\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\_2Ellist\_2Elinear\_\_order\_\_to\_\_llist\_\_eq (TMMSHuKfnQUCt4xLhkETjnvky1F8MGc6Wdx)}
\maketitle
Let $ ty\_2Ellist\_2Ellist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ellist\_2Ellist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgmgjgdhehpfcdfemgmgjgdheh}
\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:dgpfcdfecgpgpgmgpfcdfecepeffoeeefeee}
We define $\mname{c\_2Ebool\_2EBOUNDED}$ to be ${({{\uniddcgcg} V0v \in {2} .  {\mname{c\_2Ebool\_2ET}}})}$.
\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\_2Elist\_2ELIST\_\_TO\_\_SET : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a \in \\
((2^{{A\_27a}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfemejedfefpfpfefpepfpfdffeef}
\end{equation}
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 $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Ellist\_2ELNTH : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ellist\_2ELNTH~A\_27a \in (((ty\_2Eoption\_2Eoption~\\
A\_27a)^{{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgmgjgdhehpfcdfemeoeefie}
\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 $ 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}
\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: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:dgpfcdfedgpgngcgjgogpfcdfepg}
We define $\mname{c\_2Ecombin\_2Eo}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27c}}})} .  {{\uniddcgcg} V1g \in {({{A\_27c}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{V0f}}~{({{{ap}~{V1g}}~{V2x}})}})}})}}}}}}$.
\end{definition}
Let $ c\_2Epair\_2ESND : {{\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\_2Epair\_2ESND~\\
A\_27a~A\_27b \in (A\_27b^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfedfoeee}
\end{equation}
Let $ c\_2Epair\_2EFST : {{\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\_2Epair\_2EFST~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfegedfef}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfeffoedeffcfcfjf}
We define $\mname{c\_2Epair\_2EUNCURRY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1v \in {({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{V0f}}~{({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}}}}}$.
\end{definition}
Let $ c\_2Eoption\_2EOPTION\_\_BIND : {{\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\_2Eoption\_2EOPTION\_\_BIND~\\
A\_27a~A\_27b \in (((ty\_2Eoption\_2Eoption~A\_27a)^{{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{A\_27b}}})}})^{{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfepeafefjepeoepfpfcejeoeee}
\end{equation}
Let $ c\_2Earithmetic\_2EFUNPOW : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Earithmetic\_2EFUNPOW~A\_27a \in \\
(((A\_27a^{{A\_27a}})^{{ty\_2Enum\_2Enum}})^{{({{A\_27a}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfegeffoeafpehf}
\end{equation}
Let $ c\_2Eoption\_2EOPTION\_\_MAP : {{\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\_2Eoption\_2EOPTION\_\_MAP~\\
A\_27a~A\_27b \in (((ty\_2Eoption\_2Eoption~A\_27b)^{{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})^{{({{A\_27b}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfepeafefjepeoepfpfnebeaf}
\end{equation}
Let $ c\_2Ellist\_2Ellist\_\_abs : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ellist\_2Ellist\_\_abs~A\_27a \in \\
((ty\_2Ellist\_2Ellist~A\_27a)^{{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgmgjgdhehpfcdfemgmgjgdhehpfpfbgcgdh}
\end{equation}
\begin{definition}\label{def:dgpfcdfemgmgjgdhehpfcdfemeffoegepemeee}
We define $\mname{c\_2Ellist\_2ELUNFOLD}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{({{ty\_2Eoption\_2Eoption}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}^{{A\_27b}}})} .  {{\uniddcgcg} V1z \in {A\_27b} .  {({{{ap}~{({{c\_2Ellist\_2Ellist\_\_abs}~{A\_27a}})}}~{({{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{c\_2Eoption\_2EOPTION\_\_MAP}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}}~{A\_27a}})}}~{({{{c\_2Epair\_2ESND}~{A\_27b}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Earithmetic\_2EFUNPOW}~{({{ty\_2Eoption\_2Eoption}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}}~{({{\uniddcgcg} V3m \in {({{ty\_2Eoption\_2Eoption}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})} .  {({{{ap}~{({{{ap}~{({{{c\_2Eoption\_2EOPTION\_\_BIND}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}}~{V3m}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27b}}~{A\_27a}}~{({{ty\_2Eoption\_2Eoption}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}}~{({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2Eo}}~{A\_27b}}~{({{({{ty\_2Eoption\_2Eoption}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}^{{A\_27a}}})}}~{({{ty\_2Eoption\_2Eoption}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{({{ty\_2Eoption\_2Eoption}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}}~{A\_27a}})}})}}~{V0f}})}})}})}})}})}}~{V2n}})}}~{({{{ap}~{V0f}}~{V1z}})}})}})}})}})}}}}}$.
\end{definition}
Let $ c\_2Ellist\_2ELTAKE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ellist\_2ELTAKE~A\_27a \in (((ty\_2Eoption\_2Eoption~\\
(ty\_2Elist\_2Elist~A\_27a))^{{({{ty\_2Ellist\_2Ellist}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgmgjgdhehpfcdfemeefbelefe}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfemefeef}
We define $\mname{c\_2Ebool\_2ELET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{V0f}}~{V1x}})}})}})}}}$.
\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}
Let $ c\_2Epair\_2EABS\_\_prod : {{\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\_2Epair\_2EABS\_\_prod~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfebecedfpfpfahchpgeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfepfcdde}
We define $\mname{c\_2Epair\_2E\_2C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{c\_2Epair\_2EABS\_\_prod}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V2a \in {A\_27a} .  {({{\uniddcgcg} V3b \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2a}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3b}})}}~{V1y}})}})}})}})}})}}}}}$.
\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: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}
Let $ c\_2Epred\_\_set\_2EGSPEC : {{\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\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b \in ((2^{{A\_27a}})^{{({{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{2}})}^{{A\_27b}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffede}
\end{equation}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechbgoghgfg}
We define $\mname{c\_2Eset\_\_relation\_2Erange}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V1y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V1y}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27b}})}}~{({{\uniddcgcg} V2x \in {A\_27b} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27b}}~{A\_27a}})}}~{V2x}})}}~{V1y}})}})}}~{V0r}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeegpgngbgjgog}
We define $\mname{c\_2Eset\_\_relation\_2Edomain}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V1x}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27b}})}}~{({{\uniddcgcg} V2y \in {A\_27b} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27b}})}}~{V1x}})}}~{V2y}})}})}}~{V0r}})}})}})}})}})}})}}}}$.
\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:dgpfcdfeahchfgegpfpfdhfgehpfcdfeffoejepeoe}
We define $\mname{c\_2Epred\_\_set\_2EUNION}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfengjgogjgngbgmgpfpffgmgfgngfgogehdh}
We define $\mname{c\_2Eset\_\_relation\_2Eminimal\_\_elements}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0xs \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0xs}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3x\_27 \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3x\_27}})}}~{V0xs}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V3x\_27}})}}~{V2x}})}})}}~{V1r}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2x}})}}~{V3x\_27}})}})}})}})}})}})}})}})}}}}$.
\end{definition}
Let $ c\_2Epred\_\_set\_2ECHOICE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Epred\_\_set\_2ECHOICE~A\_27a \in \\
(A\_27a^{{({{2}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahchfgegpfpfdhfgehpfcdfedeiepejedefe}
\end{equation}
\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: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:dgpfcdfeahchfgegpfpfdhfgehpfcdfeeejegege}
We define $\mname{c\_2Epred\_\_set\_2EDIFF}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechchfgdhehchjgdgeh}
We define $\mname{c\_2Eset\_\_relation\_2Errestrict}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27a}}~{A\_27a}}~{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{2}})}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{2}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V3y}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V3y}})}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3y}})}}~{V1s}})}})}})}})}})}})}})}})}}}}$.
