\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\_2EsplitAtPki\_\_EQN (TMSMjJnazMtW3i9u1bkjWnMfCeMZMo7dkLJ)}
\maketitle
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
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\_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}
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 $ c\_2Elist\_2EEL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EEL~A\_27a \in ((A\_27a^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfefeme}
\end{equation}
Let $ c\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\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:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
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}
\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}
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: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\_2Earithmetic\_2E\_2D : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2D \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdee}
\end{equation}
Let $ c\_2Elist\_2ETAKE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ETAKE~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeefbelefe}
\end{equation}
Let $ c\_2Elist\_2EDROP : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EDROP~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeeecfpeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfepg}
We define $\mname{c\_2Ecombin\_2Eo}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27c}}})} .  {{\uniddcgcg} V1g \in {({{A\_27c}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{V0f}}~{({{{ap}~{V1g}}~{V2x}})}})}})}}}}}}$.
\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: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\_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\_2EsplitAtPki : {{\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\_2EsplitAtPki~\\
A\_27a~A\_27b \in (((A\_27a^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}})^{{({{({{A\_27a}^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}}})}^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}}})}})^{{({{({{2}^{{A\_27b}}})}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfedhahmgjgehbeehaflgjg}
\end{equation}
\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}
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}
Let $ c\_2Eoption\_2Eoption\_\_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\_2Eoption\_2Eoption\_\_CASE~\\
A\_27a~A\_27b \in (((A\_27b^{{({{A\_27b}^{{A\_27a}}})}})^{{A\_27b}})^{{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfepgahehjgpgogpfpfdebedffe}
\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: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}
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: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\_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: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: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: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: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\_2Ewhile\_2EWHILE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ewhile\_2EWHILE~A\_27a \in (((A\_27a^{{A\_27a}})^{{({{A\_27a}^{{A\_27a}}})}})^{{({{2}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfehhigjgmgfgpfcdfehfiejemefe}
\end{equation}
\begin{definition}\label{def:dgpfcdfehhigjgmgfgpfcdfemefebedfef}
We define $\mname{c\_2Ewhile\_2ELEAST}$ to be ${{\uniddcgcg} V0P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ewhile\_2EWHILE}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2Eo}}~{ty\_2Enum\_2Enum}}~{2}}~{2}})}}~{\mname{c\_2Ebool\_2E\_7E}}})}}~{V0P}})}})}}~{\mname{c\_2Enum\_2ESUC}}})}}~{\mname{c\_2Enum\_2E0}}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhigjgmgfgpfcdfepemefebedfef}
We define $\mname{c\_2Ewhile\_2EOLEAST}$ to be ${{\uniddcgcg} V0P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Eoption\_2Eoption}~{ty\_2Enum\_2Enum}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{V0P}}~{V1n}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{\mname{c\_2Ewhile\_2ELEAST}}}~{({{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{V0P}}~{V2n}})}})}})}})}})}}~{({{\mname{c\_2Eoption\_2ENONE}}~{ty\_2Enum\_2Enum}})}})}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((V0m=\mname{c\_2Enum\_2E0}){\unicdcdcdid}({\unicdcdaddd} V1n \in \\
