\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\_2Efmsp\_2EFMSP\_\_FEMPTY (TMdKA82fYivJ8p4SnJuLxvPVnYV9BGWwGEf)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
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: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: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:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
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\_2Efinite\_\_map\_2Efmap : {{\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\_2Efinite\_\_map\_2Efmap~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeggjgogjgehfgpfpfngbgahpfcdfeggngbgah}
\end{equation}
Let $ c\_2Efinite\_\_map\_2Efmap\_\_ABS : {{\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\_2Efinite\_\_map\_2Efmap\_\_ABS~\\
A\_27a~A\_27b \in ((ty\_2Efinite\_\_map\_2Efmap~A\_27a~A\_27b)^{{({{({{{ty\_2Esum\_2Esum}~{A\_27b}}~{ty\_2Eone\_2Eone}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggjgogjgehfgpfpfngbgahpfcdfeggngbgahpfpfbecedf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegefeneafefjf}
We define $\mname{c\_2Efinite\_\_map\_2EFEMPTY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{{ap}~{({{{c\_2Efinite\_\_map\_2Efmap\_\_ABS}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V0a \in {A\_27a} .  {({{{ap}~{({{{\mname{c\_2Esum\_2EINR}}~{A\_27b}}~{ty\_2Eone\_2Eone}})}}~{\mname{c\_2Eone\_2Eone}}})}})}})}}}$.
\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 $ c\_2Efinite\_\_map\_2Efmap\_\_REP : {{\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\_2Efinite\_\_map\_2Efmap\_\_REP~\\
A\_27a~A\_27b \in (((ty\_2Esum\_2Esum~A\_27b~ty\_2Eone\_2Eone)^{{A\_27a}})^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggjgogjgehfgpfpfngbgahpfcdfeggngbgahpfpfcffeaf}
\end{equation}
Let $ c\_2Esum\_2EOUTL : {{\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\_2EOUTL~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfhngpfcdfepeffefme}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegebeafafmejf}
We define $\mname{c\_2Efinite\_\_map\_2EFAPPLY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{c\_2Esum\_2EOUTL}~{A\_27b}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{({{{ap}~{({{{c\_2Efinite\_\_map\_2Efmap\_\_REP}~{A\_27a}}~{A\_27b}})}}~{V0f}})}}~{V1x}})}})}}}}}$.
\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\_2Esum\_2EISL : {{\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\_2EISL~\\
A\_27a~A\_27b \in (2^{{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfhngpfcdfejedfme}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegeeepene}
We define $\mname{c\_2Efinite\_\_map\_2EFDOM}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{c\_2Esum\_2EISL}~{A\_27b}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{({{{ap}~{({{{c\_2Efinite\_\_map\_2Efmap\_\_REP}~{A\_27a}}~{A\_27b}})}}~{V0f}})}}~{V1x}})}})}}}}}$.
\end{definition}
\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: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:dgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegemepepeleffaf}
We define $\mname{c\_2Efinite\_\_map\_2EFLOOKUP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V1x}})}}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFDOM}}~{A\_27a}}~{A\_27b}})}}~{V0f}})}})}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFAPPLY}}~{A\_27a}}~{A\_27b}})}}~{V0f}})}}~{V1x}})}})}})}}~{({{\mname{c\_2Eoption\_2ENONE}}~{A\_27b}})}})}}}}}$.
