\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\_2Ezero\_\_properties (TMYK3gfm1HHgr797MUbpR67NUpsJVqKbaXk)}
\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\_2Erealax\_2Ereal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Erealax\_2Ereal}}\label{hyp:ogfgpfehjhpfcdfechfgbgmgbgihpfcdfechfgbgmg}
\end{equation}
Let $ ty\_2Ebool\_2Eitself : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ebool\_2Eitself}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfecgpgpgmgpfcdfejgehdhfgmggg}
\end{equation}
Let $ c\_2Ebool\_2Ethe\_\_value : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ebool\_2Ethe\_\_value~A\_27a \in (\\
ty\_2Ebool\_2Eitself~A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfecgpgpgmgpfcdfeehigfgpfpfghbgmgfhfg}
\end{equation}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Efcp\_2Edimindex : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Efcp\_2Edimindex~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggdgahpfcdfeegjgngjgogegfgih}
\end{equation}
Let $ c\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\end{equation}
Let $ c\_2Enum\_2EABS\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EABS\_\_num} \in {({{ty\_2Enum\_2Enum}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfebecedfpfpfogfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
Let $ c\_2Enum\_2EREP\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EREP\_\_num} \in {({{omega}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfecffeafpfpfogfhng}
\end{equation}
Let $ c\_2Enum\_2ESUC\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2ESUC\_\_REP} \in {({{omega}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfedfffdepfpfcffeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfedfffde}
We define $\mname{c\_2Enum\_2ESUC}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{({{{ap}~{c\_2Enum\_2ESUC\_\_REP}}~{({{{ap}~{c\_2Enum\_2EREP\_\_num}}~{V0m}})}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2E\_2B : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2B \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdce}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefcd}
We define $\mname{c\_2Earithmetic\_2EBIT2}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
Let $ c\_2Ereal\_2Ereal\_\_of\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ereal\_2Ereal\_\_of\_\_num} \in {({{ty\_2Erealax\_2Ereal}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfechfgbgmgpfcdfechfgbgmgpfpfpgggpfpfogfhng}
\end{equation}
Let $ c\_2Ereal\_2Epow : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ereal\_2Epow \in ((ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}})^{{ty\_2Erealax\_2Ereal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgpfcdfeahpghh}
\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 $ 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 $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand : {{\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\_\_Significand~\\
A\_27t~A\_27w \in ((ty\_2Efcp\_2Ecart~2~A\_27t)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgogjgggjgdgbgogeg}
\end{equation}
Let $ ty\_2Efcp\_2Efinite\_\_image : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Efcp\_2Efinite\_\_image}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeggdgahpfcdfeggjgogjgehfgpfpfjgngbghgfg}
\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}
\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}
\begin{definition}\label{def:dgpfcdfeahchjgngpfpfchfgdgpfcdfepfddde}
We define $\mname{c\_2Eprim\_\_rec\_2E\_3C}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V2P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V2P}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V3n}})}})}})}}~{({{{ap}~{V2P}}~{V3n}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V2P}}~{V0m}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V2P}}~{V1n}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddgepfcdbd}
We define $\mname{c\_2Ebool\_2E\_3F\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{V0P}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V0P}}~{V1x}})}})}}~{({{{ap}~{V0P}}~{V2y}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggdgahpfcdfeggjgogjgehfgpfpfjgogegfgih}
We define $\mname{c\_2Efcp\_2Efinite\_\_index}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{A\_27a}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V0f \in {({{A\_27a}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V2n}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{V0f}}~{V2n}})}})}}~{V1x}})}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ c\_2Efcp\_2Edest\_\_cart : {{\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\_2Efcp\_2Edest\_\_cart~\\
A\_27a~A\_27b \in ((A\_27a^{{({{ty\_2Efcp\_2Efinite\_\_image}~{A\_27b}})}})^{{({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggdgahpfcdfeegfgdhehpfpfdgbgcheh}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggdgahpfcdfeggdgahpfpfjgogegfgih}
We define $\mname{c\_2Efcp\_2Efcp\_\_index}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{c\_2Efcp\_2Edest\_\_cart}~{A\_27a}}~{A\_27b}})}}~{V0x}})}}~{({{{ap}~{({{\mname{c\_2Efcp\_2Efinite\_\_index}}~{({{ty\_2Efcp\_2Efinite\_\_image}~{A\_27b}})}})}}~{V1i}})}})}}}}}$.
