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\begin {document}
\title {Udvidelse af S-reglerne}
\author {Maja Hanne T\o nnesen, Rune Christoffer Kildetoft Andresen \\ \& Niels Peter Meyn Milthers}
\date {\today}
\maketitle
\tableofcontents
" [ ragged right expansion ] "
\section{Introduktion}
Dette skriftlige projekt er lavet i forbindelse med
Logik-kurset 2006\footnote{Kursus 061004/202 Logik}.
Vores m\aa{}l i denne opgave var at bevise en delm\ae{}ngde af udsagnene\footnote{Eng.:
proposition} vedr\o{}rende l\ae{}res\ae{}tningerne\footnote{Eng.: theorem} fra
S-systemet, som er beskrevet i Mendelson \cite{mendelson} i kapitel 3.1. For lethedens skyld er alle nummereringer de samme som i Mendelson.
Mere pr\ae{}cist g\aa{}r opgaven ud p\aa{} at bevise 3.2(j-o), bevise 3.4, bevise
3.5, tilf\o{}je aksiomet $x < y \Leftrightarrow \exists z : z \neq 0
\land z + x = y$, bevise 3.7, tilf\o{}je aksiomer, der definere " [ math object x divides object y end math ] ",
bevise 3.10 og bevise 3.11.
I afsnittene 3 til 7 er beviserne for de p\aa{}viste lemmaer
gennemg\aa{}et. Alle trivielle hj\ae{}lpelemmaer er bevist i appendix \ref{help}.
\section{Konklusion}
Vi har ikke l\o{}st opgaven til fulde, men har dog form\aa{}et at p\aa{}vise
3.2j - 3.2o, 3.4, 3.5a -g. P\aa{} grund af problemer med definitionen af
reglen svarende til ``existensial rule'' side 77 i mendelson, har vi
som beskrevet i afsnit \ref{35} ikke v\ae{}ret i stand til at kunne bevise
lemmaerne fra 3.5h og frem.
Vi har dog som beskrevet i afsnit \ref{35h} og \ref{exist} gennemg\aa{}et hvordan
beviserne for 3.5h og ``existensial rule'' ville have set ud, hvis
problemet ikke var opst\aa{}et.
Vi har endvidere kort gennemg\aa{}et de definitioner, som ville v\ae{}re
n\o{}dvendige for at kunne p\aa{}vise Lemma 3.7
\section{S-reglerne}
S-systemet er en f\o{}rste ordens teori, som er udviklet ud fra Peanos
postulater og ved hj\ae{}lp m\ae{}ngdel\ae{}re. Det skulle v\ae{}re passende at
bruge til at bevise basis-resultaterne for tal-teori. Aksiomerne for S
er f\o{}lgende.\\
\begin{list}{}{
\setlength{\leftmargin}{5em}
\setlength{\itemindent}{-5em}}
\item \makebox[0.45\textwidth][l]{" [ math theory system s end theory end math ] "}" [ math in theory system s rule rule mp says for all terms meta a comma meta b indeed meta a imply meta b infer meta a infer meta b end rule end math ] "
\item \makebox[0.45\textwidth][l]{" [ math in theory system s rule rule gen says for all terms meta x comma meta a indeed meta a infer for all meta x indeed meta a end rule end math ] "}" [ math in theory system s rule deduction says for all terms meta a comma meta b indeed deduction meta a conclude meta b end deduction endorse meta a infer meta b end rule end math ] "
\item \makebox[0.45\textwidth][l]{}" [ math in theory system s rule axiom s two says for all terms meta a comma meta b indeed meta a equal meta b infer meta a suc equal meta b suc end rule end math ] "
\item \makebox[0.45\textwidth][l]{" [ math in theory system s rule axiom s three says for all terms meta a indeed not zero equal meta a suc end rule end math ] "}" [ math in theory system s rule axiom s four says for all terms meta a comma meta b indeed meta a suc equal meta b suc infer meta a equal meta b end rule end math ] "
\item \makebox[0.45\textwidth][l]{" [ math in theory system s rule axiom s five says for all terms meta a indeed meta a plus zero equal meta a end rule end math ] "}" [ math in theory system s rule axiom s six says for all terms meta a comma meta b indeed meta a plus meta b suc equal parenthesis meta a plus meta b end parenthesis suc end rule end math ] "
\item \makebox[0.45\textwidth][l]{" [ math in theory system s rule axiom s seven says for all terms meta a indeed meta a times zero equal zero end rule end math ] "}" [ math in theory system s rule axiom s eight says for all terms meta a comma meta b indeed meta a times parenthesis meta b suc end parenthesis equal parenthesis meta a times meta b end parenthesis plus meta a end rule end math ] "
\item " [ math in theory system s rule axiom s one says for all terms meta a comma meta b comma meta c indeed meta a equal meta b infer meta a equal meta c infer meta b equal meta c end rule end math ] "
\end{list}
Endvidere er reglen \textit{S9}, som er grundl\ae{}ggende for matematisk
induktion, ogs\aa{} en del af \textit{S}-teorien. Denne regel kan p\aa{}
almindeligt sprog udtrykkes som:
\begin{quote}\textit{Hvis en egenskab holder for 0 og egenskaben holder for efterf\o{}lgeren $x'$
til et naturligt tal $x$, som egenskaben g\ae{}lder for, s\aa{} vil egenskaben
g\ae{}lde for alle naturlige tal. }
\end{quote}
I pyk er S9 defineret som:\\
\begin{list}{}{
\setlength{\leftmargin}{5em}
\setlength{\itemindent}{-5em}}
\item " [ math in theory system s rule axiom s nine says for all terms meta x comma meta a comma meta b comma meta c indeed sub meta b is meta a where meta x is zero end sub endorse sub meta c is meta a where meta x is meta x suc end sub endorse meta b infer meta a imply meta c infer meta a end rule end math ] "
\end{list}
Reglen for bevis ved mods\ae{}tninger er den sidste grundl\ae{}ggende regel,
der skal bruges i beviserne. Reglen kan p\aa{} almindeligt sprog udtrykkes
som f\o{}lgende:
\begin{quote} \textit{Hvis man ud fra en antagelse $A$
kan p\aa{}vise at en egenskab $B$ g\ae{}lder, og at man (n\aa{}r antagelsen stadig g\ae{}lder) kan vise at den modsatte egenskab $\lnot B$ ogs\aa{} g\ae{}lder, s\aa{}
m\aa{} det modsatte af antagelsen g\ae{}lde, dvs. $\lnot A$.}
\end{quote}
I pyk er reglen defineret som:\\
\begin{list}{}{
\setlength{\leftmargin}{5em}
\setlength{\itemindent}{-5em}}
\item " [ math in theory system s rule double negation says for all terms meta a indeed for all terms meta b indeed not meta b imply not meta a infer not meta b imply meta a infer meta b end rule end math ] "
\end{list}
\section{Udsagn 3.2}
Dette udsagn indeholder de grundl\ae{}ggende regneregler for tal, som
opf\o{}rer sig som de naturlige tal. Hj\ae{}lpes\ae{}tningerne eller lemmaerne er
generelt set en direkte konsekvens af aksiomerne.
Alle lemmaerne i dette afsnit kan udledes ud fra de tidligere n\ae{}vnte og som
det er bevist i b\aa{}de check \cite{check} og Mendelson er der for ethvert
udtryk " [ math meta a comma meta b comma meta c end math ] "
f\o{}lgende velformulerede s\ae{}tninger\footnote{Eng.: Wellformed formulas} i systemet S:\\
\begin{list}{}{
\setlength{\leftmargin}{5em}
\setlength{\itemindent}{-5em}}
\item " [ math in theory system s lemma prop three two a says for all terms meta a indeed meta a equal meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three two b says for all terms meta a comma meta b indeed meta a equal meta b infer meta b equal meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three two c says for all terms meta a comma meta b comma meta c indeed meta a equal meta b infer meta b equal meta c infer meta a equal meta c end lemma end math ] "
\item " [ math in theory system s lemma prop three two d says for all terms meta a comma meta b comma meta c indeed meta a equal meta c infer meta b equal meta c infer meta a equal meta b end lemma end math ] "
\item " [ math in theory system s lemma prop three two e says for all terms meta a comma meta b comma meta c indeed meta a equal meta b infer meta a plus meta c equal meta b plus meta c end lemma end math ] "
\item " [ math in theory system s lemma prop three two f says for all terms meta a indeed meta a equal zero plus meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three two g says for all terms meta a comma meta b indeed meta a suc plus meta b equal parenthesis meta a plus meta b end parenthesis suc end lemma end math ] "
\item " [ math in theory system s lemma prop three two h says for all terms meta a comma meta b indeed meta a plus meta b equal meta b plus meta a end lemma end math ] "
\end{list}
Da 3.2i ikke er bevist i check, (men er i Mendelson),
har vi valgt for \o{}velsens skyld at indskrive beviset:\\
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three two i says for all terms meta a comma meta b comma meta c indeed meta a equal meta b infer meta c plus meta a equal meta c plus meta b end lemma end math ] "
\item " [ math system s proof of prop three two i reads any term meta a comma meta b comma meta c end line block any term macro indent meta a comma meta b comma meta c end line line ell d premise macro indent meta a equal meta b end line line ell a because prop three two e modus ponens ell d indeed macro indent meta a plus meta c equal meta b plus meta c end line line ell b because prop three two h indeed macro indent meta a plus meta c equal meta c plus meta a end line line ell c because prop three two h indeed macro indent meta b plus meta c equal meta c plus meta b end line line ell h because axiom s one modus ponens ell a modus ponens ell b indeed macro indent meta b plus meta c equal meta c plus meta a end line line ell j because prop three two b modus ponens ell h indeed macro indent meta c plus meta a equal meta b plus meta c end line because prop three two c modus ponens ell j modus ponens ell c indeed macro indent meta c plus meta a equal meta c plus meta b end line line ell g end block because deduction modus ponens ell g indeed meta a equal meta b infer meta c plus meta a equal meta c plus meta b qed end math ] "
\end{list}
De f\o{}lgende lemmaer er opskrevet, men ikke bevist i Mendelson. Disse er
blandt dem som vi \o{}nsker at bevise.
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three two j says for all terms meta a comma meta b comma meta c indeed parenthesis meta a plus meta b end parenthesis plus meta c equal meta a plus parenthesis meta b plus meta c end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma prop three two k says for all terms meta a comma meta b comma meta c indeed meta a equal meta b infer meta a times meta c equal meta b times meta c end lemma end math ] "
\item " [ math in theory system s lemma prop three two l says for all terms meta a indeed zero times meta a equal zero end lemma end math ] "
\item " [ math in theory system s lemma prop three two m says for all terms meta a comma meta b indeed meta a suc times meta b equal meta a times meta b plus meta b end lemma end math ] "
\item " [ math in theory system s lemma prop three two n says for all terms meta a comma meta b indeed meta a times meta b equal meta b times meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three two o says for all terms meta a comma meta b comma meta c indeed meta a equal meta b infer meta c times meta a equal meta c times meta b end lemma end math ] "
\end{list}
Her efter f\o{}lger beviserne.
\subsection{3.2j}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three two j one says for all terms meta a comma meta b indeed parenthesis meta a plus meta b end parenthesis plus zero equal meta a plus parenthesis meta b plus zero end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three two j one reads any term meta a comma meta b end line line ell a because axiom s five indeed parenthesis meta a plus meta b end parenthesis plus zero equal meta a plus meta b end line line ell b because axiom s five indeed meta b plus zero equal meta b end line line ell c because prop three two i modus ponens ell b indeed meta a plus parenthesis meta b plus zero end parenthesis equal meta a plus meta b end line because prop three two d modus ponens ell a modus ponens ell c indeed parenthesis meta a plus meta b end parenthesis plus zero equal meta a plus parenthesis meta b plus zero end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three two j two says for all terms meta a comma meta b comma meta c indeed parenthesis meta a plus meta b end parenthesis plus meta c equal meta a plus parenthesis meta b plus meta c end parenthesis imply parenthesis meta a plus meta b end parenthesis plus meta c suc equal meta a plus parenthesis meta b plus meta c suc end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three two j two reads any term meta a comma meta b comma meta c end line block any term macro indent meta a comma meta b comma meta c end line line ell a premise macro indent parenthesis meta a plus meta b end parenthesis plus meta c equal meta a plus parenthesis meta b plus meta c end parenthesis end line line ell b because axiom s six indeed macro indent parenthesis meta a plus meta b end parenthesis plus meta c suc equal parenthesis parenthesis meta a plus meta b end parenthesis plus meta c end parenthesis suc end line line ell c because axiom s two modus ponens ell a indeed macro indent parenthesis parenthesis meta a plus meta b end parenthesis plus meta c end parenthesis suc equal parenthesis meta a plus parenthesis meta b plus meta c end parenthesis end parenthesis suc end line line ell d because prop three two c modus ponens ell b modus ponens ell c indeed macro indent parenthesis meta a plus meta b end parenthesis plus meta c suc equal parenthesis meta a plus parenthesis meta b plus meta c end parenthesis end parenthesis suc end line line ell e because axiom s six indeed macro indent meta b plus meta c suc equal parenthesis meta b plus meta c end parenthesis suc end line line ell f because prop three two i modus ponens ell e indeed macro indent meta a plus parenthesis meta b plus meta c suc end parenthesis equal meta a plus parenthesis meta b plus meta c end parenthesis suc end line line ell g because axiom s six indeed macro indent meta a plus parenthesis meta b plus meta c end parenthesis suc equal parenthesis meta a plus parenthesis meta b plus meta c end parenthesis end parenthesis suc end line line ell h because prop three two c modus ponens ell f modus ponens ell g indeed macro indent meta a plus parenthesis meta b plus meta c suc end parenthesis equal parenthesis meta a plus parenthesis meta b plus meta c end parenthesis end parenthesis suc end line because prop three two d modus ponens ell d modus ponens ell h indeed macro indent parenthesis meta a plus meta b end parenthesis plus meta c suc equal meta a plus parenthesis meta b plus meta c suc end parenthesis end line line ell j end block because deduction modus ponens ell j indeed parenthesis meta a plus meta b end parenthesis plus meta c equal meta a plus parenthesis meta b plus meta c end parenthesis imply parenthesis meta a plus meta b end parenthesis plus meta c suc equal meta a plus parenthesis meta b plus meta c suc end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three two j says for all terms meta a comma meta b comma meta c indeed parenthesis meta a plus meta b end parenthesis plus meta c equal meta a plus parenthesis meta b plus meta c end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three two j reads any term meta a comma meta b comma meta c end line block line ell a because prop three two j one indeed macro indent parenthesis object x plus object y end parenthesis plus zero equal object x plus parenthesis object y plus zero end parenthesis end line line ell b because prop three two j two indeed macro indent parenthesis object x plus object y end parenthesis plus object z equal object x plus parenthesis object y plus object z end parenthesis imply parenthesis object x plus object y end parenthesis plus object z suc equal object x plus parenthesis object y plus object z suc end parenthesis end line because axiom s nine at object z modus ponens ell a modus ponens ell b indeed macro indent parenthesis object x plus object y end parenthesis plus object z equal object x plus parenthesis object y plus object z end parenthesis end line line ell c end block because deduction modus ponens ell c indeed parenthesis meta a plus meta b end parenthesis plus meta c equal meta a plus parenthesis meta b plus meta c end parenthesis qed end math ] "
\end{list}
\subsection{3.2k}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three two k one says for all terms meta a comma meta b indeed meta a equal meta b imply meta a times zero equal meta b times zero end lemma end math ] "
\item " [ math system s proof of prop three two k one reads any term meta a comma meta b end line block any term meta a comma meta b end line line ell f premise meta a equal meta b end line line ell a because axiom s seven indeed meta a times zero equal zero end line line ell b because axiom s seven indeed meta b times zero equal zero end line line ell c because prop three two a indeed zero equal zero end line line ell d because prop three two b modus ponens ell b indeed zero equal meta b times zero end line because prop three two c modus ponens ell a modus ponens ell d indeed meta a times zero equal meta b times zero end line line ell e end block because deduction modus ponens ell e indeed meta a equal meta b imply meta a times zero equal meta b times zero qed end math ] "
\item " [ math in theory system s lemma prop three two k two says for all terms meta a comma meta b comma meta c indeed parenthesis meta a equal meta b imply meta a times meta c equal meta b times meta c end parenthesis imply parenthesis meta a equal meta b imply meta a times meta c suc equal meta b times meta c suc end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three two k two reads any term meta a comma meta b comma meta c end line block any term macro indent meta a comma meta b comma meta c end line line ell a premise macro indent meta a equal meta b imply meta a times meta c equal meta b times meta c end line line ell b premise macro indent meta a equal meta b end line line ell c because ell a object modus ponens ell b indeed macro indent meta a times meta c equal meta b times meta c end line line ell d because axiom s eight indeed macro indent meta a times meta c suc equal meta a times meta c plus meta a end line line ell e because axiom s eight indeed macro indent meta b times meta c suc equal meta b times meta c plus meta b end line line ell f because prop three two e modus ponens ell c indeed macro indent parenthesis meta a times meta c end parenthesis plus meta a equal parenthesis meta b times meta c end parenthesis plus meta a end line line ell h because prop three two i modus ponens ell b indeed macro indent parenthesis meta b times meta c end parenthesis plus meta a equal parenthesis meta b times meta c end parenthesis plus meta b end line line ell g because prop three two c modus ponens ell f modus ponens ell h indeed macro indent meta a times meta c plus meta a equal meta b times meta c plus meta b end line line ell j because prop three two c modus ponens ell d modus ponens ell g indeed macro indent meta a times meta c suc equal meta b times meta c plus meta b end line because prop three two d modus ponens ell j modus ponens ell e indeed macro indent meta a times meta c suc equal meta b times meta c suc end line line ell i end block because deduction modus ponens ell i indeed parenthesis meta a equal meta b imply meta a times meta c equal meta b times meta c end parenthesis imply parenthesis meta a equal meta b imply meta a times meta c suc equal meta b times meta c suc end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three two k says for all terms meta a comma meta b comma meta c indeed meta a equal meta b infer meta a times meta c equal meta b times meta c end lemma end math ] "
\item " [ math system s proof of prop three two k reads any term meta a comma meta b comma meta c end line line ell e premise meta a equal meta b end line block line ell a because prop three two k one indeed macro indent object x equal object y imply object x times zero equal object y times zero end line line ell b because prop three two k two indeed macro indent parenthesis object x equal object y imply object x times object z equal object y times object z end parenthesis imply parenthesis object x equal object y imply object x times object z suc equal object y times object z suc end parenthesis end line because axiom s nine at object z modus ponens ell a modus ponens ell b indeed macro indent object x equal object y imply object x times object z equal object y times object z end line line ell c end block line ell d because deduction modus ponens ell c indeed meta a equal meta b imply meta a times meta c equal meta b times meta c end line because ell d object modus ponens ell e indeed meta a times meta c equal meta b times meta c qed end math ] "
\end{list}
\subsection{3.2l}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three two l two says for all terms meta a indeed zero times meta a equal zero imply zero times meta a suc equal zero end lemma end math ] "
