\documentclass{article}
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
\newcommand\unicdcdcdbg{\cup}
\newcommand\unicdcdbdgd{\setminus}
\newcommand\unicdcdcdjd{\cap}
\newcommand\unicdcdadad{\forall}
\newcommand\unicdbdjdcd{\Rightarrow}
\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
\newcommand\unicdcdcdhd{\land}
\newcommand\uniddcgcg{\lambda}
\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{thm\_2Elbtree\_2Elbtree\_\_cases (TMP2gkck6r9dUpkzMwqe3gzSkyfTQPYexfp)}
\maketitle
\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: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}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\end{definition}
Let $ ty\_2Eoption\_2Eoption : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Eoption\_2Eoption}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfepgahehjgpgogpfcdfepgahehjgpgog}
\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:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ ty\_2Elist\_2Elist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Elist\_2Elist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgjgdhehpfcdfemgjgdheh}
\end{equation}
Let $ ty\_2Eone\_2Eone : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eone\_2Eone}}\label{hyp:ogfgpfehjhpfcdfepgogfgpfcdfepgogfg}
\end{equation}
Let $ ty\_2Esum\_2Esum : {{\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\_2Esum\_2Esum~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfedhfhngpfcdfedhfhng}
\end{equation}
Let $ c\_2Esum\_2EABS\_\_sum : {{\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\_2Esum\_2EABS\_\_sum~\\
A\_27a~A\_27b \in ((ty\_2Esum\_2Esum~A\_27a~A\_27b)^{{({{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}^{{2}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfhngpfcdfebecedfpfpfdhfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfedhfhngpfcdfejeoeme}
We define $\mname{c\_2Esum\_2EINL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0e \in {A\_27a} .  {({{{ap}~{({{{c\_2Esum\_2EABS\_\_sum}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V1b \in {2} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2x}})}}~{V0e}})}})}}~{V1b}})}})}})}})}})}}}}$.
\end{definition}
Let $ c\_2Eoption\_2Eoption\_\_ABS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eoption\_2Eoption\_\_ABS~A\_27a \in \\
((ty\_2Eoption\_2Eoption~A\_27a)^{{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{ty\_2Eone\_2Eone}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfepgahehjgpgogpfpfbecedf}
\end{equation}
\begin{definition}\label{def:dgpfcdfepgahehjgpgogpfcdfedfpenefe}
We define $\mname{c\_2Eoption\_2ESOME}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {({{{ap}~{({{c\_2Eoption\_2Eoption\_\_ABS}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Esum\_2EINL}}~{A\_27a}}~{ty\_2Eone\_2Eone}})}}~{V0x}})}})}}}$.
\end{definition}
Let $ c\_2Elist\_2Elist\_\_CASE : {{\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\_2Elist\_2Elist\_\_CASE~\\
A\_27a~A\_27b \in (((A\_27b^{{({{({{A\_27b}^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}})}^{{A\_27a}}})}})^{{A\_27b}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfemgjgdhehpfpfdebedffe}
\end{equation}
\begin{definition}\label{def:dgpfcdfemgcgehchfgfgpfcdfeoeegchfgah}
We define $\mname{c\_2Elbtree\_2ENdrep}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0a \in {A\_27a} .  {{\uniddcgcg} V1t1 \in {({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})} .  {{\uniddcgcg} V2t2 \in {({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})} .  {({{\uniddcgcg} V3l \in {({{ty\_2Elist\_2Elist}~{2}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{c\_2Elist\_2Elist\_\_CASE}~{2}}~{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})}}~{V3l}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{A\_27a}})}}~{V0a}})}})}}~{({{\uniddcgcg} V4v \in {2} .  {({{\uniddcgcg} V5xs \in {({{ty\_2Elist\_2Elist}~{2}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})}}~{V4v}})}}~{({{{ap}~{V1t1}}~{V5xs}})}})}}~{({{{ap}~{V2t2}}~{V5xs}})}})}})}})}})}})}}}}}$.