\end{definition}
Let $ c\_2Earithmetic\_2EEXP : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EEXP \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfefeifaf}
\end{equation}
\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:dgpfcdfecgjgehpfcdfedfcejeef}
We define $\mname{c\_2Ebit\_2ESBIT}$ to be ${{\uniddcgcg} V0b \in {2} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{V0b}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EEXP}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{V1n}})}})}}~{\mname{c\_2Enum\_2E0}}})}}}$.
\end{definition}
Let $ c\_2Esum\_\_num\_2ESUM : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Esum\_\_num\_2ESUM \in ((ty\_2Enum\_2Enum^{{({{ty\_2Enum\_2Enum}^{{ty\_2Enum\_2Enum}}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfhngpfpfogfhngpfcdfedfffne}
\end{equation}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhcdog}
We define $\mname{c\_2Ewords\_2Ew2n}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{c\_2Esum\_\_num\_2ESUM}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}}~{({{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2ESBIT}}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{2}}~{A\_27a}})}}~{V0w}})}}~{V1i}})}})}}~{V1i}})}})}})}}}$.
\end{definition}
Let $ ty\_2Ehreal\_2Ehreal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ehreal\_2Ehreal}}\label{hyp:ogfgpfehjhpfcdfeigchfgbgmgpfcdfeigchfgbgmg}
\end{equation}
Let $ ty\_2Epair\_2Eprod : {{\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\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_2Erealax\_2Ereal\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Ereal\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{ty\_2Erealax\_2Ereal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfcffeafpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfcffeaf}
We define $\mname{c\_2Erealax\_2Ereal\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})}}~{({{{ap}~{c\_2Erealax\_2Ereal\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_inv : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_inv \in ((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfjgoggh}
\end{equation}
Let $ c\_2Erealax\_2Etreal\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_eq \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpffgbh}
\end{equation}
Let $ c\_2Erealax\_2Ereal\_\_ABS\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Ereal\_\_ABS\_\_CLASS \in (ty\_2Erealax\_2Ereal^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfbecedf}
We define $\mname{c\_2Erealax\_2Ereal\_\_ABS}$ to be ${{\uniddcgcg} V0r \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})} .  {({{{ap}~{c\_2Erealax\_2Ereal\_\_ABS\_\_CLASS}}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_eq}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfejgoggh}
We define $\mname{c\_2Erealax\_2Einv}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_inv}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}})}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_mul : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_mul \in (((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfngfhmg}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfngfhmg}
We define $\mname{c\_2Erealax\_2Ereal\_\_mul}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1T2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_mul}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfcdfepfcdge}
We define $\mname{c\_2Ereal\_2E\_2F}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{V0x}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Einv}}}~{V1y}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefbd}
We define $\mname{c\_2Earithmetic\_2EBIT1}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_add : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_add \in (((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfbgegeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfbgegeg}
We define $\mname{c\_2Erealax\_2Ereal\_\_add}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1T2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_add}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
Let $ c\_2Ewords\_2EINT\_\_MAX : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ewords\_2EINT\_\_MAX~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfehhpgchegdhpfcdfejeoeefpfpfnebeif}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent : {{\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\_\_Exponent~\\
A\_27t~A\_27w \in ((ty\_2Efcp\_2Ecart~2~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpffeihahpgogfgogeh}
\end{equation}
Let $ ty\_2Eone\_2Eone : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eone\_2Eone}}\label{hyp:ogfgpfehjhpfcdfepgogfgpfcdfepgogfg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign : {{\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~\\
A\_27t~A\_27w \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgog}
\end{equation}
Let $ c\_2Erealax\_2Etreal\_\_neg : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_neg \in ((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfogfghg}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfogfghg}
We define $\mname{c\_2Erealax\_2Ereal\_\_neg}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_neg}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2EDIV : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EDIV \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfeeejegf}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfeeejegfpfpfcdfeifaf}
We define $\mname{c\_2Ebit\_2EDIV\_\_2EXP}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EDIV}}~{V1n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EEXP}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{V0x}})}})}}}$.
