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\newtheorem{axm}{Axiom}[section]
\newtheorem{abr}{Abbreviation}[section]
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\newtheorem{dec}{Rule Declaration}[section]
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\title{Some more theorems of Predicate Calculus}
\author{Michael Meyling}
\date{\tt <module@qedeq.org>}


\begin{document}

\maketitle

\setlongtables



{\small
This document is part of the project ``Hilbert II''.
To get more information about this project look at:\\
\mbox{\url{http://www.qedeq.org}}.


Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2
or any later version published by the Free Software Foundation;
with no Invariant Sections, no Front-Cover Texts, and no
Back-Cover Texts. See also under \url{http://www.gnu.org/copyleft/}
}


\section*{Abstract}

This module includes first proofs of predicate calculus theorems.


\section*{Specification}

This document has the following specification:

\begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}}
Name: & predtheo2 \\
Version: & 1.00.00 \\
Rule version: & 1.02.00 \\
Orgin: & \url{http://www.qedeq.org/0_00_53/predtheo2_1.00.00_1.02.00.qedeq} \\
\end{longtable}

\medskip

Author of this module: 
\begin{longtable}[h!]{l@{\extracolsep{\fill}}l}
Michael Meyling & mime@qedeq.org \\
\end{longtable}




\section*{References}

This document uses the results of the following documents:

\begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}}
Name: & predaxiom \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{predaxiom_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{predaxiom_1.00.00_1.00.00.pdf}  \\
Name: & prophilbert3 \\
Version: & 1.00.00 \\
Rule version: & 1.02.00 \\
Orgin: & \url{prophilbert3_1.00.00_1.02.00.qedeq}  \\
pdf: & \url{prophilbert3_1.00.00_1.02.00.pdf}  \\
\end{longtable}


\section*{Content}




A simple implication:




\begin{thm}[predtheo1]
\hypertarget{predtheo1}{}
\begin{displaymath}
\forall x\, R(x)\ \rightarrow \  \exists x\, R(x)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{predtheo1:1}
  $1$ & $\forall x\, R(x)\ \rightarrow \  R(y)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{axiom5}{axiom5} } \\
\label{predtheo1:2}
  $2$ & $R(y)\ \rightarrow \  \exists x\, R(x)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{axiom6}{axiom6} } \\
\label{predtheo1:3}
  $3$ & $\forall x\, R(x)\ \rightarrow \  \exists x\, R(x)$
  & {\tiny \hyperref{prophilbert1_1.00.00_1.02.00.pdf}{}{rule11}{HS} with \hyperref[predtheo1:1]{$1$} and \hyperref[predtheo1:2]{$2$}
} \\
 & & \qedhere
\end{longtable}
\end{proof}



A well known implication:




\begin{thm}[predtheo2]
\hypertarget{predtheo2}{}
\begin{displaymath}
\exists x\, R(x)\ \rightarrow \  \neg \forall x\, \neg R(x)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{predtheo2:1}
  $1$ & $\forall x\, R(x)\ \rightarrow \  R(y)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{axiom5}{axiom5} } \\
\label{predtheo2:2}
  $2$ & $\forall x\, \neg R(x)\ \rightarrow \  \neg R(y)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule3}{replace}  $R(@S_{1})$ by $\neg R(@S_{1})$ in \hyperref[predtheo2:1]{$1$}
} \\
\label{predtheo2:3}
  $3$ & $\neg \forall x\, \neg R(x)\ \vee \  \neg R(y)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule5}{use abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[predtheo2:2]{$2$} at occurence $1$
} \\
\label{predtheo2:4}
  $4$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom3}{axiom3} } \\
\label{predtheo2:5}
  $5$ & $(\neg \forall x\, \neg R(x)\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg \forall x\, \neg R(x))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $\neg \forall x\, \neg R(x)$ in \hyperref[predtheo2:4]{$4$}
} \\
\label{predtheo2:6}
  $6$ & $(\neg \forall x\, \neg R(x)\ \vee \  \neg R(y))\ \rightarrow \  (\neg R(y)\ \vee \  \neg \forall x\, \neg R(x))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $\neg R(y)$ in \hyperref[predtheo2:5]{$5$}
} \\
\label{predtheo2:7}
  $7$ & $\neg R(y)\ \vee \  \neg \forall x\, \neg R(x)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[predtheo2:3]{$3$}, \hyperref[predtheo2:6]{$6$}} \\
\label{predtheo2:8}
  $8$ & $R(y)\ \rightarrow \  \neg \forall x\, \neg R(x)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[predtheo2:7]{$7$} at occurence $1$
} \\
\label{predtheo2:9}
  $9$ & $\exists y\, R(y)\ \rightarrow \  \neg \forall x\, \neg R(x)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule4}{Particularize} by $y$ in \hyperref[predtheo2:8]{$8$}} \\
\label{predtheo2:10}
  $10$ & $\exists x\, R(x)\ \rightarrow \  \neg \forall x\, \neg R(x)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule1}{rename}  $y$ into $x$ in \hyperref[predtheo2:9]{$9$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



