\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\_2Erat\_2ERATN\_\_RATD\_\_EQ\_\_THM (TMLzzgC2SyJ68qPQNtQ4SyStWYEuyvP9xHi)}
\maketitle
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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 $ ty\_2Einteger\_2Eint : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Einteger\_2Eint}}\label{hyp:ogfgpfehjhpfcdfejgogehfghgfgchpfcdfejgogeh}
\end{equation}
Let $ c\_2Einteger\_2Eint\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeafpfpfdemebedfdf}
\end{equation}
\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: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:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeaf}
We define $\mname{c\_2Einteger\_2Eint\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_neg : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_neg \in ((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfogfghg}
\end{equation}
Let $ c\_2Einteger\_2Etint\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_eq \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpffgbh}
\end{equation}
Let $ c\_2Einteger\_2Eint\_\_ABS\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_ABS\_\_CLASS \in (ty\_2Einteger\_2Eint^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedf}
We define $\mname{c\_2Einteger\_2Eint\_\_ABS}$ to be ${{\uniddcgcg} V0r \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})} .  {({{{ap}~{c\_2Einteger\_2Eint\_\_ABS\_\_CLASS}}~{({{{ap}~{c\_2Einteger\_2Etint\_\_eq}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfogfghg}
We define $\mname{c\_2Einteger\_2Eint\_\_neg}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{c\_2Einteger\_2Etint\_\_neg}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}})}}$.
\end{definition}
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}
Let $ c\_2Einteger\_2Eint\_\_of\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Einteger\_2Eint\_\_of\_\_num} \in {({{ty\_2Einteger\_2Eint}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfpgggpfpfogfhng}
\end{equation}
Let $ c\_2Einteger\_2Etint\_\_lt : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_lt \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfmgeh}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfmgeh}
We define $\mname{c\_2Einteger\_2Eint\_\_lt}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_lt}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def: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: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:dgpfcdfejgogehfghgfgchpfcdfebecedf}
We define $\mname{c\_2Einteger\_2EABS}$ to be ${{\uniddcgcg} V0n \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_lt}}}~{V0n}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_neg}}}~{V0n}})}})}}~{V0n}})}}$.
\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:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfmgfg}
We define $\mname{c\_2Einteger\_2Eint\_\_le}$ to be ${{\uniddcgcg} V0x \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1y \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_lt}}}~{V1y}})}}~{V0x}})}})}}}$.
\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: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: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}
Let $ ty\_2Erat\_2Erat : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Erat\_2Erat}}\label{hyp:ogfgpfehjhpfcdfechbgehpfcdfechbgeh}
\end{equation}
Let $ c\_2Erat\_2ERATD : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Erat\_2ERATD} \in {({{ty\_2Enum\_2Enum}^{{ty\_2Erat\_2Erat}}})}}\label{hyp:ngfgngpfdgpfcdfechbgehpfcdfecfbeefee}
\end{equation}
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 $ ty\_2Efrac\_2Efrac : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Efrac\_2Efrac}}\label{hyp:ogfgpfehjhpfcdfeggchbgdgpfcdfeggchbgdg}
\end{equation}
Let $ c\_2Efrac\_2Eabs\_\_frac : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Efrac\_2Eabs\_\_frac \in (ty\_2Efrac\_2Efrac^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggchbgdgpfcdfebgcgdhpfpfggchbgdg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfbd}
We define $\mname{c\_2Efrac\_2Efrac\_\_1}$ to be ${({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}$.
\end{definition}
Let $ c\_2Efrac\_2Erep\_\_frac : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Efrac\_2Erep\_\_frac \in ((ty\_2Epair\_2Eprod~ty\_2Einteger\_2Eint~\\
ty\_2Einteger\_2Eint)^{{ty\_2Efrac\_2Efrac}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggchbgdgpfcdfechfgahpfpfggchbgdg}
\end{equation}
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}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfegogng}
We define $\mname{c\_2Efrac\_2Efrac\_\_dnm}$ to be ${{\uniddcgcg} V0f \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{c\_2Epair\_2ESND}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{c\_2Efrac\_2Erep\_\_frac}}~{V0f}})}})}}$.
