\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\_2Elist\_2EisPREFIX\_\_THM (TMcLsSYBZuGES8J6Atm81thvxZpzDSFnm1H)}
\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\_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}
Let $ c\_2Elist\_2ECONS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ECONS~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfedepeoedf}
\end{equation}
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}
Let $ c\_2Elist\_2EisPREFIX : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EisPREFIX~A\_27a \in ((2^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfejgdhafcffegejeif}
\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}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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}
\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} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
({\unicdcdadad} V0v \in A\_27b .  ({\unicdcdadad} V1f \in ((A\_27b^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{A\_27a}}) .  \\
((ap~(ap~(ap~(c\_2Elist\_2Elist\_\_CASE~A\_27a~A\_27b)~(c\_2Elist\_2ENIL~\\
A\_27a))~V0v)~V1f)=V0v))){\unicdcdcdhd}({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3a1 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V4v \in A\_27b .  ({\unicdcdadad} V5f \in (\\
(A\_27b^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{A\_27a}}) .  ((ap~(ap~(ap~(c\_2Elist\_2Elist\_\_CASE~\\
A\_27a~A\_27b)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V2a0)~V3a1))~V4v)~V5f)=\\
(ap~(ap~V5f~V2a0)~V3a1)))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemgjgdhehpfpfdgbgdhfgpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0l \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((p~(ap~(ap~(c\_2Elist\_2EisPREFIX~A\_27a)~(c\_2Elist\_2ENIL~\\
A\_27a))~V0l)){\unicdbdjded}True)){\unicdcdcdhd}({\unicdcdadad} V1h \in A\_27a .  ({\unicdcdadad} V2t \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V3l \in (ty\_2Elist\_2Elist~A\_27a) .  ((p~(ap~(ap~(c\_2Elist\_2EisPREFIX~\\
A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V1h)~V2t))~V3l)){\unicdbdjded}(p~(ap~(\\
ap~(ap~(c\_2Elist\_2Elist\_\_CASE~A\_27a~2)~V3l)~\mname{c\_2Ebool\_2EF})~({\uniddcgcg} V4h\_27 \in \\
A\_27a .  ({\uniddcgcg} V5t\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  (ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~\\
(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~A\_27a)~V1h)~V4h\_27))~(ap~(ap~(c\_2Elist\_2EisPREFIX~\\
A\_27a)~V2t)~V5t\_27)))))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfejgdhafcffegejeif}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfemgjgdhehpfcdfejgdhafcffegejeifpfpfefiene}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0l \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V1h \in A\_27a .  ({\unicdcdadad} V2t \in (ty\_2Elist\_2Elist~A\_27a) .  \\
({\unicdcdadad} V3h1 \in A\_27a .  ({\unicdcdadad} V4t1 \in (ty\_2Elist\_2Elist~A\_27a) .  \\
({\unicdcdadad} V5h2 \in A\_27a .  ({\unicdcdadad} V6t2 \in (ty\_2Elist\_2Elist~A\_27a) .  \\
(((p~(ap~(ap~(c\_2Elist\_2EisPREFIX~A\_27a)~(c\_2Elist\_2ENIL~A\_27a))~\\
V0l)){\unicdbdjded}True){\unicdcdcdhd}(((p~(ap~(ap~(c\_2Elist\_2EisPREFIX~A\_27a)~(ap~(ap~(\\
c\_2Elist\_2ECONS~A\_27a)~V1h)~V2t))~(c\_2Elist\_2ENIL~A\_27a))){\unicdbdjded}False){\unicdcdcdhd}\\
((p~(ap~(ap~(c\_2Elist\_2EisPREFIX~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V3h1)~V4t1))~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V5h2)~V6t2))){\unicdbdjded}\\
((V3h1=V5h2){\unicdcdcdhd}(p~(ap~(ap~(c\_2Elist\_2EisPREFIX~A\_27a)~V4t1)~V6t2)))))))))))))\end{array}$$
\end{theorem}
\end{document}