\end{definition}
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}
Let $ c\_2Earithmetic\_2EMOD : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EMOD \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfenepeee}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfenepeeepfpfcdfeifaf}
We define $\mname{c\_2Ebit\_2EMOD\_\_2EXP}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EMOD}}~{V1n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EEXP}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfecejeefdf}
We define $\mname{c\_2Ebit\_2EBITS}$ to be ${{\uniddcgcg} V0h \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1l \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EMOD\_\_2EXP}}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V0h}})}})}}~{V1l}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EDIV\_\_2EXP}}}~{V1l}})}}~{V2n}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfecejeef}
We define $\mname{c\_2Ebit\_2EBIT}$ to be ${{\uniddcgcg} V0b \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EBITS}}}~{V0b}})}}~{V0b}})}}~{V1n}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggdgahpfcdfegedeaf}
We define $\mname{c\_2Efcp\_2EFCP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0g \in {({{A\_27a}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})}})}}~{({{\uniddcgcg} V1f \in {({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V2i}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27b}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27b}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{A\_27a}}~{A\_27b}})}}~{V1f}})}}~{V2i}})}})}}~{({{{ap}~{V0g}}~{V2i}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfeogcdhh}
We define $\mname{c\_2Ewords\_2En2w}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{\mname{c\_2Efcp\_2EFCP}}~{2}}~{A\_27a}})}}~{({{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EBIT}}}~{V1i}})}}~{V0n}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfehpgpfpfchfgbgmg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_to\_\_real}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27w}})}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_neg}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2E\_2F}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{c\_2Ewords\_2EINT\_\_MAX}~{A\_27w}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27w}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2E\_2F}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27t}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27t}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27t}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_neg}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2E\_2F}}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27w}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{c\_2Ewords\_2EINT\_\_MAX}~{A\_27w}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27w}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_add}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2E\_2F}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27t}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ereal\_2Epow}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27t}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27t}})}})}})}})}})}})}})}}}}$.
\end{definition}
Let $ ty\_2Ebinary\_\_ieee\_2Efloat\_\_value : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_value}}\label{hyp:ogfgpfehjhpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfghbgmgfhfg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2EFloat : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ebinary\_\_ieee\_2EFloat \in (ty\_2Ebinary\_\_ieee\_2Efloat\_\_value^{{ty\_2Erealax\_2Ereal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfegemgpgbgeh}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2ENaN : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ebinary\_\_ieee\_2ENaN} \in {ty\_2Ebinary\_\_ieee\_2Efloat\_\_value}}\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeoebgoe}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2EInfinity : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ebinary\_\_ieee\_2EInfinity} \in {ty\_2Ebinary\_\_ieee\_2Efloat\_\_value}}\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfejeogggjgogjgehjh}
\end{equation}
Let $ c\_2Ewords\_2EUINT\_\_MAX : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ewords\_2EUINT\_\_MAX~A\_27a \in (\\
ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfehhpgchegdhpfcdfeffjeoeefpfpfnebeif}
\end{equation}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfef}
We define $\mname{c\_2Ewords\_2Eword\_\_T}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27a}})}}~{({{{ap}~{({{c\_2Ewords\_2EUINT\_\_MAX}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfghbgmgfhfg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_value}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{\mname{c\_2Ewords\_2Eword\_\_T}}~{A\_27w}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_value}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27t}})}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{c\_2Ebinary\_\_ieee\_2EInfinity}})}}~{c\_2Ebinary\_\_ieee\_2ENaN}})}})}}~{({{{ap}~{c\_2Ebinary\_\_ieee\_2EFloat}}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_to\_\_real}}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}}}}$.
