\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\_2Epath\_2EPL\_\_take (TMGvkJ4kSi6wT8FpmQWpN6jHhTZeXrVBfbw)}
\maketitle
\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}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Earithmetic\_2EEVEN : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Earithmetic\_2EEVEN} \in {({{2}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfefegffeoe}
\end{equation}
Let $ c\_2Earithmetic\_2EODD : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Earithmetic\_2EODD} \in {({{2}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepeeeee}
\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: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: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}
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: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: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}
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}
\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}
\begin{definition}\label{def:dgpfcdfeahchjgngpfpfchfgdgpfcdfeafcffe}
We define $\mname{c\_2Eprim\_\_rec\_2EPRE}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{\mname{c\_2Enum\_2E0}}})}})}}~{\mname{c\_2Enum\_2E0}}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V1n}})}})}})}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2EEXP : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EEXP \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfefeifaf}
\end{equation}
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\_2Earithmetic\_2E\_2A : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2A \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdbe}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngfgchbgmgpfcdfejgkf}
We define $\mname{c\_2Enumeral\_2EiZ}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\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\_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:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefcd}
We define $\mname{c\_2Earithmetic\_2EBIT2}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}})}}$.
\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:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\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 $ 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}
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 $ 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\_\_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}
Let $ ty\_2Epair\_2Eprod : {{\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\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ ty\_2Ellist\_2Ellist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ellist\_2Ellist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgmgjgdhehpfcdfemgmgjgdheh}
\end{equation}
Let $ ty\_2Epath\_2Epath : {{\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\_2Epath\_2Epath~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgehigpfcdfeahbgehig}
\end{equation}
Let $ c\_2Epath\_2EfromPath : {{\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\_2Epath\_2EfromPath~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~(ty\_2Ellist\_2Ellist~(ty\_2Epair\_2Eprod~\\
A\_27b~A\_27a)))^{{({{{ty\_2Epath\_2Epath}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehigpfcdfeggchpgngafbgehig}
\end{equation}
Let $ c\_2Epair\_2ESND : {{\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\_2Epair\_2ESND~\\
A\_27a~A\_27b \in (A\_27b^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfedfoeee}
\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\_2Ellist\_2Ellist\_\_rep : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ellist\_2Ellist\_\_rep~A\_27a \in \\
(((ty\_2Eoption\_2Eoption~A\_27a)^{{ty\_2Enum\_2Enum}})^{{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgmgjgdhehpfcdfemgmgjgdhehpfpfchfgah}
\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}
\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\_2Ellist\_2Ellist\_\_abs : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ellist\_2Ellist\_\_abs~A\_27a \in \\
((ty\_2Ellist\_2Ellist~A\_27a)^{{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgmgjgdhehpfcdfemgmgjgdhehpfpfbgcgdh}
\end{equation}
\begin{definition}\label{def:dgpfcdfemgmgjgdhehpfcdfemedepeoedf}
We define $\mname{c\_2Ellist\_2ELCONS}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0h \in {A\_27a} .  {{\uniddcgcg} V1t \in {({{ty\_2Ellist\_2Ellist}~{A\_27a}})} .  {({{{ap}~{({{c\_2Ellist\_2Ellist\_\_abs}~{A\_27a}})}}~{({{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V2n}})}}~{\mname{c\_2Enum\_2E0}}})}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{A\_27a}})}}~{V0h}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Ellist\_2Ellist\_\_rep}~{A\_27a}})}}~{V1t}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{V2n}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfemgmgjgdhehpfcdfemeoejeme}
