\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\_2Ebinary\_\_ieee\_2Efloat\_\_fn\_\_updates (TMQxQgE36AzowR7ph95j2evkzRpT1niXQ7m)}
\maketitle
\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: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: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\_2Ebinary\_\_ieee\_2Efloat : {{\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\_2Ebinary\_\_ieee\_2Efloat~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgeh}
\end{equation}
Let $ ty\_2Eone\_2Eone : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eone\_2Eone}}\label{hyp:ogfgpfehjhpfcdfepgogfgpfcdfepgogfg}
\end{equation}
Let $ ty\_2Efcp\_2Ecart : {{\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\_2Efcp\_2Ecart~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeggdgahpfcdfedgbgcheh}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~\\
A\_27t~A\_27w \in (((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgogpfpfggfhaheg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}{\unicdcdadad} A\_27x .  \\
nonempty~A\_27x{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~\\
A\_27w~A\_27x \in (((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27x)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27x}})}^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpffeihahpgogfgogehpfpfggfhaheg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27t~A\_27w \in ((((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}})^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}})^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfechfgdgpgchegehjhahfgpfcdfeggmgpgbgeh}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27u .  nonempty~A\_27u{\unicdbdjdcd}{\unicdcdadad} A\_27w .  \\
nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w \in (((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27u~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27u}})}^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgogjgggjgdgbgogegpfpfggfhaheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V1c \in (ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone) .  ({\unicdcdadad} V2c0 \in \\
(ty\_2Efcp\_2Ecart~2~A\_27w) .  ({\unicdcdadad} V3c1 \in (ty\_2Efcp\_2Ecart~2~\\
A\_27t) .  ((ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~A\_27t~\\
A\_27w)~V0f)~(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27t~A\_27w)~V1c)~V2c0)~V3c1))=(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27t~A\_27w)~(ap~V0f~V1c))~V2c0)~V3c1))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgogpfpfggfhaheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}{\unicdcdadad} A\_27x .  \\
nonempty~A\_27x{\unicdbdjdcd}({\unicdcdadad} V0f \in ((ty\_2Efcp\_2Ecart~2~A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  \\
({\unicdcdadad} V1c \in (ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone) .  ({\unicdcdadad} V2c0 \in \\
(ty\_2Efcp\_2Ecart~2~A\_27w) .  ({\unicdcdadad} V3c1 \in (ty\_2Efcp\_2Ecart~2~\\
A\_27t) .  ((ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~\\
A\_27t~A\_27w~A\_27x)~V0f)~(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27t~A\_27w)~V1c)~V2c0)~V3c1))=(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27t~A\_27x)~V1c)~(ap~V0f~V2c0))~V3c1))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpffeihahpgogfgogehpfpfggfhaheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27u .  nonempty~A\_27u{\unicdbdjdcd}{\unicdcdadad} A\_27w .  \\
nonempty~A\_27w{\unicdbdjdcd}({\unicdcdadad} V0f \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V1c \in (ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone) .  ({\unicdcdadad} V2c0 \in \\
(ty\_2Efcp\_2Ecart~2~A\_27w) .  ({\unicdcdadad} V3c1 \in (ty\_2Efcp\_2Ecart~2~\\
A\_27t) .  ((ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w)~V0f)~(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27t~A\_27w)~V1c)~V2c0)~V3c1))=(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27u~A\_27w)~V1c)~V2c0)~(ap~V0f~V3c1)))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgogjgggjgdgbgogegpfpfggfhaheg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfggogpfpffhahegbgehfgdh}
$$\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27u .  nonempty~A\_27u{\unicdbdjdcd}{\unicdcdadad} A\_27w .  \\
nonempty~A\_27w{\unicdbdjdcd}{\unicdcdadad} A\_27x .  nonempty~A\_27x{\unicdbdjdcd}(({\unicdcdadad} V0f \in (\\
(ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V1c \in (ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone) .  ({\unicdcdadad} V2c0 \in \\
(ty\_2Efcp\_2Ecart~2~A\_27w) .  ({\unicdcdadad} V3c1 \in (ty\_2Efcp\_2Ecart~2~\\
A\_27t) .  ((ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~A\_27t~\\
A\_27w)~V0f)~(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27t~A\_27w)~V1c)~V2c0)~V3c1))=(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27t~A\_27w)~(ap~V0f~V1c))~V2c0)~V3c1)))))){\unicdcdcdhd}(({\unicdcdadad} V4f \in ((ty\_2Efcp\_2Ecart~\\
2~A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  ({\unicdcdadad} V5c \in (ty\_2Efcp\_2Ecart~\\
2~ty\_2Eone\_2Eone) .  ({\unicdcdadad} V6c0 \in (ty\_2Efcp\_2Ecart~2~A\_27w) .  \\
({\unicdcdadad} V7c1 \in (ty\_2Efcp\_2Ecart~2~A\_27t) .  ((ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~\\
A\_27t~A\_27w~A\_27x)~V4f)~(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27t~A\_27w)~V5c)~V6c0)~V7c1))=(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27t~A\_27x)~V5c)~(ap~V4f~V6c0))~V7c1)))))){\unicdcdcdhd}({\unicdcdadad} V8f \in ((ty\_2Efcp\_2Ecart~\\
2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  ({\unicdcdadad} V9c \in (ty\_2Efcp\_2Ecart~\\
2~ty\_2Eone\_2Eone) .  ({\unicdcdadad} V10c0 \in (ty\_2Efcp\_2Ecart~2~A\_27w) .  \\
({\unicdcdadad} V11c1 \in (ty\_2Efcp\_2Ecart~2~A\_27t) .  ((ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w)~V8f)~(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27t~A\_27w)~V9c)~V10c0)~V11c1))=(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Erecordtype\_2Efloat~\\
A\_27u~A\_27w)~V9c)~V10c0)~(ap~V8f~V11c1)))))))))\end{array}$$
\end{theorem}
\end{document}
