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\hyphenation{her-ud-over ek-si-stens-va-ri-ab-le an-dre dob-belt-im-pli-ka-ti-on ob-jekt-kvan-tor de-fi-ni-tions-lem-ma ens-be-tyd-en-de
und-er-af-snit slut-ning-en inde-hol-der for-klar-ing-en si-de-be-ting-el-sen
des-uden be-vis-check-er-en}
%\title {}}
%\author {}
%\date{}
%\maketitle
" [ ragged right expansion ] "
(*** MAKROER BEGYNDER ***)
\section{Makrodefinitioner} \label{sec:makro}
Dette afsnit indeholder d\'{e} makrodefinitioner, som vi vil g\o{}re brug af i resten af rapporten. Definitionerne drejer sig for det meste om m\ae{}ngdeteoretiske begreber, f.eks.\ ``\ae{}kvivalensklasse'' og ``partition''. Til sidst i afsnittet formulerer vi hovedresultatet --- at der til enhver \ae{}kvivalensrelation svarer en partition --- som et formelt teorem.
\subsection{Konnektiver} \label{sec:konnektiver}
Ud fra de to basale konnektiver " [ bracket not0 var x end bracket ] " og " [ bracket var x imply var y end bracket ] " definerer vi konjunktion, disjunktion og dobbeltimplikation:
\display{" [ math macro define var x and0 var y as not0 parenthesis var x imply not0 var y end parenthesis end define end math ] "
" [ math macro define var x or0 var y as not0 var x imply var y end define end math ] "
" [ math macro define var x iff var y as parenthesis var x imply var y end parenthesis and0 parenthesis var y imply var x end parenthesis end define end math ] "
}
\subsection{Negerede formler}
Det er ganske enkelt at definere negeret lighed (" [ math var x zermelo ~is var y end math ] ") og negeret medlemskab (" [ math var x zermelo ~in var y end math ] "):
\display{
" [ math macro define var x zermelo ~is var y as not0 var x zermelo is var y end define end math ] "
" [ math macro define var x zermelo ~in var y as not0 var x in0 var y end define end math ] "\footnote{H\o{}jresiderne i disse definitioner skal l\ae{}ses som hhv.\ " [ bracket not0 parenthesis var x zermelo is var y end parenthesis end bracket ] " og " [ bracket not0 parenthesis var x in0 var y end parenthesis end bracket ] ".}
}
\subsection{Delm\ae{}ngde} \label{sec:subset}
M\ae{}ngden " [ math var x end math ] " er en delm\ae{}ngde af " [ math var y end math ] " hviss ethvert medlem af " [ math var x end math ] " ogs\aa{} tilh\o{}rer " [ math var y end math ] ":
\display{
" [ math macro define var x is subset of var y as parenthesis object s in0 var x imply object s in0 var y end parenthesis end define end math ] "
}
\subsection{Singleton-m\ae{}ngde} \label{sec:single}
" [ bracket zermelo singleton var x end singleton end bracket ] " er m\ae{}ngden, der indeholder " [ math var x end math ] " som sit eneste
element. Vi definerer " [ bracket zermelo singleton var x end singleton end bracket ] " ved at parre " [ math var x end math ] " med sig selv:
\display{" [ math macro define zermelo singleton var x end singleton as zermelo pair var x comma var x end pair end define end math ] "}
\subsection{Bin\ae{}r foreningsm\ae{}ngde og f\ae{}llesm\ae{}ngde}
\label{sec:makrobin}
Vi definerer foreningsm\ae{}ngden mellem to m\ae{}ngder " [ math var x end math ] " og " [ math var y end math ] " som f\o{}lger:
\display{
" [ math macro define binary-union var x comma var y end union as union zermelo pair zermelo singleton var x end singleton comma zermelo singleton var y end singleton end pair end union end define end math ] "
}
F\ae{}llesm\ae{}ngden mellem to m\ae{}ngder " [ math var x end math ] " og " [ math var y end math ] " er en delm\ae{}ngde af deres foreningsm\ae{}ngde:
\verb| |
\display{
" [ math macro define intersection var x comma var y end intersection as the set of ph in binary-union var x comma var y end union such that placeholder-var3 in0 var x and0 placeholder-var3 in0 var y end set end define end math ] "
}
\subsection{Relation} \label{sec:relation}
Det ordnede par " [ math zermelo ordered pair var x comma var y end pair end math ] " indeholder " [ math var x end math ] " som ``f\o{}rstekomponent'' og " [ math var y end math ] " som ``andenkomponent''. Den f\o{}lgende definition af " [ math zermelo ordered pair var x comma var y end pair end math ] " er den mest udbredte i litteraturen (se f.eks.\ afsnit 4.3 i \cite{kn:gold} og afsnit 2.1 i \cite{kn:hrba}):
