=\omega', X/X'$ \BY{assumption}
      \item $\Omega_\text{params} = \Omega', X' : \sigma$ \BY{assumption}
      \item $\domof{\Omega_\text{app}} \cap \domof{\Omega'} = \emptyset$ \BY{IH on \pfref{prepcp}} \pflabel{IH1}
      \item $\hastypeP{\Omega_\text{app}}{\omega'}{\Omega'}$ \BY{IH on \pfref{prepcp}} \pflabel{IH2}
      \item $\hastsmtypePat{\Omega_\text{app}}{\Phi}{\epsilon'}{[\omega']\aeallmods{\sigma}{X'}{\rho}}$ \BY{IH on \pfref{prepcp}} \pflabel{IH3}
      \item $\domof{\Omega_\text{app}} \cap \domof{\Omega', X' : \sigma}$ \BY{\pfref{notin} and \pfref{IH1} and definition of finite set intersection}
      \item $\hastypeP{\Omega_\text{app}}{\omega', X/X'}{\Omega', X' : \sigma}$ \BY{Definition \ref{def:substitution-p} on \pfref{hassig}}
      \item $\hastsmtypePat{\Omega_\text{app}}{\Phi}{\aeapmod{X}{\epsilon'}}{(\omega', X/X')\rho}$ \BY{Rule (\ref{rule:hastsmtypePat-apmod}) on \pfref{IH3} and \pfref{hassig}}
    \end{pfsteps*}
    \resetpfcounter
\end{byCases}
\end{proof}

\begin{theorem}[Typed Pattern Expansion]\label{thm:typed-pattern-expansion-P} ~
\begin{enumerate}
  \item If $\pExpandsSP{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}}{\uPhi}{\upv}{p}{\tau}{\uOmegaEx{\uD'}{\uG'}{\uMctx'}{\Omega'}}$ then $\uMctx' = \emptyset$ and $\uD' = \emptyset$ and $\patTypePC{\Omega_\text{app}}{\Omega'}{p}{\tau}$.
  \item If $\cvalidPP{\uOmegaEx{\uD'}{\uG'}{\uMctx'}{\Omega'}}{\psceneP{\omega : \Omega_\text{params}}{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}}{\uPhi}{b}}{\cpv}{p}{\tau}$ and $\domof{\Omega_\text{params}} \cap \domof{\Omega_\text{app}} = \emptyset$ then $\uMctx' = \emptyset$ and $\uD' = \emptyset$ and $\patTypePC{\Omega_\text{params} \cup \Omega_\text{app}}{\Omega'}{p}{\tau}$.
\end{enumerate}
\end{theorem}
\begin{proof} My mutual rule induction over Rules (\ref{rules:patExpandsP}) and Rules (\ref{rules:cvalidPP}).
\begin{enumerate}
\item In the following, let $\uOmega = \uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}$ and $\uOmega' = \uOmegaEx{\uD'}{\uG'}{\uMctx'}{\Omega'}$.
  \begin{byCases}
    \item[\text{(\ref{rule:patExpandsP-subsume}) \textbf{through} (\ref{rule:patExpandsP-in})}] These cases follow by applying the IH, part 1 and applying the corresponding pattern typing rule in Rules (\ref{rules:patTypeP}).

    \item[\text{(\ref{rule:patExpandsP-apuptsm})}] We have:
    \begin{pfsteps*}
    \item $\upv=\utsmap{\uepsilon}{b}$ \BY{assumption}
    \item $\uPhi=\uAS{\uA}{\Phi}$ \BY{assumption}
    \item $\tsmexpExpandsPat{\uOmega}{\uPhi}{\uepsilon}{\epsilon}{\aetype{\tau_\text{final}}}$ \BY{assumption}
    \item $\tsmexpEvalsPat{\Omega_\text{app}}{\Phi}{\epsilon}{\epsilon_\text{normal}}$ \BY{assumption}
    \item $\tsmdefof{\epsilon_\text{normal}}=a$ \BY{assumption}
    \item $\Phi = \Phi', \pptsmdefn{a}{\rho}{\eparse}$ \BY{assumption}
    \item $\encodeBody{b}{\ebody}$ \BY{assumption}
    \item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessP}}{\ecand}}$ \BY{assumption}
    \item $\decodePCEPat{\ecand}{\pcp}$ \BY{assumption}
    \item $\prepcp{\Omega_\text{app}}{\Phi}{\pcp}{\cpv}{\epsilon_\text{normal}}{\aetype{\tau_\text{proto}}}{\omega}{\Omega_\text{params}}$ \BY{assumption} \pflabel{prepcp}
    \item $\cvalidPP{\uOmega'}{\psceneP{\omega : \Omega_\text{params}}{\uOmega}{\uPhi}{b}}{\cpv}{p}{\tau_\text{proto}}$ \BY{assumption} \pflabel{cvalidPP}
    \item $\tau = [\omega]\tau_\text{proto}$ \BY{assumption}
    \item $\domof{\Omega_\text{params}} \cap \domof{\Omega_\text{app}} = \emptyset$ \BY{Lemma \ref{lemma:pattern-deparameterization-P} on \pfref{prepcp}} \pflabel{disjoint}
    \item $\hastypeP{\Omega_\text{app}}{\omega}{\Omega_\text{params}}$ \BY{Lemma \ref{lemma:pattern-deparameterization-P} on \pfref{prepcp}} \pflabel{hastypeP}
    \item $\uMctx' = \emptyset$ and $\uD' = \emptyset$ \BY{IH, part 2 on \pfref{cvalidPP} and \pfref{disjoint}}
    \item $\patTypePC{\Omega_\text{params} \cup \Omega_\text{app}}{\Omega'}{p}{\tau_\text{proto}}$ \BY{IH, part 2 on \pfref{cvalidPP} and \pfref{disjoint}} \pflabel{patType}
    \item $\patTypePC{\Omega_\text{app}}{\Omega'}{p}{[\omega]\tau_\text{proto}}$ \BY{Substitution Lemma \ref{lemma:substitution-P} on \pfref{hastypeP} and \pfref{patType}}
    \end{pfsteps*}
    \resetpfcounter
    % \item[\text{(\ref{rule:patExpandsP-subsume})}] We have:
    %   \begin{pfsteps*}
    %   \item $\patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{\tau'}$ \BY{assumption} \pflabel{patExpandsP}
    %   \item $\issubtypeP{\Omega}{\tau'}{\tau}$ \BY{assumption} \pflabel{issubtypeP}
    %   \item $\uMctx' = \emptyset$  \BY{IH, part 1 on \pfref{patExpandsP}}
    %   \item $\uD' = \emptyset$   \BY{IH, part 1 on \pfref{patExpandsP}}
    %   \item $\patTypeP{\Omega'}{p}{\tau'}$  \BY{IH, part 1 on \pfref{patExpandsP}}
    %   \item $\patTypeP{\Omega'}{p}{\tau}$ \BY{Rule (\ref{rule:patTypeP-subsume}) on \pfref{patExpandsP} and \pfref{issubtypeP}}
    %   \end{pfsteps*}
    %   \resetpfcounter
    % \item[\text{(\ref{rule:patExpandsP-var})}] We have:
    %   \begin{pfsteps*}
    %   \item $\upv=x$ \BY{assumption}
    %   \item $p=x$ \BY{assumption}
    %   \item $\uMctx' = \emptyset$ \BY{assumption}
    %   \item $\uD' = \emptyset$ \BY{assumption}
    %   \item $\Omega' = x : \tau$ \BY{assumption}
    %   \item $\patTypeP{\Omega'}{x}{\tau}$ \BY{Rule (\ref{rule:patTypeP-var})}
    %   \end{pfsteps*}
    %    \resetpfcounter
    % \item[\text{(\ref{rule:patExpandsP-wild})}] We have:
    %   \begin{pfsteps*}
    %   \item $\upv=\wildp$ \BY{assumption}
    %   \item $p = \aewildp$ \BY{assumption}
    %   \item $\uMctx' = \emptyset$ \BY{assumption}
    %   \item $\uD' = \emptyset$ \BY{assumption}
    %   \item $\Omega' = \emptyset$ \BY{assumption}
    %   \item $\patTypeP{\Omega'}{\aewildp}{\tau}$ \BY{Rule (\ref{rule:patTypeP-wild})}
    %   \end{pfsteps*}
    %   \resetpfcounter
    % \item[\text{(\ref{rule:patExpandsP-fold})}] We have:
    %   \begin{pfsteps*}
    %   \item $\upv = \foldp{\upv'}$ \BY{assumption}
    %   \item $p = \aefoldp{p'}$ \BY{assumption}
    %   \item $\tau=\arec{t}{\tau'}$ \BY{assumption}
    %   \item $\patExpandsP{\uOmega'}{\uPhi}{\upv'}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{assumption}\pflabel{patExpandsP}
    %   \item $\uMctx' = \emptyset$ \BY{IH, part 1 on \pfref{patExpandsP}}
    %   \item $\uD' = \emptyset$ \BY{IH, part 1 on \pfref{patExpandsP}}
    %   \item $\patTypeP{\Omega'}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{IH, part 1 on \pfref{patExpandsP}} \pflabel{patTypeP}
    %   \item $\patTypeP{\Omega'}{\aefoldp{p'}}{\arec{t}{\tau'}}$ \BY{Rule (\ref{rule:patTypeP-fold}) on \pfref{patTypeP}}
    %   \end{pfsteps*}
    %   \resetpfcounter
  \end{byCases}
\item We induct on the premise. In the following, let $\uOmega = \uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}$ and $\uOmega' = \uOmegaEx{\uD'}{\uG'}{\uMctx'}{\Omega'}$.
  \begin{byCases}
    \item[\text{(\ref{rule:cvalidPP-wild}) \textbf{through} (\ref{rule:cvalidPP-in})}] These cases follow by applying the IH, part 2 and then applying the corresponding pattern rule in Rules (\ref{rules:patTypeP}).
    \item[\text{(\ref{rule:cvalidPP-spliced})}] ~
      \begin{pfsteps*}
        \item $\cpv=\acesplicedp{m}{n}{\ctau}$ \BY{assumption}
        \item $\tau=[\omega]\tau'$ \BY{assumption}
        \item $\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau}{\tau'}{\akty}$ \BY{assumption} \pflabel{cvalidC}
        \item $\parseUPat{\bsubseq{b}{m}{n}}{\upv}$ \BY{assumption} \pflabel{parseUPat}
        \item $\patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{[\omega]\tau'}$ \BY{assumption} \pflabel{patExpands}
        \item  $\uMctx' = \emptyset$ and $\uD' = \emptyset$ \BY{IH, part 1 on \pfref{patExpands}}
        \item $\patTypePC{\Omega_\text{app}}{\Omega'}{p}{[\omega]\tau'}$ \BY{IH, part 1 on \pfref{patExpands}} \pflabel{pattype}
        \item $\patTypePC{\OParams \cup \Omega_\text{app}}{\Omega'}{p}{[\omega]\tau'}$ \BY{Weakening on \pfref{pattype}}
      \end{pfsteps*}
      \resetpfcounter
    \end{byCases}
\end{enumerate}
The mutual induction can be shown to be well-founded by an argument analagous to that in the proof of Theorem \ref{thm:typed-pattern-expansion}, appealing to Condition  \ref{condition:pattern-parsing-P} and Condition \ref{condition:body-subsequences-P}.

% \begin{align*}
% \sizeof{\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}} & = \sizeof{\upv}\\
% \sizeof{{\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}}} & = \sizeof{b}
% \end{align*}
% where $\sizeof{b}$ is the length of $b$ and $\sizeof{\upv}$ is the sum of the lengths of the literal bodies in $\upv$,
% \begin{align*}
% \sizeof{\ux} & = 0\\
% \sizeof{\aufoldp{\upv}} & = \sizeof{\upv}\\
% \sizeof{\autplp{\labelset}{\mapschema{\upv}{i}{\labelset}}} & = \sum_{i \in \labelset} \sizeof{\upv_i}\\
% \sizeof{\auinjp{\ell}{\upv}} & = \sizeof{\upv}\\
% \sizeof{\auapuptsm{b}{\tsmv}} & = \sizeof{b}\\
% \sizeof{\auplit{b}} & = \sizeof{b}
% \end{align*}

% The only case in the proof of part 1 that invokes part 2 are Case (\ref{rule:patExpandsP-apuptsm}) and (\ref{rule:patExpandsP-lit}). There, we have that the metric remains stable: \begin{align*}
%  & \sizeof{\patExpands{\upctx}{\uPhi, \uShyp{\tsmv}{a}{\tau}{\eparse}}{\auapuptsm{b}{\tsmv}}{p}{\tau}}\\
% =& \sizeof{\patExpands{\upctx}{\uASI{\uA}{\Phi', \xuptsmbnd{a}{\tau}{\eparse}}{\uI', \designate{\tau}{a}}}{\auplit{b}}{p}{\tau}}\\
% =& \sizeof{{\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi, \uShyp{\tsmv}{a}{\tau}{\eparse}}{b}}{\cpv}{p}{\tau}}}\\
% =&\sizeof{b}\end{align*}

% The only case in the proof of part 2 that invokes part 1 is Case (\ref{rule:cvalidP-B-spliced}). There, we have that $\parseUPat{\bsubseq{b}{m}{n}}{\upv}$ and the IH is applied to the judgement $\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}$. Because the metric is stable when passing from part 1 to part 2, we must have that it is strictly decreasing in the other direction:
% \[\sizeof{\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}} < \sizeof{{\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\acesplicedp{m}{n}{\ctau}}{p}{\tau}}}\]
% i.e. by the definitions above, 
% \[\sizeof{\upv} < \sizeof{b}\]

% This is established by appeal to Condition \ref{condition:body-subsequences-P}, which states that subsequences of $b$ are no longer than $b$, and the following condition, which states that an unexpanded pattern constructed by parsing a textual sequence $b$ is strictly smaller, as measured by the metric defined above, than the length of $b$, because some characters must necessarily be used to delimit each literal body.
% % \begin{condition}[Pattern Parsing Monotonicity]\label{condition:pattern-parsing-B} If $\parseUPat{b}{\upv}$ then $\sizeof{\upv} < \sizeof{b}$.\end{condition}

% Combining Conditions \ref{condition:body-subsequences-P} and \ref{condition:pattern-parsing-P}, we have that $\sizeof{\ue} < \sizeof{b}$ as needed.
\end{proof}
\subsubsection{Expressions and Rules}
\begin{lemma}[Proto-Expression Deparameterization]
\label{lemma:expression-deparameterization-P}
If $\prepce{\Omega_\text{app}}{\Psi}{\pce}{\ce}{\epsilon}{\rho}{\omega}{\Omega_\text{params}}$ then $\domof{\Omega_\text{app}} \cap \domof{\Omega_\text{params}} = \emptyset$ and $\hastypeP{\Omega_\text{app}}{\omega}{\Omega_\text{params}}$ and $\hastsmtypeExp{\Omega_\text{app}}{\Psi}{\epsilon}{[\omega]\rho}$. 
\end{lemma}
\begin{proof} By rule induction over Rules (\ref{rules:prepce}).
\begin{byCases}
  \item[\text{(\ref{rule:prepce-ceexp})}] We have:
    \begin{pfsteps*}
      \item $\epsilon=\adefref{a}$ \BY{assumption}
      \item $\omega=\emptyset$ \BY{assumption}
      \item $\Psi=\Psi', \petsmdefn{a}{\rho}{\eparse}$ \BY{assumption}
      \item $\Omega_\text{params}=\emptyset$ \BY{assumption}
      \item $\istsmty{\Omega_\text{app}}{\rho}$ \BY{assumption} \pflabel{istsmty}
      \item $\domof{\Omega_\text{app}} \cap \domof{\emptyset} = \emptyset$ \BY{definition}
      \item $\hastypeP{\Omega_\text{app}}{\emptyset}{\emptyset}$ \BY{definition}
      \item $[\emptyset]\rho=\rho$ \BY{definition}
      \item $\hastsmtypeExp{\Omega_\text{app}}{\Psi', \petsmdefn{a}{\rho}{\eparse}}{a}{\rho}$ \BY{Rule (\ref{rule:hastsmtypeExp-defref}) on \pfref{istsmty}}
    \end{pfsteps*}
    \resetpfcounter
  \item[\text{(\ref{rule:prepce-allmods})}] We have:
    \begin{pfsteps*}
      \item $\epsilon=\aeapmod{X}{\epsilon'}$ \BY{assumption}
      \item $\prepce{\Omega_\text{app}}{\Psi}{\pce}{\ce}{\epsilon'}{\aeallmods{\sigma}{X'}{\rho}}{\omega'}{\Omega'}$ \BY{assumption} \pflabel{prepcp}
      \item $\hassig{\Omega_\text{app}}{X}{\sigma}$ \BY{assumption} \pflabel{hassig}
      \item $X' \notin \domof{\Omega_\text{app}}$ \BY{assumption} \pflabel{notin}
      \item $\omega=\omega', X/X'$ \BY{assumption}
      \item $\Omega_\text{params} = \Omega', X' : \sigma$ \BY{assumption}
      \item $\domof{\Omega_\text{app}} \cap \domof{\Omega'} = \emptyset$ \BY{IH on \pfref{prepcp}} \pflabel{IH1}
      \item $\hastypeP{\Omega_\text{app}}{\omega'}{\Omega'}$ \BY{IH on \pfref{prepcp}} \pflabel{IH2}
      \item $\hastsmtypeExp{\Omega_\text{app}}{\Psi}{\epsilon'}{[\omega']\aeallmods{\sigma}{X'}{\rho}}$ \BY{IH on \pfref{prepcp}} \pflabel{IH3}
      \item $\domof{\Omega_\text{app}} \cap \domof{\Omega', X' : \sigma}$ \BY{\pfref{notin} and \pfref{IH1} and definition of finite set intersection}
      \item $\hastypeP{\Omega_\text{app}}{\omega', X/X'}{\Omega', X' : \sigma}$ \BY{Definition \ref{def:substitution-p} on \pfref{hassig}}
      \item $\hastsmtypeExp{\Omega_\text{app}}{\Psi}{\aeapmod{X}{\epsilon'}}{(\omega', X/X')\rho}$ \BY{Rule (\ref{rule:hastsmtypeExp-apmod}) on \pfref{IH3} and \pfref{hassig}}
    \end{pfsteps*}
    \resetpfcounter
\end{byCases}
\end{proof}

\begin{theorem}[Typed Expression and Rule Expansion]
\label{thm:typed-expression-expansion-P}
~
\begin{enumerate}
\item \begin{enumerate}
  \item If $\expandsP{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\uPsi}{\uPhi}{\ue}{e}{\tau}$ then $\hastypeP{\Omega}{e}{\tau}$.
  \item If $\rExpandsSP{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\uPsi}{\uPhi}{\urv}{r}{\tau}{\tau'}$ then $\ruleTypeP{\Omega}{r}{\tau}{\tau'}$.
  \end{enumerate}
\item \begin{enumerate}
  \item If $\cvalidEP{\Omega}{\esceneP{\omega : \OParams}{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}$ and $\domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset$ then $\hastypeP{\Omega \cup \Omega_\text{app}}{e}{\tau}$.
  \item If $\cvalidRP{\Omega}{\esceneP{\omega : \OParams}{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}}{\uPsi}{\uPhi}{b}}{\crv}{r}{\tau}{\tau'}$ and $\domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset$ then $\ruleTypeP{\Omega \cup \Omega_\text{app}}{r}{\tau}{\tau'}$.
  \end{enumerate}
\end{enumerate}
\end{theorem}
\begin{proof} 
By mutual rule induction over Rules (\ref{rules:expandsP}), Rule (\ref{rule:rExpandsP}), Rules (\ref{rules:cvalidE-P}) and Rule (\ref{rule:cvalidR-P}).
\begin{enumerate}
  \item \begin{enumerate}
    \item In the following, let $\uOmega = \uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}$. 
    \begin{byCases}
      \item[\text{(\ref{rule:expandsP-subsume})}] ~
        \begin{pfsteps*}
          \item $\expandsPX{\ue}{e}{\tau'}$ \BY{assumption} \pflabel{expandsP}
          \item $\issubtypePX{\tau}{\tau'}$ \BY{assumption} \pflabel{issubtype}
          \item $\hastypeP{\Omega}{e}{\tau'}$ \BY{IH, part 1(a) on \pfref{expandsP}} \pflabel{hastype}
          \item $\hastypeP{\Omega}{e}{\tau}$ \BY{Rule (\ref{rule:hastypeP-subsume}) on \pfref{hastype} and \pfref{issubtype}}
        \end{pfsteps*}
        \resetpfcounter
      \item[\text{(\ref{rule:expandsP-var}) \textbf{through} (\ref{rule:expandsP-mval})}] In each of these cases, we apply the IH, part 1(a) or 1(b), over the premises and then apply the corresponding typing rule in Rules (\ref{rules:hastypeP}) and weakening as needed.
      \item[\text{(\ref{rule:expandsP-apuetsm})}] ~
        \begin{pfsteps*}
          \item $\ue=\utsmap{\uepsilon}{b}$ \BY{assumption}
          \item $e=[\omega]e'$ \BY{assumption}
          \item $\tau=[\omega]\tau_\text{proto}$ \BY{assumption}
          \item $\uPsi=\uAS{\uA}{\Psi}$ \BY{assumption}
          \item $\tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\aetype{\tau_\text{final}}}$ \BY{assumption} \pflabel{tsmexpExpands}
          \item $\tsmexpEvalsExp{\Omega}{\Psi}{\epsilon}{\epsilon_\text{normal}}$ \BY{assumption} \pflabel{tsmexpEvals}
          \item $\tsmdefof{\epsilon_\text{normal}}=a$ \BY{assumption} \pflabel{tsmdefof}
          \item $\Psi = \Psi', \petsmdefn{a}{\rho}{\eparse}$ \BY{assumption} 
          \item $\encodeBody{b}{\ebody}$ \BY{assumption}
          \item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessE}}{e_\text{pproto}}}$ \BY{assumption}
          \item $\decodePCEExp{e_\text{pproto}}{\pce}$ \BY{assumption}
          \item $\prepce{\Omega}{\Psi}{\pce}{\ce}{\epsilon_\text{normal}}{\aetype{\tau_\text{proto}}}{\omega}{\Omega_\text{params}}$ \BY{assumption} \pflabel{prepce}
          \item $\cvalidEP{\Omega_\text{params}}{\esceneP{\omega : \OParams}{\uOmega}{\uPsi}{\uPhi}{b}}{\ce}{e'}{\tau_\text{proto}}$ \BY{assumption} \pflabel{cvalidEP}
          \item $\hastypeP{\Omega}{\omega}{\Omega_\text{params}}$ \BY{Lemma \ref{lemma:expression-deparameterization-P} on \pfref{prepce}} \pflabel{hastypeP}
          \item $\hastypeP{\Omega \cup \OParams}{e'}{\tau_\text{proto}}$ \BY{IH, part 2(a) on \pfref{cvalidEP}} \pflabel{hastype}
          \item $\hastypeP{\Omega}{[\omega]e'}{[\omega]\tau_\text{proto}}$ \BY{Substitution Lemma \ref{lemma:substitution-P} on \pfref{hastypeP} and \pfref{hastype}}
        \end{pfsteps*}
        \resetpfcounter
    \end{byCases}
    \item In the following, let  $\uOmega = \uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}$. 
    \begin{byCases}
      \item[\text{(\ref{rule:rExpandsP})}] ~
        \begin{pfsteps*}
          \item $\urv=\matchrule{\upv}{\ue}$ \BY{assumption}
          \item $r=\aematchrule{p}{e}$ \BY{assumption}
          \item $\patExpandsP{\uOmegaEx{\emptyset}{\uG'}{\emptyset}{\Omega'}}{\uPhi}{\upv}{p}{\tau}$ \BY{assumption} \pflabel{patexpands}
          \item $\expandsP{\uOmegaEx{\uD}{\uG \uplus \uG'}{\uMctx}{\Omega \cup \Omega'}}{\uPsi}{\uPhi}{\ue}{e}{\tau'}$ \BY{assumption} \pflabel{expandsP}
          \item $\patTypePC{\Omega}{\Omega'}{p}{\tau}$ \BY{Theorem \ref{thm:typed-pattern-expansion-P} on \pfref{patexpands}} \pflabel{pattype}
          \item $\hastypeP{\Omega \cup \Omega'}{e}{\tau'}$ \BY{IH, part 1(a) on \pfref{expandsP}} \pflabel{hastype}
          \item $\ruleTypeP{\Omega}{r}{\tau}{\tau'}$ \BY{Rule (\ref{rule:ruleTypeP}) on \pfref{pattype} and \pfref{hastype}}
        \end{pfsteps*}
        \resetpfcounter
    \end{byCases}
  \end{enumerate}
  \item \begin{enumerate}
    \item In the following, let $\escenev=\esceneP{\omega : \OParams}{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}}{\uPsi}{\uPhi}{b}$.
     \begin{byCases}
      \item[\text{(\ref{rule:cvalidE-P-subsume})}] ~
        \begin{pfsteps*}
          \item $\cvalidEP{\Omega}{\escenev}{\ce}{e}{\tau'}$ \BY{assumption} \pflabel{cvalid}
          \item $\issubtypeP{\Omega}{\tau'}{\tau}$ \BY{assumption} \pflabel{issubtype}
          \item $\hastypeP{\Omega \cup \Omega_\text{app}}{e}{\tau'}$ \BY{IH, part 2(a) on \pfref{cvalid}} \pflabel{hastype}
          \item $\hastypeP{\Omega \cup \Omega_\text{app}}{e}{\tau}$ \BY{Rule (\ref{rule:hastypeP-subsume}) on \pfref{hastype} and \pfref{issubtype}}
        \end{pfsteps*}
        \resetpfcounter
      \item[\text{(\ref{rule:cvalidE-P-var}) \textbf{through} (\ref{rule:cvalidE-P-mval})}] In each of these cases, we apply the IH, part 2(a) or 2(b), over the premises and then apply the corresponding typing rule in Rules (\ref{rules:hastypeP}) and weakening as needed.
      \item[\text{(\ref{rule:cvalidE-P-splicede})}] ~
        \begin{pfsteps*}
          \item $\ce=\acesplicede{m}{n}{\ctau}$ \BY{assumption}
          \item $\tau=[\omega]\tau'$ \BY{assumption}
          \item $\cvalidC{\OParams}{\csfrom{\escenev}}{\ctau}{\tau'}{\akty}$ \BY{assumption} \pflabel{cvalid}
          \item $\parseUExp{\bsubseq{b}{m}{n}}{\ue}$ \BY{assumption} \pflabel{parseUExp}
          \item $\expandsP{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}}{\uPsi}{\uPhi}{\ue}{e}{[\omega]\tau'}$ \BY{assumption} \pflabel{expandsP}
          % \item $\domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset$ \BY{assumption} \pflabel{disjoint}
          \item $\hastypeP{\Omega_\text{app}}{e}{[\omega]\tau'}$ \BY{IH, part 1(a) on \pfref{expandsP}} \pflabel{hastype}
          \item $\hastypeP{\Omega_\text{app} \cup \Omega}{e}{[\omega]\tau'}$ \BY{Weakening on \pfref{hastype}}
        \end{pfsteps*}
        \resetpfcounter
    \end{byCases}
    \item \begin{byCases}
      \item[\text{(\ref{rule:cvalidR-P})}] ~
      \begin{pfsteps*}
        \item $\crv=\acematchrule{p}{\ce}$ \BY{assumption}
        \item $r=\aematchrule{p}{e}$ \BY{assumption}
        \item $\patTypePC{\Omega}{\Omega'}{p}{\tau}$ \BY{assumption} \pflabel{pattype}
        \item $\cvalidEP{\Gcons{\Omega}{\Omega'}}{\esceneP{\omega : \OParams}{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau'}$ \BY{assumption} \pflabel{cvalidEP}
        \item $(\Omega \cup \Omega') \cap \Omega_\text{app} = \emptyset$ \BY{identification convention} \pflabel{disjoint}
        \item $\hastypeP{\Omega \cup \Omega' \cup \Omega_\text{app}}{e}{\tau'}$ \BY{IH, part 2(a) on \pfref{cvalidEP} and \pfref{disjoint}} \pflabel{hastype}
        \item $\patTypePC{\Omega \cup \Omega_\text{app}}{\Omega'}{p}{\tau}$ \BY{Weakening on \pfref{pattype}} \pflabel{pattype2}
        \item $\ruleTypeP{\Omega \cup \Omega_\text{app}}{r}{\tau}{\tau'}$ \BY{Rule (\ref{rule:ruleTypeP}) on \pfref{pattype2} and \pfref{hastype}}
      \end{pfsteps*}
      \resetpfcounter
    \end{byCases}
  \end{enumerate}
\end{enumerate}

The mutual induction can be shown to be well-founded by an argument analagous to that in the proof of Theorem \ref{thm:typed-expansion-full-U}, appealing to Condition  \ref{condition:body-parsing-P} and Condition \ref{condition:body-subsequences-P}.
\end{proof}

\subsubsection{Modules}

\begin{theorem}[Module Expansion]
\label{thm:module-expansion-P}
If $\mExpandsP{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\uPsi}{\uPhi}{\uM}{M}{\sigma}$ then $\hassig{\Omega}{M}{\sigma}$.
\end{theorem}
\begin{proof} 
By rule induction over Rules (\ref{rules:mExpandsP}). In the following, let $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}$.

\begin{byCases}
  \item[\text{(\ref{rule:mExpandsP-subsumes})}] ~
    \begin{pfsteps*}
      \item $\mExpandsPX{\uM}{M}{\sigma'}$ \BY{assumption} \pflabel{mexpands}
      \item $\sigsub{\uOmega}{\sigma'}{\sigma}$ \BY{assumption} \pflabel{sigsub}
      \item $\hassig{\Omega}{M}{\sigma'}$ \BY{IH on \pfref{mexpands}} \pflabel{hassig}
      \item $\hassig{\Omega}{M}{\sigma}$ \BY{Rule (\ref{rule:hassig-subsume}) on \pfref{hassig} and \pfref{sigsub}}
    \end{pfsteps*}
    \resetpfcounter
  \item[\text{(\ref{rule:mExpandsP-var}) \textbf{through} (\ref{rule:mExpandsP-mlet})}] In each of these cases, we apply the IH over each module expansion premise, Theorem \ref{thm:typed-expression-expansion-P} over each expression expansion premise and Theorem \ref{thm:kind-and-constructor-expansion-P} over each construction expansion premise, then apply the corresponding signature matching rule in Rules (\ref{rules:hassig}) and weakening as needed.
  \item[\text{(\ref{rule:mExpandsP-syntaxpe}) \textbf{through} (\ref{rule:mExpandsP-letpptsm})}] In each of these cases, we apply the IH to the module expansion premise.
\end{byCases}
\end{proof}

\subsection{Abstract Reasoning Principles}
\begin{lemma}[Proto-Construction and Proto-Kind Decomposition]\label{lemma:proto-con-decomp}
~
\begin{enumerate}
  \item If $\cvalidC{\Omega}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cec}{c}{\kappa}$
    where $\segof{\cec} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon}$ then 
    \begin{enumerate}
      \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$
      \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$
      \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$
      \item $c = [\sseq{\kappa_i/k_i}{\nkind}, \sseq{c_i/u_i}{\ncon}, \omega]c'$ for some $c'$ and fresh $\sseq{k_i}{\nkind}$ and fresh $\sseq{u_i}{\ncon}$
      \item $\mathsf{fv}(c') \subset \sseq{k_i}{\nkind} \cup \sseq{u_i}{\ncon} \cup \domof{\Omega}$
    \end{enumerate}
  \item If $\cvalidK{\Omega}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa}{\kappa}$
    where $\segof{\cekappa} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon}$ then 
    \begin{enumerate}
      \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$
      \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$
      \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$
      \item $\kappa = [\sseq{\kappa_i/k_i}{\nkind}, \sseq{c_i/u_i}{\ncon}, \omega]\kappa'$ for some $\kappa'$ and fresh $\sseq{k_i}{\nkind}$ and fresh $\sseq{u_i}{\ncon}$
      \item $\mathsf{fv}(\kappa') \subset \sseq{k_i}{\nkind} \cup \sseq{u_i}{\ncon} \cup \domof{\Omega}$
    \end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof} % TODO this one is sloppy
  By mutual rule induction over Rules (\ref{rules:cvalidC}) and Rules (\ref{rules:cvalidK}).
  \begin{enumerate}
    \item \begin{byCases}
        \item[\text{(\ref{rule:cvalidC-subsume})}] This case follows by applying the IH.
        \item[\text{(\ref{rule:cvalidC-var}) \textbf{through} (\ref{rule:cvalidC-stat})}] These cases follow by applying the IH, gathering together the sets of conclusions and invoking the identification convention as necessary.
        \item[\text{(\ref{rule:cvalidC-spliced})}] Letting $\cscenev=\csceneP{\omega : \OParams}{\uOmega}{b}$, 
          \begin{pfsteps*}
          \item $\cec=\acesplicedc{m}{n}{\cekappa}$ \BY{assumption}
  \item $\cvalidK{\OParams}{\cscenev}{\cekappa}{\kappa}$ \BY{assumption} \pflabel{cvalidK}
  \item $\parseUCon{\bsubseq{b}{m}{n}}{\uc}$ \BY{assumption} \pflabel{parseUCon}
  \item $\cExpands{\uOmega}{\uc}{c}{[\omega]\kappa}$ \BY{assumption} \pflabel{cExpands}
  \item $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}$ \BY{assumption} \pflabel{uOmega}
  \item $\domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset$ \BY{assumption} \pflabel{domof}
  \item $\segof{\acesplicedc{m}{n}{\cekappa}} = \segof{\cekappa} \cup \{ \acesplicedc{m}{n}{\cekappa} \}$ \BY{definition} \pflabel{summaryOf1}
  \item $\segof{\cekappa} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon - 1}$ \BY{definition} \pflabel{summaryOf2}
      \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$ \BY{IH, part 2 on \pfref{cvalidK} and \pfref{summaryOf2}} \pflabel{puk}
      \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{{\ncon - 1}}$ \BY{IH, part 2 on \pfref{cvalidK} and \pfref{summaryOf2}} \pflabel{cvalidK2}
      \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{{\ncon - 1}}$ \BY{IH, part 2 on \pfref{cvalidK} and \pfref{summaryOf2}} \pflabel{cExpands2}
      % \item $\kappa = [\sseq{\kappa_i/k_i}{\nkind}, \sseq{c_i/u_i}{{\ncon - 1}}, \omega]\kappa'$ for some $\kappa'$ and fresh $\sseq{k_i}{\nkind}$ and fresh $\sseq{u_i}{\ncon}$ \BY{IH, part 2 on \pfref{cvalidK} and \pfref{summaryOf2}}
      % \item $\mathsf{fv}(\kappa') \subset \sseq{k_i}{\nkind} \cup \sseq{u_i}{{\ncon - 1}} \cup \domof{\Omega}$ \BY{IH, part 2 on \pfref{cvalidK} and \pfref{summaryOf2}}
          \end{pfsteps*}
          The conclusions hold as follows:
          \begin{enumerate}
            \item \pfref{puk}
            \item \pfref{cvalidK} and \pfref{cvalidK2}
            \item \pfref{parseUCon}, \pfref{cExpands} and \pfref{cExpands2}
            \item Choose $c'=u$ for fresh $u$. Then $c=[\omega', c/u]u$ for any $\omega'$.
            \item $\mathsf{fv}({u})=u$ and $u \subset \{u\} \cup \sseq{k_i}{\nkind} \cup \sseq{u_i}{{\ncon - 1}} \cup \domof{\Omega}$ by definition.
          \end{enumerate}
          \resetpfcounter
      \end{byCases}
    \item \begin{byCases}
        \item[\text{(\ref{rule:cvalidK-darr}) \textbf{through} (\ref{rule:cvalidK-sing})}] These cases follow by applying the IH and gathering together the sets of conclusions, invoking the identification convention as necessary.
        \item[\text{(\ref{rule:cvalidK-spliced})}] Letting $\cscenev=\csceneP{\omega : \OParams}{\uOmega}{b}$ 
          \begin{pfsteps*}
            \item $\cekappa = \acesplicedk{m}{n}$ \BY{assumption}
            \item $\parseUKind{\bsubseq{b}{m}{n}}{\ukappa}$ \BY{assumption} \pflabel{parse}
            \item $\kExpands{\uOmega}{\ukappa}{\kappa}$ \BY{assumption} \pflabel{ex}
            \item $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}$ \BY{assumption}
            \item $\domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset$ \BY{assumption}
            \item $\ncon=0$ \BY{definition of summary} \pflabel{ncon}
          \end{pfsteps*}
          The conclusions hold as follows:
          \begin{enumerate}
            \item \pfref{parse} and \pfref{ex}
            \item \pfref{ncon}
            \item \pfref{ncon}
            \item Choose $\kappa'=k$ for fresh $k$. Then $\kappa=[\omega', \kappa/k]k$ for any $\omega'$.
            \item $\mathsf{fv}(k)=k$ and $k \subset \{k\} \cup \domof{\Omega}$ by definition.
          \end{enumerate}
          \resetpfcounter
        \end{byCases}
  \end{enumerate}
\end{proof}

