\documentclass{article}
\newtheorem{axiom}{Axiom}
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\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
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\newcommand\unicdcdcdjd{\cap}
\newcommand\unicdcdadad{\forall}
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\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
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\newcommand\uniddcgcg{\lambda}
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\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{thm\_2Ebitstring\_2Eextend (TMQgkW68p6mFxD97wk9Lcmg7j5WAVCn6B6v)}
\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}
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\_2Earithmetic\_2E\_2D : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2D \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdee}
\end{equation}
\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}
Let $ c\_2Elist\_2EGENLIST : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EGENLIST~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{ty\_2Enum\_2Enum}})^{{({{A\_27a}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfehefeoemejedfef}
\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:dgpfcdfemgjgdhehpfcdfeafbeeepfpfmefegeef}
We define $\mname{c\_2Elist\_2EPAD\_\_LEFT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0c \in {A\_27a} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V2s \in {({{ty\_2Elist\_2Elist}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{c\_2Elist\_2EAPPEND}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{c\_2Elist\_2EGENLIST}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{ty\_2Enum\_2Enum}})}}~{V0c}})}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{V1n}})}}~{({{{ap}~{({{c\_2Elist\_2ELENGTH}~{A\_27a}})}}~{V2s}})}})}})}})}}~{V2s}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgehdhehchjgoghgpfcdfekhfgchpgpfpffgihehfgogeg}
We define $\mname{c\_2Ebitstring\_2Ezero\_\_extend}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1v \in {({{ty\_2Elist\_2Elist}~{2}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Elist\_2EPAD\_\_LEFT}}~{2}})}}~{\mname{c\_2Ebool\_2EF}}})}}~{V0n}})}}~{V1v}})}}}$.
\end{definition}
Let $ c\_2Elist\_2EHD : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EHD~A\_27a \in (A\_27a^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeieee}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgjgehdhehchjgoghgpfcdfedhjghgogpfpffgihehfgogeg}
We define $\mname{c\_2Ebitstring\_2Esign\_\_extend}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1v \in {({{ty\_2Elist\_2Elist}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Elist\_2EPAD\_\_LEFT}}~{A\_27a}})}}~{({{{ap}~{({{c\_2Elist\_2EHD}~{A\_27a}})}}~{V1v}})}})}}~{V0n}})}}~{V1v}})}}}}$.
\end{definition}
Let $ c\_2Ebitstring\_2Eextend : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ebitstring\_2Eextend \in ((((ty\_2Elist\_2Elist~2)^{{({{ty\_2Elist\_2Elist}~{2}})}})^{{ty\_2Enum\_2Enum}})^{{2}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgehdhehchjgoghgpfcdfefgihehfgogeg}
\end{equation}
\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}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1l \in (ty\_2Elist\_2Elist~\\
2) .  ({\unicdcdadad} V2c \in 2 .  ((ap~(ap~(ap~(\mname{c\_2Elist\_2EPAD\_\_LEFT}~2)~V2c)~\\
V0n)~V1l)=(ap~(ap~(ap~c\_2Ebitstring\_2Eextend~V2c)~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V0n)~(ap~(c\_2Elist\_2ELENGTH~2)~V1l)))~V1l)))))\end{array}
\label{hyp:ehigngpfcdfecgjgehdhehchjgoghgpfcdfeahbgegpfpfmgfgggehpfpffgihehfgogeg}
\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{\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}
\begin{theorem}\label{thm:ehigngpfcdfecgjgehdhehchjgoghgpfcdfefgihehfgogeg}
$$\begin{array}{c}(({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1v \in (ty\_2Elist\_2Elist~\\
2) .  ((ap~(ap~\mname{c\_2Ebitstring\_2Ezero\_\_extend}~V0n)~V1v)=(ap~(ap~(\\
ap~c\_2Ebitstring\_2Eextend~\mname{c\_2Ebool\_2EF})~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V0n)~(ap~(c\_2Elist\_2ELENGTH~2)~V1v)))~V1v)))){\unicdcdcdhd}({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V3v \in (ty\_2Elist\_2Elist~2) .  ((ap~(ap~(\mname{c\_2Ebitstring\_2Esign\_\_extend}~\\
2)~V2n)~V3v)=(ap~(ap~(ap~c\_2Ebitstring\_2Eextend~(ap~(c\_2Elist\_2EHD~\\
2)~V3v))~(ap~(ap~c\_2Earithmetic\_2E\_2D~V2n)~(ap~(c\_2Elist\_2ELENGTH~\\
2)~V3v)))~V3v)))))\end{array}$$
\end{theorem}
\end{document}