\verb| |
\display{" [ math macro define zermelo ordered pair var x comma var y end pair as zermelo pair zermelo singleton var x end singleton comma zermelo pair var x comma var y end pair end pair end define end math ] "}
Vi kan nu definere en ``relation'' som en m\ae{}ngde af ordnede par. Vi udtrykker denne definition ved at formalisere, hvad det vil sige, at " [ math var x end math ] " er relateret til " [ math var y end math ] " i kraft af relationen " [ math var r end math ] ":
\verb| |
\display{" [ math macro define var x is related to var y under var r as zermelo ordered pair var x comma var y end pair in0 var r end define end math ] "}
Vi kommer faktisk ikke til at bruge disse to definitioner i rapporten; beviserne vil behandle " [ bracket var x is related to var y under var r end bracket ] " som en primitiv konstruktion. Men det er alligevel betryggende at have det formelle grundlag for relationsbegrebet p\aa{} plads.
\newpage
\subsection{\AE{}kvivalensrelation} \label{sec:eqrel}
At en relation er refleksiv p\aa{} en m\ae{}ngde " [ math var x end math ] " vil sige, at alle elementer i " [ math var x end math ] " er relateret til sig selv:
\display{" [ math macro define var r is reflexive relation in var x as for all object s indeed parenthesis object s in0 var x imply object s is related to object s under var r end parenthesis end define end math ] "}
At en relation er symmetrisk p\aa{} en m\ae{}ngde " [ math var x end math ] " vil sige, at alle elementer i " [ math var x end math ] " opfylder den f\o{}lgende implikation:
\verb| |
\display{" [ math macro define var r is symmetric relation in var x as for all object s comma object t indeed parenthesis object s in0 var x imply object t in0 var x imply object s is related to object t under var r imply object t is related to object s under var r end parenthesis end define end math ] "}
At en relation er transitiv p\aa{} en m\ae{}ngde " [ math var x end math ] " vil sige, at alle elementer i " [ math var x end math ] " opfylder den f\o{}lgende implikation:
\verb| |
\display{" [ math macro define var r is transitive relation in var x as macro newline for all object s comma object t comma object u indeed parenthesis object s in0 var x imply object t in0 var x imply object u in0 var x imply object s is related to object t under var r imply object t is related to object u under var r imply object s is related to object u under var r end parenthesis end define end math ] "}
Endelig er en \ae{}kvivalensrelation det samme som en relation, der er refleksiv, symmetrisk og transitiv:
\verb| |
\display{" [ math macro define var r is equivalence relation in var x as var r is reflexive relation in var x and0 var r is symmetric relation in var x and0 var r is transitive relation in var x end define end math ] "}
\subsection{M\ae{}ngde-variable}
Mange af rapportens beviser sker i forhold til en uspecificeret m\ae{}ngde. Vi vil referere til denne m\ae{}ngde med metavariablen " [ math meta big set end math ] " og objektvariablen " [ math object big set end math ] ":
\display{
" [ math macro define meta big set as metavar var big set end metavar end define end math ] "
" [ math macro define object big set as object var var big set end var end define end math ] "\footnote{Navnene ``" [ math meta big set end math ] "'' og ``" [ math object big set end math ] "'' st\aa{}r for hhv.\ for ``big set'' og ``object big set''. Konstruktionerne " [ bracket metavar var x end metavar end bracket ] " og " [ bracket object var var x end var end bracket ] " omdanner " [ math var x end math ] " til hhv.\ en meta- og en objektvariabel. Variablen " [ bracket var big set end bracket ] " vil ogs\aa{} blive brugt i nogle af de kommende definitioner, men ikke i selve beviserne.}
}
\noindent Vi vil s\aa{} vidt muligt bruge metavariablen, men i afsnit \ref{sec:sameinter} og senere bliver det n\o{}dvendigt at g\aa{} over til objektvariablen.