\end{definition}
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:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfogngch}
We define $\mname{c\_2Efrac\_2Efrac\_\_nmr}$ to be ${{\uniddcgcg} V0f \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{c\_2Epair\_2EFST}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{c\_2Efrac\_2Erep\_\_frac}}~{V0f}})}})}}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_mul : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_mul \in (((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfngfhmg}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfngfhmg}
We define $\mname{c\_2Einteger\_2Eint\_\_mul}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_mul}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpffgbhfhjggh}
We define $\mname{c\_2Erat\_2Erat\_\_equiv}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {{\uniddcgcg} V1f2 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V1f2}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V1f2}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}})}}}$.
\end{definition}
Let $ c\_2Erat\_2Eabs\_\_rat\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Erat\_2Eabs\_\_rat\_\_CLASS} \in {({{ty\_2Erat\_2Erat}^{{({{2}^{{ty\_2Efrac\_2Efrac}}})}}})}}\label{hyp:ngfgngpfdgpfcdfechbgehpfcdfebgcgdhpfpfchbgehpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfebgcgdhpfpfchbgeh}
We define $\mname{c\_2Erat\_2Eabs\_\_rat}$ to be ${{\uniddcgcg} V0r \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{c\_2Erat\_2Eabs\_\_rat\_\_CLASS}}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_equiv}}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfbd}
We define $\mname{c\_2Erat\_2Erat\_\_1}$ to be ${({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{\mname{c\_2Efrac\_2Efrac\_\_1}}})}$.
\end{definition}
Let $ c\_2Erat\_2Erep\_\_rat\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Erat\_2Erep\_\_rat\_\_CLASS} \in {({{({{2}^{{ty\_2Efrac\_2Efrac}}})}^{{ty\_2Erat\_2Erat}}})}}\label{hyp:ngfgngpfdgpfcdfechbgehpfcdfechfgahpfpfchbgehpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechfgahpfpfchbgeh}
We define $\mname{c\_2Erat\_2Erep\_\_rat}$ to be ${{\uniddcgcg} V0a \in {ty\_2Erat\_2Erat} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Efrac\_2Efrac}})}}~{({{{ap}~{c\_2Erat\_2Erep\_\_rat\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_add : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_add \in (((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfbgegeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbgegeg}
We define $\mname{c\_2Einteger\_2Eint\_\_add}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_add}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfbgegeg}
We define $\mname{c\_2Efrac\_2Efrac\_\_add}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {{\uniddcgcg} V1f2 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_add}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V1f2}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V1f2}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V1f2}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfbgegeg}
We define $\mname{c\_2Erat\_2Erat\_\_add}$ to be ${{\uniddcgcg} V0r1 \in {ty\_2Erat\_2Erat} .  {{\uniddcgcg} V1r2 \in {ty\_2Erat\_2Erat} .  {({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_add}}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V0r1}})}})}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V1r2}})}})}})}}}$.
\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:dgpfcdfedgpgngcgjgogpfcdfedf}
We define $\mname{c\_2Ecombin\_2ES}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1g \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V2x}})}}~{({{{ap}~{V1g}}~{V2x}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfeje}
We define $\mname{c\_2Ecombin\_2EI}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2ES}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}}~{A\_27a}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}})}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{A\_27a}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2Enum\_\_CASE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Earithmetic\_2Enum\_\_CASE~A\_27a \in \\
(((A\_27a^{{({{A\_27a}^{{ty\_2Enum\_2Enum}}})}})^{{A\_27a}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfeogfhngpfpfdebedffe}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfad}
We define $\mname{c\_2Efrac\_2Efrac\_\_0}$ to be ${({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfad}
We define $\mname{c\_2Erat\_2Erat\_\_0}$ to be ${({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{\mname{c\_2Efrac\_2Efrac\_\_0}}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehfge}
We define $\mname{c\_2Erelation\_2EWF}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1B \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V2w \in {A\_27a} .  {({{{ap}~{V1B}}~{V2w}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3min \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1B}}~{V3min}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4b}})}}~{V3min}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V1B}}~{V4b}})}})}})}})}})}})}})}})}})}})}})}}}$.