\item " [ math system s proof of prop three two l two reads any term meta a end line block any term macro indent meta a end line line ell a premise macro indent zero times meta a equal zero end line line ell b because axiom s eight indeed macro indent zero times meta a suc equal zero times meta a plus zero end line line ell c because axiom s five indeed macro indent zero times meta a plus zero equal zero times meta a end line line ell d because prop three two c modus ponens ell b modus ponens ell c indeed macro indent zero times meta a suc equal zero times meta a end line because prop three two c modus ponens ell d modus ponens ell a indeed macro indent zero times meta a suc equal zero end line line ell e end block because deduction modus ponens ell e indeed zero times meta a equal zero imply zero times meta a suc equal zero qed end math ] "
\item " [ math in theory system s lemma prop three two l says for all terms meta a indeed zero times meta a equal zero end lemma end math ] "
\item " [ math system s proof of prop three two l reads any term meta a end line block line ell a because axiom s seven indeed macro indent zero times zero equal zero end line line ell b because prop three two l two indeed macro indent zero times object x equal zero imply zero times object x suc equal zero end line because axiom s nine at object x modus ponens ell a modus ponens ell b indeed macro indent zero times object x equal zero end line line ell c end block because deduction modus ponens ell c indeed zero times meta a equal zero qed end math ] "
\end{list}
\subsection{3.2m}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three two m one says for all terms meta a indeed meta a suc times zero equal meta a times zero plus zero end lemma end math ] "
\item " [ math system s proof of prop three two m one reads any term meta a end line line ell a because axiom s seven indeed meta a suc times zero equal zero end line line ell b because prop three two f indeed zero equal zero plus zero end line line ell c because axiom s seven indeed meta a times zero equal zero end line line ell i because prop three two b modus ponens ell c indeed zero equal meta a times zero end line line ell d because prop three two e modus ponens ell i indeed zero plus zero equal meta a times zero plus zero end line line ell g because prop three two c modus ponens ell b modus ponens ell d indeed zero equal meta a times zero plus zero end line because prop three two c modus ponens ell a modus ponens ell g indeed meta a suc times zero equal meta a times zero plus zero end line end math ] "
\item " [ math in theory system s lemma prop three two m two says for all terms meta a comma meta b indeed meta a suc times meta b equal meta a times meta b plus meta b imply meta a suc times meta b suc equal meta a times meta b suc plus meta b suc end lemma end math ] "
\item " [ math system s proof of prop three two m two reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell a premise macro indent meta a suc times meta b equal meta a times meta b plus meta b end line line ell b because axiom s eight indeed macro indent meta a suc times meta b suc equal meta a suc times meta b plus meta a suc end line line ell c because prop three two e modus ponens ell a indeed macro indent parenthesis meta a suc times meta b end parenthesis plus meta a suc equal parenthesis meta a times meta b plus meta b end parenthesis plus meta a suc end line line ell e because axiom s six indeed macro indent meta b plus meta a suc equal parenthesis meta b plus meta a end parenthesis suc end line line ell f because prop three two g indeed macro indent meta b suc plus meta a equal parenthesis meta b plus meta a end parenthesis suc end line line ell g because prop three two d modus ponens ell e modus ponens ell f indeed macro indent meta b plus meta a suc equal meta b suc plus meta a end line line ell i because prop three two h indeed macro indent meta b suc plus meta a equal meta a plus meta b suc end line line ell l because prop three two c modus ponens ell g modus ponens ell i indeed macro indent meta b plus meta a suc equal meta a plus meta b suc end line line ell h because prop three two i modus ponens ell l indeed macro indent meta a times meta b plus parenthesis meta b plus meta a suc end parenthesis equal meta a times meta b plus parenthesis meta a plus meta b suc end parenthesis end line line ell p because prop three two j indeed macro indent parenthesis meta a times meta b plus meta a end parenthesis plus meta b suc equal meta a times meta b plus parenthesis meta a plus meta b suc end parenthesis end line line ell q because prop three two d modus ponens ell h modus ponens ell p indeed macro indent meta a times meta b plus parenthesis meta b plus meta a suc end parenthesis equal parenthesis meta a times meta b plus meta a end parenthesis plus meta b suc end line line ell j because axiom s eight indeed macro indent meta a times meta b suc equal meta a times meta b plus meta a end line line ell k because prop three two e modus ponens ell j indeed macro indent meta a times meta b suc plus meta b suc equal parenthesis meta a times meta b plus meta a end parenthesis plus meta b suc end line line ell m because prop three two d modus ponens ell q modus ponens ell k indeed macro indent meta a times meta b plus parenthesis meta b plus meta a suc end parenthesis equal meta a times meta b suc plus meta b suc end line line ell r because prop three two j indeed macro indent parenthesis meta a times meta b plus meta b end parenthesis plus meta a suc equal meta a times meta b plus parenthesis meta b plus meta a suc end parenthesis end line line ell s because prop three two c modus ponens ell c modus ponens ell r indeed macro indent parenthesis meta a suc times meta b end parenthesis plus meta a suc equal meta a times meta b plus parenthesis meta b plus meta a suc end parenthesis end line line ell n because prop three two c modus ponens ell s modus ponens ell m indeed macro indent parenthesis meta a suc times meta b end parenthesis plus meta a suc equal meta a times meta b suc plus meta b suc end line because prop three two c modus ponens ell b modus ponens ell n indeed macro indent meta a suc times meta b suc equal meta a times meta b suc plus meta b suc end line line ell o end block because deduction modus ponens ell o indeed meta a suc times meta b equal meta a times meta b plus meta b imply meta a suc times meta b suc equal meta a times meta b suc plus meta b suc qed end math ] "
\item " [ math in theory system s lemma prop three two m says for all terms meta a comma meta b indeed meta a suc times meta b equal meta a times meta b plus meta b end lemma end math ] "
\item " [ math system s proof of prop three two m reads any term meta a comma meta b end line block line ell a because prop three two m one indeed macro indent object x suc times zero equal object x times zero plus zero end line line ell b because prop three two m two indeed macro indent parenthesis object x suc times object y equal object x times object y plus object y end parenthesis imply parenthesis object x suc times object y suc equal object x times object y suc plus object y suc end parenthesis end line because axiom s nine at object y modus ponens ell a modus ponens ell b indeed macro indent object x suc times object y equal object x times object y plus object y end line line ell c end block because deduction modus ponens ell c indeed meta a suc times meta b equal meta a times meta b plus meta b qed end math ] "
\end{list}
\subsection{3.2n}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three two n one says for all terms meta a indeed meta a times zero equal zero times meta a end lemma end math ] "
\item " [ math system s proof of prop three two n one reads any term meta a end line line ell a because axiom s seven indeed meta a times zero equal zero end line line ell b because prop three two l indeed zero times meta a equal zero end line because prop three two d modus ponens ell a modus ponens ell b indeed meta a times zero equal zero times meta a qed end math ] "
\item " [ math in theory system s lemma prop three two n two says for all terms meta a comma meta b indeed meta a times meta b equal meta b times meta a imply meta a times meta b suc equal meta b suc times meta a end lemma end math ] "
\item " [ math system s proof of prop three two n two reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell a premise macro indent meta a times meta b equal meta b times meta a end line line ell b because axiom s eight indeed macro indent meta a times meta b suc equal meta a times meta b plus meta a end line line ell c because prop three two e modus ponens ell a indeed macro indent meta a times meta b plus meta a equal meta b times meta a plus meta a end line line ell g because prop three two m indeed macro indent meta b suc times meta a equal meta b times meta a plus meta a end line line ell d because prop three two b modus ponens ell g indeed macro indent meta b times meta a plus meta a equal meta b suc times meta a end line line ell e because prop three two c modus ponens ell c modus ponens ell d indeed macro indent meta a times meta b plus meta a equal meta b suc times meta a end line because prop three two c modus ponens ell b modus ponens ell e indeed macro indent meta a times meta b suc equal meta b suc times meta a end line line ell f end block because deduction modus ponens ell f indeed meta a times meta b equal meta b times meta a imply meta a times meta b suc equal meta b suc times meta a qed end math ] "
\item " [ math in theory system s lemma prop three two n says for all terms meta a comma meta b indeed meta a times meta b equal meta b times meta a end lemma end math ] "
\item " [ math system s proof of prop three two n reads any term meta a comma meta b end line block line ell a because prop three two n one indeed macro indent object x times zero equal zero times object x end line line ell b because prop three two n two indeed macro indent parenthesis object x times object y equal object y times object x end parenthesis imply parenthesis object x times object y suc equal object y suc times object x end parenthesis end line because axiom s nine at object y modus ponens ell a modus ponens ell b indeed macro indent object x times object y equal object y times object x end line line ell c end block because deduction modus ponens ell c indeed meta a times meta b equal meta b times meta a qed end math ] "
\end{list}
\subsection{3.2o}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three two o says for all terms meta a comma meta b comma meta c indeed meta a equal meta b infer meta c times meta a equal meta c times meta b end lemma end math ] "
\item " [ math system s proof of prop three two o reads any term meta a comma meta b comma meta c end line line ell j premise meta a equal meta b end line block any term macro indent meta a comma meta b comma meta c end line line ell a premise macro indent meta a equal meta b end line line ell b because prop three two k modus ponens ell a indeed macro indent meta a times meta c equal meta b times meta c end line line ell d because prop three two n indeed macro indent meta a times meta c equal meta c times meta a end line line ell e because prop three two n indeed macro indent meta b times meta c equal meta c times meta b end line line ell f because prop three two c modus ponens ell b modus ponens ell e indeed macro indent meta a times meta c equal meta c times meta b end line because axiom s one modus ponens ell d modus ponens ell f indeed macro indent meta c times meta a equal meta c times meta b end line line ell i end block line ell t because deduction modus ponens ell i indeed meta a equal meta b imply meta c times meta a equal meta c times meta b end line because ell t object modus ponens ell j indeed meta c times meta a equal meta c times meta b qed end math ] "
\end{list}
\section{Udsagn 3.4}
De f\o{}lgende udsagn er en udvidelse af egenskaberne ved addition og
multiplikation.
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three four a says for all terms meta a comma meta b comma meta c indeed meta a times parenthesis meta b plus meta c end parenthesis equal parenthesis meta a times meta b end parenthesis plus parenthesis meta a times meta c end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma prop three four b says for all terms meta a comma meta b comma meta c indeed parenthesis meta b plus meta c end parenthesis times meta a equal parenthesis meta b times meta a end parenthesis plus parenthesis meta c times meta a end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma prop three four c says for all terms meta a comma meta b comma meta c indeed parenthesis meta a times meta b end parenthesis times meta c equal meta a times parenthesis meta b times meta c end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma prop three four d says for all terms meta a comma meta b comma meta c indeed meta a plus meta c equal meta b plus meta c imply meta a equal meta b end lemma end math ] "
\end{list}
\subsection{3.4a}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three four a one says for all terms meta a comma meta b indeed meta a times parenthesis meta b plus zero end parenthesis equal meta a times meta b plus meta a times zero end lemma end math ] "
\item " [ math system s proof of prop three four a one reads any term meta a comma meta b end line line ell a because axiom s five indeed parenthesis meta b plus zero end parenthesis equal parenthesis meta b end parenthesis end line line ell b because prop three two o modus ponens ell a indeed meta a times parenthesis meta b plus zero end parenthesis equal meta a times parenthesis meta b end parenthesis end line line ell c because axiom s five indeed meta a times meta b plus zero equal meta a times meta b end line line ell d because prop three two d modus ponens ell b modus ponens ell c indeed meta a times parenthesis meta b plus zero end parenthesis equal meta a times meta b plus zero end line line ell e because axiom s seven indeed meta a times zero equal zero end line line ell f because prop three two i modus ponens ell e indeed meta a times meta b plus meta a times zero equal meta a times meta b plus zero end line because prop three two d modus ponens ell d modus ponens ell f indeed meta a times parenthesis meta b plus zero end parenthesis equal meta a times meta b plus meta a times zero qed end math ] "
\item " [ math in theory system s lemma prop three four a two says for all terms meta a comma meta b comma meta c indeed meta a times parenthesis meta b plus meta c end parenthesis equal meta a times meta b plus meta a times meta c imply meta a times parenthesis meta b plus meta c suc end parenthesis equal meta a times meta b plus meta a times meta c suc end lemma end math ] "
\item " [ math system s proof of prop three four a two reads any term meta a comma meta b comma meta c end line block any term macro indent meta a comma meta b comma meta c end line line ell b premise macro indent meta a times parenthesis meta b plus meta c end parenthesis equal meta a times meta b plus meta a times meta c end line line ell c because axiom s six indeed macro indent meta b plus meta c suc equal parenthesis meta b plus meta c end parenthesis suc end line line ell d because prop three two o modus ponens ell c indeed macro indent meta a times parenthesis meta b plus meta c suc end parenthesis equal meta a times parenthesis meta b plus meta c end parenthesis suc end line line ell e because axiom s eight indeed macro indent meta a times parenthesis meta b plus meta c end parenthesis suc equal meta a times parenthesis meta b plus meta c end parenthesis plus meta a end line line ell f because prop three two e modus ponens ell b indeed macro indent parenthesis meta a times parenthesis meta b plus meta c end parenthesis end parenthesis plus meta a equal parenthesis meta a times meta b plus meta a times meta c end parenthesis plus meta a end line line ell j because prop three two j indeed macro indent parenthesis meta a times meta b plus meta a times meta c end parenthesis plus meta a equal meta a times meta b plus parenthesis meta a times meta c plus meta a end parenthesis end line line ell k because prop three two c modus ponens ell f modus ponens ell j indeed macro indent parenthesis meta a times parenthesis meta b plus meta c end parenthesis end parenthesis plus meta a equal meta a times meta b plus parenthesis meta a times meta c plus meta a end parenthesis end line line ell g because axiom s eight indeed macro indent meta a times meta c suc equal meta a times meta c plus meta a end line line ell h because prop three two i modus ponens ell g indeed macro indent meta a times meta b plus meta a times meta c suc equal meta a times meta b plus parenthesis meta a times meta c plus meta a end parenthesis end line line ell i because prop three two d modus ponens ell k modus ponens ell h indeed macro indent meta a times parenthesis meta b plus meta c end parenthesis plus meta a equal meta a times meta b plus meta a times meta c suc end line line ell l because prop three two c modus ponens ell e modus ponens ell i indeed macro indent meta a times parenthesis meta b plus meta c end parenthesis suc equal meta a times meta b plus meta a times meta c suc end line because prop three two c modus ponens ell d modus ponens ell l indeed macro indent meta a times parenthesis meta b plus meta c suc end parenthesis equal meta a times meta b plus meta a times meta c suc end line line ell m end block because deduction modus ponens ell m indeed parenthesis meta a times parenthesis meta b plus meta c end parenthesis equal meta a times meta b plus meta a times meta c end parenthesis imply meta a times parenthesis meta b plus meta c suc end parenthesis equal meta a times meta b plus meta a times meta c suc qed end math ] "
\item " [ math in theory system s lemma prop three four a says for all terms meta a comma meta b comma meta c indeed meta a times parenthesis meta b plus meta c end parenthesis equal parenthesis meta a times meta b end parenthesis plus parenthesis meta a times meta c end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three four a reads any term meta a comma meta b comma meta c end line block line ell a because prop three four a one indeed macro indent object x times parenthesis object y plus zero end parenthesis equal object x times object y plus object x times zero end line line ell b because prop three four a two indeed macro indent parenthesis object x times parenthesis object y plus object z end parenthesis equal object x times object y plus object x times object z end parenthesis imply parenthesis object x times parenthesis object y plus object z suc end parenthesis equal object x times object y plus object x times object z suc end parenthesis end line because axiom s nine at object z modus ponens ell a modus ponens ell b indeed macro indent object x times parenthesis object y plus object z end parenthesis equal object x times object y plus object x times object z end line line ell c end block because deduction modus ponens ell c indeed meta a times parenthesis meta b plus meta c end parenthesis equal meta a times meta b plus meta a times meta c qed end math ] "
\end{list}
\subsection{3.4b}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three four b says for all terms meta a comma meta b comma meta c indeed parenthesis meta b plus meta c end parenthesis times meta a equal parenthesis meta b times meta a end parenthesis plus parenthesis meta c times meta a end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three four b reads any term meta a comma meta b comma meta c end line line ell a because prop three four a indeed meta a times parenthesis meta b plus meta c end parenthesis equal meta a times meta b plus meta a times meta c end line line ell b because prop three two n indeed meta a times parenthesis meta b plus meta c end parenthesis equal parenthesis meta b plus meta c end parenthesis times meta a end line line ell c because prop three two n indeed meta a times meta b equal meta b times meta a end line line ell d because prop three two n indeed meta a times meta c equal meta c times meta a end line line ell e because axiom s one modus ponens ell b modus ponens ell a indeed parenthesis meta b plus meta c end parenthesis times meta a equal meta a times meta b plus meta a times meta c end line line ell f because prop three two e modus ponens ell c indeed meta a times meta b plus meta a times meta c equal meta b times meta a plus meta a times meta c end line line ell g because prop three two i modus ponens ell d indeed meta b times meta a plus meta a times meta c equal meta b times meta a plus meta c times meta a end line line ell h because prop three two c modus ponens ell f modus ponens ell g indeed meta a times meta b plus meta a times meta c equal meta b times meta a plus meta c times meta a end line because prop three two c modus ponens ell e modus ponens ell h indeed parenthesis meta b plus meta c end parenthesis times meta a equal meta b times meta a plus meta c times meta a qed end math ] "