\verb| |
\subsection{\AE{}kvivalensklasse} \label{sec:eqclass}
Lad " [ math var r end math ] " v\ae{}re en \ae{}kvivalensrelation defineret p\aa{} " [ math var big set end math ] ", og lad " [ math var x end math ] " v\ae{}re et medlem af " [ math var big set end math ] ". Vi definerer \ae{}kvivalensklassen " [ math equivalence class of var x in var big set modulo var r end math ] " som den delm\ae{}ngde af " [ math var big set end math ] ", hvis medlemmer st\aa{}r i forhold til " [ math var x end math ] ":
\display{
" [ math macro define equivalence class of var x in var big set modulo var r as the set of ph in var big set such that placeholder-var1 is related to var x under var r end set end define end math ] "
}
\AE{}kvivalenssystemet " [ math eq-system of var big set modulo var r end math ] " er m\ae{}ngden af alle de \ae{}kvivalensklasser, som " [ math var big set end math ] " definerer p\aa{} " [ math var r end math ] ". Vi definerer " [ math eq-system of var big set modulo var r end math ] " som en delm\ae{}ngde af potensm\ae{}ngden " [ math power var big set end power end math ] ":
\verb| |
\display{
" [ math macro define eq-system of var big set modulo var r as the set of ph in power var big set end power such that ex20 in0 var big set and0 equivalence class of ex20 in var big set modulo var r zermelo is placeholder-var2 end set end define end math ] "
}
\newpage
\subsection{Partition} \label{sec:partition}
En partition af en m\ae{}ngde " [ math var big set end math ] " er en m\ae{}ngde " [ math var p end math ] ", som opfylder tre krav:
%
\begin{enumerate}
\item Ingen af m\ae{}ngderne i " [ math var p end math ] " er tomme.
\item Alle m\ae{}ngderne i " [ math var p end math ] " er indbyrdes disjunkte.
\item Foreningsm\ae{}ngden af alle m\ae{}ngderne i " [ math var p end math ] " er lig med " [ math var big set end math ] ".
\end{enumerate}
%
Den formelle version af denne definition ser s\aa{}ledes ud:
\display{
" [ math macro define var p is partition of var big set as parenthesis for all object s indeed parenthesis object s in0 var p imply object s zermelo ~is zermelo empty set end parenthesis end parenthesis and0 macro newline parenthesis for all object s comma object t indeed parenthesis object s in0 var p imply object t in0 var p imply object s zermelo ~is object t imply intersection object s comma object t end intersection zermelo is zermelo empty set end parenthesis end parenthesis and0 macro newline union var p end union zermelo is var big set end define end math ] "
}
(*** MAKROER SLUTTER ***)
\section{Deduktionsreglen} \label{sec:deduktion}
Dette bilag pr\ae{}senterer d\'{e}n version af deduktionsreglen fra \cite{kn:check}, som jeg har gjort brug af. Underafsnit \ref{sec:motivering} forklarer, hvorfor jeg har \ae{}ndret p\aa{} den oprindelige regel, og underafsnit \ref{sec:kode} indeholder selve den \ae{}ndrede kode (som er skrevet i L).