\end{definition}
Let $ ty\_2Eone\_2Eone : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eone\_2Eone}}\label{hyp:ogfgpfehjhpfcdfepgogfgpfcdfepgogfg}
\end{equation}
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: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}
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: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}
\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: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}
\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}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfefeneafefjf}
We define $\mname{c\_2Epred\_\_set\_2EEMPTY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2EF}}})}}$.
\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:dgpfcdfemgmgjgdhehpfcdfemgjgogfgbgchpfpfpgchegfgchpfpfehpgpfpfmgjgdhehpfpfgg}
We define $\mname{c\_2Ellist\_2Elinear\_\_order\_\_to\_\_list\_\_f}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0lo \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Ebool\_2ELET}}~{({{2}^{{A\_27a}}})}}~{({{ty\_2Eoption\_2Eoption}~{({{{ty\_2Epair\_2Eprod}~{({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})}}~{A\_27a}})}})}})}}~{({{\uniddcgcg} V1min \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Eoption\_2Eoption}~{({{{ty\_2Epair\_2Eprod}~{({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})}}~{A\_27a}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V1min}})}}~{({{\mname{c\_2Epred\_\_set\_2EEMPTY}}~{A\_27a}})}})}})}}~{({{\mname{c\_2Eoption\_2ENONE}}~{({{{ty\_2Epair\_2Eprod}~{({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})}}~{A\_27a}})}})}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{({{{ty\_2Epair\_2Eprod}~{({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Errestrict}}~{A\_27a}})}}~{V0lo}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EDIFF}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EUNION}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Edomain}}~{A\_27a}}~{A\_27a}})}}~{V0lo}})}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Erange}}~{A\_27a}}~{A\_27a}})}}~{V0lo}})}})}})}}~{V1min}})}})}})}}~{({{{ap}~{({{c\_2Epred\_\_set\_2ECHOICE}~{A\_27a}})}}~{V1min}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Eminimal\_\_elements}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EUNION}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Edomain}}~{A\_27a}}~{A\_27a}})}}~{V0lo}})}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Erange}}~{A\_27a}}~{A\_27a}})}}~{V0lo}})}})}})}}~{V0lo}})}})}}}$.
\end{definition}
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\_2Earithmetic\_2EODD : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Earithmetic\_2EODD} \in {({{2}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepeeeee}
\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}
\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:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfddfe}
We define $\mname{c\_2Earithmetic\_2E\_3E}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V1n}})}}~{V0m}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfddfepfddee}
We define $\mname{c\_2Earithmetic\_2E\_3E\_3D}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2E\_3E}}}~{V0m}})}}~{V1n}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{V1n}})}})}}}$.
\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\_2Earithmetic\_2E\_2A : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2A \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdbe}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngfgchbgmgpfcdfejgkf}
We define $\mname{c\_2Enumeral\_2EiZ}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\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}
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}
\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}
Let $ c\_2Eoption\_2EOPTION\_\_JOIN : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eoption\_2EOPTION\_\_JOIN~A\_27a \in \\
((ty\_2Eoption\_2Eoption~A\_27a)^{{({{ty\_2Eoption\_2Eoption}~{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfepeafefjepeoepfpfkepejeoe}
\end{equation}
Let $ c\_2Eoption\_2Eoption\_\_CASE : {{\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\_2Eoption\_2Eoption\_\_CASE~\\
A\_27a~A\_27b \in (((A\_27b^{{({{A\_27b}^{{A\_27a}}})}})^{{A\_27b}})^{{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfepgahehjgpgogpfpfdebedffe}
\end{equation}
Let $ c\_2Eoption\_2EIS\_\_NONE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eoption\_2EIS\_\_NONE~A\_27a \in (\\
2^{{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfejedfpfpfoepeoefe}
\end{equation}
Let $ c\_2Eoption\_2EIS\_\_SOME : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eoption\_2EIS\_\_SOME~A\_27a \in (\\
2^{{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfejedfpfpfdfpenefe}
\end{equation}
Let $ c\_2Eoption\_2ETHE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eoption\_2ETHE~A\_27a \in (A\_27a^{{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgahehjgpgogpfcdfeefiefe}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfeahbgjgchpfpfdebedffe}
We define $\mname{c\_2Epair\_2Epair\_\_CASE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0p \in {({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27c}})} .  {{\uniddcgcg} V1f \in {({{({{A\_27a}^{{A\_27c}}})}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{V1f}}~{({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27b}}~{A\_27c}})}}~{V0p}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27b}}~{A\_27c}})}}~{V0p}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfejeoedffecfef}
We define $\mname{c\_2Epred\_\_set\_2EINSERT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2y}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2y}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2y}})}}~{V1s}})}})}})}})}})}}}}$.
\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}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeeefemefeeffe}
We define $\mname{c\_2Epred\_\_set\_2EDELETE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EDIFF}}~{A\_27a}})}}~{V0s}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EINSERT}}~{A\_27a}})}}~{V1x}})}}~{({{\mname{c\_2Epred\_\_set\_2EEMPTY}}~{A\_27a}})}})}})}}}}$.
\end{definition}
Let $ c\_2Epred\_\_set\_2ECARD : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Epred\_\_set\_2ECARD~A\_27a \in (ty\_2Enum\_2Enum^{{({{2}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahchfgegpfpfdhfgehpfcdfedebecfee}
\end{equation}
\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}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfedfffcedffeef}
We define $\mname{c\_2Epred\_\_set\_2ESUBSET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeafdfffcedffeef}
We define $\mname{c\_2Epred\_\_set\_2EPSUBSET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{A\_27a}})}}~{V0s}})}}~{V1t}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0s}})}}~{V1t}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchjgngpfpfchfgdgpfcdfeafcffe}
We define $\mname{c\_2Eprim\_\_rec\_2EPRE}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{\mname{c\_2Enum\_2E0}}})}})}}~{\mname{c\_2Enum\_2E0}}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V1n}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfegejeoejeeffe}
We define $\mname{c\_2Epred\_\_set\_2EFINITE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}})}}~{({{\uniddcgcg} V1P \in {({{2}^{{({{2}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1P}}~{({{\mname{c\_2Epred\_\_set\_2EEMPTY}}~{A\_27a}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V1P}}~{V2s}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3e \in {A\_27a} .  {({{{ap}~{V1P}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EINSERT}}~{A\_27a}})}}~{V3e}})}}~{V2s}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{V1P}}~{V0s}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfebgogehjgdhjhng}
We define $\mname{c\_2Eset\_\_relation\_2Eantisym}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2y}})}}~{V1x}})}})}}~{V0r}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeehchbgogdhjgehjgghfg}
We define $\mname{c\_2Eset\_\_relation\_2Etransitive}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2y}})}}~{V3z}})}})}}~{V0r}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V3z}})}})}}~{V0r}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfejeoeeffecf}
We define $\mname{c\_2Epred\_\_set\_2EINTER}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfedfjeoehe}
We define $\mname{c\_2Epred\_\_set\_2ESING}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0s}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EINSERT}}~{A\_27a}})}}~{V1x}})}}~{({{\mname{c\_2Epred\_\_set\_2EEMPTY}}~{A\_27a}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeggjgogjgehfgpfpfahchfgggjgihfgdh}
We define $\mname{c\_2Eset\_\_relation\_2Efinite\_\_prefixes}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V2e \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{V2e}})}}~{V1s}})}})}}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EFINITE}}~{A\_27a}})}}~{({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V3e\_27 \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V3e\_27}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27b}})}}~{V3e\_27}})}}~{V2e}})}})}}~{V0r}})}})}})}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfemgjgogfgbgchpfpfpgchegfgch}
We define $\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Edomain}}~{A\_27a}}~{A\_27a}})}}~{V0r}})}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Erange}}~{A\_27a}}~{A\_27a}})}}~{V0r}})}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Etransitive}}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Eantisym}}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3y}})}}~{V1s}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V3y}})}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V3y}})}}~{V2x}})}})}}~{V0r}})}})}})}})}})}})}})}})}})}})}})}}}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))=\\
(ap~\mname{c\_2Enum\_2ESUC}~\mname{c\_2Enum\_2E0}))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfepeoefe}
\end{equation}
Assume the following.
