\documentclass{article}
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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
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\newcommand\unicdbdjded{\Leftrightarrow}
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\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{thm\_2Epatricia\_\_casts\_2ETHE\_\_PTREE\_\_SOME\_\_PTREE (TMcGLAprL6A4QC9Wd8tG1AM6RKeWvGjxg54)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
Let $ ty\_2Epatricia\_2Eptree : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Epatricia\_2Eptree}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeahbgehchjgdgjgbgpfcdfeahehchfgfg}
\end{equation}
Let $ ty\_2Epatricia\_\_casts\_2Eword\_\_ptree : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Epatricia\_\_casts\_2Eword\_\_ptree~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfehhpgchegpfpfahehchfgfg}
\end{equation}
Let $ c\_2Epatricia\_\_casts\_2ETHE\_\_PTREE : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epatricia\_\_casts\_2ETHE\_\_PTREE~\\
A\_27a~A\_27b \in ((ty\_2Epatricia\_2Eptree~A\_27a)^{{({{{ty\_2Epatricia\_\_casts\_2Eword\_\_ptree}~{A\_27b}}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfeefiefepfpfafefcffefe}
\end{equation}
Let $ ty\_2Eone\_2Eone : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eone\_2Eone}}\label{hyp:ogfgpfehjhpfcdfepgogfgpfcdfepgogfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfedad}
We define $\mname{c\_2Emin\_2E\_40}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} P \in {{2}^{{A}}} .  {{\mathbf{if}}~{({{\unicdcdaddd} x \in {A} .  {{p}~{({{{ap}~{P}}~{x}})}}})}~{\mathbf{then}}~{({{the}~{({{\uniddcgcg} x .  {{{x} \in {A}}{\unicdcdcdhd}{{p}~{({{{ap}~{P}}~{x}})}}}})}})}~{\mathbf{else}}~{({{the}~{({{\uniddcgcg} x .  {{x} \in {A}}})}})}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfepgogfgpfcdfepgogfg}
We define $\mname{c\_2Eone\_2Eone}$ to be ${({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Eone\_2Eone}})}}~{({{\uniddcgcg} V0x \in {ty\_2Eone\_2Eone} .  {\mname{c\_2Ebool\_2ET}}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfele}
We define $\mname{c\_2Ecombin\_2EK}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1y \in {A\_27b} .  {V0x}})}})}}}$.
\end{definition}
Let $ c\_2Epatricia\_\_casts\_2EWord\_\_ptree : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epatricia\_\_casts\_2EWord\_\_ptree~\\
A\_27a~A\_27b \in (((ty\_2Epatricia\_\_casts\_2Eword\_\_ptree~A\_27a~A\_27b)^{{({{ty\_2Epatricia\_2Eptree}~{A\_27b}})}})^{{({{ty\_2Eone\_2Eone}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfehfpgchegpfpfahehchfgfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfedfpenefepfpfafefcffefe}
We define $\mname{c\_2Epatricia\_\_casts\_2ESOME\_\_PTREE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0t \in {({{ty\_2Epatricia\_2Eptree}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{({{{c\_2Epatricia\_\_casts\_2EWord\_\_ptree}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{{\mname{c\_2Ecombin\_2EK}}~{ty\_2Eone\_2Eone}}~{A\_27a}})}}~{\mname{c\_2Eone\_2Eone}}})}})}}~{V0t}})}}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0a \in (ty\_2Eone\_2Eone^{{A\_27b}}) .  ({\unicdcdadad} V1t \in (ty\_2Epatricia\_2Eptree~\\
A\_27a) .  ((ap~(c\_2Epatricia\_\_casts\_2ETHE\_\_PTREE~A\_27a~A\_27b)~\\
(ap~(ap~(c\_2Epatricia\_\_casts\_2EWord\_\_ptree~A\_27b~A\_27a)~V0a)~\\
V1t))=V1t)))\end{array}
\label{hyp:ehigngpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfeefiefepfpfafefcffefepfpfegfggg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeahbgehchjgdgjgbgpfpfdgbgdhehdhpfcdfeefiefepfpfafefcffefepfpfdfpenefepfpfafefcffefe}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0t \in (ty\_2Epatricia\_2Eptree~A\_27a) .  ((ap~(c\_2Epatricia\_\_casts\_2ETHE\_\_PTREE~\\
A\_27a~A\_27b)~(ap~(\mname{c\_2Epatricia\_\_casts\_2ESOME\_\_PTREE}~A\_27b~A\_27a)~\\
V0t))=V0t))\end{array}$$
\end{theorem}
\end{document}