\subsection{Kode} \label{sec:kode}
Funktionen " [ bracket 1deduction var p conclude var c end 1deduction end bracket ] " er en kopi af " [ bracket deduction var p conclude var c end deduction end bracket ] " fra \cite{kn:check}:
\begin{list}{}{
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\item " [ math macro define 1deduction var p conclude var c end 1deduction as lambda var x dot 1deduction zero quote var p end quote conclude quote var c end quote end 1deduction end define end math ] "
% -- aendring begynder
\item Jeg har \ae{}ndret funktionen " [ bracket deduction zero var p conclude var c end deduction end bracket ] ", s\aa{} den kalder " [ bracket 1deduction side 1deduction seven var p end 1deduction conclude var c condition true end 1deduction end bracket ] " i stedet for " [ bracket deduction one deduction seven var p end deduction conclude var c condition true end deduction end bracket ] ":
\item " [ math value define 1deduction zero var p conclude var c end 1deduction as var c tagged guard 1deduction eight var p bound true end 1deduction macro and 1deduction side 1deduction seven var p end 1deduction conclude var c condition true end 1deduction end define end math ] "
\item Funktionen " [ bracket 1deduction side var p conclude var c condition var s end 1deduction end bracket ] " giver straks kontrollen videre til " [ bracket deduction one var p conclude var c condition var s end deduction end bracket ] " --- medmindre " [ math var p end math ] " og " [ math var c end math ] " begynder med et antal identiske sidebetingelser. I s\aa{} fald flyttes disse sidebetingelser fra " [ math var p end math ] " og " [ math var c end math ] " over til listen " [ math var s end math ] ", f\o{}r kontrollen g\aa{}r videre til " [ bracket deduction one var p conclude var c condition var s end deduction end bracket ] ":
\item " [ math value define 1deduction side var p conclude var c condition var s end 1deduction as open if var p term root equal quote var x endorse var y end quote then macro newline var c term root equal quote var x endorse var y end quote and var p first term equal var c first and 1deduction side var p second conclude var c second condition var c first pair var s end 1deduction else macro newline 1deduction one var p conclude var c condition var s end 1deduction end define end math ] "
% -- aendring slutter
\item Fra og med " [ bracket deduction one var p conclude var c condition var s end deduction end bracket ] " er koden kopieret fra appendikset til \cite{kn:check}:
\item
" [ math value define 1deduction one var p conclude var c condition var s end 1deduction as open if var c term root equal quote var x endorse var y end quote then 1deduction one var p conclude var c second condition var c first pair var s end 1deduction else 1deduction two var p conclude var c condition var s end 1deduction end define end math ] "
\item " [ math value define 1deduction two var p conclude var c condition var s end 1deduction as var s tagged guard macro newline var p term root equal quote var x infer var y end quote and var c term root equal quote var x imply var y end quote select 1deduction three var p first conclude var c first condition var s bound true end 1deduction and 1deduction two var p second conclude var c second condition var s end 1deduction else 1deduction four var p conclude var c condition var s bound 1deduction six var p conclude var c exception true bound true end 1deduction end 1deduction end select end define end math ] "
\item " [ math value define 1deduction three var p conclude var c condition var s bound var b end 1deduction as open if not var c term root equal quote for all var x indeed var y end quote then 1deduction four var p conclude var c condition var s bound var b end 1deduction else macro newline open if var p term root equal quote for all var x indeed var y end quote and var p first term equal var c first then 1deduction four var p conclude var c condition var s bound var b end 1deduction else macro newline 1deduction three var p conclude var c second condition var s bound parenthesis var c first pair var c first end parenthesis pair var b end 1deduction end define end math ] "