We define $\mname{c\_2Ellist\_2ELNIL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{c\_2Ellist\_2Ellist\_\_abs}~{A\_27a}})}}~{({{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{\mname{c\_2Eoption\_2ENONE}}~{A\_27a}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfemgmgjgdhehpfcdfemgmgfgoghgehigpfpfchfgmg}
We define $\mname{c\_2Ellist\_2Ellength\_\_rel}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0a0 \in {({{ty\_2Ellist\_2Ellist}~{A\_27a}})} .  {({{\uniddcgcg} V1a1 \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{2}^{{ty\_2Enum\_2Enum}}})}^{{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}}})}})}}~{({{\uniddcgcg} V2llength\_\_rel\_27 \in {({{({{2}^{{ty\_2Enum\_2Enum}}})}^{{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})}}~{({{\uniddcgcg} V3a0 \in {({{ty\_2Ellist\_2Ellist}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V4a1 \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})}}~{V3a0}})}}~{({{\mname{c\_2Ellist\_2ELNIL}}~{A\_27a}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V4a1}})}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V5h \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V6n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})}}~{({{\uniddcgcg} V7t \in {({{ty\_2Ellist\_2Ellist}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})}}~{V3a0}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ellist\_2ELCONS}}~{A\_27a}})}}~{V5h}})}}~{V7t}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V4a1}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V6n}})}})}})}}~{({{{ap}~{({{{ap}~{V2llength\_\_rel\_27}}~{V7t}})}}~{V6n}})}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V2llength\_\_rel\_27}}~{V3a0}})}}~{V4a1}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V2llength\_\_rel\_27}}~{V0a0}})}}~{V1a1}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfemgmgjgdhehpfcdfemegejeoejeeffe}
We define $\mname{c\_2Ellist\_2ELFINITE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0a0 \in {({{ty\_2Ellist\_2Ellist}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}}})}})}}~{({{\uniddcgcg} V1LFINITE\_27 \in {({{2}^{{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})}}~{({{\uniddcgcg} V2a0 \in {({{ty\_2Ellist\_2Ellist}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})}}~{V2a0}})}}~{({{\mname{c\_2Ellist\_2ELNIL}}~{A\_27a}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3h \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})}}~{({{\uniddcgcg} V4t \in {({{ty\_2Ellist\_2Ellist}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})}}~{V2a0}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ellist\_2ELCONS}}~{A\_27a}})}}~{V3h}})}}~{V4t}})}})}})}}~{({{{ap}~{V1LFINITE\_27}}~{V4t}})}})}})}})}})}})}})}})}}~{({{{ap}~{V1LFINITE\_27}}~{V2a0}})}})}})}})}})}}~{({{{ap}~{V1LFINITE\_27}}~{V0a0}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfemgmgjgdhehpfcdfememefeoeheefie}
We define $\mname{c\_2Ellist\_2ELLENGTH}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0ll \in {({{ty\_2Ellist\_2Ellist}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Eoption\_2Eoption}~{ty\_2Enum\_2Enum}})}})}}~{({{{ap}~{({{\mname{c\_2Ellist\_2ELFINITE}}~{A\_27a}})}}~{V0ll}})}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Ellist\_2Ellength\_\_rel}}~{A\_27a}})}}~{V0ll}})}}~{V1n}})}})}})}})}})}}~{({{\mname{c\_2Eoption\_2ENONE}}~{ty\_2Enum\_2Enum}})}})}}}$.
\end{definition}
Let $ c\_2Eoption\_2ETHE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eoption\_2ETHE~A\_27a \in (A\_27a^{{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfeefiefe}
\end{equation}
Let $ c\_2Ellist\_2ELTAKE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ellist\_2ELTAKE~A\_27a \in (((ty\_2Eoption\_2Eoption~\\
(ty\_2Elist\_2Elist~A\_27a))^{{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgmgjgdhehpfcdfemeefbelefe}
\end{equation}
\begin{definition}\label{def:dgpfcdfemgmgjgdhehpfcdfeehpgmejgdheh}
We define $\mname{c\_2Ellist\_2EtoList}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0ll \in {({{ty\_2Ellist\_2Ellist}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Eoption\_2Eoption}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ellist\_2ELFINITE}}~{A\_27a}})}}~{V0ll}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Ellist\_2ELTAKE}~{A\_27a}})}}~{({{{ap}~{({{c\_2Eoption\_2ETHE}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{\mname{c\_2Ellist\_2ELLENGTH}}~{A\_27a}})}}~{V0ll}})}})}})}}~{V0ll}})}})}}~{({{\mname{c\_2Eoption\_2ENONE}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}})}}}$.