\end{list}
\subsection{3.4c}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three four c one says for all terms meta a comma meta b indeed parenthesis meta a times meta b end parenthesis times zero equal meta a times parenthesis meta b times zero end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three four c one reads any term meta a comma meta b end line line ell a because axiom s seven indeed parenthesis meta a times meta b end parenthesis times zero equal zero end line line ell b because axiom s seven indeed meta b times zero equal zero end line line ell c because prop three two o modus ponens ell b indeed meta a times parenthesis meta b times zero end parenthesis equal meta a times zero end line line ell d because axiom s seven indeed meta a times zero equal zero end line line ell e because prop three two c modus ponens ell c modus ponens ell d indeed meta a times parenthesis meta b times zero end parenthesis equal zero end line because prop three two d modus ponens ell a modus ponens ell e indeed parenthesis meta a times meta b end parenthesis times zero equal meta a times parenthesis meta b times zero end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three four c two says for all terms meta a comma meta b comma meta c indeed parenthesis meta a times meta b end parenthesis times meta c equal meta a times parenthesis meta b times meta c end parenthesis imply parenthesis meta a times meta b end parenthesis times meta c suc equal meta a times parenthesis meta b times meta c suc end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three four c two reads any term meta a comma meta b comma meta c end line block any term macro indent meta a comma meta b comma meta c end line line ell a premise macro indent parenthesis meta a times meta b end parenthesis times meta c equal meta a times parenthesis meta b times meta c end parenthesis end line line ell b because axiom s eight indeed macro indent parenthesis meta a times meta b end parenthesis times meta c suc equal parenthesis meta a times meta b end parenthesis times meta c plus parenthesis meta a times meta b end parenthesis end line line ell c because prop three two e modus ponens ell a indeed macro indent parenthesis parenthesis meta a times meta b end parenthesis times meta c end parenthesis plus parenthesis meta a times meta b end parenthesis equal parenthesis meta a times parenthesis meta b times meta c end parenthesis end parenthesis plus parenthesis meta a times meta b end parenthesis end line line ell d because prop three two c modus ponens ell b modus ponens ell c indeed macro indent parenthesis meta a times meta b end parenthesis times meta c suc equal parenthesis meta a times parenthesis meta b times meta c end parenthesis end parenthesis plus parenthesis meta a times meta b end parenthesis end line line ell e because prop three four a indeed macro indent meta a times parenthesis parenthesis meta b times meta c end parenthesis plus meta b end parenthesis equal meta a times parenthesis meta b times meta c end parenthesis plus meta a times meta b end line line ell f because prop three two d modus ponens ell d modus ponens ell e indeed macro indent parenthesis meta a times meta b end parenthesis times meta c suc equal meta a times parenthesis parenthesis meta b times meta c end parenthesis plus meta b end parenthesis end line line ell g because axiom s eight indeed macro indent meta b times meta c suc equal meta b times meta c plus meta b end line line ell h because prop three two o modus ponens ell g indeed macro indent meta a times parenthesis meta b times meta c suc end parenthesis equal meta a times parenthesis meta b times meta c plus meta b end parenthesis end line because prop three two d modus ponens ell f modus ponens ell h indeed macro indent parenthesis meta a times meta b end parenthesis times meta c suc equal meta a times parenthesis meta b times meta c suc end parenthesis end line line ell i end block because deduction modus ponens ell i indeed parenthesis meta a times meta b end parenthesis times meta c equal meta a times parenthesis meta b times meta c end parenthesis imply parenthesis meta a times meta b end parenthesis times meta c suc equal meta a times parenthesis meta b times meta c suc end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three four c says for all terms meta a comma meta b comma meta c indeed parenthesis meta a times meta b end parenthesis times meta c equal meta a times parenthesis meta b times meta c end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three four c reads any term meta a comma meta b comma meta c end line block line ell a because prop three four c one indeed macro indent parenthesis object x times object y end parenthesis times zero equal object x times parenthesis object y times zero end parenthesis end line line ell b because prop three four c two indeed macro indent parenthesis object x times object y end parenthesis times object z equal object x times parenthesis object y times object z end parenthesis imply parenthesis object x times object y end parenthesis times object z suc equal object x times parenthesis object y times object z suc end parenthesis end line because axiom s nine at object z modus ponens ell a modus ponens ell b indeed macro indent parenthesis object x times object y end parenthesis times object z equal object x times parenthesis object y times object z end parenthesis end line line ell c end block because deduction modus ponens ell c indeed parenthesis meta a times meta b end parenthesis times meta c equal meta a times parenthesis meta b times meta c end parenthesis qed end math ] "
\end{list}
\subsection{3.4d}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three four d one says for all terms meta a comma meta b indeed meta a plus zero equal meta b plus zero imply meta a equal meta b end lemma end math ] "
\item " [ math system s proof of prop three four d one reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell a premise macro indent meta a plus zero equal meta b plus zero end line line ell b because axiom s five indeed macro indent meta a plus zero equal meta a end line line ell c because axiom s five indeed macro indent meta b plus zero equal meta b end line line ell d because axiom s one modus ponens ell b modus ponens ell a indeed macro indent meta a equal meta b plus zero end line because prop three two c modus ponens ell d modus ponens ell c indeed macro indent meta a equal meta b end line line ell e end block because deduction modus ponens ell e indeed meta a plus zero equal meta b plus zero imply meta a equal meta b qed end math ] "
\item " [ math in theory system s lemma prop three four d two says for all terms meta a comma meta b comma meta c indeed parenthesis meta a plus meta c equal meta b plus meta c imply meta a equal meta b end parenthesis imply meta a plus meta c suc equal meta b plus meta c suc imply meta a equal meta b end lemma end math ] "
\item " [ math system s proof of prop three four d two reads any term meta a comma meta b comma meta c end line block any term macro indent meta a comma meta b comma meta c end line line ell a premise macro indent meta a plus meta c equal meta b plus meta c imply meta a equal meta b end line line ell b premise macro indent meta a plus meta c suc equal meta b plus meta c suc end line line ell c because axiom s six indeed macro indent meta a plus meta c suc equal parenthesis meta a plus meta c end parenthesis suc end line line ell d because axiom s six indeed macro indent meta b plus meta c suc equal parenthesis meta b plus meta c end parenthesis suc end line line ell e because axiom s one modus ponens ell c modus ponens ell b indeed macro indent parenthesis meta a plus meta c end parenthesis suc equal meta b plus meta c suc end line line ell f because prop three two c modus ponens ell e modus ponens ell d indeed macro indent parenthesis meta a plus meta c end parenthesis suc equal parenthesis meta b plus meta c end parenthesis suc end line line ell g because axiom s four modus ponens ell f indeed macro indent parenthesis meta a plus meta c end parenthesis equal parenthesis meta b plus meta c end parenthesis end line because ell a object modus ponens ell g indeed macro indent meta a equal meta b end line line ell h end block because deduction modus ponens ell h indeed parenthesis meta a plus meta c equal meta b plus meta c imply meta a equal meta b end parenthesis imply meta a plus meta c suc equal meta b plus meta c suc imply meta a equal meta b qed end math ] "
\item " [ math in theory system s lemma prop three four d says for all terms meta a comma meta b comma meta c indeed meta a plus meta c equal meta b plus meta c imply meta a equal meta b end lemma end math ] "
\item " [ math system s proof of prop three four d reads any term meta a comma meta b comma meta c end line block line ell a because prop three four d one indeed macro indent object x plus zero equal object y plus zero imply object x equal object y end line line ell b because prop three four d two indeed macro indent parenthesis object x plus object z equal object y plus object z imply object x equal object y end parenthesis imply object x plus object z suc equal object y plus object z suc imply object x equal object y end line because axiom s nine at object z modus ponens ell a modus ponens ell b indeed macro indent object x plus object z equal object y plus object z imply object x equal object y end line line ell c end block because deduction modus ponens ell c indeed meta a plus meta c equal meta b plus meta c imply meta a equal meta b qed end math ] "
\end{list}
\section{Udsagn 3.5}\label{35}
I denne del har vi kun kunne l\o{}se indtil h, da denne og de efterf\o{}lgende
kr\ae{}ver reglen ``existensial rule''. Dog er der i afsnit \ref{exist} genemg\aa{}et hvilket problem der forhindrer os i at bevise '' existencial rule'', og hvordan vi
ville have bevist reglen, hvis problemet ikke var opst\aa{}et. Endvidere er der i \ref{exist} genemg\aa{}et hvordan vi ville have l\o{}st 3.5h.
I de f\o{}lgende udsagn indg\aa{}r numeraler opf\o{}rer sig som de naturlige
tal og er defineret i forhold til 0 p\aa{} f\o{}lgende m\aa{}de:
\[
\begin{array}{lcr}
0 &=& \overline{0} \\
0' &=& \overline{1} \\
0'' & = & \overline{2} \\
\vdots & & \vdots \\
0^{n*'} & = & \overline{n} \\
\end{array}
\]
Dvs. hvis $\overline{0}$ er et numeral, og hvis $\overline{n}$ er et numeral, s\aa{} er $\overline{n}'$ ogs\aa{}.\\
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three five a says for all terms meta a indeed meta a plus numeral one equal meta a suc end lemma end math ] "
\item " [ math in theory system s lemma prop three five b says for all terms meta a indeed meta a times numeral one equal meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three five c says for all terms meta a indeed meta a times numeral two equal meta a plus meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three five d says for all terms meta a comma meta b indeed meta a plus meta b equal zero imply meta a equal zero and1 meta b equal zero end lemma end math ] "
\item " [ math in theory system s lemma prop three five e says for all terms meta a comma meta b indeed meta b neq zero imply parenthesis meta a times meta b equal zero imply meta a equal zero end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma prop three five f says for all terms meta a comma meta b indeed meta a plus meta b equal numeral one imply parenthesis meta a equal zero and1 meta b equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 meta b equal zero end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma prop three five g says for all terms meta a comma meta b indeed meta a times meta b equal numeral one imply parenthesis meta a equal numeral one and1 meta b equal numeral one end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma prop three five h says for all terms meta a indeed meta a neq zero imply exists meta b indeed meta a equal meta b suc end lemma end math ] "
\item " [ math in theory system s lemma prop three five i says for all terms meta a comma meta b comma meta c indeed meta c neq zero imply parenthesis meta a times meta c equal meta b times meta c imply meta a equal meta b end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma prop three five j says for all terms meta a indeed meta a neq zero imply meta a neq numeral one imply exists meta b indeed meta a equal meta b suc suc end lemma end math ] "
\end{list}
Efterf\o[]lgende vil vi gennemg\aa de n\o[]dvendige hj\ae{}lpes\ae{}tninger og definitioner for at kunne bevise 3.5a-3.5g.
\subsection{Definitioner i forbindelse med $\land$ og $\lor$}
Fra 3.5d bruges $\land$ og $\lor$, som begge er makrodefinerede udtryk.
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math macro define var x and1 var y as not parenthesis var x imply not var y end parenthesis end define end math ] "
\item " [ math macro define var x or1 var y as parenthesis not var x end parenthesis imply var y end define end math ] "
\end{list}
Endvidere er det klart at reglerne for introduktion og eliminering af $\land$ og
$\lor$ dermed ogs\aa{} skal bruges. Alle disse er bevist i de f\o{}lgende afsnit.
\subsubsection{Introduktion af $\land$}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma conjin says for all terms meta a comma meta b indeed meta a infer meta b infer meta a and1 meta b end lemma end math ] "
\item " [ math system s proof of conjin reads any term meta a comma meta b end line line ell a premise meta a end line line ell b premise meta b end line block any term macro indent meta a comma meta b end line line ell c premise macro indent meta a imply not meta b end line line ell d because repetition modus ponens ell a indeed macro indent meta a end line because ell c object modus ponens ell d indeed macro indent not meta b end line line ell e end block line ell f because deduction modus ponens ell e indeed parenthesis meta a imply not meta b end parenthesis imply not meta b end line line ell g because h two indeed meta b imply not not meta b end line line ell h because ell g object modus ponens ell b indeed not not meta b end line because modus tollens modus ponens ell f modus ponens ell h indeed not parenthesis meta a imply not meta b end parenthesis qed end math ] "
\end{list}
\subsubsection{Elimination af $\land$ 1}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma conjel1 says for all terms meta a comma meta b indeed meta a and1 meta b infer meta a end lemma end math ] "
\item " [ math system s proof of conjel1 reads any term meta a comma meta b end line line ell a premise not parenthesis meta a imply not meta b end parenthesis end line block any term macro indent meta a comma meta b end line line ell b premise macro indent not meta a end line line ell c because lemma a one indeed macro indent not meta a imply not not meta b imply not meta a end line line ell e because ell c object modus ponens ell b indeed macro indent not not meta b imply not meta a end line line ell g because h eight indeed macro indent parenthesis not not meta b imply not meta a end parenthesis imply meta a imply not meta b end line because ell g object modus ponens ell e indeed macro indent meta a imply not meta b end line line ell d end block line ell f because deduction modus ponens ell d indeed parenthesis not meta a end parenthesis imply parenthesis meta a imply not meta b end parenthesis end line line ell i because modus tollens modus ponens ell f modus ponens ell a indeed not not meta a end line line ell h because h one indeed not not meta a imply meta a end line because ell h object modus ponens ell i indeed meta a qed end math ] "
\end{list}
\subsubsection{Elimination af $\land$ 2}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma conjel2 says for all terms meta a comma meta b indeed meta a and1 meta b infer meta b end lemma end math ] "
\item " [ math system s proof of conjel2 reads any term meta a comma meta b end line line ell a premise not parenthesis meta a imply not meta b end parenthesis end line block any term macro indent meta a comma meta b end line line ell b premise macro indent not meta b end line line ell c because lemma a one indeed macro indent not meta b imply meta a imply not meta b end line because ell c object modus ponens ell b indeed macro indent meta a imply not meta b end line line ell d end block line ell e because deduction modus ponens ell d indeed not meta b imply meta a imply not meta b end line line ell f because modus tollens modus ponens ell e modus ponens ell a indeed not not meta b end line line ell h because h one indeed not not meta b imply meta b end line because ell h object modus ponens ell f indeed meta b qed end math ] "
\end{list}
\subsubsection{Introduktion af $\lor$ 1}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma disjin1 says for all terms meta a comma meta b indeed meta a infer meta a or1 meta b end lemma end math ] "
\item " [ math system s proof of disjin1 reads any term meta a comma meta b end line line ell a premise meta a end line line ell b because t one indeed not not meta a imply parenthesis not meta a imply meta b end parenthesis end line line ell c because h two indeed meta a imply not not meta a end line line ell d because ell c object modus ponens ell a indeed not not meta a end line because ell b object modus ponens ell d indeed not meta a imply meta b qed end math ] "
\end{list}
\subsubsection{Introduktion af $\lor$ 2}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma disjin2 says for all terms meta a comma meta b indeed meta b infer meta a or1 meta b end lemma end math ] "
\item " [ math system s proof of disjin2 reads any term meta a comma meta b end line line ell a premise meta b end line line ell b because lemma a one indeed meta b imply not meta a imply meta b end line because ell b object modus ponens ell a indeed not meta a imply meta b qed end math ] "
\end{list}
\subsubsection{Elimination af $\lor$ }
Da vi ikke har haft brug for at fjerne $\lor$, har vi ikke bevist denne regel.
\subsection{Andre hj\ae{}lpes\ae{}tninger}
Vi har til beviset af 3.5a-3.5g haft brug for nogle af reglerne fra
\cite{check}, dog med \texttt{imply} istedet for
\texttt{infer}. S\aa{}danne regler er navngivet med det $<$oprindelige
navn$>$' og beviserne for disse kan ses i Appendix \ref{help}. \\
\subsubsection{ $\mathcal{A} \neq \mathcal{B}$}
En ny hj\ae{}lperegel er \textbf{regel " [ math h three end math ] " }, som skal bruges i forbindelse med
$\mathcal{A} \neq \mathcal{B}$ for at kunne konkludere sammenh\ae{}nge i
mellem $(x = y)$ $\land$ $(x \neq 3)$ $\Rightarrow$ $(y \neq 3)$.\\
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma h three says for all terms meta a comma meta b comma meta c indeed meta a imply parenthesis meta b imply meta c end parenthesis infer meta a infer not meta c infer not meta b end lemma end math ] "
\item " [ math system s proof of h three reads any term meta a comma meta b comma meta c end line line ell a premise meta a imply parenthesis meta b imply meta c end parenthesis end line line ell b premise meta a end line line ell c premise not meta c end line line ell d because ell a object modus ponens ell b indeed meta b imply meta c end line because modus tollens modus ponens ell d modus ponens ell c indeed not meta b qed end math ] "
\end{list}
\subsubsection{$\exists$ x}\label{exist}
Selv om vi ikke har n\aa{}et at implementere fra opgave 3.5h, hvor der
g\o{}res brug af $\exists$, har vi alligevel makrodefineret kvantoren som
f\o{}lger.
" [ math macro define exists var x indeed var y as not parenthesis for all var x indeed not var y end parenthesis end define end math ] "
Vores problem har v\ae{}ret, at vi ikke har kunne udtrykke
\emph{``$\mathcal{B}(x,t)$, hvor $t$ kan inds\ae{}ttes i stedet for $x$'erne uden at
blive bundet til en alkvantor''}, i pyk.
Beviset for hj\ae{}lpes\ae{}tningen, der indf\o{}rer eksistenskvantoren, ville dog have v\ae{}ret opbygget
nogenlunde som f\o{}lger.\\
F\o{}lgende tautologi,
\[( \mathcal{A} \Rightarrow \lnot \mathcal{B})
\Rightarrow (\mathcal{B} \Rightarrow \lnot \mathcal{A})\]
antages vist nogenlunde som Modus Tollens, dog med variation.
Ved axiom " [ math lemma a four end math ] " kan vi f\aa{} instansen: \[(\forall x) \lnot
\mathcal{A}(x,t) \Rightarrow \lnot A(t,t) \]
Idet man kan f\aa{} f\o{}lgende instans af tautologien:
\[
\big( (\forall x) \lnot \mathcal{A}(x,t) \Rightarrow \lnot A(t,t) \big)
\Rightarrow
\big(A(t,t) \Rightarrow \lnot (\forall x) \lnot \mathcal{A}(x,t)\big)
\]
kan man vha. MP p\aa{} instansen og axiomet f\aa{}:
\[\big(A(t,t) \Rightarrow \lnot (\forall x) \lnot \mathcal{A}(x,t)\big)
\]
og pga. af makrodefinitionen er denne nu vist.