\begin{equation}
{({{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}={\mname{c\_2Enum\_2E0}}})}\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeoepecfnepfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((V0m=\mname{c\_2Enum\_2E0}){\unicdcdcdid}({\unicdcdaddd} V1n \in \\
ty\_2Enum\_2Enum .  (V0m=(ap~\mname{c\_2Enum\_2ESUC}~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeogfhngpfpfdebedffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(ap~\mname{c\_2Enum\_2ESUC}~V0m))~(ap~\mname{c\_2Enum\_2ESUC}~\\
V1n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0m)~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfnepeoepepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0m)~V1n)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~V0m))~(ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~V1n))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeafcffepfpfmefedfdfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(V0m=V1n){\unicdcdcdid}((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0m)~V1n)){\unicdcdcdid}(p~(ap~(ap~\\
\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1n)~V0m))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfmefedfdfpfpfdebedffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
\mname{c\_2Enum\_2E0})~V0m)){\unicdbdjded}((ap~\mname{c\_2Enum\_2ESUC}~(ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\\
V0m))=V0m)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffdepfpfafcffe}
\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}
{({{\unicdcdadad} V0t \in {2} .  {({{({{p}~{V0t}})}{\unicdcdcdid}{({\preoh{{({{p}~{V0t}})}}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifdemeffeefeeepfpfnejeeeeemefe}
\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} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1y \in A\_27a .  ((ap~({\uniddcgcg} V2x \in \\
A\_27a .  (ap~V0f~V2x))~V1y)=(ap~V0f~V1y))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecefeefbepfpfefiene}
\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} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdbdjded}((p~V1t2){\unicdcdcdhd}\\
(p~V0t1)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoekepfpfdepenene}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({\preoh{{({{p}~{V0t}})}}})}{\unicdbdjdcd}{({{({{p}~{V0t}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepfpfjeneaf}
\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} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\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} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1g \in (A\_27b^{{A\_27a}}) .  ((V0f=\\
V1g){\unicdbdjded}({\unicdcdadad} V2x \in A\_27a .  ((ap~V0f~V2x)=(ap~V1g~V2x))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegeffoepfpffebfpfpfefiene}
\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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t1 \in A\_27a .  ({\unicdcdadad} V1t2 \in \\
A\_27a .  (((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2ET})~V0t1)~\\
V1t2)=V0t1){\unicdcdcdhd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2EF})~\\
V0t1)~V1t2)=V1t2))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ((\preoh ({\unicdcdadad} V1x \in \\
A\_27a .  (p~(ap~V0P~V1x)))){\unicdbdjded}({\unicdcdaddd} V2x \in A\_27a .  (\preoh (p~(ap~V0P~V2x))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfgepecfbememepfpfefiene}
\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} V0P \in 2 .  ({\unicdcdadad} V1Q \in 2 .  \\
({\unicdcdadad} V2x \in A\_27a .  ({\unicdcdadad} V3x\_27 \in A\_27a .  ({\unicdcdadad} V4y \in A\_27a .  \\
({\unicdcdadad} V5y\_27 \in A\_27a .  ((((p~V0P){\unicdbdjded}(p~V1Q)){\unicdcdcdhd}(((p~V1Q){\unicdbdjdcd}(V2x=V3x\_27)){\unicdcdcdhd}\\
((\preoh (p~V1Q)){\unicdbdjdcd}(V4y=V5y\_27)))){\unicdbdjdcd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\\
V0P)~V2x)~V4y)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~V1Q)~V3x\_27)~\\
V5y\_27)))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdepeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  (({\unicdcdaddd} V2x \in A\_27a .  ((V2x=V1a){\unicdcdcdhd}(p~(ap~V0P~V2x)))){\unicdbdjded}(p~(\\
ap~V0P~V1a)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeffoehfjeoeeepfpfefienecd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0v \in {2} .  {({{({{p}~{({{{ap}~{\mname{c\_2Ebool\_2EBOUNDED}}}~{V0v}})}})}{\unicdbdjded}{True}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecepeffoeeefeeepfpfefiene}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1g \in (A\_27a^{{A\_27c}}) .  \\
({\unicdcdadad} V2x \in A\_27c .  ((ap~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27c~A\_27b~A\_27a)~\\
V0f)~V1g)~V2x)=(ap~V0f~(ap~V1g~V2x))))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfepgpfpfefiene}
\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} V0h \in A\_27b .  ({\unicdcdadad} V1t \in (ty\_2Elist\_2Elist~A\_27b) .  ((\\
(ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a)~(c\_2Elist\_2ENIL~A\_27a))=\\
(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a)){\unicdcdcdhd}((ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~\\
A\_27b)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27b)~V0h)~V1t))=(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~\\
A\_27b)~V0h)~(ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27b)~V1t))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemejedfefpfpfefpepfpfdffeef}
\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} V0f \in ((ty\_2Eoption\_2Eoption~(ty\_2Epair\_2Eprod~A\_27b~\\
A\_27a))^{{A\_27b}}) .  ({\unicdcdadad} V1x \in A\_27b .  ({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  \\
(((ap~(ap~(c\_2Ellist\_2ELNTH~A\_27a)~\mname{c\_2Enum\_2E0})~(ap~(ap~(\mname{c\_2Ellist\_2ELUNFOLD}~\\
A\_27a~A\_27b)~V0f)~V1x))=(ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~(ty\_2Epair\_2Eprod~\\
A\_27b~A\_27a)~A\_27a)~(c\_2Epair\_2ESND~A\_27b~A\_27a))~(ap~V0f~V1x))){\unicdcdcdhd}\\
((ap~(ap~(c\_2Ellist\_2ELNTH~A\_27a)~(ap~\mname{c\_2Enum\_2ESUC}~V2n))~(ap~(\\
ap~(\mname{c\_2Ellist\_2ELUNFOLD}~A\_27a~A\_27b)~V0f)~V1x))=(ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~\\
(ty\_2Epair\_2Eprod~A\_27b~A\_27a)~(ty\_2Eoption\_2Eoption~A\_27a))~\\
(ap~V0f~V1x))~(\mname{c\_2Eoption\_2ENONE}~A\_27a))~({\uniddcgcg} V3v \in (ty\_2Epair\_2Eprod~\\
A\_27b~A\_27a) .  (ap~(ap~(\mname{c\_2Epair\_2Epair\_\_CASE}~(ty\_2Eoption\_2Eoption~\\
A\_27a)~A\_27b~A\_27a)~V3v)~({\uniddcgcg} V4tx \in A\_27b .  ({\uniddcgcg} V5hx \in A\_27a .  (ap~\\
(ap~(c\_2Ellist\_2ELNTH~A\_27a)~V2n)~(ap~(ap~(\mname{c\_2Ellist\_2ELUNFOLD}~\\
A\_27a~A\_27b)~V0f)~V4tx))))))))))))\end{array}
\label{hyp:ehigngpfcdfemgmgjgdhehpfcdfemeoeefiepfpfmeffoegepemeee}
\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} V0f \in ((ty\_2Eoption\_2Eoption~(ty\_2Epair\_2Eprod~A\_27b~\\
A\_27a))^{{A\_27b}}) .  ({\unicdcdadad} V1x \in A\_27b .  ({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  \\
(((ap~(ap~(c\_2Ellist\_2ELTAKE~A\_27a)~\mname{c\_2Enum\_2E0})~(ap~(ap~(\mname{c\_2Ellist\_2ELUNFOLD}~\\
A\_27a~A\_27b)~V0f)~V1x))=(ap~(\mname{c\_2Eoption\_2ESOME}~(ty\_2Elist\_2Elist~\\
A\_27a))~(c\_2Elist\_2ENIL~A\_27a))){\unicdcdcdhd}((ap~(ap~(c\_2Ellist\_2ELTAKE~\\
A\_27a)~(ap~\mname{c\_2Enum\_2ESUC}~V2n))~(ap~(ap~(\mname{c\_2Ellist\_2ELUNFOLD}~A\_27a~\\
A\_27b)~V0f)~V1x))=(ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~(ty\_2Epair\_2Eprod~\\
A\_27b~A\_27a)~(ty\_2Eoption\_2Eoption~(ty\_2Elist\_2Elist~A\_27a)))~\\
(ap~V0f~V1x))~(\mname{c\_2Eoption\_2ENONE}~(ty\_2Elist\_2Elist~A\_27a)))~(\\
