\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\_2Ealist\_\_to\_\_fmap\_\_MAP (TMP6ULnBAsadwPNzCTTgBorpJ6CEcikE44z)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
Let $ ty\_2Eone\_2Eone : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eone\_2Eone}}\label{hyp:ogfgpfehjhpfcdfepgogfgpfcdfepgogfg}
\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: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:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
Let $ 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: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 $ 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\_\_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 $ 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\_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}
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 $ 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\_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}
Let $ c\_2Efinite\_\_map\_2Eo\_\_f : {{\iota}{\unicdbdjdcd}{{\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}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}c\_2Efinite\_\_map\_2Eo\_\_f~A\_27a~A\_27b~A\_27c \in (((\\
ty\_2Efinite\_\_map\_2Efmap~A\_27a~A\_27c)^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})}})^{{({{A\_27c}^{{A\_27b}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggjgogjgehfgpfpfngbgahpfcdfepgpfpfgg}
\end{equation}
Let $ c\_2Efinite\_\_map\_2EMAP\_\_KEYS : {{\iota}{\unicdbdjdcd}{{\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}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}c\_2Efinite\_\_map\_2EMAP\_\_KEYS~A\_27a~A\_27b~A\_27c \in \\
(((ty\_2Efinite\_\_map\_2Efmap~A\_27b~A\_27c)^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27c}})}})^{{({{A\_27b}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggjgogjgehfgpfpfngbgahpfcdfenebeafpfpflefejfdf}
\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: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 $ 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}
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 $ 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:dgpfcdfengbgchlgfgchpfcdfebecgcgchfggh}
We define $\mname{c\_2Emarker\_2EAbbrev}$ to be ${{\uniddcgcg} V0x \in {2} .  {V0x}}$.
\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: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: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: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}
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:dgpfcdfeahchfgegpfpfdhfgehpfcdfejeoeke}
We define $\mname{c\_2Epred\_\_set\_2EINJ}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V2t \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{({{{ap}~{V0f}}~{V3x}})}})}}~{V2t}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V4x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V5y}})}}~{V1s}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{V0f}}~{V4x}})}})}}~{({{{ap}~{V0f}}~{V5y}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V4x}})}}~{V5y}})}})}})}})}})}})}})}})}}}}}}$.
\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} 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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in 2 .  (({\unicdcdadad} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepecfbememepfpfdfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} 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} 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} 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} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0f \in (A\_27c^{{A\_27b}}) .  ({\unicdcdadad} V1g \in (ty\_2Efinite\_\_map\_2Efmap~\\
A\_27a~A\_27b) .  ((ap~(\mname{c\_2Efinite\_\_map\_2EFDOM}~A\_27a~A\_27c)~(ap~(ap~\\
(c\_2Efinite\_\_map\_2Eo\_\_f~A\_27a~A\_27b~A\_27c)~V0f)~V1g))=(ap~(\mname{c\_2Efinite\_\_map\_2EFDOM}~\\
A\_27a~A\_27b)~V1g))))\end{array}
\label{hyp:ehigngpfcdfeggjgogjgehfgpfpfngbgahpfcdfegeeepenepfpfpgpfpfgg}
\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}}) .  ((ap~(ap~(c\_2Efinite\_\_map\_2Eo\_\_f~\\
A\_27a~A\_27c~A\_27b)~V0f)~(\mname{c\_2Efinite\_\_map\_2EFEMPTY}~A\_27a~A\_27c))=\\
(\mname{c\_2Efinite\_\_map\_2EFEMPTY}~A\_27a~A\_27b)))\end{array}
\label{hyp:ehigngpfcdfeggjgogjgehfgpfpfngbgahpfcdfepgpfpfggpfpfgefeneafefjf}
\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} V1fm \in (ty\_2Efinite\_\_map\_2Efmap~\\
A\_27a~A\_27c) .  ({\unicdcdadad} V2k \in A\_27a .  ({\unicdcdadad} V3v \in A\_27c .  ((ap~(ap~\\
(c\_2Efinite\_\_map\_2Eo\_\_f~A\_27a~A\_27c~A\_27b)~V0f)~(ap~(ap~(c\_2Efinite\_\_map\_2EFUPDATE~\\
A\_27a~A\_27c)~V1fm)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27c)~V2k)~V3v)))=\\
(ap~(ap~(c\_2Efinite\_\_map\_2EFUPDATE~A\_27a~A\_27b)~(ap~(ap~(c\_2Efinite\_\_map\_2Eo\_\_f~\\
A\_27a~A\_27c~A\_27b)~V0f)~V1fm))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~\\
V2k)~(ap~V0f~V3v))))))))\end{array}
\label{hyp:ehigngpfcdfeggjgogjgehfgpfpfngbgahpfcdfepgpfpfggpfpfgeffafeebeeffe}
