\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\_2Ealist\_2EALOOKUP\_\_IN\_\_FRANGE (TMbFh3f2WqfMeSe5wQynQVvQsztiSXMhKuA)}
\maketitle
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 $ ty\_2Efinite\_\_map\_2Efmap : {{\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\_2Efinite\_\_map\_2Efmap~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeggjgogjgehfgpfpfngbgahpfcdfeggngbgah}
\end{equation}
Let $ c\_2Efinite\_\_map\_2Efmap\_\_REP : {{\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\_2Efinite\_\_map\_2Efmap\_\_REP~\\
A\_27a~A\_27b \in (((ty\_2Esum\_2Esum~A\_27b~ty\_2Eone\_2Eone)^{{A\_27a}})^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggjgogjgehfgpfpfngbgahpfcdfeggngbgahpfpfcffeaf}
\end{equation}
Let $ c\_2Esum\_2EISL : {{\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\_2EISL~\\
A\_27a~A\_27b \in (2^{{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfhngpfcdfejedfme}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegeeepene}
We define $\mname{c\_2Efinite\_\_map\_2EFDOM}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{c\_2Esum\_2EISL}~{A\_27b}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{({{{ap}~{({{{c\_2Efinite\_\_map\_2Efmap\_\_REP}~{A\_27a}}~{A\_27b}})}}~{V0f}})}}~{V1x}})}})}}}}}$.
\end{definition}
Let $ ty\_2Elist\_2Elist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Elist\_2Elist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgjgdhehpfcdfemgjgdheh}
\end{equation}
Let $ c\_2Ebool\_2EARB : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A\_27a .  {{{nonempty}~{A\_27a}}{\unicdbdjdcd}{{{c\_2Ebool\_2EARB}~{A\_27a}} \in {A\_27a}}}}\label{hyp:ngfgngpfdgpfcdfecgpgpgmgpfcdfebecfce}
\end{equation}
Let $ 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: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: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: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: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:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
Let $ ty\_2Epair\_2Eprod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_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}
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: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: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: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}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfecffedfef}
We define $\mname{c\_2Epred\_\_set\_2EREST}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EDELETE}}~{A\_27a}})}}~{V0s}})}}~{({{{ap}~{({{c\_2Epred\_\_set\_2ECHOICE}~{A\_27a}})}}~{V0s}})}})}}}$.
\end{definition}
Let $ c\_2Elist\_2ECONS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ECONS~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfedepeoedf}
\end{equation}
Let $ c\_2Elist\_2ENIL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ENIL~A\_27a \in (ty\_2Elist\_2Elist~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeoejeme}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfedad}
We define $\mname{c\_2Emin\_2E\_40}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} P \in {{2}^{{A}}} .  {{\mathbf{if}}~{({{\unicdcdaddd} x \in {A} .  {{p}~{({{{ap}~{P}}~{x}})}}})}~{\mathbf{then}}~{({{the}~{({{\uniddcgcg} x .  {{{x} \in {A}}{\unicdcdcdhd}{{p}~{({{{ap}~{P}}~{x}})}}}})}})}~{\mathbf{else}}~{({{the}~{({{\uniddcgcg} x .  {{x} \in {A}}})}})}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def: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: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:dgpfcdfedgpgngcgjgogpfcdfedf}
We define $\mname{c\_2Ecombin\_2ES}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1g \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V2x}})}}~{({{{ap}~{V1g}}~{V2x}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfeje}
We define $\mname{c\_2Ecombin\_2EI}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2ES}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}}~{A\_27a}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}})}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{A\_27a}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehfge}
We define $\mname{c\_2Erelation\_2EWF}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1B \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V2w \in {A\_27a} .  {({{{ap}~{V1B}}~{V2w}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3min \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1B}}~{V3min}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4b}})}}~{V3min}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V1B}}~{V4b}})}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfecffedfefcfjedeef}
We define $\mname{c\_2Erelation\_2ERESTRICT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{V1R}}~{V3y}})}}~{V2x}})}})}}~{({{{ap}~{V0f}}~{V3y}})}})}}~{({{c\_2Ebool\_2EARB}~{A\_27b}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeefde}