{\uniddcgcg} V3v \in (ty\_2Epair\_2Eprod~A\_27b~A\_27a) .  (ap~(ap~(\mname{c\_2Epair\_2Epair\_\_CASE}~\\
(ty\_2Eoption\_2Eoption~(ty\_2Elist\_2Elist~A\_27a))~A\_27b~A\_27a)~\\
V3v)~({\uniddcgcg} V4tx \in A\_27b .  ({\uniddcgcg} V5hx \in A\_27a .  (ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~\\
(ty\_2Elist\_2Elist~A\_27a)~(ty\_2Elist\_2Elist~A\_27a))~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V5hx))~(ap~(ap~(c\_2Ellist\_2ELTAKE~A\_27a)~V2n)~(ap~(ap~(\mname{c\_2Ellist\_2ELUNFOLD}~\\
A\_27a~A\_27b)~V0f)~V4tx)))))))))))))\end{array}
\label{hyp:ehigngpfcdfemgmgjgdhehpfcdfemeefbelefepfpfmeffoegepemeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0P \in (2^{{ty\_2Enum\_2Enum}}) .  (((p~(ap~V0P~\mname{c\_2Enum\_2E0})){\unicdcdcdhd}\\
({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ((p~(ap~V0P~V1n)){\unicdbdjdcd}(p~(ap~V0P~(ap~\mname{c\_2Enum\_2ESUC}~\\
V1n)))))){\unicdbdjdcd}({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  (p~(ap~V0P~V2n)))))\end{array}
\label{hyp:ehigngpfcdfeogfhngpfcdfejeoeeeffdeefjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
\mname{c\_2Enum\_2E0})~V0n)=V0n)){\unicdcdcdhd}(({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ((ap~\\
(ap~c\_2Earithmetic\_2E\_2B~V1n)~\mname{c\_2Enum\_2E0})=V1n)){\unicdcdcdhd}(({\unicdcdadad} V2n \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V3m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V2n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V3m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Enumeral\_2EiZ}~(ap~\\
(ap~c\_2Earithmetic\_2E\_2B~V2n)~V3m)))))){\unicdcdcdhd}(({\unicdcdadad} V4n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2E\_2A~\mname{c\_2Enum\_2E0})~V4n)=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}\\
(({\unicdcdadad} V5n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V5n)~\mname{c\_2Enum\_2E0})=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V6n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V7m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~(\\
ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V6n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V7m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V6n)~V7m))))){\unicdcdcdhd}(({\unicdcdadad} V8n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
\mname{c\_2Enum\_2E0})~V8n)=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V9n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2E\_2D~V9n)~\mname{c\_2Enum\_2E0})=V9n)){\unicdcdcdhd}(({\unicdcdadad} V10n \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V11m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V10n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V11m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V10n)~V11m))))){\unicdcdcdhd}(({\unicdcdadad} V12n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EEXP~\\
\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V12n)))=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V13n \in ty\_2Enum\_2Enum .  ((ap~\\
(ap~c\_2Earithmetic\_2EEXP~\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V13n)))=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V14n \in \\
ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EEXP~V14n)~\mname{c\_2Enum\_2E0})=\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}\\
(({\unicdcdadad} V15n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V16m \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2EEXP~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V15n))~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V16m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~(ap~c\_2Earithmetic\_2EEXP~V15n)~V16m))))){\unicdcdcdhd}(((ap~\mname{c\_2Enum\_2ESUC}~\\
\mname{c\_2Enum\_2E0})=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}(({\unicdcdadad} V17n \in ty\_2Enum\_2Enum .  (\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V17n))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Enum\_2ESUC}~V17n)))){\unicdcdcdhd}(((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\mname{c\_2Enum\_2E0})=\\
\mname{c\_2Enum\_2E0}){\unicdcdcdhd}(({\unicdcdadad} V18n \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V18n))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~V18n)))){\unicdcdcdhd}(({\unicdcdadad} V19n \in ty\_2Enum\_2Enum .  \\
(((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V19n)=\mname{c\_2Enum\_2E0}){\unicdbdjded}(V19n=\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}\\
(({\unicdcdadad} V20n \in ty\_2Enum\_2Enum .  ((\mname{c\_2Enum\_2E0}=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V20n)){\unicdbdjded}(V20n=\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}(({\unicdcdadad} V21n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V22m \in ty\_2Enum\_2Enum .  (((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V21n)=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V22m)){\unicdbdjded}(V21n=V22m)))){\unicdcdcdhd}\\
(({\unicdcdadad} V23n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V23n)~\mname{c\_2Enum\_2E0})){\unicdbdjded}False)){\unicdcdcdhd}(({\unicdcdadad} V24n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V24n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Earithmetic\_2EZERO})~\\
V24n)))){\unicdcdcdhd}(({\unicdcdadad} V25n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V26m \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V25n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V26m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V25n)~V26m))))){\unicdcdcdhd}(({\unicdcdadad} V27n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~\\
\mname{c\_2Enum\_2E0})~V27n)){\unicdbdjded}False)){\unicdcdcdhd}(({\unicdcdadad} V28n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V28n))~\mname{c\_2Enum\_2E0})){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Earithmetic\_2EZERO})~\\
V28n)))){\unicdcdcdhd}(({\unicdcdadad} V29n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V30m \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V29n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V30m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V30m)~V29n))))){\unicdcdcdhd}(({\unicdcdadad} V31n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
\mname{c\_2Enum\_2E0})~V31n)){\unicdbdjded}True)){\unicdcdcdhd}(({\unicdcdadad} V32n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V32n))~\mname{c\_2Enum\_2E0})){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V32n)~\\
\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}(({\unicdcdadad} V33n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V34m \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V33n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V34m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V33n)~V34m))))){\unicdcdcdhd}(\\
({\unicdcdadad} V35n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~\\
V35n)~\mname{c\_2Enum\_2E0})){\unicdbdjded}True)){\unicdcdcdhd}(({\unicdcdadad} V36n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~\mname{c\_2Enum\_2E0})~V36n)){\unicdbdjded}(V36n=\\
\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(({\unicdcdadad} V37n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V38m \in \\
ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V37n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V38m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V38m)~V37n))))){\unicdcdcdhd}(({\unicdcdadad} V39n \in ty\_2Enum\_2Enum .  ((p~(ap~c\_2Earithmetic\_2EODD~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V39n))){\unicdbdjded}(p~(ap~c\_2Earithmetic\_2EODD~\\
