\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\_2Einftree\_2Erelrec\_\_cases (TMcbKtRjizWy6uoXp4kJCv4ZCXqPc1z38bt)}
\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}
\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: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}
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 $ c\_2Elist\_2ETL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ETL~A\_27a \in ((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeefme}
\end{equation}
Let $ c\_2Elist\_2EHD : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EHD~A\_27a \in (A\_27a^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeieee}
\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 $ ty\_2Einftree\_2Einftree : {{\iota}{\unicdbdjdcd}{{\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}{\unicdcdadad} A2 .  nonempty~\\
A2{\unicdbdjdcd}nonempty~(ty\_2Einftree\_2Einftree~A0~A1~A2)\end{array}
\label{hyp:ogfgpfehjhpfcdfejgogggehchfgfgpfcdfejgogggehchfgfg}
\end{equation}
Let $ c\_2Einftree\_2Efrom\_\_inftree : {{\iota}{\unicdbdjdcd}{{\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}{\unicdcdadad} A\_27d .  \\
nonempty~A\_27d{\unicdbdjdcd}c\_2Einftree\_2Efrom\_\_inftree~A\_27a~A\_27b~A\_27d \in \\
(((ty\_2Esum\_2Esum~A\_27a~A\_27b)^{{({{ty\_2Elist\_2Elist}~{A\_27d}})}})^{{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27b}}~{A\_27d}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogggehchfgfgpfcdfeggchpgngpfpfjgogggehchfgfg}
\end{equation}
\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}
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: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}
Let $ c\_2Elist\_2ENIL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ENIL~A\_27a \in (ty\_2Elist\_2Elist~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeoejeme}
\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: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 $ c\_2Einftree\_2Eto\_\_inftree : {{\iota}{\unicdbdjdcd}{{\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}{\unicdcdadad} A\_27d .  \\
nonempty~A\_27d{\unicdbdjdcd}c\_2Einftree\_2Eto\_\_inftree~A\_27a~A\_27b~A\_27d \in \\
((ty\_2Einftree\_2Einftree~A\_27a~A\_27b~A\_27d)^{{({{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}^{{({{ty\_2Elist\_2Elist}~{A\_27d}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogggehchfgfgpfcdfeehpgpfpfjgogggehchfgfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogggehchfgfgpfcdfejgoeeg}
We define $\mname{c\_2Einftree\_2EiNd}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0b \in {A\_27b} .  {{\uniddcgcg} V1f \in {({{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27b}}~{A\_27c}})}^{{A\_27c}}})} .  {({{{ap}~{({{{{c\_2Einftree\_2Eto\_\_inftree}~{A\_27a}}~{A\_27b}}~{A\_27c}})}}~{({{\uniddcgcg} V2p \in {({{ty\_2Elist\_2Elist}~{A\_27c}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Elist\_2Elist}~{A\_27c}})}})}}~{V2p}})}}~{({{c\_2Elist\_2ENIL}~{A\_27c}})}})}})}}~{({{{ap}~{({{{\mname{c\_2Esum\_2EINR}}~{A\_27a}}~{A\_27b}})}}~{V0b}})}})}}~{({{{ap}~{({{{ap}~{({{{{c\_2Einftree\_2Efrom\_\_inftree}~{A\_27a}}~{A\_27b}}~{A\_27c}})}}~{({{{ap}~{V1f}}~{({{{ap}~{({{c\_2Elist\_2EHD}~{A\_27c}})}}~{V2p}})}})}})}}~{({{{ap}~{({{c\_2Elist\_2ETL}~{A\_27c}})}}~{V2p}})}})}})}})}})}}}}}}$.
\end{definition}
\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}
\begin{definition}\label{def:dgpfcdfejgogggehchfgfgpfcdfejgmegg}
We define $\mname{c\_2Einftree\_2EiLf}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0a \in {A\_27a} .  {({{{ap}~{({{{{c\_2Einftree\_2Eto\_\_inftree}~{A\_27a}}~{A\_27b}}~{A\_27c}})}}~{({{\uniddcgcg} V1p \in {({{ty\_2Elist\_2Elist}~{A\_27c}})} .  {({{{ap}~{({{{\mname{c\_2Esum\_2EINL}}~{A\_27a}}~{A\_27b}})}}~{V0a}})}})}})}}}}}$.