We define $\mname{c\_2Erelation\_2ETC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1a \in {A\_27a} .  {{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{2}^{{A\_27a}}})}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3P \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4x}})}}~{V5y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V4x}})}}~{V5y}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V7y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V8z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V7y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V7y}})}}~{V8z}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V8z}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V1a}})}}~{V2b}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfebgahahchpgih}
We define $\mname{c\_2Erelation\_2Eapprox}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{A\_27b}^{{A\_27a}}})}})}}~{V3f}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V4y \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4y}})}})}}~{V4y}})}})}})}}~{V0R}})}}~{V2x}})}})}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeehigfgpfpfggfhog}
We define $\mname{c\_2Erelation\_2Ethe\_\_fun}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{A\_27b}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Eapprox}}~{A\_27a}}~{A\_27b}})}}~{V0R}})}}~{V1M}})}}~{V2x}})}}~{V3f}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehfgecffede}
We define $\mname{c\_2Erelation\_2EWFREC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Ethe\_\_fun}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{\mname{c\_2Erelation\_2ETC}}~{A\_27a}})}}~{V0R}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V4v \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4v}})}})}}~{V4v}})}})}})}})}}~{V2x}})}})}}~{V0R}})}}~{V2x}})}})}}~{V2x}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfemgjgdhehpfcdfedffeefpfpfefpepfpfmejedfef}
We define $\mname{c\_2Elist\_2ESET\_\_TO\_\_LIST}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2EWFREC}}~{({{2}^{{A\_27a}}})}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{({{2}^{{({{2}^{{A\_27a}}})}}})}^{{({{2}^{{A\_27a}}})}}})}})}}~{({{\uniddcgcg} V0R \in {({{({{2}^{{({{2}^{{A\_27a}}})}}})}^{{({{2}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Erelation\_2EWF}}~{({{2}^{{A\_27a}}})}})}}~{V0R}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EFINITE}}~{A\_27a}})}}~{V1s}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V1s}})}}~{({{\mname{c\_2Epred\_\_set\_2EEMPTY}}~{A\_27a}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EREST}}~{A\_27a}})}}~{V1s}})}})}}~{V1s}})}})}})}})}})}})}})}})}}~{({{\uniddcgcg} V2SET\_\_TO\_\_LIST \in {({{({{ty\_2Elist\_2Elist}~{A\_27a}})}^{{({{2}^{{A\_27a}}})}}})} .  {({{\uniddcgcg} V3a \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EFINITE}}~{A\_27a}})}}~{V3a}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V3a}})}}~{({{\mname{c\_2Epred\_\_set\_2EEMPTY}}~{A\_27a}})}})}})}}~{({{c\_2Elist\_2ENIL}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2ECONS}~{A\_27a}})}}~{({{{ap}~{({{c\_2Epred\_\_set\_2ECHOICE}~{A\_27a}})}}~{V3a}})}})}}~{({{{ap}~{V2SET\_\_TO\_\_LIST}}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EREST}}~{A\_27a}})}}~{V3a}})}})}})}})}})}}~{({{c\_2Ebool\_2EARB}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ c\_2Esum\_2EOUTL : {{\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\_2EOUTL~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfhngpfcdfepeffefme}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegebeafafmejf}
We define $\mname{c\_2Efinite\_\_map\_2EFAPPLY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{c\_2Esum\_2EOUTL}~{A\_27b}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{({{{ap}~{({{{c\_2Efinite\_\_map\_2Efmap\_\_REP}~{A\_27a}}~{A\_27b}})}}~{V0f}})}}~{V1x}})}})}}}}}$.
\end{definition}
Let $ c\_2Elist\_2EMAP : {{\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\_2Elist\_2EMAP~\\
A\_27a~A\_27b \in (((ty\_2Elist\_2Elist~A\_27b)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{A\_27b}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfenebeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgmgjgdhehpfcdfeggngbgahpfpfehpgpfpfbgmgjgdheh}
We define $\mname{c\_2Ealist\_2Efmap\_\_to\_\_alist}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0s \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{({{{c\_2Elist\_2EMAP}~{A\_27a}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})}}~{({{\uniddcgcg} V1k \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27b}})}}~{V1k}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFAPPLY}}~{A\_27a}}~{A\_27b}})}}~{V0s}})}}~{V1k}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Elist\_2ESET\_\_TO\_\_LIST}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFDOM}}~{A\_27a}}~{A\_27b}})}}~{V0s}})}})}})}}}}$.
