\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\_2Efmaptree\_2Efmtree\_\_Axiom (TMRT3xKuJUXsAHkZLDYHRnVsh56PjWWsv6q)}
\maketitle
\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: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: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: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\_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\_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 $ ty\_2Eoption\_2Eoption : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Eoption\_2Eoption}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfepgahehjgpgogpfcdfepgahehjgpgog}
\end{equation}
Let $ ty\_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 $ ty\_2Efmaptree\_2Efmaptree : {{\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\_2Efmaptree\_2Efmaptree~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeggngbgahehchfgfgpfcdfeggngbgahehchfgfg}
\end{equation}
Let $ c\_2Efmaptree\_2EtoF : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27key .  nonempty~A\_27key{\unicdbdjdcd}{\unicdcdadad} A\_27value .  nonempty~\\
A\_27value{\unicdbdjdcd}c\_2Efmaptree\_2EtoF~A\_27key~A\_27value \in (((ty\_2Eoption\_2Eoption~\\
A\_27value)^{{({{ty\_2Elist\_2Elist}~{A\_27key}})}})^{{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27key}}~{A\_27value}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggngbgahehchfgfgpfcdfeehpgge}
\end{equation}
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: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 $ 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}
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\_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\_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}
\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: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: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}
Let $ c\_2Elist\_2Elist\_\_CASE : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Elist\_2Elist\_\_CASE~\\
A\_27a~A\_27b \in (((A\_27b^{{({{({{A\_27b}^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}})}^{{A\_27a}}})}})^{{A\_27b}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfemgjgdhehpfpfdebedffe}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggngbgahehchfgfgpfcdfedgpgogdhehchfhdgeh}
We define $\mname{c\_2Efmaptree\_2Econstruct}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0a \in {A\_27a} .  {{\uniddcgcg} V1kfm \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27b}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}}})}})} .  {{\uniddcgcg} V2kl \in {({{ty\_2Elist\_2Elist}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{c\_2Elist\_2Elist\_\_CASE}~{A\_27b}}~{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})}}~{V2kl}})}}~{({{{ap}~{({{\mname{c\_2Eoption\_2ESOME}}~{A\_27a}})}}~{V0a}})}})}}~{({{\uniddcgcg} V3h \in {A\_27b} .  {({{\uniddcgcg} V4t \in {({{ty\_2Elist\_2Elist}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{V3h}})}}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFDOM}}~{A\_27b}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}}})}})}}~{V1kfm}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFAPPLY}}~{A\_27b}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27a}})}^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}}})}})}}~{V1kfm}})}}~{V3h}})}}~{V4t}})}})}}~{({{\mname{c\_2Eoption\_2ENONE}}~{A\_27a}})}})}})}})}})}}}}}}$.
\end{definition}
Let $ c\_2Efmaptree\_2EfromF : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27key .  nonempty~A\_27key{\unicdbdjdcd}{\unicdcdadad} A\_27value .  nonempty~\\
A\_27value{\unicdbdjdcd}c\_2Efmaptree\_2EfromF~A\_27key~A\_27value \in ((ty\_2Efmaptree\_2Efmaptree~\\
A\_27key~A\_27value)^{{({{({{ty\_2Eoption\_2Eoption}~{A\_27value}})}^{{({{ty\_2Elist\_2Elist}~{A\_27key}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggngbgahehchfgfgpfcdfeggchpgngge}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggngbgahehchfgfgpfcdfegeefoepgegfg}
We define $\mname{c\_2Efmaptree\_2EFTNode}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {A\_27b} .  {{\uniddcgcg} V1fm \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27a}}~{A\_27b}})}})} .  {({{{ap}~{({{{c\_2Efmaptree\_2EfromF}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efmaptree\_2Econstruct}}~{A\_27b}}~{A\_27a}})}}~{V0i}})}}~{({{{ap}~{({{{ap}~{({{{{c\_2Efinite\_\_map\_2Eo\_\_f}~{A\_27a}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27a}}~{A\_27b}})}}~{({{({{ty\_2Eoption\_2Eoption}~{A\_27b}})}^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}})}})}}~{({{{c\_2Efmaptree\_2EtoF}~{A\_27a}}~{A\_27b}})}})}}~{V1fm}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggngbgahehchfgfgpfcdfechfgmgchfgdg}