ty\_2Enum\_2Enum .  (V0m=(ap~\mname{c\_2Enum\_2ESUC}~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeogfhngpfpfdebedffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((\preoh (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
\mname{c\_2Enum\_2E0})~V0n))){\unicdbdjded}(V0n=\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeoepeefpfpfmeefpfpfkffecfpepfpffebfpfpfkffecfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(ap~\mname{c\_2Enum\_2ESUC}~V0m))~(ap~\mname{c\_2Enum\_2ESUC}~\\
V1n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0m)~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfnepeoepepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~(\\
ap~\mname{c\_2Enum\_2ESUC}~V0m))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))=V0m))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffdepfpfdfffcebd}
\end{equation}
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}
{({{\unicdcdadad} V0t \in {2} .  {({{False}{\unicdbdjdcd}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegebemedfjeefjf}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({{p}~{V0t}})}{\unicdcdcdid}{({\preoh{{({{p}~{V0t}})}}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifdemeffeefeeepfpfnejeeeeemefe}
\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}
\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} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((p~V0A){\unicdcdcdhd}(p~V1B))){\unicdbdjded}((\preoh (\\
p~V0A)){\unicdcdcdid}(\preoh (p~V1B)))){\unicdcdcdhd}((\preoh ((p~V0A){\unicdcdcdid}(p~V1B))){\unicdbdjded}((\preoh (p~V0A)){\unicdcdcdhd}(\preoh (p~V1B)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeefepfpfnepecfhebeoepfpfefiene}
\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} 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 .  ({\unicdcdadad} V1Q \in 2 .  \\
({\unicdcdadad} V2x \in A\_27a .  ({\unicdcdadad} V3x\_27 \in A\_27a .  ({\unicdcdadad} V4y \in A\_27a .  \\
({\unicdcdadad} V5y\_27 \in A\_27a .  ((((p~V0P){\unicdbdjded}(p~V1Q)){\unicdcdcdhd}(((p~V1Q){\unicdbdjdcd}(V2x=V3x\_27)){\unicdcdcdhd}\\
((\preoh (p~V1Q)){\unicdbdjdcd}(V4y=V5y\_27)))){\unicdbdjdcd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\\
V0P)~V2x)~V4y)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~V1Q)~V3x\_27)~\\
V5y\_27)))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdepeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0t1 \in A\_27a .  ({\unicdcdadad} V1t2 \in \\
A\_27a .  ((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2ET})~V0t1)~\\
V1t2)=V0t1))){\unicdcdcdhd}({\unicdcdadad} V2t1 \in A\_27a .  ({\unicdcdadad} V3t2 \in A\_27a .  ((ap~\\
(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2EF})~V2t1)~V3t2)=V3t2))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecgpgpgmgpfpfdgbgdhfgpfpfehigng}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1g \in (A\_27a^{{A\_27c}}) .  \\
({\unicdcdadad} V2x \in A\_27c .  ((ap~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27c~A\_27b~A\_27a)~\\
V0f)~V1g)~V2x)=(ap~V0f~(ap~V1g~V2x))))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfepgpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0h \in A\_27a .  ({\unicdcdadad} V1t \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ((ap~(c\_2Elist\_2EHD~A\_27a)~(ap~(ap~(\\
c\_2Elist\_2ECONS~A\_27a)~V0h)~V1t))=V0h)))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfeieee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(((ap~(c\_2Elist\_2ELENGTH~A\_27a)~\\
(c\_2Elist\_2ENIL~A\_27a))=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}({\unicdcdadad} V0h \in A\_27a .  (\\
{\unicdcdadad} V1t \in (ty\_2Elist\_2Elist~A\_27a) .  ((ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V0h)~V1t))=(ap~\mname{c\_2Enum\_2ESUC}~\\
(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V1t))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemefeoeheefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}) .  \\
(((p~(ap~V0P~(c\_2Elist\_2ENIL~A\_27a))){\unicdcdcdhd}({\unicdcdadad} V1t \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((p~(ap~V0P~V1t)){\unicdbdjdcd}({\unicdcdadad} V2h \in A\_27a .  (p~(ap~V0P~(ap~(ap~(\\
