\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\_2Einteger\_\_word\_2EINT\_\_MIN (TMXM4nCXYXrmxzpDpSG4phSitYehwFEyHyQ)}
\maketitle
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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: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: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: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 $ 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}
Let $ ty\_2Einteger\_2Eint : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Einteger\_2Eint}}\label{hyp:ogfgpfehjhpfcdfejgogehfghgfgchpfcdfejgogeh}
\end{equation}
Let $ c\_2Einteger\_2Eint\_\_of\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Einteger\_2Eint\_\_of\_\_num} \in {({{ty\_2Einteger\_2Eint}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfpgggpfpfogfhng}
\end{equation}
Let $ ty\_2Ebool\_2Eitself : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ebool\_2Eitself}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfecgpgpgmgpfcdfejgehdhfgmggg}
\end{equation}
Let $ c\_2Einteger\_\_word\_2EINT\_\_MAX : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Einteger\_\_word\_2EINT\_\_MAX~\\
A\_27a \in (ty\_2Einteger\_2Eint^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfpfhhpgchegpfcdfejeoeefpfpfnebeif}
\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\_2Einteger\_2Eint\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeafpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeaf}
We define $\mname{c\_2Einteger\_2Eint\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_neg : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_neg \in ((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfogfghg}
\end{equation}
Let $ c\_2Einteger\_2Etint\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_eq \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpffgbh}
\end{equation}
Let $ c\_2Einteger\_2Eint\_\_ABS\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_ABS\_\_CLASS \in (ty\_2Einteger\_2Eint^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedf}
We define $\mname{c\_2Einteger\_2Eint\_\_ABS}$ to be ${{\uniddcgcg} V0r \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})} .  {({{{ap}~{c\_2Einteger\_2Eint\_\_ABS\_\_CLASS}}~{({{{ap}~{c\_2Einteger\_2Etint\_\_eq}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfogfghg}
We define $\mname{c\_2Einteger\_2Eint\_\_neg}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{c\_2Einteger\_2Etint\_\_neg}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}})}}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_add : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_add \in (((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfbgegeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbgegeg}
We define $\mname{c\_2Einteger\_2Eint\_\_add}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_add}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfdhfhcg}
We define $\mname{c\_2Einteger\_2Eint\_\_sub}$ to be ${{\uniddcgcg} V0x \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1y \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_add}}}~{V0x}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_neg}}}~{V1y}})}})}}}$.
\end{definition}
Let $ c\_2Einteger\_\_word\_2EINT\_\_MIN : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Einteger\_\_word\_2EINT\_\_MIN~\\
A\_27a \in (ty\_2Einteger\_2Eint^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfpfhhpgchegpfcdfejeoeefpfpfnejeoe}
\end{equation}
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}
Let $ c\_2Efcp\_2Edimindex : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Efcp\_2Edimindex~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggdgahpfcdfeegjgngjgogegfgih}
\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}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
\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:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\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\_2Ebool\_2Ethe\_\_value : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ebool\_2Ethe\_\_value~A\_27a \in (\\
ty\_2Ebool\_2Eitself~A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfecgpgpgmgpfcdfeehigfgpfpfghbgmgfhfg}
\end{equation}
Let $ c\_2Ewords\_2EINT\_\_MIN : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ewords\_2EINT\_\_MIN~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfehhpgchegdhpfcdfejeoeefpfpfnejeoe}