\end{definition}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE~\\
A\_27a \in ((((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{A\_27a}^{{ty\_2Erealax\_2Ereal}}})}})^{{ty\_2Ebinary\_\_ieee\_2Efloat\_\_value}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfghbgmgfhfgpfpfdebedffe}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfjgdhpfpfogbgog}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_nan}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE}~{2}})}}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{\uniddcgcg} V1v1 \in {ty\_2Erealax\_2Ereal} .  {\mname{c\_2Ebool\_2EF}}})}})}}~{\mname{c\_2Ebool\_2EF}}})}}~{\mname{c\_2Ebool\_2ET}}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfjgdhpfpfjgogggjgogjgehfg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_infinite}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE}~{2}})}}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{\uniddcgcg} V1v1 \in {ty\_2Erealax\_2Ereal} .  {\mname{c\_2Ebool\_2EF}}})}})}}~{\mname{c\_2Ebool\_2ET}}})}}~{\mname{c\_2Ebool\_2EF}}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfjgdhpfpfogpgchngbgmg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_normal}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27w}})}}~{\mname{c\_2Enum\_2E0}}})}})}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{\mname{c\_2Ewords\_2Eword\_\_T}}~{A\_27w}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfjgdhpfpfdhfhcgogpgchngbgmg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_subnormal}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27w}})}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27t}})}}~{\mname{c\_2Enum\_2E0}}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfjgdhpfpfkhfgchpg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_zero}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE}~{2}})}}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{\uniddcgcg} V1r \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Erealax\_2Ereal}})}}~{V1r}})}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}})}})}}~{\mname{c\_2Ebool\_2EF}}})}}~{\mname{c\_2Ebool\_2EF}}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfjgdhpfpfggjgogjgehfg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_finite}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE}~{2}})}}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{\uniddcgcg} V1v1 \in {ty\_2Erealax\_2Ereal} .  {\mname{c\_2Ebool\_2ET}}})}})}}~{\mname{c\_2Ebool\_2EF}}})}}~{\mname{c\_2Ebool\_2EF}}})}}}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_lt : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_lt \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfmgeh}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfmgeh}
We define $\mname{c\_2Erealax\_2Ereal\_\_lt}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1T2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_lt}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V1T2}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfcdfechfgbgmgpfpfmgehfg}
We define $\mname{c\_2Ereal\_2Ereal\_\_lte}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{V1y}})}}~{V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfcdfebgcgdh}
We define $\mname{c\_2Ereal\_2Eabs}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2Ereal\_\_lte}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}}~{V0x}})}})}}~{V0x}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_neg}}}~{V0x}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfejgdhpfpfjgogehfghgchbgmg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Eis\_\_integral}$ to be ${{\uniddcgcg} V0r \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{\mname{c\_2Ereal\_2Eabs}}}~{V0r}})}})}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{V1n}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfjgdhpfpfjgogehfghgchbgmg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_integral}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE}~{2}})}}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{\uniddcgcg} V1r \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Ebinary\_\_ieee\_2Eis\_\_integral}}}~{V1r}})}})}})}}~{\mname{c\_2Ebool\_2EF}}})}}~{\mname{c\_2Ebool\_2EF}}})}}}}$.
\end{definition}
Let $ c\_2Ebool\_2EARB : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A\_27a .  {{{nonempty}~{A\_27a}}{\unicdbdjdcd}{{{c\_2Ebool\_2EARB}~{A\_27a}} \in {A\_27a}}}}\label{hyp:ngfgngpfdgpfcdfecgpgpgmgpfcdfebecfce}
\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}
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\_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}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfbddgpgngah}
We define $\mname{c\_2Ewords\_2Eword\_\_1comp}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{{\mname{c\_2Efcp\_2EFCP}}~{2}}~{A\_27a}})}}~{({{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{2}}~{A\_27a}})}}~{V0w}})}}~{V1i}})}})}})}})}}}$.
\end{definition}
\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}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfogfghgbgehfg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_negate}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd}~{A\_27t}}~{A\_27w}})}}~{({{{ap}~{({{{\mname{c\_2Ecombin\_2EK}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Eword\_\_1comp}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{V0x}})}}}}$.
\end{definition}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_min : {{\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\_\_minus\_\_min~\\
A\_27t~A\_27w \in ((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w)^{{({{ty\_2Ebool\_2Eitself}~{({{{ty\_2Epair\_2Eprod}~{A\_27t}}~{A\_27w}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfngjgogfhdhpfpfngjgog}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_min : {{\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\_\_plus\_\_min~\\
A\_27t~A\_27w \in ((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w)^{{({{ty\_2Ebool\_2Eitself}~{({{{ty\_2Epair\_2Eprod}~{A\_27t}}~{A\_27w}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfahmgfhdhpfpfngjgog}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_zero : {{\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\_\_minus\_\_zero~\\
A\_27t~A\_27w \in ((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w)^{{({{ty\_2Ebool\_2Eitself}~{({{{ty\_2Epair\_2Eprod}~{A\_27t}}~{A\_27w}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfngjgogfhdhpfpfkhfgchpg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_zero : {{\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\_\_plus\_\_zero~\\
A\_27t~A\_27w \in ((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w)^{{({{ty\_2Ebool\_2Eitself}~{({{{ty\_2Epair\_2Eprod}~{A\_27t}}~{A\_27w}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfahmgfhdhpfpfkhfgchpg}
\end{equation}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfngdhcg}
We define $\mname{c\_2Ewords\_2Eword\_\_msb}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{2}}~{A\_27a}})}}~{V0w}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfjgdhpfpfdhjghgogbgmgmgjgoghg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_signalling}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_nan}}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Eword\_\_msb}}~{A\_27t}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfemefeef}
We define $\mname{c\_2Ebool\_2ELET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{V0f}}~{V1x}})}})}})}}}$.
