\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\_2Emachine\_\_ieee\_2Efp32\_\_lessThan (TMVqkF9LuGwnG6S4vfNj5wrfo6hYZ1j4ZjC)}
\maketitle
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 $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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: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: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: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: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: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}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
\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}
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}
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}
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}
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 $ 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}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\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 $ 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\_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}
\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}
Let $ ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}}\label{hyp:ogfgpfehjhpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdgpgngahbgchfg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2ELT : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ebinary\_\_ieee\_2ELT} \in {ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}}\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfemeef}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2EUN : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ebinary\_\_ieee\_2EUN} \in {ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}}\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeffoe}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2EGT : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ebinary\_\_ieee\_2EGT} \in {ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}}\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeheef}
\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\_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\_\_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}
\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}
Let $ c\_2Ebinary\_\_ieee\_2EEQ : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ebinary\_\_ieee\_2EEQ} \in {ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}}\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfefebf}
\end{equation}
Let $ ty\_2Erealax\_2Ereal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Erealax\_2Ereal}}\label{hyp:ogfgpfehjhpfcdfechfgbgmgbgihpfcdfechfgbgmg}
\end{equation}
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\_\_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}
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\_\_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}
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 $ 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}
\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 $ 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}
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 $ 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}
\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 $ 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\_2Epair\_2EABS\_\_prod : {{\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\_2Epair\_2EABS\_\_prod~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfebecedfpfpfahchpgeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfepfcdde}
We define $\mname{c\_2Epair\_2E\_2C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{c\_2Epair\_2EABS\_\_prod}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V2a \in {A\_27a} .  {({{\uniddcgcg} V3b \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2a}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3b}})}}~{V1y}})}})}})}})}})}}}}}$.
\end{definition}
Let $ c\_2Epair\_2ESND : {{\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\_2Epair\_2ESND~\\
A\_27a~A\_27b \in (A\_27b^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfedfoeee}
\end{equation}
Let $ c\_2Epair\_2EFST : {{\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\_2Epair\_2EFST~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfegedfef}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfeahbgjgchpfpfdebedffe}
We define $\mname{c\_2Epair\_2Epair\_\_CASE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0p \in {({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27c}})} .  {{\uniddcgcg} V1f \in {({{({{A\_27a}^{{A\_27c}}})}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{V1f}}~{({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27b}}~{A\_27c}})}}~{V0p}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27b}}~{A\_27c}})}}~{V0p}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdgpgngahbgchfg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_compare}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {{\uniddcgcg} V1y \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Epair\_2Epair\_\_CASE}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_value}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_value}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_value}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_value}})}}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_value}}~{A\_27t}}~{A\_27w}})}}~{V1y}})}})}})}}~{({{\uniddcgcg} V2v \in {ty\_2Ebinary\_\_ieee\_2Efloat\_\_value} .  {({{\uniddcgcg} V3v1 \in {ty\_2Ebinary\_\_ieee\_2Efloat\_\_value} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}})}}~{V2v}})}}~{({{\uniddcgcg} V4r1 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}})}}~{V3v1}})}}~{({{\uniddcgcg} V5r2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{V4r1}})}}~{V5r2}})}})}}~{c\_2Ebinary\_\_ieee\_2ELT}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Erealax\_2Ereal}})}}~{V4r1}})}}~{V5r2}})}})}}~{c\_2Ebinary\_\_ieee\_2EEQ}})}}~{c\_2Ebinary\_\_ieee\_2EGT}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{A\_27t}}~{A\_27w}})}}~{V1y}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}~{c\_2Ebinary\_\_ieee\_2EGT}})}}~{c\_2Ebinary\_\_ieee\_2ELT}})}})}}~{c\_2Ebinary\_\_ieee\_2EUN}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ebinary\_\_ieee\_2Efloat\_\_value\_\_CASE}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}})}}~{V3v1}})}}~{({{\uniddcgcg} V6v7 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}~{c\_2Ebinary\_\_ieee\_2ELT}})}}~{c\_2Ebinary\_\_ieee\_2EGT}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{A\_27t}}~{A\_27w}})}}~{V1y}})}})}})}}~{c\_2Ebinary\_\_ieee\_2EEQ}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}~{c\_2Ebinary\_\_ieee\_2ELT}})}}~{c\_2Ebinary\_\_ieee\_2EGT}})}})}})}}~{c\_2Ebinary\_\_ieee\_2EUN}})}})}}~{c\_2Ebinary\_\_ieee\_2EUN}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfmgfgdhdhpfpfehigbgog}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_less\_\_than}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {{\uniddcgcg} V1y \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Ebinary\_\_ieee\_2Efloat\_\_compare}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_compare}}~{A\_27t}}~{A\_27w}})}}~{V0x}})}}~{V1y}})}})}}~{c\_2Ebinary\_\_ieee\_2ELT}})}}}}}$.
