\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\_2EindexedLists\_2ELIST\_\_RELi\_\_ind (TMFV2sxKQChumGB5V2UjrPTvN9esQpwkckb)}
\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}
\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}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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 $ c\_2Elist\_2ELENGTH : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ELENGTH~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfemefeoeheefie}
\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\_2EAPPEND : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EAPPEND~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfebeafaffeoeee}
\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:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\end{definition}
Let $ c\_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: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}
\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:dgpfcdfejgogegfgihfgegmejgdhehdhpfcdfemejedfefpfpfcffemejg}
We define $\mname{c\_2EindexedLists\_2ELIST\_\_RELi}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0R \in {({{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}^{{ty\_2Enum\_2Enum}}})} .  {({{\uniddcgcg} V1a0 \in {({{ty\_2Elist\_2Elist}~{A\_27a}})} .  {({{\uniddcgcg} V2a1 \in {({{ty\_2Elist\_2Elist}~{A\_27b}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{2}^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}}})}^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}})}})}}~{({{\uniddcgcg} V3LIST\_\_RELi\_27 \in {({{({{2}^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}}})}^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{\uniddcgcg} V4a0 \in {({{ty\_2Elist\_2Elist}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{ty\_2Elist\_2Elist}~{A\_27b}})}})}}~{({{\uniddcgcg} V5a1 \in {({{ty\_2Elist\_2Elist}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{V4a0}})}}~{({{c\_2Elist\_2ENIL}~{A\_27a}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Elist\_2Elist}~{A\_27b}})}})}}~{V5a1}})}}~{({{c\_2Elist\_2ENIL}~{A\_27b}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V6h1 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27b}})}}~{({{\uniddcgcg} V7h2 \in {A\_27b} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{\uniddcgcg} V8l1 \in {({{ty\_2Elist\_2Elist}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{ty\_2Elist\_2Elist}~{A\_27b}})}})}}~{({{\uniddcgcg} V9l2 \in {({{ty\_2Elist\_2Elist}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{V4a0}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2EAPPEND}~{A\_27a}})}}~{V8l1}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2ECONS}~{A\_27a}})}}~{V6h1}})}}~{({{c\_2Elist\_2ENIL}~{A\_27a}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Elist\_2Elist}~{A\_27b}})}})}}~{V5a1}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2EAPPEND}~{A\_27b}})}}~{V9l2}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2ECONS}~{A\_27b}})}}~{V7h2}})}}~{({{c\_2Elist\_2ENIL}~{A\_27b}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{{ap}~{V0R}}~{({{{ap}~{({{c\_2Elist\_2ELENGTH}~{A\_27a}})}}~{V8l1}})}})}}~{V6h1}})}}~{V7h2}})}})}}~{({{{ap}~{({{{ap}~{V3LIST\_\_RELi\_27}}~{V8l1}})}}~{V9l2}})}})}})}})}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3LIST\_\_RELi\_27}}~{V4a0}})}}~{V5a1}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3LIST\_\_RELi\_27}}~{V1a0}})}}~{V2a1}})}})}})}})}})}})}})}}}$.
\end{definition}
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}
\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} V0x \in 2 .  ({\unicdcdadad} V1y \in 2 .  ({\unicdcdadad} V2z \in 2 .  ({\unicdcdadad} V3w \in \\
2 .  ((((p~V0x){\unicdbdjdcd}(p~V1y)){\unicdcdcdhd}((p~V2z){\unicdbdjdcd}(p~V3w))){\unicdbdjdcd}(((p~V0x){\unicdcdcdhd}(p~V2z)){\unicdbdjdcd}\\
((p~V1y){\unicdcdcdhd}(p~V3w))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfenepeoepepfpfbeoeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in 2 .  ({\unicdcdadad} V1y \in 2 .  ({\unicdcdadad} V2z \in 2 .  ({\unicdcdadad} V3w \in \\
2 .  ((((p~V0x){\unicdbdjdcd}(p~V1y)){\unicdcdcdhd}((p~V2z){\unicdbdjdcd}(p~V3w))){\unicdbdjdcd}(((p~V0x){\unicdcdcdid}(p~V2z)){\unicdbdjdcd}\\
((p~V1y){\unicdcdcdid}(p~V3w))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfenepeoepepfpfpecf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1Q \in \\
(2^{{A\_27a}}) .  (({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~V0P~V2x)){\unicdbdjdcd}(p~(ap~V1Q~V2x)))){\unicdbdjdcd}\\
(({\unicdcdaddd} V3x \in A\_27a .  (p~(ap~V0P~V3x))){\unicdbdjdcd}({\unicdcdaddd} V4x \in A\_27a .  (p~(\\
ap~V1Q~V4x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfenepeoepepfpffeifjedfefdf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfejgogegfgihfgegmejgdhehdhpfcdfemejedfefpfpfcffemejgpfpfjgogeg}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0R \in (((2^{{A\_27b}})^{{A\_27a}})^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1LIST\_\_RELi\_27 \in \\
((2^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}) .  ((\\
(p~(ap~(ap~V1LIST\_\_RELi\_27~(c\_2Elist\_2ENIL~A\_27a))~(c\_2Elist\_2ENIL~\\
A\_27b))){\unicdcdcdhd}({\unicdcdadad} V2h1 \in A\_27a .  ({\unicdcdadad} V3h2 \in A\_27b .  ({\unicdcdadad} V4l1 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V5l2 \in (ty\_2Elist\_2Elist~A\_27b) .  \\
(((p~(ap~(ap~(ap~V0R~(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V4l1))~V2h1)~V3h2)){\unicdcdcdhd}\\
(p~(ap~(ap~V1LIST\_\_RELi\_27~V4l1)~V5l2))){\unicdbdjdcd}(p~(ap~(ap~V1LIST\_\_RELi\_27~\\
(ap~(ap~(c\_2Elist\_2EAPPEND~A\_27a)~V4l1)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V2h1)~(c\_2Elist\_2ENIL~A\_27a))))~(ap~(ap~(c\_2Elist\_2EAPPEND~\\
A\_27b)~V5l2)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27b)~V3h2)~(c\_2Elist\_2ENIL~\\
A\_27b))))))))))){\unicdbdjdcd}({\unicdcdadad} V6a0 \in (ty\_2Elist\_2Elist~A\_27a) .  (\\
{\unicdcdadad} V7a1 \in (ty\_2Elist\_2Elist~A\_27b) .  ((p~(ap~(ap~(ap~(\mname{c\_2EindexedLists\_2ELIST\_\_RELi}~\\
A\_27a~A\_27b)~V0R)~V6a0)~V7a1)){\unicdbdjdcd}(p~(ap~(ap~V1LIST\_\_RELi\_27~V6a0)~\\
V7a1))))))))\end{array}$$
\end{theorem}
\end{document}