\begin{lemma}[Proto-Expression and Proto-Rule Decomposition]\label{lemma:proto-expr-decomp}
~
\begin{enumerate}
  \item If $\cvalidEP{\Omega}{\esceneP{\omega : \OParams}{\uOmega}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau_\text{proto}}$
    where $\segof{\ce} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon} \cup \sseq{\acesplicede{m''_i}{n''_i}{\ctau_i}}{\nexp}$ then 
    \begin{enumerate}
      \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$
      \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$
      \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$
      \item $\sseq{\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau_i}{\tau_i}{\akty}}{\nexp}$
      \item $\sseq{\expandsP{\uOmega}{\uPsi}{\uPhi}{\parseUExpF{\bsubseq{b}{m''_i}{n''_i}}}{e_i}{[\omega]\tau_i}}{\nexp}$
      \item $e = [\sseq{\kappa_i/k_i}{\nkind}, \sseq{c_i/u_i}{\ncon}, \sseq{e_i/x_i}{\nexp}, \omega]e''$ for some $e''$ and fresh $\sseq{k_i}{\nkind}$ and fresh $\sseq{u_i}{\ncon}$ and fresh $\sseq{x_i}{\nexp}$
      \item $\mathsf{fv}(e'') \subset \sseq{k_i}{\nkind} \cup \sseq{u_i}{\ncon} \cup \sseq{x_i}{\nexp} \cup \domof{\Omega}$
    \end{enumerate}
  \item If $\cvalidRP{\Omega}{\esceneP{\omega : \OParams}{\uOmega}{\uPsi}{\uPhi}{b}}{\crv}{r}{\tau}{\tau'}$
    where $\segof{\crv} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon} \cup \sseq{\acesplicede{m''_i}{n''_i}{\ctau_i}}{\nexp}$ then 
    \begin{enumerate}
      \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$
      \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$
      \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$
      \item $\sseq{\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau_i}{\tau_i}{\akty}}{\nexp}$
      \item $\sseq{\expandsP{\uOmega}{\uPsi}{\uPhi}{\parseUExpF{\bsubseq{b}{m''_i}{n''_i}}}{e_i}{[\omega]\tau_i}}{\nexp}$
      \item $r = [\sseq{\kappa_i/k_i}{\nkind}, \sseq{c_i/u_i}{\ncon}, \sseq{e_i/x_i}{\nexp}, \omega]r'$ for some $r'$ and fresh $\sseq{k_i}{\nkind}$ and fresh $\sseq{u_i}{\ncon}$ and fresh $\sseq{x_i}{\nexp}$
      \item $\mathsf{fv}(r') \subset \sseq{k_i}{\nkind} \cup \sseq{u_i}{\ncon} \cup \sseq{x_i}{\nexp} \cup \domof{\Omega}$
    \end{enumerate}
\end{enumerate}
\end{lemma} % TODO this is sloppy too
\begin{proof} By mutual rule induction over Rules (\ref{rules:cvalidE-P}) and Rule (\ref{rule:cvalidR-P}).
  \begin{enumerate}
    \item \begin{byCases}
      \item[\text{(\ref{rule:cvalidE-P-subsume})}] This case follows by applying the IH.
      \item[\text{(\ref{rule:cvalidE-P-var}) \textbf{through} (\ref{rule:cvalidE-P-mval})}] These cases follow by applying the IH or Lemma \ref{lemma:proto-con-decomp}, gathering together the sets of conclusions and invoking the identification convention as necessary. 
      \item[\text{(\ref{rule:cvalidE-P-splicede})}] Letting $\escenev = \esceneP{\omega : \OParams}{\uOmega}{\uPsi}{\uPhi}{b}$,    
      \begin{pfsteps*}
        \item $\ce = \acesplicede{m}{n}{\ctau}$ \BY{assumption}
        \item $\cvalidC{\OParams}{\csfrom{\escenev}}{\ctau}{\tau}{\akty}$ \BY{assumption} \pflabel{cvalidC}
        \item $\parseUExp{\bsubseq{b}{m}{n}}{\ue}$ \BY{assumption} \pflabel{parse}
        \item $\expandsP{\uOmega}{\uPsi}{\uPhi}{\ue}{e}{[\omega]\tau}$ \BY{assumption} \pflabel{expandsP}
        \item $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}$ \BY{assumption}
        \item $\domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset$ \BY{assumption}
        \item $\segof{\ce} = \segof{\ctau} \cup \{ \acesplicede{m}{n}{\ctau} \}$ \BY{assumption}
        \item $\segof{\ctau} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon}$  \BY{definition} \pflabel{sumC}
        \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$ \BY{Lemma \ref{lemma:proto-con-decomp} on \pfref{cvalidC} and \pfref{sumC}} \pflabel{A}
        \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$ \BY{Lemma \ref{lemma:proto-con-decomp} on \pfref{cvalidC} and \pfref{sumC}} \pflabel{B}
        \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$ \BY{Lemma \ref{lemma:proto-con-decomp} on \pfref{cvalidC} and \pfref{sumC}} \pflabel{C}
      \end{pfsteps*}
      The conclusions hold as follows:
      \begin{enumerate}
        \item \pfref{A}
        \item \pfref{B}
        \item \pfref{C}
        \item \pfref{cvalidC}
        \item \pfref{parse} and \pfref{expandsP}
            \item Choose $e''=x$ for fresh $x$. Then $e=[\omega', e/x]x$ for any $\omega'$.
            \item $\mathsf{fv}({x})=x$ and $x \subset \{x\} \cup \sseq{k_i}{\nkind} \cup \sseq{u_i}{{\ncon}} \cup \domof{\Omega}$ by definition.
      \end{enumerate}
      \resetpfcounter 
    \end{byCases}
    \item There is only one case.
    \begin{byCases}
      \item[\text{(\ref{rule:cvalidR-P})}] ~
        \begin{pfsteps*}
          \item $\crv=\acematchrule{p}{e}$ \BY{assumption}
          \item $r=\matchrule{p}{e}$ \BY{assumption}
          \item $\patTypeP{\Omega'}{p}{\tau}$ \BY{assumption} \pflabel{pattype}
          \item $\cvalidEP{\Gcons{\Omega}{\Omega'}}{\escenev}{\ce}{e}{\tau'}$ \BY{assumption} \pflabel{cvalidEP}
          \item $\segof{\crv} = \segof{e}$ \BY{definition} \pflabel{summ}
          \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$ \BY{IH, part 1 on \pfref{cvalidEP} and \pfref{summ}} \pflabel{c1}
          \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$ \BY{IH, part 1 on \pfref{cvalidEP} and \pfref{summ}} \pflabel{c2}
          \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$ \BY{IH, part 1 on \pfref{cvalidEP} and \pfref{summ}} \pflabel{c3}
          \item $\sseq{\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau_i}{\tau_i}{\akty}}{\nexp}$ \BY{IH, part 1 on \pfref{cvalidEP} and \pfref{summ}} \pflabel{c4}
          \item $\sseq{\expandsP{\uOmega}{\uPsi}{\uPhi}{\parseUExpF{\bsubseq{b}{m''_i}{n''_i}}}{e_i}{[\omega]\tau_i}}{\nexp}$ \BY{IH, part 1 on \pfref{cvalidEP} and \pfref{summ}} \pflabel{c5}
          \item $e = [\sseq{\kappa_i/k_i}{\nkind}, \sseq{c_i/u_i}{\ncon}, \sseq{e_i/x_i}{\nexp}, \omega]e''$ for some $e''$ and fresh $\sseq{k_i}{\nkind}$ and fresh $\sseq{u_i}{\ncon}$ and fresh $\sseq{x_i}{\nexp}$ \BY{IH, part 1 on \pfref{cvalidEP} and \pfref{summ}} \pflabel{c6}
          \item $\mathsf{fv}(e'') \subset \sseq{k_i}{\nkind} \cup \sseq{u_i}{\ncon} \cup \sseq{x_i}{\nexp} \cup \domof{\Omega \cup \Omega'}$ \BY{IH, part 1 on \pfref{cvalidEP} and \pfref{summ}} \pflabel{c7}
          \item $\mathsf{fv}(e'') \subset \sseq{k_i}{\nkind} \cup \sseq{u_i}{\ncon} \cup \sseq{x_i}{\nexp} \cup \domof{\Omega} \cup \domof{\Omega'}$ \BY{distributivity of union} \pflabel{c7d}
          \item $r=[\sseq{\kappa_i/k_i}{\nkind}, \sseq{c_i/u_i}{\ncon}, \sseq{e_i/x_i}{\nexp}, \omega]\matchrule{p}{e''}$ \BY{definition of substitution} \pflabel{req}
          \item $\domof{\Omega'} = \mathsf{patvars}(p)$ \BY{Lemma \ref{lemma:pattern-binding} on \pfref{pattype}} \pflabel{domeq}
          \item $\mathsf{fv}(\matchrule{p}{e''}) \subset \sseq{k_i}{\nkind} \cup \sseq{u_i}{\ncon} \cup \sseq{x_i}{\nexp} \cup \domof{\Omega}$ \BY{Definition of $\mathsf{fv}(r)$ and \pfref{domeq} and \pfref{c7d}} \pflabel{fvfinal}
        \end{pfsteps*}
    \end{byCases}
    The conclusions hold as follows:
    \begin{enumerate}
      \item \pfref{c1}
      \item \pfref{c2}
      \item \pfref{c3}
      \item \pfref{c4}
      \item \pfref{c5}
      \item Choose $r'=\matchrule{p}{e''}$, then by \pfref{req}
      \item \pfref{fvfinal}
    \end{enumerate}
    \resetpfcounter
  \end{enumerate}
\end{proof}

\begin{theorem}[peTLM Abstract Reasoning Principles]
\label{thm:petsm-abstract-reasoning-principles}
If $\expandsP{\uOmega}{\uPsi}{\uPhi}{\utsmap{\uepsilon}{b}}{e}{\tau}$ then:
\begin{enumerate}
  \item $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}$
  \item $\uPsi=\uAS{\uA}{\Psi}$
  \item (\textbf{Typing 1}) $\tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\aetype{\tau'}}$ and $\hastypeP{\Omega_\text{app}}{e}{\tau'}$ for $\tau'$ such that $\issubtypeP{\Omega_\text{app}}{\tau'}{\tau}$.
  \item $\tsmexpEvalsExp{\Omega_\text{app}}{\Psi}{\epsilon}{\epsilon_\text{normal}}$
  \item $\tsmdefof{\epsilon_\text{normal}}=a$
  \item $\Psi = \Psi', \petsmdefn{a}{\rho}{\eparse}$
  \item $\encodeBody{b}{\ebody}$
    \item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessE}}{e_\text{pproto}}}$
  \item $\decodePCEExp{e_\text{pproto}}{\pce}$
  \item $\prepce{\Omega_\text{app}}{\Psi}{\pce}{\ce}{\epsilon_\text{normal}}{\aetype{\tau_\text{proto}}}{\omega}{\Omega_\text{params}}$
  \item (\textbf{Segmentation}) $\segOKP{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\segof{\ce}}{b}$
  \item $\cvalidEP{\Omega_\text{params}}{\esceneP{\omega : \OParams}{\uOmega}{\uPsi}{\uPhi}{b}}{\ce}{e'}{\tau_\text{proto}}$
  \item $e = [\omega]e'$
  \item $\tau' = [\omega]\tau_\text{proto}$
  \item $
    \segof{\ce} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon} \cup\\
               \sseq{\acesplicede{m''_i}{n''_i}{\ctau_i}}{\nexp}
    $
  \item (\textbf{Kinding 1}) $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$ and $\sseq{\iskind{\Omega_\text{app}}{\kappa_i}}{\nkind}$
  \item (\textbf{Kinding 2}) $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$ and $\sseq{\iskind{\Omega_\text{app}}{[\omega]\kappa'_i}}{\ncon}$
  \item (\textbf{Kinding 3}) $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$ and $\sseq{\haskind{\Omega_\text{app}}{c_i}{[\omega]\kappa'_i}}{\ncon}$
  \item (\textbf{Kinding 4}) $\sseq{\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau_i}{\tau_i}{\akty}}{\nexp}$ and $\sseq{\haskind{\Omega_\text{app}}{[\omega]\tau_i}{\akty}}{\nexp}$
  \item (\textbf{Typing 2}) $\sseq{\expandsP{\uOmega}{\uPsi}{\uPhi}{\parseUExpF{\bsubseq{b}{m''_i}{n''_i}}}{e_i}{[\omega]\tau_i}}{\nexp}$ and $\sseq{\hastypeP{\Omega_\text{app}}{e_i}{[\omega]\tau_i}}{\nexp}$
  \item (\textbf{Capture Avoidance}) $e = [\sseq{\kappa_i/k_i}{\nkind}, \sseq{c_i/u_i}{\ncon}, \sseq{e_i/x_i}{\nexp}, \omega]e''$ for some $e''$ and fresh $\sseq{k_i}{\nkind}$ and fresh $\sseq{u_i}{\ncon}$ and fresh $\sseq{x_i}{\nexp}$
  \item (\textbf{Context Independence}) \[\mathsf{fv}(e'') \subset \sseq{k_i}{\nkind} \cup \sseq{u_i}{\ncon} \cup \sseq{x_i}{\nexp} \cup \domof{\OParams}\]
  % $\hastypeP{\sseq{\Khyp{k_i}}{\nkind} \cup \sseq{u_i :: [\omega]\kappa'_i}{\ncon} \cup \sseq{x_i : [\omega]\tau_i}{\nexp}}{[\omega]e''}{\tau}$\todo{maybe restate this in terms of free variables of e'' here and elsewhere, because context isn't technically well-formed here?}
\end{enumerate}
\end{theorem}
\begin{proof} By rule induction over Rules (\ref{rules:expandsP}). There are two rules that apply.
\begin{byCases}
  \item[\text{(\ref{rule:expandsP-subsume})}] ~
    \begin{pfsteps*}
      \item $\expandsPX{\utsmap{\uepsilon}{b}}{e}{\tau'}$ \BY{assumption} \pflabel{expands}
      \item $\issubtypePX{\tau'}{\tau}$ \BY{assumption} \pflabel{issubtype1}
      \item $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}$ \BY{IH on \pfref{expands}} \pflabel{A}
      \item $\uPsi=\uAS{\uA}{\Psi}$ \BY{IH on \pfref{expands}} \pflabel{B}
      \item $\tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\aetype{\tau''}}$ and $\hastypeP{\Omega_\text{app}}{e}{\tau''}$ for $\tau''$ such that $\issubtypeP{\Omega_\text{app}}{\tau''}{\tau'}$. \BY{IH on \pfref{expands}} \pflabel{C}
      \item $\tsmexpEvalsExp{\Omega_\text{app}}{\Psi}{\epsilon}{\epsilon_\text{normal}}$ \BY{IH on \pfref{expands}} \pflabel{D}
      \item $\tsmdefof{\epsilon_\text{normal}}=a$ \BY{IH on \pfref{expands}} \pflabel{E}
      \item $\Psi = \Psi', \petsmdefn{a}{\rho}{\eparse}$ \BY{IH on \pfref{expands}} \pflabel{F}
      \item $\encodeBody{b}{\ebody}$ \BY{IH on \pfref{expands}} \pflabel{G}
        \item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessE}}{e_\text{pproto}}}$ \BY{IH on \pfref{expands}} \pflabel{H}
      \item $\decodePCEExp{e_\text{pproto}}{\pce}$ \BY{IH on \pfref{expands}} \pflabel{I}
      \item $\prepce{\Omega_\text{app}}{\Psi}{\pce}{\ce}{\epsilon_\text{normal}}{\aetype{\tau_\text{proto}}}{\omega}{\Omega_\text{params}}$ \BY{IH on \pfref{expands}} \pflabel{J}
      \item $\segOKP{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\segof{\ce}}{b}$ \BY{IH on \pfref{expands}} \pflabel{K}
      \item $\cvalidEP{\Omega_\text{params}}{\esceneP{\omega : \OParams}{\uOmega}{\uPsi}{\uPhi}{b}}{\ce}{e'}{\tau_\text{proto}}$ \BY{IH on \pfref{expands}} \pflabel{L}
      \item $e = [\omega]e'$ \BY{IH on \pfref{expands}} \pflabel{M}
      \item $\tau' = [\omega]\tau_\text{proto}$ \BY{IH on \pfref{expands}} \pflabel{N}
      \item $
        \segof{\ce} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon} \cup 
                   \sseq{\acesplicede{m''_i}{n''_i}{\ctau_i}}{\nexp}
        $ \BY{IH on \pfref{expands}} \pflabel{O}
      \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$ and $\sseq{\iskind{\Omega_\text{app}}{\kappa_i}}{\nkind}$ \BY{IH on \pfref{expands}} \pflabel{P}
      \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$ and $\sseq{\iskind{\Omega_\text{app}}{[\omega]\kappa'_i}}{\ncon}$ \BY{IH on \pfref{expands}} \pflabel{Q}
      \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$ and $\sseq{\haskind{\Omega_\text{app}}{c_i}{[\omega]\kappa'_i}}{\ncon}$ \BY{IH on \pfref{expands}} \pflabel{R}
      \item $\sseq{\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau_i}{\tau_i}{\akty}}{\nexp}$ and $\sseq{\haskind{\Omega_\text{app}}{[\omega]\tau_i}{\akty}}{\nexp}$ \BY{IH on \pfref{expands}} \pflabel{S}
      \item $\sseq{\expandsP{\uOmega}{\uPsi}{\uPhi}{\parseUExpF{\bsubseq{b}{m''_i}{n''_i}}}{e_i}{[\omega]\tau_i}}{\nexp}$ and $\sseq{\hastypeP{\Omega_\text{app}}{e_i}{[\omega]\tau_i}}{\nexp}$ \BY{IH on \pfref{expands}} \pflabel{T}
      \item $e = [\sseq{\kappa_i/k_i}{\nkind}, \sseq{c_i/u_i}{\ncon}, \sseq{e_i/x_i}{\nexp}, \omega]e''$ for some $e''$ and fresh $\sseq{k_i}{\nkind}$ and fresh $\sseq{u_i}{\ncon}$ and fresh $\sseq{x_i}{\nexp}$ \BY{IH on \pfref{expands}} \pflabel{U}
      \item $\mathsf{fv}(e'') \subset \sseq{k_i}{\nkind} \cup \sseq{u_i}{\ncon} \cup \sseq{x_i}{\nexp} \cup \domof{\OParams}$ \BY{IH on \pfref{expands}} \pflabel{V}
      \item $\issubtypeP{\Omega_\text{app}}{\tau''}{\tau}$ \BY{Rule (\ref{rule:issubtypeP-trans}) on \pfref{issubtype1} and \pfref{C}} \pflabel{issubtype2}
    \end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
      \item \pfref{A}
      \item \pfref{B}
      \item Choosing $\tau''$, by \pfref{C} and \pfref{issubtype2}
      \item \pfref{D}
      \item \pfref{E}
      \item \pfref{F}
      \item \pfref{G}
      \item \pfref{H}
      \item \pfref{I}
      \item \pfref{J}
      \item \pfref{K}
      \item \pfref{L}
      \item \pfref{M}
      \item \pfref{N}
      \item \pfref{O}
      \item \pfref{P}
      \item \pfref{Q}
      \item \pfref{R}
      \item \pfref{S}
      \item \pfref{T}
      \item \pfref{U}
      \item \pfref{V}
    \end{enumerate}
    \resetpfcounter
  \item[\text{(\ref{rule:expandsP-apuetsm})}] ~
    \begin{pfsteps*}
      \item $e=[\omega]e'$ \BY{assumption} \pflabel{e}
      \item $\tau=[\omega]\tau_\text{proto}$ \BY{assumption} \pflabel{tau}
      \item $\uOmega = \uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}$ \BY{assumption} \pflabel{uomega}
      \item $\uPsi=\uAS{\uA}{\Psi}$ \BY{assumption} \pflabel{upsi}
      \item $\tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\aetype{\tau_\text{final}}}$ \BY{assumption} \pflabel{tsmexpExpands}
      \item $\tsmexpEvalsExp{\Omega_\text{app}}{\Psi}{\epsilon}{\epsilon_\text{normal}}$ \BY{assumption} \pflabel{tsmexpEvals}
      \item $\tsmdefof{\epsilon_\text{normal}}=a$ \BY{assumption} \pflabel{tsmdefof}
      \item $\Psi = \Psi', \petsmdefn{a}{\rho}{\eparse}$ \BY{assumption} \pflabel{Psi}
      \item $\encodeBody{b}{\ebody}$ \BY{assumption} \pflabel{encodeBody}
      \item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessE}}{e_\text{pproto}}}$ \BY{assumption} \pflabel{evalU}
      \item $\decodePCEExp{e_\text{pproto}}{\pce}$ \BY{assumption} \pflabel{decode}
      \item $\prepce{\Omega_\text{app}}{\Psi}{\pce}{\ce}{\epsilon_\text{normal}}{\aetype{\tau_\text{proto}}}{\omega}{\Omega_\text{params}}$ \BY{assumption} \pflabel{prepce}
      \item $\segOKP{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\segof{\ce}}{b}$ \BY{assumption} \pflabel{segOK}
      \item $\cvalidEP{\Omega_\text{params}}{\esceneP{\omega : \OParams}{\uOmega}{\uPsi}{\uPhi}{b}}{\ce}{e'}{\tau_\text{proto}}$ \BY{assumption} \pflabel{cvalid}
      \item $\hastsmtypeExp{\Omega_\text{app}}{\Psi}{\epsilon}{\aetype{\tau_\text{final}}}$ \BY{Theorem \ref{thm:peTLM-expression-expansion} on \pfref{tsmexpExpands}} \pflabel{hastsmtype}
      \item $\hastsmtypeExp{\Omega_\text{app}}{\Psi}{\epsilon_\text{normal}}{\aetype{\tau_\text{final}}}$ \BY{Corollary \ref{thm:peTLM-preservation-evaluation} on \pfref{hastsmtype} and \pfref{tsmexpEvals}} \pflabel{hastsmtypeNormal}
      \item $\hastsmtypeExp{\Omega_\text{app}}{\Psi}{\epsilon_\text{normal}}{[\omega]\aetype{\tau_\text{proto}}}$ \BY{Lemma \ref{lemma:expression-deparameterization-P} on \pfref{prepce}} \pflabel{hastsmtypeNormal2}
      \item $\aetype{\tau_\text{final}} = [\omega]\aetype{\tau_\text{proto}}$ \BY{Theorem \ref{lemma:peTLM-unicity} on \pfref{hastsmtypeNormal} and \pfref{hastsmtypeNormal2}} \pflabel{tsmtypeseq}
      \item $\tau_\text{final}=[\omega]\tau_\text{proto}=\tau$ \BY{definition of substitution and \pfref{tsmtypeseq} and \pfref{tau}} \pflabel{feqp}
      \item $\istsmty{\Omega_\text{app}}{\aetype{\tau_\text{final}}}$ \BY{Lemma \ref{lemma:peTLM-regularity} on \pfref{hastsmtype}} \pflabel{istsmty}
      \item $\haskind{\Omega_\text{app}}{\tau}{\akty}$ \BY{Inversion of Rule (\ref{rule:istsmty-type}) on \pfref{istsmty}} \pflabel{istype}
      \item $\cequal{\Omega_\text{app}}{\tau}{\tau}{\akty}$ \BY{Rule (\ref{rule:cequal-refl}) on \pfref{istype}} \pflabel{refl}
      \item $\issubtypeP{\Omega_\text{app}}{\tau}{\tau}$ \BY{Rule (\ref{rule:issubtypeP-equal}) on \pfref{refl}} \pflabel{subty}
      \item  $\expandsP{\uOmega}{\uPsi}{\uPhi}{\utsmap{\uepsilon}{b}}{e}{\tau}$ \BY{assumption} \pflabel{expands}
      \item $\hastypeP{\Omega_\text{app}}{e}{\tau}$ \BY{Theorem \ref{thm:typed-expression-expansion-P} on \pfref{expands}} \pflabel{hastype}
      \item $\hastypeP{\Omega_\text{app}}{\omega}{\Omega_\text{params}}$ \BY{Lemma \ref{lemma:expression-deparameterization-P} on \pfref{prepce}} \pflabel{omegaType}
      \item $
    \segof{\ce} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon} \cup
               \sseq{\acesplicede{m''_i}{n''_i}{\ctau_i}}{\nexp}
    $ \BY{definition} \pflabel{summaryOf}
       \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$ \BY{Lemma \ref{lemma:proto-expr-decomp} on \pfref{cvalid} and \pfref{summaryOf}} \pflabel{A}
       \item $\sseq{\iskind{\Omega_\text{app}}{\kappa_i}}{\nkind}$ \BY{Theorem \ref{thm:kind-and-constructor-expansion-P} over \pfref{A}} \pflabel{A2}
      \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$ \BY{Lemma \ref{lemma:proto-expr-decomp} on \pfref{cvalid} and \pfref{summaryOf}} \pflabel{B}
      \item $\sseq{\iskind{\Omega_\text{app} \cup \OParams}{\kappa'_i}}{\ncon}$ \BY{Theorem \ref{thm:kind-and-constructor-expansion-P} over \pfref{B}} \pflabel{B2}
      \item $\sseq{\iskind{\Omega_\text{app}}{[\omega]\kappa'_i}}{\ncon}$ \BY{Lemma \ref{lemma:substitution-P} over \pfref{B2} and \pfref{omegaType}} \pflabel{B3}
      \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$ \BY{Lemma \ref{lemma:proto-expr-decomp} on \pfref{cvalid} and \pfref{summaryOf}} \pflabel{C}
      \item $\sseq{\haskind{\Omega_\text{app}}{c_i}{[\omega]\kappa'_i}}{\ncon}$ \BY{Theorem \ref{thm:kind-and-constructor-expansion-P} over \pfref{C}} \pflabel{C2}
      \item $\sseq{\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau_i}{\tau_i}{\akty}}{\nexp}$ \BY{Lemma \ref{lemma:proto-expr-decomp} on \pfref{cvalid} and \pfref{summaryOf}} \pflabel{D}
      \item $\sseq{\haskind{\Omega_\text{app} \cup \OParams}{\tau_i}{\akty}}{\nexp}$ \BY{Theorem \ref{thm:kind-and-constructor-expansion-P} over \pfref{D}} \pflabel{D2}
      \item $\sseq{\haskind{\Omega_\text{app}}{[\omega]\tau_i}{\akty}}{\nexp}$ \BY{Lemma \ref{lemma:substitution-P} over \pfref{D2} and \pfref{omegaType}} \pflabel{D3}
      \item $\sseq{\expandsP{\uOmega}{\uPsi}{\uPhi}{\parseUExpF{\bsubseq{b}{m''_i}{n''_i}}}{e_i}{[\omega]\tau_i}}{\nexp}$ \BY{Lemma \ref{lemma:proto-expr-decomp} on \pfref{cvalid} and \pfref{summaryOf}} \pflabel{E}
      \item $\sseq{\hastypeP{\Omega_\text{app}}{e_i}{[\omega]\tau_i}}{\nexp}$ \BY{Theorem \ref{thm:typed-expression-expansion-P} over \pfref{E}} \pflabel{E2}
      \item $e = [\sseq{\kappa_i/k_i}{\nkind}, \sseq{c_i/u_i}{\ncon}, \sseq{e_i/x_i}{\nexp}, \omega]e''$ for some $e''$ and fresh $\sseq{k_i}{\nkind}$ and fresh $\sseq{u_i}{\ncon}$ and fresh $\sseq{x_i}{\nexp}$ \BY{Lemma \ref{lemma:proto-expr-decomp} on \pfref{cvalid} and \pfref{summaryOf}} \pflabel{F}
      \item $\mathsf{fv}(e'') \subset \sseq{k_i}{\nkind} \cup \sseq{u_i}{\ncon} \cup \sseq{x_i}{\nexp} \cup \domof{\OParams}$ \BY{Lemma \ref{lemma:proto-expr-decomp} on \pfref{cvalid} and \pfref{summaryOf}} \pflabel{G}
    \end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
     \item \pfref{uomega}
     \item \pfref{upsi}
     \item Choosing $\tau'$, by \pfref{tsmexpExpands}, \pfref{feqp}, \pfref{hastype} and \pfref{subty}
     \item \pfref{tsmexpEvals}
     \item \pfref{tsmdefof}
     \item \pfref{Psi}
     \item \pfref{encodeBody}
     \item \pfref{evalU}
     \item \pfref{decode}
     \item \pfref{prepce}
     \item \pfref{segOK}
     \item \pfref{cvalid}
     \item \pfref{e}
     \item \pfref{tau}
     \item \pfref{summaryOf}
     \item \pfref{A} and \pfref{A2}
     \item \pfref{B} and \pfref{B3}
     \item \pfref{C} and \pfref{C2}
     \item \pfref{D} and \pfref{D3}
     \item \pfref{E} and \pfref{E2}
     \item \pfref{F}
     \item \pfref{G}
    \end{enumerate}
    \resetpfcounter
\end{byCases}

 \end{proof}

\begin{lemma}[Proto-Pattern Decomposition]
\label{lemma:proto-pattern-decomposition-P}
If $\cvalidPP{\uOmega'}{\psceneP{\omega : \Omega_\text{params}}{\uOmega}{\uPhi}{b}}{\cpv}{p}{\tau}$ where \[
  \begin{array}{lll}
  \segof{\cpv} & = &\sseq{\acesplicedk{m_i}{n_i}}{\nkind} \\
  & \cup & \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon} \\
  & \cup & \sseq{\acesplicedp{m''_i}{n''_i}{\ctau_i}}{\npat}
  \end{array}
  \] then 
  \begin{enumerate}
      \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$
      \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$
      \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$
      \item $\sseq{\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau_i}{\tau_i}{\akty}}{\npat}$
      \item $\sseq{\patExpandsP{\uOmegaEx{\emptyset}{\uG_i}{\emptyset}{\Omega_i}}{\uPhi}{\parseUPatF{\bsubseq{b}{m''_i}{n''_i}}}{p_i}{[\omega]\tau_i}}{\npat}$
      \item $\uOmega'=\uOmegaEx{\emptyset}{\biguplus_{0 \leq i < \npat} \uG_i}{\emptyset}{\bigcup_{0 \leq i < \npat} \Omega_i}$
  \end{enumerate}
\end{lemma}
\begin{proof} By rule induction over Rules (\ref{rules:cvalidPP}). 
\begin{byCases}
  \item[\text{(\ref{rule:cvalidPP-wild}) \textbf{through} (\ref{rule:cvalidPP-in})}] In each of these cases, we apply the IH to or over each premise and then gather the sets of conclusions, applying the identification convention as necessary. 
  \item[\text{(\ref{rule:cvalidPP-spliced})}] ~
    \begin{pfsteps*}
      \item $\cpv = \acesplicedp{m}{n}{\ctau}$ \BY{assumption} 
      \item $\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau}{\tau}{\akty}$ \BY{assumption} \pflabel{cvalidC}
      \item $\parseUPat{\bsubseq{b}{m}{n}}{\upv}$ \BY{assumption} \pflabel{parseUPat}
      \item $\patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{[\omega]\tau}$ \BY{assumption} \pflabel{patExpandsP}
      \item $\segof{\cpv} = \segof{\ctau} \cup \{ \acesplicedp{m}{n}{\ctau} \}$ \BY{definition} \pflabel{summ}
      \item $\segof{\ctau} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon}$ \BY{definition} \pflabel{ctausumm}
      \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$ \BY{Lemma \ref{lemma:proto-con-decomp} on \pfref{cvalidC} and \pfref{ctausumm}} \pflabel{A}
      \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$ \BY{Lemma \ref{lemma:proto-con-decomp} on \pfref{cvalidC} and \pfref{ctausumm}} \pflabel{B}
      \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$ \BY{Lemma \ref{lemma:proto-con-decomp} on \pfref{cvalidC} and \pfref{ctausumm}} \pflabel{C}
    \end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
      \item \pfref{A}
      \item \pfref{B}
      \item \pfref{C}
      \item \pfref{cvalidC}
      \item \pfref{parseUPat} and \pfref{patExpandsP}
      \item \pfref{patExpandsP} because $\npat = 1$
    \end{enumerate}
    \resetpfcounter
\end{byCases}

 \end{proof}

\begin{theorem}[ppTLM Abstract Reasoning Principles]
\label{thm:pptsm-abstract-reasoning-principles}
If $\patExpandsP{\uOmega'}{\uPhi}{\utsmap{\uepsilon}{b}}{p}{\tau}$ then:
\begin{enumerate}
  \item $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}$
  \item $\uPhi=\uAS{\uA}{\Phi}$
  \item (\textbf{Typing 1}) $\tsmexpExpandsPat{\uOmega}{\uPhi}{\uepsilon}{\epsilon}{\aetype{\tau'}}$ and $\patTypePC{\Omega_\text{app}}{\uOmega'}{p}{\tau'}$ for $\tau'$ such that $\issubtypeP{\Omega_\text{app}}{\tau'}{\tau}$
  \item $\tsmexpEvalsPat{\Omega_\text{app}}{\Phi}{\epsilon}{\epsilon_\text{normal}}$
  \item $\tsmdefof{\epsilon_\text{normal}}=a$
  \item $\Phi = \Phi', \pptsmdefn{a}{\rho}{\eparse}$
  \item $\encodeBody{b}{\ebody}$
  \item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessP}}{e_\text{pproto}}}$
  \item $\decodePCEPat{e_\text{pproto}}{\pcp}$
  \item $\prepcp{\Omega_\text{app}}{\Phi}{\pcp}{\cpv}{\epsilon_\text{normal}}{\aetype{\tau_\text{proto}}}{\omega}{\Omega_\text{params}}$
  \item (\textbf{Segmentation}) $\segOKP{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\segof{\cpv}}{b}$
  \item $\cvalidPP{\uOmega'}{\psceneP{\omega : \Omega_\text{params}}{\uOmega}{\uPhi}{b}}{\cpv}{p}{\tau_\text{proto}}$
  \item $\tau'=[\omega]\tau_\text{proto}$
  \item $
  \segof{\cpv} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon} \cup\\
             \sseq{\acesplicedp{m''_i}{n''_i}{\ctau_i}}{\npat}
  $
  \item (\textbf{Kinding 1}) $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$ and $\sseq{\iskind{\Omega_\text{app}}{\kappa_i}}{\nkind}$
  \item (\textbf{Kinding 2}) $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$ and $\sseq{\iskind{\Omega_\text{app}}{[\omega]\kappa'_i}}{\ncon}$
  \item (\textbf{Kinding 3}) $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$ and $\sseq{\haskind{\Omega_\text{app}}{c_i}{[\omega]\kappa'_i}}{\ncon}$
  \item (\textbf{Kinding 4}) $\sseq{\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau_i}{\tau_i}{\akty}}{\npat}$ and $\sseq{\haskind{\Omega_\text{app}}{[\omega]\tau_i}{\akty}}{\npat}$
  \item (\textbf{Typing 2}) $\sseq{\patExpandsP{\uOmegaEx{\emptyset}{\uG_i}{\emptyset}{\Omega_i}}{\uPhi}{\parseUPatF{\bsubseq{b}{m''_i}{n''_i}}}{p_i}{[\omega]\tau_i}}{\npat}$ and $\sseq{\patTypePC{\Omega_\text{app}}{\Omega_i}{p_i}{[\omega]\tau_i}}{\npat}$
      \item (\textbf{Visibility}) $\uOmega'=\uOmegaEx{\emptyset}{\biguplus_{0 \leq i < \npat} \uG_i}{\emptyset}{\bigcup_{0 \leq i < \npat} \Omega_i}$