\end{definition}
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 $ c\_2Ealist\_2EALOOKUP : {{\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\_2Ealist\_2EALOOKUP~\\
A\_27a~A\_27b \in (((ty\_2Eoption\_2Eoption~A\_27a)^{{A\_27b}})^{{({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgmgjgdhehpfcdfebemepepeleffaf}
\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}
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}
\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}
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: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}
Let $ c\_2Efinite\_\_map\_2Efmap\_\_ABS : {{\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\_2Efinite\_\_map\_2Efmap\_\_ABS~\\
A\_27a~A\_27b \in ((ty\_2Efinite\_\_map\_2Efmap~A\_27a~A\_27b)^{{({{({{{ty\_2Esum\_2Esum}~{A\_27b}}~{ty\_2Eone\_2Eone}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggjgogjgehfgpfpfngbgahpfcdfeggngbgahpfpfbecedf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegefeneafefjf}
We define $\mname{c\_2Efinite\_\_map\_2EFEMPTY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{{ap}~{({{{c\_2Efinite\_\_map\_2Efmap\_\_ABS}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V0a \in {A\_27a} .  {({{{ap}~{({{{\mname{c\_2Esum\_2EINR}}~{A\_27b}}~{ty\_2Eone\_2Eone}})}}~{\mname{c\_2Eone\_2Eone}}})}})}})}}}$.
\end{definition}
Let $ c\_2Efinite\_\_map\_2EFUPDATE : {{\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\_2Efinite\_\_map\_2EFUPDATE~\\
A\_27a~A\_27b \in (((ty\_2Efinite\_\_map\_2Efmap~A\_27a~A\_27b)^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegeffafeebeeffe}
\end{equation}
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}
\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\_2Elist\_2EFOLDR : {{\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\_2Elist\_2EFOLDR~\\
A\_27a~A\_27b \in (((A\_27b^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{A\_27b}})^{{({{({{A\_27b}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfegepemeeecf}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgmgjgdhehpfcdfebgmgjgdhehpfpfehpgpfpfggngbgah}
We define $\mname{c\_2Ealist\_2Ealist\_\_to\_\_fmap}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0s \in {({{ty\_2Elist\_2Elist}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{c\_2Elist\_2EFOLDR}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}~{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27a}}~{A\_27b}}~{({{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})}^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})}}})}})}}~{({{\uniddcgcg} V1k \in {A\_27a} .  {({{\uniddcgcg} V2v \in {A\_27b} .  {({{\uniddcgcg} V3f \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{({{{c\_2Efinite\_\_map\_2EFUPDATE}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27b}})}}~{V1k}})}}~{V2v}})}})}})}})}})}})}})}}~{({{{\mname{c\_2Efinite\_\_map\_2EFEMPTY}}~{A\_27a}}~{A\_27b}})}})}}~{V0s}})}}}}$.
\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 $ 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: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: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}
\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:dgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegemepepeleffaf}
We define $\mname{c\_2Efinite\_\_map\_2EFLOOKUP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V1x}})}}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFDOM}}~{A\_27a}}~{A\_27b}})}}~{V0f}})}})}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFAPPLY}}~{A\_27a}}~{A\_27b}})}}~{V0f}})}}~{V1x}})}})}})}}~{({{\mname{c\_2Eoption\_2ENONE}}~{A\_27b}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeffoejegf}
We define $\mname{c\_2Epred\_\_set\_2EUNIV}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfedepeneafme}
We define $\mname{c\_2Epred\_\_set\_2ECOMPL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EDIFF}}~{A\_27a}})}}~{({{\mname{c\_2Epred\_\_set\_2EUNIV}}~{A\_27a}})}})}}~{V0P}})}}}$.
