\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\_2EquantHeuristics\_2ELIST\_\_LENGTH\_\_COMPARE\_\_SUC (TMby6N9jZnaph6ctnTAUyCSH5uysTLEpxA5)}
\maketitle
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Enum\_2EREP\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EREP\_\_num} \in {({{omega}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfecffeafpfpfogfhng}
\end{equation}
Let $ c\_2Enum\_2ESUC\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2ESUC\_\_REP} \in {({{omega}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfedfffdepfpfcffeaf}
\end{equation}
Let $ c\_2Enum\_2EABS\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EABS\_\_num} \in {({{ty\_2Enum\_2Enum}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfebecedfpfpfogfhng}
\end{equation}
\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:dgpfcdfeogfhngpfcdfedfffde}
We define $\mname{c\_2Enum\_2ESUC}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{({{{ap}~{c\_2Enum\_2ESUC\_\_REP}}~{({{{ap}~{c\_2Enum\_2EREP\_\_num}}~{V0m}})}})}})}}$.
\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}
Let $ c\_2Earithmetic\_2E\_2B : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2B \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdce}
\end{equation}
\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}
\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: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: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:dgpfcdfeahchjgngpfpfchfgdgpfcdfepfddde}
We define $\mname{c\_2Eprim\_\_rec\_2E\_3C}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V2P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V2P}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V3n}})}})}})}}~{({{{ap}~{V2P}}~{V3n}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V2P}}~{V0m}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V2P}}~{V1n}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfddfe}
We define $\mname{c\_2Earithmetic\_2E\_3E}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V1n}})}}~{V0m}})}}}$.
\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:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfddfepfddee}
We define $\mname{c\_2Earithmetic\_2E\_3E\_3D}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2E\_3E}}}~{V0m}})}}~{V1n}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{V1n}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfdddepfddee}
We define $\mname{c\_2Earithmetic\_2E\_3C\_3D}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V0m}})}}~{V1n}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{V1n}})}})}}}$.
\end{definition}
Let $ c\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\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\_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\_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}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefbd}
We define $\mname{c\_2Earithmetic\_2EBIT1}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
Let $ c\_2Elist\_2ELENGTH : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ELENGTH~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfemefeoeheefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Enum\_2ESUC}~V0m)=(ap~(ap~\\
c\_2Earithmetic\_2E\_2B~V0m)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeebd}
\end{equation}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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} 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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0l \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V1x \in ty\_2Enum\_2Enum .  ((((ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V0l)=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO}))){\unicdbdjded}({\unicdcdaddd} V2e1 \in A\_27a .  (V0l=(ap~(ap~\\
(c\_2Elist\_2ECONS~A\_27a)~V2e1)~(c\_2Elist\_2ENIL~A\_27a))))){\unicdcdcdhd}((\\
((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))=\\
(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l)){\unicdbdjded}({\unicdcdaddd} V3e1 \in A\_27a .  (V0l=\\
(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V3e1)~(c\_2Elist\_2ENIL~A\_27a))))){\unicdcdcdhd}\\
(((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V0l))){\unicdbdjded}({\unicdcdaddd} V4l\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V5e1 \in \\
A\_27a .  (V0l=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V5e1)~V4l\_27))))){\unicdcdcdhd}\\
(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~(ap~(c\_2Elist\_2ELENGTH~A\_27a)~\\
V0l))~\mname{c\_2Enum\_2E0})){\unicdbdjded}({\unicdcdaddd} V6l\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  \\
({\unicdcdaddd} V7e1 \in A\_27a .  (V0l=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V7e1)~\\
V6l\_27))))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V0l))){\unicdbdjded}({\unicdcdaddd} V8l\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V9e1 \in \\
A\_27a .  (V0l=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V9e1)~V8l\_27))))){\unicdcdcdhd}\\
(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~(ap~(c\_2Elist\_2ELENGTH~A\_27a)~\\
V0l))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))){\unicdbdjded}({\unicdcdaddd} V10l\_27 \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdaddd} V11e1 \in A\_27a .  (V0l=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~\\
V11e1)~V10l\_27))))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\\
(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))~V1x))~(ap~(c\_2Elist\_2ELENGTH~A\_27a)~\\
V0l))){\unicdbdjded}({\unicdcdaddd} V12l\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V13e1 \in \\
A\_27a .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1x)~(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V12l\_27))){\unicdcdcdhd}(V0l=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V13e1)~\\