\end{definition}
Let $ ty\_2Efcp\_2Ebit0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Efcp\_2Ebit0}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeggdgahpfcdfecgjgehad}
\end{equation}
Let $ ty\_2Efcp\_2Ebit1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Efcp\_2Ebit1}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeggdgahpfcdfecgjgehbd}
\end{equation}
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}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfenejeoe}
We define $\mname{c\_2Earithmetic\_2EMIN}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V0m}})}}~{V1n}})}})}}~{V0m}})}}~{V1n}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfdddepfddee}
We define $\mname{c\_2Earithmetic\_2E\_3C\_3D}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V0m}})}}~{V1n}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{V1n}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfcgjgehdh}
We define $\mname{c\_2Ewords\_2Eword\_\_bits}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0h \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1l \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V2w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{{\mname{c\_2Efcp\_2EFCP}}~{2}}~{A\_27a}})}}~{({{\uniddcgcg} V3i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2E\_3C\_3D}}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V3i}})}}~{V1l}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2EMIN}}}~{V0h}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{2}}~{A\_27a}})}}~{V2w}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V3i}})}}~{V1l}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhcdhh}
We define $\mname{c\_2Ewords\_2Ew2w}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27b}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27a}})}}~{V0w}})}})}}}}$.
\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}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpffgihehchbgdgeh}
We define $\mname{c\_2Ewords\_2Eword\_\_extract}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0h \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1l \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2Eo}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27b}})}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})}})}}~{({{{\mname{c\_2Ewords\_2Ew2w}}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ewords\_2Eword\_\_bits}}~{A\_27a}})}}~{V0h}})}}~{V1l}})}})}}}}}$.
\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}
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:dgpfcdfengbgdgigjgogfgpfpfjgfgfgfgpfcdfeggahddcdpfpfehpgpfpfggmgpgbgeh}
We define $\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_to\_\_float}$ to be ${{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}})} .  {({{{ap}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{{ap}~{({{{\mname{c\_2Ecombin\_2EK}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Ewords\_2Eword\_\_extract}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}})}})}}~{V0w}})}})}})}}~{({{{ap}~{({{{ap}~{({{{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{{ap}~{({{{\mname{c\_2Ecombin\_2EK}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Ewords\_2Eword\_\_extract}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}})}}~{V0w}})}})}})}}~{({{{ap}~{({{{ap}~{({{{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{{ap}~{({{{\mname{c\_2Ecombin\_2EK}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Ewords\_2Eword\_\_extract}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}})}}~{\mname{c\_2Enum\_2E0}}})}}~{V0w}})}})}})}}~{({{c\_2Ebool\_2EARB}~{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengbgdgigjgogfgpfpfjgfgfgfgpfcdfeggahddcdpfpfmgfgdhdhefigbgog}
We define $\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_lessThan}$ to be ${{\uniddcgcg} V0a \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}})} .  {{\uniddcgcg} V1b \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_less\_\_than}}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{{ap}~{\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_to\_\_float}}}~{V0a}})}})}}~{({{{ap}~{\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_to\_\_float}}}~{V1b}})}})}}}$.
\end{definition}
Let $ ty\_2Esum\_2Esum : {{\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\_2Esum\_2Esum~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfedhfhngpfcdfedhfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfmgdhmg}
We define $\mname{c\_2Ewords\_2Eword\_\_lsl}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{\mname{c\_2Efcp\_2EFCP}}~{2}}~{A\_27a}})}}~{({{\uniddcgcg} V2i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V2i}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2E\_3C\_3D}}}~{V1n}})}}~{V2i}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{2}}~{A\_27a}})}}~{V0w}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{V2i}})}}~{V1n}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfpgch}
We define $\mname{c\_2Ewords\_2Eword\_\_or}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0v \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {{\uniddcgcg} V1w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{{\mname{c\_2Efcp\_2EFCP}}~{2}}~{A\_27a}})}}~{({{\uniddcgcg} V2i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{2}}~{A\_27a}})}}~{V0v}})}}~{V2i}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{2}}~{A\_27a}})}}~{V1w}})}}~{V2i}})}})}})}})}}}}$.