  % \item $\cvalidPP{\uOmega'}{\psceneP{\omega : \Omega_\text{params}}{\uOmega}{\uPhi}{b}}{\cpv}{p}{\tau_\text{proto}}$
  % \item $\tau = [\omega]\tau_\text{proto}$
  % \item (\textbf{Typing}) $\tau_\text{final} = [\omega]\tau_\text{proto}$
\end{enumerate}
\end{theorem}
\begin{proof} 
By rule induction over Rules (\ref{rules:patExpandsP}). There are two rules that apply.
\begin{byCases}
  \item[\text{(\ref{rule:patExpandsP-subsume})}] ~
    \begin{pfsteps*}
      \item $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}$ \BY{assumption}
      \item $\patExpandsP{\uOmega'}{\uPhi}{\utsmap{\uepsilon}{b}}{p}{\tau'}$ \BY{assumption} \pflabel{patExpands}
      \item $\issubtypeP{\Omega}{\tau'}{\tau}$ \BY{assumption} \pflabel{issubtypeP}
      \item $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}$ \BY{IH on \pfref{patExpands}} \pflabel{A}
      \item $\uPhi=\uAS{\uA}{\Phi}$ \BY{IH on \pfref{patExpands}} \pflabel{B}
      \item $\tsmexpExpandsPat{\uOmega}{\uPhi}{\uepsilon}{\epsilon}{\aetype{\tau''}}$ and $\patTypePC{\Omega_\text{app}}{\uOmega'}{p}{\tau''}$ for $\tau''$ such that $\issubtypeP{\Omega_\text{app}}{\tau''}{\tau'}$ \BY{IH on \pfref{patExpands}} \pflabel{C}
      \item $\tsmexpEvalsPat{\Omega_\text{app}}{\Phi}{\epsilon}{\epsilon_\text{normal}}$ \BY{IH on \pfref{patExpands}} \pflabel{D}
      \item $\tsmdefof{\epsilon_\text{normal}}=a$ \BY{IH on \pfref{patExpands}} \pflabel{E}
      \item $\Phi = \Phi', \pptsmdefn{a}{\rho}{\eparse}$\BY{IH on \pfref{patExpands}} \pflabel{F}
      \item $\encodeBody{b}{\ebody}$\BY{IH on \pfref{patExpands}} \pflabel{G}
      \item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessP}}{e_\text{pproto}}}$\BY{IH on \pfref{patExpands}} \pflabel{H}
      \item $\decodePCEPat{e_\text{pproto}}{\pcp}$\BY{IH on \pfref{patExpands}} \pflabel{I}
      \item $\prepcp{\Omega_\text{app}}{\Phi}{\pcp}{\cpv}{\epsilon_\text{normal}}{\aetype{\tau_\text{proto}}}{\omega}{\Omega_\text{params}}$\BY{IH on \pfref{patExpands}} \pflabel{J}
      \item $\segOKP{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\segof{\cpv}}{b}$ \BY{IH on \pfref{patExpands}} \pflabel{K}
      \item $\cvalidPP{\uOmega'}{\psceneP{\omega : \Omega_\text{params}}{\uOmega}{\uPhi}{b}}{\cpv}{p}{\tau_\text{proto}}$\BY{IH on \pfref{patExpands}} \pflabel{L}
      \item $\tau'=[\omega]\tau_\text{proto}$ \BY{IH on \pfref{patExpands}}\pflabel{L2}
      \item $
      \segof{\cpv} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon} \cup
                 \sseq{\acesplicedp{m''_i}{n''_i}{\ctau_i}}{\npat}
      $\BY{IH on \pfref{patExpands}} \pflabel{M}
      \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$ and $\sseq{\iskind{\Omega_\text{app}}{\kappa_i}}{\nkind}$\BY{IH on \pfref{patExpands}} \pflabel{N}
      \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$ and $\sseq{\iskind{\Omega_\text{app}}{[\omega]\kappa'_i}}{\ncon}$\BY{IH on \pfref{patExpands}} \pflabel{O}
      \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$ and $\sseq{\haskind{\Omega_\text{app}}{c_i}{[\omega]\kappa'_i}}{\ncon}$\BY{IH on \pfref{patExpands}} \pflabel{P}
      \item $\sseq{\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau_i}{\tau_i}{\akty}}{\npat}$ and $\sseq{\haskind{\Omega_\text{app}}{[\omega]\tau_i}{\akty}}{\npat}$\BY{IH on \pfref{patExpands}} \pflabel{Q}
      \item $\sseq{\patExpandsP{\uOmegaEx{\emptyset}{\uG_i}{\emptyset}{\Omega_i}}{\uPhi}{\parseUPatF{\bsubseq{b}{m''_i}{n''_i}}}{p_i}{[\omega]\tau_i}}{\npat}$ and $\sseq{\patTypePC{\Omega_\text{app}}{\Omega_i}{p_i}{[\omega]\tau_i}}{\npat}$\BY{IH on \pfref{patExpands}} \pflabel{R}
      \item $\uOmega'=\uOmegaEx{\emptyset}{\biguplus_{0 \leq i < \npat} \uG_i}{\emptyset}{\bigcup_{0 \leq i < \npat} \Omega_i}$\BY{IH on \pfref{patExpands}} \pflabel{S}
      \item $\issubtypeP{\Omega_\text{app}}{\tau''}{\tau}$ \BY{Rule (\ref{rule:issubtypeP-trans}) on \pfref{C} and \pfref{issubtypeP}} \pflabel{issubtypeF}
    \end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
      \item \pfref{A}
      \item \pfref{B}
      \item Choosing $\tau''$, by \pfref{C} and \pfref{issubtypeF}
      \item \pfref{D}
      \item \pfref{E}
      \item \pfref{F}
      \item \pfref{G}
      \item \pfref{H}
      \item \pfref{I}
      \item \pfref{J}
      \item \pfref{K}
      \item \pfref{L}
      \item \pfref{L2}
      \item \pfref{M}
      \item \pfref{N}
      \item \pfref{O}
      \item \pfref{P}
      \item \pfref{Q}
      \item \pfref{R}
      \item \pfref{S}
    \end{enumerate}
    \resetpfcounter
  \item[\text{(\ref{rule:patExpandsP-apuptsm})}] ~
    \begin{pfsteps*}
      \item $\uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}$ \BY{assumption} \pflabel{uOmega}
      \item $\uPhi=\uAS{\uA}{\Phi}$ \BY{assumption} \pflabel{uPhi}
      \item $\tsmexpExpandsPat{\uOmega}{\uPhi}{\uepsilon}{\epsilon}{\aetype{\tau_\text{final}}}$ \BY{assumption} \pflabel{tsmexpExpands}
      \item $\tsmexpEvalsPat{\Omega_\text{app}}{\Phi}{\epsilon}{\epsilon_\text{normal}}$ \BY{assumption} \pflabel{tsmexpEvals}
      \item $\tsmdefof{\epsilon_\text{normal}}=a$ \BY{assumption} \pflabel{tsmdefof}
      \item $\Phi = \Phi', \pptsmdefn{a}{\rho}{\eparse}$ \BY{assumption} \pflabel{Phi}
      \item $\encodeBody{b}{\ebody}$ \BY{assumption} \pflabel{encodeBody}
      \item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessP}}{e_\text{pproto}}}$ \BY{assumption} \pflabel{eval}
      \item $\decodePCEPat{e_\text{pproto}}{\pcp}$ \BY{assumption} \pflabel{decode}
      \item $\prepcp{\Omega_\text{app}}{\Phi}{\pcp}{\cpv}{\epsilon_\text{normal}}{\aetype{\tau_\text{proto}}}{\omega}{\Omega_\text{params}}$ \BY{assumption} \pflabel{prep}
      \item $\segOKP{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\segof{\cpv}}{b}$ \BY{assumption} \pflabel{segOK}
      \item $\cvalidPP{\uOmega'}{\psceneP{\omega : \Omega_\text{params}}{\uOmega}{\uPhi}{b}}{\cpv}{p}{\tau_\text{proto}}$ \BY{assumption} \pflabel{cvalidPP}
        \item $\tau=[\omega]\tau_\text{proto}$ \BY{assumption} \pflabel{tau}
      \item $\hastsmtypePat{\Omega_\text{app}}{\Psi}{\epsilon}{\aetype{\tau_\text{final}}}$ \BY{Theorem \ref{thm:ppTLM-expression-expansion} on \pfref{tsmexpExpands}} \pflabel{hastsmtype}
      \item $\hastsmtypePat{\Omega_\text{app}}{\Psi}{\epsilon_\text{normal}}{\aetype{\tau_\text{final}}}$ \BY{Corollary \ref{thm:ppTLM-preservation-evaluation} on \pfref{hastsmtype} and \pfref{tsmexpEvals}} \pflabel{hastsmtypeNormal}
      \item $\hastsmtypePat{\Omega_\text{app}}{\Psi}{\epsilon_\text{normal}}{[\omega]\aetype{\tau_\text{proto}}}$ \BY{Lemma \ref{lemma:pattern-deparameterization-P} on \pfref{prep}} \pflabel{hastsmtypeNormal2}
      \item $\aetype{\tau_\text{final}} = [\omega]\aetype{\tau_\text{proto}}$ \BY{Theorem \ref{lemma:ppTLM-unicity} on \pfref{hastsmtypeNormal} and \pfref{hastsmtypeNormal2}} \pflabel{tsmtypeseq}
      \item $\tau_\text{final}=[\omega]\tau_\text{proto}=\tau$ \BY{definition of substitution and \pfref{tsmtypeseq} and \pfref{tau}} \pflabel{feqp}
      \item $\istsmty{\Omega_\text{app}}{\aetype{\tau_\text{final}}}$ \BY{Lemma \ref{lemma:ppTLM-regularity} on \pfref{hastsmtype}} \pflabel{istsmty}
      \item $\haskind{\Omega_\text{app}}{\tau}{\akty}$ \BY{Inversion of Rule (\ref{rule:istsmty-type}) on \pfref{istsmty}} \pflabel{istype}
      \item $\cequal{\Omega_\text{app}}{\tau}{\tau}{\akty}$ \BY{Rule (\ref{rule:cequal-refl}) on \pfref{istype}} \pflabel{refl}
      \item $\issubtypeP{\Omega_\text{app}}{\tau}{\tau}$ \BY{Rule (\ref{rule:issubtypeP-equal}) on \pfref{refl}} \pflabel{subty}
      \item $\patExpandsP{\uOmega'}{\uPhi}{\utsmap{\uepsilon}{b}}{p}{\tau}$ \BY{assumption} \pflabel{patExpandsA}
      \item $\patTypePC{\Omega_\text{app}}{\uOmega'}{p}{\tau}$ \BY{Theorem \ref{thm:typed-pattern-expansion-P} on \pfref{patExpandsA}} \pflabel{patType}
      \item $\hastypeP{\Omega_\text{app}}{\omega}{\Omega_\text{params}}$ \BY{Lemma \ref{lemma:pattern-deparameterization-P} on \pfref{prep}} \pflabel{omegaType}
      \item $\segof{\cpv} = \sseq{\acesplicedk{m_i}{n_i}}{\nkind} \cup \sseq{\acesplicedc{m'_i}{n'_i}{\cekappa'_i}}{\ncon} \cup
             \sseq{\acesplicedp{m''_i}{n''_i}{\ctau_i}}{\npat}
  $ \BY{definition} \pflabel{summ}
      \item $\sseq{\kExpands{\uOmega}{\parseUKindF{\bsubseq{b}{m_i}{n_i}}}{\kappa_i}}{\nkind}$ \BY{Lemma \ref{lemma:proto-pattern-decomposition-P} on \pfref{cvalidPP} and \pfref{summ}} \pflabel{A}
      \item $\sseq{\iskind{\Omega_\text{app}}{\kappa_i}}{\nkind}$ \BY{Lemma \ref{thm:kind-and-constructor-expansion-P} over \pfref{A}} \pflabel{A2}
      \item $\sseq{\cvalidK{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\cekappa'_i}{\kappa'_i}}{\ncon}$ \BY{Lemma \ref{lemma:proto-pattern-decomposition-P} on \pfref{cvalidPP} and \pfref{summ}} \pflabel{B}
      \item $\sseq{\iskind{\Omega_\text{app} \cup \Omega_\text{params}}{\kappa'_i}}{\ncon}$ \BY{Theorem \ref{thm:kind-and-constructor-expansion-P} over \pfref{B}} \pflabel{B2}
      \item $\sseq{\iskind{\Omega_\text{app}}{[\omega]\kappa'_i}}{\ncon}$ \BY{Lemma \ref{lemma:substitution-P} over \pfref{B2} and \pfref{omegaType}} \pflabel{B3}
      \item $\sseq{\cExpands{\uOmega}{\parseUConF{\bsubseq{b}{m'_i}{n'_i}}}{c_i}{[\omega]\kappa'_i}}{\ncon}$ \BY{Lemma \ref{lemma:proto-pattern-decomposition-P} on \pfref{cvalidPP} and \pfref{summ}} \pflabel{C}
      \item $\sseq{\haskind{\Omega_\text{app}}{[\omega]\tau_i}{\akty}}{\npat}$ \BY{Theorem \ref{thm:kind-and-constructor-expansion-P} over \pfref{C}} \pflabel{C2}
      \item $\sseq{\cvalidC{\OParams}{\csceneP{\omega : \OParams}{\uOmega}{b}}{\ctau_i}{\tau_i}{\akty}}{\npat}$ \BY{Lemma \ref{lemma:proto-pattern-decomposition-P} on \pfref{cvalidPP} and \pfref{summ}} \pflabel{D}
      \item $\sseq{\haskind{\Omega_\text{app} \cup \OParams}{\tau_i}{\akty}}{\npat}$ \BY{Theorem \ref{thm:kind-and-constructor-expansion-P} over \pfref{D}} \pflabel{D2}
      \item $\sseq{\haskind{\Omega_\text{app}}{[\omega]\tau_i}{\akty}}{\npat}$ \BY{Lemma \ref{lemma:substitution-P} over \pfref{D2} and \pfref{omegaType}} \pflabel{D3}
      \item $\sseq{\patExpandsP{\uOmegaEx{\emptyset}{\uG_i}{\emptyset}{\Omega_i}}{\uPhi}{\parseUPatF{\bsubseq{b}{m''_i}{n''_i}}}{p_i}{[\omega]\tau_i}}{\npat}$ \BY{Lemma \ref{lemma:proto-pattern-decomposition-P} on \pfref{cvalidPP} and \pfref{summ}} \pflabel{E}
      \item $\sseq{\patTypePC{\Omega_\text{app}}{\Omega_i}{p_i}{[\omega]\tau_i}}{\npat}$ \BY{Theorem \ref{thm:typed-pattern-expansion-P} over \pfref{E}} \pflabel{E2}
      \item $\uOmega'=\uOmegaEx{\emptyset}{\biguplus_{0 \leq i < \npat} \uG_i}{\emptyset}{\bigcup_{0 \leq i < \npat} \Omega_i}$ \BY{Lemma \ref{lemma:proto-pattern-decomposition-P} on \pfref{cvalidPP} and \pfref{summ}} \pflabel{F}
    \end{pfsteps*}
    The conclusions hold as follows:
    \begin{enumerate}
      \item \pfref{uOmega}
      \item \pfref{uPhi}
      \item Choosing $\tau$, by \pfref{tsmexpExpands} and \pfref{patType} and \pfref{subty}
      \item \pfref{tsmexpEvals}
      \item \pfref{tsmdefof}
      \item \pfref{Phi}
      \item \pfref{encodeBody}
      \item \pfref{eval}
      \item \pfref{decode}
      \item \pfref{prep}
      \item \pfref{segOK}
      \item \pfref{cvalidPP}
      \item \pfref{tau}
      \item \pfref{summ}
      \item \pfref{A} and \pfref{A2}
      \item \pfref{B} and \pfref{B3}
      \item \pfref{C} and \pfref{C2}
      \item \pfref{D} and \pfref{D3}
      \item \pfref{E} and \pfref{E2}
      \item \pfref{F}
    \end{enumerate}
    \resetpfcounter
\end{byCases}
\end{proof}

% \begin{byCases}
%   \item[\text{(\ref{rule:patExpandsP-subsume})}] \todo{subsumption}
%   \item[\text{(\ref{rule:patExpandsP-var}) \textbf{through} (\ref{rule:patExpandsP-in})}] \todo{inductive cases}
%   \item[\text{(\ref{rule:patExpandsP-apuptsm})}] ~
%     \begin{pfsteps*}
%       \item 
%     \end{pfsteps*}
%     \resetpfcounter
%     The conclusions hold as follows:
%     \begin{enumerate}
%      \item \todo{1}
%      \item \todo{2}
%      \item \todo{3}
%      \item \todo{4}
%      \item \todo{5}
%      \item \todo{6}
%      \item \todo{7}
%      \item \todo{8}
%      \item \todo{9}
%      \item \todo{10}
%      \item \todo{11}
%      \item \todo{12}
%      \item \todo{13}
%      \item \todo{14}
%      \item \todo{15}
%      \item \todo{16}
%      \item \todo{17}
%      \item \todo{18}
%     \end{enumerate}
% \end{byCases}

% \end{proof}

\ificfp 
\else

\chapter{\texorpdfstring{Bidirectional $\miniVersePat$}{Bidirectional miniVerseS}}\label{appendix:simple-implicits}

\section{Expanded Language (XL)}
The Bidirectional $\miniVersePat$ expanded language (XL) is the same as the  $\miniVersePat$ XL, which was detailed in Appendix \ref{appendix:SES-XL}. %It consists of types, $\tau$, expanded expressions, $e$, expanded rules, $r$, and expanded patterns, $p$.

\section{Unexpanded Language (UL)}
\subsection{Syntax}
\subsubsection{Stylized Syntax}
The stylized syntax extends the stylized syntax of the $\miniVersePat$ UL given in Sec. \ref{appendix:SES-syntax}.

\[\begin{array}{lllllll}
\textbf{Sort} & & 
%& \textbf{Operational Form} 
& \textbf{Stylized Form} & \textbf{Description}\\
\mathsf{UTyp} & \utau & ::= 
%& \cdots 
& \cdots & \text{(as in $\miniVersePat$)}\\
\mathsf{UExp} & \ue & ::= 
%& \cdots 
& \cdots & \text{(as in $\miniVersePat$)}\\
% &&
%& \auasc{\utau}{\ue} 
% & \asc{\ue}{\utau} & \text{ascription}\\
% &&
%& \auletsyn{\ux}{\ue}{\ue} 
% & \letsyn{\ux}{\ue}{\ue} & \text{value binding}\\
&&& \implicite{\tsmv}{\ue} & \text{seTLM designation}\\
&&& \implicitp{\tsmv}{\ue} & \text{spTLM designation}\\
&&& \lit{b} & \text{seTLM unadorned literal}\\
% &&& \auanalam{\ux}{\ue} & \analam{\ux}{\ue} & \text{abstraction (unannotated)}\\
% &&& \aulam{\utau}{\ux}{\ue} & \lam{\ux}{\utau}{\ue} & \text{abstraction (annotated)}\\
% &&& \auap{\ue}{\ue} & \ap{\ue}{\ue} & \text{application}\\
% &&& \autlam{\ut}{\ue} & \Lam{\ut}{\ue} & \text{type abstraction}\\
% &&& \autap{\ue}{\utau} & \App{\ue}{\utau} & \text{type application}\\
% &&& \auanafold{\ue} & \fold{\ue} & \text{fold}\\
% &&& \auunfold{\ue} & \unfold{\ue} & \text{unfold}\\
% &&& \autpl{\labelset}{\mapschema{\ue}{i}{\labelset}} & \tpl{\mapschema{\ue}{i}{\labelset}} & \text{labeled tuple}\\
% &&& \aupr{\ell}{\ue} & \prj{\ue}{\ell} & \text{projection}\\
% &&& \auanain{\ell}{\ue} & \inj{\ell}{\ue} & \text{injection}\\
% &&& \aumatchwithb{n}{\ue}{\seqschemaX{\urv}} & \matchwith{\ue}{\seqschemaX{\urv}} & \text{match}\\
% &&& \audefuetsm{\utau}{e}{\tsmv}{\ue} & \texttt{syntax}~\tsmv~\texttt{at}~\utau~\texttt{for} & \text{ueTLM definition}\\
% &&&                                    & \texttt{expressions}~\{e\}~\texttt{in}~\ue\\
% \LCC &&& \color{light-gray} & \color{light-gray} & \color{light-gray} \\
% &&& \auimplicite{\tsmv}{\ue} & \texttt{implicit\,syntax}~\tsmv~\texttt{for} & \text{ueTLM designation}\\
% &&&                          & \texttt{expressions\,in}~\ue\\ \ECC
% &&& \autsmap{b}{\tsmv} & \utsmap{\tsmv}{b} & \text{ueTLM application}\\%\ECC
% \LCC &&& \color{light-gray} & \color{light-gray} & \color{light-gray} \\
% &&& \auelit{b} & {\lit{b}}  & \text{ueTLM unadorned literal}\\\ECC
% &&& \audefuptsm{\utau}{e}{\tsmv}{\ue} & \texttt{syntax}~\tsmv~\texttt{at}~\utau~\texttt{for} & \text{upTLM definition}\\
% &&&                                    & \texttt{patterns}~\{e\}~\texttt{in}~\ue\\
% \LCC &&& \color{light-gray} & \color{light-gray} & \color{light-gray} \\
% &&& \auimplicitp{\tsmv}{\ue} & \texttt{implicit\,syntax}~\tsmv~\texttt{for} & \text{upTLM designation}\\
% &&&                          & \texttt{patterns\,in}~\ue\\ \ECC
\mathsf{URule} & \urv & ::= 
%& \aumatchrule{\upv}{\ue} 
& \cdots & \text{(as in $\miniVersePat$)}\\
\mathsf{UPat} & \upv & ::= 
%& \ux 
& \cdots & \text{(as in $\miniVersePat$)}\\
&&& \lit{b} & \text{spTLM unadorned literal}
% &&& \auwildp & \wildp & \text{wildcard pattern}\\
% &&& \aufoldp{\upv} & \foldp{\upv} & \text{fold pattern}\\
% &&& \autplp{\labelset}{\mapschema{\upv}{i}{\labelset}} & \tplp{\mapschema{\upv}{i}{\labelset}} & \text{labeled tuple pattern}\\
% &&& \auinjp{\ell}{\upv} & \injp{\ell}{\upv} & \text{injection pattern}\\
% &&& \auapuptsm{b}{\tsmv} & \utsmap{\tsmv}{b} & \text{upTLM application}\\
% \LCC &&& \color{light-gray} & \color{light-gray} & \color{light-gray}\\
% &&& \auplit{b} & \lit{b} & \text{upTLM unadorned literal}\ECC
\end{array}\]
\subsubsection{Body Lengths}
We write $\sizeof{b}$ for the length of $b$. The metafunction $\sizeof{\ue}$ computes the sum of the lengths of expression literal bodies in $\ue$. It is defined by extending the definition given in Sec. \ref{appendix:SES-syntax} with the following additional cases:
\[
\begin{array}{ll}
% \sizeof{\asc{\ue}{\utau}} & = \sizeof{\ue}\\
% \sizeof{\letsyn{\ux}{\ue_1}{\ue_2}} & = \sizeof{\ue_1} + \sizeof{\ue_2}\\
\sizeof{\implicite{\tsmv}{\ue}} & = \sizeof{\ue}\\
\sizeof{\implicitp{\tsmv}{\ue}} & = \sizeof{\ue}\\
\sizeof{\lit{b}} & = \sizeof{b}
\end{array}
\]

Similarly, the metafunction $\sizeof{\upv}$ computes the sum of the lengths of the pattern literal bodies in $\upv$. It is defined by extending the definition given in Sec. \ref{appendix:SES-syntax} with the following additional case:
\begin{align*}
\sizeof{\lit{b}} & = \sizeof{b}
\end{align*}

\subsubsection{Textual Syntax}
In addition to the stylized syntax, there is also a context-free textual syntax for the UL. We need only posit the existence of the following partial metafunctions.

\begin{condition}[Textual Representability]\label{condition:textual-representability-BS} ~
\begin{enumerate}
\item For each $\utau$, there exists $b$ such that $\parseUTyp{b}{\utau}$. 
\item For each $\ue$, there exists $b$ such that $\parseUExp{b}{\ue}$.
% \item For each $\urv$, there exists $b$ such that $\parseURule{b}{\urv}$.
\item {For each $\upv$, there exists $b$ such that $\parseUPat{b}{\upv}$.}
\end{enumerate}
\end{condition}

We also impose the following technical conditions.

\begin{condition}[Expression Parsing Monotonicity]\label{condition:body-parsing-BS} If $\parseUExp{b}{\ue}$ then $\sizeof{\ue} < \sizeof{b}$.\end{condition}

\begin{condition}[Pattern Parsing Monotonicity]\label{condition:pattern-parsing-BS} If $\parseUPat{b}{\upv}$ then $\sizeof{\upv} < \sizeof{b}$.\end{condition}

\subsection{Bidirectionally Typed Expansion}
\subsubsection{Contexts}
\emph{Unexpanded type formation contexts}, $\uDelta$, and \emph{unexpanded typing contexts}, $\uGamma$, were defined in Sec. \ref{appendix:typed-expression-expansion-S}.

\subsubsection{Body Encoding and Decoding}
The type $\tBody$ and the judgements $\encodeBody{b}{e}$ and $\decodeBody{e}{b}$ are characterized in Sec. \ref{appendix:typed-expression-expansion-S}.

\subsubsection{Parse Results}
The types $\tParseResultExp$ and $\tParseResultPat$ are defined as in Sec. \ref{appendix:typed-expression-expansion-S}.

\subsubsection{TLM Contexts}

\emph{seTLM contexts}, $\uPsi$, are of the form $\uASI{\uA}{\Psi}{\uI}$, where $\uA$ is a \emph{TLM identifier expansion context}, $\Psi$ is a \emph{seTLM definition context} and $\uI$ is a \emph{TLM implicit designation context}.

\emph{spTLM contexts}, $\uPhi$, are of the form $\uASI{\uA}{\Phi}{\uI}$, where $\uA$ is a {TLM identifier expansion context}, defined above, and $\Phi$ is a \emph{spTLM definition context}. 

A \emph{TLM identifier expansion context}, $\uA$, is a finite function mapping each TLM identifier $\tsmv \in \domof{\uA}$ to the \emph{TLM identifier expansion}, $\vExpands{\tsmv}{a}$, for some \emph{TLM name}, $a$. We write $\ctxUpdate{\uA}{\tsmv}{a}$ for the TLM identifier expansion context that maps $\tsmv$ to $\vExpands{\tsmv}{a}$, and defers to $\uA$ for all other TLM identifiers (i.e. the previous mapping is \emph{updated}.)

An \emph{seTLM definition context}, $\Psi$, is a finite function mapping each TLM name $a \in \domof{\Psi}$ to an \emph{expanded seTLM definition}, $\xuetsmbnd{a}{\tau}{\eparse}$, where $\tau$ is the seTLM's type annotation, and $\eparse$ is its parse function. We write $\Psi, \xuetsmbnd{a}{\tau}{\eparse}$ when $a \notin \domof{\Psi}$ for the extension of $\Psi$ that maps $a$ to $\xuetsmbnd{a}{\tau}{\eparse}$. We write $\uetsmenv{\Delta}{\Psi}$  when all the type annotations in $\Psi$ are well-formed assuming $\Delta$, and the parse functions in $\Psi$ are closed and of the appropriate type.

\begin{definition}[seTLM Definition Context Formation]\label{def:seTLM-def-ctx-formation-B} $\uetsmenv{\Delta}{\Psi}$ iff for each $\xuetsmbnd{a}{\tau}{\eparse} \in \Psi$, we have $\istypeU{\Delta}{\tau}$ and $\hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultExp}}$.\end{definition}

An \emph{spTLM definition context}, $\Phi$, is a finite function mapping each TLM name $a \in \domof{\Phi}$ to an \emph{expanded seTLM definition}, $\xuptsmbnd{a}{\tau}{\eparse}$, where $\tau$ is the spTLM's type annotation, and $\eparse$ is its parse function. We write $\Phi, \xuptsmbnd{a}{\tau}{\eparse}$ when $a \notin \domof{\Phi}$ for the extension of $\Phi$ that maps $a$ to $\xuptsmbnd{a}{\tau}{\eparse}$. We write $\uptsmenv{\Delta}{\Phi}$  when all the type annotations in $\Phi$ are well-formed assuming $\Delta$, and the parse functions in $\Phi$ are closed and of the appropriate type.

\begin{definition}[spTLM Definition Context Formation]\label{def:spTLM-def-ctx-formation-B} $\uptsmenv{\Delta}{\Phi}$ iff for each $\xuptsmbnd{a}{\tau}{\eparse} \in \Phi$, we have $\istypeU{\Delta}{\tau}$ and $\hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPat}}$.\end{definition}

A \emph{TLM implicit designation context}, $\uI$, is a finite function that maps each type $\tau \in \domof{\uI}$ to the \emph{TLM designation} $\designate{\tau}{a}$, for some TLM name $a$. We write $\uI \uplus \designate{\tau}{a}$ for the TLM implicit designation context that maps $\tau$ to $\designate{\tau}{a}$ and defers to $\uI$ for all other types (i.e. the previous designation, if any, is updated.)

\begin{definition}[TLM Implicit Designation Context Formation]\label{def:implicit-designation-ctx-formation-S} $\uIOK{\Delta}{\uI}$ iff for each $\designate{\tau}{a} \in \uI$, we have $\istypeU{\Delta}{\tau}$.
\end{definition}

\begin{definition}[seTLM Context Formation] $\uetsmctx{\Delta}{\uASI{\uA}{\Psi}{\uI}}$ iff
\begin{enumerate}
\item $\uetsmenv{\Delta}{\Psi}$; and
\item for each $\vExpands{\tsmv}{a} \in \uA$ we have $a \in \domof{\Psi}$; and
\item $\uIOK{\Delta}{\uI}$; and
\item for each $\designate{\tau}{a} \in \uI$, we have $a \in \domof{\Psi}$.
\end{enumerate}
\end{definition}

\begin{definition}[spTLM Context Formation] $\uptsmctx{\Delta}{\uASI{\uA}{\Phi}{\uI}}$ iff 
\begin{enumerate}
\item $\uptsmenv{\Delta}{\Phi}$; and
\item for each $\vExpands{\tsmv}{a} \in \uA$ we have $a \in \domof{\Phi}$; and
\item $\uIOK{\Delta}{\uI}$; and
\item for each $\designate{\tau}{a} \in \uI$ we have $a \in \domof{\Phi}$.
\end{enumerate}
\end{definition}

We define $\uPsi, \uShyp{\tsmv}{a}{\tau}{\eparse}$, when $\uPsi=\uASI{\uA}{\Phi}{\uI}$, as an abbreviation of \[\uASI{\ctxUpdate{\uA}{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}\]
%\vspace{10px}

We define $\uPhi, \uPhyp{\tsmv}{a}{\tau}{\eparse}$, when $\uPhi=\uASI{\uA}{\Phi}{\uI}$, as an abbreviation of \[\uASI{\ctxUpdate{\uA}{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI}\]

\subsubsection{Type Expansion}
The \emph{type expansion judgement}, $\expandsTU{\uDelta}{\utau}{\tau}$, is inductively defined as in $\miniVersePat$ by Rules (\ref{rules:expandsTU}).

\subsubsection{Bidirectionally Typed Expression and Rule Expansion}
\noindent\fbox{$\esyn{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}$}~~$\ue$ has expansion $e$ synthesizing type $\tau$

\begin{subequations}\label{rules:esyn-S}
\begin{equation}\label{rule:esyn-S-var}
  \inferrule{ }{ 
    \esyn{\uDelta}{\uGamma, \uGhyp{\ux}{x}{\tau}}{\uPsi}{\uPhi}{\ux}{x}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:esyn-S-asc}
  \inferrule{
    \expandsTU{\uDelta}{\utau}{\tau}\\
    \eanaX{\ue}{e}{\tau}
  }{
    \esynX{\asc{\ue}{\utau}}{e}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:esyn-S-let}
  \inferrule{
    \esynX{\ue}{e}{\tau}\\
    \esyn{\uDelta}{\uGamma, \uGhyp{\ux}{x}{\tau}}{\uPsi}{\uPhi}{\ue'}{e'}{\tau'}
  }{
    \esynX{\letsyn{\ux}{\ue}{\ue'}}{\aeap{\aelam{\tau}{x}{e'}}{e}}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:esyn-S-lam}
  \inferrule{
    \expandsTU{\uDelta}{\utau_1}{\tau_1}\\
    \esyn{\uDelta}{\uGamma, \uGhyp{\ux}{x}{\tau_1}}{\uPsi}{\uPhi}{\ue}{e}{\tau_2}
  }{
    \esynX{\lam{\ux}{\utau_1}{\ue}}{\aelam{\tau_1}{x}{e}}{\aparr{\tau_1}{\tau_2}}
  }
\end{equation}
\begin{equation}\label{rule:esyn-S-ap}
  \inferrule{
    \esynX{\ue_1}{e_1}{\aparr{\tau_2}{\tau}}\\
    \eanaX{\ue_2}{e_2}{\tau_2}
  }{
    \esynX{\ap{\ue_1}{\ue_2}}{\aeap{e_1}{e_2}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:esyn-S-tlam}
  \inferrule{
    \esyn{\uDelta, \uDhyp{\ut}{t}}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}
  }{
    \esynX{\Lam{\ut}{\ue}}{\aetlam{t}{e}}{\aall{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:esyn-S-tap}
  \inferrule{
    \esynX{\ue}{e}{\aall{t}{\tau}}\\
    \expandsTU{\uDelta}{\utau'}{\tau'}
  }{
    \esynX{\App{\ue}{\utau'}}{\aetap{e}{\tau'}}{[\tau'/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:esyn-S-unfold}
  \inferrule{
    \esynX{\ue}{e}{\arec{t}{\tau}}
  }{
    \esynX{\unfold{\ue}}{\aeunfold{e}}{[\arec{t}{\tau}/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:esyn-S-tpl}
  \inferrule{
    \{\esynX{\ue_i}{e_i}{\tau_i}\}_{i \in \labelset}
  }{
    \esynX{\tpl{\mapschema{\ue}{i}{\labelset}}}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}
\begin{equation}\label{rule:esyn-S-pr}
  \inferrule{
    \esynX{\ue}{e}{\aprod{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
  }{
    \esynX{\prj{\ue}{\ell}}{\aepr{\ell}{e}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:esyn-defuetsm}
\inferrule{
  \expandsTU{\uDelta}{\utau}{\tau}\\
  \hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultExp}}\\\\
  \evalU{\eparse}{\eparse'}\\
  \esyn{\uDelta}{\uGamma}{\uPsi, \uShyp{\tsmv}{a}{\tau}{\eparse'}}{\uPhi}{\ue}{e}{\tau'}
}{
  \esynX{\usyntaxueP{\tsmv}{\utau}{\eparse}{\ue}}{e}{\tau'}
}
\end{equation}
\begin{equation}\label{rule:esyn-S-apuetsm}
\inferrule{
  \uPsi = \uPsi', \uShyp{\tsmv}{a}{\tau}{\eparse}\\\\
  \encodeBody{b}{\ebody}\\
  \evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessE}}{\ecand}}\\
  \decodeCondE{\ecand}{\ce}\\\\
    \segOK{\segof{\ce}}{b}\\
  \cana{\emptyset}{\emptyset}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}
}{
  \esyn{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\utsmap{\tsmv}{b}}{e}{\tau}
}
\end{equation}
\begin{equation}\label{rule:esyn-S-implicite}
  \inferrule{
    \uPsi = \uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}\\\\
    \esyn{\uDelta}{\uGamma}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\uPhi}{\ue}{e}{\tau'}
  }{
    \esyn{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\implicite{\tsmv}{\ue}}{e}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:esyn-S-defuptsm}
\inferrule{
  \expandsTU{\uDelta}{\utau}{\tau}\\
  \hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPat}}\\\\
  \evalU{\eparse}{\eparse'}\\
  \esyn{\uDelta}{\uGamma}{\uPsi}{\uPhi, \uPhyp{\tsmv}{a}{\tau}{\eparse'}}{\ue}{e}{\tau'}
}{
  \esynX{\usyntaxup{\tsmv}{\utau}{\eparse}{\ue}}{e}{\tau'}
}
\end{equation}
\begin{equation}\label{rule:esyn-S-implicitp}
  \inferrule{
    \uPhi = \uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI}\\\\
    \esyn{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\ue}{e}{\tau'}
  }{
    \esyn{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\implicitp{\tsmv}{\ue}}{e}{\tau'}
  }
\end{equation}
\end{subequations}

\noindent\fbox{$\eana{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}$}~~$\ue$ has expansion $e$ when analyzed against type $\tau$

\begin{subequations}\label{rules:eana-S}
\begin{equation}\label{rule:eana-S-subsume}
  \inferrule{
    \esynX{\ue}{e}{\tau}
  }{
    \eanaX{\ue}{e}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:eana-S-let}
  \inferrule{
    \esynX{\ue}{e}{\tau}\\
    \eana{\uDelta}{\uGamma, \uGhyp{\ux}{x}{\tau}}{\uPsi}{\uPhi}{\ue'}{e'}{\tau'}
  }{
    \eanaX{\letsyn{\ux}{\ue}{\ue'}}{\aeap{\aelam{\tau}{x}{e'}}{e}}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:eana-S-tlam}
  \inferrule{
    \eana{\uDelta, \uDhyp{\ut}{t}}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}
  }{
    \eanaX{\Lam{\ut}{\ue}}{\aetlam{t}{e}}{\aall{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:eana-S-fold}
  \inferrule{
    \eanaX{\ue}{e}{[\arec{t}{\tau}/t]\tau}
  }{
    \eanaX{\fold{\ue}}{\aefold{e}}{\arec{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:eana-S-tpl}
  \inferrule{
    \{\eanaX{\ue_i}{e_i}{\tau_i}\}_{i \in \labelset}
  }{
    \eanaX{\tpl{\mapschema{\ue}{i}{\labelset}}}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
  }
\end{equation}
\begin{equation}\label{rule:eana-S-in}
  \inferrule{
    \eanaX{\ue}{e}{\tau'}
  }{
    \eanaX{\inj{\ell}{\ue}}{\aein{\ell}{e}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}}
  }
\end{equation}
\begin{equation}\label{rule:eana-S-match}
  \inferrule{
    \esynX{\ue}{e}{\tau}\\
    \{\ranaX{\urv_i}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}
  }{
    \eanaX{\matchwith{\ue}{\seqschemaX{\urv}}}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:eana-S-defuetsm}
\inferrule{
  \expandsTU{\uDelta}{\utau}{\tau}\\
  \hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultExp}}\\\\
  \evalU{\eparse}{\eparse'}\\
  \eana{\uDelta}{\uGamma}{\uPsi, \uShyp{\tsmv}{a}{\tau}{\eparse'}}{\uPhi}{\ue}{e}{\tau'}
}{
  \eanaX{\usyntaxueP{\tsmv}{\utau}{\eparse}{\ue}}{e}{\tau'}
}
\end{equation}
\begin{equation}\label{rule:eana-S-implicite}
  \inferrule{
    \uPsi = \uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}\\\\
    \eana{\uDelta}{\uGamma}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\uPhi}{\ue}{e}{\tau'}
  }{
    \eana{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\implicite{\tsmv}{\ue}}{e}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:eana-S-lit}
  \inferrule{
    \uPsi = \uASI{\uA}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}\\\\
  \encodeBody{b}{\ebody}\\
  \evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessE}}{\ecand}}\\
  \decodeCondE{\ecand}{\ce}\\\\
    \segOK{\segof{\ce}}{b}\\
  \cana{\emptyset}{\emptyset}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}
  }{
    \eana{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\lit{b}}{e}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:eana-S-defuptsm}
\inferrule{
  \expandsTU{\uDelta}{\utau}{\tau}\\
  \hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPat}}\\\\
  \evalU{\eparse}{\eparse'}\\
  \eana{\uDelta}{\uGamma}{\uPsi}{\uPhi, \uPhyp{\tsmv}{a}{\tau}{\eparse'}}{\ue}{e}{\tau'}
}{
  \eanaX{\usyntaxup{\tsmv}{\utau}{\eparse}{\ue}}{e}{\tau'}
}
\end{equation}
\begin{equation}\label{rule:eana-S-implicitp}
  \inferrule{
    \uPhi = \uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI}\\\\
    \eana{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\ue}{e}{\tau'}
  }{
    \eana{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\implicitp{\tsmv}{\ue}}{e}{\tau'}
  }
\end{equation}
\end{subequations}

\noindent\fbox{$\rana{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\urv}{r}{\tau}{\tau'}$}~~$\urv$ has expansion $r$ taking values of type $\tau$ to values of type $\tau'$

\begin{equation}\label{rule:rana-S}
  \inferrule{
    \patExpands{\uGG{\uG'}{\Gamma'}}{\uPhi}{\upv}{p}{\tau}\\
    \eana{\uDD{\uD}{\Delta}}{\uGG{\uG \uplus \uG'}{\Gamma \cup \Gamma'}}{\uPsi}{\uPhi}{\ue}{e}{\tau'}
  }{
    \rana{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\matchrule{\upv}{\ue}}{\aematchrule{p}{e}}{\tau}{\tau'}
  }
\end{equation}

\subsubsection{Pattern Expansion}
\noindent\fbox{$\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}$}~~$\upv$ has expansion $p$ matching against $\tau$ generating hypotheses $\upctx$

\begin{subequations}\label{rules:patExpands-B}
\begin{equation}\label{rule:patExpands-B-var}
\inferrule{ }{
  \patExpands{\uGG{\vExpands{\ux}{x}}{\Ghyp{x}{\tau}}}{\uPhi}{\ux}{x}{\tau}
}
\end{equation}
\begin{equation}\label{rule:patExpands-B-wild}
\inferrule{ }{
  \patExpands{\uGG{\emptyset}{\emptyset}}{\uPhi}{\wildp}{\aewildp}{\tau}
}
\end{equation}
\begin{equation}\label{rule:patExpands-B-fold}
\inferrule{ 
  \patExpands{\upctx}{\uPhi}{\upv}{p}{[\arec{t}{\tau}/t]\tau}
}{
  \patExpands{\upctx}{\uPhi}{\foldp{\upv}}{\aefoldp{p}}{\arec{t}{\tau}}
}
\end{equation}
\begin{equation}\label{rule:patExpands-B-tpl}
\inferrule{
    \tau = \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}\\\\
  \{\patExpands{{\upctx_i}}{\uPhi}{\upv_i}{p_i}{\tau_i}\}_{i \in \labelset}\\
}{
    \patExpands{\GIconsi{i \in \labelset}{\upctx_i}}{\uPhi}{\tplp{\mapschema{\upv}{i}{\labelset}}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\tau}
}
\end{equation}
\begin{equation}\label{rule:patExpands-B-in}
\inferrule{
  \patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}
}{
  \patExpands{\upctx}{\uPhi}{\injp{\ell}{\upv}}{\aeinjp{\ell}{p}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
}
\end{equation}
\begin{equation}\label{rule:patExpands-B-apuptsm}
\inferrule{
  \uPhi = \uPhi', \uPhyp{\tsmv}{a}{\tau}{\eparse}\\\\
  \encodeBody{b}{\ebody}\\
  \evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessP}}{\ecand}}\\
  \decodeCEPat{\ecand}{\cpv}\\\\
    \segOK{\segof{\cpv}}{b}\\
  \cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}
}{
  \patExpands{\upctx}{\uPhi}{\utsmap{\tsmv}{b}}{p}{\tau}
}
\end{equation}
\begin{equation}\label{rule:patExpands-B-lit}
\inferrule{
  \uPhi = \uASI{\uA}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI, \designate{\tau}{a}}\\\\
  \encodeBody{b}{\ebody}\\
  \evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessP}}{\ecand}}\\
  \decodeCEPat{\ecand}{\cpv}\\\\
    \segOK{\segof{\cpv}}{b}\\
  \cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}
}{
  \patExpands{\upctx}{\uPhi}{\lit{b}}{p}{\tau}
}
\end{equation}
\end{subequations}

\section{Proto-Expansion Validation}
\subsection{Syntax of Proto-Expansions}
The syntax of proto-expansions was defined in Sec. \ref{appendix:proto-expansions-SES}.