\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\_2D~V0m)~V1n)=\mname{c\_2Enum\_2E0}){\unicdbdjded}(p~(ap~(\\
ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0m)~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffcepfpffebfpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0b1 \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1b2 \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V2x \in ty\_2Enum\_2Enum .  (((ap~(ap~c\_2Earithmetic\_2EEXP~\\
V0b1)~V2x)=(ap~(ap~c\_2Earithmetic\_2EEXP~V1b2)~V2x)){\unicdbdjded}((V2x=\mname{c\_2Enum\_2E0}){\unicdcdcdid}\\
(V0b1=V1b2))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfefeifafpfpffeifafpfpfjeoekefedeefjegffe}
\end{equation}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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} 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} V0x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
((ap~\mname{c\_2Einteger\_2Eint\_\_neg}~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_sub}~V0x)~\\
V1y))=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_sub}~V1y)~V0x))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfoefehepfpfdfffce}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
((ap~c\_2Einteger\_2Eint\_\_of\_\_num~V0m)=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
V1n)){\unicdbdjded}(V0m=V1n))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfjeoeke}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_sub}~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_sub}~\\
V0x)~V1y))~V0x)=(ap~\mname{c\_2Einteger\_2Eint\_\_neg}~V1y))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfdfffcepfpfdfffce}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  \\
(((ap~c\_2Einteger\_2Eint\_\_of\_\_num~V0m)=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
V1n)){\unicdbdjded}(V0m=V1n)))){\unicdcdcdhd}(({\unicdcdadad} V2x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V3y \in \\
ty\_2Einteger\_2Eint .  (((ap~\mname{c\_2Einteger\_2Eint\_\_neg}~V2x)=(ap~\mname{c\_2Einteger\_2Eint\_\_neg}~\\
V3y)){\unicdbdjded}(V2x=V3y)))){\unicdcdcdhd}({\unicdcdadad} V4n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V5m \in \\
ty\_2Enum\_2Enum .  ((((ap~c\_2Einteger\_2Eint\_\_of\_\_num~V4n)=(ap~\\
\mname{c\_2Einteger\_2Eint\_\_neg}~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~V5m))){\unicdbdjded}\\
((V4n=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}(V5m=\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(((ap~\mname{c\_2Einteger\_2Eint\_\_neg}~\\
(ap~c\_2Einteger\_2Eint\_\_of\_\_num~V4n))=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
V5m)){\unicdbdjded}((V4n=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}(V5m=\mname{c\_2Enum\_2E0}))))))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpffebfpfpfdebemedeffmebeeffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}((ap~(c\_2Einteger\_\_word\_2EINT\_\_MAX~\\
A\_27a)~(c\_2Ebool\_2Ethe\_\_value~A\_27a))=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_sub}~\\
(ap~c\_2Einteger\_2Eint\_\_of\_\_num~(ap~(c\_2Ewords\_2EINT\_\_MIN~A\_27a)~\\
(c\_2Ebool\_2Ethe\_\_value~A\_27a))))~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfpfhhpgchegpfcdfejeoeefpfpfnebeifpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}((ap~(c\_2Einteger\_\_word\_2EINT\_\_MIN~\\
A\_27a)~(c\_2Ebool\_2Ethe\_\_value~A\_27a))=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_sub}~\\
(ap~\mname{c\_2Einteger\_2Eint\_\_neg}~(ap~(c\_2Einteger\_\_word\_2EINT\_\_MAX~\\
A\_27a)~(c\_2Ebool\_2Ethe\_\_value~A\_27a))))~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfpfhhpgchegpfcdfejeoeefpfpfnejeoepfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}((ap~(c\_2Ewords\_2EINT\_\_MIN~A\_27a)~\\
(c\_2Ebool\_2Ethe\_\_value~A\_27a))=(ap~(ap~c\_2Earithmetic\_2EEXP~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))~\\
(ap~(ap~c\_2Earithmetic\_2E\_2D~(ap~(c\_2Efcp\_2Edimindex~A\_27a)~(\\
c\_2Ebool\_2Ethe\_\_value~A\_27a)))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfejeoeefpfpfnejeoepfpfegfggg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfejgogehfghgfgchpfpfhhpgchegpfcdfejeoeefpfpfnejeoe}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}((ap~(c\_2Einteger\_\_word\_2EINT\_\_MIN~\\
A\_27a)~(c\_2Ebool\_2Ethe\_\_value~A\_27a))=(ap~\mname{c\_2Einteger\_2Eint\_\_neg}~\\
(ap~c\_2Einteger\_2Eint\_\_of\_\_num~(ap~(c\_2Ewords\_2EINT\_\_MIN~A\_27a)~\\
(c\_2Ebool\_2Ethe\_\_value~A\_27a)))))\end{array}$$
\end{theorem}
\end{document}