V39n)))){\unicdcdcdhd}(({\unicdcdadad} V40n \in ty\_2Enum\_2Enum .  ((p~(ap~c\_2Earithmetic\_2EEVEN~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V40n))){\unicdbdjded}(p~(ap~c\_2Earithmetic\_2EEVEN~\\
V40n)))){\unicdcdcdhd}((\preoh (p~(ap~c\_2Earithmetic\_2EODD~\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(p~(\\
ap~c\_2Earithmetic\_2EEVEN~\mname{c\_2Enum\_2E0})))))))))))))))))))))))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfegjgdhehchjgcg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\mname{c\_2Earithmetic\_2EZERO})=\mname{c\_2Earithmetic\_2EZERO}){\unicdcdcdhd}\\
(((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))=\\
\mname{c\_2Earithmetic\_2EZERO}){\unicdcdcdhd}(({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~\\
\mname{c\_2Eprim\_\_rec\_2EPRE}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~(ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~(ap~\\
\mname{c\_2Earithmetic\_2EBIT1}~V0n))))){\unicdcdcdhd}(({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  \\
((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V1n)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V1n)))){\unicdcdcdhd}({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V2n))=(ap~\mname{c\_2Earithmetic\_2EBIT1}~V2n)))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfahchfg}
\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 .  (((ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V0x)=(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V1y)){\unicdbdjded}(V0x=V1y))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfedfpenefepfpfbdbd}
\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} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1x \in A\_27a .  ((ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~\\
A\_27a~A\_27b)~V0f)~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V1x))=(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27b)~(ap~V0f~V1x))))){\unicdcdcdhd}({\unicdcdadad} V2f \in (A\_27b^{{A\_27a}}) .  ((ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~\\
A\_27a~A\_27b)~V2f)~(\mname{c\_2Eoption\_2ENONE}~A\_27a))=(\mname{c\_2Eoption\_2ENONE}~\\
A\_27b))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfepeafefjepeoepfpfnebeafpfpfeefege}
\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} V0e \in A\_27b .  ({\unicdcdadad} V1f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V2e \in (\\
ty\_2Eoption\_2Eoption~A\_27a) .  (({\unicdcdadad} V3x \in A\_27a .  ({\unicdcdadad} V4y \in \\
A\_27a .  (((ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V3x)=(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V4y)){\unicdbdjded}(V3x=V4y)))){\unicdcdcdhd}(({\unicdcdadad} V5x \in A\_27a .  ((ap~(c\_2Eoption\_2ETHE~\\
A\_27a)~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V5x))=V5x)){\unicdcdcdhd}(({\unicdcdadad} V6x \in \\
A\_27a .  (\preoh ((\mname{c\_2Eoption\_2ENONE}~A\_27a)=(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V6x)))){\unicdcdcdhd}(({\unicdcdadad} V7x \in A\_27a .  (\preoh ((ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V7x)=(\mname{c\_2Eoption\_2ENONE}~A\_27a)))){\unicdcdcdhd}(({\unicdcdadad} V8x \in A\_27a .  \\
((p~(ap~(c\_2Eoption\_2EIS\_\_SOME~A\_27a)~(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V8x))){\unicdbdjded}True)){\unicdcdcdhd}(((p~(ap~(c\_2Eoption\_2EIS\_\_SOME~A\_27a)~\\
(\mname{c\_2Eoption\_2ENONE}~A\_27a))){\unicdbdjded}False){\unicdcdcdhd}(({\unicdcdadad} V9x \in (ty\_2Eoption\_2Eoption~\\
A\_27a) .  ((p~(ap~(c\_2Eoption\_2EIS\_\_NONE~A\_27a)~V9x)){\unicdbdjded}(V9x=(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a)))){\unicdcdcdhd}(({\unicdcdadad} V10x \in (ty\_2Eoption\_2Eoption~A\_27a) .  ((\preoh \\
(p~(ap~(c\_2Eoption\_2EIS\_\_SOME~A\_27a)~V10x))){\unicdbdjded}(V10x=(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a)))){\unicdcdcdhd}(({\unicdcdadad} V11x \in (ty\_2Eoption\_2Eoption~A\_27a) .  ((p~\\
(ap~(c\_2Eoption\_2EIS\_\_SOME~A\_27a)~V11x)){\unicdbdjdcd}((ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~(ap~(c\_2Eoption\_2ETHE~A\_27a)~V11x))=V11x))){\unicdcdcdhd}(({\unicdcdadad} V12x \in \\
(ty\_2Eoption\_2Eoption~A\_27a) .  ((ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~\\
A\_27a~(ty\_2Eoption\_2Eoption~A\_27a))~V12x)~(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a))~(\mname{c\_2Eoption\_2ESOME}~A\_27a))=V12x)){\unicdcdcdhd}(({\unicdcdadad} V13x \in (\\
ty\_2Eoption\_2Eoption~A\_27a) .  ((ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~\\
A\_27a~(ty\_2Eoption\_2Eoption~A\_27a))~V13x)~V13x)~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a))=V13x)){\unicdcdcdhd}(({\unicdcdadad} V14x \in (ty\_2Eoption\_2Eoption~A\_27a) .  \\
((p~(ap~(c\_2Eoption\_2EIS\_\_NONE~A\_27a)~V14x)){\unicdbdjdcd}((ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~\\
A\_27a~A\_27b)~V14x)~V0e)~V1f)=V0e))){\unicdcdcdhd}(({\unicdcdadad} V15x \in (ty\_2Eoption\_2Eoption~\\
A\_27a) .  ((p~(ap~(c\_2Eoption\_2EIS\_\_SOME~A\_27a)~V15x)){\unicdbdjdcd}((ap~(ap~\\
(ap~(c\_2Eoption\_2Eoption\_\_CASE~A\_27a~A\_27b)~V15x)~V0e)~V1f)=(\\
ap~V1f~(ap~(c\_2Eoption\_2ETHE~A\_27a)~V15x))))){\unicdcdcdhd}(({\unicdcdadad} V16x \in \\
(ty\_2Eoption\_2Eoption~A\_27a) .  ((p~(ap~(c\_2Eoption\_2EIS\_\_SOME~\\
A\_27a)~V16x)){\unicdbdjdcd}((ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~A\_27a~(\\
ty\_2Eoption\_2Eoption~A\_27a))~V16x)~V2e)~(\mname{c\_2Eoption\_2ESOME}~A\_27a))=\\
V16x))){\unicdcdcdhd}(({\unicdcdadad} V17v \in A\_27b .  ({\unicdcdadad} V18f \in (A\_27b^{{A\_27a}}) .  (\\
(ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~A\_27a~A\_27b)~(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a))~V17v)~V18f)=V17v))){\unicdcdcdhd}(({\unicdcdadad} V19x \in A\_27a .  ({\unicdcdadad} V20v \in \\
A\_27b .  ({\unicdcdadad} V21f \in (A\_27b^{{A\_27a}}) .  ((ap~(ap~(ap~(c\_2Eoption\_2Eoption\_\_CASE~\\
A\_27a~A\_27b)~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V19x))~V20v)~V21f)=\\
(ap~V21f~V19x))))){\unicdcdcdhd}(({\unicdcdadad} V22f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V23x \in \\
A\_27a .  ((ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~A\_27a~A\_27b)~V22f)~(\\
ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V23x))=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27b)~\\
(ap~V22f~V23x))))){\unicdcdcdhd}(({\unicdcdadad} V24f \in (A\_27b^{{A\_27a}}) .  ((ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~\\
A\_27a~A\_27b)~V24f)~(\mname{c\_2Eoption\_2ENONE}~A\_27a))=(\mname{c\_2Eoption\_2ENONE}~\\
A\_27b))){\unicdcdcdhd}(((ap~(c\_2Eoption\_2EOPTION\_\_JOIN~A\_27a)~(\mname{c\_2Eoption\_2ENONE}~\\
(ty\_2Eoption\_2Eoption~A\_27a)))=(\mname{c\_2Eoption\_2ENONE}~A\_27a)){\unicdcdcdhd}\\
({\unicdcdadad} V25x \in (ty\_2Eoption\_2Eoption~A\_27a) .  ((ap~(c\_2Eoption\_2EOPTION\_\_JOIN~\\
A\_27a)~(ap~(\mname{c\_2Eoption\_2ESOME}~(ty\_2Eoption\_2Eoption~A\_27a))~\\
V25x))=V25x))))))))))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfepgahehjgpgogpfpfdemebeffdffedf}
\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} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1x \in (ty\_2Eoption\_2Eoption~\\
A\_27a) .  ({\unicdcdadad} V2y \in A\_27b .  (((ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~\\
A\_27a~A\_27b)~V0f)~V1x)=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27b)~V2y)){\unicdbdjded}({\unicdcdaddd} V3z \in \\
A\_27a .  ((V1x=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V3z)){\unicdcdcdhd}(V2y=(ap~V0f~\\
V3z))))))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfepeafefjepeoepfpfnebeafpfpffebfpfpfdfpenefe}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0f \in (A\_27b^{{A\_27c}}) .  ({\unicdcdadad} V1g \in (A\_27c^{{A\_27a}}) .  \\