\end{definition}
Let $ ty\_2Ebinary\_\_ieee\_2Efp\_\_op : {{\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\_2Efp\_\_op~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggahpfpfpgah}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdhpgngfgpfpfbhogbgog}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_some\_\_qnan}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0fp\_\_op \in {({{{ty\_2Ebinary\_\_ieee\_2Efp\_\_op}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efp\_\_op}~{A\_27t}}~{A\_27w}})}}})}})}}~{({{\uniddcgcg} V1f \in {({{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efp\_\_op}~{A\_27t}}~{A\_27w}})}}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Ebool\_2ELET}}~{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}~{2}})}}~{({{\uniddcgcg} V2qnan \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_nan}}~{A\_27t}}~{A\_27w}})}}~{V2qnan}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_signalling}}~{A\_27t}}~{A\_27w}})}}~{V2qnan}})}})}})}})}})}}~{({{{ap}~{V1f}}~{V0fp\_\_op}})}})}})}})}}~{V0fp\_\_op}})}}}}$.
\end{definition}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_infinity : {{\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\_\_minus\_\_infinity~\\
A\_27t~A\_27w \in ((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w)^{{({{ty\_2Ebool\_2Eitself}~{({{{ty\_2Epair\_2Eprod}~{A\_27t}}~{A\_27w}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfngjgogfhdhpfpfjgogggjgogjgehjh}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_infinity : {{\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\_\_plus\_\_infinity~\\
A\_27t~A\_27w \in ((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w)^{{({{ty\_2Ebool\_2Eitself}~{({{{ty\_2Epair\_2Eprod}~{A\_27t}}~{A\_27w}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfahmgfhdhpfpfjgogggjgogjgehjh}
\end{equation}
\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:dgpfcdfedgpgngcgjgogpfcdfepg}
We define $\mname{c\_2Ecombin\_2Eo}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27c}}})} .  {{\uniddcgcg} V1g \in {({{A\_27c}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{V0f}}~{({{{ap}~{V1g}}~{V2x}})}})}})}}}}}}$.
\end{definition}
Let $ c\_2Ewords\_2Edimword : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ewords\_2Edimword~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfehhpgchegdhpfcdfeegjgnghhpgcheg}
\end{equation}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfcddgpgngah}
We define $\mname{c\_2Ewords\_2Eword\_\_2comp}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{({{{ap}~{({{c\_2Ewords\_2Edimword}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27a}})}}~{V0w}})}})}})}}}$.
\end{definition}
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} A\_27x .  nonempty~A\_27x{\unicdbdjdcd}(({\unicdcdadad} V0f0 \in \\
((ty\_2Efcp\_2Ecart~2~A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  ({\unicdcdadad} V1f \in \\
(ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27x)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~\\
A\_27t~A\_27w~A\_27x)~V0f0)~V1f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w)~V1f)))){\unicdcdcdhd}(({\unicdcdadad} V2f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V3f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~A\_27u~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w)~V2f0)~V3f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w)~V3f)))){\unicdcdcdhd}(({\unicdcdadad} V4f0 \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V5f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~A\_27t~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~\\
A\_27t~A\_27w)~V4f0)~V5f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w)~V5f)))){\unicdcdcdhd}(({\unicdcdadad} V6f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V7f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~A\_27u~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w)~V6f0)~V7f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w)~V7f)))){\unicdcdcdhd}(({\unicdcdadad} V8f0 \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V9f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~A\_27w)~(ap~(ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~A\_27t~A\_27w)~V8f0)~V9f))=\\
(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~A\_27w)~V9f)))){\unicdcdcdhd}\\
(({\unicdcdadad} V10f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  \\
({\unicdcdadad} V11f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~A\_27x)~(ap~(ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~A\_27w~A\_27x)~\\
V10f0)~V11f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~\\
A\_27w)~V11f)))){\unicdcdcdhd}(({\unicdcdadad} V12f0 \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V13f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~A\_27t~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~\\
A\_27t~A\_27w)~V12f0)~V13f))=(ap~V12f0~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w)~V13f))))){\unicdcdcdhd}(({\unicdcdadad} V14f0 \in ((ty\_2Efcp\_2Ecart~2~\\
A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  ({\unicdcdadad} V15f \in (ty\_2Ebinary\_\_ieee\_2Efloat~\\