\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}
\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:dgpfcdfejgogggehchfgfgpfcdfechfgmgchfgdg}
We define $\mname{c\_2Einftree\_2Erelrec}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} A\_27d : {\iota} .  {({{\uniddcgcg} V0a0 \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V1a1 \in {({{({{A\_27b}^{{({{A\_27b}^{{A\_27d}}})}}})}^{{A\_27c}}})} .  {({{\uniddcgcg} V2a2 \in {({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27c}}~{A\_27d}})} .  {({{\uniddcgcg} V3a3 \in {A\_27b} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{({{({{2}^{{A\_27b}}})}^{{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27c}}~{A\_27d}})}}})}^{{({{({{A\_27b}^{{({{A\_27b}^{{A\_27d}}})}}})}^{{A\_27c}}})}}})}^{{({{A\_27b}^{{A\_27a}}})}}})}})}}~{({{\uniddcgcg} V4relrec\_27 \in {({{({{({{({{2}^{{A\_27b}}})}^{{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27c}}~{A\_27d}})}}})}^{{({{({{A\_27b}^{{({{A\_27b}^{{A\_27d}}})}}})}^{{A\_27c}}})}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{A\_27b}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V5a0 \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{A\_27b}^{{({{A\_27b}^{{A\_27d}}})}}})}^{{A\_27c}}})}})}}~{({{\uniddcgcg} V6a1 \in {({{({{A\_27b}^{{({{A\_27b}^{{A\_27d}}})}}})}^{{A\_27c}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27c}}~{A\_27d}})}})}}~{({{\uniddcgcg} V7a2 \in {({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27c}}~{A\_27d}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V8a3 \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V9a \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27c}}~{A\_27d}})}})}}~{V7a2}})}}~{({{{ap}~{({{{{\mname{c\_2Einftree\_2EiLf}}~{A\_27a}}~{A\_27c}}~{A\_27d}})}}~{V9a}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V8a3}})}}~{({{{ap}~{V5a0}}~{V9a}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27c}})}}~{({{\uniddcgcg} V10b \in {A\_27c} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27c}}~{A\_27d}})}^{{A\_27d}}})}})}}~{({{\uniddcgcg} V11df \in {({{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27c}}~{A\_27d}})}^{{A\_27d}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{A\_27b}^{{A\_27d}}})}})}}~{({{\uniddcgcg} V12g \in {({{A\_27b}^{{A\_27d}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27c}}~{A\_27d}})}})}}~{V7a2}})}}~{({{{ap}~{({{{ap}~{({{{{\mname{c\_2Einftree\_2EiNd}}~{A\_27a}}~{A\_27c}}~{A\_27d}})}}~{V10b}})}}~{V11df}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V8a3}})}}~{({{{ap}~{({{{ap}~{V6a1}}~{V10b}})}}~{V12g}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27d}})}}~{({{\uniddcgcg} V13d \in {A\_27d} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{V4relrec\_27}}~{V5a0}})}}~{V6a1}})}}~{({{{ap}~{V11df}}~{V13d}})}})}}~{({{{ap}~{V12g}}~{V13d}})}})}})}})}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{V4relrec\_27}}~{V5a0}})}}~{V6a1}})}}~{V7a2}})}}~{V8a3}})}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{V4relrec\_27}}~{V0a0}})}}~{V1a1}})}}~{V2a2}})}}~{V3a3}})}})}})}})}})}})}})}})}}}}}$.