\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}}) .  ((ap~(ap~(c\_2Efinite\_\_map\_2EMAP\_\_KEYS~\\
A\_27a~A\_27b~A\_27c)~V0f)~(\mname{c\_2Efinite\_\_map\_2EFEMPTY}~A\_27a~A\_27c))=\\
(\mname{c\_2Efinite\_\_map\_2EFEMPTY}~A\_27b~A\_27c)))\end{array}
\label{hyp:ehigngpfcdfeggjgogjgehfgpfpfngbgahpfcdfenebeafpfpflefejfdfpfpfgefeneafefjf}
\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} V1fm \in (ty\_2Efinite\_\_map\_2Efmap~\\
A\_27a~A\_27c) .  ({\unicdcdadad} V2k \in A\_27a .  ({\unicdcdadad} V3v \in A\_27c .  ((p~(ap~(\\
ap~(ap~(\mname{c\_2Epred\_\_set\_2EINJ}~A\_27a~A\_27b)~V0f)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~\\
A\_27a)~V2k)~(ap~(\mname{c\_2Efinite\_\_map\_2EFDOM}~A\_27a~A\_27c)~V1fm)))~(\\
\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27b))){\unicdbdjdcd}((ap~(ap~(c\_2Efinite\_\_map\_2EMAP\_\_KEYS~\\
A\_27a~A\_27b~A\_27c)~V0f)~(ap~(ap~(c\_2Efinite\_\_map\_2EFUPDATE~A\_27a~\\
A\_27c)~V1fm)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27c)~V2k)~V3v)))=(\\
ap~(ap~(c\_2Efinite\_\_map\_2EFUPDATE~A\_27b~A\_27c)~(ap~(ap~(c\_2Efinite\_\_map\_2EMAP\_\_KEYS~\\
A\_27a~A\_27b~A\_27c)~V0f)~V1fm))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27b~A\_27c)~\\
(ap~V0f~V2k))~V3v))))))))\end{array}
\label{hyp:ehigngpfcdfeggjgogjgehfgpfpfngbgahpfcdfenebeafpfpflefejfdfpfpfgeffafeebeeffe}
\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}}) .  ((ap~(ap~(c\_2Elist\_2EMAP~A\_27a~A\_27b)~\\
V0f)~(c\_2Elist\_2ENIL~A\_27a))=(c\_2Elist\_2ENIL~A\_27b))){\unicdcdcdhd}({\unicdcdadad} V1f \in \\
(A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V2h \in A\_27a .  ({\unicdcdadad} V3t \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((ap~(ap~(c\_2Elist\_2EMAP~A\_27a~A\_27b)~V1f)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V2h)~V3t))=(ap~(ap~(c\_2Elist\_2ECONS~A\_27b)~(ap~V1f~V2h))~\\
(ap~(ap~(c\_2Elist\_2EMAP~A\_27a~A\_27b)~V1f)~V3t)))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfenebeaf}
\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} 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} 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 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} 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} V0x \in A\_27a .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27a))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfffoejegf}
\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} V1s \in (2^{{A\_27b}}) .  (p~(ap~(ap~\\
(ap~(\mname{c\_2Epred\_\_set\_2EINJ}~A\_27a~A\_27b)~V0f)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~\\
A\_27a))~V1s))){\unicdcdcdhd}({\unicdcdadad} V2s \in (2^{{A\_27a}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINJ}~\\
A\_27a~A\_27b)~V0f)~V2s)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27b))){\unicdbdjded}(V2s=\\
(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoekepfpffeneafefjf}
\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 .  ({\unicdcdadad} V2s \in (\\
2^{{A\_27a}}) .  ({\unicdcdadad} V3t \in (2^{{A\_27b}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINJ}~\\
A\_27a~A\_27b)~V0f)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V1x)~V2s))~\\
V3t)){\unicdbdjded}((p~(ap~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINJ}~A\_27a~A\_27b)~V0f)~V2s)~\\
V3t)){\unicdcdcdhd}((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27b)~(ap~V0f~V1x))~V3t)){\unicdcdcdhd}({\unicdcdadad} V4y \in \\
A\_27a .  (((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V4y)~V2s)){\unicdcdcdhd}((ap~V0f~V1x)=\\
(ap~V0f~V4y))){\unicdbdjdcd}(V1x=V4y))))))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoekepfpfjeoedffecfef}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfebgmgjgdhehpfcdfebgmgjgdhehpfpfehpgpfpfggngbgahpfpfnebeaf}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0f1 \in (\\
A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1f2 \in (A\_27d^{{A\_27c}}) .  ({\unicdcdadad} V2al \in (ty\_2Elist\_2Elist~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27c)) .  ((p~(ap~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINJ}~\\
A\_27a~A\_27b)~V0f1)~(ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a)~(ap~(ap~\\
(c\_2Elist\_2EMAP~(ty\_2Epair\_2Eprod~A\_27a~A\_27c)~A\_27a)~(c\_2Epair\_2EFST~\\
A\_27a~A\_27c))~V2al)))~(\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27b))){\unicdbdjdcd}((ap~(\mname{c\_2Ealist\_2Ealist\_\_to\_\_fmap}~\\
A\_27b~A\_27d)~(ap~(ap~(c\_2Elist\_2EMAP~(ty\_2Epair\_2Eprod~A\_27a~A\_27c)~\\
(ty\_2Epair\_2Eprod~A\_27b~A\_27d))~(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~\\
A\_27c~(ty\_2Epair\_2Eprod~A\_27b~A\_27d))~({\uniddcgcg} V3x \in A\_27a .  ({\uniddcgcg} V4y \in \\
A\_27c .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27b~A\_27d)~(ap~V0f1~V3x))~(ap~V1f2~\\
V4y))))))~V2al))=(ap~(ap~(c\_2Efinite\_\_map\_2EMAP\_\_KEYS~A\_27a~\\
A\_27b~A\_27d)~V0f1)~(ap~(ap~(c\_2Efinite\_\_map\_2Eo\_\_f~A\_27a~A\_27c~\\
A\_27d)~V1f2)~(ap~(\mname{c\_2Ealist\_2Ealist\_\_to\_\_fmap}~A\_27a~A\_27c)~V2al))))))))\end{array}$$
\end{theorem}
\end{document}