\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}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfecffedfefcfjedeef}
We define $\mname{c\_2Erelation\_2ERESTRICT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{V1R}}~{V3y}})}}~{V2x}})}})}}~{({{{ap}~{V0f}}~{V3y}})}})}}~{({{c\_2Ebool\_2EARB}~{A\_27b}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeefde}
We define $\mname{c\_2Erelation\_2ETC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1a \in {A\_27a} .  {{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{2}^{{A\_27a}}})}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3P \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4x}})}}~{V5y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V4x}})}}~{V5y}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V7y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V8z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V7y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V7y}})}}~{V8z}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V8z}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V1a}})}}~{V2b}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfebgahahchpgih}
We define $\mname{c\_2Erelation\_2Eapprox}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{A\_27b}^{{A\_27a}}})}})}}~{V3f}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V4y \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4y}})}})}}~{V4y}})}})}})}}~{V0R}})}}~{V2x}})}})}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeehigfgpfpfggfhog}
We define $\mname{c\_2Erelation\_2Ethe\_\_fun}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{A\_27b}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Eapprox}}~{A\_27a}}~{A\_27b}})}}~{V0R}})}}~{V1M}})}}~{V2x}})}}~{V3f}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehfgecffede}
We define $\mname{c\_2Erelation\_2EWFREC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Ethe\_\_fun}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{\mname{c\_2Erelation\_2ETC}}~{A\_27a}})}}~{V0R}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V4v \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4v}})}})}}~{V4v}})}})}})}})}}~{V2x}})}})}}~{V0R}})}}~{V2x}})}})}}~{V2x}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfpgggpfpfogfhng}
We define $\mname{c\_2Erat\_2Erat\_\_of\_\_num}$ to be ${({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2EWFREC}}~{ty\_2Enum\_2Enum}}~{ty\_2Erat\_2Erat}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{({{2}^{{ty\_2Enum\_2Enum}}})}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V0R \in {({{({{2}^{{ty\_2Enum\_2Enum}}})}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Erelation\_2EWF}}~{ty\_2Enum\_2Enum}})}}~{V0R}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{V0R}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V1n}})}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V1n}})}})}})}})}})}})}})}})}})}}~{({{\uniddcgcg} V2rat\_\_of\_\_num \in {({{ty\_2Erat\_2Erat}^{{ty\_2Enum\_2Enum}}})} .  {({{\uniddcgcg} V3a \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Earithmetic\_2Enum\_\_CASE}~{ty\_2Erat\_2Erat}})}}~{V3a}})}}~{({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{ty\_2Erat\_2Erat}})}}~{\mname{c\_2Erat\_2Erat\_\_0}}})}})}}~{({{\uniddcgcg} V4v \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Earithmetic\_2Enum\_\_CASE}~{ty\_2Erat\_2Erat}})}}~{V4v}})}}~{({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{ty\_2Erat\_2Erat}})}}~{\mname{c\_2Erat\_2Erat\_\_1}}})}})}}~{({{\uniddcgcg} V5n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{ty\_2Erat\_2Erat}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_add}}}~{({{{ap}~{V2rat\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V5n}})}})}})}}~{\mname{c\_2Erat\_2Erat\_\_1}}})}})}})}})}})}})}})}})}})}$.
