\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=}
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\newcommand\unicdcdbdgd{\setminus}
\newcommand\unicdcdcdjd{\cap}
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\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\_2EdefCNF\_2EFINAL\_\_DEF (TMTCrRoCkqP8sikcNvg4GXuNuqwopVH27rF)}
\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\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Enum\_2EREP\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EREP\_\_num} \in {({{omega}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfecffeafpfpfogfhng}
\end{equation}
Let $ c\_2Enum\_2ESUC\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2ESUC\_\_REP} \in {({{omega}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfedfffdepfpfcffeaf}
\end{equation}
Let $ c\_2Enum\_2EABS\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EABS\_\_num} \in {({{ty\_2Enum\_2Enum}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfebecedfpfpfogfhng}
\end{equation}
\begin{definition}\label{def: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 $ 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 $ 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 $ c\_2EdefCNF\_2EUNIQUE : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EdefCNF\_2EUNIQUE \in (((2^{{({{{ty\_2Epair\_2Eprod}~{({{({{2}^{{2}}})}^{{2}}})}}~{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{2}})}}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{2}})}})}})}})^{{ty\_2Enum\_2Enum}})^{{({{2}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeegfgggdeoegepfcdfeffoejebffffe}
\end{equation}
Let $ ty\_2Elist\_2Elist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Elist\_2Elist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgjgdhehpfcdfemgjgdheh}
\end{equation}
Let $ c\_2Elist\_2ECONS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ECONS~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfedepeoedf}
\end{equation}
Let $ c\_2Elist\_2ENIL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ENIL~A\_27a \in (ty\_2Elist\_2Elist~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeoejeme}
\end{equation}
Let $ c\_2EdefCNF\_2EDEF : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EdefCNF\_2EDEF \in (((2^{{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{({{({{2}^{{2}}})}^{{2}}})}}~{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{2}})}}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{2}})}})}})}})}})^{{ty\_2Enum\_2Enum}})^{{({{2}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeegfgggdeoegepfcdfeeefege}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def: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}
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} V0t \in 2 .  (({\unicdcdadad} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepecfbememepfpfdfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} 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} V0v \in (2^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~(ap~c\_2EdefCNF\_2EDEF~V0v)~V1n)~(c\_2Elist\_2ENIL~(ty\_2Epair\_2Eprod~\\
((2^{{2}})^{{2}})~(ty\_2Epair\_2Eprod~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~2)~\\
(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~2)))))){\unicdbdjded}True))){\unicdcdcdhd}({\unicdcdadad} V2v \in \\
(2^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V3n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V4x \in \\
(ty\_2Epair\_2Eprod~((2^{{2}})^{{2}})~(ty\_2Epair\_2Eprod~(ty\_2Esum\_2Esum~\\
ty\_2Enum\_2Enum~2)~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~2))) .  ({\unicdcdadad} V5xs \in \\
(ty\_2Elist\_2Elist~(ty\_2Epair\_2Eprod~((2^{{2}})^{{2}})~(ty\_2Epair\_2Eprod~\\
(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~2)~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~\\
2)))) .  ((p~(ap~(ap~(ap~c\_2EdefCNF\_2EDEF~V2v)~V3n)~(ap~(ap~(c\_2Elist\_2ECONS~\\
(ty\_2Epair\_2Eprod~((2^{{2}})^{{2}})~(ty\_2Epair\_2Eprod~(ty\_2Esum\_2Esum~\\
ty\_2Enum\_2Enum~2)~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~2))))~V4x)~\\
V5xs))){\unicdbdjded}((p~(ap~(ap~(ap~c\_2EdefCNF\_2EUNIQUE~V2v)~V3n)~V4x)){\unicdcdcdhd}(p~\\
(ap~(ap~(ap~c\_2EdefCNF\_2EDEF~V2v)~(ap~\mname{c\_2Enum\_2ESUC}~V3n))~V5xs)))))))))\end{array}
\label{hyp:ehigngpfcdfeegfgggdeoegepfcdfeeefegepfpfegfggg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeegfgggdeoegepfcdfegejeoebemepfpfeefege}
$$\begin{array}{c}({\unicdcdadad} V0v \in (2^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V2x \in 2 .  (((p~(ap~V0v~V1n)){\unicdbdjded}(p~V2x)){\unicdbdjded}(((p~(ap~V0v~V1n)){\unicdbdjded}\\
(p~V2x)){\unicdcdcdhd}(p~(ap~(ap~(ap~c\_2EdefCNF\_2EDEF~V0v)~(ap~\mname{c\_2Enum\_2ESUC}~\\
V1n))~(c\_2Elist\_2ENIL~(ty\_2Epair\_2Eprod~((2^{{2}})^{{2}})~(ty\_2Epair\_2Eprod~\\
(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~2)~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~\\
2)))))))))))\end{array}$$
\end{theorem}
\end{document}