\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:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfhngpfcdfejeoecf}
We define $\mname{c\_2Esum\_2EINR}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0e \in {A\_27b} .  {({{{ap}~{({{{c\_2Esum\_2EABS\_\_sum}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V1b \in {2} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3y}})}}~{V0e}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{V1b}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfepgahehjgpgogpfcdfeoepeoefe}
We define $\mname{c\_2Eoption\_2ENONE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{c\_2Eoption\_2Eoption\_\_ABS}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Esum\_2EINR}}~{A\_27a}}~{ty\_2Eone\_2Eone}})}}~{\mname{c\_2Eone\_2Eone}}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfemgcgehchfgfgpfcdfemeggchfgah}
We define $\mname{c\_2Elbtree\_2ELfrep}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0l \in {A\_27a} .  {({{\mname{c\_2Eoption\_2ENONE}}~{A\_27b}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfemgcgehchfgfgpfcdfejgdhpfpfmgcgehchfgfg}
We define $\mname{c\_2Elbtree\_2Eis\_\_lbtree}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0t \in {({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})}}})}})}}~{({{\uniddcgcg} V1P \in {({{2}^{{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})}})}}~{({{\uniddcgcg} V2t \in {({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V1P}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})}})}}~{V2t}})}}~{({{{\mname{c\_2Elbtree\_2ELfrep}}~{({{ty\_2Elist\_2Elist}~{2}})}}~{A\_27a}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3a \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})}})}}~{({{\uniddcgcg} V4t1 \in {({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})}})}}~{({{\uniddcgcg} V5t2 \in {({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1P}}~{V4t1}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1P}}~{V5t2}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})}})}}~{V2t}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Elbtree\_2ENdrep}}~{A\_27a}})}}~{V3a}})}}~{V4t1}})}}~{V5t2}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{V1P}}~{V0t}})}})}})}})}}}$.
\end{definition}
Let $ ty\_2Elbtree\_2Elbtree : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Elbtree\_2Elbtree}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgcgehchfgfgpfcdfemgcgehchfgfg}
\end{equation}
Let $ c\_2Elbtree\_2Elbtree\_\_abs : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elbtree\_2Elbtree\_\_abs~A\_27a \in \\
((ty\_2Elbtree\_2Elbtree~A\_27a)^{{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{2}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgcgehchfgfgpfcdfemgcgehchfgfgpfpfbgcgdh}
\end{equation}
\begin{definition}\label{def:dgpfcdfemgcgehchfgfgpfcdfemegg}
We define $\mname{c\_2Elbtree\_2ELf}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{c\_2Elbtree\_2Elbtree\_\_abs}~{A\_27a}})}}~{({{{\mname{c\_2Elbtree\_2ELfrep}}~{({{ty\_2Elist\_2Elist}~{2}})}}~{A\_27a}})}})}}$.
\end{definition}
Let $ c\_2Elbtree\_2Elbtree\_\_rep : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elbtree\_2Elbtree\_\_rep~A\_27a \in \\
(((ty\_2Eoption\_2Eoption~A\_27a)^{{({{ty\_2Elist\_2Elist}~{2}})}})^{{({{ty\_2Elbtree\_2Elbtree}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgcgehchfgfgpfcdfemgcgehchfgfgpfpfchfgah}
\end{equation}
\begin{definition}\label{def:dgpfcdfemgcgehchfgfgpfcdfeoeeg}
We define $\mname{c\_2Elbtree\_2ENd}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0a \in {A\_27a} .  {{\uniddcgcg} V1t1 \in {({{ty\_2Elbtree\_2Elbtree}~{A\_27a}})} .  {{\uniddcgcg} V2t2 \in {({{ty\_2Elbtree\_2Elbtree}~{A\_27a}})} .  {({{{ap}~{({{c\_2Elbtree\_2Elbtree\_\_abs}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Elbtree\_2ENdrep}}~{A\_27a}})}}~{V0a}})}}~{({{{ap}~{({{c\_2Elbtree\_2Elbtree\_\_rep}~{A\_27a}})}}~{V1t1}})}})}}~{({{{ap}~{({{c\_2Elbtree\_2Elbtree\_\_rep}~{A\_27a}})}}~{V2t2}})}})}})}}}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjdcd}(p~V1t2)){\unicdbdjdcd}(((p~\\
