\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\_2Estring\_2EIMPLODE\_\_EXPLODE (TMKaQdvapyBMLvk28WyqeLLqsJQeyNRj6fy)}
\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\_2Estring\_2Echar : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Estring\_2Echar}}\label{hyp:ogfgpfehjhpfcdfedhehchjgoghgpfcdfedgigbgch}
\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\_2Estring\_2EIMPLODE : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Estring\_2EIMPLODE \in ((ty\_2Elist\_2Elist~ty\_2Estring\_2Echar)^{{({{ty\_2Elist\_2Elist}~{ty\_2Estring\_2Echar}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhehchjgoghgpfcdfejeneafmepeeefe}
\end{equation}
Let $ c\_2Elist\_2ECONS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ECONS~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfedepeoedf}
\end{equation}
Let $ c\_2Elist\_2ENIL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ENIL~A\_27a \in (ty\_2Elist\_2Elist~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeoejeme}
\end{equation}
Let $ c\_2Estring\_2EEXPLODE : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Estring\_2EEXPLODE \in ((ty\_2Elist\_2Elist~ty\_2Estring\_2Echar)^{{({{ty\_2Elist\_2Elist}~{ty\_2Estring\_2Echar}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhehchjgoghgpfcdfefeifafmepeeefe}
\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}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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} 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} 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} 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} V0a0 \in A\_27a .  ({\unicdcdadad} V1a1 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2a0\_27 \in A\_27a .  ({\unicdcdadad} V3a1\_27 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  (((ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V0a0)~\\
V1a1)=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V2a0\_27)~V3a1\_27)){\unicdbdjded}((V0a0=\\
V2a0\_27){\unicdcdcdhd}(V1a1=V3a1\_27)))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfedepeoedfpfpfbdbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(((ap~c\_2Estring\_2EIMPLODE~(c\_2Elist\_2ENIL~ty\_2Estring\_2Echar))=\\
(c\_2Elist\_2ENIL~ty\_2Estring\_2Echar)){\unicdcdcdhd}({\unicdcdadad} V0c \in ty\_2Estring\_2Echar .  \\
({\unicdcdadad} V1cs \in (ty\_2Elist\_2Elist~ty\_2Estring\_2Echar) .  ((ap~c\_2Estring\_2EIMPLODE~\\
(ap~(ap~(c\_2Elist\_2ECONS~ty\_2Estring\_2Echar)~V0c)~V1cs))=(ap~\\
(ap~(c\_2Elist\_2ECONS~ty\_2Estring\_2Echar)~V0c)~(ap~c\_2Estring\_2EIMPLODE~\\
V1cs))))))\end{array}
\label{hyp:ehigngpfcdfedhehchjgoghgpfcdfejeneafmepeeefepfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(((ap~c\_2Estring\_2EEXPLODE~(c\_2Elist\_2ENIL~ty\_2Estring\_2Echar))=\\
(c\_2Elist\_2ENIL~ty\_2Estring\_2Echar)){\unicdcdcdhd}({\unicdcdadad} V0c \in ty\_2Estring\_2Echar .  \\
({\unicdcdadad} V1s \in (ty\_2Elist\_2Elist~ty\_2Estring\_2Echar) .  ((ap~c\_2Estring\_2EEXPLODE~\\
(ap~(ap~(c\_2Elist\_2ECONS~ty\_2Estring\_2Echar)~V0c)~V1s))=(ap~(\\
ap~(c\_2Elist\_2ECONS~ty\_2Estring\_2Echar)~V0c)~(ap~c\_2Estring\_2EEXPLODE~\\
V1s))))))\end{array}
\label{hyp:ehigngpfcdfedhehchjgoghgpfcdfefeifafmepeeefepfpfegfggg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedhehchjgoghgpfcdfejeneafmepeeefepfpffeifafmepeeefe}
$$\begin{array}{c}({\unicdcdadad} V0s \in (ty\_2Elist\_2Elist~ty\_2Estring\_2Echar) .  ((ap~c\_2Estring\_2EIMPLODE~\\
(ap~c\_2Estring\_2EEXPLODE~V0s))=V0s))\end{array}$$
\end{theorem}
\end{document}