% \[\begin{array}{lllllll}
% \textbf{Sort} & & & \textbf{Operational Form} & \textbf{Stylized Form} & \textbf{Description}\\
% \mathsf{PrTyp} & \ctau & ::= & \cdots & \cdots & \text{(as in $\miniVersePat$)}\\
% % &&& \aceparr{\ctau}{\ctau} & \parr{\ctau}{\ctau} & \text{partial function}\\
% % &&& \aceall{t}{\ctau} & \forallt{t}{\ctau} & \text{polymorphic}\\
% % &&& \acerec{t}{\ctau} & \rect{t}{\ctau} & \text{recursive}\\
% % &&& \aceprod{\labelset}{\mapschema{\ctau}{i}{\labelset}} & \prodt{\mapschema{\ctau}{i}{\labelset}} & \text{labeled product}\\
% % &&& \acesum{\labelset}{\mapschema{\ctau}{i}{\labelset}} & \sumt{\mapschema{\ctau}{i}{\labelset}} & \text{labeled sum}\\
% %\LCC &&& \gray & \gray & \gray\\
% % &&& \acesplicedt{m}{n} & \splicedt{m}{n} & \text{spliced}\\%\ECC
% \mathsf{PrExp} & \ce & ::= & \cdots & \cdots & \text{(as in $\miniVersePat$)}\\
% &&& \aceasc{\ctau}{\ce} & \asc{\ce}{\ctau} & \text{ascription}\\
% &&& \aceletsyn{x}{\ce}{\ce} & \letsyn{x}{\ce}{\ce} & \text{value binding}\\
% % &&& \aceanalam{x}{\ce} & \analam{x}{\ce} & \text{abstraction (unannotated)}\\
% % &&& \acelam{\ctau}{x}{\ce} & \lam{x}{\ctau}{\ce} & \text{abstraction (annotated)}\\
% % &&& \aceap{\ce}{\ce} & \ap{\ce}{\ce} & \text{application}\\
% % &&& \acetlam{t}{\ce} & \Lam{t}{\ce} & \text{type abstraction}\\
% % &&& \acetap{\ce}{\ctau} & \App{\ce}{\ctau} & \text{type application}\\
% % &&& \aceanafold{\ce} & \fold{\ce} & \text{fold}\\
% % &&& \aceunfold{\ce} & \unfold{\ce} & \text{unfold}\\
% % &&& \acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}} & \tpl{\mapschema{\ce}{i}{\labelset}} & \text{labeled tuple}\\
% % &&& \acepr{\ell}{\ce} & \prj{\ce}{\ell} & \text{projection}\\
% % &&& \aceanain{\ell}{\ce} & \inj{\ell}{\ce} & \text{injection}\\
% % &&& \acematchwithb{n}{\ce}{\seqschemaX{\urv}} & \matchwith{\ce}{\seqschemaX{\crv}} & \text{match}\\%\LCC &&& \gray & \gray & \gray\\
% % &&& \acesplicede{m}{n} & \splicede{m}{n} & \text{spliced}\\%\ECC
% % &&& \acesplicedet{m}{n}{\ctau} & \splicedet{m}{n}{\ctau} & \text{spliced (analytic)}\\
% \mathsf{PrRule} & \crv & ::= & \cdots & \cdots & \text{(as in $\miniVersePat$)}\\
% \mathsf{PrPat} & \cpv & ::= & \cdots & \cdots & \text{(as in $\miniVersePat$)}\\
% % &&& \acefoldp{p} & \foldp{p} & \text{fold pattern}\\
% % &&& \acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}} & \tplp{\mapschema{\cpv}{i}{\labelset}} & \text{labeled tuple pattern}\\
% % &&& \aceinjp{\ell}{\cpv} & \injp{\ell}{\cpv} & \text{injection pattern}\\
% % &&& \acesplicedp{m}{n} & \splicedp{m}{n} & \text{spliced}
% \end{array}\]

\subsubsection{Common Proto-Expansion Terms}
Each expanded term, except variable patterns, maps onto a proto-expansion term. We refer to these as the \emph{common proto-expansion terms}. In particular:
\begin{itemize}
  \item Each type, $\tau$, maps onto a proto-type, $\Cof{\tau}$, as follows:
  \[\arraycolsep=1pt\begin{array}{rl}
  \Cof{t} & = t\\
  \Cof{\aparr{\tau_1}{\tau_2}} & = \aceparr{\Cof{\tau_1}}{\Cof{\tau_2}}\\
  \Cof{\aall{t}{\tau}} & = \aceall{t}{\Cof{\tau}}\\
  \Cof{\arec{t}{\tau}} & = \acerec{t}{\Cof{\tau}}\\
  \Cof{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}} & = \aceprod{\labelset}{\mapschemax{\Cofv}{\tau}{i}{\labelset}}\\
  \Cof{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}} & = \acesum{\labelset}{\mapschemax{\Cofv}{\tau}{i}{\labelset}}
  \end{array}\]
  \item Each expanded expression, $e$, maps onto a proto-expression, $\Cof{e}$, as follows:
  \[\arraycolsep=1pt\hspace{-15px}\begin{array}{rl}
  \Cof{x} & = x\\
  \Cof{\aelam{\tau}{x}{e}} & = \acelam{\Cof{\tau}}{x}{\Cof{e}}\\
  \Cof{\aeap{e_1}{e_2}} & = \aceap{\Cof{e_1}}{\Cof{e_2}}\\
  \Cof{\aetlam{t}{e}} & = \acetlam{t}{\Cof{e}}\\
  \Cof{\aetap{e}{\tau}} & = \acetap{\Cof{e}}{\Cof{\tau}}\\
  \Cof{\aefold{e}} & = \aceasc{\acerec{t}{\Cof{\tau}}}{\acefold{\Cof e}}\\
  \Cof{\aeunfold{e}} & = \aceunfold{\Cof{e}}\\
  \Cof{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}} & = \acetpl{\labelset}{\mapschemax{\Cofv}{e}{i}{\labelset}}\\
  \Cof{\aein{\ell}{e}} &= \aceasc
    {
      \acesum{\labelset}{\mapschemax{\Cofv}{\tau}{i}{\labelset}}
    }{\acein{\ell}{\Cof{e}}}\\
  \Cof{\aematchwith{n}{e}{\seqschemaX{r}}} & = \aceasc{\Cof{\tau}}{\acematchwith{n}{\Cof{e}}{\seqschemaXx{\Cofv}{r}}}
  \end{array}\]
  \item Each expanded rule, $r$, maps onto the proto-rule, $\Cof{r}$, as follows:
  \begin{align*}
  \Cof{\aematchrule{p}{e}} & = \acematchrule{p}{\Cof{e}}
  \end{align*}
  \item Each expanded pattern, $p$, except for the variable patterns, maps onto a proto-pattern, $\Cof{p}$, as follows:
  \begin{align*}
  \Cof{\aewildp} & = \acewildp\\
  \Cof{\aefoldp{p}} & = \acefoldp{\Cof{p}}\\
  \Cof{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}} & = \acetplp{\labelset}{\mapschemax{\Cofv}{p}{i}{\labelset}}\\
  \Cof{\aeinjp{\ell}{p}} & = \aceinjp{\ell}{\Cof{p}}
  \end{align*}
\end{itemize}

These definitions differ from those given in Sec. \ref{appendix:proto-expansions-SES} in that they include the type information necessary for bidirectional typechecking.

\subsubsection{Proto-Expression Encoding and Decoding}
The type $\tCEExp$ and the judgements $\encodeCondE{\ce}{e}$ and $\decodeCondE{e}{\ce}$ are characterized as described in Sec. \ref{appendix:proto-expansions-SES}.

\subsubsection{Proto-Pattern Encoding and Decoding}
The type $\tCEPat$ and the judgements $\encodeCEPat{\cpv}{e}$ and $\decodeCEPat{e}{\cpv}$ are characterized as described in Sec. \ref{appendix:proto-expansions-SES}.

\subsubsection{Splice Summaries}
The \emph{splice summary} of a proto-expression, $\segof{\ce}$, or proto-pattern, $\segof{\cpv}$, is the finite set of references to spliced types, expressions {and patterns} that it mentions.

\subsubsection{Segmentations}
A \emph{segment set}, $\psi$, is a finite set of pairs of natural numbers indicating the locations of spliced terms. The \emph{segmentation} of a proto-expression, $\segof{\ce}$, or proto-pattern, $\segof{\cpv}$, is the segment set implied by its splice summary.

\subsection{Proto-Expansion Validation}\label{appendix:proto-expansion-validation-BS}
\subsubsection{Proto-Type Validation}
The \emph{proto-type validation judgement}, $\cvalidT{\Delta}{\tscenev}{\ctau}{\tau}$, is inductively defined by Rules (\ref{rules:cvalidT-U}), which were defined in Sec. \ref{appendix:proto-type-validation-SES}.

\subsubsection{Bidirectional Proto-Expression and Proto-Rule Validation}
\emph{Expression splicing scenes}, $\escenev$, are of the form $\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}$. We write $\tsfrom{\escenev}$ for the type splicing scene constructed by dropping unnecessary contexts from $\escenev$:
\[\tsfrom{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}} = \tsceneUP{\uDelta}{b}\]

\noindent\fbox{$\strut\csynX{\ce}{e}{\tau}$}~~$\ce$ has expansion $e$ synthesizing type $\tau$
\begin{subequations}\label{rules:csyn}
\begin{equation}\label{rule:csyn-var}
  \inferrule{ }{ 
    \csyn{\Delta}{\Gamma, \Ghyp{x}{\tau}}{\escenev}{x}{x}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:csyn-asc}
  \inferrule{
    \cvalidT{\Delta}{\tsfrom{\escenev}}{\ctau}{\tau}\\
    \canaX{\ce}{e}{\tau}
  }{
    \csynX{\aceasc{\ctau}{\ce}}{e}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:csyn-let}
  \inferrule{
    \csynX{\ce}{e}{\tau}\\
    \csyn{\Delta}{\Gamma, \Ghyp{x}{\tau}}{\escenev}{\ce'}{e'}{\tau'}
  }{
    \csynX{\aceletsyn{x}{\ce}{\ce'}}{\aeap{\aelam{\tau}{x}{e'}}{e}}{\tau'}
  }
\end{equation}
\begin{equation}\label{rule:csyn-lam}
  \inferrule{
    \cvalidT{\Delta}{\tsfrom{\escenev}}{\ctau_1}{\tau_1}\\
    \csyn{\Delta}{\Gamma, \Ghyp{x}{\tau_1}}{\escenev}{\ce}{e}{\tau_2}
  }{
    \csynX{\acelam{\ctau_1}{x}{\ce}}{\aelam{\tau_1}{x}{e}}{\aparr{\tau_1}{\tau_2}}
  }
\end{equation}
\begin{equation}\label{rule:csyn-ap}
  \inferrule{
    \csynX{\ce_1}{e_1}{\aparr{\tau_2}{\tau}}\\
    \canaX{\ce_2}{e_2}{\tau_2}
  }{
    \csynX{\aceap{\ce_1}{\ce_2}}{\aeap{e_1}{e_2}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:csyn-tlam}
  \inferrule{
    \csyn{\Delta, \Dhyp{t}}{\Gamma}{\escenev}{\ce}{e}{\tau}
  }{
    \csynX{\acetlam{t}{\ce}}{\aetlam{t}{e}}{\aall{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:csyn-tap}
  \inferrule{
    \csynX{\ce}{e}{\aall{t}{\tau}}\\
    \cvalidT{\Delta}{\tsfrom{\escenev}}{\ctau'}{\tau'}
  }{
    \csynX{\acetap{\ce}{\ctau'}}{\aetap{e}{\tau'}}{[\tau'/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:csyn-unfold}
  \inferrule{
    \csynX{\ce}{e}{\arec{t}{\tau}}
  }{
    \csynX{\aceunfold{\ce}}{\aeunfold{e}}{[\arec{t}{\tau}/t]\tau}
  }
\end{equation}
\begin{equation}\label{rule:csyn-tpl}
  \inferrule{
    \tau = \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}\\\\
    \{\csynX{\ce_i}{e_i}{\tau_i}\}_{i \in \labelset}
  }{
    \csynX{\acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}}}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:csyn-pr}
  \inferrule{
    \csynX{\ce}{e}{\aprod{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
  }{
    \csynX{\acepr{\ell}{\ce}}{\aepr{\ell}{e}}{\tau}
  }
\end{equation}
% \begin{equation}\label{rule:csyn-match}
%   \inferrule{
%     n > 0\\
%     \csynX{\ce}{e}{\tau}\\
%     \{\crsynX{\crv_i}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}
%   }{
%     \csynX{\acematchwithb{n}{\ce}{\seqschemaX{\crv}}}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}
%   }
% \end{equation}
\begin{equation}\label{rule:csyn-splicede}
\inferrule{
  \cvalidT{\emptyset}{\tsfrom{\escenev}}{\ctau}{\tau}\\
  \escenev=\esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}\\
  \parseUExp{\bsubseq{b}{m}{n}}{\ue}\\
  \eana{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{\ue}{e}{\tau}\\\\
  \Delta \cap \Delta_\text{app} = \emptyset\\
  \domof{\Gamma} \cap \domof{\Gamma_\text{app}} = \emptyset
}{
  \csyn{\Delta}{\Gamma}{\escenev}{\acesplicede{m}{n}{\ctau}}{e}{\tau}
}
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\canaX{\ce}{e}{\tau}$}~~$\ce$ has expansion $e$ when analyzed against type $\tau$
\begin{subequations}\label{rules:cana}
\begin{equation}\label{rule:cana-subsume}
  \inferrule{
    \csynX{\ce}{e}{\tau}
  }{
    \canaX{\ce}{e}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:cana-let}
  \inferrule{
    \csynX{\ce}{e}{\tau}\\
    \cana{\Delta}{\Gamma, \Ghyp{x}{\tau}}{\escenev}{\ce'}{e'}{\tau'}
  }{
    \canaX{\aceletsyn{x}{\ce}{\ce'}}{\aeap{\aelam{\tau}{x}{e'}}{e}}{\tau'}
  }
\end{equation}
% \begin{equation}\label{rule:cana-analam}
%   \inferrule{
%     \cana{\Delta}{\Gamma, \Ghyp{x}{\tau_1}}{\escenev}{\ce}{e}{\tau_2}
%   }{
%     \canaX{\aceanalam{x}{\ue}}{\aelam{\tau_1}{x}{e}}{\aparr{\tau_1}{\tau_2}}
%   }
% \end{equation}
\begin{equation}\label{rule:cana-tlam}
  \inferrule{
    \cana{\Delta, \Dhyp{t}}{\Gamma}{\escenev}{\ce}{e}{\tau}
  }{
    \canaX{\acetlam{t}{\ce}}{\aetlam{t}{e}}{\aall{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:cana-fold}
  \inferrule{
    \canaX{\ce}{e}{[\arec{t}{\tau}/t]\tau}
  }{
    \canaX{\aceanafold{\ce}}{\aefold{e}}{\arec{t}{\tau}}
  }
\end{equation}
\begin{equation}\label{rule:cana-tpl}
  \inferrule{
    \tau = \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}\\\\
    \{\canaX{\ce_i}{e_i}{\tau_i}\}_{i \in \labelset}
  }{
    \canaX{\acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}}}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\tau}
  }
\end{equation}
\begin{equation}\label{rule:cana-in}
  \inferrule{
    \canaX{\ce}{e}{\tau'}
  }{
    % \left(\shortstack{$\Delta~\Gamma \vdash^{\escenev} \aceanain{\ell}{\ce}$\\$\leadsto$\\$\aein{\ell} \Leftarrow \asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}$\vspace{-1.2em}}\right)
    \canaX{\acein{\ell}{\ce}}{\aein{\ell}{e}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}}
  }
\end{equation}
\begin{equation}\label{rule:cana-match}
  \inferrule{
    \csynX{\ce}{e}{\tau}\\
    \{\cranaX{\crv_i}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}
  }{
    \canaX{\acematchwithb{n}{\ce}{\seqschemaX{\crv}}}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}
  }
\end{equation}
\end{subequations}

\noindent\fbox{$\strut\crana{\Delta}{\Gamma}{\escenev}{\crv}{r}{\tau}{\tau'}$}~~$\crv$ has expansion $r$ taking values of type $\tau$ to values of type $\tau'$
\begin{equation}\label{rule:crana}
\inferrule{
  \patType{\pctx}{p}{\tau}\\
  \cana{\Delta}{\Gcons{\Gamma}{\pctx}}{\escenev}{\ce}{e}{\tau'}
}{
  \crana{\Delta}{\Gamma}{\escenev}{\acematchrule{p}{\ce}}{\aematchrule{p}{e}}{\tau}{\tau'}
}
\end{equation}

\subsubsection{Proto-Pattern Validation}
\emph{Pattern splicing scenes}, $\pscenev$, are of the form $\pscene{\uDelta}{\uPhi}{b}$.

\vspace{10px}\noindent\fbox{\strut$\cvalidP{\upctx}{\pscenev}{\cpv}{p}{\tau}$}~~$\cpv$ has expansion $p$ matching against $\tau$ generating hypotheses $\upctx$
\begin{subequations}\label{rules:cvalidP-B}
\begin{equation}\label{rule:cvalidP-B-wild}
\inferrule{ }{
  \cvalidP{\uGG{\emptyset}{\emptyset}}{\pscenev}{\acewildp}{\aewildp}{\tau}
}
\end{equation}
\begin{equation}\label{rule:cvalidP-B-fold}
\inferrule{
  \cvalidP{\upctx}{\pscenev}{\cpv}{p}{[\arec{t}{\tau}/t]\tau}
}{
  \cvalidP{\upctx}{\pscenev}{\acefoldp{\cpv}}{\aefoldp{p}}{\arec{t}{\tau}}
}
\end{equation}
\begin{equation}\label{rule:cvalidP-B-tpl}
\inferrule{
  \tau = \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}\\\\
  \{\cvalidP{\upctx_i}{\pscenev}{\cpv_i}{p_i}{\tau_i}\}_{i \in \labelset}
}{
% \left(\shortstack{$\vdash^{\pscenev} \acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}}$\\$\leadsto$\\$\aetplp{\labelset}{\mapschema{p}{i}{\labelset}} : \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}~\dashVx^{\,\Gconsi{i \in \labelset}{\upctx_i}}$\vspace{-1.2em}}\right)
  \cvalidP{\GIconsi{i \in \labelset}{\upctx_i}}{\pscenev}{\acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\tau}
}
\end{equation}
\begin{equation}\label{rule:cvalidP-B-in}
\inferrule{
  \cvalidP{\upctx}{\pscenev}{\cpv}{p}{\tau}
}{
  \cvalidP{\upctx}{\pscenev}{\aceinjp{\ell}{\cpv}}{\aeinjp{\ell}{p}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
}
\end{equation}
\begin{equation}\label{rule:cvalidP-B-spliced}
\inferrule{
  \cvalidT{\emptyset}{\tsceneUP{\uDelta}{b}}{\ctau}{\tau}\\
  \parseUPat{\bsubseq{b}{m}{n}}{\upv}\\
  \patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}
}{
  \cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\acesplicedp{m}{n}{\ctau}}{p}{\tau}
}
\end{equation}
\end{subequations}
\section{Metatheory}\label{appendix:B-metatheory}
\subsection{Typed Pattern Expansion}\label{appendix:B-typed-pattern-expansion}
\begin{theorem}[Typed Pattern Expansion]\label{thm:typed-pattern-expansion-B} ~
\begin{enumerate}
  \item If $\pExpandsSP{\uDD{\uD}{\Delta}}{\uASI{\uA}{\Phi}{\uI}}{\upv}{p}{\tau}{\uGG{\uG}{\pctx}}$ then $\patType{\pctx}{p}{\tau}$.
  \item If $\cvalidP{\uGG{\uG}{\pctx}}{\pscene{\uDD{\uD}{\Delta}}{\uAP{\uA}{\Phi}}{b}}{\cpv}{p}{\tau}$ then $\patType{\pctx}{p}{\tau}$.
\end{enumerate}
\end{theorem}
\begin{proof}
  By mutual rule induction over Rules (\ref{rules:patExpands-B}) and Rules (\ref{rules:cvalidP-B}).
  \begin{enumerate}
  \item We induct on the premise. In the following, let $\uDelta=\uDD{\uD}{\Delta}$ and $\upctx=\uGG{\uG}{\pctx}$ and $\uPhi=\uASI{\uA}{\Phi}{\uI}$.
  \begin{byCases}
    \item[\text{(\ref{rule:patExpands-B-var}) \textbf{through} (\ref{rule:patExpands-B-apuptsm})}] These cases follow by identical argument to the corresponding cases of Theorem \ref{thm:typed-pattern-expansion}.

    \item[\text{(\ref{rule:patExpands-B-lit})}]~
    \resetpfcounter
    \begin{pfsteps}
      \item \upv = \lit{b} \BY{assumption}
      \item \Phi = \Phi', \xuptsmbnd{a}{\tau}{\eparse} \BY{assumption}
      \item \uI = \uI', \designate{\tau}{a} \BY{assumption}
      \item   \encodeBody{b}{\ebody} \BY{assumption}
      \item   \evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessP}}{\ecand}} \BY{assumption}
      \item  \decodeCEPat{\ecand}{\cpv} \BY{assumption}
      \item  \cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau} \BY{assumption} \pflabel{cvalidP}
      \item \patType{\pctx}{p}{\tau} \BY{IH, part 2 on \pfref{cvalidP}}
    \end{pfsteps}
    \resetpfcounter
%     \item[\text{(\ref{rule:patExpands-var})}] ~
%       \begin{pfsteps*}
%         \item $\upv=\ux$ \BY{assumption}
%         \item $p=x$ \BY{assumption}
%         \item $\pctx=\Ghyp{x}{\tau}$ \BY{assumption}
%         \item $\patType{\Ghyp{x}{\tau}}{x}{\tau}$ \BY{Rule (\ref{rule:patType-var})}
%       \end{pfsteps*}
%       \resetpfcounter
%     \item[\text{(\ref{rule:patExpands-wild})}] ~
%       \begin{pfsteps*}
%         \item $p=\aewildp$ \BY{assumption}
%         \item $\pctx = \emptyset$ \BY{assumption}
%         \item $\patType{\emptyset}{\aewildp}{\tau}$ \BY{Rule (\ref{rule:patType-wild})}
%       \end{pfsteps*}
%       \resetpfcounter
%     \item[\text{(\ref{rule:patExpands-fold})}] ~
%       \begin{pfsteps*}
%         \item $\upv=\foldp{\upv'}$ \BY{assumption}
%         \item $p=\aefoldp{p'}$ \BY{assumption}
%         \item $\tau=\arec{t}{\tau'}$ \BY{assumption}
%         %\item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
%         \item $\patExpands{\upctx}{\uPhi}{\upv'}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{assumption} \pflabel{patExpands}
%         \item $\patType{\pctx}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{IH, part 1 on \pfref{patExpands}} \pflabel{patType}
%         \item $\patType{\pctx}{\aefoldp{p'}}{\arec{t}{\tau'}}$ \BY{Rule (\ref{rule:patType-fold}) on \pfref{patType}}
%       \end{pfsteps*}
%       \resetpfcounter
%     \item[\text{(\ref{rule:patExpands-tpl})}] ~
%       \begin{pfsteps*}
%         \item $\upv=\tplp{\mapschema{\upv}{i}{\labelset}}$ \BY{assumption}
%         \item $p=\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}$ \BY{assumption}
%         \item $\tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$ \BY{assumption}
%         \item $\{\patExpands{\uGG{\uG_i}{\pctx_i}}{\uPhi}{\upv_i}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{assumption} \pflabel{patExpands}
%         \item $\pctx = \Gconsi{i \in \labelset}{\pctx_i}$ \BY{assumption}
%         %\item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
%         \item $\{\patType{\pctx_i}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{IH, part 1 over \pfref{patExpands}}\pflabel{patType}
%         \item $\patType{\Gconsi{i \in \labelset}{\pctx_i}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}$ \BY{Rule (\ref{rule:patType-tpl}) on \pfref{patType}}
%       \end{pfsteps*}
%       \resetpfcounter
%     \item[\text{(\ref{rule:patExpands-in})}] ~
%       \begin{pfsteps*}
%         \item $\upv=\injp{\ell}{\upv'}$ \BY{assumption}
%         \item $p=\aeinjp{\ell}{p'}$ \BY{assumption}
%         \item $\tau=\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}$ \BY{assumption}
%         \item $\patExpands{\upctx}{\uPhi}{\upv'}{p'}{\tau'}$ \BY{assumption} \pflabel{patExpands}
% %        \item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
%         \item $\patType{\pctx}{p'}{\tau'}$ \BY{IH, part 1 on \pfref{patExpands}} \pflabel{patType}
%         \item $\patType{\pctx}{\aeinjp{\ell}{p'}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}}$ \BY{Rule (\ref{rule:patType-inj}) on \pfref{patType}}
%       \end{pfsteps*}
%       \resetpfcounter
%     \item[\text{(\ref{rule:patExpands-apuptsm})}] ~
%       \begin{pfsteps*}
%         \item $\upv=\utsmap{\tsmv}{b}$ \BY{assumption}
%         \item $\uA=\uA', \vExpands{\tsmv}{a}$ \BY{assumption}
%         \item $\Phi=\Phi', \xuptsmbnd{a}{\tau}{\eparse}$ \BY{assumption}
%         \item $\encodeBody{b}{\ebody}$ \BY{assumption}
%         \item $\evalU{\eparse(\ebody)}{{\lbltxt{SuccessP}}\cdot{\ecand}}$ \BY{assumption}
%         \item $\decodeCEPat{\ecand}{\cpv}$ \BY{assumption}
%         \item $\cvalidP{\uGG{\uG}{\pctx}}{\pscene{\uDelta}{\uAP{\uA}{\Phi}}{b}}{\cpv}{p}{\tau}$ \BY{assumption} \pflabel{cvalidP}
% %        \item $\uptsmenv{\Delta}{\Phi', \xuptsmbnd{a}{\tau}{\eparse}}$ \BY{assumption} \pflabel{env}
%         \item $\patType{\pctx}{p}{\tau}$ \BY{IH, part 2 on \pfref{cvalidP}}
%       \end{pfsteps*}
%       \resetpfcounter
  \end{byCases}

  \item We induct on the premise. All cases follow by identical argument to the corresponding cases of Theorem \ref{thm:typed-pattern-expansion}.
%   \begin{byCases}
%     \item[\text{(\ref{rule:cvalidP-B-wild})}] ~
%       \begin{pfsteps*}
%         \item $p=\aewildp$ \BY{assumption}
%         \item $\pctx=\emptyset$ \BY{assumption}
%         \item $\patType{\emptyset}{\aewildp}{\tau}$ \BY{Rule (\ref{rule:patType-wild})}
%       \end{pfsteps*}
%       \resetpfcounter
%     \item[\text{(\ref{rule:cvalidP-UP-fold})}] ~
%       \begin{pfsteps*}
%         \item $\cpv=\acefoldp{\cpv'}$ \BY{assumption}
%         \item $p=\aefoldp{p'}$ \BY{assumption}
%         \item $\tau=\arec{t}{\tau'}$ \BY{assumption}
%         % \item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
%         \item $\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv'}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{assumption} \pflabel{cvalidP}
%         \item $\patType{\pctx}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{IH, part 2 on \pfref{cvalidP}} \pflabel{patType}
%         \item $\patType{\pctx}{\aefoldp{p'}}{\arec{t}{\tau'}}$ \BY{Rule (\ref{rule:patType-fold}) on \pfref{patType}}
%       \end{pfsteps*}
%       \resetpfcounter
%     \item[\text{(\ref{rule:cvalidP-UP-tpl})}] ~
%       \begin{pfsteps*}
%         \item $\cpv=\acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}}$ \BY{assumption}
%         \item $p=\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}$ \BY{assumption}
%         \item $\tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$ \BY{assumption}
%         \item $\{\cvalidP{\uGG{\uG_i}{\pctx_i}}{\pscene{\uDelta}{\uPhi}{b}}{\cpv_i}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{assumption} \pflabel{cvalidP}
%         \item $\pctx = \Gconsi{i \in \labelset}{\pctx_i}$ \BY{assumption}
%         %\item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
%         \item $\{\patType{\pctx_i}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{IH, part 2 over \pfref{cvalidP}}\pflabel{patType}
%         \item $\patType{\Gconsi{i \in \labelset}{\pctx_i}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}$ \BY{Rule (\ref{rule:patType-tpl}) on \pfref{patType}}
%       \end{pfsteps*}
%       \resetpfcounter
%     \item[\text{(\ref{rule:cvalidP-UP-in})}] ~
%       \begin{pfsteps*}
%         \item $\cpv=\aceinjp{\ell}{\cpv'}$ \BY{assumption}
%         \item $p=\aeinjp{\ell}{p'}$ \BY{assumption}
%         \item $\tau=\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}$ \BY{assumption}
%         \item $\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv'}{p'}{\tau'}$ \BY{assumption} \pflabel{cvalidP}
% %        \item $\uptsmenv{\Delta}{\Phi}$ \BY{assumption} \pflabel{env}
%         \item $\patType{\pctx}{p'}{\tau'}$ \BY{IH, part 2 on \pfref{cvalidP}} \pflabel{patType}
%         \item $\patType{\pctx}{\aeinjp{\ell}{p'}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}}$ \BY{Rule (\ref{rule:patType-inj}) on \pfref{patType}}
%       \end{pfsteps*}
%       \resetpfcounter
%     \item[\text{(\ref{rule:cvalidP-UP-spliced})}] ~
%       \begin{pfsteps*}
%         \item $\cpv=\acesplicedp{m}{n}{\ctau}$ \BY{assumption}
%         \item $\cvalidT{\emptyset}{\tsceneUP{\uDelta}{b}}{\ctau}{\tau}$ \BY{assumption}
%         \item $\parseUExp{\bsubseq{b}{m}{n}}{\upv}$ \BY{assumption}
%         \item $\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}$ \BY{assumption} \pflabel{patExpands}
%         \item $\patType{\pctx}{p}{\tau}$ \BY{IH, part 1 on \pfref{patExpands}}
%       \end{pfsteps*}
%       \resetpfcounter
%   \end{byCases}
  \end{enumerate}
The mutual induction can be shown to be well-founded by an argument nearly identical to that that given in the proof of Theorem \ref{thm:typed-pattern-expansion}, differing only in that the appeal to Condition \ref{condition:pattern-parsing} is replaced by an appeal to the analagous Condition \ref{condition:pattern-parsing-BS}.

% showing that the following numeric metric on the judgements that we induct on is decreasing:
% \begin{align*}
% \sizeof{\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}} & = \sizeof{\upv}\\
% \sizeof{{\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}}} & = \sizeof{b}
% \end{align*}
% where $\sizeof{b}$ is the length of $b$ and $\sizeof{\upv}$ is the sum of the lengths of the literal bodies in $\upv$, as defined in Sec. \ref{appendix:SES-syntax}.

% The only case in the proof of part 1 that invokes part 2 is Case (\ref{rule:patExpands-apuptsm}). There, we have that the metric remains stable: \begin{align*}
%  & \sizeof{\patExpands{\upctx}{\uPhi}{\utsmap{\tsmv}{b}}{p}{\tau}}\\
% =& \sizeof{{\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}}}\\
% =&\sizeof{b}\end{align*}

% The only case in the proof of part 2 that invokes part 1 is Case (\ref{rule:cvalidP-UP-spliced}). There, we have that $\parseUPat{\bsubseq{b}{m}{n}}{\upv}$ and the IH is applied to the judgement $\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}$. Because the metric is stable when passing from part 1 to part 2, we must have that it is strictly decreasing in the other direction:
% \[\sizeof{\patExpands{\upctx}{\uPhi}{\upv}{p}{\tau}} < \sizeof{{\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\acesplicedp{m}{n}{\ctau}}{p}{\tau}}}\]
% i.e. by the definitions above, 
% \[\sizeof{\upv} < \sizeof{b}\]

% This is established by appeal to Condition \ref{condition:body-subsequences}, which states that subsequences of $b$ are no longer than $b$, and the Condition \ref{condition:pattern-parsing}, which states that an unexpanded pattern constructed by parsing a textual sequence $b$ is strictly smaller, as measured by the metric defined above, than the length of $b$, because some characters must necessarily be used to apply the pattern TLM and delimit each literal body. Combining Conditions \ref{condition:body-subsequences} and \ref{condition:pattern-parsing}, we have that $\sizeof{\upv} < \sizeof{b}$ as needed.
\end{proof}
\subsection{Typed Expression and Rule Expansion}\label{appendix:P-typed-expression-expansion}
\begin{theorem}[Typed Expression and Rule Expansion]\label{thm:typed-expansion-full-B} ~
\begin{enumerate}
  \item \begin{enumerate}
    \item If $\esyn{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\ue}{e}{\tau}$ then $\hastypeU{\Delta}{\Gamma}{e}{\tau}$.
    \item If $\eana{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\ue}{e}{\tau}$ and $\istypeU{\Delta}{\tau}$ then $\hastypeU{\Delta}{\Gamma}{e}{\tau}$.
    \item If $\rana{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\urv}{r}{\tau}{\tau'}$ and $\istypeU{\Delta}{\tau'}$ then $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$.
  \end{enumerate}
  \item \begin{enumerate}
    \item If $\csyn{\Delta}{\Gamma}{\esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}$ and $\Delta \cap \Delta_\text{app}=\emptyset$ and $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ then $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e}{\tau}$. 
    \item If $\cana{\Delta}{\Gamma}{\esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}$ and $\istypeU{\Delta}{\tau}$ and $\Delta \cap \Delta_\text{app}=\emptyset$ and $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ then $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e}{\tau}$. 
    \item If $\crana{\Delta}{\Gamma}{\esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}}{\crv}{r}{\tau}{\tau'}$ and $\istypeU{\Delta}{\tau'}$ and $\Delta \cap \Delta_\text{app}=\emptyset$ and $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ then $\ruleType{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{r}{\tau}{\tau'}$.
  \end{enumerate}
\end{enumerate}
\end{theorem}
\begin{proof}
By mutual rule induction over Rules (\ref{rules:esyn-S}), Rules (\ref{rules:eana-S}), Rule (\ref{rule:rana-S}), Rules (\ref{rules:csyn}), Rules (\ref{rules:cana}) and Rule (\ref{rule:crana}). The proof follows the proof of Theorem \ref{thm:typed-expansion-full-U}.

% \begin{enumerate}
% \item In the following, let $\uDelta=\uDD{\uD}{\Delta}$ and $\uGamma=\uGG{\uG}{\Gamma}$.
%   \begin{enumerate}
%     \item \todo{1a}
%     \item \todo{1b}
%     \item \todo{1c}
%   \end{enumerate}
% \item \todo{2}
%   \begin{enumerate}
%     \item \todo{2a}
%     \item \todo{2b}
%     \item \todo{2c}
%   \end{enumerate}
% \end{enumerate}
\end{proof}

% % \subsection{Expressibility}
% The following lemma establishes that each type can be expressed as a well-formed proto-type, under the same type formation context and any type splicing scene.
% \begin{lemma}[Proto-Expansion Type Expressibility]\label{lemma:proto-type-expressibility-U} If $\istypeU{\Delta}{\tau}$ then $\cvalidT{\Delta}{\tscenev}{\Cof{\tau}}{\tau}$. \end{lemma}
% \begin{proof}
% By rule induction over Rules (\ref{rules:istypeU}). In each case, we apply the IH on or over each premise, then apply the corresponding proto-type validation rule in Rules (\ref{rules:cvalidT-U}).
% \end{proof}

% The Type Expressibility Lemma establishes that every well-formed type, $\tau$, can be expressed as a well-formed unexpanded type, $\Uof{\tau}$. This requires defining the metafunction $\Uof{\Delta}$ which maps $\Delta$ onto an unexpanded type formation context as follows:
% \begin{align*}
% \Uof{\emptyset} &= \uDD{\emptyset}{\emptyset}\\
% \Uof{\Delta, \Dhyp{t}} &= \Uof{\Delta}, \uDhyp{\sigilof{t}}{t}
% \end{align*}
% \begin{lemma}[Type Expressibility]\label{lemma:type-expressibility} If $\istypeU{\Delta}{\tau}$ then $\expandsTU{\Uof{\Delta}}{\Uof{\tau}}{\tau}$.\end{lemma}
% \begin{proof} By rule induction over Rules (\ref{rules:istypeU}) using the definitions of $\Uof{\tau}$ and $\Uof{\Delta}$ above. In each case, we apply the IH to or over each premise, then apply the corresponding type expansion rule in Rules (\ref{rules:expandsTU}).\end{proof}


% The following lemma establishes that each well-typed expanded expression, $e$, can be expressed as a valid proto-expression, $\Cof{e}$, that is assigned the same type under any expression splicing scene.
% \begin{theorem}[Proto-Expansion Expression Expressibility]\label{theorem:proto-expressions-expressibility-U} If $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ then $\cvalidE{\Delta}{\Gamma}{\escenev}{\Cof{e}}{e}{\tau}$.\end{theorem}
% \begin{proof} By rule induction over Rules (\ref{rules:hastypeU}). The rule transformation above guarantees that this lemma holds by construction. In particular, in each case, we apply Lemma \ref{lemma:proto-type-expressibility-U} to or over each type formation premise, the IH to or over each typing premise, then apply the corresponding proto-expression validation rule in Rules (\ref{rule:cvalidE-U-var}) through (\ref{rule:cvalidE-U-case}).
% \end{proof}

% The following lemma establishes that each well-typed expanded expression, $e$, can be expressed as a valid ce-expression, $\Cof{e}$, that is assigned the same type under any expression splicing scene.
% \begin{theorem}[Candidate Expansion Expression Expressibility]\label{lemma:ce-expressions-expressibility-UP} Both of the following hold:
% \begin{enumerate}
% \item If $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ then $\cvalidE{\Delta}{\Gamma}{\escenev}{\Cof{e}}{e}{\tau}$.
% \item If $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$ then $\cvalidR{\Delta}{\Gamma}{\escenev}{\Cof{r}}{r}{\tau}{\tau'}$.
% \end{enumerate}
% \end{theorem}
% \begin{proof} By mutual rule induction over Rules (\ref{rules:hastypeUP}) and Rule (\ref{rule:ruleType}). 