c\_2Elist\_2ECONS~A\_27a)~V2h)~V1t))))))){\unicdbdjdcd}({\unicdcdadad} V3l \in (ty\_2Elist\_2Elist~\\
A\_27a) .  (p~(ap~V0P~V3l)))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemgjgdhehpfpfjeoeeeffdeef}
\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} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1l \in A\_27b .  ({\unicdcdadad} V2ls \in \\
(ty\_2Elist\_2Elist~A\_27b) .  (((ap~(c\_2Elist\_2EEL~A\_27a)~\mname{c\_2Enum\_2E0})=\\
(c\_2Elist\_2EHD~A\_27a)){\unicdcdcdhd}((ap~(ap~(c\_2Elist\_2EEL~A\_27b)~(ap~\mname{c\_2Enum\_2ESUC}~\\
V0n))~(ap~(ap~(c\_2Elist\_2ECONS~A\_27b)~V1l)~V2ls))=(ap~(ap~(c\_2Elist\_2EEL~\\
A\_27b)~V0n)~V2ls))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfefemepfpfchfgdhehchjgdgehfgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\\
(ap~(ap~(c\_2Elist\_2ETAKE~A\_27a)~V0n)~(c\_2Elist\_2ENIL~A\_27a))=\\
(c\_2Elist\_2ENIL~A\_27a)))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfeefbelefepfpfogjgmg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V1x \in A\_27a .  ({\unicdcdadad} V2xs \in (ty\_2Elist\_2Elist~A\_27a) .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~V0n)){\unicdbdjdcd}((ap~(ap~(c\_2Elist\_2ETAKE~\\
A\_27a)~V0n)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V1x)~V2xs))=(ap~(ap~\\
(c\_2Elist\_2ECONS~A\_27a)~V1x)~(ap~(ap~(c\_2Elist\_2ETAKE~A\_27a)~(\\
ap~(ap~c\_2Earithmetic\_2E\_2D~V0n)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~V2xs)))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfeefbelefepfpfdgpgogdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\\
(ap~(ap~(c\_2Elist\_2EDROP~A\_27a)~V0n)~(c\_2Elist\_2ENIL~A\_27a))=\\
(c\_2Elist\_2ENIL~A\_27a)))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfeeecfpeafpfpfogjgmg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V1x \in A\_27a .  ({\unicdcdadad} V2xs \in (ty\_2Elist\_2Elist~A\_27a) .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~V0n)){\unicdbdjdcd}((ap~(ap~(c\_2Elist\_2EDROP~\\
A\_27a)~V0n)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V1x)~V2xs))=(ap~(ap~\\
(c\_2Elist\_2EDROP~A\_27a)~(ap~(ap~c\_2Earithmetic\_2E\_2D~V0n)~(ap~\\
\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~\\
V2xs))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfeeecfpeafpfpfdgpgogdh}
\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) .  ((ap~(ap~(c\_2Elist\_2ETAKE~A\_27a)~\mname{c\_2Enum\_2E0})~V0l)=(c\_2Elist\_2ENIL~\\
A\_27a)))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfeefbelefepfpfad}
\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) .  ((ap~(ap~(c\_2Elist\_2EDROP~A\_27a)~\mname{c\_2Enum\_2E0})~V0l)=V0l))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfeeecfpeafpfpfad}
\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} V0P \in ((2^{{A\_27b}})^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1k \in ((A\_27a^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}}) .  \\
((ap~(ap~(ap~(c\_2Elist\_2EsplitAtPki~A\_27a~A\_27b)~V0P)~V1k)~(c\_2Elist\_2ENIL~\\
A\_27b))=(ap~(ap~V1k~(c\_2Elist\_2ENIL~A\_27b))~(c\_2Elist\_2ENIL~A\_27b))))){\unicdcdcdhd}\\
({\unicdcdadad} V2P \in ((2^{{A\_27b}})^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V3k \in ((A\_27a^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}}) .  \\
({\unicdcdadad} V4h \in A\_27b .  ({\unicdcdadad} V5t \in (ty\_2Elist\_2Elist~A\_27b) .  (\\
(ap~(ap~(ap~(c\_2Elist\_2EsplitAtPki~A\_27a~A\_27b)~V2P)~V3k)~(ap~(ap~\\
(c\_2Elist\_2ECONS~A\_27b)~V4h)~V5t))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~\\
A\_27a)~(ap~(ap~V2P~\mname{c\_2Enum\_2E0})~V4h))~(ap~(ap~V3k~(c\_2Elist\_2ENIL~\\
A\_27b))~(ap~(ap~(c\_2Elist\_2ECONS~A\_27b)~V4h)~V5t)))~(ap~(ap~(ap~(\\
c\_2Elist\_2EsplitAtPki~A\_27a~A\_27b)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~ty\_2Enum\_2Enum~\\
(2^{{A\_27b}})~ty\_2Enum\_2Enum)~V2P)~\mname{c\_2Enum\_2ESUC}))~({\uniddcgcg} V6p \in (ty\_2Elist\_2Elist~\\
A\_27b) .  ({\uniddcgcg} V7s \in (ty\_2Elist\_2Elist~A\_27b) .  (ap~(ap~V3k~(ap~(ap~\\