\item " [ math value define 1deduction four var p conclude var c condition var s bound var b end 1deduction as var s tagged guard var b tagged guard macro newline open if var p term root equal quote object x end quote then lookup var p stack var b default true end lookup term equal var c else macro newline open if not var p term root equal var c then false else macro newline open if var p term root equal quote for all var x indeed var y end quote then var p first term equal var c first and 1deduction four var p second conclude var c second condition var s bound parenthesis var p first pair var p first end parenthesis pair var b end 1deduction else macro newline open if not var p term root equal quote meta x end quote then 1deduction four star var p tail conclude var c tail condition var s bound var b end 1deduction else macro newline var p first term equal var c first and 1deduction five var p condition var s bound var b end 1deduction end define end math ] "
\item " [ math value define 1deduction four star var p conclude var c condition var s bound var b end 1deduction as var c tagged guard var s tagged guard var b tagged guard open if var p then true else 1deduction four var p head conclude var c head condition var s bound var b end 1deduction and 1deduction four star var p tail conclude var c tail condition var s bound var b end 1deduction end define end math ] "
\item " [ math value define 1deduction five var p condition var s bound var b end 1deduction as var p tagged guard var s tagged guard open if var b then true else macro newline tuple quote var x avoid var y end quote head comma tuple quote quote x end quote end quote head comma var b head head end tuple comma tuple quote quote var x end quote end quote head comma var p end tuple end tuple term in var s and 1deduction five var p condition var s bound var b tail end 1deduction end define end math ] "
\item " [ math value define 1deduction six var p conclude var c exception var e bound var b end 1deduction as var p tagged guard var c tagged guard var b tagged guard var e tagged guard macro newline open if var p term root equal quote object x end quote then var p term in var e select var b else parenthesis var p pair var c end parenthesis pair var b end select else macro newline open if not var p term root equal var c then true else macro newline open if var p term root equal quote meta a end quote then var b else macro newline open if var p term root equal quote for all var x indeed var y end quote then 1deduction six var p second conclude var c second exception var c first pair var e bound var b end 1deduction else macro newline 1deduction six star var p tail conclude var c tail exception var e bound var b end 1deduction end define end math ] "
\item " [ math value define 1deduction six star var p conclude var c exception var e bound var b end 1deduction as var p tagged guard var c tagged guard var b tagged guard var e tagged guard open if var p then var b else 1deduction six star var p tail conclude var c tail exception var e bound 1deduction six var p head conclude var c head exception var e bound var b end 1deduction end 1deduction end define end math ] "
\item " [ math value define 1deduction seven var p end 1deduction as var p term root equal quote for all terms var x indeed var y end quote select 1deduction seven var p second end 1deduction else var p end select end define end math ] "
\item " [ math value define 1deduction eight var p bound var b end 1deduction as macro newline open if var p term root equal quote for all terms var x indeed var y end quote then 1deduction eight var p second bound var p first pair var b end 1deduction else macro newline open if var p term root equal quote meta a end quote then var p term in var b else 1deduction eight star var p tail bound var b end 1deduction end define end math ] "
\item " [ math value define 1deduction eight star var p bound var b end 1deduction as var b tagged guard open if var p then true else 1deduction eight var p head bound var b end 1deduction macro and 1deduction eight star var p tail bound var b end 1deduction end define end math ] "
\end{list}
(*** EKSISTENS-VARIABLE ***)
\display{" [ math value define var x is existential var as var x term root equal quote existential var var x end var end quote end define end math ] "
}
\noindent Vi kan da definere de fire eksistens-variable, som denne rapport vil g\o{}re brug af (jvf.\ bilag \ref{sec:variable}):
\verb| |
\display{
" [ math macro define ex1 as existential var var a end var end define end math ] "
}
" [ math macro define ex2 as existential var var b end var end define end math ] "
" [ math macro define ex10 as existential var var j end var end define end math ] "
" [ math macro define ex20 as existential var var t end var end define end math ] "
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math macro define exist-sub var a is var b where var x is var t end sub as exist-sub0 quote var a end quote is quote var b end quote where quote var x end quote is quote var t end quote end sub end define end math ] "