A\_27t~A\_27w) .  ((ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~A\_27t~\\
A\_27x)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~\\
A\_27w~A\_27x)~V14f0)~V15f))=(ap~V14f0~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w)~V15f))))){\unicdcdcdhd}({\unicdcdadad} V16f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V17f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27u~A\_27w)~(ap~(ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~A\_27t~A\_27u~A\_27w)~\\
V16f0)~V17f))=(ap~V16f0~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~\\
A\_27t~A\_27w)~V17f)))))))))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfbgdgdgggfhahegdh}
\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\_27v .  \\
nonempty~A\_27v{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}{\unicdcdadad} A\_27x .  nonempty~\\
A\_27x{\unicdbdjdcd}{\unicdcdadad} A\_27y .  nonempty~A\_27y{\unicdbdjdcd}(({\unicdcdadad} V0g \in ((ty\_2Efcp\_2Ecart~\\
2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  ({\unicdcdadad} V1f0 \in \\
((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V2f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~A\_27t~A\_27w)~V1f0)~\\
(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~A\_27t~A\_27w)~V0g)~\\
V2f))=(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~A\_27t~A\_27w)~\\
(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~(ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)~(\\
ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)~(ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone))~\\
V1f0)~V0g))~V2f))))){\unicdcdcdhd}(({\unicdcdadad} V3g \in ((ty\_2Efcp\_2Ecart~2~A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  \\
({\unicdcdadad} V4f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27y)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27x}})}}) .  \\
({\unicdcdadad} V5f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~A\_27x~A\_27y)~\\
V4f0)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~\\
A\_27w~A\_27x)~V3g)~V5f))=(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~\\
A\_27t~A\_27w~A\_27y)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~(ty\_2Efcp\_2Ecart~2~\\
A\_27w)~(ty\_2Efcp\_2Ecart~2~A\_27y)~(ty\_2Efcp\_2Ecart~2~A\_27x))~\\
V4f0)~V3g))~V5f))))){\unicdcdcdhd}({\unicdcdadad} V6g \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V7f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27v)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27u}})}}) .  \\
({\unicdcdadad} V8f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~A\_27u~A\_27v~\\
A\_27w)~V7f0)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w)~V6g)~V8f))=(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27v~A\_27w)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~(ty\_2Efcp\_2Ecart~2~\\
A\_27t)~(ty\_2Efcp\_2Ecart~2~A\_27v)~(ty\_2Efcp\_2Ecart~2~A\_27u))~\\
V7f0)~V6g))~V8f)))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfggfhahegggfhahegdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0a \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V1f \in (A\_27a^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V2v \in A\_27a .  \\
({\unicdcdadad} V3v1 \in A\_27a .  ((ap~(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE~\\
A\_27a)~(ap~c\_2Ebinary\_\_ieee\_2EFloat~V0a))~V1f)~V2v)~V3v1)=(ap~\\
V1f~V0a)))))){\unicdcdcdhd}(({\unicdcdadad} V4f \in (A\_27a^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V5v \in \\
A\_27a .  ({\unicdcdadad} V6v1 \in A\_27a .  ((ap~(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE~\\
A\_27a)~c\_2Ebinary\_\_ieee\_2EInfinity)~V4f)~V5v)~V6v1)=V5v)))){\unicdcdcdhd}\\
({\unicdcdadad} V7f \in (A\_27a^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V8v \in A\_27a .  \\
({\unicdcdadad} V9v1 \in A\_27a .  ((ap~(ap~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE~\\
A\_27a)~c\_2Ebinary\_\_ieee\_2ENaN)~V7f)~V8v)~V9v1)=V9v1))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfghbgmgfhfgpfpfdgbgdhfgpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}(\\
(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_zero~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))=(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~\\
A\_27t~A\_27w)~(ap~(\mname{c\_2Ecombin\_2EK}~(ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)~\\
(ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone))~(ap~(\mname{c\_2Ewords\_2En2w}~ty\_2Eone\_2Eone)~\\
\mname{c\_2Enum\_2E0})))~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~\\
A\_27t~A\_27w~A\_27w)~(ap~(\mname{c\_2Ecombin\_2EK}~(ty\_2Efcp\_2Ecart~2~A\_27w)~\\
(ty\_2Efcp\_2Ecart~2~A\_27w))~(ap~(\mname{c\_2Ewords\_2En2w}~A\_27w)~\mname{c\_2Enum\_2E0})))~\\
(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~A\_27t~\\
A\_27t~A\_27w)~(ap~(\mname{c\_2Ecombin\_2EK}~(ty\_2Efcp\_2Ecart~2~A\_27t)~(ty\_2Efcp\_2Ecart~\\
2~A\_27t))~(ap~(\mname{c\_2Ewords\_2En2w}~A\_27t)~\mname{c\_2Enum\_2E0})))~(c\_2Ebool\_2EARB~\\
(ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfahmgfhdhpfpfkhfgchpgpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}(\\