\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}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfkgpgjgog}
We define $\mname{c\_2Ewords\_2Eword\_\_join}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0v \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {{\uniddcgcg} V1w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Ebool\_2ELET}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Ebool\_2ELET}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})}}~{({{({{{ty\_2Efcp\_2Ecart}~{2}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})}^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})}}})}})}}~{({{\uniddcgcg} V2cv \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})} .  {({{\uniddcgcg} V3cw \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Ewords\_2Eword\_\_or}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ewords\_2Eword\_\_lsl}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})}}~{V2cv}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27b}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27b}})}})}})}})}}~{V3cw}})}})}})}})}}~{({{{ap}~{({{{\mname{c\_2Ewords\_2Ew2w}}~{A\_27a}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})}}~{V0v}})}})}})}}~{({{{ap}~{({{{\mname{c\_2Ewords\_2Ew2w}}~{A\_27b}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})}}~{V1w}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfdgpgogdgbgeh}
We define $\mname{c\_2Ewords\_2Eword\_\_concat}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0v \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {{\uniddcgcg} V1w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27b}})} .  {({{{ap}~{({{{\mname{c\_2Ewords\_2Ew2w}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}}~{A\_27c}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Ewords\_2Eword\_\_join}}~{A\_27a}}~{A\_27b}})}}~{V0v}})}}~{V1w}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengbgdgigjgogfgpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfehpgpfpfggahddcd}
We define $\mname{c\_2Emachine\_\_ieee\_2Efloat\_\_to\_\_fp32}$ to be ${{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ewords\_2Eword\_\_concat}}~{ty\_2Eone\_2Eone}}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ewords\_2Eword\_\_concat}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{ty\_2Eone\_2Eone}})}})}})}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{V0x}})}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit1}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{({{ty\_2Efcp\_2Ebit0}~{ty\_2Eone\_2Eone}})}})}})}})}}~{V0x}})}})}})}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in (ty\_2Ebinary\_\_ieee\_2Efloat~(ty\_2Efcp\_2Ebit1~(\\
ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone)))) .  \\
((ap~\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_to\_\_float}~(ap~\mname{c\_2Emachine\_\_ieee\_2Efloat\_\_to\_\_fp32}~\\
V0x))=V0x))\end{array}
\label{hyp:ehigngpfcdfengbgdgigjgogfgpfpfjgfgfgfgpfcdfeggahddcdpfpfehpgpfpfggmgpgbgehpfpfggmgpgbgehpfpfehpgpfpfggahddcd}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfengbgdgigjgogfgpfpfjgfgfgfgpfcdfeggahddcdpfpfmgfgdhdhefigbgog}
$$\begin{array}{c}(({\unicdcdadad} V0b \in (ty\_2Ebinary\_\_ieee\_2Efloat~(ty\_2Efcp\_2Ebit1~\\
(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone)))) .  \\
({\unicdcdadad} V1a \in (ty\_2Ebinary\_\_ieee\_2Efloat~(ty\_2Efcp\_2Ebit1~(\\
ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone)))) .  \\
((p~(ap~(ap~\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_lessThan}~(ap~\mname{c\_2Emachine\_\_ieee\_2Efloat\_\_to\_\_fp32}~\\
V1a))~(ap~\mname{c\_2Emachine\_\_ieee\_2Efloat\_\_to\_\_fp32}~V0b))){\unicdbdjded}(p~(ap~\\
(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_less\_\_than}~(ty\_2Efcp\_2Ebit1~\\
(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))~\\
V1a)~V0b))))){\unicdcdcdhd}(({\unicdcdadad} V2b \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V3a \in (ty\_2Ebinary\_\_ieee\_2Efloat~\\
(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit0~\\
ty\_2Eone\_2Eone))))~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
ty\_2Eone\_2Eone)))) .  ((p~(ap~(ap~\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_lessThan}~\\
(ap~\mname{c\_2Emachine\_\_ieee\_2Efloat\_\_to\_\_fp32}~V3a))~(ap~(\mname{c\_2Ewords\_2En2w}~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))))~V2b))){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_less\_\_than}~\\
(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit0~\\
ty\_2Eone\_2Eone))))~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
ty\_2Eone\_2Eone))))~V3a)~(ap~\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_to\_\_float}~\\
(ap~(\mname{c\_2Ewords\_2En2w}~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))))~V2b))))))){\unicdcdcdhd}\\
(({\unicdcdadad} V4b \in (ty\_2Ebinary\_\_ieee\_2Efloat~(ty\_2Efcp\_2Ebit1~\\
(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone)))) .  \\
({\unicdcdadad} V5a \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_lessThan}~\\
(ap~(\mname{c\_2Ewords\_2En2w}~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))))~V5a))~\\
(ap~\mname{c\_2Emachine\_\_ieee\_2Efloat\_\_to\_\_fp32}~V4b))){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_less\_\_than}~\\
(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit0~\\
ty\_2Eone\_2Eone))))~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
ty\_2Eone\_2Eone))))~(ap~\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_to\_\_float}~\\
(ap~(\mname{c\_2Ewords\_2En2w}~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))))~V5a)))~\\
V4b))))){\unicdcdcdhd}({\unicdcdadad} V6b \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V7a \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_lessThan}~(ap~(\mname{c\_2Ewords\_2En2w}~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))))~V7a))~(ap~(\mname{c\_2Ewords\_2En2w}~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))))~V6b))){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_less\_\_than}~\\
(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit1~(ty\_2Efcp\_2Ebit0~\\
ty\_2Eone\_2Eone))))~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
ty\_2Eone\_2Eone))))~(ap~\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_to\_\_float}~\\
(ap~(\mname{c\_2Ewords\_2En2w}~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))))~V7a)))~\\
(ap~\mname{c\_2Emachine\_\_ieee\_2Efp32\_\_to\_\_float}~(ap~(\mname{c\_2Ewords\_2En2w}~\\
(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~(ty\_2Efcp\_2Ebit0~\\
(ty\_2Efcp\_2Ebit0~ty\_2Eone\_2Eone))))))~V6b))))))))))\end{array}$$
\end{theorem}
\end{document}