\end{definition}
Let $ c\_2Efinite\_\_map\_2EDRESTRICT : {{\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\_2Efinite\_\_map\_2EDRESTRICT~\\
A\_27a~A\_27b \in (((ty\_2Efinite\_\_map\_2Efmap~A\_27a~A\_27b)^{{({{2}^{{A\_27a}}})}})^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggjgogjgehfgpfpfngbgahpfcdfeeecffedfefcfjedeef}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggjgogjgehfgpfpfngbgahpfcdfeggegpgngdhfhcg}
We define $\mname{c\_2Efinite\_\_map\_2Efdomsub}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0fm \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V1k \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{c\_2Efinite\_\_map\_2EDRESTRICT}~{A\_27a}}~{A\_27b}})}}~{V0fm}})}}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ECOMPL}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EINSERT}}~{A\_27a}})}}~{V1k}})}}~{({{\mname{c\_2Epred\_\_set\_2EEMPTY}}~{A\_27a}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegecfbeoehefe}
We define $\mname{c\_2Efinite\_\_map\_2EFRANGE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27b}}~{A\_27b}})}}~{({{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27b}}~{2}})}}~{V1y}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFDOM}}~{A\_27a}}~{A\_27b}})}}~{V0f}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFAPPLY}}~{A\_27a}}~{A\_27b}})}}~{V0f}})}}~{V2x}})}})}}~{V1y}})}})}})}})}})}})}})}}}}$.
\end{definition}
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} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0k \in A\_27c .  \\
({\unicdcdadad} V1v \in A\_27d .  ({\unicdcdadad} V2t \in (ty\_2Elist\_2Elist~(ty\_2Epair\_2Eprod~\\
A\_27c~A\_27d)) .  (((ap~(\mname{c\_2Ealist\_2Ealist\_\_to\_\_fmap}~A\_27a~A\_27b)~\\
(c\_2Elist\_2ENIL~(ty\_2Epair\_2Eprod~A\_27a~A\_27b)))=(\mname{c\_2Efinite\_\_map\_2EFEMPTY}~\\
A\_27a~A\_27b)){\unicdcdcdhd}((ap~(\mname{c\_2Ealist\_2Ealist\_\_to\_\_fmap}~A\_27c~A\_27d)~\\
(ap~(ap~(c\_2Elist\_2ECONS~(ty\_2Epair\_2Eprod~A\_27c~A\_27d))~(ap~(\\
ap~(\mname{c\_2Epair\_2E\_2C}~A\_27c~A\_27d)~V0k)~V1v))~V2t))=(ap~(ap~(c\_2Efinite\_\_map\_2EFUPDATE~\\
A\_27c~A\_27d)~(ap~(\mname{c\_2Ealist\_2Ealist\_\_to\_\_fmap}~A\_27c~A\_27d)~V2t))~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27c~A\_27d)~V0k)~V1v)))))))\end{array}
\label{hyp:ehigngpfcdfebgmgjgdhehpfcdfebgmgjgdhehpfpfehpgpfpfggngbgahpfpfehigng}
\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} V0q \in A\_27b .  ((ap~(ap~(c\_2Ealist\_2EALOOKUP~A\_27a~A\_27b)~\\
(c\_2Elist\_2ENIL~(ty\_2Epair\_2Eprod~A\_27b~A\_27a)))~V0q)=(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a))){\unicdcdcdhd}({\unicdcdadad} V1y \in A\_27a .  ({\unicdcdadad} V2x \in A\_27b .  ({\unicdcdadad} V3t \in \\
(ty\_2Elist\_2Elist~(ty\_2Epair\_2Eprod~A\_27b~A\_27a)) .  ({\unicdcdadad} V4q \in \\
A\_27b .  ((ap~(ap~(c\_2Ealist\_2EALOOKUP~A\_27a~A\_27b)~(ap~(ap~(c\_2Elist\_2ECONS~\\
(ty\_2Epair\_2Eprod~A\_27b~A\_27a))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27b~\\
A\_27a)~V2x)~V1y))~V3t))~V4q)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~(ty\_2Eoption\_2Eoption~\\
A\_27a))~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~A\_27b)~V2x)~V4q))~(ap~(\mname{c\_2Eoption\_2ESOME}~\\
A\_27a)~V1y))~(ap~(ap~(c\_2Ealist\_2EALOOKUP~A\_27a~A\_27b)~V3t)~V4q))))))))\end{array}
\label{hyp:ehigngpfcdfebgmgjgdhehpfcdfebemepepeleffafpfpfegfggg}
\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} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0al \in (\\
ty\_2Elist\_2Elist~(ty\_2Epair\_2Eprod~A\_27a~A\_27b)) .  ({\unicdcdadad} V1fm \in \\