\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} 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} V0x \in 2 .  ({\unicdcdadad} V1y \in 2 .  ({\unicdcdadad} V2z \in 2 .  ({\unicdcdadad} V3w \in \\
2 .  ((((p~V0x){\unicdbdjdcd}(p~V1y)){\unicdcdcdhd}((p~V2z){\unicdbdjdcd}(p~V3w))){\unicdbdjdcd}(((p~V0x){\unicdcdcdhd}(p~V2z)){\unicdbdjdcd}\\
((p~V1y){\unicdcdcdhd}(p~V3w))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfenepeoepepfpfbeoeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in 2 .  ({\unicdcdadad} V1y \in 2 .  ({\unicdcdadad} V2z \in 2 .  ({\unicdcdadad} V3w \in \\
2 .  ((((p~V0x){\unicdbdjdcd}(p~V1y)){\unicdcdcdhd}((p~V2z){\unicdbdjdcd}(p~V3w))){\unicdbdjdcd}(((p~V0x){\unicdcdcdid}(p~V2z)){\unicdbdjdcd}\\
((p~V1y){\unicdcdcdid}(p~V3w))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfenepeoepepfpfpecf}
\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^{{A\_27a}}) .  (({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~V0P~V2x)){\unicdbdjdcd}(p~(ap~V1Q~V2x)))){\unicdbdjdcd}\\
(({\unicdcdadad} V3x \in A\_27a .  (p~(ap~V0P~V3x))){\unicdbdjdcd}({\unicdcdadad} V4x \in A\_27a .  (p~(\\
ap~V1Q~V4x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfenepeoepepfpfbememe}
\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^{{A\_27a}}) .  (({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~V0P~V2x)){\unicdbdjdcd}(p~(ap~V1Q~V2x)))){\unicdbdjdcd}\\
(({\unicdcdaddd} V3x \in A\_27a .  (p~(ap~V0P~V3x))){\unicdbdjdcd}({\unicdcdaddd} V4x \in A\_27a .  (p~(\\
ap~V1Q~V4x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfenepeoepepfpffeifjedfefdf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfejgogggehchfgfgpfcdfechfgmgchfgdgpfpfdgbgdhfgdh}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0a0 \in (\\
A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1a1 \in ((A\_27b^{{({{A\_27b}^{{A\_27d}}})}})^{{A\_27c}}) .  ({\unicdcdadad} V2a2 \in \\
(ty\_2Einftree\_2Einftree~A\_27a~A\_27c~A\_27d) .  ({\unicdcdadad} V3a3 \in A\_27b .  \\
((p~(ap~(ap~(ap~(ap~(\mname{c\_2Einftree\_2Erelrec}~A\_27a~A\_27b~A\_27c~A\_27d)~\\
V0a0)~V1a1)~V2a2)~V3a3)){\unicdbdjded}(({\unicdcdaddd} V4a \in A\_27a .  ((V2a2=(ap~(\mname{c\_2Einftree\_2EiLf}~\\
A\_27a~A\_27c~A\_27d)~V4a)){\unicdcdcdhd}(V3a3=(ap~V0a0~V4a)))){\unicdcdcdid}({\unicdcdaddd} V5b \in \\
A\_27c .  ({\unicdcdaddd} V6df \in ((ty\_2Einftree\_2Einftree~A\_27a~A\_27c~A\_27d)^{{A\_27d}}) .  \\
({\unicdcdaddd} V7g \in (A\_27b^{{A\_27d}}) .  ((V2a2=(ap~(ap~(\mname{c\_2Einftree\_2EiNd}~\\
A\_27a~A\_27c~A\_27d)~V5b)~V6df)){\unicdcdcdhd}((V3a3=(ap~(ap~V1a1~V5b)~V7g)){\unicdcdcdhd}(\\
{\unicdcdadad} V8d \in A\_27d .  (p~(ap~(ap~(ap~(ap~(\mname{c\_2Einftree\_2Erelrec}~A\_27a~\\
A\_27b~A\_27c~A\_27d)~V0a0)~V1a1)~(ap~V6df~V8d))~(ap~V7g~V8d)))))))))))))))\end{array}$$
\end{theorem}
\end{document}