\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}
\begin{definition}\label{def:dgpfcdfeahbgehigpfcdfeggjgogjgehfg}
We define $\mname{c\_2Epath\_2Efinite}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0sigma \in {({{{ty\_2Epath\_2Epath}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{\mname{c\_2Ellist\_2ELFINITE}}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27a}}~{({{ty\_2Ellist\_2Ellist}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}}~{({{{ap}~{({{{c\_2Epath\_2EfromPath}~{A\_27a}}~{A\_27b}})}}~{V0sigma}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahbgehigpfcdfemgfgoghgehig}
We define $\mname{c\_2Epath\_2Elength}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0p \in {({{{ty\_2Epath\_2Epath}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Eoption\_2Eoption}~{ty\_2Enum\_2Enum}})}})}}~{({{{ap}~{({{{\mname{c\_2Epath\_2Efinite}}~{A\_27a}}~{A\_27b}})}}~{V0p}})}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{({{{ap}~{({{c\_2Elist\_2ELENGTH}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}}~{({{{ap}~{({{c\_2Eoption\_2ETHE}~{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ellist\_2EtoList}}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27a}}~{({{ty\_2Ellist\_2Ellist}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}}~{({{{ap}~{({{{c\_2Epath\_2EfromPath}~{A\_27a}}~{A\_27b}})}}~{V0p}})}})}})}})}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}~{({{\mname{c\_2Eoption\_2ENONE}}~{ty\_2Enum\_2Enum}})}})}}}}$.
\end{definition}
Let $ c\_2Epair\_2EABS\_\_prod : {{\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\_2Epair\_2EABS\_\_prod~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfebecedfpfpfahchpgeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfepfcdde}
We define $\mname{c\_2Epair\_2E\_2C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{c\_2Epair\_2EABS\_\_prod}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V2a \in {A\_27a} .  {({{\uniddcgcg} V3b \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2a}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3b}})}}~{V1y}})}})}})}})}})}}}}}$.
\end{definition}
Let $ c\_2Epred\_\_set\_2EGSPEC : {{\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\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b \in ((2^{{A\_27a}})^{{({{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{2}})}^{{A\_27b}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffede}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgehigpfcdfeafme}
We define $\mname{c\_2Epath\_2EPL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0p \in {({{{ty\_2Epath\_2Epath}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Enum\_2Enum}}~{2}})}}~{V1i}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{\mname{c\_2Epath\_2Efinite}}~{A\_27a}}~{A\_27b}})}}~{V0p}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V1i}})}}~{({{{ap}~{({{c\_2Eoption\_2ETHE}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{\mname{c\_2Epath\_2Elength}}~{A\_27a}}~{A\_27b}})}}~{V0p}})}})}})}})}})}})}})}}}}$.
\end{definition}
Let $ c\_2Epath\_2Etail : {{\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\_2Epath\_2Etail~\\
A\_27a~A\_27b \in ((ty\_2Epath\_2Epath~A\_27a~A\_27b)^{{({{{ty\_2Epath\_2Epath}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehigpfcdfeehbgjgmg}
\end{equation}
Let $ c\_2Epath\_2Efirst\_\_label : {{\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\_2Epath\_2Efirst\_\_label~\\
A\_27a~A\_27b \in (A\_27b^{{({{{ty\_2Epath\_2Epath}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehigpfcdfeggjgchdhehpfpfmgbgcgfgmg}
\end{equation}
Let $ c\_2Epair\_2EFST : {{\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\_2Epair\_2EFST~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfegedfef}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgehigpfcdfeggjgchdheh}
We define $\mname{c\_2Epath\_2Efirst}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0p \in {({{{ty\_2Epath\_2Epath}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27a}}~{({{ty\_2Ellist\_2Ellist}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}}~{({{{ap}~{({{{c\_2Epath\_2EfromPath}~{A\_27a}}~{A\_27b}})}}~{V0p}})}})}}}}$.