(ty\_2Efinite\_\_map\_2Efmap~A\_27c~A\_27d) .  (((ap~(\mname{c\_2Efinite\_\_map\_2EFLOOKUP}~\\
A\_27a~A\_27b)~(ap~(\mname{c\_2Ealist\_2Ealist\_\_to\_\_fmap}~A\_27a~A\_27b)~V0al))=\\
(ap~(c\_2Ealist\_2EALOOKUP~A\_27b~A\_27a)~V0al)){\unicdcdcdhd}((ap~(c\_2Ealist\_2EALOOKUP~\\
A\_27d~A\_27c)~(ap~(\mname{c\_2Ealist\_2Efmap\_\_to\_\_alist}~A\_27c~A\_27d)~V1fm))=\\
(ap~(\mname{c\_2Efinite\_\_map\_2EFLOOKUP}~A\_27c~A\_27d)~V1fm)))))\end{array}
\label{hyp:ehigngpfcdfebgmgjgdhehpfcdfebemepepeleffafpfpffebfpfpfgemepepeleffaf}
\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} V0al \in (ty\_2Elist\_2Elist~(ty\_2Epair\_2Eprod~A\_27a~A\_27b)) .  \\
((ap~(\mname{c\_2Efinite\_\_map\_2EFDOM}~A\_27a~A\_27b)~(ap~(\mname{c\_2Ealist\_2Ealist\_\_to\_\_fmap}~\\
A\_27a~A\_27b)~V0al))=(ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a)~(ap~\\
(ap~(c\_2Elist\_2EMAP~(ty\_2Epair\_2Eprod~A\_27a~A\_27b)~A\_27a)~(c\_2Epair\_2EFST~\\
A\_27a~A\_27b))~V0al))))\end{array}
\label{hyp:ehigngpfcdfebgmgjgdhehpfcdfegeeepenepfpfbgmgjgdhehpfpfehpgpfpfggngbgah}
\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} V0t \in 2 .  (({\unicdcdadad} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepecfbememepfpfdfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\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}
{{\unicdcdadad} A\_27a .  {{{nonempty}~{A\_27a}}{\unicdbdjdcd}{({{\unicdcdadad} V0x \in {A\_27a} .  {({{V0x}={V0x}})}})}}}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfcffegeme}
\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} 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 .  ({\unicdcdadad} V1Q \in (\\
2^{{A\_27a}}) .  (((p~V0P){\unicdcdcdhd}({\unicdcdadad} V2x \in A\_27a .  (p~(ap~V1Q~V2x)))){\unicdbdjded}({\unicdcdadad} V3x \in \\
A\_27a .  ((p~V0P){\unicdcdcdhd}(p~(ap~V1Q~V3x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecfjeheieefpfpfbeoeeepfpfgepecfbememepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  ({\unicdcdadad} V2C \in 2 .  (((p~V0A){\unicdcdcdid}(\\
(p~V1B){\unicdcdcdid}(p~V2C))){\unicdbdjded}(((p~V0A){\unicdcdcdid}(p~V1B)){\unicdcdcdid}(p~V2C))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeejedfkepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((p~V0A){\unicdcdcdid}(p~V1B)){\unicdbdjded}((p~V1B){\unicdcdcdid}\\
(p~V0A)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeejedfkepfpfdepenene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  ({\unicdcdadad} V2C \in 2 .  (((p~V0A){\unicdcdcdid}(\\
(p~V1B){\unicdcdcdhd}(p~V2C))){\unicdbdjded}(((p~V0A){\unicdcdcdid}(p~V1B)){\unicdcdcdhd}((p~V0A){\unicdcdcdid}(p~V2C)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemefegeefpfpfpecfpfpfpegffecfpfpfbeoeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((p~V0A){\unicdbdjdcd}(p~V1B)){\unicdbdjded}((\preoh (p~V0A)){\unicdcdcdid}\\
(p~V1B)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfeejedfkepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdbdjdcd}\\
((p~V1t2){\unicdbdjdcd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdbdjdcd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfjeneafpfpfjeoeefcfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0b \in 2 .  ({\unicdcdadad} V1f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V2g \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V3x \in A\_27a .  ((ap~(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~(A\_27b^{{A\_27a}}))~\\
V0b)~V1f)~V2g)~V3x)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27b)~V0b)~(ap~\\
V1f~V3x))~(ap~V2g~V3x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfcfbeefpecf}