({\unicdcdadad} V2x \in (ty\_2Eoption\_2Eoption~A\_27a) .  ((ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~\\
A\_27c~A\_27b)~V0f)~(ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~A\_27a~A\_27c)~\\
V1g)~V2x))=(ap~(ap~(c\_2Eoption\_2EOPTION\_\_MAP~A\_27a~A\_27b)~(ap~\\
(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27a~A\_27b~A\_27c)~V0f)~V1g))~V2x)))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfepeafefjepeoepfpfnebeafpfpfdepeneafpedffe}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2a \in A\_27a .  ({\unicdcdadad} V3b \in \\
A\_27b .  (((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y)=(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V2a)~V3b)){\unicdbdjded}((V0x=V2a){\unicdcdcdhd}(V1y=V3b)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeafbejecfpfpffebf}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2a \in A\_27a .  ({\unicdcdadad} V3b \in \\
A\_27b .  (((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y)=(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V2a)~V3b)){\unicdbdjded}((V0x=V2a){\unicdcdcdhd}(V1y=V3b)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfedemepedffeeepfpfafbejecfpfpffebf}
\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} V0x \in (ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  ({\unicdcdaddd} V1q \in A\_27a .  \\
({\unicdcdaddd} V2r \in A\_27b .  (V0x=(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~\\
V1q)~V2r)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfebecedfpfpfafbejecfpfpfefiene}
\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} V0x \in (ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  ((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~A\_27b)~(ap~(c\_2Epair\_2EFST~A\_27a~A\_27b)~V0x))~(ap~(c\_2Epair\_2ESND~\\
A\_27a~A\_27b)~V0x))=V0x))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeafbejecf}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ((ap~(c\_2Epair\_2EFST~A\_27a~\\
A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y))=V0x)))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfegedfef}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ((ap~(c\_2Epair\_2ESND~A\_27a~\\
A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y))=V1y)))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfedfoeee}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0f \in ((A\_27c^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1x \in \\
A\_27a .  ({\unicdcdadad} V2y \in A\_27b .  ((ap~(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~\\
A\_27b~A\_27c)~V0f)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V1x)~V2y))=\\
(ap~(ap~V0f~V1x)~V2y)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeffoedeffcfcfjfpfpfeefege}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27b .  ({\unicdcdadad} V1y \in A\_27c .  ({\unicdcdadad} V2f \in \\
((A\_27a^{{A\_27c}})^{{A\_27b}}) .  ((ap~(ap~(\mname{c\_2Epair\_2Epair\_\_CASE}~A\_27a~A\_27b~\\
A\_27c)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27b~A\_27c)~V0x)~V1y))~V2f)=(ap~\\
(ap~V2f~V0x)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeahbgjgchpfpfdgbgdhfgpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ((V0s=V1t){\unicdbdjded}({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V0s)){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfefeifeffeoedfjepeoe}
\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} V0f \in ((ty\_2Epair\_2Eprod~A\_27a~2)^{{A\_27b}}) .  ({\unicdcdadad} V1v \in \\
A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1v)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b)~V0f))){\unicdbdjded}({\unicdcdaddd} V2x \in A\_27b .  ((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~2)~V1v)~\mname{c\_2Ebool\_2ET})=(ap~V0f~V2x))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffedejegejedebeefjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0y \in A\_27a .  ({\unicdcdadad} V1P \in \\
(2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0y)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27a)~({\uniddcgcg} V2x \in A\_27a .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~\\
V2x)~(ap~V1P~V2x)))))){\unicdbdjded}(p~(ap~V1P~V0y)))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfhedfaffedepfpfjegege}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2P \in ((2^{{A\_27b}})^{{A\_27a}}) .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Epair\_2Eprod~A\_27a~A\_27b))~(ap~\\
(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y))~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b)~(ty\_2Epair\_2Eprod~A\_27a~A\_27b))~\\
(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~A\_27b~(ty\_2Epair\_2Eprod~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27b)~2))~({\uniddcgcg} V3x \in A\_27a .  ({\uniddcgcg} V4y \in A\_27b .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b)~2)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~\\
A\_27b)~V3x)~V4y))~(ap~(ap~V2P~V3x)~V4y)))))))){\unicdbdjded}(p~(ap~(ap~V2P~V0x)~\\
V1y))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfeafbejecfpfpfjeoepfpfhedfaffedepfpfjegege}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  (\preoh (p~(ap~(ap~\\
(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a)))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfeoepeefpfpfjeoepfpffeneafefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  (p~(ap~(\\
ap~(\mname{c\_2Epred\_\_set\_2ESUBSET}~A\_27a)~V0s)~V0s)))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfedfffcedffeefpfpfcffegeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V2x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EUNION}~A\_27a)~V0s)~V1t))){\unicdbdjded}((p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V0s)){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfffoejepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V2x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~A\_27a)~V0s)~V1t))){\unicdbdjded}((p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V0s)){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfjeoeeffecf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  (p~(ap~(ap~(\mname{c\_2Epred\_\_set\_2ESUBSET}~A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~\\
A\_27a)~V0s)~V1t))~V0s)))){\unicdcdcdhd}({\unicdcdadad} V2s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V3t \in \\
(2^{{A\_27a}}) .  (p~(ap~(ap~(\mname{c\_2Epred\_\_set\_2ESUBSET}~A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~\\
A\_27a)~V3t)~V2s))~V2s)))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoeeffecfpfpfdfffcedffeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V2x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EDIFF}~A\_27a)~V0s)~V1t))){\unicdbdjded}((p~(ap~(\\
ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V0s)){\unicdcdcdhd}(\preoh (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V1t))))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfeejegege}
\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} V2s \in (2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V0x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V1y)~V2s))){\unicdbdjded}((V0x=\\
V1y){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~V2s)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfjeoedffecfef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1s \in \\
(2^{{A\_27a}}) .  (\preoh ((ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V0x)~V1s)=\\
(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a)))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfeoepeefpfpfjeoedffecfefpfpffeneafefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1x \in \\