\item " [ math value define exist-sub0 var a is var b where var x is var t end sub as lambda var c dot var x is existential var and exist-sub1 var a is var b where var x is var t end sub end define end math ] "
\item " [ math value define exist-sub1 var a is var b where var x is var t end sub as var a tagged guard var x tagged guard var t tagged guard newline open if var b term root equal quote for all var u indeed var v end quote then false else newline open if var b is existential var and var b term equal var x then var a term equal var t else newline var a term root equal var b macro and exist-sub* var a tail is var b tail where var x is var t end sub end define end math ] "
\item " [ math value define exist-sub* var a is var b where var x is var t end sub as var b tagged guard var x tagged guard var t tagged guard tagged if var a then true else exist-sub1 var a head is var b head where var x is var t end sub macro and exist-sub* var a tail is var b tail where var x is var t end sub end if end define end math ] "
\end{list}
(*** AKSIOMATISK SYSTEM ***)
" [ math theory system Q end theory end math ] "
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system Q rule 1rule 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 " [ math in theory system Q rule 1rule 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 ] "
\item " [ math in theory system Q rule 1rule repetition says for all terms meta a indeed meta a infer meta a end rule end math ] "
\item " [ math in theory system Q rule 1rule ad absurdum says for all terms meta a comma meta b indeed not0 meta b imply meta a infer not0 meta b imply not0 meta a infer meta b end rule end math ] "
\item " [ math in theory system Q rule 1rule deduction says for all terms meta a comma meta b indeed 1deduction meta a conclude meta b end 1deduction endorse meta a infer meta b end rule end math ] "
\item " [ math in theory system Q rule 1rule exist intro says for all terms meta x comma meta t comma meta a comma meta b indeed exist-sub meta a is meta b where meta x is meta t end sub endorse meta a infer meta b end rule end math ] "
\item " [ math in theory system Q rule axiom extensionality says for all terms meta x comma meta y indeed meta x zermelo is meta y iff for all object s indeed parenthesis object s in0 meta x iff object s in0 meta y end parenthesis end rule end math ] "
\item " [ math in theory system Q rule axiom empty set says for all terms meta s indeed not0 meta s in0 zermelo empty set end rule end math ] "
\item " [ math in theory system Q rule axiom pair definition says for all terms meta s comma meta x comma meta y indeed meta s in0 zermelo pair meta x comma meta y end pair iff meta s zermelo is meta x or0 meta s zermelo is meta y end rule end math ] "
\item " [ math in theory system Q rule axiom union definition says for all terms meta s comma meta x indeed meta s in0 union meta x end union iff parenthesis meta s in0 ex10 and0 ex10 in0 meta x end parenthesis end rule end math ] "
\item " [ math in theory system Q rule axiom power definition says for all terms meta s comma meta x indeed meta s in0 power meta x end power iff for all object s indeed parenthesis object s in0 meta s imply object s in0 meta x end parenthesis end rule end math ] "
\item " [ math in theory system Q rule axiom separation definition says for all terms meta a comma meta b comma meta p comma meta x comma meta z indeed meta p is placeholder-var and ph-sub meta b is meta a where meta p is meta z end sub endorse macro newline meta z in0 the set of ph in meta x such that meta a end set iff meta z in0 meta x and0 meta b end rule end math ] "
\end{list}
\section{Udsagnslogisk bibliotek} \label{sec:udsagn}
I dette afsnit vil jeg bevise en samling af udsagnslogiske sandheder ( eller ``tautologier''), som vil blive brugt i de f\o{}lgende afsnit. De fleste af disse tautologier har mange andre anvendelser end lige netop m\ae{}ngdel\ae{}re. Beviserne er fordelt p\aa{} syv underafsnit; figur 1 giver et overblik over, hvordan beviserne forholder sig til hinanden. Jeg vil kommentere de fleste af beviserne; dog er nogle af dem s\aa{} tekniske, at jeg har ladet dem st\aa{} alene.
\subsection{MP-lemmaer} \label{sec:mplemmaer}
Man f\aa{}r ofte brug for at anvende slutningsreglen " [ math 1rule mp end math ] " flere gange i tr\ae{}k. Derfor vil jeg begynde med at vise fire lemmaer, der kan klare mellem 2 og 5 anvendelser af " [ math 1rule mp end math ] "\footnote{I afsnit \ref{sec:transdef} f\aa{}r vi faktisk brug for at anvende " [ math 1rule mp end math ] " 6 gange i tr\ae{}k; men et eller andet sted skal man jo stoppe.}:
