\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\_2Econtainer\_2EBAG\_\_TO\_\_LIST\_\_CARD (TMJfXxR7ScMrRu4M37MMpAweaZC88wvuckR)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\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\_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\_2Ebool\_2EARB : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A\_27a .  {{{nonempty}~{A\_27a}}{\unicdbdjdcd}{{{c\_2Ebool\_2EARB}~{A\_27a}} \in {A\_27a}}}}\label{hyp:ngfgngpfdgpfcdfecgpgpgmgpfcdfebecfce}
\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\_2Ebag\_2EBAG\_\_CHOICE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ebag\_2EBAG\_\_CHOICE~A\_27a \in (\\
A\_27a^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgbghgpfcdfecebehepfpfdeiepejedefe}
\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}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfele}
We define $\mname{c\_2Ecombin\_2EK}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1y \in {A\_27b} .  {V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfefeneafefjfpfpfcebehe}
We define $\mname{c\_2Ebag\_2EEMPTY\_\_BAG}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{\mname{c\_2Ecombin\_2EK}}~{ty\_2Enum\_2Enum}}~{A\_27a}})}}~{\mname{c\_2Enum\_2E0}}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
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: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}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfedad}
We define $\mname{c\_2Emin\_2E\_40}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} P \in {{2}^{{A}}} .  {{\mathbf{if}}~{({{\unicdcdaddd} x \in {A} .  {{p}~{({{{ap}~{P}}~{x}})}}})}~{\mathbf{then}}~{({{the}~{({{\uniddcgcg} x .  {{{x} \in {A}}{\unicdcdcdhd}{{p}~{({{{ap}~{P}}~{x}})}}}})}})}~{\mathbf{else}}~{({{the}~{({{\uniddcgcg} x .  {{x} \in {A}}})}})}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfecebehepfpfjeoedffecfef}
We define $\mname{c\_2Ebag\_2EBAG\_\_INSERT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0e \in {A\_27a} .  {{\uniddcgcg} V1b \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2x}})}}~{V0e}})}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{({{{ap}~{V1b}}~{V0e}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{V1b}}~{V2x}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfefemepfpfcebehe}
We define $\mname{c\_2Ebag\_2EEL\_\_BAG}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0e \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Ebag\_2EBAG\_\_INSERT}}~{A\_27a}})}}~{V0e}})}}~{({{\mname{c\_2Ebag\_2EEMPTY\_\_BAG}}~{A\_27a}})}})}}}$.
\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}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfecebehepfpfeejegege}
We define $\mname{c\_2Ebag\_2EBAG\_\_DIFF}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0b1 \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {{\uniddcgcg} V1b2 \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{({{{ap}~{V0b1}}~{V2x}})}})}}~{({{{ap}~{V1b2}}~{V2x}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfecebehepfpfcffedfef}
We define $\mname{c\_2Ebag\_2EBAG\_\_REST}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0b \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Ebag\_2EBAG\_\_DIFF}}~{A\_27a}})}}~{V0b}})}}~{({{{ap}~{({{\mname{c\_2Ebag\_2EEL\_\_BAG}}~{A\_27a}})}}~{({{{ap}~{({{c\_2Ebag\_2EBAG\_\_CHOICE}~{A\_27a}})}}~{V0b}})}})}})}}}$.
\end{definition}
Let $ c\_2Elist\_2ECONS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ECONS~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfedepeoedf}
\end{equation}
Let $ c\_2Elist\_2ENIL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ENIL~A\_27a \in (ty\_2Elist\_2Elist~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeoejeme}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfegejeoejeeffepfpfcebehe}
We define $\mname{c\_2Ebag\_2EFINITE\_\_BAG}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0b \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}}})}})}}~{({{\uniddcgcg} V1P \in {({{2}^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1P}}~{({{\mname{c\_2Ebag\_2EEMPTY\_\_BAG}}~{A\_27a}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2b \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V1P}}~{V2b}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3e \in {A\_27a} .  {({{{ap}~{V1P}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebag\_2EBAG\_\_INSERT}}~{A\_27a}})}}~{V3e}})}}~{V2b}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{V1P}}~{V0b}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfedf}
We define $\mname{c\_2Ecombin\_2ES}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1g \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V2x}})}}~{({{{ap}~{V1g}}~{V2x}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfeje}
We define $\mname{c\_2Ecombin\_2EI}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2ES}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}}~{A\_27a}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}})}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{A\_27a}})}})}}$.