A\_27a .  ({\unicdcdadad} V2y \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1x)~\\
(ap~(ap~(\mname{c\_2Epred\_\_set\_2EDELETE}~A\_27a)~V0s)~V2y))){\unicdbdjded}((p~(ap~(ap~\\
(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1x)~V0s)){\unicdcdcdhd}(\preoh (V1x=V2y)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfeefemefeeffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1s \in \\
(2^{{A\_27a}}) .  (p~(ap~(ap~(\mname{c\_2Epred\_\_set\_2ESUBSET}~A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EDELETE}~\\
A\_27a)~V1s)~V0x))~V1s))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfeeefemefeeffepfpfdfffcedffeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1s \in \\
(2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~V1s)){\unicdbdjdcd}((ap~(ap~\\
(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V0x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EDELETE}~\\
A\_27a)~V1s)~V0x))=V1s))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoedffecfefpfpfeefemefeeffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ((\preoh (V0s=\\
(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
(ap~(c\_2Epred\_\_set\_2ECHOICE~A\_27a)~V0s))~V0s))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfedeiepejedefepfpfeefege}
\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 .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~\\
A\_27a)~V1y)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a)))){\unicdbdjded}(V0x=V1y))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfdfjeoehe}
\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 .  (((ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~\\
A\_27a))=(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V1y)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~\\
A\_27a))){\unicdbdjded}(V0x=V1y))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfefebfffbemepfpfdfjeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((ap~(c\_2Epred\_\_set\_2ECHOICE~\\
A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~\\
A\_27a)))=V0x))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfedeiepejedefepfpfdfjeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(p~(ap~(\mname{c\_2Epred\_\_set\_2EFINITE}~\\
A\_27a)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a)))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfegejeoejeeffepfpffeneafefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1s \in \\
(2^{{A\_27a}}) .  ((p~(ap~(\mname{c\_2Epred\_\_set\_2EFINITE}~A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EDELETE}~\\
A\_27a)~V1s)~V0x))){\unicdbdjded}(p~(ap~(\mname{c\_2Epred\_\_set\_2EFINITE}~A\_27a)~V1s)))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfegejeoejeeffepfpfeefemefeeffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}((ap~(c\_2Epred\_\_set\_2ECARD~A\_27a)~\\
(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a))=\mname{c\_2Enum\_2E0})\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfedebecfeepfpffeneafefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ((p~(ap~\\
(\mname{c\_2Epred\_\_set\_2EFINITE}~A\_27a)~V0s)){\unicdbdjdcd}({\unicdcdadad} V1x \in A\_27a .  ((\\
ap~(c\_2Epred\_\_set\_2ECARD~A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~\\
A\_27a)~V1x)~V0s))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~ty\_2Enum\_2Enum)~\\
(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1x)~V0s))~(ap~(c\_2Epred\_\_set\_2ECARD~\\
A\_27a)~V0s))~(ap~\mname{c\_2Enum\_2ESUC}~(ap~(c\_2Epred\_\_set\_2ECARD~A\_27a)~\\
V0s)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfedebecfeepfpfjeoedffecfef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ((p~(ap~\\
(\mname{c\_2Epred\_\_set\_2EFINITE}~A\_27a)~V0s)){\unicdbdjdcd}({\unicdcdadad} V1x \in A\_27a .  ((\\
ap~(c\_2Epred\_\_set\_2ECARD~A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EDELETE}~\\
A\_27a)~V0s)~V1x))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~ty\_2Enum\_2Enum)~\\
(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1x)~V0s))~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
(ap~(c\_2Epred\_\_set\_2ECARD~A\_27a)~V0s))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~(ap~(c\_2Epred\_\_set\_2ECARD~\\
A\_27a)~V0s))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfedebecfeepfpfeefemefeeffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ((p~(ap~\\
(\mname{c\_2Epred\_\_set\_2EFINITE}~A\_27a)~V0s)){\unicdbdjdcd}({\unicdcdadad} V1t \in (2^{{A\_27a}}) .  \\
((p~(ap~(ap~(\mname{c\_2Epred\_\_set\_2ESUBSET}~A\_27a)~V1t)~V0s)){\unicdbdjdcd}(p~(ap~(ap~\\
\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~(c\_2Epred\_\_set\_2ECARD~A\_27a)~V1t))~\\
(ap~(c\_2Epred\_\_set\_2ECARD~A\_27a)~V0s)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfedebecfeepfpfdfffcedffeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{({{2}^{{A\_27a}}})}}) .  ((\\
{\unicdcdadad} V1x \in (2^{{A\_27a}}) .  (({\unicdcdadad} V2y \in (2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EPSUBSET}~\\
A\_27a)~V2y)~V1x)){\unicdbdjdcd}(p~(ap~V0P~V2y)))){\unicdbdjdcd}((p~(ap~(\mname{c\_2Epred\_\_set\_2EFINITE}~\\
A\_27a)~V1x)){\unicdbdjdcd}(p~(ap~V0P~V1x))))){\unicdbdjdcd}({\unicdcdadad} V3x \in (2^{{A\_27a}}) .  ((p~(\\
ap~(\mname{c\_2Epred\_\_set\_2EFINITE}~A\_27a)~V3x)){\unicdbdjdcd}(p~(ap~V0P~V3x))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfegejeoejeeffepfpfdepeneafmefeeffepfpfjeoeeeffdeefjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
((ap~\mname{c\_2Enum\_2ESUC}~V0m)=(ap~\mname{c\_2Enum\_2ESUC}~V1n)){\unicdbdjded}(V0m=V1n))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfejeoegfpfpfdfffdepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\mname{c\_2Enum\_2E0})=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}({\unicdcdadad} V0m \in \\
ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~(ap~\mname{c\_2Enum\_2ESUC}~V0m))=\\
V0m)))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfeafcffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\preoh (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V0n)~\mname{c\_2Enum\_2E0}))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfeoepeefpfpfmefedfdfpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0n))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfemefedfdfpfpfad}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Epair\_2Eprod~A\_27a~A\_27b))~(ap~\\
(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y))~V2r)){\unicdbdjdcd}((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V0x)~(ap~(\mname{c\_2Eset\_\_relation\_2Edomain}~A\_27a~A\_27b)~V2r))){\unicdcdcdhd}\\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27b)~V1y)~(ap~(\mname{c\_2Eset\_\_relation\_2Erange}~\\
A\_27b~A\_27a)~V2r))))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfejgogpfpfegpgngpfpfchhg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  (p~(ap~(ap~(\mname{c\_2Epred\_\_set\_2ESUBSET}~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27a))~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Errestrict}~A\_27a)~V0r)~\\
V1s))~V0r))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechchfgdhehchjgdgehpfpfdfffcedffeef}
\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} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  ({\unicdcdadad} V1s \in \\
(2^{{A\_27b}}) .  ({\unicdcdadad} V2r\_27 \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  \\
({\unicdcdadad} V3s\_27 \in (2^{{A\_27b}}) .  ((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Efinite\_\_prefixes}~\\