The reverse is also true:




\begin{thm}[predtheo3]
\hypertarget{predtheo3}{}
\begin{displaymath}
\neg \forall x\, \neg R(x)\ \rightarrow \  \exists x\, R(x)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{predtheo3:1}
  $1$ & $R(y)\ \rightarrow \  \exists x\, R(x)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{axiom6}{axiom6} } \\
\label{predtheo3:2}
  $2$ & $\neg \exists x\, R(x)\ \rightarrow \  \neg R(y)$
  & {\tiny \hyperref{prophilbert1_1.00.00_1.02.00.pdf}{}{rule10}{apply} \hyperref{prophilbert1_1.00.00_1.02.00.pdf}{}{hilb7}{hilb7} in \hyperref[predtheo3:1]{$1$}
} \\
\label{predtheo3:3}
  $3$ & $\neg \exists x\, R(x)\ \rightarrow \  \forall y\, \neg R(y)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule5}{Generalize} by $y$ in \hyperref[predtheo3:2]{$2$}} \\
\label{predtheo3:4}
  $4$ & $\neg \forall y\, \neg R(y)\ \rightarrow \  \neg \neg \exists x\, R(x)$
  & {\tiny \hyperref{prophilbert1_1.00.00_1.02.00.pdf}{}{rule10}{apply} \hyperref{prophilbert1_1.00.00_1.02.00.pdf}{}{hilb7}{hilb7} in \hyperref[predtheo3:3]{$3$}
} \\
\label{predtheo3:5}
  $5$ & $\neg \forall y\, \neg R(y)\ \rightarrow \  \exists x\, R(x)$
  & {\tiny \hyperref{prophilbert1_1.00.00_1.02.00.pdf}{}{rule30}{elementary equivalence} in \hyperref[predtheo3:4]{$4$} at \hyperref[predtheo3:1]{$1$} of \hyperref{prophilbert1_1.00.00_1.02.00.pdf}{}{hilb6}{hilb6}  with \hyperref{prophilbert1_1.00.00_1.02.00.pdf}{}{hilb6}{hilb6} 
} \\
\label{predtheo3:6}
  $6$ & $\neg \forall x\, \neg R(x)\ \rightarrow \  \exists x\, R(x)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule1}{rename}  $y$ into $x$ in \hyperref[predtheo3:5]{$5$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



Exchange of universal quantors:




\begin{thm}[predtheo4]
\hypertarget{predtheo4}{}
\begin{displaymath}
\forall x\, \forall y\, R(x, y)\ \rightarrow \  \forall y\, \forall x\, R(x, y)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{predtheo4:1}
  $1$ & $\forall x\, R(x)\ \rightarrow \  R(y)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{axiom5}{axiom5} } \\
\label{predtheo4:2}
  $2$ & $\forall y\, R(y)\ \rightarrow \  R(u)$
  & {\tiny \hyperref{subst_1.00.00_1.01.00.pdf}{}{rule8}{Substitute Variables} in \hyperref[predtheo4:1]{$1$}
} \\
\label{predtheo4:3}
  $3$ & $\forall y\, R(z, y)\ \rightarrow \  R(z, u)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule3}{replace}  $R(@S_{1})$ by $R(z, @S_{1})$ in \hyperref[predtheo4:2]{$2$}
} \\
\label{predtheo4:4}
  $4$ & $\forall v\, R(v)\ \rightarrow \  R(z)$
  & {\tiny \hyperref{subst_1.00.00_1.01.00.pdf}{}{rule8}{Substitute Variables} in \hyperref[predtheo4:1]{$1$}
} \\
\label{predtheo4:5}
  $5$ & $\forall v\, \forall w\, R(v, w)\ \rightarrow \  \forall w\, R(z, w)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule3}{replace}  $R(@S_{1})$ by $\forall w\, R(@S_{1}, w)$ in \hyperref[predtheo4:4]{$4$}
} \\
\label{predtheo4:6}
  $6$ & $\forall x\, \forall y\, R(x, y)\ \rightarrow \  \forall y\, R(z, y)$
  & {\tiny \hyperref{subst_1.00.00_1.01.00.pdf}{}{rule8}{Substitute Variables} in \hyperref[predtheo4:5]{$5$}
} \\
\label{predtheo4:7}
  $7$ & $\forall x\, \forall y\, R(x, y)\ \rightarrow \  R(z, u)$
  & {\tiny \hyperref{prophilbert1_1.00.00_1.02.00.pdf}{}{rule11}{HS} with \hyperref[predtheo4:6]{$6$} and \hyperref[predtheo4:3]{$3$}
} \\
\label{predtheo4:8}
  $8$ & $\forall x\, \forall y\, R(x, y)\ \rightarrow \  \forall z\, R(z, u)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule5}{Generalize} by $z$ in \hyperref[predtheo4:7]{$7$}} \\
\label{predtheo4:9}
  $9$ & $\forall x\, \forall y\, R(x, y)\ \rightarrow \  \forall u\, \forall z\, R(z, u)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule5}{Generalize} by $u$ in \hyperref[predtheo4:8]{$8$}} \\
\label{predtheo4:10}
  $10$ & $\forall x\, \forall y\, R(x, y)\ \rightarrow \  \forall y\, \forall z\, R(z, y)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule1}{rename}  $u$ into $y$ in \hyperref[predtheo4:9]{$9$} at occurence $1$
} \\
\label{predtheo4:11}
  $11$ & $\forall x\, \forall y\, R(x, y)\ \rightarrow \  \forall y\, \forall x\, R(x, y)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule1}{rename}  $z$ into $x$ in \hyperref[predtheo4:10]{$10$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