V1t2){\unicdbdjdcd}(p~V0t1)){\unicdbdjdcd}((p~V0t1){\unicdbdjded}(p~V1t2))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfbeoeefjedfjfnepfpfbeif}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in 2 .  (({\unicdcdadad} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepecfbememepfpfdfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} A\_27a .  {{{nonempty}~{A\_27a}}{\unicdbdjdcd}{({{\unicdcdadad} V0x \in {A\_27a} .  {({{V0x}={V0x}})}})}}}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfcffegeme}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in 2 .  ({\unicdcdadad} V1Q \in (\\
2^{{A\_27a}}) .  (((p~V0P){\unicdcdcdhd}({\unicdcdadad} V2x \in A\_27a .  (p~(ap~V1Q~V2x)))){\unicdbdjded}({\unicdcdadad} V3x \in \\
A\_27a .  ((p~V0P){\unicdcdcdhd}(p~(ap~V1Q~V3x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecfjeheieefpfpfbeoeeepfpfgepecfbememepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1Q \in \\
2 .  (({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~V0P~V2x)){\unicdbdjdcd}(p~V1Q))){\unicdbdjded}(({\unicdcdaddd} V3x \in \\
A\_27a .  (p~(ap~V0P~V3x))){\unicdbdjdcd}(p~V1Q)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemefegeefpfpfgepecfbememepfpfjeneafpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  ({\unicdcdadad} V2C \in 2 .  (((p~V0A){\unicdcdcdid}(\\
(p~V1B){\unicdcdcdhd}(p~V2C))){\unicdbdjded}(((p~V0A){\unicdcdcdid}(p~V1B)){\unicdcdcdhd}((p~V0A){\unicdcdcdid}(p~V2C)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemefegeefpfpfpecfpfpfpegffecfpfpfbeoeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  ({\unicdcdadad} V2C \in 2 .  ((((p~V1B){\unicdcdcdhd}\\
(p~V2C)){\unicdcdcdid}(p~V0A)){\unicdbdjded}(((p~V1B){\unicdcdcdid}(p~V0A)){\unicdcdcdhd}((p~V2C){\unicdcdcdid}(p~V0A)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecfjeheieefpfpfpecfpfpfpegffecfpfpfbeoeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0P \in 2 .  ({\unicdcdadad} V1Q \in 2 .  ({\unicdcdadad} V2R \in 2 .  ((((p~V0P){\unicdcdcdid}\\
(p~V1Q)){\unicdbdjdcd}(p~V2R)){\unicdbdjded}(((p~V0P){\unicdbdjdcd}(p~V2R)){\unicdcdcdhd}((p~V1Q){\unicdbdjdcd}(p~V2R)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeejedfkepfpfjeneafpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdbdjdcd}\\
((p~V1t2){\unicdbdjdcd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdbdjdcd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfjeneafpfpfjeoeefcfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjded}(p~V1t2)){\unicdbdjded}(((p~\\
V0t1){\unicdbdjdcd}(p~V1t2)){\unicdcdcdhd}((p~V1t2){\unicdbdjdcd}(p~V0t1))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfjeneafpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in 2 .  ({\unicdcdadad} V1x\_27 \in 2 .  ({\unicdcdadad} V2y \in 2 .  ({\unicdcdadad} V3y\_27 \in \\
2 .  ((((p~V0x){\unicdbdjded}(p~V1x\_27)){\unicdcdcdhd}((p~V1x\_27){\unicdbdjdcd}((p~V2y){\unicdbdjded}(p~V3y\_27)))){\unicdbdjdcd}\\
(((p~V0x){\unicdbdjdcd}(p~V2y)){\unicdbdjded}((p~V1x\_27){\unicdbdjdcd}(p~V3y\_27))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdepeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  (({\unicdcdaddd} V2x \in A\_27a .  ((V2x=V1a){\unicdcdcdhd}(p~(ap~V0P~V2x)))){\unicdbdjded}(p~(\\
ap~V0P~V1a)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeffoehfjeoeeepfpfefienecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0a \in (ty\_2Elbtree\_2Elbtree~\\
A\_27a) .  ((ap~(c\_2Elbtree\_2Elbtree\_\_abs~A\_27a)~(ap~(c\_2Elbtree\_2Elbtree\_\_rep~\\
A\_27a)~V0a))=V0a)){\unicdcdcdhd}({\unicdcdadad} V1r \in ((ty\_2Eoption\_2Eoption~A\_27a)^{{({{ty\_2Elist\_2Elist}~{2}})}}) .  \\
((p~(ap~(\mname{c\_2Elbtree\_2Eis\_\_lbtree}~A\_27a)~V1r)){\unicdbdjded}((ap~(c\_2Elbtree\_2Elbtree\_\_rep~\\
A\_27a)~(ap~(c\_2Elbtree\_2Elbtree\_\_abs~A\_27a)~V1r))=V1r))))\end{array}
\label{hyp:ehigngpfcdfemgcgehchfgfgpfcdfemgcgehchfgfgpfpfbgcgdhchfgah}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({\preoh{{({\preoh{{({{p}~{V0t}})}}})}}})}{\unicdbdjded}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeoepeefpfpfoepeef}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0A \in {2} .  {({{({{p}~{V0A}})}{\unicdbdjdcd}{({{({\preoh{{({{p}~{V0A}})}}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfebeoeeepfpfjeoegfpfpfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((p~V0A){\unicdcdcdid}(p~V1B))){\unicdbdjdcd}False){\unicdbdjded}\\