Herefter vises Lemmaerne 3.5a-3.5g
\subsection{3.5a}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three five a says for all terms meta a indeed meta a plus numeral one equal meta a suc end lemma end math ] "
\item " [ math system s proof of prop three five a reads any term meta a end line line ell a because axiom s six indeed meta a plus zero suc equal parenthesis meta a plus zero end parenthesis suc end line line ell b because axiom s five indeed meta a plus zero equal meta a end line line ell c because axiom s two modus ponens ell b indeed parenthesis meta a plus zero end parenthesis suc equal meta a suc end line line ell d because prop three two c modus ponens ell a modus ponens ell c indeed meta a plus zero suc equal meta a suc end line line ell e because prop three two a indeed meta a plus zero suc equal meta a plus numeral one end line because axiom s one modus ponens ell e modus ponens ell d indeed meta a plus numeral one equal meta a suc qed end math ] "
\end{list}
\subsection{3.5b}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three five b says for all terms meta a indeed meta a times numeral one equal meta a end lemma end math ] "
\item " [ math system s proof of prop three five b reads any term meta a end line line ell a because axiom s eight indeed meta a times zero suc equal meta a times zero plus meta a end line line ell b because axiom s seven indeed meta a times zero equal zero end line line ell c because prop three two e modus ponens ell b indeed meta a times zero plus meta a equal zero plus meta a end line line ell d because prop three two c modus ponens ell a modus ponens ell c indeed meta a times zero suc equal zero plus meta a end line line ell h because prop three two f indeed meta a equal zero plus meta a end line line ell e because prop three two b modus ponens ell h indeed zero plus meta a equal meta a end line line ell f because prop three two c modus ponens ell d modus ponens ell e indeed meta a times zero suc equal meta a end line line ell g because prop three two a indeed meta a times zero suc equal meta a times numeral one end line because axiom s one modus ponens ell g modus ponens ell f indeed meta a times numeral one equal meta a qed end math ] "
\end{list}
\subsection{3.5c}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three five c says for all terms meta a indeed meta a times numeral two equal meta a plus meta a end lemma end math ] "
\item " [ math system s proof of prop three five c reads any term meta a end line line ell a because axiom s eight indeed meta a times numeral one suc equal meta a times numeral one plus meta a end line line ell b because prop three five b indeed meta a times numeral one equal meta a end line line ell c because prop three two e modus ponens ell b indeed meta a times numeral one plus meta a equal meta a plus meta a end line line ell d because prop three two c modus ponens ell a modus ponens ell c indeed meta a times numeral one suc equal meta a plus meta a end line line ell e because prop three two a indeed meta a times numeral one suc equal meta a times numeral two end line because axiom s one modus ponens ell e modus ponens ell d indeed meta a times numeral two equal meta a plus meta a qed end math ] "
\end{list}
\subsection{3.5d}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three five d one says for all terms meta a indeed meta a plus zero equal zero imply parenthesis meta a equal zero end parenthesis and1 parenthesis zero equal zero end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five d one reads any term meta a end line block any term macro indent meta a end line line ell a premise macro indent meta a plus zero equal zero end line line ell b because axiom s five indeed macro indent meta a plus zero equal meta a end line line ell c because axiom s one modus ponens ell b modus ponens ell a indeed macro indent meta a equal zero end line line ell d because prop three two a indeed macro indent zero equal zero end line because conjin modus ponens ell c modus ponens ell d indeed macro indent meta a equal zero and1 zero equal zero end line line ell e end block because deduction modus ponens ell e indeed meta a plus zero equal zero imply parenthesis meta a equal zero end parenthesis and1 parenthesis zero equal zero end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three five d two says for all terms meta a comma meta b indeed parenthesis meta a plus meta b equal zero imply meta a equal zero and1 meta b equal zero end parenthesis imply meta a plus meta b suc equal zero imply meta a equal zero and1 meta b suc equal zero end lemma end math ] "
\item " [ math system s proof of prop three five d two reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell a premise macro indent meta a plus meta b equal zero imply meta a equal zero and1 meta b equal zero end line block any term macro indent meta a comma meta b end line line ell b premise macro indent meta a plus meta b suc equal zero end line line ell c because axiom s three indeed macro indent zero neq parenthesis meta a plus meta b end parenthesis suc end line line ell i because h nine indeed macro indent parenthesis meta a plus meta b end parenthesis suc equal zero imply zero equal parenthesis meta a plus meta b end parenthesis suc end line line ell j because h seven indeed macro indent parenthesis parenthesis meta a plus meta b end parenthesis suc equal zero imply zero equal parenthesis meta a plus meta b end parenthesis suc end parenthesis imply zero neq parenthesis meta a plus meta b end parenthesis suc imply parenthesis meta a plus meta b end parenthesis suc neq zero end line line ell k because ell j object modus ponens ell i indeed macro indent zero neq parenthesis meta a plus meta b end parenthesis suc imply parenthesis meta a plus meta b end parenthesis suc neq zero end line line ell l because ell k object modus ponens ell c indeed macro indent parenthesis meta a plus meta b end parenthesis suc neq zero end line line ell d because axiom s six indeed macro indent parenthesis meta a plus meta b suc end parenthesis equal parenthesis meta a plus meta b end parenthesis suc end line line ell h because h four mark indeed macro indent meta a plus meta b suc equal parenthesis meta a plus meta b end parenthesis suc imply meta a plus meta b suc equal zero imply parenthesis meta a plus meta b end parenthesis suc equal zero end line line ell e because h three modus ponens ell h modus ponens ell d modus ponens ell l indeed macro indent meta a plus meta b suc neq zero end line line ell m because t one indeed macro indent meta a plus meta b suc neq zero imply meta a plus meta b suc equal zero imply meta a equal zero and1 meta b suc equal zero end line line ell n because ell m object modus ponens ell e indeed macro indent meta a plus meta b suc equal zero imply meta a equal zero and1 meta b suc equal zero end line because ell n object modus ponens ell b indeed macro indent meta a equal zero and1 meta b suc equal zero end line line ell f end block because deduction modus ponens ell f indeed macro indent meta a plus meta b suc equal zero imply meta a equal zero and1 meta b suc equal zero end line line ell g end block because deduction modus ponens ell g indeed parenthesis meta a plus meta b equal zero imply meta a equal zero and1 meta b equal zero end parenthesis imply meta a plus meta b suc equal zero imply meta a equal zero and1 meta b suc equal zero qed end math ] "
\item " [ math in theory system s lemma prop three five d says for all terms meta a comma meta b indeed meta a plus meta b equal zero imply meta a equal zero and1 meta b equal zero end lemma end math ] "
\item " [ math system s proof of prop three five d reads any term meta a comma meta b end line block line ell a because prop three five d one indeed macro indent object x plus zero equal zero imply object x equal zero and1 zero equal zero end line line ell b because prop three five d two indeed macro indent parenthesis object x plus object y equal zero imply object x equal zero and1 object y equal zero end parenthesis imply object x plus object y suc equal zero imply object x equal zero and1 object y suc equal zero end line because axiom s nine at object y modus ponens ell a modus ponens ell b indeed macro indent object x plus object y equal zero imply object x equal zero and1 object y equal zero end line line ell c end block because deduction modus ponens ell c indeed meta a plus meta b equal zero imply meta a equal zero and1 meta b equal zero qed end math ] "
\end{list}
\subsection{3.5e}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three five e one says for all terms meta a indeed zero neq zero imply parenthesis meta a times zero equal zero imply meta a equal zero end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five e one reads any term meta a end line block any term macro indent meta a end line line ell a premise macro indent zero neq zero end line line ell b because prop three two a indeed macro indent zero equal zero end line line ell c because t one indeed macro indent not parenthesis zero equal zero end parenthesis imply parenthesis parenthesis zero equal zero end parenthesis imply parenthesis meta a times zero equal zero imply meta a equal zero end parenthesis end parenthesis end line line ell d because ell c object modus ponens ell a indeed macro indent parenthesis zero equal zero end parenthesis imply parenthesis meta a times zero equal zero imply meta a equal zero end parenthesis end line because ell d object modus ponens ell b indeed macro indent meta a times zero equal zero imply meta a equal zero end line line ell e end block because deduction modus ponens ell e indeed zero neq zero imply meta a times zero equal zero imply meta a equal zero qed end math ] "
\item " [ math in theory system s lemma prop three five e two says for all terms meta a comma meta b indeed parenthesis meta b neq zero imply parenthesis meta a times meta b equal zero imply meta a equal zero end parenthesis end parenthesis imply meta b suc neq zero imply parenthesis meta a times meta b suc equal zero imply meta a equal zero end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five e two reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell a premise macro indent meta b neq zero imply parenthesis meta a times meta b equal zero imply meta a equal zero end parenthesis end line block any term macro indent meta a comma meta b end line line ell b premise macro indent meta b suc neq zero end line block any term macro indent meta a comma meta b end line line ell c premise macro indent meta a times meta b suc equal zero end line line ell d because axiom s eight indeed macro indent meta a times meta b suc equal meta a times meta b plus meta a end line line ell e because axiom s one modus ponens ell d modus ponens ell c indeed macro indent parenthesis meta a times meta b end parenthesis plus meta a equal zero end line line ell f because prop three five d indeed macro indent parenthesis meta a times meta b end parenthesis plus meta a equal zero imply parenthesis meta a times meta b end parenthesis equal zero and1 meta a equal zero end line line ell j because ell f object modus ponens ell e indeed macro indent parenthesis meta a times meta b end parenthesis equal zero and1 meta a equal zero end line because conjel2 modus ponens ell j indeed macro indent meta a equal zero end line line ell g end block because deduction modus ponens ell g indeed macro indent meta a times meta b suc equal zero imply meta a equal zero end line line ell h end block because deduction modus ponens ell h indeed macro indent meta b suc neq zero imply parenthesis meta a times meta b suc equal zero imply meta a equal zero end parenthesis end line line ell i end block because deduction modus ponens ell i indeed parenthesis meta b neq zero imply parenthesis meta a times meta b equal zero imply meta a equal zero end parenthesis end parenthesis imply meta b suc neq zero imply parenthesis meta a times meta b suc equal zero imply meta a equal zero end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three five e says for all terms meta a comma meta b indeed meta b neq zero imply parenthesis meta a times meta b equal zero imply meta a equal zero end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five e reads any term meta a comma meta b end line block line ell a because prop three five e one indeed macro indent zero neq zero imply parenthesis object x times zero equal zero imply object x equal zero end parenthesis end line line ell b because prop three five e two indeed macro indent macro indent parenthesis object y neq zero imply parenthesis object x times object y equal zero imply object x equal zero end parenthesis end parenthesis imply object y suc neq zero imply parenthesis object x times object y suc equal zero imply object x equal zero end parenthesis end line because axiom s nine at object y modus ponens ell a modus ponens ell b indeed macro indent object y neq zero imply parenthesis object x times object y equal zero imply object x equal zero end parenthesis end line line ell c end block because deduction modus ponens ell c indeed meta b neq zero imply parenthesis meta a times meta b equal zero imply meta a equal zero end parenthesis qed end math ] "
\end{list}
\subsection{3.5f}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three five f one says for all terms meta a indeed meta a plus zero equal numeral one imply parenthesis meta a equal zero and1 zero equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 zero equal zero end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five f one reads any term meta a end line block any term macro indent meta a end line line ell a premise macro indent meta a plus zero equal numeral one end line line ell b because axiom s five indeed macro indent meta a plus zero equal meta a end line line ell c because axiom s one modus ponens ell b modus ponens ell a indeed macro indent meta a equal numeral one end line line ell d because prop three two a indeed macro indent zero equal zero end line line ell e because conjin modus ponens ell c modus ponens ell d indeed macro indent meta a equal numeral one and1 zero equal zero end line because disjin2 modus ponens ell e indeed macro indent macro indent parenthesis meta a equal zero and1 zero equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 zero equal zero end parenthesis end line line ell f end block because deduction modus ponens ell f indeed meta a plus zero equal numeral one imply parenthesis meta a equal zero and1 zero equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 zero equal zero end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three five f two says for all terms meta a comma meta b indeed parenthesis meta a plus meta b equal numeral one imply parenthesis parenthesis meta a equal zero and1 meta b equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 meta b equal zero end parenthesis end parenthesis end parenthesis imply parenthesis meta a plus meta b suc equal numeral one imply parenthesis parenthesis meta a equal zero and1 meta b suc equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 meta b suc equal zero end parenthesis end parenthesis end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five f two reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell a premise macro indent meta a plus meta b equal numeral one imply parenthesis parenthesis meta a equal zero and1 meta b equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 meta b equal zero end parenthesis end parenthesis end line block any term macro indent meta a comma meta b end line line ell b premise macro indent macro indent meta a plus meta b suc equal numeral one end line line ell c because axiom s six indeed macro indent meta a plus meta b suc equal parenthesis meta a plus meta b end parenthesis suc end line line ell d because axiom s one modus ponens ell c modus ponens ell b indeed macro indent parenthesis meta a plus meta b end parenthesis suc equal numeral one end line line ell e because axiom s four modus ponens ell d indeed macro indent meta a plus meta b equal zero end line line ell f because prop three five d indeed macro indent meta a plus meta b equal zero imply meta a equal zero and1 meta b equal zero end line line ell m because ell f object modus ponens ell e indeed macro indent meta a equal zero and1 meta b equal zero end line line ell g because conjel1 modus ponens ell m indeed macro indent meta a equal zero end line line ell h because conjel2 modus ponens ell m indeed macro indent meta b equal zero end line line ell i because axiom s two modus ponens ell h indeed macro indent meta b suc equal numeral one end line line ell j because conjin modus ponens ell g modus ponens ell i indeed macro indent meta a equal zero and1 meta b suc equal numeral one end line because disjin1 modus ponens ell j indeed macro indent parenthesis meta a equal zero and1 meta b suc equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 meta b suc equal zero end parenthesis end line line ell l end block because deduction modus ponens ell l indeed macro indent meta a plus meta b suc equal numeral one imply parenthesis meta a equal zero and1 meta b suc equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 meta b suc equal zero end parenthesis end line line ell k end block because deduction modus ponens ell k indeed parenthesis meta a plus meta b equal numeral one imply parenthesis parenthesis meta a equal zero and1 meta b equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 meta b equal zero end parenthesis end parenthesis end parenthesis imply meta a plus meta b suc equal numeral one imply parenthesis meta a equal zero and1 meta b suc equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 meta b suc equal zero end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three five f says for all terms meta a comma meta b indeed meta a plus meta b equal numeral one imply parenthesis meta a equal zero and1 meta b equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 meta b equal zero end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five f reads any term meta a comma meta b end line block line ell a because prop three five f one indeed macro indent object x plus zero equal numeral one imply parenthesis object x equal zero and1 zero equal numeral one end parenthesis or1 parenthesis object x equal numeral one and1 zero equal zero end parenthesis end line line ell b because prop three five f two indeed macro indent parenthesis object x plus object y equal numeral one imply parenthesis parenthesis object x equal zero and1 object y equal numeral one end parenthesis or1 parenthesis object x equal numeral one and1 object y equal zero end parenthesis end parenthesis end parenthesis imply object x plus object y suc equal numeral one imply parenthesis object x equal zero and1 object y suc equal numeral one end parenthesis or1 parenthesis object x equal numeral one and1 object y suc equal zero end parenthesis end line because axiom s nine at object y modus ponens ell a modus ponens ell b indeed macro indent object x plus object y equal numeral one imply parenthesis parenthesis object x equal zero and1 object y equal numeral one end parenthesis or1 parenthesis object x equal numeral one and1 object y equal zero end parenthesis end parenthesis end line line ell c end block because deduction modus ponens ell c indeed meta a plus meta b equal numeral one imply parenthesis meta a equal zero and1 meta b equal numeral one end parenthesis or1 parenthesis meta a equal numeral one and1 meta b equal zero end parenthesis qed end math ] "
\end{list}
\subsection{3.5g}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three five g one says for all terms meta a indeed meta a times zero equal numeral one imply parenthesis meta a equal numeral one end parenthesis and1 parenthesis zero equal numeral one end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five g one reads any term meta a end line block any term macro indent meta a end line line ell a premise macro indent meta a times zero equal numeral one end line line ell b because axiom s seven indeed macro indent meta a times zero equal zero end line line ell c because axiom s one modus ponens ell b modus ponens ell a indeed macro indent zero equal numeral one end line line ell d because axiom s three indeed macro indent zero neq numeral one end line line ell e because t one indeed macro indent not parenthesis zero equal numeral one end parenthesis imply parenthesis parenthesis zero equal numeral one end parenthesis imply parenthesis meta a equal numeral one and1 zero equal numeral one end parenthesis end parenthesis end line line ell f because ell e object modus ponens ell d indeed macro indent parenthesis zero equal numeral one end parenthesis imply parenthesis meta a equal numeral one and1 zero equal numeral one end parenthesis end line because ell f object modus ponens ell c indeed macro indent parenthesis meta a equal numeral one and1 zero equal numeral one end parenthesis end line line ell g end block because deduction modus ponens ell g indeed meta a times zero equal numeral one imply parenthesis meta a equal numeral one and1 zero equal numeral one end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three five g three says for all terms meta a comma meta b indeed parenthesis meta a times meta b equal zero and1 meta a equal numeral one end parenthesis imply parenthesis meta a equal numeral one and1 meta b suc equal numeral one end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five g three reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell e premise macro indent meta a times meta b equal zero and1 meta a equal numeral one end line line ell f because conjel2 modus ponens ell e indeed macro indent meta a equal numeral one end line line ell g because conjel1 modus ponens ell e indeed macro indent meta a times meta b equal zero end line line ell big b because axiom s three indeed macro indent zero neq numeral one end line line ell big c because h ten modus ponens ell big b indeed macro indent numeral one neq zero end line line ell big d because h four mark indeed macro indent meta a equal numeral one imply meta a equal zero imply numeral one equal zero end line line ell big e because h three modus ponens ell big d modus ponens ell f modus ponens ell big c indeed macro indent meta a neq zero end line line ell big f because prop three two n indeed macro indent meta a times meta b equal meta b times meta a end line line ell big g because axiom s one modus ponens ell big f modus ponens ell g indeed macro indent meta b times meta a equal zero end line line ell h because prop three five e indeed macro indent meta a neq zero imply meta b times meta a equal zero imply meta b equal zero end line line ell big i because ell h object modus ponens ell big e indeed macro indent meta b times meta a equal zero imply meta b equal zero end line line ell big h because ell big i object modus ponens ell big g indeed macro indent macro indent macro indent meta b equal zero end line line ell i because axiom s two modus ponens ell big h indeed macro indent meta b suc equal numeral one end line because conjin modus ponens ell f modus ponens ell i indeed macro indent meta a equal numeral one and1 meta b suc equal numeral one end line line ell j end block because deduction modus ponens ell j indeed parenthesis meta a times meta b equal zero and1 meta a equal numeral one end parenthesis imply parenthesis meta a equal numeral one and1 meta b suc equal numeral one end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three five g four says for all terms meta a comma meta b indeed parenthesis meta a times meta b equal numeral one imply parenthesis meta a equal numeral one and1 meta b equal numeral one end parenthesis end parenthesis infer parenthesis meta a times meta b equal numeral one and1 meta a equal zero end parenthesis imply parenthesis meta a equal numeral one and1 meta b suc equal numeral one end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five g four reads any term meta a comma meta b end line line ell y premise meta a times meta b equal numeral one imply parenthesis meta a equal numeral one and1 meta b equal numeral one end parenthesis end line block any term macro indent meta a comma meta b end line line ell l premise macro indent meta a times meta b equal numeral one and1 meta a equal zero end line line ell m because conjel1 modus ponens ell l indeed macro indent meta a times meta b equal numeral one end line line ell n because ell y object modus ponens ell m indeed macro indent parenthesis meta a equal numeral one and1 meta b equal numeral one end parenthesis end line line ell o because conjel1 modus ponens ell n indeed macro indent meta a equal numeral one end line line ell p because conjel2 modus ponens ell l indeed macro indent meta a equal zero end line line ell q because axiom s one modus ponens ell p modus ponens ell o indeed macro indent zero equal numeral one end line line ell r because axiom s three indeed macro indent zero neq numeral one end line line ell s because t one indeed macro indent zero neq numeral one imply parenthesis parenthesis zero equal numeral one end parenthesis imply parenthesis meta a equal numeral one and1 meta b suc equal numeral one end parenthesis end parenthesis end line line ell t because ell s object modus ponens ell r indeed macro indent parenthesis zero equal numeral one end parenthesis imply parenthesis meta a equal numeral one and1 meta b suc equal numeral one end parenthesis end line because ell t object modus ponens ell q indeed macro indent parenthesis meta a equal numeral one and1 meta b suc equal numeral one end parenthesis end line line ell u end block because deduction modus ponens ell u indeed parenthesis meta a times meta b equal numeral one and1 meta a equal zero end parenthesis imply parenthesis meta a equal numeral one and1 meta b suc equal numeral one end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three five g two says for all terms meta a comma meta b indeed parenthesis meta a times meta b equal numeral one imply parenthesis meta a equal numeral one and1 meta b equal numeral one end parenthesis end parenthesis imply parenthesis meta a times meta b suc equal numeral one imply parenthesis meta a equal numeral one and1 meta b suc equal numeral one end parenthesis end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five g two reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell y premise macro indent meta a times meta b equal numeral one imply parenthesis meta a equal numeral one and1 meta b equal numeral one end parenthesis end line line ell a premise macro indent meta a times meta b suc equal numeral one end line line ell b because axiom s eight indeed macro indent meta a times meta b suc equal meta a times meta b plus meta a end line line ell c because axiom s one modus ponens ell b modus ponens ell a indeed macro indent meta a times meta b plus meta a equal numeral one end line line ell z because prop three five f indeed macro indent parenthesis meta a times meta b plus meta a equal numeral one end parenthesis imply parenthesis meta a times meta b equal zero and1 meta a equal numeral one end parenthesis or1 parenthesis meta a times meta b equal numeral one and1 meta a equal zero end parenthesis end line line ell d because ell z object modus ponens ell c indeed macro indent parenthesis meta a times meta b equal zero and1 meta a equal numeral one end parenthesis or1 parenthesis meta a times meta b equal numeral one and1 meta a equal zero end parenthesis end line line ell k because prop three five g three indeed macro indent parenthesis meta a times meta b equal zero and1 meta a equal numeral one end parenthesis imply parenthesis meta a equal numeral one and1 meta b suc equal numeral one end parenthesis end line line ell v because prop three five g four modus ponens ell y indeed macro indent macro indent parenthesis meta a times meta b equal numeral one and1 meta a equal zero end parenthesis imply parenthesis meta a equal numeral one and1 meta b suc equal numeral one end parenthesis end line because h eleven modus ponens ell d modus ponens ell k modus ponens ell v indeed macro indent meta a equal numeral one and1 meta b suc equal numeral one end line line ell x end block because deduction modus ponens ell x indeed parenthesis meta a times meta b equal numeral one imply parenthesis meta a equal numeral one and1 meta b equal numeral one end parenthesis end parenthesis imply meta a times meta b suc equal numeral one imply parenthesis meta a equal numeral one and1 meta b suc equal numeral one end parenthesis qed end math ] "
\item " [ math in theory system s lemma prop three five g says for all terms meta a comma meta b indeed meta a times meta b equal numeral one imply parenthesis meta a equal numeral one and1 meta b equal numeral one end parenthesis end lemma end math ] "
\item " [ math system s proof of prop three five g reads any term meta a comma meta b end line block line ell a because prop three five g one indeed macro indent object x times zero equal numeral one imply parenthesis object x equal numeral one and1 zero equal numeral one end parenthesis end line line ell b because prop three five g two indeed macro indent parenthesis object x times object y equal numeral one imply parenthesis object x equal numeral one and1 object y equal numeral one end parenthesis end parenthesis imply parenthesis object x times object y suc equal numeral one imply parenthesis object x equal numeral one and1 object y suc equal numeral one end parenthesis end parenthesis end line because axiom s nine at object y modus ponens ell a modus ponens ell b indeed macro indent object x times object y equal numeral one imply parenthesis object x equal numeral one and1 object y equal numeral one end parenthesis end line line ell c end block because deduction modus ponens ell c indeed meta a times meta b equal numeral one imply parenthesis meta a equal numeral one and1 meta b equal numeral one end parenthesis qed end math ] "
\end{list}
\subsection{3.5h}\label{35h}
" [ math in theory system s lemma prop three five h says for all terms meta a indeed meta a neq zero imply exists meta b indeed meta a equal meta b suc end lemma end math ] "
Beviset for 3.5.h ville i pyk have set ud nogenlunde som f\o{}lgende:
\[
t \neq 0 \Rightarrow (\exists y) (t = y')
\]
\subsubsection*{Part I}
\[
\begin{array}{lll}
L01 & prem & 0 \neq 0 \\
L02 & abbr. L01 & \lnot ( 0=0)\\
L03 & 3.2.a & 0 = 0 \\
L04 & Lemma 1.11.c & \lnot (0=0) \Rightarrow ((0=0) \Rightarrow (\exists w)(0=w')) \\
L05 & MP, L02, L03 & (\exists w)(0=w')\\
L06 & Ded, L05 & (0 \neq 0) \Rightarrow (\exists w)(0 = w') \\
\end{array}
\]
\subsubsection*{Part II}
\[
\begin{array}{llr}
L01: & premise & (x \neq 0) \Rightarrow (\exists w) (x = w')\\
L02: & S3'& x' \neq 0 \\
L03: & Taut.:\ \mathcal{A} \lor \lnot \mathcal{A} & (x=0)\lor \lnot(x = 0) \\
L04: &prem.& (x= 0) \\
L05: &S2'& 0'= x'\\
L06: &E4& (x=0)\Rightarrow(\exists w)(x'= w')\\
L07: &prem& \lnot(x = 0)\\
L08: & MP,L01,L07 & (\exists w) (x = w')\\
L09: & rule c& x = b'\\
L10: &S2' &(\exists w) (x' = w')\\
L11: &E4& \lnot (x = 0) \Rightarrow (\exists w)(x' = w')\\
L12 & Lemma\ H11,L06. L11 & (x' \neq 0) \Rightarrow (\exists w)(x' = w') \\
\end{array}
\]
Tilbage er blot at bruge s\ae{}tning " [ math axiom s nine end math ] " for at fuldende induktionsbeviset.