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma mp2 says for all terms meta a comma meta b comma meta c indeed meta a imply meta b imply meta c infer meta a infer meta b infer meta c end lemma end math ] "
\item " [ math in theory system Q lemma prop lemma mp3 says for all terms meta a comma meta b comma meta c comma meta d indeed meta a imply meta b imply meta c imply meta d infer meta a infer meta b infer meta c infer meta d end lemma end math ] "
\item " [ math in theory system Q lemma prop lemma mp4 says for all terms meta a comma meta b comma meta c comma meta d comma meta e indeed macro newline meta a imply meta b imply meta c imply meta d imply meta e infer meta a infer meta b infer meta c infer meta d infer meta e end lemma end math ] "
\item " [ math in theory system Q lemma prop lemma mp5 says for all terms meta a comma meta b comma meta c comma meta d comma meta e comma meta f indeed macro newline meta a imply meta b imply meta c imply meta d imply meta e imply meta f infer meta a infer meta b infer meta c infer meta d infer meta e infer meta f end lemma end math ] "
\end{list}
\subsubsection{Det f\o{}rste bevis}
Vi begynder med at bevise " [ math prop lemma mp2 end math ] ":
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma mp2 says for all terms meta a comma meta b comma meta c indeed meta a imply meta b imply meta c infer meta a infer meta b infer meta c end lemma end math ] "
\item " [ math system Q proof of prop lemma mp2 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 a end line line ell c premise meta b end line line ell d because 1rule mp modus ponens ell a modus ponens ell b indeed meta b imply meta c end line because 1rule mp modus ponens ell d modus ponens ell c indeed meta c qed end math ] "
\end{list}
\noindent Da dette er rapportens f\o{}rste bevis, vil jeg bringe nogle ekstra kommentarer\footnote{Denne beskrivelse er en revideret udgave af afsnit 5.1 i \cite{kn:peano}.}. Oven over beviset har jeg gentaget definitionen af det, der skal bevises; dette er kun for overblikkets skyld --- det er ikke en formel n\o{}dvendighed. Selve beviset for " [ math prop lemma mp2 end math ] " best\aa{}r af seks linier, nummereret fra 1 til 6. En bevislinie kan have to former. Den f\o{}rste form er:
%
\begin{eqnarray*}
\texttt{Argumentation} & \gg & \texttt{Konklusion} %\label{eq:linie}
\end{eqnarray*}
%
hvor \texttt{Konklusion} er det som linien beviser, mens teksten i \texttt{Argumentation} udg\o{}r en begrundelse for, at \texttt{Konklusion} g\ae{}lder. F.eks.\ siger linie 5, at meta-formlen \mbox{" [ bracket meta b imply meta c end bracket ] "} g\ae{}lder, fordi den kan udledes fra slutningsreglen " [ math 1rule mp end math ] " ved substitution. Argumentationen skal l\ae{}ses p\aa{} den m\aa{}de, at konklusionerne fra linie 2 og 3 bliver brugt som pr\ae{}misser til " [ math 1rule mp end math ] ". Den generelle betydning af konstruktionen " [ bracket var x modus ponens var y end bracket ] " er, at konklusionen fra linie " [ math var y end math ] " bliver brugt som pr\ae{}mis i forhold til " [ math var x end math ] ".
Den anden form, en bevislinie kan have, er:
%
\begin{eqnarray*}
\texttt{N\o{}gleord} & \gg & \texttt{Konklusion} %\label{eq:linie2}
\end{eqnarray*}
%
hvor \texttt{N\o{}gleord} er et af de tre ord ``Arbitrary'', ``Premise'' eller ``Side-condition''. Betydningen af ordene ``Premise'' og ``Side-condition'' er \aa{}benlys: De angiver, at liniens konklusion indg\aa{}r som en pr\ae{}mis (hhv.\ sidebetingelse) i den s\ae{}tning, der skal bevises. F.eks.\ siger bevisets linie 2, at " [ math prop lemma mp2 end math ] " bruger meta-formlen \\ \mbox{" [ bracket meta a imply meta b imply meta c end bracket ] "}
som pr\ae{}mis. N\aa{}r ordet ``Arbitrary'' bruges, best\aa{}r konklusionen af en liste af meta-variable (f.eks. " [ bracket meta a comma meta b comma meta c end bracket ] " i linie 1). Ideen hermed er at udtrykke, at vi ikke antager noget om de p\aa{}g\ae{}ldende meta-variable, og at vi derfor har ret til at binde dem med en meta-alkvantor i den s\ae{}tning, der skal bevises. I det forh\aa{}ndenv\ae{}rende bevis berettiger linien med ``Arbitrary'' alts\aa{}, at " [ math prop lemma mp2 end math ] " er kvantificeret med " [ bracket for all terms meta a comma meta b comma meta c indeed cdots end bracket ] ".
Alt dette har drejet sig om den formelle syntaks for et Logiweb bevis. Der er ikke s\aa{} meget at sige om selve beviset for " [ math prop lemma mp2 end math ] "; vi indkapsler simpelthen to p\aa{} hinanden f\o{}lgende anvendelser af " [ math 1rule mp end math ] ".