Implication of changing sequence of existence and universal quantor:




\begin{thm}[predtheo5]
\hypertarget{predtheo5}{}
\begin{displaymath}
\exists x\, \forall y\, R(x, y)\ \rightarrow \  \forall y\, \exists x\, R(x, y)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{predtheo5:1}
  $1$ & $\forall x\, R(x)\ \rightarrow \  R(y)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{axiom5}{axiom5} } \\
\label{predtheo5:2}
  $2$ & $\forall y\, R(y)\ \rightarrow \  R(u)$
  & {\tiny \hyperref{subst_1.00.00_1.01.00.pdf}{}{rule8}{Substitute Variables} in \hyperref[predtheo5:1]{$1$}
} \\
\label{predtheo5:3}
  $3$ & $\forall y\, R(x, y)\ \rightarrow \  R(x, u)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule3}{replace}  $R(@S_{1})$ by $R(x, @S_{1})$ in \hyperref[predtheo5:2]{$2$}
} \\
\label{predtheo5:4}
  $4$ & $R(y)\ \rightarrow \  \exists x\, R(x)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{axiom6}{axiom6} } \\
\label{predtheo5:5}
  $5$ & $R(x)\ \rightarrow \  \exists z\, R(z)$
  & {\tiny \hyperref{subst_1.00.00_1.01.00.pdf}{}{rule8}{Substitute Variables} in \hyperref[predtheo5:4]{$4$}
} \\
\label{predtheo5:6}
  $6$ & $R(x, u)\ \rightarrow \  \exists z\, R(z, u)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule3}{replace}  $R(@S_{1})$ by $R(@S_{1}, u)$ in \hyperref[predtheo5:5]{$5$}
} \\
\label{predtheo5:7}
  $7$ & $\forall y\, R(x, y)\ \rightarrow \  \exists z\, R(z, u)$
  & {\tiny \hyperref{prophilbert1_1.00.00_1.02.00.pdf}{}{rule11}{HS} with \hyperref[predtheo5:3]{$3$} and \hyperref[predtheo5:6]{$6$}
} \\
\label{predtheo5:8}
  $8$ & $\exists x\, \forall y\, R(x, y)\ \rightarrow \  \exists z\, R(z, u)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule4}{Particularize} by $x$ in \hyperref[predtheo5:7]{$7$}} \\
\label{predtheo5:9}
  $9$ & $\exists x\, \forall y\, R(x, y)\ \rightarrow \  \forall u\, \exists z\, R(z, u)$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule5}{Generalize} by $u$ in \hyperref[predtheo5:8]{$8$}} \\
\label{predtheo5:10}
  $10$ & $\exists x\, \forall y\, R(x, y)\ \rightarrow \  \forall y\, \exists x\, R(x, y)$
  & {\tiny \hyperref{subst_1.00.00_1.01.00.pdf}{}{rule8}{Substitute Variables} in \hyperref[predtheo5:9]{$9$}
} \\
 & & \qedhere
\end{longtable}
\end{proof}


\end{document}