% For part 1, we induct on the assumption. 
% \begin{byCases}
% \item[\text{(\ref{rule:hastypeUP-var}) through (\ref{rule:hastypeUP-in})}] In each of these cases, we apply Lemma \ref{lemma:ce-type-expressibility-U} to or over each type formation premise, the IH (part 1) to or over each typing premise, then apply the corresponding ce-expression validation rule in Rules (\ref{rule:cvalidE-UP-var}) through (\ref{rule:cvalidE-UP-in}).
% \item[\text{(\ref{rule:hastypeUP-match})}] ~
%   \begin{pfsteps}
%   \item e = \aematchwith{n}{e'}{\seqschemaX{r}} \BY{assumption}
%   \item \Cof{e} = \acematchwith{n}{\Cof{\tau}}{\Cof{e'}}{\seqschemaXx{\Cofv}{r}} \BY{definition of $\Cof{e}$}
%   \item \hastypeU{\Delta}{\Gamma}{e'}{\tau'} \BY{assumption} \pflabel{hasType}
%   \item \istypeU{\Delta}{\tau} \BY{assumption} \pflabel{isType}
%   \item \{\ruleType{\Delta}{\Gamma}{r_i}{\tau'}{\tau}\}_{1 \leq i \leq n} \BY{assumption} \pflabel{ruleType}
%   \item \cvalidE{\Delta}{\Gamma}{\escenev}{\Cof{e'}}{e'}{\tau'} \BY{IH, part 1 on \pfref{hasType}} \pflabel{cvalidE}
%   \item \cvalidT{\Delta}{\tsfrom{\escenev}}{\Cof{\tau}}{\tau} \BY{Lemma \ref{lemma:candidate-expansion-type-validation} on \pfref{isType}} \pflabel{cvalidT}
%   \item \{\cvalidR{\Delta}{\Gamma}{\escenev}{\Cof{r_i}}{r_i}{\tau'}{\tau}\}_{1 \leq i \leq n} \BY{IH, part 2 over \pfref{ruleType}} \pflabel{cvalidR}
%   \item \cvalidE{\Delta}{\Gamma}{\escenev}{\acematchwith{n}{\Cof{\tau}}{\Cof{e'}}{\seqschemaXx{\Cofv}{r}}}{\aematchwith{n}{e'}{\seqschemaX{r}}}{\tau} \BY{Rule (\ref{rule:cvalidE-UP-match}) on \pfref{cvalidE}, \pfref{cvalidT} and \pfref{cvalidR}}
%   \end{pfsteps}
% \end{byCases}
% \resetpfcounter

% For part 2, we induct on the assumption. There is only one case.
% \begin{byCases}
% \item[\text{(\ref{rule:ruleType})}] ~
%   \begin{pfsteps}
%     \item r = \aematchrule{p}{e} \BY{assumption}
%     \item \Cof{r} = \acematchrule{p}{\Cof{e}} \BY{definition of $\Cof{r}$}
%     \item \patType{\pctx}{p}{\tau} \BY{assumption} \pflabel{patType}
%     \item \hastypeU{\Delta}{\Gcons{\Gamma}{\pctx}}{e}{\tau'} \BY{assumption} \pflabel{hasType}
%     \item \cvalidE{\Delta}{\Gcons{\Gamma}{\pctx}}{\escenev}{\Cof{e}}{e}{\tau'} \BY{IH, part 1 on \pfref{hasType}} \pflabel{cvalidE}
%     \item \cvalidR{\Delta}{\Gamma}{\escenev}{\acematchrule{p}{\Cof{e}}}{\aematchrule{p}{e}}{\tau}{\tau'} \BY{Rule (\ref{rule:cvalidR-UP}) on \pfref{patType} and \pfref{cvalidE}}
%   \end{pfsteps}
%   \resetpfcounter
% \end{byCases}
% \end{proof}

% The following lemma establishes that every well-typed expanded pattern that generates no hypotheses can be expressed as a ce-pattern.
% \begin{lemma}[Candidate Expansion Pattern Expressibility]\label{lemma:ce-pattern-expressibility-U} If $\patType{\emptyset}{p}{\tau}$ then $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\Cof{p}}{p}{\tau}$.\end{lemma}
% \begin{proof} By rule induction over Rules (\ref{rules:patType}).
% \begin{byCases}
% \item[\text{(\ref{rule:patType-var})}] This case does not apply.
% \item[\text{(\ref{rule:patType-wild})}] ~
%   \begin{pfsteps*}
%     \item $p=\aewildp$ \BY{assumption}
%     \item $\Cof{p}=\acewildp$ \BY{definition of $\Cof{p}$}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\acewildp}{\aewildp}{\tau}$ \BY{Rule (\ref{rule:cvalidP-UP-wild})}
%   \end{pfsteps*}
%   \resetpfcounter
% \item[\text{(\ref{rule:patType-fold})}] ~
%   \begin{pfsteps*}
%     \item $p=\aefoldp{p'}$ \BY{assumption}
%     \item $\Cof{p}=\acefoldp{\Cof{p'}}$ \BY{definition of $\Cof{p}$}
%     \item $\tau=\arec{t}{\tau'}$ \BY{assumption}
%     \item $\patType{\emptyset}{p'}{[\arec{t}{\tau'}/t]\tau'}$ \BY{assumption} \pflabel{patType}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\Cof{p'}}{p}{[\arec{t}{\tau'}/t]\tau'}$ \BY{IH on \pfref{patType}} \pflabel{cvalidP}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\acefoldp{\Cof{p'}}}{\aefoldp{p'}}{\arec{t}{\tau'}}$ \BY{Rule (\ref{rule:cvalidP-UP-fold}) on \pfref{cvalidP}}
%   \end{pfsteps*}
%   \resetpfcounter
% \item[\text{(\ref{rule:patType-tpl})}] ~
%   \begin{pfsteps*}
%     \item $p=\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}$ \BY{assumption}
%     \item $\Cof{p}=\acetpl{\labelset}{\mapschemax{\Cofv}{p}{i}{\labelset}}$ \BY{definition of $\Cof{p}$}
%     \item $\tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}$ \BY{assumption}
%     \item $\{\patType{\emptyset}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{assumption} \pflabel{patType}
%     \item $\{\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\Cof{p_i}}{p_i}{\tau_i}\}_{i \in \labelset}$ \BY{IH over \pfref{patType}} \pflabel{cvalidP}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\acetpl{\labelset}{\mapschemax{\Cofv}{p}{i}{\labelset}}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}$ \BY{Rule (\ref{rule:cvalidP-UP-tpl}) on \pfref{cvalidP}}
%   \end{pfsteps*}
%   \resetpfcounter
% \item[\text{(\ref{rule:patType-inj})}] ~
%   \begin{pfsteps*}
%     \item $p=\aeinjp{\ell}{p'}$ \BY{assumption}
%     \item $\Cof{p}=\aceinjp{\ell}{\Cof{p'}}$ \BY{definition of $\Cof{p}$}
%     \item $\tau=\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}$ \BY{assumption}
%     \item $\patType{\emptyset}{p'}{\tau'}$ \BY{assumption}\pflabel{patType}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\Cof{p'}}{p'}{\tau'}$ \BY{IH on \pfref{patType}}\pflabel{cvalidP}
%     \item $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\uDelta}{\uPhi}{b}}{\aceinjp{\ell}{\Cof{p'}}}{\aeinjp{\ell}{p'}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}}$ \BY{Rule (\ref{rule:cvalidP-UP-in}) on \pfref{cvalidP}}
%   \end{pfsteps*}
%   \resetpfcounter
% \end{byCases}
% \end{proof}

% \subsubsection{Expressibility}
% The following lemma establishes that each well-typed expanded pattern can be expressed as an unexpanded pattern matching values of the same type and generating the same hypotheses and corresponding identifier updates. The metafunction $\Uof{\pctx}$ maps $\pctx$ to an unexpanded typing context as follows:
% \begin{align*}
% \Uof{\emptyset} & = \uGG{\emptyset}{\emptyset}\\
% \Uof{\pctx, x : \tau} & = \Uof{\pctx}, \uGhyp{\sigilof{x}}{x}{\tau}\\
% \Uof{\Gconsi{i \in \labelset}{\pctx_i}} & = \Gconsi{i \in \labelset}{\Uof{\pctx_i}}
% \end{align*}
% \begin{lemma}[Pattern Expressibility]\label{lemma:pattern-expressibility} If $\patType{\pctx}{p}{\tau}$ then $\patExpands{\Uof{\pctx}}{\uPhi}{\Uof{p}}{p}{\tau}$.\end{lemma}
% \begin{proof} By rule induction over Rules (\ref{rules:patType}), using the definitions of $\Uof{\pctx}$ and $\Uof{p}$ given above. In each case, we can apply the IH to or over each premise, then apply the corresponding rule in Rules (\ref{rules:patExpands}).\end{proof}

% We can now establish the Expressibility Theorem -- that each well-typed expanded expression, $e$, can be expressed as an unexpanded expression, $\ue$, and assigned the same type under the corresponding contexts.

% \begin{theorem}[Expressibility] Both of the following hold:
% \begin{enumerate}
% \item If $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ then $\expandsUP{\Uof{\Delta}}{\Uof{\Gamma}}{\uPsi}{\uPhi}{\Uof{e}}{e}{\tau}$.
% \item If $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$ then $\ruleExpands{\Uof{\Delta}}{\Uof{\Gamma}}{\uPsi}{\uPhi}{\Uof{r}}{r}{\tau}{\tau'}$.
% \end{enumerate}
% \end{theorem}
% \begin{proof} By mutual rule induction over Rules (\ref{rules:hastypeUP}) and Rule (\ref{rule:ruleType}). 

% For part 1, we induct on the assumption. The rule transformation defined above guarantees that this part holds by its construction. In particular, in each case, we can apply Lemma \ref{lemma:type-expressibility} to or over each type formation premise, the IH (part 1) to or over each typing premise, the IH (part 2) over each rule typing premise, then apply the corresponding rule in Rules (\ref{rules:expandsUP}).

% For part 2, we induct on the assumption. There is only one case:
% \begin{byCases}
% \item[(\ref{rule:ruleType})] ~
% \begin{pfsteps*}
% \item $r = \aematchrule{p}{e}$ \BY{assumption}
% \item $\patType{\pctx}{p}{\tau}$ \BY{assumption} \pflabel{patType}
% \item $\hastypeU{\Delta}{\Gamma \cup \pctx}{e}{\tau'}$ \BY{assumption} \pflabel{hasType}
% \item $\Uof{\Gamma}=\uGG{\uG}{\Gamma}$, for some $\uG$ \BY{definition of $\Uof{\Gamma}$}
% \item $\Uof{\pctx} =\uGG{\uG'}{\pctx}$, for some $\uG'$ \BY{definition of $\Uof{\pctx}$}
% \item $\Uof{\Gamma \cup \pctx} = \uGG{\uG \uplus \uG'}{\Gamma \cup \pctx}$ \BY{definition of $\Uof{\pctx}$}
% \item $\Uof{r} = \aumatchrule{\Uof{p}}{\Uof{e}}$ \BY{definition of $\Uof{r}$}
% \item $\patExpands{\uGG{\uG'}{\pctx}}{\uPhi}{\Uof{p}}{p}{\tau}$ \BY{Lemma \ref{lemma:pattern-expressibility} on \pfref{patType}} \pflabel{patExpands}
% \item $\expandsUP{\uDelta}{\uGG{\uGcons{\uG}{\uG'}}{\Gcons{\Gamma}{\pctx}}}{\uPsi}{\uPhi}{\Uof{e}}{e}{\tau'}$ \BY{IH, part 1 on \pfref{hasType}} \pflabel{expandsUP}
% \item $\ruleExpands{\Uof{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\aumatchrule{\Uof{p}}{\Uof{e}}}{\aematchrule{p}{e}}{\tau}{\tau'}$ \BY{Rule (\ref{rule:ruleExpands}) on \pfref{patExpands} and \pfref{expandsUP}}
% \end{pfsteps*}
% \resetpfcounter
% \end{byCases}
% \end{proof}

\subsection{Abstract Reasoning Principles}
\begin{lemma}[Proto-Expression and Proto-Rule Expansion Decomposition] 
\label{thm:proto-expression-expansion-decomposition-B} ~
\begin{enumerate}
\item If ($\cana
  {\Delta}{\Gamma}
  {\esceneUP
    {\uDD{\uD}{\Delta_\text{app}}}
    {\uGG{\uG}{\Gamma_\text{app}}}
    {\uPsi}{\uPhi}{b}
  }{\ce}{e}{\tau}$ or $\csyn
  {\Delta}{\Gamma}
  {\esceneUP
    {\uDD{\uD}{\Delta_\text{app}}}
    {\uGG{\uG}{\Gamma_\text{app}}}
    {\uPsi}{\uPhi}{b}
  }{\ce}{e}{\tau}$) where $\segof{\ce} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicede{m_i}{n_i}{\ctau_i}}{\nexp}$ then all of the following hold:
  \begin{enumerate}
    \item $\sseq{
          \expandsTU{\uDD{\uD}{\Delta_\text{app}}}
          {
            \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
          }{\tau'_i}
        }{\nty}$
    \item $\sseq{
      \cvalidT{\emptyset}{
        \tsceneUP
          {\uDD
            {\uD}{\Delta_\text{app}}
          }{b}
      }{
        \ctau_i
      }{\tau_i}
    }{\nexp}$
    \item $\sseq{
      \eana
        {\uDD{\uD}{\Delta_\text{app}}}
        {\uGG{\uG}{\Gamma_\text{app}}}
        {\uPsi}
        {\uPhi}
        {\parseUExpF{\bsubseq{b}{m_i}{n_i}}}
        {e_i}
        {\tau_i}
    }{\nexp}$
    \item $e = [\sseq{\tau'_i/t_i}{\nty}, \sseq{e_i/x_i}{\nexp}]e'$ for some $e'$ and $\sseq{t_i}{\nty}$ and $\sseq{x_i}{\nexp}$ such that $\sseq{t_i}{\nty}$ fresh (i.e.  $\sseq{t_i \notin \domof{\Delta}}{\nty}$ and $\sseq{t_i \notin \domof{\Delta_\text{app}}}{\nty}$) and $\sseq{x_i}{\nexp}$ fresh (i.e.  $\sseq{x_i \notin \domof{\Gamma}}{\nexp}$ and $\sseq{x_i \notin \domof{\Gamma_\text{app}}}{\nty}$)
    \item $\mathsf{fv}(e') \subset \domof{\Delta} \cup \domof{\Gamma} \cup \sseq{t_i}{\nty} \cup \sseq{x_i}{\nexp}$
    % \item $\hastypeU
    %   {\Delta \cup \sseq{\Dhyp{t_i}}{\nty}}
    %   {\Gamma \cup \sseq{x_i : \tau_i}{\nexp}}
    %   {e'}{\tau}$
  \end{enumerate}
\item If $\cvalidR{\Delta}{\Gamma}{\esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}}{\crv}{r}{\tau}{\tau'}$ where \[\segof{\crv} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicede{m_i}{n_i}{\ctau_i}}{\nexp}\] then all of the following hold:
  \begin{enumerate}
    \item $\sseq{
          \expandsTU{\uDD{\uD}{\Delta_\text{app}}}
          {
            \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
          }{\tau'_i}
        }{\nty}$
    \item $\sseq{
      \cvalidT{\emptyset}{
        \tsceneUP
          {\uDD
            {\uD}{\Delta_\text{app}}
          }{b}
      }{
        \ctau_i
      }{\tau_i}
    }{\nexp}$
    \item $\sseq{
      \eana
        {\uDD{\uD}{\Delta_\text{app}}}
        {\uGG{\uG}{\Gamma_\text{app}}}
        {\uPsi}
        {\uPhi}
        {\parseUExpF{\bsubseq{b}{m_i}{n_i}}}
        {e_i}
        {\tau_i}
    }{\nexp}$
    \item $r = [\sseq{\tau'_i/t_i}{\nty}, \sseq{e_i/x_i}{\nexp}]r'$ for some $e'$ and fresh $\sseq{t_i}{\nty}$ and fresh $\sseq{x_i}{\nexp}$ 
    \item $\mathsf{fv}(r') \subset \domof{\Delta} \cup \domof{\Gamma} \cup \sseq{t_i}{\nty} \cup \sseq{x_i}{\nexp}$
    % \item $\ruleType
    %   {\Delta \cup \sseq{\Dhyp{t_i}}{\nty}}
    %   {\Gamma \cup \sseq{x_i : \tau_i}{\nexp}}
    %   {r'}{\tau}{\tau'}$
  \end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof} By rule induction over Rules (\ref{rules:cana}) and Rule (\ref{rule:crana}). The proof follows the proof of Theorem \ref{thm:proto-expression-expansion-decomposition}. 
\end{proof}
% \begin{equation}\label{rule:cvalidR-UP}
% \inferrule{
%   \patType{\pctx}{p}{\tau}\\
%   \cvalidE{\Delta}{\Gcons{\Gamma}{\pctx}}{\escenev}{\ce}{e}{\tau'}
% }{
%   \cvalidR{\Delta}{\Gamma}{\escenev}{\acematchrule{p}{\ce}}{\aematchrule{p}{e}}{\tau}{\tau'}
% }
% \end{equation}

\begin{theorem}[seTLM Abstract Reasoning Principles - Explicit Application]
\label{thm:tsc-implicit-B}
If \[\esyn{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\utsmap{\tsmv}{b}}{e}{\tau}\] then:
\begin{enumerate}
\item (\textbf{Typing 1}) $\uPsi = \uPsi', \uShyp{\tsmv}{a}{\tau}{\eparse}$ and $\hastypeU{\Delta}{\Gamma}{e}{\tau}$
\item $\encodeBody{b}{\ebody}$
\item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessE}}{\ecand}}$
\item $\decodeCondE{\ecand}{\ce}$
\item (\textbf{Segmentation}) $\segOK{\segof{\ce}}{b}$
\item $\segof{\ce} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicede{m_i}{n_i}{\ctau_i}}{\nexp}$
\item \textbf{(Typing 2)} $\sseq{
      \expandsTU{\uDD{\uD}{\Delta}}
      {
        \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
      }{\tau'_i}
    }{\nty}$ and $\sseq{\istypeU{\Delta}{\tau'_i}}{\nty}$
\item \textbf{(Typing 3)} $\sseq{
  \cvalidT{\emptyset}{
    \tsceneUP
      {\uDD
        {\uD}{\Delta}
      }{b}
  }{
    \ctau_i
  }{\tau_i}
}{\nexp}$ and $\sseq{\istypeU{\Delta}{\tau_i}}{\nexp}$
\item \textbf{(Typing 4)} $\sseq{
  \eana
    {\uDD{\uD}{\Delta}}
    {\uGG{\uG}{\Gamma}}
    {\uPsi}
    {\uPhi}
    {\parseUExpF{\bsubseq{b}{m_i}{n_i}}}
    {e_i}
    {\tau_i}
}{\nexp}$ and $\sseq{\hastypeU{\Delta}{\Gamma}{e_i}{\tau_i}}{\nexp}$
\item (\textbf{Capture Avoidance}) $e = [\sseq{\tau'_i/t_i}{\nty}, \sseq{e_i/x_i}{\nexp}]e'$ for some $\sseq{t_i}{\nty}$ and $\sseq{x_i}{\nexp}$ and $e'$
\item (\textbf{Context Independence}) $\mathsf{fv}(e') \subset \sseq{t_i}{\nty} \cup \sseq{x_i}{\nexp}$
  % $\hastypeU
  % {\sseq{\Dhyp{t_i}}{\nty}}
  % {\sseq{x_i : \tau_i}{\nexp}}
  % {e'}{\tau}$
\end{enumerate}
\end{theorem}
\begin{proof} By rule induction over Rules (\ref{rules:esyn-S}). The proof follows the proof of Theorem \ref{thm:tsc-SES}.
\end{proof}

\begin{theorem}[seTLM Abstract Reasoning Principles - Implicit Application]
\label{thm:tsc-B}
If \[\eana{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\lit{b}}{e}{\tau}\] then:
\begin{enumerate}
\item (\textbf{Typing 1}) $\uPsi = \uASI{\uA}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}$ and $\hastypeU{\Delta}{\Gamma}{e}{\tau}$
\item $\encodeBody{b}{\ebody}$
\item $\evalU{\ap{\eparse}{\ebody}}{\aein{\mathtt{SuccessE}}{\ecand}}$
\item $\decodeCondE{\ecand}{\ce}$
\item (\textbf{Segmentation}) $\segOK{\segof{\ce}}{b}$
\item $\segof{\ce} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicede{m_i}{n_i}{\ctau_i}}{\nexp}$
\item \textbf{(Typing 2)} $\sseq{
      \expandsTU{\uDD{\uD}{\Delta}}
      {
        \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
      }{\tau'_i}
    }{\nty}$ and $\sseq{\istypeU{\Delta}{\tau'_i}}{\nty}$
\item \textbf{(Typing 3)} $\sseq{
  \cvalidT{\emptyset}{
    \tsceneUP
      {\uDD
        {\uD}{\Delta}
      }{b}
  }{
    \ctau_i
  }{\tau_i}
}{\nexp}$ and $\sseq{\istypeU{\Delta}{\tau_i}}{\nexp}$
\item \textbf{(Typing 4)} $\sseq{
  \eana
    {\uDD{\uD}{\Delta}}
    {\uGG{\uG}{\Gamma}}
    {\uPsi}
    {\uPhi}
    {\parseUExpF{\bsubseq{b}{m_i}{n_i}}}
    {e_i}
    {\tau_i}
}{\nexp}$ and $\sseq{\hastypeU{\Delta}{\Gamma}{e_i}{\tau_i}}{\nexp}$
\item (\textbf{Capture Avoidance}) $e = [\sseq{\tau'_i/t_i}{\nty}, \sseq{e_i/x_i}{\nexp}]e'$ for some $\sseq{t_i}{\nty}$ and $\sseq{x_i}{\nexp}$ and $e'$
\item (\textbf{Context Independence}) $\mathsf{fv}(e') \subset \sseq{t_i}{\nty} \cup \sseq{x_i}{\nexp}$
  % $\hastypeU
  % {\sseq{\Dhyp{t_i}}{\nty}}
  % {\sseq{x_i : \tau_i}{\nexp}}
  % {e'}{\tau}$
\end{enumerate}
\end{theorem}
\begin{proof} By rule induction over Rules (\ref{rules:eana-S}). The proof follows the proof of Theorem \ref{thm:tsc-SES}, differing only in how the TLM definition is looked up.
\end{proof}


% The following theorem establishes a prohibition on \textbf{Shadowing} as discussed in Sec. \ref{sec:uetsms-validation}.

% \begin{theorem}[Shadowing Prohibition]
% \label{thm:shadowing-prohibition-SES} ~
% \begin{enumerate}
% \item If $\cvalidT{\Delta}{\tsceneU{\uDD{\uD}{\Delta_\text{app}}}{b}}{\acesplicedt{m}{n}}{\tau}$ then:\begin{enumerate}
% \item $\parseUTyp{\bsubseq{b}{m}{n}}{\utau}$
% \item $\expandsTU{\uDD{\uD}{\Delta_\text{app}}}{\utau}{\tau}$
% \item $\Delta \cap \Delta_\text{app} = \emptyset$
% \end{enumerate}
% \item If $\cvalidE{\Delta}{\Gamma}{\escenev}{\acesplicede{m}{n}{\ctau}}{e}{\tau}$ then:
% \begin{enumerate}
% \item $\cvalidT{\emptyset}{\tsfrom{\escenev}}{\ctau}{\tau}$
% \item $  \escenev=\esceneU{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{b}$
% \item $\parseUExp{\bsubseq{b}{m}{n}}{\ue}$
% \item $\expandsU{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\ue}{e}{\tau}$
% \item $\Delta \cap \Delta_\text{app} = \emptyset$
% \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}} = \emptyset$
% \end{enumerate}
% \end{enumerate}
% \end{theorem}
% \begin{proof} ~
% \begin{enumerate}
% \item By rule induction over Rules (\ref{rules:cvalidT-U}). The only rule that applies is Rule (\ref{rule:cvalidT-U-splicedt}). The conclusions are the premises of this rule.
% \item By rule induction over Rules (\ref{rules:cvalidE-U}). The only rule that applies is Rule (\ref{rule:cvalidE-U-splicede}). The conclusions are the premises of this rule.
% \end{enumerate}
% \end{proof}

\begin{lemma}[Proto-Pattern Expansion Decomposition]
\label{lemma:proto-pattern-expansion-decomposition-B}
If $\cvalidP{\upctx}{\pscene{\uDelta}{\uPhi}{b}}{\cpv}{p}{\tau}$ where  
\[ 
\segof{\cpv} = \sseq{\acesplicedt{m'_i}{n'_i}}{\nty} \cup \sseq{\acesplicedp{m_i}{n_i}{\ctau_i}}{\npat}
\]
then all of the following hold:
\begin{enumerate}
    \item $\sseq{
          \expandsTU{\uDelta}
          {
            \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
          }{\tau'_i}
        }{\nty}$
    \item $\sseq{
      \cvalidT{\emptyset}{
        \tsceneUP
          {\uDelta}{b}
      }{
        \ctau_i
      }{\tau_i}
    }{\npat}$
    \item $\sseq{
      \patExpands
        {\upctx_i}
        {\uPhi}
        {\parseUPatF{\bsubseq{b}{m_i}{n_i}}}
        {p_i}
        {\tau_i}
    }{\npat}$
  \item $\upctx = \biguplus_{0 \leq i < \npat} \upctx_i$
\end{enumerate}
\end{lemma}
\begin{proof} By rule induction over Rules (\ref{rules:cvalidP-B}). The proof follows the proof of Theorem \ref{lemma:proto-pattern-expansion-decomposition-S}.
\end{proof}

\begin{theorem}[spTLM Abstract Reasoning Principles - Explicit Application]
\label{thm:spTLM-Typing-Segmentation-B}
If \[\patExpands{\upctx}{\uPhi}{\utsmap{\tsmv}{b}}{p}{\tau}\] where $\uDelta=\uDD{\uD}{\Delta}$ and $\uGamma=\uGG{\uG}{\Gamma}$ then all of the following hold:
\begin{enumerate}
        \item (\textbf{Typing 1}) $\uPhi=\uPhi', \uPhyp{\tsmv}{a}{\tau}{\eparse}$ and $\patType{\pctx}{p}{\tau}$
        \item $\encodeBody{b}{\ebody}$
        \item $\evalU{\eparse(\ebody)}{\aein{\mathtt{SuccessP}}{\ecand}}$
        \item $\decodeCEPat{\ecand}{\cpv}$
        \item (\textbf{Segmentation}) $\segOK{\segof{\cpv}}{b}$
        \item $\segof{\cpv} = \sseq{\acesplicedt{n'_i}{m'_i}}{\nty} \cup \sseq{\acesplicedp{m_i}{n_i}{\ctau_i}}{\npat}$
        \item (\textbf{Typing 2}) $\sseq{
              \expandsTU{\uDelta}
              {
                \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
              }{\tau'_i}
            }{\nty}$ and $\sseq{\istypeU{\Delta}{\tau'_i}}{\nty}$
        \item (\textbf{Typing 3}) $\sseq{
          \cvalidT{\emptyset}{
            \tsceneUP
              {\uDelta}{b}
          }{
            \ctau_i
          }{\tau_i}
        }{\npat}$ and $\sseq{\istypeU{\Delta}{\tau_i}}{\npat}$
        \item (\textbf{Typing 4}) $\sseq{
          \patExpands
            {\upctx_i}
            {\uPhi}
            {\parseUPatF{\bsubseq{b}{m_i}{n_i}}}
            {p_i}
            {\tau_i}
        }{\npat}$ 
      \item (\textbf{Visibility}) $\upctx = \biguplus_{0 \leq i < \npat} \upctx_i$
\end{enumerate}
\end{theorem}
\begin{proof} By rule induction over Rules (\ref{rules:patExpands-B}). The proof follows the proof of Theorem \ref{thm:spTLM-Typing-Segmentation}.
\end{proof}

\begin{theorem}[spTLM Abstract Reasoning Principles - Implicit Application]
\label{thm:spTLM-Typing-Segmentation-Implicit-B}
If \[\patExpands{\upctx}{\uPhi}{\lit{b}}{p}{\tau}\] where $\uDelta=\uDD{\uD}{\Delta}$ and $\uGamma=\uGG{\uG}{\Gamma}$ then all of the following hold:
\begin{enumerate}
        \item (\textbf{Typing 1}) $\uPhi = \uASI{\uA}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI, \designate{\tau}{a}}$ and $\patType{\pctx}{p}{\tau}$
        \item $\encodeBody{b}{\ebody}$
        \item $\evalU{\eparse(\ebody)}{\aein{\mathtt{SuccessP}}{\ecand}}$
        \item $\decodeCEPat{\ecand}{\cpv}$
        \item (\textbf{Segmentation}) $\segOK{\segof{\cpv}}{b}$
        \item $\segof{\cpv} = \sseq{\acesplicedt{n'_i}{m'_i}}{\nty} \cup \sseq{\acesplicedp{m_i}{n_i}{\ctau_i}}{\npat}$
        \item (\textbf{Typing 2}) $\sseq{
              \expandsTU{\uDelta}
              {
                \parseUTypF{\bsubseq{b}{m'_i}{n'_i}}
              }{\tau'_i}
            }{\nty}$ and $\sseq{\istypeU{\Delta}{\tau'_i}}{\nty}$
        \item (\textbf{Typing 3}) $\sseq{
          \cvalidT{\emptyset}{
            \tsceneUP
              {\uDelta}{b}
          }{
            \ctau_i
          }{\tau_i}
        }{\npat}$ and $\sseq{\istypeU{\Delta}{\tau_i}}{\npat}$
        \item (\textbf{Typing 4}) $\sseq{
          \patExpands
            {\upctx_i}
            {\uPhi}
            {\parseUPatF{\bsubseq{b}{m_i}{n_i}}}
            {p_i}
            {\tau_i}
        }{\npat}$ 
      \item (\textbf{Visibility}) $\upctx = \biguplus_{0 \leq i < \npat} \upctx_i$
\end{enumerate}
\end{theorem}
\begin{proof} By rule induction over Rules (\ref{rules:patExpands-B}). The proof follows the proof of Theorem \ref{thm:spTLM-Typing-Segmentation}, differing only in how the parse function is looked up.
\end{proof}

\fi
% \subsubsection{Candidate Expansion Expressibility}
% The following lemma establishes that each well-typed expanded expression, $e$, can be expressed as a valid ce-expression, $\Cof{e}$, that synthesizes the same type under the same contexts and any expression splicing scene.
% \begin{theorem}[Candidate Expansion Expression Expressibility]\label{lemma:ce-expressions-expressibility-B} Both of the following hold:
% \begin{enumerate}
% \item If $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ then $\csyn{\Delta}{\Gamma}{\escenev}{\Cof{e}}{e}{\tau}$.
% \item If $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$ then $\crsyn{\Delta}{\Gamma}{\escenev}{\Cof{r}}{r}{\tau}{\tau'}$.
% \end{enumerate}
% \end{theorem}
% \begin{proof} By mutual rule induction over Rules (\ref{rules:hastypeUP}) and Rule (\ref{rule:ruleType}). In each case, we apply the IH, part 1 to or over each typing premise, the IH, part 2 over each rule typing premise, Lemma \ref{lemma:ce-type-expressibility-U} to or over each type formation premise and then derive the conclusion by applying Rules (\ref{rules:csyn}) and Rule (\ref{rule:crsyn}) as needed.
% \end{proof}

% The following lemma establishes that every well-typed expanded pattern that generates no hypotheses can be expressed as a ce-pattern.
% \begin{lemma}[Candidate Expansion Pattern Expressibility]\label{lemma:ce-pattern-expressibility-B} If $\patType{\emptyset}{p}{\tau}$ then $\cvalidP{\uGG{\emptyset}{\emptyset}}{\pscene{\Delta}{\uPhi}{b}}{\Cof{p}}{p}{\tau}$.\end{lemma}
% \begin{proof} The proof is nearly identical to the proof of Lemma \ref{lemma:ce-pattern-expressibility-U}, differing only in that each mention of a rule in Rules (\ref{rules:cvalidP-UP}) is replaced by a mention of the corresponding rule in Rules (\ref{rules:cvalidP-B}).
% \end{proof}

% \subsubsection{Outer Surface Expressibility}
% The following lemma establishes that each well-typed expanded pattern can be expressed as an unexpanded pattern matching values of the same type and generating the same hypotheses and corresponding sigil updates. The metafunction $\Uof{\pctx}$ was defined in \ref{sec:typed-expansion-UP}.
% \begin{lemma}[Pattern Expressibility]\label{lemma:pattern-expressibility-B} If $\patType{\pctx}{p}{\tau}$ then $\patExpands{\Uof{\pctx}}{\uPhi}{\Uof{p}}{p}{\tau}$.\end{lemma}
% \begin{proof} By rule induction over Rules (\ref{rules:patType}), using the definitions of $\Uof{\pctx}$ and $\Uof{p}$. In each case, we can apply the IH to or over each premise, then apply the corresponding rule in Rules (\ref{rules:patExpands-B}).\end{proof}

% We can now establish the Expressibility Theorem -- that each well-typed expanded expression, $e$, can be expressed as an unexpanded expression, $\ue$, which synthesizes the same type under the corresponding contexts.

% \begin{theorem}[Expressibility] Both of the following hold:
% \begin{enumerate}
% \item If $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ then $\esyn{\Uof{\Delta}}{\Uof{\Gamma}}{\uPsi}{\uPhi}{\Uof{e}}{e}{\tau}$.
% \item If $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$ then $\rsyn{\Uof{\Delta}}{\Uof{\Gamma}}{\uPsi}{\uPhi}{\Uof{r}}{r}{\tau}{\tau'}$.
% \end{enumerate}
% \end{theorem}
% \begin{proof} By mutual rule induction over Rules (\ref{rules:hastypeUP}) and Rule (\ref{rule:ruleType}) using the definitions of $\Uof{\Delta}$, $\Uof{\Gamma}$, $\Uof{e}$ and $\Uof{r}$. In each case, we apply the IH, part 1 to or over each typing premise, the IH, part 2 over each rule typing premise, Lemma \ref{lemma:type-expressibility} to or over each type formation premise, Lemma \ref{lemma:pattern-expressibility-B} to each pattern typing premise, then derive the conclusion by applying Rules (\ref{rules:esyn}) and Rule (\ref{rule:rsyn}).  
% \end{proof} 

% \chapter{Parametric Implicits}
% ...