\end{definition}
Let $ c\_2Epath\_2EtoPath : {{\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\_2Epath\_2EtoPath~\\
A\_27a~A\_27b \in ((ty\_2Epath\_2Epath~A\_27a~A\_27b)^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{({{ty\_2Ellist\_2Ellist}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehigpfcdfeehpgafbgehig}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgehigpfcdfeahdgpgogdh}
We define $\mname{c\_2Epath\_2Epcons}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1r \in {A\_27b} .  {{\uniddcgcg} V2p \in {({{{ty\_2Epath\_2Epath}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{c\_2Epath\_2EtoPath}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{({{ty\_2Ellist\_2Ellist}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}}~{V0x}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ellist\_2ELCONS}}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27b}}~{A\_27a}})}}~{V1r}})}}~{({{{ap}~{({{{\mname{c\_2Epath\_2Efirst}}~{A\_27a}}~{A\_27b}})}}~{V2p}})}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27a}}~{({{ty\_2Ellist\_2Ellist}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}}~{({{{ap}~{({{{c\_2Epath\_2EfromPath}~{A\_27a}}~{A\_27b}})}}~{V2p}})}})}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahbgehigpfcdfedhehpgahahfgegpfpfbgeh}
We define $\mname{c\_2Epath\_2Estopped\_\_at}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {({{{ap}~{({{{c\_2Epath\_2EtoPath}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{({{ty\_2Ellist\_2Ellist}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}}~{V0x}})}}~{({{\mname{c\_2Ellist\_2ELNIL}}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}})}}}}$.
\end{definition}
Let $ c\_2Epath\_2Etake : {{\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\_2Epath\_2Etake~\\
A\_27a~A\_27b \in (((ty\_2Epath\_2Epath~A\_27a~A\_27b)^{{({{{ty\_2Epath\_2Epath}~{A\_27a}}~{A\_27b}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgehigpfcdfeehbglgfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfejeoe}
We define $\mname{c\_2Ebool\_2EIN}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1f \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V1f}}~{V0x}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfejenebehefe}
We define $\mname{c\_2Epred\_\_set\_2EIMAGE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27b}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27b}}~{2}})}}~{({{{ap}~{V0f}}~{V2x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfefeneafefjf}
We define $\mname{c\_2Epred\_\_set\_2EEMPTY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2EF}}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfejeoedffecfef}
We define $\mname{c\_2Epred\_\_set\_2EINSERT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2y}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2y}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2y}})}}~{V1s}})}})}})}})}})}}}}$.
\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 .  ({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Enum\_2ESUC}~V0n))~(ap~\\
\mname{c\_2Enum\_2ESUC}~V1m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0n)~\\
V1m)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebfpfpfnepeoepe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
\mname{c\_2Enum\_2E0})~V0n)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpepfpfmefedfdfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0n)~\mname{c\_2Enum\_2E0})){\unicdbdjded}(V0n=\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebfpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
((ap~(ap~c\_2Earithmetic\_2E\_2A~\mname{c\_2Enum\_2E0})~V0m)=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}\\
(((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~\mname{c\_2Enum\_2E0})=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}\\
(((ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~V0m)=V0m){\unicdcdcdhd}\\
(((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))=V0m){\unicdcdcdhd}(\\
((ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Enum\_2ESUC}~V0m))~V1n)=(ap~\\
(ap~c\_2Earithmetic\_2E\_2B~(ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~V1n))~\\
V1n)){\unicdcdcdhd}((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~(ap~\mname{c\_2Enum\_2ESUC}~V1n))=\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~(ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V0m)~V1n))))))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeneffmeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  (((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0m)~V1n)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1n)~V2p))){\unicdbdjdcd}(p~(\\
ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0m)~V2p))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebfpfpfefcfbeoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n))~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0m)~V2p))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1n)~V2p))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfnepeoepepfpfmefedfdfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(\preoh (p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0m)~V1n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Enum\_2ESUC}~V1n))~V0m)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeoepeefpfpfmefebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Enum\_2ESUC}~V0n)=(ap~(ap~\\
c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))~V0n)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffdepfpfpeoefepfpfbeeeee}
\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}
\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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in 2 .  (({\unicdcdaddd} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifjedfefdfpfpfdfjeneaf}
\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}
{({{\unicdcdadad} V0t \in {2} .  {({{({{({{p}~{V0t}})}{\unicdbdjdcd}{False}})}{\unicdbdjded}{({{({{p}~{V0t}})}{\unicdbdjded}{False}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfgepfpffebfpfpfge}
\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} 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} V0P \in (2^{{ty\_2Enum\_2Enum}}) .  (((p~(ap~V0P~\mname{c\_2Enum\_2E0})){\unicdcdcdhd}\\
({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ((p~(ap~V0P~V1n)){\unicdbdjdcd}(p~(ap~V0P~(ap~\mname{c\_2Enum\_2ESUC}~\\
V1n)))))){\unicdbdjdcd}({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  (p~(ap~V0P~V2n)))))\end{array}
\label{hyp:ehigngpfcdfeogfhngpfcdfejeoeeeffdeefjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
\mname{c\_2Enum\_2E0})~V0n)=V0n)){\unicdcdcdhd}(({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ((ap~\\
(ap~c\_2Earithmetic\_2E\_2B~V1n)~\mname{c\_2Enum\_2E0})=V1n)){\unicdcdcdhd}(({\unicdcdadad} V2n \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V3m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V2n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V3m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Enumeral\_2EiZ}~(ap~\\
(ap~c\_2Earithmetic\_2E\_2B~V2n)~V3m)))))){\unicdcdcdhd}(({\unicdcdadad} V4n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2E\_2A~\mname{c\_2Enum\_2E0})~V4n)=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}\\
(({\unicdcdadad} V5n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V5n)~\mname{c\_2Enum\_2E0})=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V6n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V7m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~(\\
ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V6n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V7m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V6n)~V7m))))){\unicdcdcdhd}(({\unicdcdadad} V8n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
\mname{c\_2Enum\_2E0})~V8n)=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V9n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2E\_2D~V9n)~\mname{c\_2Enum\_2E0})=V9n)){\unicdcdcdhd}(({\unicdcdadad} V10n \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V11m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V10n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V11m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V10n)~V11m))))){\unicdcdcdhd}(({\unicdcdadad} V12n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EEXP~\\
\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V12n)))=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V13n \in ty\_2Enum\_2Enum .  ((ap~\\
(ap~c\_2Earithmetic\_2EEXP~\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V13n)))=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V14n \in \\
ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EEXP~V14n)~\mname{c\_2Enum\_2E0})=\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}\\