\section{Udsagn 3.7}
Vi har kun lavet de indledende definitioner, da hele 3.7 kr\ae{}ver ''existential rule''.
\subsection{Definitioner af " [ math object x ist object y end math ] "}
Vi har makrodefineret f\o{}lgende definitioner:
" [ math macro define var x ist var y as exists var z indeed parenthesis var z neq zero and1 var z plus var x equal var y end parenthesis end define end math ] "
" [ math macro define var x istq var y as var x ist var y or1 var x equal var y end define end math ] "
" [ math macro define var x igt var y as var y ist var x end define end math ] "
" [ math macro define var x igtq var y as var y istq var x end define end math ] "
" [ math macro define var x inst var y as not parenthesis var x ist var y end parenthesis end define end math ] "
" [ math macro define var x ingt var y as var y inst var x end define end math ] "
" [ math macro define var x divides var y as exists var z indeed var y equal var x times var z end define end math ] "
Ved hj\ae{}lp af disse definitioner og ''existential rule'' ville 3.7 kunne bevises.
\appendix
\section{Hj\ae{}lpe lemmaer}\label{help}
Til flere af vores beviser har vi brug for nogle hj\ae{}lpe-lemmaer, disse er bevist i det f\o{}lgende afsnit
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma t one says for all terms meta a comma meta b indeed not meta a imply parenthesis meta a imply meta b end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma h zero a says for all terms meta a comma meta b comma meta c indeed parenthesis meta a imply meta b end parenthesis infer parenthesis meta b imply meta c end parenthesis infer meta a imply meta c end lemma end math ] "
\item " [ math in theory system s lemma h zero b says for all terms meta a comma meta b comma meta c indeed meta a imply parenthesis meta b imply meta c end parenthesis infer meta b infer meta a imply meta c end lemma end math ] "
\item " [ math in theory system s lemma h one says for all terms meta a indeed not not meta a imply meta a end lemma end math ] "
\item " [ math in theory system s lemma h two says for all terms meta a indeed meta a imply not not meta a end lemma end math ] "
\item " [ math in theory system s lemma h four says for all terms meta a comma meta b comma meta c indeed meta a equal meta b imply meta b equal meta c imply meta a equal meta c end lemma end math ] "
\item " [ math in theory system s lemma h four mark says for all terms meta a comma meta b comma meta c indeed meta a equal meta b imply meta a equal meta c imply meta b equal meta c end lemma end math ] "
\item " [ math in theory system s lemma h five says for all terms meta a comma meta b indeed parenthesis not meta b imply not meta a end parenthesis imply parenthesis not meta b imply meta a end parenthesis imply meta b end lemma end math ] "
\item " [ math in theory system s lemma h six says for all terms meta a indeed meta a imply meta a end lemma end math ] "
\item " [ math in theory system s lemma h seven says for all terms meta a comma meta b indeed parenthesis meta a imply meta b end parenthesis imply parenthesis not meta b imply not meta a end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma h eight says for all terms meta a comma meta b indeed parenthesis not meta b imply not meta a end parenthesis imply parenthesis meta a imply meta b end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma h nine says for all terms meta a comma meta b indeed meta a equal meta b imply meta b equal meta a end lemma end math ] "
\item " [ math in theory system s lemma h ten says for all terms meta a comma meta b indeed meta b neq meta a infer meta a neq meta b end lemma end math ] "
\item " [ math in theory system s lemma h eleven says for all terms meta a comma meta b comma meta c indeed meta a or1 meta b infer meta a imply meta c infer meta b imply meta c infer meta c end lemma end math ] "
\item " [ math in theory system s lemma h twelwe says for all terms meta a comma meta b indeed parenthesis meta a imply meta b end parenthesis infer parenthesis not meta a imply meta b end parenthesis infer meta b end lemma end math ] "
\item " [ math in theory system s lemma modus tollens says for all terms meta a comma meta b indeed parenthesis meta a imply meta b end parenthesis infer not meta b infer not meta a end lemma end math ] "
\end{list}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma t one says for all terms meta a comma meta b indeed not meta a imply parenthesis meta a imply meta b end parenthesis end lemma end math ] "
\item " [ math system s proof of t one reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell a premise macro indent not meta a end line block any term macro indent meta a comma meta b end line line ell b premise macro indent meta a end line line ell i because repetition modus ponens ell a indeed macro indent not meta a end line line ell c because lemma a one indeed macro indent meta a imply parenthesis not meta b imply meta a end parenthesis end line line ell g because ell c object modus ponens ell b indeed macro indent not meta b imply meta a end line line ell d because lemma a one indeed macro indent not meta a imply parenthesis not meta b imply not meta a end parenthesis end line line ell h because ell d object modus ponens ell i indeed macro indent not meta b imply not meta a end line because double negation modus ponens ell h modus ponens ell g indeed macro indent meta b end line line ell e end block because deduction modus ponens ell e indeed macro indent meta a imply meta b end line line ell f end block because deduction modus ponens ell f indeed not meta a imply parenthesis meta a imply meta b end parenthesis qed end math ] "
\item " [ math in theory system s lemma h zero a says for all terms meta a comma meta b comma meta c indeed parenthesis meta a imply meta b end parenthesis infer parenthesis meta b imply meta c end parenthesis infer meta a imply meta c end lemma end math ] "
\item " [ math system s proof of h zero a reads any term meta a comma meta b comma meta c end line line ell g premise meta a imply meta b end line line ell h premise meta b imply meta c end line block any term macro indent meta a comma meta b comma meta c end line line ell a premise macro indent meta a imply meta b end line line ell b premise macro indent meta b imply meta c end line line ell c premise macro indent meta a end line line ell d because ell a object modus ponens ell c indeed macro indent meta b end line because ell b object modus ponens ell d indeed macro indent meta c end line line ell e end block line ell f because deduction modus ponens ell e indeed parenthesis meta a imply meta b end parenthesis imply parenthesis meta b imply meta c end parenthesis imply meta a imply meta c end line line ell i because ell f object modus ponens ell g indeed parenthesis meta b imply meta c end parenthesis imply meta a imply meta c end line because ell i object modus ponens ell h indeed meta a imply meta c qed end math ] "
\item " [ math in theory system s lemma h zero b says for all terms meta a comma meta b comma meta c indeed meta a imply parenthesis meta b imply meta c end parenthesis infer meta b infer meta a imply meta c end lemma end math ] "
\item " [ math system s proof of h zero b reads any term meta a comma meta b comma meta c end line line ell a premise meta a imply meta b imply meta c end line line ell b premise meta b end line line ell c because lemma a two indeed parenthesis meta a imply meta b imply meta c end parenthesis imply parenthesis meta a imply meta b end parenthesis imply meta a imply meta c end line line ell d because ell c object modus ponens ell a indeed parenthesis meta a imply meta b end parenthesis imply meta a imply meta c end line line ell e because lemma a one indeed meta b imply meta a imply meta b end line line ell f because ell e object modus ponens ell b indeed meta a imply meta b end line because ell d object modus ponens ell f indeed meta a imply meta c qed end math ] "
\item " [ math in theory system s lemma h one says for all terms meta a indeed not not meta a imply meta a end lemma end math ] "
\item " [ math system s proof of h one reads any term meta a end line line ell a because h five indeed parenthesis not parenthesis meta a end parenthesis imply not parenthesis not meta a end parenthesis end parenthesis imply parenthesis parenthesis not parenthesis meta a end parenthesis imply parenthesis not meta a end parenthesis end parenthesis imply meta a end parenthesis end line line ell b because h six indeed not meta a imply not meta a end line line ell c because h zero b modus ponens ell a modus ponens ell b indeed parenthesis not meta a imply not parenthesis not meta a end parenthesis end parenthesis imply meta a end line line ell d because lemma a one indeed not parenthesis not meta a end parenthesis imply parenthesis not meta a imply not parenthesis not meta a end parenthesis end parenthesis end line because h zero a modus ponens ell d modus ponens ell c indeed not not meta a imply meta a qed end math ] "
\item " [ math in theory system s lemma h two says for all terms meta a indeed meta a imply not not meta a end lemma end math ] "
\item " [ math system s proof of h two reads any term meta a end line line ell a because h five indeed parenthesis not not not meta a imply not meta a end parenthesis imply parenthesis parenthesis not not not meta a imply meta a end parenthesis imply not not meta a end parenthesis end line line ell b because h one indeed not not parenthesis not meta a end parenthesis imply parenthesis not meta a end parenthesis end line line ell c because ell a object modus ponens ell b indeed parenthesis not not not meta a imply meta a end parenthesis imply not not meta a end line line ell d because lemma a one indeed meta a imply parenthesis not not not meta a imply meta a end parenthesis end line because h zero a modus ponens ell d modus ponens ell c indeed meta a imply not not meta a qed end math ] "
\item " [ math in theory system s lemma h four says for all terms meta a comma meta b comma meta c indeed meta a equal meta b imply meta b equal meta c imply meta a equal meta c end lemma end math ] "
\item " [ math system s proof of h four reads any term meta a comma meta b comma meta c end line block any term macro indent meta a comma meta b comma meta c end line line ell a premise macro indent meta a equal meta b end line block any term macro indent meta a comma meta b comma meta c end line line ell b premise macro indent meta b equal meta c end line because prop three two c modus ponens ell a modus ponens ell b indeed macro indent meta a equal meta c end line line ell c end block because deduction modus ponens ell c indeed macro indent meta b equal meta c imply meta a equal meta c end line line ell d end block because deduction modus ponens ell d indeed meta a equal meta b imply meta b equal meta c imply meta a equal meta c qed end math ] "
\item " [ math in theory system s lemma h four mark says for all terms meta a comma meta b comma meta c indeed meta a equal meta b imply meta a equal meta c imply meta b equal meta c end lemma end math ] "
\item " [ math system s proof of h four mark reads any term meta a comma meta b comma meta c end line block any term macro indent meta a comma meta b comma meta c end line line ell a premise macro indent meta a equal meta b end line block any term macro indent meta a comma meta b comma meta c end line line ell b premise macro indent meta a equal meta c end line because axiom s one modus ponens ell a modus ponens ell b indeed macro indent meta b equal meta c end line line ell c end block because deduction modus ponens ell c indeed macro indent meta a equal meta c imply meta b equal meta c end line line ell d end block because deduction modus ponens ell d indeed meta a equal meta b imply meta a equal meta c imply meta b equal meta c qed end math ] "
\item " [ math in theory system s lemma h five says for all terms meta a comma meta b indeed parenthesis not meta b imply not meta a end parenthesis imply parenthesis not meta b imply meta a end parenthesis imply meta b end lemma end math ] "
\item " [ math system s proof of h five reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell a premise macro indent not meta b imply not meta a end line block any term macro indent meta a comma meta b end line line ell b premise macro indent not meta b imply meta a end line because double negation modus ponens ell a modus ponens ell b indeed macro indent macro indent meta b end line line ell c end block because deduction modus ponens ell c indeed macro indent parenthesis not meta b imply meta a end parenthesis imply meta b end line line ell d end block because deduction modus ponens ell d indeed parenthesis not meta b imply not meta a end parenthesis imply parenthesis not meta b imply meta a end parenthesis imply meta b qed end math ] "
\item " [ math in theory system s lemma h six says for all terms meta a indeed meta a imply meta a end lemma end math ] "
\item " [ math system s proof of h six reads any term meta a end line block any term macro indent meta a end line line ell a premise meta a end line because repetition modus ponens ell a indeed macro indent meta a end line line ell b end block because deduction modus ponens ell b indeed meta a imply meta a qed end math ] "
\item " [ math in theory system s lemma h ten says for all terms meta a comma meta b indeed meta b neq meta a infer meta a neq meta b end lemma end math ] "
\item " [ math system s proof of h ten reads any term meta a comma meta b end line line ell a premise meta b neq meta a end line line ell e because h nine indeed meta a equal meta b imply meta b equal meta a end line line ell f because h seven indeed parenthesis meta a equal meta b imply meta b equal meta a end parenthesis imply meta b neq meta a imply meta a neq meta b end line line ell g because ell f object modus ponens ell e indeed meta b neq meta a imply meta a neq meta b end line because ell g object modus ponens ell a indeed meta a neq meta b qed end math ] "
\item " [ math in theory system s lemma modus tollens says for all terms meta a comma meta b indeed parenthesis meta a imply meta b end parenthesis infer not meta b infer not meta a end lemma end math ] "
\item " [ math system s proof of modus tollens reads any term meta a comma meta b end line line ell a premise meta a imply meta b end line line ell b premise not meta b end line line ell c because h one indeed not not meta a imply meta a end line line ell d because h zero a modus ponens ell c modus ponens ell a indeed not not meta a imply meta b end line block any term macro indent meta a comma meta b end line line ell e premise macro indent macro indent not not meta a imply meta b end line line ell f premise macro indent not meta b end line line ell g because lemma a one indeed macro indent not meta b imply not not meta a imply not meta b end line line ell h because ell g object modus ponens ell f indeed macro indent not not meta a imply not meta b end line because double negation modus ponens ell h modus ponens ell e indeed macro indent not meta a end line line ell i end block line ell j because deduction modus ponens ell i indeed parenthesis not not meta a imply meta b end parenthesis imply not meta b imply not meta a end line line ell k because ell j object modus ponens ell d indeed not meta b imply not meta a end line because ell k object modus ponens ell b indeed not meta a qed end math ] "
\item " [ math in theory system s lemma h seven says for all terms meta a comma meta b indeed parenthesis meta a imply meta b end parenthesis imply parenthesis not meta b imply not meta a end parenthesis end lemma end math ] "
\item " [ math system s proof of h seven reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell a premise macro indent meta a imply meta b end line line ell b because h one indeed macro indent not not meta a imply meta a end line line ell c because h zero a modus ponens ell b modus ponens ell a indeed macro indent not not meta a imply meta b end line line ell d because h two indeed macro indent meta b imply not not meta b end line line ell e because h zero a modus ponens ell c modus ponens ell d indeed macro indent not not meta a imply not not meta b end line line ell f because h eight indeed macro indent parenthesis not not meta a imply not not meta b end parenthesis imply not meta b imply not meta a end line because ell f object modus ponens ell e indeed macro indent not meta b imply not meta a end line line ell g end block because deduction modus ponens ell g indeed parenthesis meta a imply meta b end parenthesis imply not meta b imply not meta a qed end math ] "
\item " [ math in theory system s lemma h eight says for all terms meta a comma meta b indeed parenthesis not meta b imply not meta a end parenthesis imply parenthesis meta a imply meta b end parenthesis end lemma end math ] "
\item " [ math system s proof of h eight reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell a premise macro indent not meta b imply not meta a end line line ell b because h five indeed macro indent parenthesis not meta b imply not meta a end parenthesis imply parenthesis parenthesis not meta b imply meta a end parenthesis imply meta b end parenthesis end line line ell c because lemma a one indeed macro indent macro indent meta a imply parenthesis not meta b imply meta a end parenthesis end line line ell d because ell b object modus ponens ell a indeed macro indent parenthesis not meta b imply meta a end parenthesis imply meta b end line because h zero a modus ponens ell c modus ponens ell d indeed macro indent meta a imply meta b end line line ell e end block because deduction modus ponens ell e indeed parenthesis not meta b imply not meta a end parenthesis imply parenthesis meta a imply meta b end parenthesis qed end math ] "
\item " [ math in theory system s lemma h nine says for all terms meta a comma meta b indeed meta a equal meta b imply meta b equal meta a end lemma end math ] "
\item " [ math system s proof of h nine reads any term meta a comma meta b end line block any term macro indent meta a comma meta b end line line ell a premise macro indent meta a equal meta b end line because prop three two b modus ponens ell a indeed macro indent meta b equal meta a end line line ell b end block because deduction modus ponens ell b indeed meta a equal meta b imply meta b equal meta a qed end math ] "
\item " [ math in theory system s lemma h twelwe says for all terms meta a comma meta b indeed parenthesis meta a imply meta b end parenthesis infer parenthesis not meta a imply meta b end parenthesis infer meta b end lemma end math ] "
\item " [ math system s proof of h twelwe reads any term meta a comma meta b end line line ell a premise meta a imply meta b end line line ell b premise not meta a imply meta b end line line ell c because h seven indeed parenthesis meta a imply meta b end parenthesis imply not meta b imply not meta a end line line ell d because ell c object modus ponens ell a indeed not meta b imply not meta a end line line ell e because h seven indeed parenthesis not meta a imply meta b end parenthesis imply not meta b imply not not meta a end line line ell f because ell e object modus ponens ell b indeed not meta b imply not not meta a end line because double negation modus ponens ell f modus ponens ell d indeed meta b qed end math ] "
\item " [ math in theory system s lemma h eleven says for all terms meta a comma meta b comma meta c indeed meta a or1 meta b infer meta a imply meta c infer meta b imply meta c infer meta c end lemma end math ] "
\item " [ math system s proof of h eleven reads any term meta a comma meta b comma meta c end line line ell a premise meta a or1 meta b end line line ell b premise meta a imply meta c end line line ell c premise meta b imply meta c end line line ell d because h zero a modus ponens ell a modus ponens ell c indeed not meta a imply meta c end line because h twelwe modus ponens ell b modus ponens ell d indeed meta c qed end math ] "
\end{list}
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\bibliography{./page}
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\everymath{}
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File page.bib
@article {grue92,
author = {K. Grue},
title = {Map Theory},
journal = TCS,
year = {1992},
volume = {102},
number = {1},
pages = {1--133},
month = {jul}}
@book {MathComp,
author = {Klaus Grue},
year = {2001},
title = {Mathematics and Computation},
publisher = {DIKU},
address = {Universitetsparken 1, DK-2100 Copenhagen},
volume = {1--3},
edition = {7},
keywords = {Logic}}
@techreport{grue02b,
author = {K. Grue},
year = {2002},
title = {Map Theory with Classical Maps},
institution
= {DIKU},
note = {\url{http://www.diku.dk/publikationer/tekniske.rapporter/2002/}},
number = {02/21}}
@inproceedings{grue04,
author = {K. Grue},
title = {Logiweb},
editor = {Fairouz Kamareddine},
booktitle = {Mathematical Knowledge Management Symposium 2003},
publisher = {Elsevier},
series = {Electronic Notes in Theoretical Computer Science},
volume = {93},
year = {2004},
pages = {70--101}}
@InProceedings{grue05,
author = {K. Grue},
year = {2005},
title = {The implementation of Logiweb},
booktitle = {Empirically Successful Classical Automated Reasoning
(ESCAR)},
editor = {Bernd Fischer and Stephan Schulz and Geoff Sutcliffe},
keywords = {Automated Reasoning, Electronic Publishing}}
@techreport{base,
author = {K. Grue},
year = {2006},
title = {A Logiweb base page},
institution={Logiweb},
note = {{\small \url{\lgwBaseUrl}}}}
@techreport{check,
author = {K. Grue},
year = {2006},
title = {A Logiweb base page},
institution={Logiweb},
note = {{\small \url{\lgwCheckUrl}}}}
@techreport{ijcar,
author = {K. Grue},
year = {2006},
title = {A Logiweb base page},
institution={Logiweb},
note = {{\small \url{\lgwIjcarUrl}}}}
@book {mendelson,
author = {E. Mendelson},
title = {Introduction to Mathematical Logic},
publisher = {Chapmann \& Hall},
year = {1997},
edition = {4.}}
End of file
latex page
bibtex page
latex page
latex page
dvipdfm page
File appendix.tex
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\hypersetup{pdftitle=Logiweb sequent calculus - Appendix}
\hypersetup{colorlinks=true}
\begin {document}
\title {Udvidedlse af S-reglerne, Appendix}