% \subsubsection{Kinds and Constructors}
% Kind expansion

% \begin{subequations}\label{rules:kExpands}
% \begin{equation}\label{rule:kExpands-darr}
% \inferrule{
%   \kExpandsX{\ukappa_1}{\kappa_1}\\
%   \kExpands{\uOmega, \uKhyp{\uu}{u}{\kappa_1}}{\ukappa_2}{\kappa_2}
% }{
%   \kExpandsX{\kdarr{\uu}{\ukappa_1}{\ukappa_2}}{\akdarr{\kappa_1}{u}{\kappa_2}}
% }
% \end{equation}
% \begin{equation}\label{rule:kExpands-unit}
% \inferrule{ }{
%   \kExpandsX{\kunit}{\akunit}
% }
% \end{equation}
% \begin{equation}\label{rule:kExpands-dprod}
% \inferrule{
%   \kExpandsX{\ukappa_1}{\kappa_1}\\
%   \kExpands{\uOmega, \uKhyp{\uu}{u}{\kappa_1}}{\ukappa_2}{\kappa_2}
% }{
%   \kExpandsX{\kdbprod{\uu}{\ukappa_1}{\ukappa_2}}{\akdbprod{\kappa_1}{u}{\kappa_2}}
% }
% \end{equation}
% \begin{equation}\label{rule:kExpands-ty}
% \inferrule{ }{
%   \kExpandsX{\kty}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:kExpands-sing}
% \inferrule{
%   \kanaX{\utau}{\tau}{\akty}
% }{
%   \kExpandsX{\ksing{\utau}}{\aksing{\tau}}
% }
% \end{equation}
% \end{subequations}

% Synthetic constructor expansion
% \begin{subequations}\label{rules:ksyn}
% \begin{equation}\label{rule:ksyn-var}
% \inferrule{ }{\ksyn{\uOmega, \uKhyp{\uu}{u}{\kappa}}{\uu}{u}{\kappa}}
% \end{equation}
% \begin{equation}\label{rule:ksyn-asc}
% \inferrule{
%   \kExpandsX{\ukappa}{\kappa}\\
%   \kanaX{\uc}{c}{\kappa}
% }{
%   \ksynX{\casc{\uc}{\ukappa}}{c}{\kappa}
% }
% \end{equation}
% \begin{equation}\label{rule:ksyn-app}
% \inferrule{
%   \ksynX{\uc_1}{c_1}{\akdarr{\kappa_2}{u}{\kappa}}\\
%   \kanaX{\uc_2}{c_2}{\kappa_2}
% }{
%   \ksynX{\capp{\uc_1}{\uc_2}}{\acapp{c_1}{c_2}}{[c_1/u]\kappa}
% }
% \end{equation}
% \begin{equation}\label{rule:ksyn-unit}
% \inferrule{ }{
%   \ksynX{\ctriv}{\actriv}{\akunit}
% }
% \end{equation}
% \begin{equation}\label{rule:ksyn-prl}
% \inferrule{
%   \ksynX{\uc}{c}{\akdbprod{\kappa_1}{u}{\kappa_2}}
% }{
%   \ksynX{\cprl{\uc}}{\acprl{c}}{\kappa_1}
% }
% \end{equation}
% \begin{equation}\label{rule:ksyn-prr}
% \inferrule{
%   \ksynX{\uc}{c}{\akdbprod{\kappa_1}{u}{\kappa_2}}
% }{
%   \ksynX{\cprr{\uc}}{\acprr{c}}{[\acprl{c}/u]\kappa_2}
% }
% \end{equation}
% \begin{equation}\label{rule:ksyn-parr}
% \inferrule{
%   \kanaX{\utau_1}{\tau_1}{\akty}\\
%   \kanaX{\utau_2}{\tau_2}{\akty}
% }{
%   \ksynX{\parr{\utau_1}{\utau_2}}{\aparr{\tau_1}{\tau_2}}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:ksyn-all}
% \inferrule{
%   \kExpandsX{\ukappa}{\kappa}\\
%   \kana{\uOmega, \uKhyp{\uu}{u}{\kappa}}{\utau}{\tau}{\akty}
% }{
%   \ksynX{\forallu{\uu}{\ukappa}{\utau}}{\aallu{\kappa}{u}{\tau}}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:ksyn-rec}
% \inferrule{
%   \kana{\uOmega, \uKhyp{\ut}{t}{\akty}}{\utau}{\tau}{\akty}
% }{
%   \ksynX{\rect{\ut}{\utau}}{\arec{t}{\tau}}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:ksyn-prod}
% \inferrule{
%   \{\kanaX{\utau_i}{\tau_i}{\akty}\}_{1 \leq i \leq n}
% }{
%   \ksynX{\prodt{\mapschema{\utau}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:ksyn-sum}
% \inferrule{
%   \{\kanaX{\utau_i}{\tau_i}{\akty}\}_{1 \leq i \leq n}
% }{
%   \ksynX{\sumt{\labelset}{\mapschema{\utau}{i}{\labelset}}}{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:ksyn-stat}
% \inferrule{ }{
%   \ksyn{\uOmega, \uMhyp{\uX}{X}{\asignature{\kappa}{u}{\tau}}}{\mcon{\uX}}{\amcon{X}}{\kappa}
% }
% \end{equation}
% \end{subequations}

% Analytic constructor expansion
% \begin{subequations}\label{rules:kana}
% \begin{equation}\label{rule:kana-subsume}
% \inferrule{
%   \ksynX{\uc}{c}{\kappa_1}\\
%   \ksubX{\kappa_1}{\kappa_2}
% }{
%   \kanaX{\uc}{c}{\kappa_2}
% }
% \end{equation}
% \begin{equation}\label{rule:kana-sing}
% \inferrule{
%   \kanaX{\uc}{c}{\akty}
% }{
%   \kanaX{\uc}{c}{\aksing{c}}
% }
% \end{equation}
% \begin{equation}\label{rule:kana-abs}
% \inferrule{
%   \kana{\uOmega, \uKhyp{\uu}{u}{\kappa_1}}{\uc_2}{c_2}{\kappa_2}
% }{
%   \kanaX{\cabs{\uu}{\uc_2}}{\acabs{u}{c_2}}{\akdarr{\kappa_1}{u}{\kappa_2}}
% }
% \end{equation}
% \begin{equation}\label{rule:kana-pair}
% \inferrule{
%   \kanaX{\uc_1}{c_1}{\kappa_1}\\
%   \kanaX{\uc_2}{c_2}{[c_1/u]\kappa_2}
% }{
%   \kanaX{\cpair{\uc_1}{\uc_2}}{\acpair{c_1}{c_2}}{\akdbprod{\kappa_1}{u}{\kappa_2}}
% }
% \end{equation}
% \end{subequations}


% \subsubsection{Types, Expressions, Rules and Patterns}
% Type expansion
% \begin{equation}\label{rule:tExpandsP-B}
% \inferrule{
%   \kanaX{\utau}{\tau}{\akty}
% }{
%   \cExpandsX{\utau}{\tau}{\akty}
% }
% \end{equation}

% Synthetic typed expression expansion
% \begin{subequations}\label{rules:esynP}
% \begin{equation}\label{rule:esynP-var}
%   \inferrule{ }{ 
%     \esynP{\uOmega, \uGhyp{\ux}{x}{\tau}}{\uPsi}{\uPhi}{\ux}{x}{\tau}
%   }
% \end{equation}

% %A \emph{type ascription} can be placed on an unexpanded expression to specify the type that it should be analyzed against. The ascribed type is synthesized if type analysis succeeds.
% \begin{equation}\label{rule:esynP-asc}
%   \inferrule{
%     \cExpandsX{\utau}{\tau}{\akty}\\
%     \eanaPX{\ue}{e}{\tau}
%   }{
%     \esynPX{\asc{\ue}{\utau}}{e}{\tau}
%   }
% \end{equation}

% %We define let-binding of a value in synthetic position primitively in $\miniVerseUB$. The following rule governs such bindings in synthetic position.
% \begin{equation}\label{rule:esynP-let}
%   \inferrule{
%     \esynPX{\ue}{e}{\tau}\\
%     \esynP{\uOmega, \uGhyp{\ux}{x}{\tau}}{\uPsi}{\uPhi}{\ue'}{e'}{\tau'}
%   }{
%     \esynPX{\letsyn{\ux}{\ue}{\ue'}}{\aeap{\aelam{\tau}{x}{e'}}{e}}{\tau'}
%   }
% \end{equation}

% %Functions with an argument type annotation can appear in synthetic position.
% \begin{equation}\label{rule:esynP-lam}
%   \inferrule{
%     \cExpandsX{\utau_1}{\tau_1}{\akty}\\
%     \esynP{\uOmega, \uGhyp{\ux}{x}{\tau_1}}{\uPsi}{\uPhi}{\ue}{e}{\tau_2}
%   }{
%     \esynPX{\lam{\ux}{\utau_1}{\ue}}{\aelam{\tau_1}{x}{e}}{\aparr{\tau_1}{\tau_2}}
%   }
% \end{equation}

% %Function applications can appear in synthetic position. The argument is analyzed against the argument type synthesized by the function.
% \begin{equation}\label{rule:esynP-ap}
%   \inferrule{
%     \esynPX{\ue_1}{e_1}{\aparr{\tau_2}{\tau}}\\
%     \eanaPX{\ue_2}{e_2}{\tau_2}
%   }{
%     \esynPX{\ap{\ue_1}{\ue_2}}{\aeap{e_1}{e_2}}{\tau}
%   }
% \end{equation}

% %Type lambdas and type applications can appear in synthetic position.
% \begin{equation}\label{rule:esynP-tlam}
%   \inferrule{
%     \kExpandsX{\ukappa}{\kappa}\\
%     \esynP{\uOmega, \uKhyp{\uu}{u}{\kappa}}{\uPsi}{\uPhi}{\ue}{e}{\tau}
%   }{
%     \esynPX{\clam{\uu}{\ukappa}{\ue}}{\aeclam{\kappa}{u}{e}}{\aallu{\kappa}{u}{\tau}}
%   }
% \end{equation}
% \begin{equation}\label{rule:esynP-tap}
%   \inferrule{
%     \esynPX{\ue}{e}{\aallu{\kappa}{u}{\tau}}\\
%     \ksynX{\uc}{c}{\kappa}
%   }{
%     \esynPX{\cAp{\ue}{\uc}}{\aecap{e}{c}}{[c/t]\tau}
%   }
% \end{equation}

% %Unfoldings can appear in synthetic position.
% \begin{equation}\label{rule:esynP-unfold}
%   \inferrule{
%     \esynPX{\ue}{e}{\arec{t}{\tau}}
%   }{
%     \esynPX{\unfold{\ue}}{\aeunfold{e}}{[\arec{t}{\tau}/t]\tau}
%   }
% \end{equation}

% %Labeled tuples can appear in synthetic position. Each of the field values are then in synthetic position. 
% \begin{equation}\label{rule:esynP-tpl}
%   \inferrule{
%     \{\esynPX{\ue_i}{e_i}{\tau_i}\}_{i \in \labelset}
%   }{
%     \esynPX{\tpl{\mapschema{\ue}{i}{\labelset}}}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
%   }
% \end{equation}

% %Fields can be projected out of a labeled tuple in synthetic position.
% \begin{equation}\label{rule:esynP-pr}
%   \inferrule{
%     \esynPX{\ue}{e}{\aprod{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
%   }{
%     \esynPX{\prj{\ue}{\ell}}{\aepr{\ell}{e}}{\tau}
%   }
% \end{equation}

% %Match expressions can appear in synthetic position.
% \begin{equation}\label{rule:esynP-match}
%   \inferrule{
%     n > 0\\
%     \esynPX{\ue}{e}{\tau}\\
%     \{\rsynPX{\urv_i}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}
%   }{
%     \esynPX{\matchwith{\ue}{\seqschemaX{\urv}}}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}
%   }
% \end{equation}

% \begin{equation}\label{rule:esynP-mval}
%   \inferrule{ }{
%     \esynP{\uOmega, \uMhyp{\uX}{X}{\asignature{\kappa}{u}{\tau}}}{\uPsi}{\uPhi}{\mval{\uX}}{\amval{X}}{[\amcon{X}/u]\tau}
%   }
% \end{equation}

% % ueTLMs can be defined and applied in synthetic position.
% % \begin{equation}\label{rule:esynP-defpetsm}
% % \inferrule{
% %   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
% %   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPCEExp}}\\\\
% %   \esynP{\uOmega}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\uI}}{\uPhi}{\ue}{e}{\tau}
% % }{
% %   \esynP{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\usyntaxueP{\tsmv}{\urho}{\eparse}{\ue}}{e}{\tau}
% % }
% % \end{equation}

% % \begin{equation}\label{rule:esynP-letpetsm}
% % \inferrule{
% %   \tsmexpExpandsExp{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
% %   \esynP{\uOmega}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Psi}{\uI}}{\uPhi}{\ue}{e}{\tau}
% % }{
% %   \esynP{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\uletpetsm{\tsmv}{\uepsilon}{\ue}}{e}{\tau}
% % }
% % \end{equation}

% \begin{equation}\label{rule:esynP-apuetsm}
% \inferrule{
%   \uOmega = \uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
%   \uPsi=\uAS{\uA}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\uI}\\\\
%   \tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\aetype{\tau_\text{final}}}\\
%   \tsmexpEvalsExp{\Omega_\text{app}}{\Psi}{\epsilon}{\epsilon_\text{normal}}\\\\
%   \tsmdefof{\epsilon_\text{normal}}=a\\
%   \encodeBody{b}{\ebody}\\
%   \evalU{\ap{\eparse}{\ebody}}{{\lbltxt{SuccessE}}\cdot{e_\text{pproto}}}\\\\
%   \decodePCEExp{e_\text{pproto}}{\pce}\\\\
%   \prepce{\Omega_\text{app}}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\pce}{\ce}{\epsilon_\text{normal}}{\aetype{\tau_\text{proto}}}{\omega}{\Omega_\text{params}}\\\\
%   \segOK{\segof{\ce}}{b}\\
%   \canaP{\Omega_\text{params}}{\esceneP{\omega : \OParams}{\uOmega}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau_\text{proto}}
% }{
%   \esynP{\uOmega}{\uPsi}{\uPhi}{\utsmap{\uepsilon}{b}}{[\omega]e}{[\omega]\tau_\text{proto}}
% }
% \end{equation}

% % These rules are nearly identical to Rules (\ref{rule:expandsUP-syntax}) and (\ref{rule:expandsUP-tsmap}), differing only in that the typed expansion premises have been replaced by corresponding synthetic typed expansion premises. The premises of these rules can be understood as described in Sections \ref{sec:U-uetsm-definition} and \ref{sec:U-uetsm-application}. The body encoding judgement and candidate expansion expression decoding judgements were characterized in Sec. \ref{sec:typed-expansion-UP}. We discuss candidate expansion validation in Sec. \ref{sec:ce-validation-B} below.

% % To support ueTLM implicits, ueTLM contexts, $\uPsi$, are redefined to take the form $\uASI{\uA}{\Psi}{\uI}$. TLM naming contexts, $\uA$, and ueTLM definition contexts, $\Psi$, were defined in Sec. \ref{sec:typed-expansion-UP}. We write $\uPsi, \uShyp{\tsmv}{a}{\tau}{\eparse}$ when $\uPsi=\uASI{\uA}{\Psi}{\uI}$ as shorthand for \[\uASI{\ctxUpdate{\uA}{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}\]

% % \emph{TLM designation contexts}, $\uI$, are finite functions that map each type $\tau \in \domof{\uI}$ to the \emph{TLM designation} $\designate{\tau}{a}$, for some symbol $a$. We write $\uI \uplus \designate{\tau}{a}$ for the TLM designation context that maps $\tau$ to $\designate{\tau}{a}$ and defers to $\uI$ for all other types (i.e. the previous designation, if any, is updated). 

% % The TLM designation context in the ueTLM context is updated by expressions of ueTLM designation form. Such expressions can appear in synthetic position, where they are governed by the following rule:% We write $\uIOK{\Delta}{\uI}$ when each type in $\uI$ is well-formed assuming $\Delta$.
% %\begin{definition}[TLM Designation Context Well-Formedness] $\uIOK{\Delta}{{\uI}$ iff for each $\designate{\tau}{a}$ we have $\istypeU{\Delta}{\tau}$.\end{definition}

% % \todo{peTLM implicit designation}
% % \begin{equation}\label{rule:esynP-implicite}
% %   \inferrule{
% %     \esyn{\uDelta}{\uGamma}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\uPhi}{\ue}{e}{\tau'}
% %   }{
% %     \esyn{\uDelta}{\uGamma}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}}{\uPhi}{\implicite{\tsmv}{\ue}}{e}{\tau'}
% %   }
% % \end{equation}

% % % Like ueTLMs, upTLMs can be defined in synthetic position.
% % \begin{equation}\label{rule:esynP-syntaxup}
% % \inferrule{
% %   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
% %   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultCEPat}}\\\\
% %   \esynP{\uOmega}{\uPsi}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\uI}}{\ue}{e}{\tau}
% % }{
% %   \esynP{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\usyntaxup{\tsmv}{\urho}{\eparse}{\ue}}{e}{\tau}
% % }
% % \end{equation}


% % \begin{equation}\label{rule:esynP-letpptsm}
% % \inferrule{
% %   \tsmexpExpandsPat{\uOmega}{\uASI{\uA}{\Phi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
% %   \esynP{\uOmega}{\uPsi}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Phi}{\uI}}{\ue}{e}{\tau}
% % }{
% %   \esynP{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\uletpptsm{\tsmv}{\uepsilon}{\ue}}{e}{\tau}
% % }
% % \end{equation}

% % % This rule is nearly identical to Rule (\ref{rule:expandsUP-defuptsm}), differing only in that the typed expansion premise has been replaced by the corresponding synthetic typed expansion premise. The premises can be understood as described in Section \ref{sec:uptsm-definition}.

% % % To support upTLM implicits, upTLM contexts, $\uPhi$, are redefined to take the form $\uASI{\uA}{\Phi}{\uI}$. upTLM definition contexts, $\Phi$, were defined in Sec. \ref{sec:uptsm-definition}. We write $\uPhi, \uPhyp{\tsmv}{a}{\tau}{\eparse}$ when $\uPhi=\uASI{\uA}{\Phi}{\uI}$ as shorthand for \[\uASI{\ctxUpdate{\uA}{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI}\]

% % % The TLM designation context in the upTLM context is updated by expressions of upTLM designation form. Such expressions can appear in synthetic position, where they are governed by the following rule:% We write $\uIOK{\Delta}{\uI}$ when each type in $\uI$ is well-formed assuming $\Delta$.
% % %\begin{definition}[TLM Designation Context Well-Formedness] $\uIOK{\Delta}{{\uI}$ iff for each $\designate{\tau}{a}$ we have $\istypeU{\Delta}{\tau}$.\end{definition}
% % \todo{ppTLM implicit designation}
% % \begin{equation}\label{rule:esynP-implicitp}
% %   \inferrule{
% %     \esyn{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\ue}{e}{\tau'}
% %   }{
% %     \esyn{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}}{\implicitp{\tsmv}{\ue}}{e}{\tau'}
% %   }
% % \end{equation}
% \end{subequations}


% Analytic typed expression expansion
% \begin{subequations}\label{rules:eanaP}
% % Type analysis subsumes type synthesis, in that when a type can be synthesized for an unexpanded expression, that unexpanded expression can also be analyzed against that type, producing the same expansion. This is expressed by the following \emph{subsumption rule} for unexpanded expressions.
% \begin{equation}\label{rule:eanaP-subsume}
%   \inferrule{
%     \esynPX{\ue}{e}{\tau}\\
%     \issubtypePX{\tau}{\tau'}
%   }{
%     \eanaPX{\ue}{e}{\tau'}
%   }
% \end{equation}

% % Additional rules are needed for certain forms in order to propagate types for analysis into subexpressions, and for forms that can appear only in analytic position.

% % Rule (\ref{rule:esyn-let}) governed value bindings in synthetic position. The following rule governs value bindings in analytic position.
% \begin{equation}\label{rule:eanaP-let}
%   \inferrule{
%     \esynPX{\ue}{e}{\tau}\\
%     \eanaP{\uOmega, \uGhyp{\ux}{x}{\tau}}{\uPsi}{\uPhi}{\ue'}{e'}{\tau'}
%   }{
%     \eanaPX{\letsyn{\ux}{\ue}{\ue'}}{\aeap{\aelam{\tau}{x}{e'}}{e}}{\tau'}
%   }
% \end{equation}

% % An unannotated function can appear only in analytic position. The argument type is determined from the type that the unannotated function is being analyzed against. 
% \begin{equation}\label{rule:eanaP-analam}
%   \inferrule{
%     \eanaP{\uOmega, \uGhyp{\ux}{x}{\tau_1}}{\uPsi}{\uPhi}{\ue}{e}{\tau_2}
%   }{
%     \eanaPX{\analam{\ux}{\ue}}{\aelam{\tau_1}{x}{e}}{\aparr{\tau_1}{\tau_2}}
%   }
% \end{equation}

% % Rule (\ref{rule:esyn-tlam}) governed type lambdas in synthetic position. The following rule governs type lambdas in analytic position.
% % \begin{equation}\label{rule:eanaP-tlam}
% %   \inferrule{
% %     \eana{\uDelta, \uDhyp{\ut}{t}}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}
% %   }{
% %     \eanaPX{\clam{\uu}{\ue}}{\aetlam{t}{e}}{\aall{t}{\tau}}
% %   }
% % \end{equation}

% % Values of recursive types can be introduced only in analytic position.
% \begin{equation}\label{rule:eanaP-fold}
%   \inferrule{
%     \eanaPX{\ue}{e}{[\arec{t}{\tau}/t]\tau}
%   }{
%     \eanaPX{\fold{\ue}}{\aefold{e}}{\arec{t}{\tau}}
%   }
% \end{equation}

% % Rule (\ref{rule:esyn-tpl}) governed labeled tuples in synthetic position. The following rule governs labeled tuples in analytic position.
% \begin{equation}\label{rule:eanaP-tpl}
%   \inferrule{
%     \{\eanaPX{\ue_i}{e_i}{\tau_i}\}_{i \in \labelset}
%   }{
%     \eanaPX{\tpl{\mapschema{\ue}{i}{\labelset}}}{\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
%   }
% \end{equation}

% % Values of labeled sum type can appear only in analytic position.
% \begin{equation}\label{rule:eanaP-in}
%   \inferrule{
%     \tau = \asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau'}}\\\\
%     \eanaPX{\ue'}{e'}{\tau'}
%   }{
%     \eanaPX{\inj{\ell}{\ue}}{\aein{\ell}{e'}}{\tau}
%     % \uOmega \vdash_{\uPsi; \uPhi} \left(\shortstack{$\ue \leadsto $\\$\Leftarrow$\vspace{-1.2em}}\right)
%     %\eanaPX{\auanain{\ell}{\ue}}{\aein{\ell}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
%   }
% \end{equation}

% % Rule (\ref{rule:esyn-match}) governed match expressions in synthetic position. The following rule governs match expressions in analytic position.
% \begin{equation}\label{rule:eanaP-match}
%   \inferrule{
%     \esynPX{\ue}{e}{\tau}\\
%     \{\ranaPX{\urv_i}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}
%   }{
%     \eanaPX{\matchwith{\ue}{\seqschemaX{\urv}}}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}
%   }
% \end{equation}

% % Rule (\ref{rule:esyn-defuetsm}) governed ueTLM definitions in synthetic position. The following rule governs ueTLM definitions in analytic position.
% % \begin{equation}\label{rule:eanaP-defpetsm}
% % \inferrule{
% %   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
% %   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPCEExp}}\\\\
% %   \eanaP{\uOmega}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\uI}}{\uPhi}{\ue}{e}{\tau}
% % }{
% %   \eanaP{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\usyntaxueP{\tsmv}{\urho}{\eparse}{\ue}}{e}{\tau}
% % }
% % \end{equation}

% % \begin{equation}\label{rule:eanaP-letpetsm}
% % \inferrule{
% %   \tsmexpExpandsExp{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
% %   \eanaP{\uOmega}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Psi}{\uI}}{\uPhi}{\ue}{e}{\tau}
% % }{
% %   \eanaP{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\uletpetsm{\tsmv}{\uepsilon}{\ue}}{e}{\tau}
% % }
% % \end{equation}

% % \todo{peTLM implicit designation}
% % Rule (\ref{rule:esyn-implicite}) governed ueTLM designations in synthetic position. The following rule governs ueTLM designations in analytic position.
% % \begin{equation}\label{rule:eanaP-implicite}
% %   \inferrule{
% %     \eana{\uDelta}{\uGamma}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\uPhi}{\ue}{e}{\tau'}
% %   }{
% %     \eana{\uDelta}{\uGamma}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}}{\uPhi}{\implicite{\tsmv}{\ue}}{e}{\tau'}
% %   }
% % \end{equation}

% % \todo{peTLM implicit application}
% % % An expression of unadorned literal form can appear only in analytic position. The following rule extracts the TLM designated at the type that the expression is being analyzed against from the TLM designation context in the ueTLM context and applies it implicitly, i.e. the premises correspond to those of Rule (\ref{rule:esyn-apuetsm}).
% \begin{equation}\label{rule:eanaP-lit}
%   \inferrule{
%     \encodeBody{b}{\ebody}\\
%     \evalU{\ap{\eparse}{\ebody}}{\inj{\lbltxt{Success}}{\ecand}}\\
%     \decodeCondE{\ecand}{\ce}\\\\
%     \cana{\emptyset}{\emptyset}{\esceneUP{\uDelta}{\uGamma}{\uASI{\uA}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\uPhi}{b}}{\ce}{e}{\tau}
%   }{
%     \eana{\uDelta}{\uGamma}{\uASI{\uA}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\uPhi}{\auelit{b}}{e}{\tau}
%   }
% \end{equation}

% % Rule (\ref{rule:esyn-defuptsm}) governed upTLM definitions in synthetic position. The following rule governs upTLM definitions in analytic position.
% % \begin{equation}\label{rule:eanaP-syntaxup}
% % \inferrule{
% %   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
% %   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultCEPat}}\\\\
% %   \eanaP{\uOmega}{\uPsi}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\uI}}{\ue}{e}{\tau}
% % }{
% %   \eanaP{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\usyntaxup{\tsmv}{\urho}{\eparse}{\ue}}{e}{\tau}
% % }
% % \end{equation}


% % \begin{equation}\label{rule:eanaP-letpptsm}
% % \inferrule{
% %   \tsmexpExpandsPat{\uOmega}{\uASI{\uA}{\Phi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
% %   \eanaP{\uOmega}{\uPsi}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Phi}{\uI}}{\ue}{e}{\tau}
% % }{
% %   \eanaP{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\uletpptsm{\tsmv}{\uepsilon}{\ue}}{e}{\tau}
% % }
% % \end{equation}


% % \todo{ppTLM implicit designation}
% % % Rule (\ref{rule:esyn-implicitp}) governed upTLM designations in synthetic position. The following rule governs upTLM designations in analytic position.
% % \begin{equation}\label{rule:eanaP-implicitp}
% %   \inferrule{
% %     \eana{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\ue}{e}{\tau'}
% %   }{
% %     \eana{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA\uplus\vExpands{\tsmv}{a}}{\Phi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI}}{\implicitp{\tsmv}{\ue}}{e}{\tau'}
% %   }
% % \end{equation}

% \end{subequations}

% Synthetic rule expansion
% %The synthetic typed rule expansion judgement is invoked iteratively by Rule (\ref{rule:esyn-match}) to synthesize a type, $\tau'$, from the branch expressions in the rule sequence. This judgement is defined mutually inductively with Rules (\ref{rules:esyn}) and Rules (\ref{rules:eana}) by the following rule. 
% \begin{equation}\label{rule:rsynP}
%   \inferrule{
%     \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}\\
%     \patExpandsP{\uOmegaEx{\emptyset}{\uG'}{\emptyset}{\Omega'}}{\uPhi}{\upv}{p}{\tau}\\
%     \esynP{\uOmegaEx{\uD}{\uG \uplus \uG'}{\uMctx}{\Omega \cup \Omega'}}{\uPsi}{\uPhi}{\ue}{e}{\tau'}
%   }{
%     \rsynP{\uOmega}{\uPsi}{\uPhi}{\matchrule{\upv}{\ue}}{\aematchrule{p}{e}}{\tau}{\tau'}
%   }
% \end{equation}

% Analytic rule expansion
% %The analytic typed rule expansion judgement is invoked iteratively by Rule (\ref{rule:eana-match}). This judgement is defined mutually inductively with Rules (\ref{rules:esyn}), Rules (\ref{rules:eana}), and Rule (\ref{rule:rsyn}) by the following rule, which is the analytic analag of Rule (\ref{rule:rsyn}).
% \begin{equation}\label{rule:ranaP}
%   \inferrule{
%     \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}\\
%     \patExpandsP{\uOmegaEx{\emptyset}{\uG'}{\emptyset}{\Omega'}}{\uPhi}{\upv}{p}{\tau}\\
%     \eanaP{\uOmegaEx{\uD}{\uG \uplus \uG'}{\uMctx}{\Omega \cup \Omega'}}{\uPsi}{\uPhi}{\ue}{e}{\tau'}
%   }{
%     \ranaP{\uOmega}{\uPsi}{\uPhi}{\matchrule{\upv}{\ue}}{\aematchrule{p}{e}}{\tau}{\tau'}
%   }
% \end{equation}

% %The premises of these rules can be understood as described in Sec. \ref{sec:typed-expansion-UP}.% We will define typed pattern expansion below.

% Typed pattern expansion
% % The typed pattern expansion judgement is inductively defined by Rules (\ref{rules:patExpandsP}) as follows. %As in $\miniVersePat$, \emph{unexpanded pattern typing contexts}, $\upctx$, are defined identically to unexpanded typing contexts (i.e. we only use a distinct metavariable to emphasize their distinct roles in the judgements above). 

% % The following rules are written identically to the typed pattern expansion rules for shared pattern forms in $\miniVersePat$, i.e. Rules (\ref{rule:patExpands-var}) through (\ref{rule:patExpands-in}).
% \begin{subequations}\label{rules:patExpandsP-B}
% \begin{equation}\label{rule:patExpandsP-B-subsume}
% \inferrule{
%   \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}\\\\
%   \patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{\tau}\\
%   \issubtypeP{\Omega}{\tau}{\tau'}
% }{
%   \patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{\tau'}
% }
% \end{equation}
% \begin{equation}\label{rule:patExpandsP-B-var}
% \inferrule{ }{
%   \patExpandsP{\uOmegaEx{\emptyset}{\vExpands{\ux}{x}}{\emptyset}{\Ghyp{x}{\tau}}}{\uPhi}{\ux}{x}{\tau}
% }
% \end{equation}
% \begin{equation}\label{rule:patExpandsP-B-wild}
% \inferrule{ }{
%   \patExpandsP{\uOmegaEx{\emptyset}{\emptyset}{\emptyset}{\emptyset}}{\uPhi}{\wildp}{\aewildp}{\tau}
% }
% \end{equation}
% \begin{equation}\label{rule:patExpandsP-B-fold}
% \inferrule{ 
%   \patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{[\arec{t}{\tau}/t]\tau}
% }{
%   \patExpandsP{\uOmega'}{\uPhi}{\foldp{\upv}}{\aefoldp{p}}{\arec{t}{\tau}}
% }
% \end{equation}
% \begin{equation}\label{rule:patExpandsP-B-tpl}
% \inferrule{
%   \tau=\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}\\\\
%   \{\patExpandsP{{\uOmega_i}}{\uPhi}{\upv_i}{p_i}{\tau_i}\}_{i \in \labelset}
% }{
%   %\patExpandsP{\Gconsi{i \in \labelset}{\upctx_i}}{A}{B}{C}
%   \patExpandsP{\Gconsi{i \in \labelset}{\uOmega_i}}{\uPhi}{\tplp{\mapschema{\upv}{i}{\labelset}}}{\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}}{\tau}
%   % \patExpands{\Gconsi{i \in \labelset}{\pctx_i}}{\Phi}{
%   %   \autplp{\labelset}{\mapschema{\upv}{i}{\labelset}}
%   % }{
%   %   \aetplp{\labelset}{\mapschema{p}{i}{\labelset}}
%   % }{
%   %   \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}
%   % } %{\autplp{\labelset}{\mapschema{\upv}{i}{\labelset}}}{\aetplp{\labelset}{\mapschema}{p}{i}{\labelset}}{...}
%   %\left(\shortstack{$\Delta \vdash_{\uPhi} \autplp{\labelset}{\mapschema{\upv}{i}{\labelset}}$\\$\leadsto$\\$\aetplp{\labelset}{\mapschema{p}{i}{\labelset}} : \aprod{\labelset}{\mapschema{\tau}{i}{\labelset}} \dashV \Gconsi{i \in \labelset}{\upctx_i}$\vspace{-1.2em}}\right)
% }
% \end{equation}
% \begin{equation}\label{rule:patExpandsP-B-in}
% \inferrule{
%   \patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{\tau}
% }{
%   \patExpandsP{\uOmega'}{\uPhi}{\injp{\ell}{\upv}}{\aeinjp{\ell}{p}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
% }
% \end{equation}

% \begin{equation}\label{rule:patExpandsP-B-apuptsm}
% \inferrule{
%   \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
%   \uPhi=\uASI{\uA}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\uI}\\\\
%   \tsmexpExpandsPat{\uOmega}{\uPhi}{\uepsilon}{\epsilon}{\aetype{\tau_\text{final}}}\\
%   \tsmdefof{\epsilon}=a\\\\
%   \encodeBody{b}{\ebody}\\
%   \evalU{\ap{\eparse}{\ebody}}{{\lbltxt{Success}}\cdot{\ecand}}\\
%   \decodeCEPat{\ecand}{\cpv}\\\\
%   \prepcp{\Omega_\text{app}}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\pcp}{\cpv}{\epsilon}{\aetype{\tau_\text{cand}}}{\omega}{\Omega_\text{params}}\\\\
%   \cvalidPP{\uOmega'}{\psceneP{\uOmega}{\uPhi}{b}}{\cpv}{p}{\tau_\text{cand}}
% }{
%   \patExpandsP{\uOmega'}{\uPhi}{\utsmap{\uepsilon}{b}}{p}{[\omega]\tau_\text{cand}}
% }
% \end{equation}

% \todo{ppTLM implicit application}
% % Unexpanded patterns of unadorned literal form are governed by the following rule, which extracts the designated upTLM from the upTLM context and applies it implicitly, i.e. the premises correspond to those of Rule (\ref{rule:patExpandsP-apuptsm}).
% \begin{equation}\label{rule:patExpandsP-B-lit}
% \inferrule{
%   \encodeBody{b}{\ebody}\\
%   \evalU{\ap{\eparse}{\ebody}}{\inj{\lbltxt{Success}}{\ecand}}\\
%   \decodeCEPat{\ecand}{\cpv}\\\\
%   \cvalidPP{\uOmega}{\pscene{\uDelta}{\uASI{\uA}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI, \designate{\tau}{a}}}{b}}{\cpv}{p}{\tau}
% }{
%   \patExpands{\uOmega}{\uASI{\uA}{\Phi, \xuptsmbnd{a}{\tau}{\eparse}}{\uI, \designate{\tau}{a}}}{\lit{b}}{p}{\tau}
% }
% \end{equation}

% \end{subequations}


% \subsubsection{Unexpanded Signatures and Module Expressions}
% Signature expansion
% \begin{equation}\label{rule:sigExpandsP-B}
% \inferrule{
%   \kExpandsX{\ukappa}{\kappa}\\
%   \cExpands{\uOmega, \uKhyp{\uu}{u}{\kappa}}{\utau}{\tau}{\akty}
% }{
%   \sigExpandsPX{\signature{\uu}{\ukappa}{\utau}}{\asignature{\kappa}{u}{\tau}}
% }
% \end{equation}

% Synthetic module expression expansion
% \begin{subequations}\label{rules:msyn}
% \begin{equation}\label{rule:msyn-var}
% \inferrule{ }{
%   \msyn{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\uPsi}{\uPhi}{\uX}{X}{\sigma}
% }
% \end{equation}
% \begin{equation}\label{rule:msyn-seal}
% \inferrule{
%   \sigExpandsPX{\usigma}{\sigma}\\
%   \manaX{\uM}{M}{\sigma}
% }{
%   \msynX{\seal{\uM}{\usigma}}{\aseal{\sigma}{M}}{\sigma} 
% }
% \end{equation}
% \begin{equation}\label{rule:msyn-mlet}
% \inferrule{
%   \msynX{\uM}{M}{\sigma}\\
%   \sigExpandsPX{\usigma'}{\sigma'}\\\\
%   \mana{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\uPsi}{\uPhi}{\uM'}{M'}{\sigma'}
% }{
%   \msynX{\mlet{\uX}{\uM}{\uM'}{\usigma'}}{\amlet{\sigma'}{M}{X}{M'}}{\sigma'}
% }
% \end{equation}
% \begin{equation}\label{rule:msyn-syntaxpe}
% \inferrule{
%   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
%   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPCEExp}}\\\\
%   \msyn{\uOmega}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\uI}}{\uPhi}{\uM}{M}{\sigma}
% }{
%   \msyn{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\usyntaxueP{\tsmv}{\urho}{\eparse}{\uM}}{M}{\sigma}
% }
% \end{equation}
% \begin{equation}\label{rule:msyn-letpetsm}
% \inferrule{
%   \tsmexpExpandsExp{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
%   \msyn{\uOmega}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Psi}{\uI}}{\uPhi}{\uM}{M}{\sigma}
% }{
%   \msyn{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\uletpetsm{\tsmv}{\uepsilon}{\uM}}{M}{\sigma}
% }
% \end{equation}
% \todo{peTLM implicit designation at module level}
% \begin{equation}\label{rule:msyn-implicitpe}
% \inferrule{
%   ...
% }{
%   ...
% }
% \end{equation}
% \begin{equation}\label{rule:msyn-syntaxpp}
% \inferrule{
%   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
%   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultCEPat}}\\\\
%   \msyn{\uOmega}{\uPsi}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\uI}}{\uM}{M}{\sigma}
% }{
%   \msyn{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\usyntaxup{\tsmv}{\urho}{\eparse}{\uM}}{M}{\sigma}
% }
% \end{equation}
% \begin{equation}\label{rule:msyn-letpptsm}
% \inferrule{
%   \tsmexpExpandsPat{\uOmega}{\uASI{\uA}{\Phi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
%   \msyn{\uOmega}{\uPsi}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Phi}{\uI}}{\uM}{M}{\sigma}
% }{
%   \msyn{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\uletpptsm{\tsmv}{\uepsilon}{\uM}}{M}{\sigma}
% }
% \end{equation}
% \todo{ppTLM implicit designation at module level}
% \begin{equation}\label{rule:msyn-implicitpp}
% \inferrule{
%   ...
% }{
%   ...
% }
% \end{equation}
% \end{subequations}

% Analytic module expression expansion
% \begin{subequations}\label{rules:mana}
% \begin{equation}\label{rule:mana-subsumes}
% \inferrule{
%   \msynX{\uM}{M}{\sigma}\\
%   \sigsub{\uOmega}{\sigma}{\sigma'}
% }{
%   \manaX{\uM}{M}{\sigma'}
% }
% \end{equation}
% \begin{equation}\label{rule:mana-struct}
% \inferrule{
%   \kanaX{\uc}{c}{\kappa}\\
%   \eanaPX{\ue}{e}{[c/u]\tau}
% }{
%   \manaX{\struct{\uc}{\ue}}{\astruct{c}{e}}{\asignature{\kappa}{u}{\tau}}
% }
% \end{equation}
% \begin{equation}\label{rule:mana-syntaxpe}
% \inferrule{
%   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
%   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultPCEExp}}\\\\
%   \mana{\uOmega}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\uI}}{\uPhi}{\uM}{M}{\sigma}
% }{
%   \mana{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\usyntaxueP{\tsmv}{\urho}{\eparse}{\uM}}{M}{\sigma}
% }
% \end{equation}
% \begin{equation}\label{rule:mana-letpetsm}
% \inferrule{
%   \tsmexpExpandsExp{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
%   \mana{\uOmega}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Psi}{\uI}}{\uPhi}{\uM}{M}{\sigma}
% }{
%   \mana{\uOmega}{\uASI{\uA}{\Psi}{\uI}}{\uPhi}{\uletpetsm{\tsmv}{\uepsilon}{\uM}}{M}{\sigma}
% }
% \end{equation}
% \todo{peTLM implicit designation at module level}
% \begin{equation}\label{rule:mana-implicitpe}
% \inferrule{
%   ...
% }{
%   ...
% }
% \end{equation}
% \begin{equation}\label{rule:mana-syntaxpp}
% \inferrule{
%   \tsmtyExpands{\uOmega}{\urho}{\rho}\\
%   \hastypeP{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultCEPat}}\\\\
%   \mana{\uOmega}{\uPsi}{\uASI{\uA \uplus \mapitem{\tsmv}{\adefref{a}}}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\uI}}{\uM}{M}{\sigma}
% }{
%   \mana{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\usyntaxup{\tsmv}{\urho}{\eparse}{\uM}}{M}{\sigma}
% }
% \end{equation}
% \begin{equation}\label{rule:mana-letpptsm}
% \inferrule{
%   \tsmexpExpandsPat{\uOmega}{\uASI{\uA}{\Phi}{\uI}}{\uepsilon}{\epsilon}{\rho}\\
%   \mana{\uOmega}{\uPsi}{\uASI{\uA\uplus\mapitem{\tsmv}{\epsilon}}{\Phi}{\uI}}{\uM}{M}{\sigma}
% }{
%   \mana{\uOmega}{\uPsi}{\uASI{\uA}{\Phi}{\uI}}{\uletpptsm{\tsmv}{\uepsilon}{\uM}}{M}{\sigma}
% }
% \end{equation}
% \todo{ppTLM implicit designation at module level}
% \begin{equation}\label{rule:mana-implicitpp}
% \inferrule{
%   ...
% }{
%   ...
% }
% \end{equation}
% \end{subequations}

% \subsubsection{TLM Types and Expressions}
% TLM Expression Typing

% \vspace{10px}
% $\begin{array}{ll}
% \textbf{Judgement Form} & \textbf{Description}\\
% \istsmty{\Omega}{\rho} & \text{$\rho$ is a well-formed TLM type}\\
% \hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho} & \text{peTLM expression $\epsilon$ has TLM type $\rho$}\\
% \hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho} & \text{ppTLM expression $\epsilon$ has TLM type $\rho$}
% \end{array}$
% \vspace{10px}

% peTLM Expression Evaluation

% \vspace{10px}
% $\begin{array}{ll}
% \textbf{Judgement Form} & \textbf{Description}\\
% \tsmexpNormalExp{\Omega}{\Psi}{\epsilon} & \text{peTLM expression $\epsilon$ is in normal form}\\
% \tsmexpStepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'} & \text{peTLM expression $\epsilon$ transitions to $\epsilon'$}\\
% \end{array}$
% \vspace{10px}

% + auxiliary judgements for multi-step transitions and evaluation

% unexpanded TLM types and expressions

% \vspace{10px}
% $\begin{array}{ll}
% \textbf{Judgement Form} & \textbf{Description}\\
% \tsmtyExpands{\uOmega}{\urho}{\rho} & \text{$\urho$ has expansion $\rho$}\\
% \tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\rho} & \text{unexpanded peTLM expression $\uepsilon$ has expansion $\epsilon$ and type $\rho$}\\
% \tsmexpExpandsPat{\uOmega}{\uPhi}{\uepsilon}{\epsilon}{\rho} & \text{unexpanded ppTLM expression $\uepsilon$ has expansion $\epsilon$ and type $\rho$}
% \end{array}$
% \vspace{10px}

% TLM type formation
% \begin{subequations}\label{rules:istsmty-B}
% \begin{equation}\label{rule:istsmty-B-type}
% \inferrule{
%   \haskindX{\tau}{\akty}
% }{
%   \istsmty{\Omega}{\aetype{\tau}}
% }
% \end{equation}
% \begin{equation}\label{rule:istsmty-B-alltypes}
% \inferrule{
%   \istsmty{\Omega, t :: \akty}{\rho}
% }{
%   \istsmty{\Omega}{\aealltypes{t}{\rho}}
% }
% \end{equation}
% \begin{equation}\label{rule:istsmty-B-allmods}
% \inferrule{
%   \issig{\Omega}{\sigma}\\
%   \istsmty{\Omega, X : \sigma}{\rho}
% }{
%   \istsmty{\Omega}{\aeallmods{\sigma}{X}{\rho}}
% }
% \end{equation}
% \end{subequations}