(({\unicdcdadad} V15n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V16m \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2EEXP~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V15n))~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V16m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~(ap~c\_2Earithmetic\_2EEXP~V15n)~V16m))))){\unicdcdcdhd}(((ap~\mname{c\_2Enum\_2ESUC}~\\
\mname{c\_2Enum\_2E0})=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}(({\unicdcdadad} V17n \in ty\_2Enum\_2Enum .  (\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V17n))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Enum\_2ESUC}~V17n)))){\unicdcdcdhd}(((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\mname{c\_2Enum\_2E0})=\\
\mname{c\_2Enum\_2E0}){\unicdcdcdhd}(({\unicdcdadad} V18n \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V18n))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~V18n)))){\unicdcdcdhd}(({\unicdcdadad} V19n \in ty\_2Enum\_2Enum .  \\
(((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V19n)=\mname{c\_2Enum\_2E0}){\unicdbdjded}(V19n=\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}\\
(({\unicdcdadad} V20n \in ty\_2Enum\_2Enum .  ((\mname{c\_2Enum\_2E0}=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V20n)){\unicdbdjded}(V20n=\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}(({\unicdcdadad} V21n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V22m \in ty\_2Enum\_2Enum .  (((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V21n)=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V22m)){\unicdbdjded}(V21n=V22m)))){\unicdcdcdhd}\\
(({\unicdcdadad} V23n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V23n)~\mname{c\_2Enum\_2E0})){\unicdbdjded}False)){\unicdcdcdhd}(({\unicdcdadad} V24n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V24n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Earithmetic\_2EZERO})~\\
V24n)))){\unicdcdcdhd}(({\unicdcdadad} V25n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V26m \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V25n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V26m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V25n)~V26m))))){\unicdcdcdhd}(({\unicdcdadad} V27n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~\\
\mname{c\_2Enum\_2E0})~V27n)){\unicdbdjded}False)){\unicdcdcdhd}(({\unicdcdadad} V28n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V28n))~\mname{c\_2Enum\_2E0})){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Earithmetic\_2EZERO})~\\
V28n)))){\unicdcdcdhd}(({\unicdcdadad} V29n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V30m \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V29n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V30m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V30m)~V29n))))){\unicdcdcdhd}(({\unicdcdadad} V31n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
\mname{c\_2Enum\_2E0})~V31n)){\unicdbdjded}True)){\unicdcdcdhd}(({\unicdcdadad} V32n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V32n))~\mname{c\_2Enum\_2E0})){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V32n)~\\
\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}(({\unicdcdadad} V33n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V34m \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V33n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V34m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V33n)~V34m))))){\unicdcdcdhd}(\\
({\unicdcdadad} V35n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~\\
V35n)~\mname{c\_2Enum\_2E0})){\unicdbdjded}True)){\unicdcdcdhd}(({\unicdcdadad} V36n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~\mname{c\_2Enum\_2E0})~V36n)){\unicdbdjded}(V36n=\\
\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(({\unicdcdadad} V37n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V38m \in \\
ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V37n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V38m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V38m)~V37n))))){\unicdcdcdhd}(({\unicdcdadad} V39n \in ty\_2Enum\_2Enum .  ((p~(ap~c\_2Earithmetic\_2EODD~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V39n))){\unicdbdjded}(p~(ap~c\_2Earithmetic\_2EODD~\\
V39n)))){\unicdcdcdhd}(({\unicdcdadad} V40n \in ty\_2Enum\_2Enum .  ((p~(ap~c\_2Earithmetic\_2EEVEN~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V40n))){\unicdbdjded}(p~(ap~c\_2Earithmetic\_2EEVEN~\\