(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_zero~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))=(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_negate}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_zero~A\_27t~\\
A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfngjgogfhdhpfpfkhfgchpgpfpfegfggg}
\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\_27t .  \\
nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}(((ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_infinity~\\
A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~\\
A\_27w))))=c\_2Ebinary\_\_ieee\_2EInfinity){\unicdcdcdhd}(((ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_infinity~\\
A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~\\
A\_27w))))=c\_2Ebinary\_\_ieee\_2EInfinity){\unicdcdcdhd}(({\unicdcdadad} V0fp\_\_op \in \\
(ty\_2Ebinary\_\_ieee\_2Efp\_\_op~A\_27a~A\_27b) .  ((ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}~\\
A\_27a~A\_27b)~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_some\_\_qnan}~A\_27a~\\
A\_27b)~V0fp\_\_op))=c\_2Ebinary\_\_ieee\_2ENaN)){\unicdcdcdhd}(((ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_zero~A\_27t~\\
A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~A\_27w))))=\\
(ap~c\_2Ebinary\_\_ieee\_2EFloat~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}\\
(((ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}~A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_zero~\\
A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~\\
A\_27w))))=(ap~c\_2Ebinary\_\_ieee\_2EFloat~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(((ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}~A\_27t~\\
A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_min~A\_27t~A\_27w)~\\
(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~A\_27w))))=(\\
ap~c\_2Ebinary\_\_ieee\_2EFloat~(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO}))))~\\
(ap~(ap~c\_2Ereal\_2Epow~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO}))))~(ap~(ap~\\
c\_2Earithmetic\_2E\_2B~(ap~(c\_2Ewords\_2EINT\_\_MAX~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
A\_27w)))~(ap~(c\_2Efcp\_2Edimindex~A\_27t)~(c\_2Ebool\_2Ethe\_\_value~\\
A\_27t))))))){\unicdcdcdhd}((ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}~A\_27t~A\_27w)~\\
(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_min~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w))))=(ap~c\_2Ebinary\_\_ieee\_2EFloat~\\
(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))))~\\
(ap~(ap~c\_2Ereal\_2Epow~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO}))))~(ap~(ap~\\
c\_2Earithmetic\_2E\_2B~(ap~(c\_2Ewords\_2EINT\_\_MAX~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
A\_27w)))~(ap~(c\_2Efcp\_2Edimindex~A\_27t)~(c\_2Ebool\_2Ethe\_\_value~\\
A\_27t)))))))))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfghbgmgfhfgdh}
\end{equation}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\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} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((p~V0A){\unicdcdcdhd}(p~V1B))){\unicdbdjded}((\preoh (\\
p~V0A)){\unicdcdcdid}(\preoh (p~V1B)))){\unicdcdcdhd}((\preoh ((p~V0A){\unicdcdcdid}(p~V1B))){\unicdbdjded}((\preoh (p~V0A)){\unicdcdcdhd}(\preoh (p~V1B)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeefepfpfnepecfhebeoepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a \in A\_27a .  ({\unicdcdaddd} V1x \in \\
A\_27a .  (V1x=V0a)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifjedfefdfpfpfcffegeme}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ((ap~(ap~(\mname{c\_2Ecombin\_2EK}~\\
A\_27a~A\_27b)~V0x)~V1y)=V0x)))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfelepfpfefiene}
\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} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}{\unicdcdadad} A\_27e .  nonempty~\\
A\_27e{\unicdbdjdcd}{\unicdcdadad} A\_27f .  nonempty~A\_27f{\unicdbdjdcd}(({\unicdcdadad} V0f \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V1v \in A\_27c .  ((ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27a~A\_27c~A\_27b)~\\
(ap~(\mname{c\_2Ecombin\_2EK}~A\_27c~A\_27b)~V1v))~V0f)=(ap~(\mname{c\_2Ecombin\_2EK}~\\
A\_27c~A\_27a)~V1v)))){\unicdcdcdhd}({\unicdcdadad} V2f \in (A\_27e^{{A\_27d}}) .  ({\unicdcdadad} V3v \in \\
A\_27d .  ((ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27f~A\_27e~A\_27d)~V2f)~(ap~(\mname{c\_2Ecombin\_2EK}~\\
A\_27d~A\_27f)~V3v))=(ap~(\mname{c\_2Ecombin\_2EK}~A\_27e~A\_27f)~(ap~V2f~V3v))))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfelepfpfpgpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((ap~\mname{c\_2Ereal\_2Eabs}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))=\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfbecedfpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  (\\
(((ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V0n)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
V1m)){\unicdbdjded}(V0n=V1m)){\unicdcdcdhd}((((ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
