\documentclass{article}
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\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
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\begin{document}
\title{thm\_2EOmega\_2Eeval\_\_step\_\_extra1 (TMYd4v5WoXruP8jLJAkvjFyuXqCS1T6cCpA)}
\maketitle
Let $ ty\_2Einteger\_2Eint : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Einteger\_2Eint}}\label{hyp:ogfgpfehjhpfcdfejgogehfghgfgchpfcdfejgogeh}
\end{equation}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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 $ 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\_2EOmega\_2Eevallower : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EOmega\_2Eevallower \in ((2^{{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Einteger\_2Eint}})}})}})^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepengfghgbgpfcdfefgghbgmgmgpghhfgch}
\end{equation}
Let $ c\_2EOmega\_2Eevalupper : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EOmega\_2Eevalupper \in ((2^{{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Einteger\_2Eint}})}})}})^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepengfghgbgpfcdfefgghbgmgfhahahfgch}
\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: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} 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}
\begin{theorem}\label{thm:ehigngpfcdfepengfghgbgpfcdfefgghbgmgpfpfdhehfgahpfpffgihehchbgbd}
$$\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1ups \in (ty\_2Elist\_2Elist~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Einteger\_2Eint)) .  ({\unicdcdadad} V2lows \in \\
(ty\_2Elist\_2Elist~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Einteger\_2Eint)) .  \\
({\unicdcdadad} V3ex\_27 \in 2 .  (((((p~(ap~(ap~c\_2EOmega\_2Eevalupper~V0x)~\\
V1ups)){\unicdcdcdhd}(p~(ap~(ap~c\_2EOmega\_2Eevallower~V0x)~V2lows))){\unicdcdcdhd}True){\unicdcdcdhd}\\
(p~V3ex\_27)){\unicdbdjded}(((p~(ap~(ap~c\_2EOmega\_2Eevalupper~V0x)~V1ups)){\unicdcdcdhd}\\
(p~(ap~(ap~c\_2EOmega\_2Eevallower~V0x)~V2lows))){\unicdcdcdhd}(p~V3ex\_27)))))))\end{array}$$
\end{theorem}
\end{document}