V40n)))){\unicdcdcdhd}((\preoh (p~(ap~c\_2Earithmetic\_2EODD~\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(p~(\\
ap~c\_2Earithmetic\_2EEVEN~\mname{c\_2Enum\_2E0})))))))))))))))))))))))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfegjgdhehchjgcg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  (\\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\mname{c\_2Earithmetic\_2EZERO})~V0n)){\unicdbdjded}\\
True){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~\mname{c\_2Earithmetic\_2EZERO})){\unicdbdjded}False){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V0n))~\mname{c\_2Earithmetic\_2EZERO})){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0n)~V1m))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0n)~V1m))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m))){\unicdbdjded}(\preoh (p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V1m)~V0n)))){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0n)~V1m)))))))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfmgehfg}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2a \in A\_27a .  ({\unicdcdadad} V3b \in \\
A\_27b .  (((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y)=(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V2a)~V3b)){\unicdbdjded}((V0x=V2a){\unicdcdcdhd}(V1y=V3b)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeafbejecfpfpffebf}
\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^{{({{{ty\_2Epath\_2Epath}~{A\_27a}}~{A\_27b}})}}) .  (({\unicdcdadad} V1p \in \\
(ty\_2Epath\_2Epath~A\_27a~A\_27b) .  (p~(ap~V0P~V1p))){\unicdbdjded}(({\unicdcdadad} V2x \in \\
A\_27a .  (p~(ap~V0P~(ap~(\mname{c\_2Epath\_2Estopped\_\_at}~A\_27a~A\_27b)~V2x)))){\unicdcdcdhd}\\
({\unicdcdadad} V3x \in A\_27a .  ({\unicdcdadad} V4r \in A\_27b .  ({\unicdcdadad} V5p \in (ty\_2Epath\_2Epath~\\
A\_27a~A\_27b) .  (p~(ap~V0P~(ap~(ap~(ap~(\mname{c\_2Epath\_2Epcons}~A\_27a~A\_27b)~\\
V3x)~V4r)~V5p)))))))))\end{array}
\label{hyp:ehigngpfcdfeahbgehigpfcdfegepecfbememepfpfahbgehig}
\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} V0x \in A\_27a .  ((ap~(\mname{c\_2Epath\_2Efirst}~A\_27a~A\_27b)~(ap~(\mname{c\_2Epath\_2Estopped\_\_at}~\\
A\_27a~A\_27b)~V0x))=V0x)){\unicdcdcdhd}({\unicdcdadad} V1x \in A\_27a .  ({\unicdcdadad} V2r \in A\_27b .  \\
({\unicdcdadad} V3p \in (ty\_2Epath\_2Epath~A\_27a~A\_27b) .  ((ap~(\mname{c\_2Epath\_2Efirst}~\\
A\_27a~A\_27b)~(ap~(ap~(ap~(\mname{c\_2Epath\_2Epcons}~A\_27a~A\_27b)~V1x)~V2r)~\\
V3p))=V1x)))))\end{array}
\label{hyp:ehigngpfcdfeahbgehigpfcdfeggjgchdhehpfpfehigng}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1r \in A\_27b .  ({\unicdcdadad} V2p \in (ty\_2Epath\_2Epath~\\
A\_27a~A\_27b) .  ((ap~(c\_2Epath\_2Etail~A\_27a~A\_27b)~(ap~(ap~(ap~(\mname{c\_2Epath\_2Epcons}~\\
A\_27a~A\_27b)~V0x)~V1r)~V2p))=V2p))))\end{array}
\label{hyp:ehigngpfcdfeahbgehigpfcdfeehbgjgmgpfpfegfggg}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1r \in A\_27b .  ({\unicdcdadad} V2p \in (ty\_2Epath\_2Epath~\\
A\_27a~A\_27b) .  ((ap~(c\_2Epath\_2Efirst\_\_label~A\_27a~A\_27b)~(ap~(\\
ap~(ap~(\mname{c\_2Epath\_2Epcons}~A\_27a~A\_27b)~V0x)~V1r)~V2p))=V1r))))\end{array}
\label{hyp:ehigngpfcdfeahbgehigpfcdfeggjgchdhehpfpfmgbgcgfgmgpfpfegfggg}
\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} V0x \in A\_27a .  ((ap~(\mname{c\_2Epath\_2EPL}~A\_27a~A\_27b)~(ap~(\mname{c\_2Epath\_2Estopped\_\_at}~\\
A\_27a~A\_27b)~V0x))=(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~ty\_2Enum\_2Enum)~\\
\mname{c\_2Enum\_2E0})~(\mname{c\_2Epred\_\_set\_2EEMPTY}~ty\_2Enum\_2Enum)))){\unicdcdcdhd}({\unicdcdadad} V1x \in \\
A\_27a .  ({\unicdcdadad} V2r \in A\_27b .  ({\unicdcdadad} V3q \in (ty\_2Epath\_2Epath~A\_27a~\\
A\_27b) .  ((ap~(\mname{c\_2Epath\_2EPL}~A\_27a~A\_27b)~(ap~(ap~(ap~(\mname{c\_2Epath\_2Epcons}~\\
A\_27a~A\_27b)~V1x)~V2r)~V3q))=(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~ty\_2Enum\_2Enum)~\\
\mname{c\_2Enum\_2E0})~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EIMAGE}~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)~\\
\mname{c\_2Enum\_2ESUC})~(ap~(\mname{c\_2Epath\_2EPL}~A\_27a~A\_27b)~V3q))))))))\end{array}
\label{hyp:ehigngpfcdfeahbgehigpfcdfeafmepfpfehigng}
\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 (ty\_2Epath\_2Epath~A\_27a~A\_27b) .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
ty\_2Enum\_2Enum)~\mname{c\_2Enum\_2E0})~(ap~(\mname{c\_2Epath\_2EPL}~A\_27a~A\_27b)~\\