\author {Maja Hanne T\o nnesen, Rune Christoffer Kildetoft Andresen \\ \& Niels Peter Meyn Milthers}
\date {\today}
\maketitle
\tableofcontents
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three seven a says for all terms meta a indeed meta a inst meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven b says for all terms meta a comma meta b comma meta c indeed meta a ist meta b imply meta b ist meta c imply meta a ist meta c end lemma end math ] "
\item " [ math in theory system s lemma prop three seven c says for all terms meta a comma meta b indeed meta a ist meta b imply meta b inst meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven d says for all terms meta a comma meta b comma meta c indeed meta a ist meta b imply meta a plus meta c ist meta b plus meta c end lemma end math ] "
\item " [ math in theory system s lemma prop three seven e says for all terms meta a indeed meta a istq meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven f says for all terms meta a comma meta b comma meta c indeed meta a istq meta b imply meta b istq meta c imply meta a istq meta c end lemma end math ] "
\item " [ math in theory system s lemma prop three seven g says for all terms meta a comma meta b comma meta c indeed meta a istq meta b imply meta a plus meta c istq meta b plus meta c end lemma end math ] "
\item " [ math in theory system s lemma prop three seven g mark says for all terms meta a comma meta b comma meta c indeed meta a plus meta c istq meta b plus meta c imply meta a istq meta b end lemma end math ] "
\item " [ math in theory system s lemma prop three seven h says for all terms meta a comma meta b comma meta c indeed meta a istq meta b imply meta b ist meta c imply meta a ist meta c end lemma end math ] "
\item " [ math in theory system s lemma prop three seven i says for all terms meta a indeed zero istq meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven j says for all terms meta a indeed zero ist meta a suc end lemma end math ] "
\item " [ math in theory system s lemma prop three seven k says for all terms meta a comma meta b indeed meta a ist meta b imply meta a suc istq meta b end lemma end math ] "
\item " [ math in theory system s lemma prop three seven k mark says for all terms meta a comma meta b indeed meta a suc istq meta b imply meta a ist meta b end lemma end math ] "
\item " [ math in theory system s lemma prop three seven l says for all terms meta a comma meta b indeed meta a istq meta b imply meta a ist meta b suc end lemma end math ] "
\item " [ math in theory system s lemma prop three seven l mark says for all terms meta a comma meta b indeed meta a ist meta b suc imply meta a istq meta b end lemma end math ] "
\item " [ math in theory system s lemma prop three seven m says for all terms meta a indeed meta a ist meta a suc end lemma end math ] "
\item " [ math in theory system s lemma prop three seven o says for all terms meta a comma meta b indeed meta a neq meta b imply parenthesis meta a ist meta b or1 meta b ist meta a end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma prop three seven p says for all terms meta a comma meta b indeed meta a equal meta b or1 meta a ist meta b or1 meta b ist meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven q says for all terms meta a comma meta b indeed meta a istq meta b or1 meta b istq meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven r says for all terms meta a comma meta b indeed meta a plus meta b igtq meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven s says for all terms meta a comma meta b indeed meta b neq zero imply meta a plus meta b igt meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven t says for all terms meta a comma meta b indeed meta b neq zero imply meta a times meta b igtq meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven u says for all terms meta a indeed meta a neq zero imply meta a igt zero end lemma end math ] "
\item " [ math in theory system s lemma prop three seven u mark says for all terms meta a indeed meta a igt zero imply meta a neq zero end lemma end math ] "
\item " [ math in theory system s lemma prop three seven v says for all terms meta a comma meta b indeed meta a igt zero imply meta b igt zero imply meta a times meta b igt zero end lemma end math ] "
\item " [ math in theory system s lemma prop three seven w says for all terms meta a comma meta b indeed meta a neq zero imply meta b igt numeral one imply meta b times meta a igt meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven x says for all terms meta a comma meta b comma meta c indeed meta a neq zero imply meta b ist meta c imply meta b times meta a ist meta c times meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven x mark says for all terms meta a comma meta b comma meta c indeed meta a neq zero imply meta b times meta a ist meta c times meta a imply meta b ist meta c end lemma end math ] "
\item " [ math in theory system s lemma prop three seven y says for all terms meta a comma meta b comma meta c indeed meta a neq zero imply meta b istq meta c imply meta b times meta a istq meta c times meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three seven y mark says for all terms meta a comma meta b comma meta c indeed meta a neq zero imply meta b times meta a istq meta c times meta a imply meta b istq meta c end lemma end math ] "
\item " [ math in theory system s lemma prop three seven z says for all terms meta a indeed meta a inst zero end lemma end math ] "
\item " [ math in theory system s lemma prop three seven z mark says for all terms meta a comma meta b indeed meta a istq meta b and1 meta b istq meta a imply meta a equal meta b end lemma end math ] "
\end{list}
\subsection{proof of " [ math prop three seven end math ] "}
\section{Lemma " [ math prop three ten end math ] "}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three ten a says for all terms meta a indeed meta a divides meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three ten b says for all terms meta a indeed numeral one divides meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three ten c says for all terms meta a indeed meta a divides zero end lemma end math ] "
\item " [ math in theory system s lemma prop three ten d says for all terms meta a comma meta b comma meta c indeed meta a divides meta b and meta b divides meta c imply meta a divides meta c end lemma end math ] "
\item " [ math in theory system s lemma prop three ten e says for all terms meta a comma meta b indeed meta a neq zero and1 meta b divides meta b imply meta a istq meta a end lemma end math ] "
\item " [ math in theory system s lemma prop three ten f says for all terms meta a comma meta b indeed meta a divides meta b and1 meta b divides meta a imply meta a equal meta b end lemma end math ] "
\item " [ math in theory system s lemma prop three ten g says for all terms meta a comma meta b comma meta c indeed meta a divides meta b imply meta a divides parenthesis meta b times meta c end parenthesis end lemma end math ] "
\item " [ math in theory system s lemma prop three ten h says for all terms meta a comma meta b comma meta c indeed meta a divides meta b and meta a divides meta c imply meta a divides parenthesis meta b plus meta c end parenthesis end lemma end math ] "
\end{list}
\section{Lemma " [ math prop three eleven end math ] "}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system s lemma prop three eleven says for all terms meta a comma meta b indeed meta a neq zero imply exists meta c comma meta d indeed parenthesis meta b equal meta a times meta c plus meta d and1 meta d ist meta a and1 for all meta e comma meta f indeed parenthesis parenthesis meta b equal meta a times meta e plus meta f and1 meta f ist meta a end parenthesis imply meta c equal meta e and1 meta d equal meta f end parenthesis end parenthesis end lemma end math ] "
\end{list}
%------------------------------------------------------------------------------
\bibliographystyle{alpha}
\bibliography{./page}
%------------------------------------------------------------------------------
\everymath{}
\everydisplay{}
\end{document}
End of file
latex appendix
bibtex appendix
latex appendix
latex appendix
dvipdfm appendix
File chores.tex
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\input{lgwinclude}
\usepackage[danish]{babel}
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\usepackage{amssymb}
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\hypersetup{pdftitle=Logiweb sequent calculus - Chores}
\hypersetup{colorlinks=true}
\begin {document}
\title {Logiweb sequent calculus, Chores}
\author {Klaus Grue}
\date {\today}
\maketitle
\tableofcontents
\section{Test cases}
\section{Pyk definitions}
\begin{flushleft}
" [ math protect define pyk of numeral zero as text "numeral zero" end text end define linebreak define pyk of numeral one as text "numeral one" end text end define linebreak define pyk of numeral two as text "numeral two" end text end define linebreak define pyk of numeral three as text "numeral three" end text end define linebreak define pyk of numeral four as text "numeral four" end text end define linebreak define pyk of numeral five as text "numeral five" end text end define linebreak define pyk of numeral six as text "numeral six" end text end define linebreak define pyk of numeral seven as text "numeral seven" end text end define linebreak define pyk of numeral eight as text "numeral eight" end text end define linebreak define pyk of numeral nine as text "numeral nine" end text end define linebreak define pyk of numeral n as text "numeral n" end text end define linebreak define pyk of rule div as text "rule div" end text end define linebreak define pyk of rule r as text "rule r" end text end define linebreak define pyk of rule r one as text "rule r one" end text end define linebreak define pyk of rule r two as text "rule r two" end text end define linebreak define pyk of rule r three as text "rule r three" end text end define linebreak define pyk of rule r four as text "rule r four" end text end define linebreak define pyk of rule r five as text "rule r five" end text end define linebreak define pyk of rule r six as text "rule r six" end text end define linebreak define pyk of conjel1 as text "conjel1" end text end define linebreak define pyk of conjel2 as text "conjel2" end text end define linebreak define pyk of conjin as text "conjin" end text end define linebreak define pyk of disjin1 as text "disjin1" end text end define linebreak define pyk of disjin2 as text "disjin2" end text end define linebreak define pyk of t one as text "t one" end text end define linebreak define pyk of h zero a as text "h zero a" end text end define linebreak define pyk of h zero b as text "h zero b" end text end define linebreak define pyk of h one as text "h one" end text end define linebreak define pyk of h two as text "h two" end text end define linebreak define pyk of h three as text "h three" end text end define linebreak define pyk of h four as text "h four" end text end define linebreak define pyk of h four mark as text "h four mark" end text end define linebreak define pyk of h five as text "h five" end text end define linebreak define pyk of h six as text "h six" end text end define linebreak define pyk of h seven as text "h seven" end text end define linebreak define pyk of h eight as text "h eight" end text end define linebreak define pyk of h nine as text "h nine" end text end define linebreak define pyk of h ten as text "h ten" end text end define linebreak define pyk of h eleven as text "h eleven" end text end define linebreak define pyk of h twelwe as text "h twelwe" end text end define linebreak define pyk of modus tollens as text "modus tollens" end text end define linebreak define pyk of axiom s ten as text "axiom s ten" end text end define linebreak define pyk of prop three two as text "prop three two" end text end define linebreak define pyk of prop three two i as text "prop three two i" end text end define linebreak define pyk of prop three two j one as text "prop three two j one" end text end define linebreak define pyk of prop three two j two as text "prop three two j two" end text end define linebreak define pyk of prop three two j as text "prop three two j" end text end define linebreak define pyk of prop three two k one as text "prop three two k one" end text end define linebreak define pyk of prop three two k two as text "prop three two k two" end text end define linebreak define pyk of prop three two k as text "prop three two k" end text end define linebreak define pyk of prop three two l one as text "prop three two l one" end text end define linebreak define pyk of prop three two l two as text "prop three two l two" end text end define linebreak define pyk of prop three two l as text "prop three two l" end text end define linebreak define pyk of prop three two m one as text "prop three two m one" end text end define linebreak define pyk of prop three two m two as text "prop three two m two" end text end define linebreak define pyk of prop three two m as text "prop three two m" end text end define linebreak define pyk of prop three two n one as text "prop three two n one" end text end define linebreak define pyk of prop three two n two as text "prop three two n two" end text end define linebreak define pyk of prop three two n as text "prop three two n" end text end define linebreak define pyk of prop three two o as text "prop three two o" end text end define linebreak define pyk of prop three four as text "prop three four" end text end define linebreak define pyk of prop three four a one as text "prop three four a one" end text end define linebreak define pyk of prop three four a two as text "prop three four a two" end text end define linebreak define pyk of prop three four a as text "prop three four a" end text end define linebreak define pyk of prop three four b as text "prop three four b" end text end define linebreak define pyk of prop three four c one as text "prop three four c one" end text end define linebreak define pyk of prop three four c two as text "prop three four c two" end text end define linebreak define pyk of prop three four c as text "prop three four c" end text end define linebreak define pyk of prop three four d one as text "prop three four d one" end text end define linebreak define pyk of prop three four d two as text "prop three four d two" end text end define linebreak define pyk of prop three four d as text "prop three four d" end text end define linebreak define pyk of prop three five as text "prop three five" end text end define linebreak define pyk of prop three five a as text "prop three five a" end text end define linebreak define pyk of prop three five b as text "prop three five b" end text end define linebreak define pyk of prop three five c as text "prop three five c" end text end define linebreak define pyk of prop three five d one as text "prop three five d one" end text end define linebreak define pyk of prop three five d two as text "prop three five d two" end text end define linebreak define pyk of prop three five d as text "prop three five d" end text end define linebreak define pyk of prop three five e one as text "prop three five e one" end text end define linebreak define pyk of prop three five e two as text "prop three five e two" end text end define linebreak define pyk of prop three five e as text "prop three five e" end text end define linebreak define pyk of prop three five f one as text "prop three five f one" end text end define linebreak define pyk of prop three five f two as text "prop three five f two" end text end define linebreak define pyk of prop three five f as text "prop three five f" end text end define linebreak define pyk of prop three five g one as text "prop three five g one" end text end define linebreak define pyk of prop three five g two as text "prop three five g two" end text end define linebreak define pyk of prop three five g three as text "prop three five g three" end text end define linebreak define pyk of prop three five g four as text "prop three five g four" end text end define linebreak define pyk of prop three five g as text "prop three five g" end text end define linebreak define pyk of prop three five h one as text "prop three five h one" end text end define linebreak define pyk of prop three five h two as text "prop three five h two" end text end define linebreak define pyk of prop three five h as text "prop three five h" end text end define linebreak define pyk of prop three five i one as text "prop three five i one" end text end define linebreak define pyk of prop three five i two as text "prop three five i two" end text end define linebreak define pyk of prop three five i as text "prop three five i" end text end define linebreak define pyk of prop three five j one as text "prop three five j one" end text end define linebreak define pyk of prop three five j two as text "prop three five j two" end text end define linebreak define pyk of prop three five j as text "prop three five j" end text end define linebreak define pyk of prop three seven as text "prop three seven" end text end define linebreak define pyk of prop three seven a as text "prop three seven a" end text end define linebreak define pyk of prop three seven b as text "prop three seven b" end text end define linebreak define pyk of prop three seven c as text "prop three seven c" end text end define linebreak define pyk of prop three seven d as text "prop three seven d" end text end define linebreak define pyk of prop three seven e as text "prop three seven e" end text end define linebreak define pyk of prop three seven f as text "prop three seven f" end text end define linebreak define pyk of prop three seven g as text "prop three seven g" end text end define linebreak define pyk of prop three seven g mark as text "prop three seven g mark" end text end define linebreak define pyk of prop three seven h as text "prop three seven h" end text end define linebreak define pyk of prop three seven i as text "prop three seven i" end text end define linebreak define pyk of prop three seven j as text "prop three seven j" end text end define linebreak define pyk of prop three seven k as text "prop three seven k" end text end define linebreak define pyk of prop three seven k mark as text "prop three seven k mark" end text end define linebreak define pyk of prop three seven l as text "prop three seven l" end text end define linebreak define pyk of prop three seven l mark as text "prop three seven l mark" end text end define linebreak define pyk of prop three seven m as text "prop three seven m" end text end define linebreak define pyk of prop three seven n as text "prop three seven n" end text end define linebreak define pyk of prop three seven o as text "prop three seven o" end text end define linebreak define pyk of prop three seven p as text "prop three seven p" end text end define linebreak define pyk of prop three seven q as text "prop three seven q" end text end define linebreak define pyk of prop three seven r as text "prop three seven r" end text end define linebreak define pyk of prop three seven s as text "prop three seven s" end text end define linebreak define pyk of prop three seven t as text "prop three seven t" end text end define linebreak define pyk of prop three seven u as text "prop three seven u" end text end define linebreak define pyk of prop three seven u mark as text "prop three seven u mark" end text end define linebreak define pyk of prop three seven v as text "prop three seven v" end text end define linebreak define pyk of prop three seven w as text "prop three seven w" end text end define linebreak define pyk of prop three seven x as text "prop three seven x" end text end define linebreak define pyk of prop three seven x mark as text "prop three seven x mark" end text end define linebreak define pyk of prop three seven y as text "prop three seven y" end text end define linebreak define pyk of prop three seven y mark as text "prop three seven y mark" end text end define linebreak define pyk of prop three seven z as text "prop three seven z" end text end define linebreak define pyk of prop three seven z mark as text "prop three seven z mark" end text end define linebreak define pyk of prop three ten as text "prop three ten" end text end define linebreak define pyk of prop three ten a as text "prop three ten a" end text end define linebreak define pyk of prop three ten b as text "prop three ten b" end text end define linebreak define pyk of prop three ten c as text "prop three ten c" end text end define linebreak define pyk of prop three ten d as text "prop three ten d" end text end define linebreak define pyk of prop three ten e as text "prop three ten e" end text end define linebreak define pyk of prop three ten f as text "prop three ten f" end text end define linebreak define pyk of prop three ten g as text "prop three ten g" end text end define linebreak define pyk of prop three ten h as text "prop three ten h" end text end define linebreak define pyk of prop three eleven as text "prop three eleven" end text end define linebreak define pyk of x ist x as text ""! ist "!" end text end define linebreak define pyk of x istq x as text ""! istq "!" end text end define linebreak define pyk of x inst x as text ""! inst "!" end text end define linebreak define pyk of x igt x as text ""! igt "!" end text end define linebreak define pyk of x igtq x as text ""! igtq "!" end text end define linebreak define pyk of x ingt x as text ""! ingt "!" end text end define linebreak define pyk of x neq x as text ""! neq "!" end text end define linebreak define pyk of x and1 x as text ""! and1 "!" end text end define linebreak define pyk of x or1 x as text ""! or1 "!" end text end define linebreak define pyk of exists x indeed x as text "exists "! indeed "!" end text end define linebreak define pyk of x divides x as text ""! divides "!" end text end define linebreak define pyk of x ldots as text ""! ldots" end text end define linebreak define pyk of opgave as text "opgave" end text end define linebreak unicode end of text end protect end math ] "
\end{flushleft}
\section{\TeX\ definitions}
\begin{list}{}{
\setlength{\leftmargin}{5em}
\setlength{\itemindent}{-5em}}
\item " [ math tex define exists var x indeed var y as "
\exists #1.