\end{definition}
Let $ c\_2Erat\_2ERATN : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Erat\_2ERATN} \in {({{ty\_2Einteger\_2Eint}^{{ty\_2Erat\_2Erat}}})}}\label{hyp:ngfgngpfdgpfcdfechbgehpfcdfecfbeefoe}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfeoefhng}
We define $\mname{c\_2Einteger\_2ENum}$ to be ${{\uniddcgcg} V0i \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Einteger\_2Eint}})}}~{V0i}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{V1n}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfbgjgoggh}
We define $\mname{c\_2Efrac\_2Efrac\_\_ainv}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_neg}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfbgjgoggh}
We define $\mname{c\_2Erat\_2Erat\_\_ainv}$ to be ${{\uniddcgcg} V0r1 \in {ty\_2Erat\_2Erat} .  {({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_ainv}}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V0r1}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfpgggpfpfjgogeh}
We define $\mname{c\_2Erat\_2Erat\_\_of\_\_int}$ to be ${{\uniddcgcg} V0i \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Erat\_2Erat}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_lt}}}~{V0i}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_ainv}}}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_of\_\_num}}}~{({{{ap}~{\mname{c\_2Einteger\_2ENum}}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_neg}}}~{V0i}})}})}})}})}})}}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_of\_\_num}}}~{({{{ap}~{\mname{c\_2Einteger\_2ENum}}}~{V0i}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfeihehfgogdhjgpgogpfcdfedfheoe}
We define $\mname{c\_2EintExtension\_2ESGN}$ to be ${{\uniddcgcg} V0x \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Einteger\_2Eint}})}}~{V0x}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_lt}}}~{V0x}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_neg}}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfdhhgog}
We define $\mname{c\_2Efrac\_2Efrac\_\_sgn}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{\mname{c\_2EintExtension\_2ESGN}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfngjgoggh}
We define $\mname{c\_2Efrac\_2Efrac\_\_minv}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_sgn}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2EABS}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfngfhmg}
We define $\mname{c\_2Efrac\_2Efrac\_\_mul}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {{\uniddcgcg} V1f2 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V1f2}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V1f2}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfegjggh}
We define $\mname{c\_2Efrac\_2Efrac\_\_div}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {{\uniddcgcg} V1f2 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_mul}}}~{V0f1}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_minv}}}~{V1f2}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfegjggh}
We define $\mname{c\_2Erat\_2Erat\_\_div}$ to be ${{\uniddcgcg} V0r1 \in {ty\_2Erat\_2Erat} .  {{\uniddcgcg} V1r2 \in {ty\_2Erat\_2Erat} .  {({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_div}}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V0r1}})}})}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V1r2}})}})}})}}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r \in ty\_2Erat\_2Erat .  ((V0r=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_div}~\\
(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_int}~(ap~c\_2Erat\_2ERATN~V0r)))~(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\\
(ap~c\_2Erat\_2ERATD~V0r)))){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~\\
(ap~c\_2Erat\_2ERATD~V0r))){\unicdcdcdhd}((((ap~c\_2Erat\_2ERATN~V0r)=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})){\unicdbdjdcd}((ap~c\_2Erat\_2ERATD~V0r)=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}({\unicdcdadad} V1n\_27 \in \\
ty\_2Einteger\_2Eint .  ({\unicdcdadad} V2d\_27 \in ty\_2Enum\_2Enum .  (((V0r=\\
(ap~(ap~\mname{c\_2Erat\_2Erat\_\_div}~(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_int}~V1n\_27))~\\
(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~V2d\_27))){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
\mname{c\_2Enum\_2E0})~V2d\_27))){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_le}~(ap~\mname{c\_2Einteger\_2EABS}~\\
(ap~c\_2Erat\_2ERATN~V0r)))~(ap~\mname{c\_2Einteger\_2EABS}~V1n\_27))))))))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefoeeepfpfefiene}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfechbgehpfcdfecfbeefoepfpfcfbeefeepfpffebfpfpfefiene}
$$\begin{array}{c}({\unicdcdadad} V0r \in ty\_2Erat\_2Erat .  (V0r=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_div}~\\
(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_int}~(ap~c\_2Erat\_2ERATN~V0r)))~(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\\
(ap~c\_2Erat\_2ERATD~V0r)))))\end{array}$$
\end{theorem}
\end{document}