\end{definition}
Let $ ty\_2Esptree\_2Espt : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Esptree\_2Espt}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfedhahehchfgfgpfcdfedhaheh}
\end{equation}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Esptree\_2Elookup : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esptree\_2Elookup~A\_27a \in (((\\
ty\_2Eoption\_2Eoption~A\_27a)^{{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhahehchfgfgpfcdfemgpgpglgfhah}
\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}
\begin{definition}\label{def:dgpfcdfepgahehjgpgogpfcdfepeafefcffeme}
We define $\mname{c\_2Eoption\_2EOPTREL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1x \in {({{ty\_2Eoption\_2Eoption}~{A\_27a}})} .  {{\uniddcgcg} V2y \in {({{ty\_2Eoption\_2Eoption}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})}}~{V1x}})}}~{({{\mname{c\_2Eoption\_2ENONE}}~{A\_27a}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}})}}~{V2y}})}}~{({{\mname{c\_2Eoption\_2ENONE}}~{A\_27b}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3x0 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27b}})}}~{({{\uniddcgcg} V4y0 \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})}}~{V1x}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{A\_27a}})}}~{V3x0}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}})}}~{V2y}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{A\_27b}})}}~{V4y0}})}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V3x0}})}}~{V4y0}})}})}})}})}})}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggngdhahpfcdfegenedfaf}
We define $\mname{c\_2Efmsp\_2EFMSP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0AN \in {({{({{2}^{{ty\_2Enum\_2Enum}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1BC \in {({{({{2}^{{A\_27c}}})}^{{A\_27b}}})} .  {{\uniddcgcg} V2fm \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V3sp \in {({{ty\_2Esptree\_2Espt}~{A\_27c}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4a \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V5n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0AN}}~{V4a}})}}~{V5n}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Eoption\_2EOPTREL}}~{A\_27b}}~{A\_27c}})}}~{V1BC}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFLOOKUP}}~{A\_27a}}~{A\_27b}})}}~{V2fm}})}}~{V4a}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Esptree\_2Elookup}~{A\_27c}})}}~{V5n}})}}~{V3sp}})}})}})}})}})}})}})}}}}}}}}$.
\end{definition}
Let $ c\_2Esptree\_2EBS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esptree\_2EBS~A\_27a \in ((((ty\_2Esptree\_2Espt~\\
A\_27a)^{{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})^{{A\_27a}})^{{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhahehchfgfgpfcdfecedf}
\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\_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:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\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: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: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:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefbd}
We define $\mname{c\_2Earithmetic\_2EBIT1}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
Let $ c\_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\_2EDIV : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EDIV \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfeeejegf}
\end{equation}
Let $ c\_2Esptree\_2EBN : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esptree\_2EBN~A\_27a \in (((ty\_2Esptree\_2Espt~\\
A\_27a)^{{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})^{{({{ty\_2Esptree\_2Espt}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhahehchfgfgpfcdfeceoe}
\end{equation}
Let $ c\_2Esptree\_2ELS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esptree\_2ELS~A\_27a \in ((ty\_2Esptree\_2Espt~\\
A\_27a)^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhahehchfgfgpfcdfemedf}
\end{equation}
Let $ c\_2Esptree\_2ELN : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esptree\_2ELN~A\_27a \in (ty\_2Esptree\_2Espt~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfedhahehchfgfgpfcdfemeoe}
\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} 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} 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} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0k \in A\_27b .  ((ap~(ap~(\mname{c\_2Efinite\_\_map\_2EFLOOKUP}~A\_27b~\\
A\_27a)~(\mname{c\_2Efinite\_\_map\_2EFEMPTY}~A\_27b~A\_27a))~V0k)=(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a)))\end{array}
\label{hyp:ehigngpfcdfeggjgogjgehfgpfpfngbgahpfcdfegemepepeleffafpfpffeneafefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  (\preoh ((\mname{c\_2Eoption\_2ENONE}~\\
A\_27a)=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V0x))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfeoepeefpfpfoepeoefepfpfdfpenefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0k \in ty\_2Enum\_2Enum .  \\
((ap~(ap~(c\_2Esptree\_2Elookup~A\_27a)~V0k)~(c\_2Esptree\_2ELN~A\_27a))=\\
(\mname{c\_2Eoption\_2ENONE}~A\_27a))){\unicdcdcdhd}(({\unicdcdadad} V1k \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V2a \in A\_27a .  ((ap~(ap~(c\_2Esptree\_2Elookup~A\_27a)~V1k)~\\
(ap~(c\_2Esptree\_2ELS~A\_27a)~V2a))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~\\
(ty\_2Eoption\_2Eoption~A\_27a))~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~ty\_2Enum\_2Enum)~\\
V1k)~\mname{c\_2Enum\_2E0}))~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V2a))~(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a))))){\unicdcdcdhd}(({\unicdcdadad} V3t2 \in (ty\_2Esptree\_2Espt~A\_27a) .  ({\unicdcdadad} V4t1 \in \\
(ty\_2Esptree\_2Espt~A\_27a) .  ({\unicdcdadad} V5k \in ty\_2Enum\_2Enum .  ((ap~\\
(ap~(c\_2Esptree\_2Elookup~A\_27a)~V5k)~(ap~(ap~(c\_2Esptree\_2EBN~\\
A\_27a)~V4t1)~V3t2))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~(ty\_2Eoption\_2Eoption~\\
A\_27a))~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~ty\_2Enum\_2Enum)~V5k)~\mname{c\_2Enum\_2E0}))~\\
(\mname{c\_2Eoption\_2ENONE}~A\_27a))~(ap~(ap~(c\_2Esptree\_2Elookup~A\_27a)~\\
(ap~(ap~c\_2Earithmetic\_2EDIV~(ap~(ap~c\_2Earithmetic\_2E\_2D~V5k)~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO}))))~\\
(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~(ty\_2Esptree\_2Espt~A\_27a))~(ap~c\_2Earithmetic\_2EEVEN~\\
V5k))~V4t1)~V3t2))))))){\unicdcdcdhd}({\unicdcdadad} V6t2 \in (ty\_2Esptree\_2Espt~A\_27a) .  \\
({\unicdcdadad} V7t1 \in (ty\_2Esptree\_2Espt~A\_27a) .  ({\unicdcdadad} V8k \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V9a \in A\_27a .  ((ap~(ap~(c\_2Esptree\_2Elookup~A\_27a)~V8k)~\\
(ap~(ap~(ap~(c\_2Esptree\_2EBS~A\_27a)~V7t1)~V9a)~V6t2))=(ap~(ap~(ap~\\
(\mname{c\_2Ebool\_2ECOND}~(ty\_2Eoption\_2Eoption~A\_27a))~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~\\
ty\_2Enum\_2Enum)~V8k)~\mname{c\_2Enum\_2E0}))~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~\\
V9a))~(ap~(ap~(c\_2Esptree\_2Elookup~A\_27a)~(ap~(ap~c\_2Earithmetic\_2EDIV~\\
(ap~(ap~c\_2Earithmetic\_2E\_2D~V8k)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO}))))~(ap~(ap~\\
(ap~(\mname{c\_2Ebool\_2ECOND}~(ty\_2Esptree\_2Espt~A\_27a))~(ap~c\_2Earithmetic\_2EEVEN~\\
V8k))~V7t1)~V6t2)))))))))))\end{array}
\label{hyp:ehigngpfcdfedhahehchfgfgpfcdfemgpgpglgfhahpfpfegfggg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeggngdhahpfcdfegenedfafpfpfgefeneafefjf}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0AN \in ((2^{{ty\_2Enum\_2Enum}})^{{A\_27a}}) .  ({\unicdcdadad} V1BC \in \\
((2^{{A\_27c}})^{{A\_27b}}) .  (p~(ap~(ap~(ap~(ap~(\mname{c\_2Efmsp\_2EFMSP}~A\_27a~A\_27b~\\
A\_27c)~V0AN)~V1BC)~(\mname{c\_2Efinite\_\_map\_2EFEMPTY}~A\_27a~A\_27b))~(c\_2Esptree\_2ELN~\\
A\_27c)))))\end{array}$$
\end{theorem}
\end{document}