We define $\mname{c\_2Efmaptree\_2Erelrec}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0h \in {({{({{({{A\_27c}^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27b}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}})}}})}^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27b}}~{A\_27c}})}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1a0 \in {({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})} .  {({{\uniddcgcg} V2a1 \in {A\_27c} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{2}^{{A\_27c}}})}^{{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}}})}})}}~{({{\uniddcgcg} V3relrec\_27 \in {({{({{2}^{{A\_27c}}})}^{{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}})}}~{({{\uniddcgcg} V4a0 \in {({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27c}})}}~{({{\uniddcgcg} V5a1 \in {A\_27c} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V6i \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27b}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}})}})}}~{({{\uniddcgcg} V7fm \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27b}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27b}}~{A\_27c}})}})}}~{({{\uniddcgcg} V8rfm \in {({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27b}}~{A\_27c}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}})}}~{V4a0}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efmaptree\_2EFTNode}}~{A\_27b}}~{A\_27a}})}}~{V6i}})}}~{V7fm}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27c}})}}~{V5a1}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{V0h}}~{V6i}})}}~{V8rfm}})}}~{V7fm}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27b}}})}})}}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFDOM}}~{A\_27b}}~{A\_27c}})}}~{V8rfm}})}})}}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFDOM}}~{A\_27b}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}})}}~{V7fm}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V9d \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{V9d}})}}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFDOM}}~{A\_27b}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}})}}~{V7fm}})}})}})}}~{({{{ap}~{({{{ap}~{V3relrec\_27}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFAPPLY}}~{A\_27b}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}})}}~{V7fm}})}}~{V9d}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efinite\_\_map\_2EFAPPLY}}~{A\_27b}}~{A\_27c}})}}~{V8rfm}})}}~{V9d}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3relrec\_27}}~{V4a0}})}}~{V5a1}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3relrec\_27}}~{V1a0}})}}~{V2a1}})}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggngbgahehchfgfgpfcdfeggngehchfgfgchfgdg}
We define $\mname{c\_2Efmaptree\_2Efmtreerec}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0h \in {({{({{({{A\_27a}^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27c}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27c}}~{A\_27b}})}})}}})}^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27c}}~{A\_27a}})}}})}^{{A\_27b}}})} .  {{\uniddcgcg} V1ft \in {({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27c}}~{A\_27b}})} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V2r \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{{\mname{c\_2Efmaptree\_2Erelrec}}~{A\_27b}}~{A\_27c}}~{A\_27a}})}}~{V0h}})}}~{V1ft}})}}~{V2r}})}})}})}}}}}}$.
\end{definition}
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} 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} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0h \in (((A\_27a^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27c}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27c}}~{A\_27b}})}})}})^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27c}}~{A\_27a}})}})^{{A\_27b}}) .  \\
({\unicdcdadad} V1i \in A\_27b .  ({\unicdcdadad} V2fm \in (ty\_2Efinite\_\_map\_2Efmap~\\
A\_27c~(ty\_2Efmaptree\_2Efmaptree~A\_27c~A\_27b)) .  ((ap~(ap~(\mname{c\_2Efmaptree\_2Efmtreerec}~\\
A\_27a~A\_27b~A\_27c)~V0h)~(ap~(ap~(\mname{c\_2Efmaptree\_2EFTNode}~A\_27c~A\_27b)~\\
V1i)~V2fm))=(ap~(ap~(ap~V0h~V1i)~(ap~(ap~(c\_2Efinite\_\_map\_2Eo\_\_f~\\
A\_27c~(ty\_2Efmaptree\_2Efmaptree~A\_27c~A\_27b)~A\_27a)~(ap~(\mname{c\_2Efmaptree\_2Efmtreerec}~\\
A\_27a~A\_27b~A\_27c)~V0h))~V2fm))~V2fm)))))\end{array}
\label{hyp:ehigngpfcdfeggngbgahehchfgfgpfcdfeggngehchfgfgchfgdgpfpfehigng}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeggngbgahehchfgfgpfcdfeggngehchfgfgpfpfbeihjgpgng}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0h \in (((A\_27c^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27b}}~{A\_27c}})}})^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27b}}~{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}})}})^{{A\_27a}}) .  \\
({\unicdcdaddd} V1f \in (A\_27c^{{({{{ty\_2Efmaptree\_2Efmaptree}~{A\_27b}}~{A\_27a}})}}) .  \\
({\unicdcdadad} V2i \in A\_27a .  ({\unicdcdadad} V3fm \in (ty\_2Efinite\_\_map\_2Efmap~\\
A\_27b~(ty\_2Efmaptree\_2Efmaptree~A\_27b~A\_27a)) .  ((ap~V1f~(ap~(ap~\\
(\mname{c\_2Efmaptree\_2EFTNode}~A\_27b~A\_27a)~V2i)~V3fm))=(ap~(ap~(ap~V0h~\\
V2i)~V3fm)~(ap~(ap~(c\_2Efinite\_\_map\_2Eo\_\_f~A\_27b~(ty\_2Efmaptree\_2Efmaptree~\\
A\_27b~A\_27a)~A\_27c)~V1f)~V3fm)))))))\end{array}$$
\end{theorem}
\end{document}