\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} V1b \in 2 .  ({\unicdcdadad} V2x \in A\_27a .  \\
({\unicdcdadad} V3y \in A\_27a .  ((ap~V0f~(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\\
V1b)~V2x)~V3y))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27b)~V1b)~(ap~V0f~\\
V2x))~(ap~V0f~V3y)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfcfbeoeee}
\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} V0f \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1v \in \\
A\_27a .  (({\unicdcdadad} V2x \in A\_27a .  ((V2x=V1v){\unicdbdjdcd}(p~(ap~V0f~V2x)))){\unicdbdjded}(p~(\\
ap~V0f~V1v)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeffoehfjeoeeepfpfgepecfbememepfpfefienecd}
\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} V0fm \in (ty\_2Efinite\_\_map\_2Efmap~A\_27a~A\_27b) .  ({\unicdcdadad} V1k \in \\
A\_27a .  ((ap~(\mname{c\_2Efinite\_\_map\_2EFDOM}~A\_27a~A\_27b)~(ap~(ap~(\mname{c\_2Efinite\_\_map\_2Efdomsub}~\\
A\_27a~A\_27b)~V0fm)~V1k))=(ap~(ap~(\mname{c\_2Epred\_\_set\_2EDELETE}~A\_27a)~\\
(ap~(\mname{c\_2Efinite\_\_map\_2EFDOM}~A\_27a~A\_27b)~V0fm))~V1k))))\end{array}
\label{hyp:ehigngpfcdfeggjgogjgehfgpfpfngbgahpfcdfegeeepenepfpfeepenedfffce}
\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} V0fm \in (ty\_2Efinite\_\_map\_2Efmap~A\_27a~A\_27b) .  ({\unicdcdadad} V1k1 \in \\
A\_27a .  ({\unicdcdadad} V2k2 \in A\_27a .  ((ap~(ap~(\mname{c\_2Efinite\_\_map\_2EFAPPLY}~\\
A\_27a~A\_27b)~(ap~(ap~(\mname{c\_2Efinite\_\_map\_2Efdomsub}~A\_27a~A\_27b)~V0fm)~\\
V1k1))~V2k2)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27b)~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~\\
A\_27a)~V1k1)~V2k2))~(ap~(ap~(\mname{c\_2Efinite\_\_map\_2EFAPPLY}~A\_27a~A\_27b)~\\
(\mname{c\_2Efinite\_\_map\_2EFEMPTY}~A\_27a~A\_27b))~V2k2))~(ap~(ap~(\mname{c\_2Efinite\_\_map\_2EFAPPLY}~\\
A\_27a~A\_27b)~V0fm)~V2k2))))))\end{array}
\label{hyp:ehigngpfcdfeggjgogjgehfgpfpfngbgahpfcdfeeepenedfffcepfpfgebeafafmejfpfpfefiene}
\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} V0fm \in (ty\_2Efinite\_\_map\_2Efmap~A\_27a~A\_27b) .  ({\unicdcdadad} V1k \in \\
A\_27a .  ({\unicdcdadad} V2v \in A\_27b .  ((ap~(\mname{c\_2Efinite\_\_map\_2EFRANGE}~A\_27a~\\
A\_27b)~(ap~(ap~(c\_2Efinite\_\_map\_2EFUPDATE~A\_27a~A\_27b)~V0fm)~(\\
ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V1k)~V2v)))=(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~\\
A\_27b)~V2v)~(ap~(\mname{c\_2Efinite\_\_map\_2EFRANGE}~A\_27a~A\_27b)~(ap~(ap~\\
(\mname{c\_2Efinite\_\_map\_2Efdomsub}~A\_27a~A\_27b)~V0fm)~V1k)))))))\end{array}
\label{hyp:ehigngpfcdfeggjgogjgehfgpfpfngbgahpfcdfegecfbeoehefepfpfgeffafeebeeffepfpfeepenedfffce}
\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\_2Efinite\_\_map\_2Efmap~A\_27a~A\_27b) .  ({\unicdcdadad} V1v \in \\
A\_27b .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27b)~V1v)~(ap~(\mname{c\_2Efinite\_\_map\_2EFRANGE}~\\
A\_27a~A\_27b)~V0f))){\unicdbdjded}({\unicdcdaddd} V2k \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2k)~(ap~(\mname{c\_2Efinite\_\_map\_2EFDOM}~A\_27a~A\_27b)~V0f))){\unicdcdcdhd}(\\
(ap~(ap~(\mname{c\_2Efinite\_\_map\_2EFAPPLY}~A\_27a~A\_27b)~V0f)~V2k)=V1v))))))\end{array}
\label{hyp:ehigngpfcdfeggjgogjgehfgpfpfngbgahpfcdfejeoepfpfgecfbeoehefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}) .  \\
(((p~(ap~V0P~(c\_2Elist\_2ENIL~A\_27a))){\unicdcdcdhd}({\unicdcdadad} V1t \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((p~(ap~V0P~V1t)){\unicdbdjdcd}({\unicdcdadad} V2h \in A\_27a .  (p~(ap~V0P~(ap~(ap~(\\
c\_2Elist\_2ECONS~A\_27a)~V2h)~V1t))))))){\unicdbdjdcd}({\unicdcdadad} V3l \in (ty\_2Elist\_2Elist~\\