A\_27a~A\_27b)~V0r)~V1s)){\unicdbdjdcd}((p~(ap~(ap~(\mname{c\_2Epred\_\_set\_2ESUBSET}~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27b))~V2r\_27)~V0r)){\unicdbdjdcd}((p~(ap~(ap~(\mname{c\_2Epred\_\_set\_2ESUBSET}~\\
A\_27b)~V3s\_27)~V1s)){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Efinite\_\_prefixes}~\\
A\_27a~A\_27b)~V2r\_27)~V3s\_27)))))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeggjgogjgehfgpfpfahchfgggjgihfgdhpfpfdhfhcgdhfgehpfpfchdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1lo \in \\
(2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  (p~(ap~(ap~(\mname{c\_2Epred\_\_set\_2ESUBSET}~\\
A\_27a)~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Eminimal\_\_elements}~A\_27a)~\\
V0s)~V1lo))~V0s))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfengjgogjgngbgmgpfpffgmgfgngfgogehdhpfpfdhfhcgdhfgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1r \in \\
(2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  ({\unicdcdadad} V2s\_27 \in (2^{{A\_27a}}) .  \\
((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}~A\_27a)~V1r)~V0s)){\unicdbdjdcd}\\
(p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}~A\_27a)~(ap~(ap~(\\
\mname{c\_2Eset\_\_relation\_2Errestrict}~A\_27a)~V1r)~V2s\_27))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~\\
A\_27a)~V0s)~V2s\_27)))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfemgjgogfgbgchpfpfpgchegfgchpfpfchfgdhehchjgdgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0lo \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
({\unicdcdadad} V1X \in (2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}~\\
A\_27a)~V0lo)~V1X)){\unicdbdjdcd}((ap~(ap~(\mname{c\_2Epred\_\_set\_2EUNION}~A\_27a)~(ap~\\
(\mname{c\_2Eset\_\_relation\_2Edomain}~A\_27a~A\_27a)~V0lo))~(ap~(\mname{c\_2Eset\_\_relation\_2Erange}~\\
A\_27a~A\_27a)~V0lo))=V1X))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfemgjgogfgbgchpfpfpgchegfgchpfpfegpgngpfpfchhg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0lo \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
({\unicdcdadad} V1X \in (2^{{A\_27a}}) .  ({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}~\\
A\_27a)~V0lo)~V1X)){\unicdbdjdcd}((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V1X)){\unicdbdjdcd}\\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Epair\_2Eprod~A\_27a~A\_27a))~(ap~(\\
ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27a)~V2x)~V2x))~V0lo)))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfemgjgogfgbgchpfpfpgchegfgchpfpfchfgggmg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0lo \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
({\unicdcdadad} V1X \in (2^{{A\_27a}}) .  ({\unicdcdadad} V2x \in A\_27a .  ({\unicdcdadad} V3y \in A\_27a .  \\
((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}~A\_27a)~V0lo)~V1X)){\unicdbdjdcd}\\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Epair\_2Eprod~A\_27a~A\_27a))~(ap~\\
(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27a)~V2x)~V3y))~V0lo)){\unicdbdjdcd}((p~(ap~(ap~(\\
\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V1X)){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V3y)~V1X)))))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfemgjgogfgbgchpfpfpgchegfgchpfpfjgogpfpfdhfgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1X \in \\
(2^{{A\_27a}}) .  ({\unicdcdadad} V2lo \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
({\unicdcdadad} V3y \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~V1X)){\unicdbdjdcd}\\
((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}~A\_27a)~V2lo)~V1X)){\unicdbdjdcd}\\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V3y)~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Eminimal\_\_elements}~\\
A\_27a)~V1X)~V2lo))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27a))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27a)~V3y)~V0x))~V2lo)))))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfejeoepfpfnejeoepfpfmepe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1r \in \\
(2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  ({\unicdcdadad} V2s \in (2^{{A\_27a}}) .  \\
({\unicdcdadad} V3s\_27 \in (2^{{A\_27a}}) .  (((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}~\\
A\_27a)~V1r)~V2s)){\unicdcdcdhd}((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Efinite\_\_prefixes}~\\
A\_27a~A\_27a)~V1r)~V2s)){\unicdcdcdhd}((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~V3s\_27)){\unicdcdcdhd}\\
(p~(ap~(ap~(\mname{c\_2Epred\_\_set\_2ESUBSET}~A\_27a)~V3s\_27)~V2s))))){\unicdbdjdcd}(p~\\
(ap~(\mname{c\_2Epred\_\_set\_2ESING}~A\_27a)~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Eminimal\_\_elements}~\\
A\_27a)~V3s\_27)~V1r))))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeggjgogjgehfgpfpfahchfgggjgihpfpfmgjgogfgbgchpfpfpgchegfgchpfpfigbgdhpfpffhogjgbhfhfgpfpfngjgogjgngbgmg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfemgmgjgdhehpfcdfemgjgogfgbgchpfpfpgchegfgchpfpfehpgpfpfmgmgjgdhehpfpffgbh}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0lo \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
({\unicdcdadad} V1X \in (2^{{A\_27a}}) .  (((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}~\\
A\_27a)~V0lo)~V1X)){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Efinite\_\_prefixes}~\\
A\_27a~A\_27a)~V0lo)~V1X))){\unicdbdjdcd}({\unicdcdaddd} V2ll \in (ty\_2Ellist\_2Ellist~\\
A\_27a) .  ((V1X=(ap~(c\_2Epred\_\_set\_2EGSPEC~A\_27a~A\_27a)~({\uniddcgcg} V3x \in \\
A\_27a .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~V3x)~(ap~(\mname{c\_2Ebool\_2E\_3F}~\\
ty\_2Enum\_2Enum)~({\uniddcgcg} V4i \in ty\_2Enum\_2Enum .  (ap~(ap~(\mname{c\_2Emin\_2E\_3D}~\\
(ty\_2Eoption\_2Eoption~A\_27a))~(ap~(ap~(c\_2Ellist\_2ELNTH~A\_27a)~\\
V4i)~V2ll))~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V3x)))))))){\unicdcdcdhd}((V0lo=\\
(ap~(c\_2Epred\_\_set\_2EGSPEC~(ty\_2Epair\_2Eprod~A\_27a~A\_27a)~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27a))~(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~A\_27a~(ty\_2Epair\_2Eprod~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27a)~2))~({\uniddcgcg} V5x \in A\_27a .  ({\uniddcgcg} V6y \in \\
A\_27a .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Epair\_2Eprod~A\_27a~A\_27a)~\\
2)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27a)~V5x)~V6y))~(ap~(\mname{c\_2Ebool\_2E\_3F}~\\
ty\_2Enum\_2Enum)~({\uniddcgcg} V7i \in ty\_2Enum\_2Enum .  (ap~(\mname{c\_2Ebool\_2E\_3F}~\\
ty\_2Enum\_2Enum)~({\uniddcgcg} V8j \in ty\_2Enum\_2Enum .  (ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~\\
(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V7i)~V8j))~(ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~\\
(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~(ty\_2Eoption\_2Eoption~A\_27a))~(ap~(ap~(\\
c\_2Ellist\_2ELNTH~A\_27a)~V7i)~V2ll))~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~\\
V5x)))~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~(ty\_2Eoption\_2Eoption~A\_27a))~(\\
ap~(ap~(c\_2Ellist\_2ELNTH~A\_27a)~V8j)~V2ll))~(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V6y)))))))))))))){\unicdcdcdhd}({\unicdcdadad} V9i \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V10j \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V11x \in A\_27a .  ((((ap~(ap~(c\_2Ellist\_2ELNTH~\\
A\_27a)~V9i)~V2ll)=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V11x)){\unicdcdcdhd}((ap~(\\
ap~(c\_2Ellist\_2ELNTH~A\_27a)~V10j)~V2ll)=(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V11x))){\unicdbdjdcd}(V9i=V10j)))))))))))\end{array}$$
\end{theorem}
\end{document}