(c\_2Elist\_2ECONS~A\_27b)~V4h)~V6p))~V7s))))~V5t))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfedhahmgjgehbeehaflgjgpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\preoh ((ap~\mname{c\_2Enum\_2ESUC}~V0n)=\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfeogfhngpfcdfeoepeefpfpfdfffde}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0opt \in (ty\_2Eoption\_2Eoption~\\
A\_27a) .  ((V0opt=(\mname{c\_2Eoption\_2ENONE}~A\_27a)){\unicdcdcdid}({\unicdcdaddd} V1x \in A\_27a .  \\
(V0opt=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V1x)))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfepgahehjgpgogpfpfogdgigpgehpgngjh}
\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^{{A\_27a}}) .  ((ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~\\
A\_27a~A\_27b)~(\mname{c\_2Eoption\_2ENONE}~A\_27a))~V0v)~V1f)=V0v))){\unicdcdcdhd}({\unicdcdadad} V2x \in \\
A\_27a .  ({\unicdcdadad} V3v \in A\_27b .  ({\unicdcdadad} V4f \in (A\_27b^{{A\_27a}}) .  ((ap~(ap~\\
(ap~(c\_2Eoption\_2Eoption\_\_CASE~A\_27a~A\_27b)~(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V2x))~V3v)~V4f)=(ap~V4f~V2x))))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfepgahehjgpgogpfpfdgbgdhfgpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\preoh (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V0n)~\mname{c\_2Enum\_2E0}))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfeoepeefpfpfmefedfdfpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0n))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfemefedfdfpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Enum\_2Enum .  (((ap~\mname{c\_2Ewhile\_2EOLEAST}~({\uniddcgcg} V1n \in \\
ty\_2Enum\_2Enum .  (ap~(ap~(\mname{c\_2Emin\_2E\_3D}~ty\_2Enum\_2Enum)~V1n)~V0x)))=\\
(ap~(\mname{c\_2Eoption\_2ESOME}~ty\_2Enum\_2Enum)~V0x)){\unicdcdcdhd}(((ap~\mname{c\_2Ewhile\_2EOLEAST}~\\
({\uniddcgcg} V2n \in ty\_2Enum\_2Enum .  (ap~(ap~(\mname{c\_2Emin\_2E\_3D}~ty\_2Enum\_2Enum)~\\
V0x)~V2n)))=(ap~(\mname{c\_2Eoption\_2ESOME}~ty\_2Enum\_2Enum)~V0x)){\unicdcdcdhd}((\\
(ap~\mname{c\_2Ewhile\_2EOLEAST}~({\uniddcgcg} V3n \in ty\_2Enum\_2Enum .  \mname{c\_2Ebool\_2EF}))=\\
(\mname{c\_2Eoption\_2ENONE}~ty\_2Enum\_2Enum)){\unicdcdcdhd}((ap~\mname{c\_2Ewhile\_2EOLEAST}~\\
({\uniddcgcg} V4n \in ty\_2Enum\_2Enum .  \mname{c\_2Ebool\_2ET}))=(ap~(\mname{c\_2Eoption\_2ESOME}~\\
ty\_2Enum\_2Enum)~\mname{c\_2Enum\_2E0}))))))\end{array}
\label{hyp:ehigngpfcdfehhigjgmgfgpfcdfepemefebedfefpfpffebfoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0P \in (2^{{ty\_2Enum\_2Enum}}) .  (((ap~\mname{c\_2Ewhile\_2EOLEAST}~V0P)=\\
(\mname{c\_2Eoption\_2ENONE}~ty\_2Enum\_2Enum)){\unicdbdjded}({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  \\
(\preoh (p~(ap~V0P~V1n))))))\end{array}
\label{hyp:ehigngpfcdfehhigjgmgfgpfcdfepemefebedfefpfpffebfpfpfoepeoefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0P \in (2^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  \\
(((ap~\mname{c\_2Ewhile\_2EOLEAST}~V0P)=(ap~(\mname{c\_2Eoption\_2ESOME}~ty\_2Enum\_2Enum)~\\
V1n)){\unicdbdjded}((p~(ap~V0P~V1n)){\unicdcdcdhd}({\unicdcdadad} V2m \in ty\_2Enum\_2Enum .  ((p~(ap~(\\
ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V2m)~V1n)){\unicdbdjdcd}(\preoh (p~(ap~V0P~V2m)))))))))\end{array}
\label{hyp:ehigngpfcdfehhigjgmgfgpfcdfepemefebedfefpfpffebfpfpfdfpenefe}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfemgjgdhehpfcdfedhahmgjgehbeehaflgjgpfpffebfoe}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0P \in ((2^{{A\_27b}})^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1k \in ((A\_27a^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}}) .  \\
({\unicdcdadad} V2l \in (ty\_2Elist\_2Elist~A\_27b) .  ((ap~(ap~(ap~(c\_2Elist\_2EsplitAtPki~\\
A\_27a~A\_27b)~V0P)~V1k)~V2l)=(ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~\\
ty\_2Enum\_2Enum~A\_27a)~(ap~\mname{c\_2Ewhile\_2EOLEAST}~({\uniddcgcg} V3i \in ty\_2Enum\_2Enum .  \\
(ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V3i)~(ap~\\
(c\_2Elist\_2ELENGTH~A\_27b)~V2l)))~(ap~(ap~V0P~V3i)~(ap~(ap~(c\_2Elist\_2EEL~\\
A\_27b)~V3i)~V2l))))))~(ap~(ap~V1k~V2l)~(c\_2Elist\_2ENIL~A\_27b)))~\\
({\uniddcgcg} V4i \in ty\_2Enum\_2Enum .  (ap~(ap~V1k~(ap~(ap~(c\_2Elist\_2ETAKE~\\
A\_27b)~V4i)~V2l))~(ap~(ap~(c\_2Elist\_2EDROP~A\_27b)~V4i)~V2l))))))))\end{array}$$
\end{theorem}
\end{document}