A\_27a) .  (p~(ap~V0P~V3l)))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemgjgdhehpfpfjgogegfhdgehjgpgog}
\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} V0x \in A\_27a .  (\preoh ((\mname{c\_2Eoption\_2ENONE}~\\
A\_27a)=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V0x))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfeoepeefpfpfoepeoefepfpfdfpenefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in 2 .  ({\unicdcdadad} V1x \in A\_27a .  \\
({\unicdcdadad} V2y \in A\_27a .  ((((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~(ty\_2Eoption\_2Eoption~\\
A\_27a))~V0P)~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V1x))~(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a))=(\mname{c\_2Eoption\_2ENONE}~A\_27a)){\unicdbdjded}(\preoh (p~V0P))){\unicdcdcdhd}((((ap~(ap~(\\
ap~(\mname{c\_2Ebool\_2ECOND}~(ty\_2Eoption\_2Eoption~A\_27a))~V0P)~(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a))~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V1x))=(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a)){\unicdbdjded}(p~V0P)){\unicdcdcdhd}((((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~(ty\_2Eoption\_2Eoption~\\
A\_27a))~V0P)~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V1x))~(\mname{c\_2Eoption\_2ENONE}~\\
A\_27a))=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V2y)){\unicdbdjded}((p~V0P){\unicdcdcdhd}(V1x=V2y))){\unicdcdcdhd}\\
(((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~(ty\_2Eoption\_2Eoption~A\_27a))~\\
V0P)~(\mname{c\_2Eoption\_2ENONE}~A\_27a))~(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~\\
V1x))=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27a)~V2y)){\unicdbdjded}((\preoh (p~V0P)){\unicdcdcdhd}(V1x=\\
V2y)))))))))\end{array}
\label{hyp:ehigngpfcdfepgahehjgpgogpfcdfejegepfpffebfffbemedfpfpfpeafefjepeoe}
\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} 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} 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}
{({{\unicdcdadad} V0t \in {2} .  {({{({\preoh{{({\preoh{{({{p}~{V0t}})}}})}}})}{\unicdbdjded}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeoepeefpfpfoepeef}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0A \in {2} .  {({{({{p}~{V0A}})}{\unicdbdjdcd}{({{({\preoh{{({{p}~{V0A}})}}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfebeoeeepfpfjeoegfpfpfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((p~V0A){\unicdcdcdid}(p~V1B))){\unicdbdjdcd}False){\unicdbdjded}\\
(((p~V0A){\unicdbdjdcd}False){\unicdbdjdcd}((\preoh (p~V1B)){\unicdbdjdcd}False)))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfepecfpfpfeeffbemecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((\preoh (p~V0A)){\unicdcdcdid}(p~V1B))){\unicdbdjdcd}False){\unicdbdjded}\\
((p~V0A){\unicdbdjdcd}((\preoh (p~V1B)){\unicdbdjdcd}False)))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfepecfpfpfeeffbemedd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0A \in {2} .  {({{({{({\preoh{{({{p}~{V0A}})}}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{({{({{({{p}~{V0A}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfebeoeeepfpfjeoegfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdbdjded}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}((p~V1q){\unicdcdcdid}(p~V2r))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}((\preoh (\\
p~V2r)){\unicdcdcdid}(\preoh (p~V1q)))){\unicdcdcdhd}(((p~V1q){\unicdcdcdid}((\preoh (p~V2r)){\unicdcdcdid}(\preoh (p~V0p)))){\unicdcdcdhd}((p~V2r){\unicdcdcdid}\\