\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:dgpfcdfechfgmgbgehjgpgogpfcdfehfge}
We define $\mname{c\_2Erelation\_2EWF}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1B \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V2w \in {A\_27a} .  {({{{ap}~{V1B}}~{V2w}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3min \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1B}}~{V3min}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4b}})}}~{V3min}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V1B}}~{V4b}})}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfecffedfefcfjedeef}
We define $\mname{c\_2Erelation\_2ERESTRICT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{V1R}}~{V3y}})}}~{V2x}})}})}}~{({{{ap}~{V0f}}~{V3y}})}})}}~{({{c\_2Ebool\_2EARB}~{A\_27b}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeefde}
We define $\mname{c\_2Erelation\_2ETC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1a \in {A\_27a} .  {{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{2}^{{A\_27a}}})}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3P \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4x}})}}~{V5y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V4x}})}}~{V5y}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V7y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V8z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V7y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V7y}})}}~{V8z}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V8z}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V1a}})}}~{V2b}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfebgahahchpgih}
We define $\mname{c\_2Erelation\_2Eapprox}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{A\_27b}^{{A\_27a}}})}})}}~{V3f}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V4y \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4y}})}})}}~{V4y}})}})}})}}~{V0R}})}}~{V2x}})}})}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeehigfgpfpfggfhog}
We define $\mname{c\_2Erelation\_2Ethe\_\_fun}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{A\_27b}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Eapprox}}~{A\_27a}}~{A\_27b}})}}~{V0R}})}}~{V1M}})}}~{V2x}})}}~{V3f}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehfgecffede}
We define $\mname{c\_2Erelation\_2EWFREC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Ethe\_\_fun}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{\mname{c\_2Erelation\_2ETC}}~{A\_27a}})}}~{V0R}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V4v \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4v}})}})}}~{V4v}})}})}})}})}}~{V2x}})}})}}~{V0R}})}}~{V2x}})}})}}~{V2x}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgogehbgjgogfgchpfcdfecebehepfpfefpepfpfmejedfef}
We define $\mname{c\_2Econtainer\_2EBAG\_\_TO\_\_LIST}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2EWFREC}}~{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{({{2}^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}}})}^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}}})}})}}~{({{\uniddcgcg} V0R \in {({{({{2}^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}}})}^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Erelation\_2EWF}}~{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}})}}~{V0R}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1bag \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebag\_2EFINITE\_\_BAG}}~{A\_27a}})}}~{V1bag}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}})}}~{V1bag}})}}~{({{\mname{c\_2Ebag\_2EEMPTY\_\_BAG}}~{A\_27a}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{({{{ap}~{({{\mname{c\_2Ebag\_2EBAG\_\_REST}}~{A\_27a}})}}~{V1bag}})}})}}~{V1bag}})}})}})}})}})}})}})}})}}~{({{\uniddcgcg} V2BAG\_\_TO\_\_LIST \in {({{({{ty\_2Elist\_2Elist}~{A\_27a}})}^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}}})} .  {({{\uniddcgcg} V3a \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{{ap}~{({{\mname{c\_2Ebag\_2EFINITE\_\_BAG}}~{A\_27a}})}}~{V3a}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}})}}~{V3a}})}}~{({{\mname{c\_2Ebag\_2EEMPTY\_\_BAG}}~{A\_27a}})}})}})}}~{({{c\_2Elist\_2ENIL}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2ECONS}~{A\_27a}})}}~{({{{ap}~{({{c\_2Ebag\_2EBAG\_\_CHOICE}~{A\_27a}})}}~{V3a}})}})}}~{({{{ap}~{V2BAG\_\_TO\_\_LIST}}~{({{{ap}~{({{\mname{c\_2Ebag\_2EBAG\_\_REST}}~{A\_27a}})}}~{V3a}})}})}})}})}})}}~{({{c\_2Ebool\_2EARB}~{({{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}
Let $ c\_2Econtainer\_2ELIST\_\_TO\_\_BAG : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Econtainer\_2ELIST\_\_TO\_\_BAG~\\
A\_27a \in ((ty\_2Enum\_2Enum^{{A\_27a}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedgpgogehbgjgogfgchpfcdfemejedfefpfpfefpepfpfcebehe}
\end{equation}
Let $ c\_2Ebag\_2EBAG\_\_CARD : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ebag\_2EBAG\_\_CARD~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgbghgpfcdfecebehepfpfdebecfee}
\end{equation}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ({\unicdcdadad} V2z \in A\_27a .  (((V0x=V1y){\unicdcdcdhd}(V1y=V2z)){\unicdbdjdcd}(V0x=V2z)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfefcfbeoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} 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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0b \in (ty\_2Enum\_2Enum^{{A\_27a}}) .  \\
((p~(ap~(\mname{c\_2Ebag\_2EFINITE\_\_BAG}~A\_27a)~V0b)){\unicdbdjdcd}((ap~(c\_2Econtainer\_2ELIST\_\_TO\_\_BAG~\\
A\_27a)~(ap~(\mname{c\_2Econtainer\_2EBAG\_\_TO\_\_LIST}~A\_27a)~V0b))=V0b)))\end{array}
\label{hyp:ehigngpfcdfedgpgogehbgjgogfgchpfcdfecebehepfpfefpepfpfmejedfefpfpfjeoegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0ls \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((ap~(c\_2Ebag\_2EBAG\_\_CARD~A\_27a)~(ap~(c\_2Econtainer\_2ELIST\_\_TO\_\_BAG~\\
A\_27a)~V0ls))=(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0ls)))\end{array}
\label{hyp:ehigngpfcdfedgpgogehbgjgogfgchpfcdfedebecfeepfpfmejedfefpfpfefpepfpfcebehe}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedgpgogehbgjgogfgchpfcdfecebehepfpfefpepfpfmejedfefpfpfdebecfee}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0b \in (ty\_2Enum\_2Enum^{{A\_27a}}) .  \\
((p~(ap~(\mname{c\_2Ebag\_2EFINITE\_\_BAG}~A\_27a)~V0b)){\unicdbdjdcd}((ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~(ap~(\mname{c\_2Econtainer\_2EBAG\_\_TO\_\_LIST}~A\_27a)~V0b))=(ap~(\\
c\_2Ebag\_2EBAG\_\_CARD~A\_27a)~V0b))))\end{array}$$
\end{theorem}
\end{document}