\colon #2." end define end math ] "
\item " [ math tex define axiom s ten as "
S10" end define end math ] "
\item " [ math tex define var x divides var y as "#1.
\mathrel{|} #2." end define end math ] "
\item " [ math tex define prop three two as "
Prop\ 3.2" end define end math ] "
\item " [ math tex define prop three two i as "
Prop\ 3.2i" end define end math ] "
\item " [ math tex define prop three two j as "
Prop\ 3.2j" end define end math ] "
\item " [ math tex define prop three two j one as "
Prop\ 3.2j_1" end define end math ] "
\item " [ math tex define prop three two j two as "
Prop\ 3.2j_2" end define end math ] "
\item " [ math tex define prop three two k as "
Prop\ 3.2k" end define end math ] "
\item " [ math tex define prop three two k one as "
Prop\ 3.2k_1" end define end math ] "
\item " [ math tex define prop three two k two as "
Prop\ 3.2k_2" end define end math ] "
\item " [ math tex define prop three two l as "
Prop\ 3.2l" end define end math ] "
\item " [ math tex define prop three two l one as "
Prop\ 3.2l_1" end define end math ] "
\item " [ math tex define prop three two l two as "
Prop\ 3.2l_2" end define end math ] "
\item " [ math tex define prop three two m as "
Prop\ 3.2m" end define end math ] "
\item " [ math tex define prop three two m one as "
Prop\ 3.2m_1" end define end math ] "
\item " [ math tex define prop three two m two as "
Prop\ 3.2m_2" end define end math ] "
\item " [ math tex define prop three two n as "
Prop\ 3.2n" end define end math ] "
\item " [ math tex define prop three two n one as "
Prop\ 3.2n_1" end define end math ] "
\item " [ math tex define prop three two n two as "
Prop\ 3.2n_2" end define end math ] "
\item " [ math tex define prop three two o as "
Prop\ 3.2o" end define end math ] "
\item " [ math tex define prop three four as "
Prop\ 3.4" end define end math ] "
\item " [ math tex define prop three four a one as "
Prop\ 3.4a_1" end define end math ] "
\item " [ math tex define prop three four a two as "
Prop\ 3.4a_2" end define end math ] "
\item " [ math tex define prop three four a as "
Prop\ 3.4a" end define end math ] "
\item " [ math tex define prop three four b as "
Prop\ 3.4b" end define end math ] "
\item " [ math tex define prop three four c one as "
Prop\ 3.4c_1" end define end math ] "
\item " [ math tex define prop three four c two as "
Prop\ 3.4c_2" end define end math ] "
\item " [ math tex define prop three four c as "
Prop\ 3.4c" end define end math ] "
\item " [ math tex define prop three four d one as "
Prop\ 3.4d_1" end define end math ] "
\item " [ math tex define prop three four d two as "
Prop\ 3.4d_2" end define end math ] "
\item " [ math tex define prop three four d as "
Prop\ 3.4d" end define end math ] "
\item " [ math tex define prop three five as "
Prop\ 3.5" end define end math ] "
\item " [ math tex define prop three five a as "
Prop\ 3.5a" end define end math ] "
\item " [ math tex define prop three five b as "
Prop\ 3.5b" end define end math ] "
\item " [ math tex define prop three five c as "
Prop\ 3.5c" end define end math ] "
\item " [ math tex define prop three five d one as "
Prop\ 3.5d_1" end define end math ] "
\item " [ math tex define prop three five d two as "
Prop\ 3.5d_2" end define end math ] "
\item " [ math tex define prop three five d as "
Prop\ 3.5d" end define end math ] "
\item " [ math tex define prop three five e one as "
Prop\ 3.5e_1" end define end math ] "
\item " [ math tex define prop three five e two as "
Prop\ 3.5e_2" end define end math ] "
\item " [ math tex define prop three five e as "
Prop\ 3.5e" end define end math ] "
\item " [ math tex define prop three five f one as "
Prop\ 3.5f_1" end define end math ] "
\item " [ math tex define prop three five f two as "
Prop\ 3.5f_2" end define end math ] "
\item " [ math tex define prop three five f as "
Prop\ 3.5f" end define end math ] "
\item " [ math tex define prop three five g one as "
Prop\ 3.5g_1" end define end math ] "
\item " [ math tex define prop three five g two as "
Prop\ 3.5g_4" end define end math ] "
\item " [ math tex define prop three five g three as "
Prop\ 3.5g_2" end define end math ] "
\item " [ math tex define prop three five g four as "
Prop\ 3.5g_3" end define end math ] "
\item " [ math tex define prop three five g as "
Prop\ 3.5g" end define end math ] "
\item " [ math tex define prop three five h one as "
Prop\ 3.5h_1" end define end math ] "
\item " [ math tex define prop three five h two as "
Prop\ 3.5h_2" end define end math ] "
\item " [ math tex define prop three five h as "
Prop\ 3.5h" end define end math ] "
\item " [ math tex define prop three five i one as "
Prop\ 3.5i_1" end define end math ] "
\item " [ math tex define prop three five i two as "
Prop\ 3.5i_2" end define end math ] "
\item " [ math tex define prop three five i as "
Prop\ 3.5i" end define end math ] "
\item " [ math tex define prop three five j one as "
Prop\ 3.5j_1" end define end math ] "
\item " [ math tex define prop three five j two as "
Prop\ 3.5j_2" end define end math ] "
\item " [ math tex define prop three five j as "
Prop\ 3.5j" end define end math ] "
\item " [ math tex define prop three seven as "
Prop\ 3.7" end define end math ] "
\item " [ math tex define prop three seven a as "
Prop\ 3.7a" end define end math ] "
\item " [ math tex define prop three seven b as "
Prop\ 3.7b" end define end math ] "
\item " [ math tex define prop three seven c as "
Prop\ 3.7c" end define end math ] "
\item " [ math tex define prop three seven d as "
Prop\ 3.7d" end define end math ] "
\item " [ math tex define prop three seven e as "
Prop\ 3.7e" end define end math ] "
\item " [ math tex define prop three seven f as "
Prop\ 3.7f" end define end math ] "
\item " [ math tex define prop three seven g as "
Prop\ 3.7g" end define end math ] "
\item " [ math tex define prop three seven g mark as "
Prop\ 3.7g'" end define end math ] "
\item " [ math tex define prop three seven h as "
Prop\ 3.7h" end define end math ] "
\item " [ math tex define prop three seven i as "
Prop\ 3.7i" end define end math ] "
\item " [ math tex define prop three seven j as "
Prop\ 3.7j" end define end math ] "
\item " [ math tex define prop three seven k as "
Prop\ 3.7k" end define end math ] "
\item " [ math tex define prop three seven k mark as "
Prop\ 3.7k'" end define end math ] "
\item " [ math tex define prop three seven l as "
Prop\ 3.7l" end define end math ] "
\item " [ math tex define prop three seven l mark as "
Prop\ 3.7l'" end define end math ] "
\item " [ math tex define prop three seven m as "
Prop\ 3.7m" end define end math ] "
\item " [ math tex define prop three seven n as "
Prop\ 3.7n" end define end math ] "
\item " [ math tex define prop three seven o as "
Prop\ 3.7o" end define end math ] "
\item " [ math tex define prop three seven p as "
Prop\ 3.7p" end define end math ] "
\item " [ math tex define prop three seven q as "
Prop\ 3.7q" end define end math ] "
\item " [ math tex define prop three seven r as "
Prop\ 3.7r" end define end math ] "
\item " [ math tex define prop three seven s as "
Prop\ 3.7s" end define end math ] "
\item " [ math tex define prop three seven t as "
Prop\ 3.7t" end define end math ] "
\item " [ math tex define prop three seven u as "
Prop\ 3.7u" end define end math ] "
\item " [ math tex define prop three seven u mark as "
Prop\ 3.7u'" end define end math ] "
\item " [ math tex define prop three seven v as "
Prop\ 3.7v" end define end math ] "
\item " [ math tex define prop three seven w as "
Prop\ 3.7w" end define end math ] "
\item " [ math tex define prop three seven x as "
Prop\ 3.7x" end define end math ] "
\item " [ math tex define prop three seven x mark as "
Prop\ 3.7x'" end define end math ] "
\item " [ math tex define prop three seven y as "
Prop\ 3.7y" end define end math ] "
\item " [ math tex define prop three seven y mark as "
Prop\ 3.7y'" end define end math ] "
\item " [ math tex define prop three seven z as "
Prop\ 3.7z" end define end math ] "
\item " [ math tex define prop three seven z mark as "
Prop\ 3.7z'" end define end math ] "
\item " [ math tex define prop three ten as "
Prop\ 3.10" end define end math ] "
\item " [ math tex define prop three ten a as "
Prop\ 3.10a" end define end math ] "
\item " [ math tex define prop three ten b as "
Prop\ 3.10b" end define end math ] "
\item " [ math tex define prop three ten c as "
Prop\ 3.10c" end define end math ] "
\item " [ math tex define prop three ten d as "
Prop\ 3.10d" end define end math ] "
\item " [ math tex define prop three ten e as "
Prop\ 3.10e" end define end math ] "
\item " [ math tex define prop three ten f as "
Prop\ 3.10f" end define end math ] "
\item " [ math tex define prop three ten g as "
Prop\ 3.10g" end define end math ] "
\item " [ math tex define prop three ten h as "
Prop\ 3.10h" end define end math ] "
\item " [ math tex define prop three eleven as "
Prop\ 3.11" end define end math ] "
\item " [ math tex define rule r as "
R" end define end math ] "
\item " [ math tex define rule r one as "
R1" end define end math ] "
\item " [ math tex define rule r two as "
R2" end define end math ] "
\item " [ math tex define rule r three as "
R3" end define end math ] "
\item " [ math tex define rule r four as "
R4" end define end math ] "
\item " [ math tex define rule r five as "
R5" end define end math ] "
\item " [ math tex define rule r six as "
R6" end define end math ] "
\item " [ math tex define conjel1 as "
Con1" end define end math ] "
\item " [ math tex define conjel2 as "
Con2" end define end math ] "
\item " [ math tex define disjin1 as "
Dis1" end define end math ] "
\item " [ math tex define disjin2 as "
Dis2" end define end math ] "
\item " [ math tex define conjin as "
Con" end define end math ] "
\item " [ math tex define t one as "
Lem 1.11c" end define end math ] "
\item " [ math tex define h one as "
Lem 1.11a" end define end math ] "
\item " [ math tex define h two as "
Lem 1.11b" end define end math ] "
\item " [ math tex define h three as "
H3" end define end math ] "
\item " [ math tex define h four as "
Prop 3.2c'" end define end math ] "
\item " [ math tex define h four mark as "
S1''" end define end math ] "
\item " [ math tex define h five as "
Neg'" end define end math ] "
\item " [ math tex define h six as "
Repetition'" end define end math ] "
\item " [ math tex define h seven as "
Lem 1.11e" end define end math ] "
\item " [ math tex define h eight as "
Lem 1.11d" end define end math ] "
\item " [ math tex define h nine as "
Prop 3.2b'" end define end math ] "
\item " [ math tex define h ten as "
H10" end define end math ] "
\item " [ math tex define h eleven as "
H11" end define end math ] "
\item " [ math tex define h twelwe as "
Lem 1.11g" end define end math ] "
\item " [ math tex define h zero a as "
Cor 1.10a" end define end math ] "
\item " [ math tex define h zero b as "
Cor 1.10b" end define end math ] "
\item " [ math tex define modus tollens as "
MT" end define end math ] "
\item " [ math tex define var x ist var y as "#1.
< #2." end define end math ] "
\item " [ math tex define var x istq var y as "#1.
\leq #2." end define end math ] "
\item " [ math tex define var x inst var y as "#1.
\not < #2." end define end math ] "
\item " [ math tex define var x igt var y as "#1.
> #2." end define end math ] "
\item " [ math tex define var x igtq var y as "#1.
\geq #2." end define end math ] "
\item " [ math tex define var x ingt var y as "#1.
\not > #2." end define end math ] "
\item " [ math tex define var x neq var y as "#1.
\neq #2." end define end math ] "
\item " [ math tex define var x ldots as "#1.
\ldots" end define end math ] "
\item " [ math tex define var x and1 var y as "#1.
\wedge #2." end define end math ] "
\item " [ math tex define var x or1 var y as "#1.