V12l\_27)))))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V0l))~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~V1x))){\unicdbdjded}\\
({\unicdcdaddd} V14l\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V15e1 \in A\_27a .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1x)~(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V14l\_27))){\unicdcdcdhd}(V0l=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V15e1)~\\
V14l\_27)))))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V1x)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO}))))~(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l))){\unicdbdjded}\\
({\unicdcdaddd} V16l\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V17e1 \in A\_27a .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1x)~(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V16l\_27))){\unicdcdcdhd}(V0l=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V17e1)~\\
V16l\_27)))))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V0l))~(ap~(ap~c\_2Earithmetic\_2E\_2B~V1x)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))){\unicdbdjded}({\unicdcdaddd} V18l\_27 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V19e1 \in A\_27a .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V1x)~(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V18l\_27))){\unicdcdcdhd}(V0l=(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V19e1)~V18l\_27)))))){\unicdcdcdhd}((((ap~(c\_2Elist\_2ELENGTH~A\_27a)~\\
V0l)=(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~V1x)){\unicdbdjded}(\\
{\unicdcdaddd} V20l\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V21e1 \in A\_27a .  \\
(((ap~(c\_2Elist\_2ELENGTH~A\_27a)~V20l\_27)=V1x){\unicdcdcdhd}(V0l=(ap~(ap~(\\
c\_2Elist\_2ECONS~A\_27a)~V21e1)~V20l\_27)))))){\unicdcdcdhd}((((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~\\
V1x)=(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l)){\unicdbdjded}({\unicdcdaddd} V22l\_27 \in (\\
ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V23e1 \in A\_27a .  (((ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V22l\_27)=V1x){\unicdcdcdhd}(V0l=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V23e1)~\\
V22l\_27)))))){\unicdcdcdhd}((((ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l)=(ap~(ap~\\
c\_2Earithmetic\_2E\_2B~V1x)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))){\unicdbdjded}({\unicdcdaddd} V24l\_27 \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdaddd} V25e1 \in A\_27a .  (((ap~(c\_2Elist\_2ELENGTH~A\_27a)~\\
V24l\_27)=V1x){\unicdcdcdhd}(V0l=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V25e1)~V24l\_27)))))){\unicdcdcdhd}\\
((((ap~(ap~c\_2Earithmetic\_2E\_2B~V1x)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))=(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V0l)){\unicdbdjded}({\unicdcdaddd} V26l\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V27e1 \in \\
A\_27a .  (((ap~(c\_2Elist\_2ELENGTH~A\_27a)~V26l\_27)=V1x){\unicdcdcdhd}(V0l=(\\
ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V27e1)~V26l\_27)))))){\unicdcdcdhd}((((ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V0l)=\mname{c\_2Enum\_2E0}){\unicdbdjded}(V0l=(c\_2Elist\_2ENIL~A\_27a))){\unicdcdcdhd}((\\
(\mname{c\_2Enum\_2E0}=(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l)){\unicdbdjded}(V0l=(c\_2Elist\_2ENIL~\\
A\_27a))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V0l))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))){\unicdbdjded}(V0l=(c\_2Elist\_2ENIL~A\_27a))){\unicdcdcdhd}\\
(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V0l))){\unicdbdjded}(V0l=(c\_2Elist\_2ENIL~A\_27a))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~\\
\mname{c\_2Enum\_2E0})~(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l))){\unicdbdjded}(V0l=(c\_2Elist\_2ENIL~\\
A\_27a))){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V0l))~\mname{c\_2Enum\_2E0})){\unicdbdjded}(V0l=(c\_2Elist\_2ENIL~A\_27a))))))))))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfemejedfefpfpfmefeoeheefiepfpfbd}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfemejedfefpfpfmefeoeheefiepfpfdepeneafbecffepfpfdfffde}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V1l \in (ty\_2Elist\_2Elist~A\_27a) .  (((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0x))~(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V1l))){\unicdbdjded}\\
({\unicdcdaddd} V2l\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V3e1 \in A\_27a .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0x)~(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V2l\_27))){\unicdcdcdhd}(V1l=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V3e1)~V2l\_27)))))){\unicdcdcdhd}\\
(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~(ap~(c\_2Elist\_2ELENGTH~A\_27a)~\\
V1l))~(ap~\mname{c\_2Enum\_2ESUC}~V0x))){\unicdbdjded}({\unicdcdaddd} V4l\_27 \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdaddd} V5e1 \in A\_27a .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0x)~(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V4l\_27))){\unicdcdcdhd}(V1l=(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V5e1)~V4l\_27)))))){\unicdcdcdhd}((((ap~(c\_2Elist\_2ELENGTH~A\_27a)~V1l)=\\
(ap~\mname{c\_2Enum\_2ESUC}~V0x)){\unicdbdjded}({\unicdcdaddd} V6l\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  \\
({\unicdcdaddd} V7e1 \in A\_27a .  (((ap~(c\_2Elist\_2ELENGTH~A\_27a)~V6l\_27)=\\
V0x){\unicdcdcdhd}(V1l=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V7e1)~V6l\_27)))))){\unicdcdcdhd}\\
(((ap~\mname{c\_2Enum\_2ESUC}~V0x)=(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V1l)){\unicdbdjded}\\
({\unicdcdaddd} V8l\_27 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdaddd} V9e1 \in A\_27a .  \\
(((ap~(c\_2Elist\_2ELENGTH~A\_27a)~V8l\_27)=V0x){\unicdcdcdhd}(V1l=(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V9e1)~V8l\_27)))))))))))\end{array}$$
\end{theorem}
\end{document}