% Unexpanded TLM type expansion
% \begin{subequations}\label{rules:tsmtyExpands-B}
% \begin{equation}\label{rule:tsmtyExpands-B-type}
% \inferrule{
%   \cExpandsX{\utau}{\tau}{\akty}
% }{
%   \tsmtyExpands{\uOmega}{{\utau}}{\aetype{\tau}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmtyExpands-B-alltypes}
% \inferrule{
%   \tsmtyExpands{\uOmega, \uKhyp{\ut}{t}{\akty}}{\urho}{\rho}
% }{
%   \tsmtyExpands{\uOmega}{\alltypes{\ut}{\urho}}{\aealltypes{t}{\rho}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmtyExpands-B-allmods}
% \inferrule{
%   \sigExpandsPX{\usigma}{\sigma}\\
%   \tsmtyExpands{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\urho}{\rho}
% }{
%   \tsmtyExpands{\uOmega}{\allmods{\uX}{\usigma}{\urho}}{\aeallmods{\sigma}{X}{\rho}}
% }
% \end{equation}
% \end{subequations}
% peTLM Expression Typing
% \begin{subequations}\label{rules:hastsmtypeExp-B}
% \begin{equation}\label{rule:hastsmtypeExp-B-defref}
% \inferrule{ }{
%   \hastsmtypeExp{\Omega}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\adefref{a}}{\rho}
% }
% \end{equation}
% \begin{equation}\label{rule:hastsmtypeExp-B-abstype}
% \inferrule{
%   \hastsmtypeExp{\Omega, t :: \akty}{\Psi}{\epsilon}{\rho}
% }{
%   \hastsmtypeExp{\Omega}{\Psi}{\aeabstype{t}{\epsilon}}{\aealltypes{t}{\rho}}
% }
% \end{equation}
% \begin{equation}\label{rule:hastsmtypeExp-B-absmod}
% \inferrule{
%   \issigX{\sigma}\\
%   \hastsmtypeExp{\Omega, X : \sigma}{\Psi}{\epsilon}{\rho}
% }{
%   \hastsmtypeExp{\Omega}{\Psi}{\aeabsmod{\sigma}{X}{\epsilon}}{\aeallmods{\sigma}{X}{\rho}}
% }
% \end{equation}
% \begin{equation}\label{rule:hastsmtypeExp-B-aptype}
% \inferrule{
%   \hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\aealltypes{t}{\rho}}\\
%   \haskindX{\tau}{\akty}
% }{
%   \hastsmtypeExp{\Omega}{\Psi}{\aeaptype{\tau}{\epsilon}}{[\tau/t]\rho}
% }
% \end{equation}
% \begin{equation}\label{rule:hastsmtypeExp-B-apmod}
% \inferrule{
%   \hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\aeallmods{\sigma}{X'}{\rho}}\\
%   \hassig{\Omega}{X}{\sigma}
% }{
%   \hastsmtypeExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon}}{[X/X']\rho}
% }
% \end{equation}
% \end{subequations}

% ppTLM Expression Typing
% \begin{subequations}\label{rules:hastsmtypePat-B}
% \begin{equation}\label{rule:hastsmtypePat-B-defref}
% \inferrule{ }{
%   \hastsmtypePat{\Omega}{\Phi, \pptsmdefn{a}{\rho}{\eparse}}{\adefref{a}}{\rho}
% }
% \end{equation}
% \begin{equation}\label{rule:hastsmtypePat-B-abstype}
% \inferrule{
%   \hastsmtypePat{\Omega, t :: \akty}{\Phi}{\epsilon}{\rho}
% }{
%   \hastsmtypePat{\Omega}{\Phi}{\aeabstype{t}{\epsilon}}{\aealltypes{t}{\rho}}
% }
% \end{equation}
% \begin{equation}\label{rule:hastsmtypePat-B-absmod}
% \inferrule{
%   \issigX{\sigma}\\
%   \hastsmtypePat{\Omega, X : \sigma}{\Phi}{\epsilon}{\rho}
% }{
%   \hastsmtypePat{\Omega}{\Phi}{\aeabsmod{\sigma}{X}{\epsilon}}{\aeallmods{\sigma}{X}{\rho}}
% }
% \end{equation}
% \begin{equation}\label{rule:hastsmtypePat-B-aptype}
% \inferrule{
%   \hastsmtypePat{\Omega}{\Phi}{\epsilon}{\aealltypes{t}{\rho}}\\
%   \haskindX{\tau}{\akty}
% }{
%   \hastsmtypePat{\Omega}{\Phi}{\aeaptype{\tau}{\epsilon}}{[\tau/t]\rho}
% }
% \end{equation}
% \begin{equation}\label{rule:hastsmtypePat-B-apmod}
% \inferrule{
%   \hastsmtypePat{\Omega}{\Phi}{\epsilon}{\aeallmods{\sigma}{X'}{\rho}}\\
%   \hassig{\Omega}{X}{\sigma}
% }{
%   \hastsmtypePat{\Omega}{\Phi}{\aeapmod{X}{\epsilon}}{[X/X']\rho}
% }
% \end{equation}

% \end{subequations}

% peTLM Expression Expansion
% \begin{subequations}\label{rules:tsmexpExpandsExp-B}
% \begin{equation}\label{rule:tsmexpExpandsExp-B-bindref}
% \inferrule{
%   \hastsmtypeExp{\Omega}{\Psi}{\epsilon}{\rho}  
% }{
%   \tsmexpExpandsExp{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\uASI{\uA, \mapitem{\tsmv}{\epsilon}}{\Psi}{\uI}}{{\tsmv}}{\epsilon}{\rho}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpExpandsExp-B-abstype}
% \inferrule{
%   \tsmexpExpandsExp{\uOmega, \uKhyp{\ut}{t}{\akty}}{\uPsi}{\uepsilon}{\epsilon}{\rho}
% }{
%   \tsmexpExpandsExp{\uOmega}{\uPsi}{\abstype{\ut}{\uepsilon}}{\aeabstype{t}{\epsilon}}{\aealltypes{t}{\rho}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpExpandsExp-B-absmod}
% \inferrule{
%   \sigExpandsPX{\usigma}{\sigma}\\
%   \tsmexpExpandsExp{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\uPsi}{\uepsilon}{\epsilon}{\rho}
% }{
%   \tsmexpExpandsExp{\uOmega}{\uPsi}{\absmod{\uX}{\usigma}{\uepsilon}}{\aeabsmod{\sigma}{X}{\epsilon}}{\aeallmods{\sigma}{X}{\rho}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpExpandsExp-B-aptype}
% \inferrule{
%   \tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\aealltypes{t}{\rho}}\\
%   \cExpandsX{\utau}{\tau}{\akty}
% }{
%   \tsmexpExpandsExp{\uOmega}{\uPsi}{\aptype{\uepsilon}{\utau}}{\aeaptype{\tau}{\epsilon}}{[\tau/t]\rho} 
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpExpandsExp-B-apmod}
% \inferrule{
%   \tsmexpExpandsExp{\uOmega}{\uPsi}{\uepsilon}{\epsilon}{\aeallmods{\sigma}{X'}{\rho}}\\
%   \manaX{\uX}{X}{\sigma}
% }{
%   \tsmexpExpandsExp{\uOmega}{\uPsi}{\apmod{\uepsilon}{\uX}}{\aeapmod{X}{\epsilon}}{[X/X']\rho}
% }
% \end{equation}
% \end{subequations}

% ppTLM expression expansion
% \begin{subequations}\label{rules:tsmexpExpandsPat-B}
% \begin{equation}\label{rule:tsmexpExpandsPat-B-bindref}
% \inferrule{
%   \hastsmtypePat{\Omega}{\Phi}{\epsilon}{\rho}  
% }{
%   \tsmexpExpandsPat{\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega}}{\uASI{\uA, \mapitem{\tsmv}{\epsilon}}{\Phi}{\uI}}{{\tsmv}}{\epsilon}{\rho}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpExpandsPat-B-abstype}
% \inferrule{
%   \tsmexpExpandsPat{\uOmega, \uKhyp{\ut}{t}{\akty}}{\uPhi}{\uepsilon}{\epsilon}{\rho}
% }{
%   \tsmexpExpandsPat{\uOmega}{\uPhi}{\abstype{\ut}{\uepsilon}}{\aeabstype{t}{\epsilon}}{\aealltypes{t}{\rho}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpExpandsPat-B-absmod}
% \inferrule{
%   \sigExpandsPX{\usigma}{\sigma}\\
%   \tsmexpExpandsPat{\uOmega, \uMhyp{\uX}{X}{\sigma}}{\uPhi}{\uepsilon}{\epsilon}{\rho}
% }{
%   \tsmexpExpandsPat{\uOmega}{\uPhi}{\absmod{\uX}{\usigma}{\uepsilon}}{\aeabsmod{\sigma}{X}{\epsilon}}{\aeallmods{\sigma}{X}{\rho}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpExpandsPat-B-aptype}
% \inferrule{
%   \tsmexpExpandsPat{\uOmega}{\uPhi}{\uepsilon}{\epsilon}{\aealltypes{t}{\rho}}\\
%   \cExpandsX{\utau}{\tau}{\akty}
% }{
%   \tsmexpExpandsPat{\uOmega}{\uPhi}{\aptype{\uepsilon}{\utau}}{\aeaptype{\tau}{\epsilon}}{[\tau/t]\rho} 
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpExpandsPat-B-apmod}
% \inferrule{
%   \tsmexpExpandsPat{\uOmega}{\uPhi}{\uepsilon}{\epsilon}{\aeallmods{\sigma}{X'}{\rho}}\\
%   \manaX{\uX}{X}{\sigma}
% }{
%   \tsmexpExpandsPat{\uOmega}{\uPhi}{\apmod{\uepsilon}{\uX}}{\aeapmod{X}{\epsilon}}{[X/X']\rho}
% }
% \end{equation}
% \end{subequations}

% peTLM expression normal forms
% \begin{subequations}\label{rules:tsmexpNormalExp-B}
% \begin{equation}\label{rule:tsmexpNormalExp-B-defref}
% \inferrule{ }{
%   \tsmexpNormalExp{\Omega}{\Psi, \petsmdefn{a}{\rho}{\eparse}}{\adefref{a}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpNormalExp-B-abstype}
% \inferrule{ }{
%   \tsmexpNormalExp{\Omega}{\Psi}{\aeabstype{t}{\epsilon}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpNormalExp-B-absmod}
% \inferrule{ }{
%   \tsmexpNormalExp{\Omega}{\Psi}{\aeabsmod{\sigma}{X}{\epsilon}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpNormalExp-B-aptype}
% \inferrule{
%   \epsilon \neq \aeabstype{t}{\epsilon'}\\
%   \tsmexpNormalExp{\Omega}{\Psi}{\epsilon}
% }{
%   \tsmexpNormalExp{\Omega}{\Psi}{\aeaptype{\tau}{\epsilon}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpNormalExp-B-apmod}
% \inferrule{
%   \epsilon \neq \aeabsmod{\sigma}{X'}{\epsilon'}\\
%   \tsmexpNormalExp{\Omega}{\Psi}{\epsilon}
% }{
%   \tsmexpNormalExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon}}
% }
% \end{equation}
% \end{subequations}

% peTLM transitions
% \begin{subequations}\label{rules:tsmexpStepsExp-B}
% \begin{equation}\label{rule:tsmexpStepsExp-B-aptype-1}
% \inferrule{
%   \tsmexpStepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}
% }{
%   \tsmexpStepsExp{\Omega}{\Psi}{\aeaptype{\tau}{\epsilon}}{\aeaptype{\tau}{\epsilon'}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpStepsExp-B-aptype-2}
% \inferrule{ }{
%   \tsmexpStepsExp{\Omega}{\Psi}{\aeaptype{\tau}{\aeabstype{t}{\epsilon}}}{[\tau/t]\epsilon}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpStepsExp-B-apmod-1}
% \inferrule{
%   \tsmexpStepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}
% }{
%   \tsmexpStepsExp{\Omega}{\Psi}{\aeapmod{X}{\epsilon}}{\aeapmod{X}{\epsilon'}}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpStepsExp-B-apmod-2}
% \inferrule{ }{
%   \tsmexpStepsExp{\Omega}{\Psi}{\aeapmod{X}{\aeabsmod{\sigma}{X'}{\epsilon}}}{[X/X']\epsilon}
% }
% \end{equation}
% \end{subequations}

% peTLM reflexive, transitive transitions
% \begin{subequations}\label{rules:tsmexpMultistepsExp-B}
% \begin{equation}\label{rule:tsmexpMultistepsExp-B-refl}
% \inferrule{ }{
%   \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpMultistepsExp-B-steps}
% \inferrule{
%   \tsmexpStepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}
% }{
%   \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}
% }
% \end{equation}
% \begin{equation}\label{rule:tsmexpMultistepsExp-B-trans}
% \inferrule{
%   \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}\\
%   \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon'}{\epsilon''}
% }{
%   \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon''}
% }
% \end{equation}
% \end{subequations}

% peTLM normalization
% \begin{equation}\label{rule:tsmexpEvalsExp-B}
% \inferrule{
%   \tsmexpMultistepsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}\\
%   \tsmexpNormalExp{\Omega}{\Psi}{\epsilon'}
% }{
%   \tsmexpEvalsExp{\Omega}{\Psi}{\epsilon}{\epsilon'}
% }
% \end{equation}

% TLM expression definition extraction

% \begin{subequations}
% \begin{align}
% \tsmdefof{\adefref{a}} & = a\\
% \tsmdefof{\aeabstype{t}{\epsilon}} & = \tsmdefof{\epsilon}\\
% \tsmdefof{\aeabsmod{\sigma}{X}{\epsilon}} & = \tsmdefof{\epsilon}\\
% \tsmdefof{\aeaptype{\tau}{\epsilon}} & = \tsmdefof{\epsilon}\\
% \tsmdefof{\aeapmod{X}{\epsilon}} & = \tsmdefof{\epsilon}
% \end{align}
% \end{subequations}

% \subsubsection{Candidate Expansion Kind and Constructor Validation}
% %The \emph{ce-type validation judgement}, $\cvalidT{\Delta}{\tscenev}{\ctau}{\tau}$, is inductively defined by Rules (\ref{rules:cvalidT-U}), which were defined in Sec. \ref{sec:ce-validation-U}.

% ce-kind validation
% \begin{subequations}\label{rules:cvalidK-B}
% \begin{equation}\label{rule:cvalidK-B-darr}
% \inferrule{
%   \cvalidKX{\cekappa_1}{\kappa_1}\\
%   \cvalidK{\Omega, u :: \kappa_1}{\cscenev}{\cekappa_2}{\kappa_2}
% }{
%   \cvalidKX{\acekdarr{\cekappa_1}{u}{\cekappa_2}}{\akdarr{\kappa_1}{u}{\kappa_2}}
% }
% \end{equation}
% \begin{equation}\label{rule:cvalidK-B-unit}
% \inferrule{ }{
%   \cvalidKX{\acekunit}{\akunit}
% }
% \end{equation}
% \begin{equation}\label{rule:cvalidK-B-dprod}
% \inferrule{
%   \cvalidKX{\cekappa_1}{\kappa_1}\\
%   \cvalidK{\Omega, u :: \kappa_1}{\cscenev}{\cekappa_2}{\kappa_2}
% }{
%   \cvalidKX{\acekdbprod{\cekappa_1}{u}{\cekappa_2}}{\akdbprod{\kappa_1}{u}{\kappa_2}}
% }
% \end{equation}
% \begin{equation}\label{rule:cvalidK-B-ty}
% \inferrule{ }{
%   \cvalidKX{\acekty}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:cvalidK-B-sing}
% \inferrule{
%   \ccanaX{\ctau}{\tau}{\akty}
% }{
%   \cvalidKX{\aceksing{\ctau}}{\aksing{\tau}}
% }
% \end{equation}
% \begin{equation}\label{rule:cvalidK-B-spliced}
% \inferrule{
%   \parseUKind{\bsubseq{b}{m}{n}}{\ukappa}\\
%   \kExpands{\uOmega}{\ukappa}{\kappa}\\\\
%   \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
%   \domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset
% }{
%   \cvalidK{\Omega}{\tsceneP{\uOmega}{b}}{\acesplicedk{m}{n}}{\kappa}
% }
% \end{equation}
% \end{subequations}

% Synthetic ce-constructor validation
% \begin{subequations}\label{rules:ccsyn}
% \begin{equation}\label{rule:ccsyn-var}
% \inferrule{ }{\ccsyn{\Omega, {u} :: {\kappa}}{\cscenev}{u}{u}{\kappa}}
% \end{equation}
% \begin{equation}\label{rule:ccsyn-asc}
% \inferrule{
%   \cvalidKX{\cekappa}{\kappa}\\
%   \ccanaX{\cec}{c}{\kappa}
% }{
%   \ccsynX{\acecasc{\cekappa}{\cec}}{c}{\kappa}
% }
% \end{equation}
% \begin{equation}\label{rule:ccsyn-app}
% \inferrule{
%   \ccsynX{\cec_1}{c_1}{\akdarr{\kappa_2}{u}{\kappa}}\\
%   \ccsynX{\cec_2}{c_2}{\kappa_2}
% }{
%   \ccsynX{\acecapp{\cec_1}{\cec_2}}{\acapp{c_1}{c_2}}{[c_1/u]\kappa}
% }
% \end{equation}
% \begin{equation}\label{rule:ccsyn-unit}
% \inferrule{ }{
%   \ccsynX{\acectriv}{\actriv}{\akunit}
% }
% \end{equation}
% \begin{equation}\label{rule:ccsyn-prl}
% \inferrule{
%   \ccsynX{\cec}{c}{\akdbprod{\kappa_1}{u}{\kappa_2}}
% }{
%   \ccsynX{\acecprl{\cec}}{\acprl{c}}{\kappa_1}
% }
% \end{equation}
% \begin{equation}\label{rule:ccsyn-prr}
% \inferrule{
%   \ccsynX{\cec}{c}{\akdbprod{\kappa_1}{u}{\kappa_2}}
% }{
%   \ccsynX{\acecprr{\cec}}{\acprr{c}}{[\acprl{c}/u]\kappa_2}
% }
% \end{equation}
% \begin{equation}\label{rule:ccsyn-parr}
% \inferrule{
%   \ccanaX{\ctau_1}{\tau_1}{\akty}\\
%   \ccanaX{\ctau_2}{\tau_2}{\akty}
% }{
%   \ccsynX{\aceparr{\ctau_1}{\ctau_2}}{\aparr{\tau_1}{\tau_2}}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:ccsyn-all}
% \inferrule{
%   \cvalidKX{\cekappa}{\kappa}\\
%   \ccana{\Omega, u :: \kappa}{\cscenev}{\ctau}{\tau}{\akty}
% }{
%   \ccsynX{\aceallu{\cekappa}{u}{\ctau}}{\aallu{\kappa}{u}{\tau}}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:ccsyn-rec}
% \inferrule{
%   \ccana{\Omega, t :: \akty}{\cscenev}{\ctau}{\tau}{\akty}
% }{
%   \ccsynX{\acerec{t}{\ctau}}{\arec{t}{\tau}}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:ccsyn-prod}
% \inferrule{
%   \{\ccanaX{\ctau_i}{\tau_i}{\akty}\}_{1 \leq i \leq n}
% }{
%   \ccsynX{\aceprod{\labelset}{\mapschema{\ctau}{i}{\labelset}}}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:ccsyn-sum}
% \inferrule{
%   \{\ccanaX{\ctau_i}{\tau_i}{\akty}\}_{1 \leq i \leq n}
% }{
%   \ccsynX{\acesum{\labelset}{\mapschema{\ctau}{i}{\labelset}}}{\asum{\labelset}{\mapschema{\tau}{i}{\labelset}}}{\akty}
% }
% \end{equation}
% \begin{equation}\label{rule:ccsyn-stat}
% \inferrule{ }{
%   \ccsyn{\Omega, X : {\asignature{\kappa}{u}{\tau}}}{\cscenev}{\acemcon{X}}{\amcon{X}}{\kappa}
% }
% \end{equation}
% \begin{equation}\label{rule:ccsyn-spliced}
% \inferrule{
%   \parseUCon{\bsubseq{b}{m}{n}}{\uc}\\
%   \ksyn{\uOmega}{\uc}{c}{\kappa}\\\\
%   \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
%   \domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset
% }{
%   \ccsyn{\Omega}{\tsceneP{\uOmega}{b}}{\acesplicedc{m}{n}{\cekappa}}{c}{\kappa}
% }
% \end{equation}
% \end{subequations}

% Analytic constructor expansion
% \begin{subequations}\label{rules:ccana}
% \begin{equation}\label{rule:ccana-subsume}
% \inferrule{
%   \ccsynX{\cec}{c}{\kappa_1}\\
%   \ksubX{\kappa_1}{\kappa_2}
% }{
%   \ccanaX{\cec}{c}{\kappa_2}
% }
% \end{equation}
% \begin{equation}\label{rule:ccana-sing}
% \inferrule{
%   \kanaX{\cec}{c}{\akty}
% }{
%   \kanaX{\cec}{c}{\aksing{c}}
% }
% \end{equation}
% \begin{equation}\label{rule:ccana-abs}
% \inferrule{
%   \ccana{\Omega, u :: \kappa_1}{\cscenev}{\cec_2}{c_2}{\kappa_2}
% }{
%   \ccanaX{\acecabs{u}{\cec_2}}{\acabs{u}{c_2}}{\akdarr{\kappa_1}{u}{\kappa_2}}
% }
% \end{equation}
% \begin{equation}\label{rule:ccana-pair}
% \inferrule{
%   \ccanaX{\cec_1}{c_1}{\kappa_1}\\
%   \ccanaX{\cec_2}{c_2}{[c_1/u]\kappa_2}
% }{
%   \ccanaX{\acecpair{\cec_1}{\cec_2}}{\acpair{c_1}{c_2}}{\akdbprod{\kappa_1}{u}{\kappa_2}}
% }
% \end{equation}
% \begin{equation}\label{rule:ccana-spliced}
% \inferrule{
%   \parseUCon{\bsubseq{b}{m}{n}}{\uc}\\
%   \kana{\uOmega}{\uc}{c}{\kappa}\\\\
%   \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
%   \domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset
% }{
%   \ccana{\Omega}{\tsceneP{\uOmega}{b}}{\acesplicedc{m}{n}{\cekappa}}{c}{\kappa}
% }
% \end{equation}
% \end{subequations}

% \subsubsection{Bidirectional Candidate Expansion Expression Validation}
% Like the bidirectionally typed expression expansion judgements, the bidirectional ce-expression validation judgements distinguish type synthesis from type analysis. The \emph{synthetic ce-expression validation judgement}, $\csynX{\ce}{e}{\tau}$, and the \emph{analytic ce-expression validation judgement}, $\canaX{\ce}{e}{\tau}$, are defined mutually inductively with Rules (\ref{rules:esyn}) and Rules (\ref{rules:eana}) by Rules (\ref{rules:csyn}) and Rules (\ref{rules:cana}), respectively, as follows.


% \begin{equation}
% \inferrule{
%   \ccanaX{\ctau}{\tau}{\akty}
% }{
%   \cvalidTP{\Omega}{\cscenev}{\ctau}{\tau}
% }
% \end{equation}

% \paragraph{Type Synthesis} \begin{subequations}\label{rules:csynP}
% Synthetic ce-expression validation is governed by the following rules.
% \begin{equation}\label{rule:csynP-var}
%   \inferrule{ }{ 
%     \csynP{\Omega, \Ghyp{x}{\tau}}{\escenev}{x}{x}{\tau}
%   }
% \end{equation}
% \begin{equation}\label{rule:csynP-asc}
%   \inferrule{
%     \cvalidTP{\Omega}{\csfrom{\escenev}}{\ctau}{\tau}\\
%     \canaPX{\ce}{e}{\tau}
%   }{
%     \csynPX{\aceasc{\ctau}{\ce}}{e}{\tau}
%   }
% \end{equation}
% \begin{equation}\label{rule:csynP-let}
%   \inferrule{
%     \csynPX{\ce}{e}{\tau}\\
%     \csynP{\Omega, \Ghyp{x}{\tau}}{\escenev}{\ce'}{e'}{\tau'}
%   }{
%     \csynPX{\aceletsyn{x}{\ce}{\ce'}}{\aeap{\aelam{\tau}{x}{e'}}{e}}{\tau'}
%   }
% \end{equation}
% \begin{equation}\label{rule:csynP-lam}
%   \inferrule{
%     \cvalidTP{\Omega}{\csfrom{\escenev}}{\ctau_1}{\tau_1}\\
%     \csynP{\Omega, \Ghyp{x}{\tau_1}}{\escenev}{\ce}{e}{\tau_2}
%   }{
%     \csynPX{\acelam{\ctau_1}{x}{\ce}}{\aelam{\tau_1}{x}{e}}{\aparr{\tau_1}{\tau_2}}
%   }
% \end{equation}
% \begin{equation}\label{rule:csynP-ap}
%   \inferrule{
%     \csynPX{\ce_1}{e_1}{\aparr{\tau_2}{\tau}}\\
%     \canaPX{\ce_2}{e_2}{\tau_2}
%   }{
%     \csynPX{\aceap{\ce_1}{\ce_2}}{\aeap{e_1}{e_2}}{\tau}
%   }
% \end{equation}
% \begin{equation}\label{rule:csynP-clam}
%   \inferrule{
%     \cvalidK{\Omega}{\csfrom{\escenev}}{\cekappa}{\kappa}\\
%     \csynP{\Omega, u :: \kappa}{\escenev}{\ce}{e}{\tau}
%   }{
%     \csynX{\aceclam{\cekappa}{u}{\ce}}{\aeclam{\kappa}{u}{e}}{\aallu{\kappa}{u}{\tau}}
%   }
% \end{equation}
% \begin{equation}\label{rule:csynP-cap}
%   \inferrule{
%     \csynPX{\ce}{e}{\aallu{\kappa}{u}{\tau}}\\
%     \ccana{\Omega}{\csfrom{\escenev}}{\cec}{c}{\kappa}
%   }{
%     \csynPX{\acecap{\ce}{\cec}}{\aecap{e}{c}}{[c/u]\tau}
%   }
% \end{equation}
% \begin{equation}\label{rule:csynP-unfold}
%   \inferrule{
%     \csynPX{\ce}{e}{\arec{t}{\tau}}
%   }{
%     \csynPX{\aceunfold{\ce}}{\aeunfold{e}}{[\arec{t}{\tau}/t]\tau}
%   }
% \end{equation}
% \begin{equation}\label{rule:csynP-tpl}
%   \inferrule{
%     \ce=\acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}}\\
%     e=\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}\\\\
%     \{\csynPX{\ce_i}{e_i}{\tau_i}\}_{i \in \labelset}
%   }{
%     \csynPX{\ce}{e}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
%   }
% \end{equation}
% \begin{equation}\label{rule:csynP-pr}
%   \inferrule{
%     \csynPX{\ce}{e}{\aprod{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
%   }{
%     \csynPX{\acepr{\ell}{\ce}}{\aepr{\ell}{e}}{\tau}
%   }
% \end{equation}
% \begin{equation}\label{rule:csynP-match}
%   \inferrule{
%     n > 0\\
%     \csynPX{\ce}{e}{\tau}\\
%     \{\crsynPX{\crv_i}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}
%   }{
%     \csynPX{\acematchwithb{n}{\ce}{\seqschemaX{\crv}}}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}
%   }
% \end{equation}
% \begin{equation}\label{rule:csynP-mval}
% \inferrule{ }{
%   \csynP{\Omega, X : \asignature{\kappa}{u}{\tau}}{\escenev}{\acemval{X}}{\amval{X}}{[\amcon{X}/u]\tau}
% }
% \end{equation}
% \begin{equation}\label{rule:csynP-splicede}
% \inferrule{
%   \parseUExp{\bsubseq{b}{m}{n}}{\ue}\\
%   \esynP{\uOmega}{\uPsi}{\uPhi}{\ue}{e}{\tau}\\\\
%   \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
%   \domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset
% }{
%   \csynP{\Omega}{\esceneP{\OParams}{\uOmega}{\uPsi}{\uPhi}{b}}{\acesplicede{m}{n}{\ctau}}{e}{\tau}
% }
% \end{equation}
% \end{subequations}

% Rules (\ref{rule:csyn-var}) through (\ref{rule:csyn-match}) are analagous to Rules (\ref{rule:esyn-var}) through (\ref{rule:esyn-match}). Rule (\ref{rule:csyn-splicede}) governs references to spliced unexpanded expressions in synthetic position, and can be understood as described in Sec. \ref{sec:ce-validation-U}.


% \paragraph{Type Analysis} \begin{subequations}\label{rules:canaP}
% Analytic ce-expression validation is governed by the following rules.
% \begin{equation}\label{rule:canaP-subsume}
%   \inferrule{
%     \csynPX{\ce}{e}{\tau}\\
%     \issubtypePX{\tau}{\tau'}
%   }{
%     \canaPX{\ce}{e}{\tau'}
%   }
% \end{equation}
% \begin{equation}\label{rule:canaP-let}
%   \inferrule{
%     \csynPX{\ce}{e}{\tau}\\
%     \canaP{\Omega, \Ghyp{x}{\tau}}{\escenev}{\ce'}{e'}{\tau'}
%   }{
%     \canaPX{\aceletsyn{x}{\ce}{\ce'}}{\aeap{\aelam{\tau}{x}{e'}}{e}}{\tau'}
%   }
% \end{equation}
% \begin{equation}\label{rule:canaP-analam}
%   \inferrule{
%     \canaP{\Gamma, \Ghyp{x}{\tau_1}}{\escenev}{\ce}{e}{\tau_2}
%   }{
%     \canaPX{\aceanalam{x}{\ue}}{\aelam{\tau_1}{x}{e}}{\aparr{\tau_1}{\tau_2}}
%   }
% \end{equation}
% \begin{equation}\label{rule:canaP-clam}
%   \inferrule{
%     \cvalidKX{\cekappa}{\kappa}\\
%     \canaP{\Omega, u :: \kappa}{\escenev}{\ce}{e}{\tau}
%   }{
%     \canaPX{\aceclam{\cekappa}{u}{\ce}}{\aeclam{\kappa}{u}{e}}{\aallu{\kappa}{u}{\tau}}
%   }
% \end{equation}
% \begin{equation}\label{rule:canaP-fold}
%   \inferrule{
%     \canaPX{\ce}{e}{[\arec{t}{\tau}/t]\tau}
%   }{
%     \canaPX{\aceanafold{\ce}}{\aefold{e}}{\arec{t}{\tau}}
%   }
% \end{equation}
% \begin{equation}\label{rule:canaP-tpl}
%   \inferrule{
%     \ce=\acetpl{\labelset}{\mapschema{\ce}{i}{\labelset}}\\
%     e=\aetpl{\labelset}{\mapschema{e}{i}{\labelset}}\\\\
%     \{\canaPX{\ce_i}{e_i}{\tau_i}\}_{i \in \labelset}
%   }{
%     \canaPX{\ce}{e}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
%   }
% \end{equation}
% \begin{equation}\label{rule:canaP-in}
%   \inferrule{
%     \ce=\aceanain{\ell}{\ce'}\\
%     e=\aein{\labelset, \ell}{\ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}{e'}\\\\
%     \canaX{\ce'}{e'}{\tau}
%   }{
%     \canaPX{\ce}{e}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
%     %\left(\shortstack{$\Delta~\Gamma \vdash^{\escenev} $\\$\leadsto$\\$ \Leftarrow $\vspace{-1.2em}}\right)
%     %\eanaX{\auanain{\ell}{\ue}}{\aein{\ell}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
%   }
% \end{equation}
% \begin{equation}\label{rule:canaP-match}
%   \inferrule{
%     \csynPX{\ce}{e}{\tau}\\
%     \{\cranaPX{\crv_i}{r_i}{\tau}{\tau'}\}_{1 \leq i \leq n}
%   }{
%     \canaPX{\acematchwithb{n}{\ce}{\seqschemaX{\crv}}}{\aematchwith{n}{e}{\seqschemaX{r}}}{\tau'}
%   }
% \end{equation}
% \begin{equation}\label{rule:canaP-splicede}
% \inferrule{
%   \parseUExp{\bsubseq{b}{m}{n}}{\ue}\\
%   \eanaP{\uOmega}{\uPsi}{\uPhi}{\ue}{e}{\tau}\\\\
%   \uOmega=\uOmegaEx{\uD}{\uG}{\uMctx}{\Omega_\text{app}}\\
%   \domof{\Omega} \cap \domof{\Omega_\text{app}} = \emptyset
% }{
%   \canaP{\Omega}{\esceneP{\OParams}{\uOmega}{\uPsi}{\uPhi}{b}}{\acesplicede{m}{n}{\ctau}}{e}{\tau}
% }
% \end{equation}
% \end{subequations}

% Rules (\ref{rule:cana-subsume}) through (\ref{rule:cana-match}) are analagous to Rules (\ref{rule:eana-subsume}) through (\ref{rule:eana-match}). Rule (\ref{rule:cana-splicede}) governs references to spliced unexpanded expressions in analytic position. 

% \subsubsection{Bidirectional Candidate Expansion Rule Validation}
% The \emph{synthetic ce-rule validation judgement} is defined mutually inductively with Rules (\ref{rules:esyn}) by the following rule.
% \begin{equation}\label{rule:crsynP}
% \inferrule{
%   \patTypeP{\Omega'}{p}{\tau}\\
%   \csynP{\Gcons{\Omega}{\Omega'}}{\escenev}{\ce}{e}{\tau'}
% }{
%   \crsynPX{\acematchrule{p}{\ce}}{\aematchrule{p}{e}}{\tau}{\tau'}
% }
% \end{equation}

% The \emph{analytic ce-rule validation judgement} is defined mutually inductively with Rules (\ref{rules:eana}) by the following rule.
% \begin{equation}\label{rule:cranaP}
% \inferrule{
%   \patType{\Omega'}{p}{\tau}\\
%   \canaP{\Gcons{\Omega}{\Omega'}}{\escenev}{\ce}{e}{\tau'}
% }{
%   \cranaPX{\acematchrule{p}{\ce}}{\aematchrule{p}{e}}{\tau}{\tau'}
% }
% \end{equation}

% \subsubsection{Candidate Expansion Pattern Validation}
% The \emph{ce-pattern validation judgement} is inductively defined by the following rules, which are written identically to Rules (\ref{rules:cvalidP-UP}).
% \begin{subequations}\label{rules:cvalidP-P}
% \begin{equation}\label{rule:cvalidP-P-wild}
% \inferrule{ }{
%   \cvalidPP{\uOmegaEx{\emptyset}{\emptyset}{\emptyset}{\emptyset}}{\pscenev}{\acewildp}{\aewildp}{\tau}
% }
% \end{equation}
% \begin{equation}\label{rule:cvalidP-P-fold}
% \inferrule{
%   \cvalidPP{\uOmega}{\pscenev}{\cpv}{p}{[\arec{t}{\tau}/t]\tau}
% }{
%   \cvalidPP{\uOmega}{\pscenev}{\acefoldp{\cpv}}{\aefoldp{p}}{\arec{t}{\tau}}
% }
% \end{equation}
% \begin{equation}\label{rule:cvalidP-P-tpl}
% \inferrule{
%   \cpv=\acetplp{\labelset}{\mapschema{\cpv}{i}{\labelset}}\\
%   p=\aetplp{\labelset}{\mapschema{p}{i}{\labelset}}\\\\
%   \{\cvalidPP{\upctx_i}{\pscenev}{\cpv_i}{p_i}{\tau_i}\}_{i \in \labelset}
% }{
%   \cvalidPP{\Gconsi{i \in \labelset}{\uOmega_i}}{\pscenev}{\cpv}{p}{\aprod{\labelset}{\mapschema{\tau}{i}{\labelset}}}
%   %\cvalidPP{}{\cpv}{p}{}
% %\left(\shortstack{$\vdash^{\pscenev} $\\$\leadsto$\\$ :~\dashVx^{\,\Gconsi{i \in \labelset}{\upctx_i}}$\vspace{-1.2em}}\right)
% }
% \end{equation}
% \begin{equation}\label{rule:cvalidP-P-in}
% \inferrule{
%   \cvalidPP{\uOmega}{\pscenev}{\cpv}{p}{\tau}
% }{
%   \cvalidPP{\uOmega}{\pscenev}{\aceinjp{\ell}{\cpv}}{\aeinjp{\ell}{p}}{\asum{\labelset, \ell}{\mapschema{\tau}{i}{\labelset}; \mapitem{\ell}{\tau}}}
% }
% \end{equation}
% \begin{equation}\label{rule:cvalidP-P-spliced}
% \inferrule{
%   \parseUPat{\bsubseq{b}{m}{n}}{\upv}\\
%   \patExpandsP{\uOmega'}{\uPhi}{\upv}{p}{\tau}
% }{
%   \cvalidPP{\uOmega'}{\pscene{\uOmega}{\uPhi}{b}}{\acesplicedp{m}{n}{\ctau}}{p}{\tau}
% }
% \end{equation}
% \end{subequations}

% Finally, the following theorem establishes that bidirectionally typed expression and rule expansion produces expanded expressions and rules of the appropriate type under the appropriate contexts. These statements must be stated mutually with the corresponding statements about birectional ce-expression and ce-rule validation because the judgements are mutually defined. 

% \begin{theorem}[Typed Expansion] Letting $\uPsi=\uASI{\uA}{\Psi}{\uI}$, if $\uetsmenv{\Delta}{\Psi}$ then all of the following hold:
% \begin{enumerate}
%   \item \begin{enumerate}
%     \item \begin{enumerate}
%       \item If $\esyn{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\ue}{e}{\tau}$ then $\hastypeU{\Delta}{\Gamma}{e}{\tau}$.
%       \item If $\rsyn{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\urv}{r}{\tau}{\tau'}$  then $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$.
%     \end{enumerate}
%     \item \begin{enumerate}
%       \item If $\eana{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\ue}{e}{\tau}$ and $\istypeU{\Delta}{\tau}$ then $\hastypeU{\Delta}{\Gamma}{e}{\tau}$.
%       \item If $\rana{\uDD{\uD}{\Delta}}{\uGG{\uG}{\Gamma}}{\uPsi}{\uPhi}{\urv}{r}{\tau}{\tau'}$ and $\istypeU{\Delta}{\tau'}$ then $\ruleType{\Delta}{\Gamma}{r}{\tau}{\tau'}$.
%     \end{enumerate}
%   \end{enumerate}
%   \item \begin{enumerate}
%     \item \begin{enumerate}
%       \item If $\csyn{\Delta}{\Gamma}{\esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}$ and $\Delta \cap \Delta_\text{app}=\emptyset$ and $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ then $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e}{\tau}$. 
%       \item If $\crsyn{\Delta}{\Gamma}{\esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}}{\crv}{r}{\tau}{\tau'}$ and $\Delta \cap \Delta_\text{app}=\emptyset$ and $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ then $\ruleType{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{r}{\tau}{\tau'}$.
%     \end{enumerate}
%     \item \begin{enumerate}
%       \item If $\cana{\Delta}{\Gamma}{\esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}$ and $\Delta \cap \Delta_\text{app}=\emptyset$ and $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ and $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\tau}$ then $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e}{\tau}$. 
%       \item If $\crana{\Delta}{\Gamma}{\esceneUP{\uDD{\uD}{\Delta_\text{app}}}{\uGG{\uG}{\Gamma_\text{app}}}{\uPsi}{\uPhi}{b}}{\crv}{r}{\tau}{\tau'}$ and $\Delta \cap \Delta_\text{app}=\emptyset$ and $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ and $\istypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\tau'}$ then $\ruleType{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{r}{\tau}{\tau'}$.
%     \end{enumerate}
%   \end{enumerate}
% \end{enumerate}
% \end{theorem}
% \begin{proof} By mutual rule induction over Rules (\ref{rules:esyn}), Rules (\ref{rules:eana}), Rule (\ref{rule:rsyn}), Rule (\ref{rule:rana}), Rules (\ref{rules:csyn}), Rules (\ref{rules:cana}), Rule (\ref{rule:crsyn}) and Rule (\ref{rule:crana}). In the following, we refer to the induction hypothesis applied to the assumption $\uetsmenv{\Delta}{\Psi}$ as simply the ``IH''. When we apply the induction hypothesis to a different argument, we refer to it as the ``Outer IH''.