\vee #2." end define end math ] "
\end{list}
\subsection{Variables}
\section{Numerals}
" [ math macro define var x neq var y as not parenthesis var x equal var y end parenthesis end define end math ] "
" [ math macro define numeral zero as zero end define end math ] "
" [ math macro define numeral one as zero suc end define end math ] "
" [ math macro define numeral two as zero suc suc end define end math ] "
" [ math macro define numeral three as zero suc suc suc end define end math ] "
" [ math macro define numeral four as zero suc suc suc suc end define end math ] "
" [ math macro define numeral five as zero suc suc suc suc suc end define end math ] "
" [ math macro define numeral six as zero suc suc suc suc suc suc end define end math ] "
" [ math macro define numeral seven as zero suc suc suc suc suc suc suc end define end math ] "
" [ math macro define numeral eight as zero suc suc suc suc suc suc suc suc end define end math ] "
" [ math macro define numeral nine as zero suc suc suc suc suc suc suc suc suc end define end math ] "
" [ math tex define numeral zero as "
\overline{0}" end define end math ] "
" [ math tex define numeral one as "
\overline{1}" end define end math ] "
" [ math tex define numeral two as "
\overline{2}" end define end math ] "
" [ math tex define numeral three as "
\overline{3}" end define end math ] "
" [ math tex define numeral four as "
\overline{4}" end define end math ] "
" [ math tex define numeral five as "
\overline{5}" end define end math ] "
" [ math tex define numeral six as "
\overline{6}" end define end math ] "
" [ math tex define numeral seven as "
\overline{7}" end define end math ] "
" [ math tex define numeral eight as "
\overline{8}" end define end math ] "
" [ math tex define numeral nine as "
\overline{9}" end define end math ] "
" [ math tex define numeral n as "
\overline{n}" end define end math ] "
\section{Priority table}
" [ flush left math priority table preassociative priority opgave equal priority base equal priority bracket x end bracket equal priority big bracket x end bracket equal priority math x end math equal priority flush left x end left equal priority var x equal priority var y equal priority var z equal priority proclaim x as x end proclaim equal priority define x of x as x end define equal priority pyk equal priority tex equal priority tex name equal priority priority equal priority x equal priority true equal priority if x then x else x end if equal priority introduce x of x as x end introduce equal priority value equal priority claim equal priority bottom equal priority function f of x end function equal priority identity x end identity equal priority false equal priority untagged zero equal priority untagged one equal priority untagged two equal priority untagged three equal priority untagged four equal priority untagged five equal priority untagged six equal priority untagged seven equal priority untagged eight equal priority untagged nine equal priority zero equal priority one equal priority two equal priority three equal priority four equal priority five equal priority six equal priority seven equal priority eight equal priority nine equal priority var a equal priority var b equal priority var c equal priority var d equal priority var e equal priority var f equal priority var g equal priority var h equal priority var i equal priority var j equal priority var k equal priority var l equal priority var m equal priority var n equal priority var o equal priority var p equal priority var q equal priority var r equal priority var s equal priority var t equal priority var u equal priority var v equal priority var w equal priority tagged parenthesis x end tagged equal priority tagged if x then x else x end if equal priority array x is x end array equal priority left equal priority center equal priority right equal priority empty equal priority substitute x set x to x end substitute equal priority map tag x end tag equal priority raw map untag x end untag equal priority map untag x end untag equal priority normalizing untag x end untag equal priority apply x to x end apply equal priority apply one x to x end apply equal priority identifier x end identifier equal priority identifier one x plus id x end identifier equal priority array plus x and x end plus equal priority array remove x array x level x end remove equal priority array put x value x array x level x end put equal priority array add x value x index x value x level x end add equal priority bit x of x end bit equal priority bit one x of x end bit equal priority example rack equal priority vector hook equal priority bibliography hook equal priority dictionary hook equal priority body hook equal priority codex hook equal priority expansion hook equal priority code hook equal priority cache hook equal priority diagnose hook equal priority pyk aspect equal priority tex aspect equal priority texname aspect equal priority value aspect equal priority message aspect equal priority macro aspect equal priority definition aspect equal priority unpack aspect equal priority claim aspect equal priority priority aspect equal priority lambda identifier equal priority apply identifier equal priority true identifier equal priority if identifier equal priority quote identifier equal priority proclaim identifier equal priority define identifier equal priority introduce identifier equal priority hide identifier equal priority pre identifier equal priority post identifier equal priority eval x stack x cache x end eval equal priority eval two x ref x id x stack x cache x end eval equal priority eval three x function x stack x cache x end eval equal priority eval four x arguments x stack x cache x end eval equal priority lookup x stack x default x end lookup equal priority abstract x term x stack x cache x end abstract equal priority quote x end quote equal priority expand x state x cache x end expand equal priority expand two x definition x state x cache x end expand equal priority expand list x state x cache x end expand equal priority macro equal priority macro state equal priority zip x with x end zip equal priority assoc one x address x index x end assoc equal priority protect x end protect equal priority self equal priority macro define x as x end define equal priority value define x as x end define equal priority intro define x as x end define equal priority pyk define x as x end define equal priority tex define x as x end define equal priority tex name define x as x end define equal priority priority table x end table equal priority macro define one equal priority macro define two x end define equal priority macro define three x end define equal priority macro define four x state x cache x definition x end define equal priority state expand x state x cache x end expand equal priority quote expand x term x stack x end expand equal priority quote expand two x term x stack x end expand equal priority quote expand three x term x stack x value x end expand equal priority quote expand star x term x stack x end expand equal priority parenthesis x end parenthesis equal priority big parenthesis x end parenthesis equal priority display x end display equal priority statement x end statement equal priority spying test x end test equal priority false spying test x end test equal priority aspect x subcodex x end aspect equal priority aspect x term x cache x end aspect equal priority tuple x end tuple equal priority tuple one x end tuple equal priority tuple two x end tuple equal priority let two x apply x end let equal priority let one x apply x end let equal priority claim define x as x end define equal priority checker equal priority check x cache x end check equal priority check two x cache x def x end check equal priority check three x cache x def x end check equal priority check list x cache x end check equal priority check list two x cache x value x end check equal priority test x end test equal priority false test x end test equal priority raw test x end test equal priority message equal priority message define x as x end define equal priority the statement aspect equal priority statement equal priority statement define x as x end define equal priority example axiom equal priority example scheme equal priority example rule equal priority absurdity equal priority contraexample equal priority example theory primed equal priority example lemma equal priority metavar x end metavar equal priority meta a equal priority meta b equal priority meta c equal priority meta d equal priority meta e equal priority meta f equal priority meta g equal priority meta h equal priority meta i equal priority meta j equal priority meta k equal priority meta l equal priority meta m equal priority meta n equal priority meta o equal priority meta p equal priority meta q equal priority meta r equal priority meta s equal priority meta t equal priority meta u equal priority meta v equal priority meta w equal priority meta x equal priority meta y equal priority meta z equal priority sub x set x to x end sub equal priority sub star x set x to x end sub equal priority the empty set equal priority example remainder equal priority make visible x end visible equal priority intro x index x pyk x tex x end intro equal priority intro x pyk x tex x end intro equal priority error x term x end error equal priority error two x term x end error equal priority proof x term x cache x end proof equal priority proof two x term x end proof equal priority sequent eval x term x end eval equal priority seqeval init x term x end eval equal priority seqeval modus x term x end eval equal priority seqeval modus one x term x sequent x end eval equal priority seqeval verify x term x end eval equal priority seqeval verify one x term x sequent x end eval equal priority sequent eval plus x term x end eval equal priority seqeval plus one x term x sequent x end eval equal priority seqeval minus x term x end eval equal priority seqeval minus one x term x sequent x end eval equal priority seqeval deref x term x end eval equal priority seqeval deref one x term x sequent x end eval equal priority seqeval deref two x term x sequent x def x end eval equal priority seqeval at x term x end eval equal priority seqeval at one x term x sequent x end eval equal priority seqeval infer x term x end eval equal priority seqeval infer one x term x premise x sequent x end eval equal priority seqeval endorse x term x end eval equal priority seqeval endorse one x term x side x sequent x end eval equal priority seqeval est x term x end eval equal priority seqeval est one x term x name x sequent x end eval equal priority seqeval est two x term x name x sequent x def x end eval equal priority seqeval all x term x end eval equal priority seqeval all one x term x variable x sequent x end eval equal priority seqeval cut x term x end eval equal priority seqeval cut one x term x forerunner x end eval equal priority seqeval cut two x term x forerunner x sequent x end eval equal priority computably true x end true equal priority claims x cache x ref x end claims equal priority claims two x cache x ref x end claims equal priority the proof aspect equal priority proof equal priority lemma x says x end lemma equal priority proof of x reads x end proof equal priority in theory x lemma x says x end lemma equal priority in theory x antilemma x says x end antilemma equal priority in theory x rule x says x end rule equal priority in theory x antirule x says x end antirule equal priority verifier equal priority verify one x end verify equal priority verify two x proofs x end verify equal priority verify three x ref x sequents x diagnose x end verify equal priority verify four x premises x end verify equal priority verify five x ref x array x sequents x end verify equal priority verify six x ref x list x sequents x end verify equal priority verify seven x ref x id x sequents x end verify equal priority cut x and x end cut equal priority head x end head equal priority tail x end tail equal priority rule one x theory x end rule equal priority rule x subcodex x end rule equal priority rule tactic equal priority plus x and x end plus equal priority theory x end theory equal priority theory two x cache x end theory equal priority theory three x name x end theory equal priority theory four x name x sum x end theory equal priority example axiom lemma primed equal priority example scheme lemma primed equal priority example rule lemma primed equal priority contraexample lemma primed equal priority example axiom lemma equal priority example scheme lemma equal priority example rule lemma equal priority contraexample lemma equal priority example theory equal priority ragged right equal priority ragged right expansion equal priority parameter term x stack x seed x end parameter equal priority parameter term star x stack x seed x end parameter equal priority instantiate x with x end instantiate equal priority instantiate star x with x end instantiate equal priority occur x in x substitution x end occur equal priority occur star x in x substitution x end occur equal priority unify x with x substitution x end unify equal priority unify star x with x substitution x end unify equal priority unify two x with x substitution x end unify equal priority ell a equal priority ell b equal priority ell c equal priority ell d equal priority ell e equal priority ell f equal priority ell g equal priority ell h equal priority ell i equal priority ell j equal priority ell k equal priority ell l equal priority ell m equal priority ell n equal priority ell o equal priority ell p equal priority ell q equal priority ell r equal priority ell s equal priority ell t equal priority ell u equal priority ell v equal priority ell w equal priority ell x equal priority ell y equal priority ell z equal priority ell big a equal priority ell big b equal priority ell big c equal priority ell big d equal priority ell big e equal priority ell big f equal priority ell big g equal priority ell big h equal priority ell big i equal priority ell big j equal priority ell big k equal priority ell big l equal priority ell big m equal priority ell big n equal priority ell big o equal priority ell big p equal priority ell big q equal priority ell big r equal priority ell big s equal priority ell big t equal priority ell big u equal priority ell big v equal priority ell big w equal priority ell big x equal priority ell big y equal priority ell big z equal priority ell dummy equal priority sequent reflexivity equal priority tactic reflexivity equal priority sequent commutativity equal priority tactic commutativity equal priority the tactic aspect equal priority tactic equal priority tactic define x as x end define equal priority proof expand x state x cache x end expand equal priority proof expand list x state x cache x end expand equal priority proof state equal priority conclude one x cache x end conclude equal priority conclude two x proves x cache x end conclude equal priority conclude three x proves x lemma x substitution x end conclude equal priority conclude four x lemma x end conclude equal priority check equal priority general macro define x as x end define equal priority make root visible x end visible equal priority sequent example axiom equal priority sequent example rule equal priority sequent example contradiction equal priority sequent example theory equal priority sequent example lemma equal priority set x end set equal priority object var x end var equal priority object a equal priority object b equal priority object c equal priority object d equal priority object e equal priority object f equal priority object g equal priority object h equal priority object i equal priority object j equal priority object k equal priority object l equal priority object m equal priority object n equal priority object o equal priority object p equal priority object q equal priority object r equal priority object s equal priority object t equal priority object u equal priority object v equal priority object w equal priority object x equal priority object y equal priority object z equal priority sub x is x where x is x end sub equal priority sub zero x is x where x is x end sub equal priority sub one x is x where x is x end sub equal priority sub star x is x where x is x end sub equal priority deduction x conclude x end deduction equal priority deduction zero x conclude x end deduction equal priority deduction one x conclude x condition x end deduction equal priority deduction two x conclude x condition x end deduction equal priority deduction three x conclude x condition x bound x end deduction equal priority deduction four x conclude x condition x bound x end deduction equal priority deduction four star x conclude x condition x bound x end deduction equal priority deduction five x condition x bound x end deduction equal priority deduction six x conclude x exception x bound x end deduction equal priority deduction six star x conclude x exception x bound x end deduction equal priority deduction seven x end deduction equal priority deduction eight x bound x end deduction equal priority deduction eight star x bound x end deduction equal priority system s equal priority double negation equal priority rule mp equal priority rule gen equal priority deduction equal priority axiom s one equal priority axiom s two equal priority axiom s three equal priority axiom s four equal priority axiom s five equal priority axiom s six equal priority axiom s seven equal priority axiom s eight equal priority axiom s nine equal priority repetition equal priority lemma a one equal priority lemma a two equal priority lemma a four equal priority lemma a five equal priority prop three two a equal priority prop three two b equal priority prop three two c equal priority prop three two d equal priority prop three two e one equal priority prop three two e two equal priority prop three two e equal priority prop three two f one equal priority prop three two f two equal priority prop three two f equal priority prop three two g one equal priority prop three two g two equal priority prop three two g equal priority prop three two h one equal priority prop three two h two equal priority prop three two h equal priority block one x state x cache x end block equal priority block two x end block equal priority numeral zero equal priority numeral one equal priority numeral two equal priority numeral three equal priority numeral four equal priority numeral five equal priority numeral six equal priority numeral seven equal priority numeral eight equal priority numeral nine equal priority numeral n equal priority rule div equal priority rule r equal priority rule r one equal priority rule r two equal priority rule r three equal priority rule r four equal priority rule r five equal priority rule r six equal priority conjel1 equal priority conjel2 equal priority conjin equal priority disjin1 equal priority disjin2 equal priority t one equal priority h zero a equal priority h zero b equal priority h one equal priority h two equal priority h three equal priority h four equal priority h four mark equal priority h five equal priority h six equal priority h seven equal priority h eight equal priority h nine equal priority h ten equal priority h eleven equal priority h twelwe equal priority modus tollens equal priority axiom s ten equal priority prop three two equal priority prop three two i equal priority prop three two j one equal priority prop three two j two equal priority prop three two j equal priority prop three two k one equal priority prop three two k two equal priority prop three two k equal priority prop three two l one equal priority prop three two l two equal priority prop three two l equal priority prop three two m one equal priority prop three two m two equal priority prop three two m equal priority prop three two n one equal priority prop three two n two equal priority prop three two n equal priority prop three two o equal priority prop three four equal priority prop three four a one equal priority prop three four a two equal priority prop three four a equal priority prop three four b equal priority prop three four c one equal priority prop three four c two equal priority prop three four c equal priority prop three four d one equal priority prop three four d two equal priority prop three four d equal priority prop three five equal priority prop three five a equal priority prop three five b equal priority prop three five c equal priority prop three five d one equal priority prop three five d two equal priority prop three five d equal priority prop three five e one equal priority prop three five e two equal priority prop three five e equal priority prop three five f one equal priority prop three five f two equal priority prop three five f equal priority prop three five g one equal priority prop three five g two equal priority prop three five g three equal priority prop three five g four equal priority prop three five g equal priority prop three five h one equal priority prop three five h two equal priority prop three five h equal priority prop three five i one equal priority prop three five i two equal priority prop three five i equal priority prop three five j one equal priority prop three five j two equal priority prop three five j equal priority prop three seven equal priority prop three seven a equal priority prop three seven b equal priority prop three seven c equal priority prop three seven d equal priority prop three seven e equal priority prop three seven f equal priority prop three seven g equal priority prop three seven g mark equal priority prop three seven h equal priority prop three seven i equal priority prop three seven j equal priority prop three seven k equal priority prop three seven k mark equal priority prop three seven l equal priority prop three seven l mark equal priority prop three seven m equal priority prop three seven n equal priority prop three seven o equal priority prop three seven p equal priority prop three seven q equal priority prop three seven r equal priority prop three seven s equal priority prop three seven t equal priority prop three seven u equal priority prop three seven u mark equal priority prop three seven v equal priority prop three seven w equal priority prop three seven x equal priority prop three seven x mark equal priority prop three seven y equal priority prop three seven y mark equal priority prop three seven z equal priority prop three seven z mark equal priority prop three ten equal priority prop three ten a equal priority prop three ten b equal priority prop three ten c equal priority prop three ten d equal priority prop three ten e equal priority prop three ten f equal priority prop three ten g equal priority prop three ten h equal priority prop three eleven end priority greater than preassociative priority x sub x end sub equal priority x intro x index x pyk x tex x end intro equal priority x intro x pyk x tex x end intro equal priority x intro x index x pyk x tex x name x end intro equal priority x intro x pyk x tex x name x end intro equal priority x prime equal priority x assoc x end assoc equal priority x set x to x end set equal priority x set multi x to x end set equal priority x bit nil equal priority x bit one equal priority binary equal priority x color x end color equal priority x color star x end color equal priority x raw head equal priority x raw tail equal priority x cardinal untag equal priority x head equal priority x tail equal priority x is singular equal priority x is cardinal equal priority x is data equal priority x is atomic equal priority x cardinal retract equal priority x tagged retract equal priority x boolean retract equal priority x ref equal priority x id equal priority x debug equal priority x root equal priority x zeroth equal priority x first equal priority x second equal priority x third equal priority x fourth equal priority x fifth equal priority x sixth equal priority x seventh equal priority x eighth equal priority x ninth equal priority x is error equal priority x is metavar equal priority x is metaclosed equal priority x is metaclosed star equal priority x hide end priority greater than preassociative priority unicode start of text x end unicode text equal priority unicode end of text equal priority text x end text equal priority text x plus x equal priority text x plus indent x equal priority unicode newline x equal priority unicode space x equal priority unicode exclamation mark x equal priority unicode quotation mark x equal priority unicode number sign x equal priority unicode dollar sign x equal priority unicode percent x equal priority unicode ampersand x equal priority unicode apostrophe x equal priority unicode left parenthesis x equal priority unicode right parenthesis x equal priority unicode asterisk x equal priority unicode plus sign x equal priority unicode comma x equal priority unicode hyphen x equal priority unicode period x equal priority unicode slash x equal priority unicode zero x equal priority unicode one x equal priority unicode two x equal priority unicode three x equal priority unicode four x equal priority unicode five x equal priority unicode six x equal priority unicode seven x equal priority unicode eight x equal priority unicode nine x equal priority unicode colon x equal priority unicode semicolon x equal priority unicode less than x equal priority unicode equal sign x equal priority unicode greater than x equal priority unicode question mark x equal priority unicode commercial at x equal priority unicode capital a x equal priority unicode capital b x equal priority unicode capital c x equal priority unicode capital d x equal priority unicode capital e x equal priority unicode capital f x equal priority unicode capital g x equal priority unicode capital h x equal priority unicode capital i x equal priority unicode capital j x equal priority unicode capital k x equal priority unicode capital l x equal priority unicode capital m x equal priority unicode capital n x equal priority unicode capital o x equal priority unicode capital p x equal priority unicode capital q x equal priority unicode capital r x equal priority unicode capital s x equal priority unicode capital t x equal priority unicode capital u x equal priority unicode capital v x equal priority unicode capital w x equal priority unicode capital x x equal priority unicode capital y x equal priority unicode capital z x equal priority unicode left bracket x equal priority unicode backslash x equal priority unicode right bracket x equal priority unicode circumflex x equal priority unicode underscore x equal priority unicode grave accent x equal priority unicode small a x equal priority unicode small b x equal priority unicode small c x equal priority unicode small d x equal priority unicode small e x equal priority unicode small f x equal priority unicode small g x equal priority unicode small h x equal priority unicode small i x equal priority unicode small j x equal priority unicode small k x equal priority unicode small l x equal priority unicode small m x equal priority unicode small n x equal priority unicode small o x equal priority unicode small p x equal priority unicode small q x equal priority unicode small r x equal priority unicode small s x equal priority unicode small t x equal priority unicode small u x equal priority unicode small v x equal priority unicode small w x equal priority unicode small x x equal priority unicode small y x equal priority unicode small z x equal priority unicode left brace x equal priority unicode vertical line x equal priority unicode right brace x equal priority unicode tilde x equal priority preassociative x greater than x equal priority postassociative x greater than x equal priority priority x equal x equal priority priority x end priority equal priority newline x equal priority macro newline x equal priority macro indent x end priority greater than preassociative priority x apply x equal priority x tagged apply x end priority greater than preassociative priority x suc end priority greater than preassociative priority x times x equal priority x times zero x end priority greater than preassociative priority x plus x equal priority x plus zero x equal priority x plus one x equal priority x minus x equal priority x minus zero x equal priority x minus one x end priority greater than preassociative priority x term plus x end plus equal priority x term union x equal priority x term minus x end minus end priority greater than postassociative priority x raw pair x equal priority x eager pair x equal priority x tagged pair x equal priority x untagged double x equal priority x pair x equal priority x double x end priority greater than postassociative priority x comma x end priority greater than preassociative priority x boolean equal x equal priority x data equal x equal priority x cardinal equal x equal priority x peano equal x equal priority x tagged equal x equal priority x math equal x equal priority x reduce to x equal priority x term equal x equal priority x term list equal x equal priority x term root equal x equal priority x term in x equal priority x term subset x equal priority x term set equal x equal priority x sequent equal x equal priority x free in x equal priority x free in star x equal priority x free for x in x equal priority x free for star x in x equal priority x claim in x equal priority x less x equal priority x less zero x equal priority x less one x equal priority x equal x equal priority x unequal x equal priority x is object var equal priority x avoid zero x equal priority x avoid one x equal priority x avoid star x equal priority x ist x equal priority x istq x equal priority x inst x equal priority x igt x equal priority x igtq x equal priority x ingt x equal priority x neq x end priority greater than preassociative priority not x end priority greater than preassociative priority x and x equal priority x macro and x equal priority x simple and x equal priority x claim and x equal priority x and1 x end priority greater than preassociative priority x or x equal priority x parallel x equal priority x macro or x equal priority x or1 x end priority greater than preassociative priority exist x indeed x equal priority for all x indeed x equal priority for all objects x indeed x equal priority exists x indeed x end priority greater than postassociative priority x macro imply x equal priority x imply x equal priority x if and only if x end priority greater than postassociative priority x guard x equal priority x spy x equal priority x tagged guard x end priority greater than preassociative priority x select x else x end select end priority greater than preassociative priority lambda x dot x equal priority tagged lambda x dot x equal priority tagging x equal priority open if x then x else x equal priority let x be x in x equal priority let x abbreviate x in x end priority greater than preassociative priority x avoid x end priority greater than preassociative priority x init equal priority x modus equal priority x verify equal priority x curry plus equal priority x curry minus equal priority x dereference end priority greater than preassociative priority x at x equal priority x modus ponens x equal priority x modus probans x equal priority x conclude x equal priority x object modus ponens x end priority greater than postassociative priority x infer x equal priority x endorse x equal priority x id est x end priority greater than preassociative priority all x indeed x equal priority for all terms x indeed x end priority greater than postassociative priority x rule plus x end priority greater than postassociative priority x cut x end priority greater than preassociative priority x proves x end priority greater than preassociative priority x proof of x reads x equal priority line x because x indeed x end line x equal priority because x indeed x qed equal priority line x premise x end line x equal priority line x side condition x end line x equal priority arbitrary x end line x equal priority locally define x as x end line x equal priority block x line x end block x equal priority because x indeed x end line equal priority any term x end line x end priority greater than postassociative priority x alternative x end priority greater than postassociative priority x , x equal priority x [ x ] x end priority greater than preassociative priority x tab x equal priority evaluates to end priority greater than preassociative priority x row x equal priority x linebreak x equal priority x safe row x equal priority x divides x equal priority x ldots end priority greater than unicode end of text end table end math end left ] "
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The pyk compiler, version 0.grue.20060417+ by Klaus Grue,