V0n))=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V1m)){\unicdbdjded}((V0n=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}\\
(V1m=\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}((((ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V0n)=\\
(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V1m))){\unicdbdjded}\\
((V0n=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}(V1m=\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(((ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V0n))=(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V1m))){\unicdbdjded}(V0n=V1m)))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfefgbhpfpfjgogehdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (((ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~V0m)=\\
(ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~V1n)){\unicdbdjded}((ap~(ap~c\_2Earithmetic\_2EMOD~\\
V0m)~(ap~(c\_2Ewords\_2Edimword~A\_27a)~(c\_2Ebool\_2Ethe\_\_value~\\
A\_27a)))=(ap~(ap~c\_2Earithmetic\_2EMOD~V1n)~(ap~(c\_2Ewords\_2Edimword~\\
A\_27a)~(c\_2Ebool\_2Ethe\_\_value~A\_27a)))))))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfeogcdhhpfpfbdbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}((ap~(\mname{c\_2Ewords\_2Eword\_\_2comp}~\\
A\_27a)~(ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))=(\mname{c\_2Ewords\_2Eword\_\_T}~\\
A\_27a))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfehfpecfeepfpfoefehepfpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}((ap~(\mname{c\_2Ewords\_2Eword\_\_1comp}~\\
A\_27a)~(ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~\mname{c\_2Enum\_2E0}))=(\mname{c\_2Ewords\_2Eword\_\_T}~\\
A\_27a))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfehfpecfeepfpfoepeefpfpfad}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfekhfgchpgpfpfahchpgahfgchehjgfgdh}
$$\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}(\\
(p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_zero}~A\_27t~A\_27w)~(ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_zero~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w))))){\unicdcdcdhd}((p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_zero}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_zero~A\_27t~\\
A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~A\_27w))))){\unicdcdcdhd}\\
((p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_finite}~A\_27t~A\_27w)~(\\
ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_zero~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w))))){\unicdcdcdhd}((p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_finite}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_zero~A\_27t~\\
A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~A\_27w))))){\unicdcdcdhd}\\
((p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_integral}~A\_27t~A\_27w)~\\
(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_zero~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w))))){\unicdcdcdhd}((p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_integral}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_zero~A\_27t~\\
A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~A\_27w))))){\unicdcdcdhd}\\
((\preoh (p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_nan}~A\_27t~A\_27w)~(ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_zero~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))))){\unicdcdcdhd}((\preoh (p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_nan}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_zero~A\_27t~\\
A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))))){\unicdcdcdhd}\\
((\preoh (p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_normal}~A\_27t~A\_27w)~\\
(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_zero~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))))){\unicdcdcdhd}((\preoh (p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_normal}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_zero~A\_27t~\\
A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))))){\unicdcdcdhd}\\
((\preoh (p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_subnormal}~A\_27t~A\_27w)~\\
(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_zero~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))))){\unicdcdcdhd}((\preoh (p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_subnormal}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_zero~A\_27t~\\
A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))))){\unicdcdcdhd}\\
((\preoh (p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_infinite}~A\_27t~A\_27w)~\\
(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_plus\_\_zero~A\_27t~A\_27w)~(c\_2Ebool\_2Ethe\_\_value~\\
(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))))){\unicdcdcdhd}(\preoh (p~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_is\_\_infinite}~\\
A\_27t~A\_27w)~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_minus\_\_zero~A\_27t~\\
A\_27w)~(c\_2Ebool\_2Ethe\_\_value~(ty\_2Epair\_2Eprod~A\_27t~A\_27w)))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