(((p~V0A){\unicdbdjdcd}False){\unicdbdjdcd}((\preoh (p~V1B)){\unicdbdjdcd}False)))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfepecfpfpfeeffbemecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((\preoh (p~V0A)){\unicdcdcdid}(p~V1B))){\unicdbdjdcd}False){\unicdbdjded}\\
((p~V0A){\unicdbdjdcd}((\preoh (p~V1B)){\unicdbdjdcd}False)))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfepecfpfpfeeffbemedd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0A \in {2} .  {({{({{({\preoh{{({{p}~{V0A}})}}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{({{({{({{p}~{V0A}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfebeoeeepfpfjeoegfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdbdjded}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}((p~V1q){\unicdcdcdid}(p~V2r))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}((\preoh (\\
p~V2r)){\unicdcdcdid}(\preoh (p~V1q)))){\unicdcdcdhd}(((p~V1q){\unicdcdcdid}((\preoh (p~V2r)){\unicdcdcdid}(\preoh (p~V0p)))){\unicdcdcdhd}((p~V2r){\unicdcdcdid}\\
((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V0p)))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpffgbh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdcdcdhd}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V2r)))){\unicdcdcdhd}(((p~V1q){\unicdcdcdid}\\
(\preoh (p~V0p))){\unicdcdcdhd}((p~V2r){\unicdcdcdid}(\preoh (p~V0p)))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfdgpgogkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdcdcdid}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}(\preoh (p~V1q))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}(\preoh (p~V2r))){\unicdcdcdhd}\\
((p~V1q){\unicdcdcdid}((p~V2r){\unicdcdcdid}(\preoh (p~V0p))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfegjgdhkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdbdjdcd}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}(p~V1q)){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}(\preoh (p~V2r))){\unicdcdcdhd}((\\
\preoh (p~V1q)){\unicdcdcdid}((p~V2r){\unicdcdcdid}(\preoh (p~V0p))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfjgngah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  (((p~V0p){\unicdbdjded}(\preoh (p~V1q))){\unicdbdjded}(((p~V0p){\unicdcdcdid}\\
(p~V1q)){\unicdcdcdhd}((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V0p)))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfogfghg}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdbdjdcd}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({{p}~{V0p}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogjgbd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdbdjdcd}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V1q}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogjgcd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdcdcdid}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V0p}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogpgbd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdcdcdid}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V1q}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogpgcd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{({\preoh{{({\preoh{{({{p}~{V0p}})}}})}}})}{\unicdbdjdcd}{({{p}~{V0p}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogog}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfemgcgehchfgfgpfcdfemgcgehchfgfgpfpfdgbgdhfgdh}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in (ty\_2Elbtree\_2Elbtree~\\
A\_27a) .  ((V0t=(\mname{c\_2Elbtree\_2ELf}~A\_27a)){\unicdcdcdid}({\unicdcdaddd} V1a \in A\_27a .  \\
({\unicdcdaddd} V2t1 \in (ty\_2Elbtree\_2Elbtree~A\_27a) .  ({\unicdcdaddd} V3t2 \in (\\
ty\_2Elbtree\_2Elbtree~A\_27a) .  (V0t=(ap~(ap~(ap~(\mname{c\_2Elbtree\_2ENd}~\\
A\_27a)~V1a)~V2t1)~V3t2)))))))\end{array}$$
\end{theorem}
\end{document}