V0p))))\end{array}
\label{hyp:ehigngpfcdfeahbgehigpfcdfeafmepfpfad}
\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 (ty\_2Epath\_2Epath~A\_27a~A\_27b) .  ((ap~(ap~(c\_2Epath\_2Etake~\\
A\_27a~A\_27b)~\mname{c\_2Enum\_2E0})~V0p)=(ap~(\mname{c\_2Epath\_2Estopped\_\_at}~A\_27a~\\
A\_27b)~(ap~(\mname{c\_2Epath\_2Efirst}~A\_27a~A\_27b)~V0p)))){\unicdcdcdhd}({\unicdcdadad} V1n \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V2p \in (ty\_2Epath\_2Epath~A\_27a~A\_27b) .  \\
((ap~(ap~(c\_2Epath\_2Etake~A\_27a~A\_27b)~(ap~\mname{c\_2Enum\_2ESUC}~V1n))~\\
V2p)=(ap~(ap~(ap~(\mname{c\_2Epath\_2Epcons}~A\_27a~A\_27b)~(ap~(\mname{c\_2Epath\_2Efirst}~\\
A\_27a~A\_27b)~V2p))~(ap~(c\_2Epath\_2Efirst\_\_label~A\_27a~A\_27b)~V2p))~\\
(ap~(ap~(c\_2Epath\_2Etake~A\_27a~A\_27b)~V1n)~(ap~(c\_2Epath\_2Etail~\\
A\_27a~A\_27b)~V2p)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgehigpfcdfeehbglgfgpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ((V0s=V1t){\unicdbdjded}({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V0s)){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfefeifeffeoedfjepeoe}
\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} V0f \in ((ty\_2Epair\_2Eprod~A\_27a~2)^{{A\_27b}}) .  ({\unicdcdadad} V1v \in \\
A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1v)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b)~V0f))){\unicdbdjded}({\unicdcdaddd} V2x \in A\_27b .  ((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~2)~V1v)~\mname{c\_2Ebool\_2ET})=(ap~V0f~V2x))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffedejegejedebeefjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  (\preoh (p~(ap~(ap~\\
(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a)))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfeoepeefpfpfjeoepfpffeneafefjf}
\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 .  ({\unicdcdadad} V2s \in (2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V0x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V1y)~V2s))){\unicdbdjded}((V0x=\\
V1y){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~V2s)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfjeoedffecfef}
\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 .  (((ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~\\
A\_27a))=(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V1y)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~\\
A\_27a))){\unicdbdjded}(V0x=V1y))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfefebfffbemepfpfdfjeoehe}
\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} V0y \in A\_27b .  ({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V2f \in (A\_27b^{{A\_27a}}) .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27b)~V0y)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EIMAGE}~\\
A\_27a~A\_27b)~V2f)~V1s))){\unicdbdjded}({\unicdcdaddd} V3x \in A\_27a .  ((V0y=(ap~V2f~V3x)){\unicdcdcdhd}\\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V3x)~V1s))))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfjenebehefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0y \in A\_27a .  ((ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27a)~({\uniddcgcg} V1x \in A\_27a .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~\\
V1x)~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~A\_27a)~V1x)~V0y))))=(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~\\
A\_27a)~V0y)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffedepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
((ap~\mname{c\_2Enum\_2ESUC}~V0m)=(ap~\mname{c\_2Enum\_2ESUC}~V1n)){\unicdbdjded}(V0m=V1n))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfejeoegfpfpfdfffdepfpffebf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeahbgehigpfcdfeafmepfpfehbglgfg}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0p \in (ty\_2Epath\_2Epath~A\_27a~A\_27b) .  ({\unicdcdadad} V1i \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Enum\_2Enum)~V1i)~(ap~(\mname{c\_2Epath\_2EPL}~\\
A\_27a~A\_27b)~V0p))){\unicdbdjdcd}((ap~(\mname{c\_2Epath\_2EPL}~A\_27a~A\_27b)~(ap~(ap~(c\_2Epath\_2Etake~\\
A\_27a~A\_27b)~V1i)~V0p))=(ap~(c\_2Epred\_\_set\_2EGSPEC~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)~({\uniddcgcg} V2n \in ty\_2Enum\_2Enum .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~2)~V2n)~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V2n)~V1i))))))))\end{array}$$
\end{theorem}
\end{document}