((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V0p)))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpffgbh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdcdcdhd}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V2r)))){\unicdcdcdhd}(((p~V1q){\unicdcdcdid}\\
(\preoh (p~V0p))){\unicdcdcdhd}((p~V2r){\unicdcdcdid}(\preoh (p~V0p)))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfdgpgogkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdcdcdid}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}(\preoh (p~V1q))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}(\preoh (p~V2r))){\unicdcdcdhd}\\
((p~V1q){\unicdcdcdid}((p~V2r){\unicdcdcdid}(\preoh (p~V0p))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfegjgdhkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdbdjdcd}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}(p~V1q)){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}(\preoh (p~V2r))){\unicdcdcdhd}((\\
\preoh (p~V1q)){\unicdcdcdid}((p~V2r){\unicdcdcdid}(\preoh (p~V0p))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfjgngah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  (((p~V0p){\unicdbdjded}(\preoh (p~V1q))){\unicdbdjded}(((p~V0p){\unicdcdcdid}\\
(p~V1q)){\unicdcdcdhd}((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V0p)))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfogfghg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  ({\unicdcdadad} V3s \in \\
2 .  (((p~V0p){\unicdbdjded}(p~(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~2)~V1q)~V2r)~V3s))){\unicdbdjded}\\
(((p~V0p){\unicdcdcdid}((p~V1q){\unicdcdcdid}(\preoh (p~V3s)))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}((\preoh (p~V2r)){\unicdcdcdid}(\preoh (p~V1q)))){\unicdcdcdhd}\\
(((p~V0p){\unicdcdcdid}((\preoh (p~V2r)){\unicdcdcdid}(\preoh (p~V3s)))){\unicdcdcdhd}(((\preoh (p~V1q)){\unicdcdcdid}((p~V2r){\unicdcdcdid}(\preoh (\\
p~V0p)))){\unicdcdcdhd}((p~V1q){\unicdcdcdid}((p~V3s){\unicdcdcdid}(\preoh (p~V0p)))))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfdgpgogeg}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdbdjdcd}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({{p}~{V0p}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogjgbd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdbdjdcd}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V1q}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogjgcd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdcdcdid}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V0p}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogpgbd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdcdcdid}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V1q}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogpgcd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{({\preoh{{({\preoh{{({{p}~{V0p}})}}})}}})}{\unicdbdjdcd}{({{p}~{V0p}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogog}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfebgmgjgdhehpfcdfebemepepeleffafpfpfjeoepfpfgecfbeoehefe}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0ls \in (ty\_2Elist\_2Elist~(ty\_2Epair\_2Eprod~A\_27a~A\_27b)) .  \\
({\unicdcdadad} V1k \in A\_27a .  ({\unicdcdadad} V2v \in A\_27b .  (((ap~(ap~(c\_2Ealist\_2EALOOKUP~\\
A\_27b~A\_27a)~V0ls)~V1k)=(ap~(\mname{c\_2Eoption\_2ESOME}~A\_27b)~V2v)){\unicdbdjdcd}(\\
p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27b)~V2v)~(ap~(\mname{c\_2Efinite\_\_map\_2EFRANGE}~\\
A\_27a~A\_27b)~(ap~(\mname{c\_2Ealist\_2Ealist\_\_to\_\_fmap}~A\_27a~A\_27b)~V0ls))))))))\end{array}$$
\end{theorem}
\end{document}