% \begin{enumerate}
%   \item In the following, let $\uDelta=\uDD{\uD}{\Delta}$ and $\uGamma=\uGG{\uG}{\Gamma}$. We have:
%   \begin{enumerate}
%     \item \begin{enumerate}
%       \item We induct on the assumption.
%         \begin{byCases}
%           \item[\text{(\ref{rule:esyn-var})}] We have:
%             \begin{pfsteps*}
%               \item $e=x$ \BY{assumption}
%               \item $\Gamma=\Gamma', \Ghyp{x}{\tau}$ \BY{assumption}
%               \item $\hastypeU{\Delta}{\Gamma', \Ghyp{x}{\tau}}{x}{\tau}$ \BY{Rule (\ref{rule:hastypeUP-var})}
%             \end{pfsteps*}
%             \resetpfcounter
%           \item[\text{(\ref{rule:esyn-asc})}] We have:
%             \begin{pfsteps*}
%                \item $\ue=\auasc{\utau}{\ue'}$ \BY{assumption}
%                \item $\expandsTU{\uDelta}{\utau}{\tau}$ \BY{assumption}\pflabel{expandsTU}
%                \item $\eanaX{\ue'}{e}{\tau}$ \BY{assumption}\pflabel{eanaX}
%                \item $\istypeU{\Delta}{\tau}$ \BY{Lemma \ref{lemma:type-expansion-U} on \pfref{expandsTU}}\pflabel{istype}
%                \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{IH, part 1(b)(i) to \pfref{eanaX} and \pfref{istype}}
%              \end{pfsteps*}
%              \resetpfcounter
%           \item[\text{(\ref{rule:esyn-let}) through (\ref{rule:esyn-match})}] In each of these cases, we apply:
%             \begin{itemize}
%               \item Lemma \ref{lemma:type-expansion-U} to or over all type expansion premises.
%               \item The IH, part 1(a)(i) to or over all synthetic typed expression expansion premises.
%               \item The IH, part 1(a)(ii) to or over all synthetic rule expansion premises.
%               \item The IH, part 1(b)(i) to or over all analytic typed expression expansion premises.
%             \end{itemize}
%             We then derive the conclusion by applying Rules (\ref{rules:hastypeUP}) and Rule (\ref{rule:ruleType}) as needed.
%           \item[\text{(\ref{rule:esyn-defuetsm})}] We have:
%             \begin{pfsteps*}
%               \item $\ue=\audefuetsm{\utau'}{\eparse}{\tsmv}{\ue'}$ \BY{assumption}
%               \item $\expandsTU{\uDelta}{\utau'}{\tau'}$ \BY{assumption} \pflabel{expandsTU}
%               \item $\hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultExp}}$ \BY{assumption}\pflabel{eparse}
%               \item $\esyn{\uDelta}{\uGamma}{\uASI{\ctxUpdate{\uA}{\tsmv}{a}}{\Psi, \xuetsmbnd{a}{\tau'}{\eparse}}{\uI}}{\uPhi}{\ue'}{e}{\tau}$ \BY{assumption}\pflabel{expandsU}
%               \item $\uetsmenv{\Delta}{\Psi}$ \BY{assumption}\pflabel{uetsmenv1}
%               \item $\istypeU{\Delta}{\tau'}$ \BY{Lemma \ref{lemma:type-expansion-U} to \pfref{expandsTU}} \pflabel{istype}
%               \item $\uetsmenv{\Delta}{\Psi, \xuetsmbnd{\tsmv}{\tau'}{\eparse}}$ \BY{Definition \ref{def:ueTLM-def-ctx-formation-UP} on \pfref{uetsmenv1}, \pfref{istype} and \pfref{eparse}}\pflabel{uetsmenv3}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{Outer IH, part 1(a)(i) on \pfref{uetsmenv3} and \pfref{expandsU}}
%             \end{pfsteps*}
%             \resetpfcounter
%           \item[\text{(\ref{rule:esyn-apuetsm})}] We have:
%             \begin{pfsteps*}
%               \item $\ue=\autsmap{b}{\tsmv}$ \BY{assumption}
%               \item $\uPsi = \uASI{\ctxUpdate{\uA'}{\tsmv}{a}}{\Psi', \xuetsmbnd{a}{\tau}{\eparse}}{\uI}$ \BY{assumption}
%               \item $\encodeBody{b}{\ebody}$ \BY{assumption}
%               \item $\evalU{\eparse(\ebody)}{\inj{\lbltxt{Success}}{\ecand}}$ \BY{assumption}
%               \item $\decodeCondE{\ecand}{\ce}$ \BY{assumption}
%               \item $\cana{\emptyset}{\emptyset}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}$ \BY{assumption}\pflabel{cvalidE}
%               \item $\uetsmenv{\Delta}{\Psi}$ \BY{assumption} \pflabel{uetsmenv}
%               \item $\istypeU{\Delta}{\tau}$ \BY{Definition \ref{def:ueTLM-def-ctx-formation-UP} on \pfref{uetsmenv}} \pflabel{istype}
%               \item $\emptyset \cap \Delta = \emptyset$ \BY{finite set intersection identity} \pflabel{delta-cap}
%               \item ${\emptyset} \cap \domof{\Gamma} = \emptyset$ \BY{finite set intersection identity} \pflabel{gamma-cap}
%               \item $\hastypeU{\emptyset \cup \Delta}{\emptyset \cup \Gamma}{e}{\tau}$ \BY{IH, part 2(a)(i) on \pfref{cvalidE}, \pfref{delta-cap}, \pfref{gamma-cap} and \pfref{istype}} \pflabel{penultimate}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{definition of finite set and finite function union over \pfref{penultimate}}               
%              \end{pfsteps*} 
%              \resetpfcounter
%           \item[\text{(\ref{rule:esyn-implicite})}] We have:
%             \begin{pfsteps*}
%               \item $\ue=\auimplicite{\tsmv}{\ue}$ \BY{assumption}
%               \item $\uPsi=\uASI{\uA' \uplus \vExpands{\tsmv}{a}}{\Psi', \xuetsmbnd{a}{\tau'}{\eparse}}{\uI}$ \BY{assumption}
%               \item $\esyn{\uDelta}{\uGamma}{\uASI{\uA' \uplus \vExpands{\tsmv}{a}}{\Psi', \xuetsmbnd{a}{\tau'}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\uPhi}{\ue}{e}{\tau}$ \BY{assumption} \pflabel{esyn}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{IH, part 1(a)(i) on \pfref{esyn}}
%             \end{pfsteps*}
%             \resetpfcounter
%           \item[\text{(\ref{rule:esyn-defuptsm})}] We have:
%             \begin{pfsteps*}
%               \item $\ue=\audefuptsm{\utau'}{\eparse}{\tsmv}{\ue'}$ \BY{assumption}
%               \item $\expandsTU{\uDelta}{\utau'}{\tau'}$ \BY{assumption} \pflabel{expandsTU}
%             %  \item \hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultExp}} \BY{assumption}\pflabel{eparse}
%               \item $\esyn{\uDelta}{\uGamma}{\uPsi}{\uPhi, \uPhyp{\tsmv}{a}{\tau'}{\eparse}}{\ue'}{e}{\tau}$ \BY{assumption}\pflabel{expandsU}
%             %  \item \uetsmenv{\Delta}{\Psi} \BY{assumption}\pflabel{uetsmenv1}
%             %  \item \istypeU{\Delta}{\tau'} \BY{Lemma \ref{lemma:type-expansion-U} to \pfref{expandsTU}} \pflabel{istype}
%             %  \item \uetsmenv{\Delta}{\Psi, \xuetsmbnd{\tsmv}{\tau'}{\eparse}} \BY{Definition \ref{def:ueTLM-def-ctx-formation} on \pfref{uetsmenv1}, \pfref{istype} and \pfref{eparse}}\pflabel{uetsmenv3}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{IH, part 1(a)(i) on \pfref{expandsU}}
%             \end{pfsteps*}
%             \resetpfcounter
%           \item[\text{(\ref{rule:esyn-implicitp})}] We have:
%             \begin{pfsteps*}
%               \item $\ue=\auimplicitp{\tsmv}{\ue}$ \BY{assumption}
%               \item $\uPhi=\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau'}{\eparse}}{\uI}$ \BY{assumption}
%               \item $\esyn{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau'}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\ue}{e}{\tau}$ \BY{assumption} \pflabel{esyn}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{IH, part 1(a)(i) on \pfref{esyn}}
%             \end{pfsteps*}
%             \resetpfcounter
%         \end{byCases}
%       \item We induct on the assumption. There is one case.
%         \begin{byCases}
%           \item[\text{(\ref{rule:rsyn})}] We have:
%             \begin{pfsteps*}
%               \item $\urv=\aumatchrule{\upv}{\ue}$ \BY{assumption}
%               \item $r=\aematchrule{p}{e}$ \BY{assumption}
%               \item $\patExpands{\uGG{\uA'}{\pctx}}{\uPhi}{\upv}{p}{\tau}$ \BY{assumption} \pflabel{patExpands}
%               \item $\esyn{\uDelta}{\uGG{{\uA}\uplus{\uA'}}{\Gcons{\Gamma}{\pctx}}}{\uPsi}{\uPhi}{\ue}{e}{\tau'}$ \BY{assumption} \pflabel{expandsUP}
%               \item $\patType{\pctx}{p}{\tau}$ \BY{Theorem \ref{thm:typed-pattern-expansion-B}, part 1 on \pfref{patExpands}}\pflabel{patType}
%               \item $\hastypeU{\Delta}{\Gcons{\Gamma}{\pctx}}{e}{\tau'}$ \BY{IH, part 1(a)(i) on \pfref{expandsUP}} \pflabel{hasType}
%               \item $\ruleType{\Delta}{\Gamma}{\aematchrule{p}{e}}{\tau}{\tau'}$ \BY{Rule (\ref{rule:ruleType}) on \pfref{patType} and \pfref{hasType}}
%             \end{pfsteps*}
%             \resetpfcounter
%         \end{byCases}
%     \end{enumerate}
%     \item \begin{enumerate}
%       \item We induct on the assumption.
%         \begin{byCases}
%           \item[\text{(\ref{rule:eana-subsume})}] We have:
%             \begin{pfsteps*}
%               \item $\esynX{\ue}{e}{\tau}$ \BY{assumption} \pflabel{esyn}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{IH, part 1(a)(i) on \pfref{esyn}}
%             \end{pfsteps*}
%           \item[\text{(\ref{rule:eana-let}) through (\ref{rule:eana-match})}] In each of these cases, we apply:
%             \begin{itemize}
%               \item Lemma \ref{lemma:type-expansion-U} to or over all type expansion premises.
%               \item The IH, part 1(a)(i) to or over all synthetic typed expression expansion premises.
%               \item The IH, part 1(a)(ii) to or over all synthetic rule expansion premises.
%               \item The IH, part 1(b)(i) to or over all analytic typed expression expansion premises.
%             \end{itemize}
%             We then derive the conclusion by applying Rules (\ref{rules:hastypeUP}) and Rule (\ref{rule:ruleType}) as needed. 
%           \item[\text{(\ref{rule:eana-defuetsm})}] We have:
%             \begin{pfsteps*}
%               \item $\ue=\audefuetsm{\utau'}{\eparse}{\tsmv}{\ue'}$ \BY{assumption}
%               \item $\expandsTU{\uDelta}{\utau'}{\tau'}$ \BY{assumption} \pflabel{expandsTU}
%               \item $,$ \BY{assumption}\pflabel{eparse}
%               \item $\eana{\uDelta}{\uGamma}{\uPsi, \uShyp{\tsmv}{a}{\tau'}{\eparse}}{\uPhi}{\ue'}{e}{\tau}$ \BY{assumption}\pflabel{expandsU}
%               \item $\uetsmenv{\Delta}{\Psi}$ \BY{assumption}\pflabel{uetsmenv1}
%               \item $\istypeU{\Delta}{\tau'}$ \BY{Lemma \ref{lemma:type-expansion-U} to \pfref{expandsTU}} \pflabel{istype}
%               \item $\uetsmenv{\Delta}{\Psi, \xuetsmbnd{\tsmv}{\tau'}{\eparse}}$ \BY{Definition \ref{def:ueTLM-def-ctx-formation-UP} on \pfref{uetsmenv1}, \pfref{istype} and \pfref{eparse}}\pflabel{uetsmenv3}
%             %  \item \uetsmenv{\Delta}{\Psi} \BY{assumption}\pflabel{uetsmenv1}
%             %  \item \istypeU{\Delta}{\tau'} \BY{Lemma \ref{lemma:type-expansion-U} to \pfref{expandsTU}} \pflabel{istype}
%             %  \item \uetsmenv{\Delta}{\Psi, \xuetsmbnd{\tsmv}{\tau'}{\eparse}} \BY{Definition \ref{def:ueTLM-def-ctx-formation} on \pfref{uetsmenv1}, \pfref{istype} and \pfref{eparse}}\pflabel{uetsmenv3}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{IH, part 1(b)(i) on \pfref{expandsU}}
%             \end{pfsteps*}
%             \resetpfcounter
%           \item[\text{(\ref{rule:eana-implicite})}] We have:
%             \begin{pfsteps*}
%               \item $\ue=\autsmap{b}{\tsmv}$ \BY{assumption}
%               \item $\uPsi = \uPsi', \uShyp{\tsmv}{a}{\tau}{\eparse}$ \BY{assumption}
%               \item $\encodeBody{b}{\ebody}$ \BY{assumption}
%               \item $\evalU{\eparse(\ebody)}{\inj{\lbltxt{Success}}{\ecand}}$ \BY{assumption}
%               \item $\decodeCondE{\ecand}{\ce}$ \BY{assumption}
%               \item $\cana{\emptyset}{\emptyset}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}$ \BY{assumption}\pflabel{cvalidE}
%             %  \item \uetsmenv{\Delta}{\Psi} \BY{assumption} \pflabel{uetsmenv}
%               \item $\emptyset \cap \Delta = \emptyset$ \BY{finite set intersection identity} \pflabel{delta-cap}
%               \item ${\emptyset} \cap \domof{\Gamma} = \emptyset$ \BY{finite set intersection identity} \pflabel{gamma-cap}
%               \item $\hastypeU{\emptyset \cup \Delta}{\emptyset \cup \Gamma}{e}{\tau}$ \BY{IH, part 2(b)(i) on \pfref{cvalidE}, \pfref{delta-cap}, and \pfref{gamma-cap}} \pflabel{penultimate}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{definition of finite set union over \pfref{penultimate}}               
%              \end{pfsteps*} 
%              \resetpfcounter
%           \item[\text{(\ref{rule:eana-lit})}] We have:
%             \begin{pfsteps*}
%               \item $\ue=\auelit{b}$ \BY{assumption}
%               \item $\uPsi=\uASI{\uA}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}$ \BY{assumption}
%               \item $\encodeBody{b}{\ebody}$ \BY{assumption}
%               \item $\evalU{\ap{\eparse}{\ebody}}{\inj{\lbltxt{Success}}{\ecand}}$ \BY{assumption}
%               \item $\decodeCondE{\ecand}{\ce}$ \BY{assumption}
%               \item $\cana{\emptyset}{\emptyset}{\esceneUP{\uDelta}{\uGamma}{\uASI{\uA}{\Psi, \xuetsmbnd{a}{\tau}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\uPhi}{b}}{\ce}{e}{\tau}$ \BY{assumption} \pflabel{cvalidE}
%               \item $\emptyset \cap \Delta = \emptyset$ \BY{finite set intersection identity} \pflabel{delta-cap}
%               \item ${\emptyset} \cap \domof{\Gamma} = \emptyset$ \BY{finite set intersection identity} \pflabel{gamma-cap}
%               \item $\hastypeU{\emptyset \cup \Delta}{\emptyset \cup \Gamma}{e}{\tau}$ \BY{IH, part 2(a)(i) on \pfref{cvalidE}, \pfref{delta-cap}, and \pfref{gamma-cap}} \pflabel{penultimate}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{definition of finite set union over \pfref{penultimate}}
%             \end{pfsteps*}
%             \resetpfcounter
%           \item[\text{(\ref{rule:eana-defuptsm})}] We have:
%             \begin{pfsteps*}
%               \item $\ue=\audefuptsm{\utau'}{\eparse}{\tsmv}{\ue'}$ \BY{assumption}
%               \item $\expandsTU{\uDelta}{\utau'}{\tau'}$ \BY{assumption} \pflabel{expandsTU}
%             %  \item \hastypeU{\emptyset}{\emptyset}{\eparse}{\aparr{\tBody}{\tParseResultExp}} \BY{assumption}\pflabel{eparse}
%               \item $\eana{\uDelta}{\uGamma}{\uPsi}{\uPhi, \uPhyp{\tsmv}{a}{\tau'}{\eparse}}{\ue'}{e}{\tau}$ \BY{assumption}\pflabel{expandsU}
%             %  \item \uetsmenv{\Delta}{\Psi} \BY{assumption}\pflabel{uetsmenv1}
%             %  \item \istypeU{\Delta}{\tau'} \BY{Lemma \ref{lemma:type-expansion-U} to \pfref{expandsTU}} \pflabel{istype}
%             %  \item \uetsmenv{\Delta}{\Psi, \xuetsmbnd{\tsmv}{\tau'}{\eparse}} \BY{Definition \ref{def:ueTLM-def-ctx-formation} on \pfref{uetsmenv1}, \pfref{istype} and \pfref{eparse}}\pflabel{uetsmenv3}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{IH, part 1(b)(i) on \pfref{expandsU}}
%             \end{pfsteps*}
%             \resetpfcounter
%           \item[\text{(\ref{rule:eana-implicitp})}] We have:
%             \begin{pfsteps*}
%               \item $\ue=\auimplicitp{\tsmv}{\ue}$ \BY{assumption}
%               \item $\uPhi=\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau'}{\eparse}}{\uI}$ \BY{assumption}
%               \item $\eana{\uDelta}{\uGamma}{\uPsi}{\uASI{\uA \uplus \vExpands{\tsmv}{a}}{\Phi, \xuptsmbnd{a}{\tau'}{\eparse}}{\uI \uplus \designate{\tau}{a}}}{\ue}{e}{\tau}$ \BY{assumption} \pflabel{esyn}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{IH, part 1(b)(i) on \pfref{esyn}}
%             \end{pfsteps*}
%             \resetpfcounter
%         \end{byCases}
%       \item We induct on the assumption. There is one case.
%         \begin{byCases}
%           \item[\text{(\ref{rule:rana})}] We have:
%             \begin{pfsteps*}
%               \item $\urv=\aumatchrule{\upv}{\ue}$ \BY{assumption}
%               \item $r=\aematchrule{p}{e}$ \BY{assumption}
%               \item $\patExpands{\uGG{\uA'}{\pctx}}{\uPhi}{\upv}{p}{\tau}$ \BY{assumption} \pflabel{patExpands}
%               \item $\eana{\uDelta}{\uGG{{\uA}\uplus{\uA'}}{\Gcons{\Gamma}{\pctx}}}{\uPsi}{\uPhi}{\ue}{e}{\tau'}$ \BY{assumption} \pflabel{expandsUP}
%               \item $\patType{\pctx}{p}{\tau}$ \BY{Theorem \ref{thm:typed-pattern-expansion-B}, part 1 on \pfref{patExpands}}\pflabel{patType}
%               \item $\hastypeU{\Delta}{\Gcons{\Gamma}{\pctx}}{e}{\tau'}$ \BY{IH, part 1(b)(i) on \pfref{expandsUP}} \pflabel{hasType}
%               \item $\ruleType{\Delta}{\Gamma}{\aematchrule{p}{e}}{\tau}{\tau'}$ \BY{Rule (\ref{rule:ruleType}) on \pfref{patType} and \pfref{hasType}}
%             \end{pfsteps*}
%             \resetpfcounter
%         \end{byCases}
%     \end{enumerate}
%   \end{enumerate}
%   \item In the following, let $\uDelta=\uDD{\uD}{\Delta_\text{app}}$ and $\uGamma=\uGG{\uG}{\Gamma_\text{app}}$ and $\escenev=\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}$.
%   \begin{enumerate}
%     \item \begin{enumerate}
%       \item We induct on the assumption.
%         \begin{byCases}
%           \item[\text{(\ref{rule:csyn-var})}] We have:
%             \begin{pfsteps*}
%               \item $e=x$ \BY{assumption}
%               \item $\Gamma=\Gamma', \Ghyp{x}{\tau}$ \BY{assumption}
%               \item $\hastypeU{\Delta}{\Gamma', \Ghyp{x}{\tau}}{x}{\tau}$ \BY{Rule (\ref{rule:hastypeUP-var})}
%             \end{pfsteps*}
%             \resetpfcounter 
%           \item[\text{(\ref{rule:csyn-asc})}] We have:
%             \begin{pfsteps*}
%                \item $\ce=\aceasc{\ctau}{\ce'}$ \BY{assumption}
%                \item $\Delta \cap \Delta_\text{app}=\emptyset$ \BY{assumption} \pflabel{delta-disjoint}
%                \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ \BY{assumption} \pflabel{gamma-disjoint}
%                \item $\cvalidT{\Delta}{\tsfrom{\escenev}}{\ctau}{\tau}$ \BY{assumption}\pflabel{expandsTU}
%                \item $\canaX{\ce'}{e}{\tau}$ \BY{assumption}\pflabel{eanaX}
%                \item $\istypeU{\Delta \cup \Delta_\text{app}}{\tau}$ \BY{Lemma \ref{lemma:candidate-expansion-type-validation} on \pfref{expandsTU}}\pflabel{istype}
%                \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{IH, part 2(b)(i) to \pfref{eanaX}, \pfref{delta-disjoint}, \pfref{gamma-disjoint} and  \pfref{istype}}
%              \end{pfsteps*}
%              \resetpfcounter
%           \item[\text{(\ref{rule:csyn-let}) through (\ref{rule:csyn-match})}] In each of these cases, we apply:
%             \begin{itemize}
%               \item Lemma \ref{lemma:candidate-expansion-type-validation} to or over all ce-type validation premises.
%               \item The IH, part 2(a)(i) to or over all synthetic ce-expression validation premises.
%               \item The IH, part 2(a)(ii) to or over all synthetic ce-rule validation premises.
%               \item The IH, part 2(b)(i) to or over all analytic ce-expression validation premises.
%             \end{itemize}
%             We then derive the conclusion by applying Rules (\ref{rules:hastypeUP}), Rule (\ref{rule:ruleType}), Lemma \ref{lemma:weakening-UP},  the identification convention and exchange as needed.
%           \item[\text{(\ref{rule:csyn-splicede})}] We have:
%             \begin{pfsteps*}
%               \item $\ce=\acesplicede{m}{n}{\ctau}$ \BY{assumption}
%               \item $\parseUExp{\bsubseq{b}{m}{n}}{\ue}$ \BY{assumption}
%               \item $\esyn{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}$ \BY{assumption} \pflabel{expands}
%             %  \item $\uetsmenv{\Delta_\text{app}}{\Psi}$ \BY{assumption} \pflabel{uetsmenv}
%               \item $\Delta \cap \Delta_\text{app}=\emptyset$ \BY{assumption} \pflabel{delta-disjoint}
%               \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ \BY{assumption} \pflabel{gamma-disjoint}
%               \item $\hastypeU{\Delta_\text{app}}{\Gamma_\text{app}}{e}{\tau}$ \BY{IH, part 1(a)(i) on \pfref{expands}} \pflabel{hastype}
%               \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e}{\tau}$ \BY{Lemma \ref{lemma:weakening-UP} over $\Delta$ and $\Gamma$ and exchange on \pfref{hastype}}
%             \end{pfsteps*}
%             \resetpfcounter
%         \end{byCases}
%       \item We induct on the assumption. There is one case.
%         \begin{byCases}
%           \item[\text{(\ref{rule:crsyn})}] We have:
%             \begin{pfsteps*}
%               \item $\crv=\acematchrule{p}{\ce}$ \BY{assumption}
%               \item $r=\aematchrule{p}{e}$ \BY{assumption}
%               \item $\patType{\pctx}{p}{\tau}$ \BY{assumption} \pflabel{patType}
%               \item $\csyn{\Delta}{\Gcons{\Gamma}{\pctx}}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau'}$ \BY{assumption} \pflabel{cvalidE}
%               \item $\Delta \cap \Delta_\text{app} = \emptyset$ \BY{assumption}\pflabel{delta-disjoint}
%               \item $\domof{\Gamma} \cap \domof{\pctx} = \emptyset$ \BY{identification convention}\pflabel{gamma-disjoint1}
%               \item $\domof{\Gamma_\text{app}} \cap \domof{\pctx} = \emptyset$ \BY{identification convention}\pflabel{gamma-disjoint2}
%               \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}} = \emptyset$ \BY{assumption}\pflabel{gamma-disjoint3}
%               \item $\domof{\Gcons{\Gamma}{\pctx}} \cap \domof{\Gamma_\text{app}} = \emptyset$ \BY{standard finite set definitions and identities on \pfref{gamma-disjoint1}, \pfref{gamma-disjoint2} and \pfref{gamma-disjoint3}}\pflabel{gamma-disjoint4}
%               \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gcons{\Gamma}{\pctx}}{\Gamma_\text{app}}}{e}{\tau'}$ \BY{IH, part 2(a)(i) on \pfref{cvalidE}, \pfref{delta-disjoint} and \pfref{gamma-disjoint4}}\pflabel{hastype}
%               \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gcons{\Gamma}{\Gamma_\text{app}}}{\pctx}}{e}{\tau'}$ \BY{exchange of $\pctx$ and $\Gamma_\text{app}$ on \pfref{hastype}}\pflabel{hastype2}
%               \item $\ruleType{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{\aematchrule{p}{e}}{\tau}{\tau'}$ \BY{Rule (\ref{rule:ruleType}) on \pfref{patType} and \pfref{hastype2}}
%             \end{pfsteps*}
%             \resetpfcounter
%         \end{byCases}
%     \end{enumerate}
%     \item  \begin{enumerate}
%       \item We induct on the assumption.
%         \begin{byCases}
%           \item[\text{(\ref{rule:cana-subsume})}] We have:
%             \begin{pfsteps*}
%               \item $\csynX{\ce}{e}{\tau}$ \BY{assumption} \pflabel{esyn}
%               \item $\hastypeU{\Delta}{\Gamma}{e}{\tau}$ \BY{IH, part 2(a)(i) on \pfref{esyn}}
%             \end{pfsteps*}
%           \item[\text{(\ref{rule:cana-let}) through (\ref{rule:eana-match})}] In each of these cases, we apply:
%             \begin{itemize}
%               \item Lemma \ref{lemma:candidate-expansion-type-validation} to or over all ce-type validation premises.
%               \item The IH, part 2(a)(i) to or over all synthetic ce-expression validation premises.
%               \item The IH, part 2(a)(ii) to or over all synthetic ce-rule validation premises.
%               \item The IH, part 2(b)(i) to or over all analytic ce-expression validation premises.
%             \end{itemize}
%             We then derive the conclusion by applying Rules (\ref{rules:hastypeUP}), Rule (\ref{rule:ruleType}), Lemma \ref{lemma:weakening-UP},  the identification convention and exchange as needed.
%           \item[\text{(\ref{rule:cana-splicede})}] We have:
%             \begin{pfsteps*}
%               \item $\ce=\acesplicede{m}{n}{\ctau}$ \BY{assumption}
%               \item $\parseUExp{\bsubseq{b}{m}{n}}{\ue}$ \BY{assumption}
%               \item $\eana{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}$ \BY{assumption} \pflabel{expands}
%               \item $\istypeU{\Delta \cup \Delta_\text{app}}{\tau}$ \BY{assumption} \pflabel{istype}
%             %  \item $\uetsmenv{\Delta_\text{app}}{\Psi}$ \BY{assumption} \pflabel{uetsmenv}
%               \item $\Delta \cap \Delta_\text{app}=\emptyset$ \BY{assumption} \pflabel{delta-disjoint}
%               \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}}=\emptyset$ \BY{assumption} \pflabel{gamma-disjoint}
%               \item $\hastypeU{\Delta_\text{app}}{\Gamma_\text{app}}{e}{\tau}$ \BY{IH, part 1(b)(i) on \pfref{expands}, \pfref{delta-disjoint}, \pfref{gamma-disjoint} and \pfref{istype}} \pflabel{hastype}
%               \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{e}{\tau}$ \BY{Lemma \ref{lemma:weakening-UP} over $\Delta$ and $\Gamma$ and exchange on \pfref{hastype}}
%             \end{pfsteps*}
%             \resetpfcounter
%         \end{byCases}
%       \item We induct on the assumption. There is one case.
%         \begin{byCases}
%           \item[\text{(\ref{rule:crana})}] We have:    
%             \begin{pfsteps*}
%                 \item $\crv=\acematchrule{p}{\ce}$ \BY{assumption}
%                 \item $r=\aematchrule{p}{e}$ \BY{assumption}
%                 \item $\patType{\pctx}{p}{\tau}$ \BY{assumption} \pflabel{patType}
%                 \item $\cana{\Delta}{\Gcons{\Gamma}{\pctx}}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau'}$ \BY{assumption} \pflabel{cvalidE}
%                 \item $\istypeU{\Delta \cup \Delta_\text{app}}{\tau'}$ \BY{assumption} \pflabel{istype}
%                 \item $\domof{\Gamma} \cap \domof{\Gamma_\text{app}} = \emptyset$ \BY{assumption}\pflabel{gamma-disjoint3}
%                 \item $\Delta \cap \Delta_\text{app} = \emptyset$ \BY{assumption}\pflabel{delta-disjoint}
%                 \item $\domof{\Gamma} \cap \domof{\pctx} = \emptyset$ \BY{identification convention}\pflabel{gamma-disjoint1}
%                 \item $\domof{\Gamma_\text{app}} \cap \domof{\pctx} = \emptyset$ \BY{identification convention}\pflabel{gamma-disjoint2}
%                 \item $\domof{\Gcons{\Gamma}{\pctx}} \cap \domof{\Gamma_\text{app}} = \emptyset$ \BY{standard finite set definitions and identities on \pfref{gamma-disjoint1}, \pfref{gamma-disjoint2} and \pfref{gamma-disjoint3}}\pflabel{gamma-disjoint4}
%                 \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gcons{\Gamma}{\pctx}}{\Gamma_\text{app}}}{e}{\tau'}$ \BY{IH, part 2(b)(i) on \pfref{cvalidE}, \pfref{delta-disjoint}, \pfref{gamma-disjoint4} and \pfref{istype}}\pflabel{hastype}
%                 \item $\hastypeU{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gcons{\Gamma}{\Gamma_\text{app}}}{\pctx}}{e}{\tau'}$ \BY{exchange of $\pctx$ and $\Gamma_\text{app}$ on \pfref{hastype}}\pflabel{hastype2}
%                 \item $\ruleType{\Dcons{\Delta}{\Delta_\text{app}}}{\Gcons{\Gamma}{\Gamma_\text{app}}}{\aematchrule{p}{e}}{\tau}{\tau'}$ \BY{Rule (\ref{rule:ruleType}) on \pfref{patType} and \pfref{hastype2}}
%               \end{pfsteps*}
%               \resetpfcounter

%         \end{byCases}
%     \end{enumerate}
%   \end{enumerate}
% \end{enumerate}

% We must now show that the induction is well-founded. All applications of the IH are on subterms except the following.  

% \begin{itemize}
% \item The only cases in the proof of part 1 that invoke the IH, part 2 are Case (\ref{rule:esyn-apuetsm}) in the proof of part 1(a)(i) and Case (\ref{rule:eana-lit}) in the proof of part 1(b)(i). The only cases in the proof of part 2 that invoke the IH, part 1 are Case (\ref{rule:csyn-splicede}) in the proof of part 2(a)(i) and Case (\ref{rule:cana-splicede}) in the proof of part 2(b)(i). We can show that the following metric on the judgements that we induct on is stable in one direction and strictly decreasing in the other direction:
% \begin{align*}
% \sizeof{\esyn{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}} & = \sizeof{\ue}\\
% \sizeof{\eana{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}} & = \sizeof{\ue}\\
% \sizeof{\csyn{\Delta}{\Gamma}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}} & = \sizeof{b}\\
% \sizeof{\cana{\Delta}{\Gamma}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}} & = \sizeof{b}
% \end{align*}
% where $\sizeof{b}$ is the length of $b$ and $\sizeof{\ue}$ is the sum of the lengths of the ueTLM literal bodies in $\ue$,
% \begin{align*}
% \sizeof{\ux} & = 0\\
% \sizeof{\auasc{\utau}{\ue}} & = \sizeof{\ue}\\
% \sizeof{\auletsyn{\ux}{\ue}{\ue'}} & = \sizeof{\ue} + \sizeof{\ue'}\\
% \sizeof{\auanalam{\ux}{\ue}} & = \sizeof{\ue}\\
% \sizeof{\aulam{\utau}{\ux}{\ue}} &= \sizeof{\ue}\\
% \sizeof{\auap{\ue_1}{\ue_2}} & = \sizeof{\ue_1} + \sizeof{\ue_2}\\
% \sizeof{\autlam{\ut}{\ue}} & = \sizeof{\ue}\\
% \sizeof{\autap{\ue}{\utau}} & = \sizeof{\ue}\\
% \sizeof{\auanafold{\ue}} & = \sizeof{\ue}\\
% \sizeof{\auunfold{\ue}} & = \sizeof{\ue}\\
% %\end{align*}
% %\begin{align*}
% \sizeof{\autpl{\labelset}{\mapschema{\ue}{i}{\labelset}}} & = \sum_{i \in \labelset} \sizeof{\ue_i}\\
% \sizeof{\aupr{\ell}{\ue}} & = \sizeof{\ue}\\
% \sizeof{\auanain{\ell}{\ue}} & = \sizeof{\ue}\\
% %\sizeof{\aucase{\labelset}{\utau}{\ue}{\mapschemab{\ux}{\ue}{i}{\labelset}}} & = \sizeof{\ue} + \sum_{i \in \labelset} \sizeof{\ue_i}\\
% \sizeof{\aumatchwithb{n}{\ue}{\seqschemaX{\urv}}} & = \sizeof{\ue} + \sum_{1 \leq i \leq n} \sizeof{r_i}\\
% \sizeof{\audefuetsm{\utau}{\eparse}{\tsmv}{\ue}} & = \sizeof{\ue}\\
% \sizeof{\auimplicite{\tsmv}{\ue}} & = \sizeof{\ue}\\
% \sizeof{\autsmap{b}{\tsmv}} & = \sizeof{b}\\
% \sizeof{\auelit{b}} & = \sizeof{b}\\
% \sizeof{\audefuptsm{\utau}{\eparse}{\tsmv}{\ue}} & = \sizeof{\ue}\\
% \sizeof{\auimplicitp{\tsmv}{\ue}} & = \sizeof{\ue}
% \end{align*}
% and $\sizeof{r}$ is defined as follows:
% \begin{align*}
% \sizeof{\aumatchrule{\upv}{\ue}} & = \sizeof{\ue}
% \end{align*}

% Going from part 1 to part 2, the metric remains stable:
% \begin{align*}
%  & \sizeof{\esyn{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\autsmap{b}{\tsmv}}{e}{\tau}}\\
% =& \sizeof{\eana{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\auelit{b}}{e}{\tau}}\\
% =& \sizeof{\cana{\emptyset}{\emptyset}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\ce}{e}{\tau}}\\
% =&\sizeof{b}\end{align*}

% Going from part 2 to part 1, in each case we have that $\parseUExp{\bsubseq{b}{m}{n}}{\ue}$ and the IH is applied to the judgements $\esyn{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}$ and $\eana{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}$, respectively. Because the metric is stable when passing from part 1 to part 2, we must have that it is strictly decreasing in the other direction:
% \[\sizeof{\esyn{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}} < \sizeof{\csyn{\Delta}{\Gamma}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\acesplicede{m}{n}{\ctau}}{e}{\tau}}\]
% and
% \[\sizeof{\eana{\uDelta}{\uGamma}{\uPsi}{\uPhi}{\ue}{e}{\tau}} < \sizeof{\cana{\Delta}{\Gamma}{\esceneUP{\uDelta}{\uGamma}{\uPsi}{\uPhi}{b}}{\acesplicede{m}{n}{\ctau}}{e}{\tau}}\]
% i.e. by the definitions above, 
% \[\sizeof{\ue} < \sizeof{b}\]

% This is established by appeal to Condition \ref{condition:body-subsequences}, which states that subsequences of $b$ are no longer than $b$, and the following condition, which states that an unexpanded expression constructed by parsing a textual sequence $b$ is strictly smaller, as measured by the metric defined above, than the length of $b$, because some characters must necessarily be used to delimit each literal body.
% \begin{condition}[Expression Parsing Monotonicity]\label{condition:body-parsing-B} If $\parseUExp{b}{\ue}$ then $\sizeof{\ue} < \sizeof{b}$.\end{condition}

% Combining Conditions \ref{condition:body-subsequences} and \ref{condition:body-parsing-B}, we have that $\sizeof{\ue} < \sizeof{b}$ as needed.
% \item In Case (\ref{rule:eana-subsume}) of the proof of part 1(b)(i), we apply the IH, part 1(a)(i), with $\ue=\ue$. This is well-founded because all applications of the IH, part 1(b)(i) elsewhere in the proof are on strictly smaller terms.
% \item Similarly, in Case (\ref{rule:cana-subsume}) of the proof of part 2(b)(i), we apply the IH, part 2(a)(i), with $\ce=\ce$. This is well-founded because all applications of the IH, part 2(b)(i) elsewhere in the proof are on strictly smaller terms.
% \end{itemize}
% \end{proof}