\subsubsection{Beviser for de andre MP-lemmaer}
Beviserne for de \o{}vrige MP-lemmaer er lige ud ad landevejen:
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma mp3 says for all terms meta a comma meta b comma meta c comma meta d indeed meta a imply meta b imply meta c imply meta d infer meta a infer meta b infer meta c infer meta d end lemma end math ] "
\item
" [ math system Q proof of prop lemma mp3 reads any term meta a comma meta b comma meta c comma meta d end line line ell q premise meta a imply meta b imply meta c imply meta d end line line ell a premise meta a end line line ell b premise meta b end line line ell c premise meta c end line line ell d because prop lemma mp2 modus ponens ell q modus ponens ell a modus ponens ell b indeed meta c imply meta d end line because 1rule mp modus ponens ell d modus ponens ell c indeed meta d qed end math ] "
" [ math in theory system Q lemma prop lemma mp4 says for all terms meta a comma meta b comma meta c comma meta d comma meta e indeed macro newline meta a imply meta b imply meta c imply meta d imply meta e infer meta a infer meta b infer meta c infer meta d infer meta e end lemma end math ] "
" [ math system Q proof of prop lemma mp4 reads any term meta a comma meta b comma meta c comma meta d comma meta e end line line ell q premise meta a imply meta b imply meta c imply meta d imply meta e end line line ell a premise meta a end line line ell b premise meta b end line line ell c premise meta c end line line ell d premise meta d end line line ell r because prop lemma mp2 modus ponens ell q modus ponens ell a modus ponens ell b indeed meta c imply meta d imply meta e end line because prop lemma mp2 modus ponens ell r modus ponens ell c modus ponens ell d indeed meta e qed end math ] "
" [ math in theory system Q lemma prop lemma mp5 says for all terms meta a comma meta b comma meta c comma meta d comma meta e comma meta f indeed macro newline meta a imply meta b imply meta c imply meta d imply meta e imply meta f infer meta a infer meta b infer meta c infer meta d infer meta e infer meta f end lemma end math ] "
" [ math system Q proof of prop lemma mp5 reads any term meta a comma meta b comma meta c comma meta d comma meta e comma meta f end line line ell q premise meta a imply meta b imply meta c imply meta d imply meta e imply meta f end line line ell a premise meta a end line line ell b premise meta b end line line ell c premise meta c end line line ell d premise meta d end line line ell e premise meta e end line line ell r because prop lemma mp3 modus ponens ell q modus ponens ell a modus ponens ell b modus ponens ell c indeed meta d imply meta e imply meta f end line because prop lemma mp2 modus ponens ell r modus ponens ell d modus ponens ell e indeed meta f qed end math ] "
\end{list}
\subsection{ Implikation }
Dette afsnit indeholder en r\ae{}kke lemmaer vedr.\ implikation, grupperet i fire under-underafsnit.
\subsubsection{Refleksivitet; blok-konstruktionen}
Lemmaet " [ math prop lemma auto imply end math ] " udsiger, at implikations-relationen er refleksiv:
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma auto imply says for all terms meta a indeed meta a imply meta a end lemma end math ] "
\item " [ math system Q proof of prop lemma auto imply reads block any term macro indent meta a end line line ell a premise macro indent meta a end line because 1rule repetition modus ponens ell a indeed macro indent meta a end line line ell b end block any term meta a end line because 1rule deduction modus ponens ell b indeed meta a imply meta a qed end math ] "
\item Beviset for " [ math prop lemma auto imply end math ] " indeholder to nye ting i forhold til de hidtidige beviser: En bevisblok, og en anvendelse af deduktions-reglen. En bevisblok er selvst\ae{}ndig enhed i et bevis; den afh\ae{}nger ikke af den \o{}vrige del af beviset. Den ovenst\aa{}ende bevisblok indeholder et bevis for lemmaet " [ bracket for all terms meta a indeed meta a infer meta a end bracket ] ". Pointen er nu, at blokkens sidste linie (linie 5) fungerer som en forkortelse for dette lemma. Vi kan da anvende deduktionsreglen p\aa{} denne linie til at omdanne inferensen " [ bracket for all terms meta a indeed meta a infer meta a end bracket ] " til implikationen " [ bracket meta a imply meta a end bracket ] ". Det vigtigste form\aa{}l med deduktionsreglen er netop, at vi let kan skifte fra inferens til implikation.
\end{list}
\subsubsection{Transitivitet}
Lemmaet " [ math prop lemma imply transitivity end math ] " udsiger, at implikations-relationen er transitiv:
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma imply transitivity says for all terms meta a comma meta b comma meta c indeed meta a imply meta b infer meta b imply meta c infer meta a imply meta c end lemma end math ] "
\item Vi viser " [ math prop lemma imply transitivity end math ] " ved hj\ae{}lp af " [ math 1rule mp end math ] " og deduktionsreglen:
\item " [ math system Q proof of prop lemma imply transitivity reads 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 1rule mp modus ponens ell a modus ponens ell c indeed macro indent meta b end line because 1rule mp modus ponens ell b modus ponens ell d indeed macro indent meta c end line line ell e end block any term meta a comma meta b comma meta c end line line ell f premise meta a imply meta b end line line ell g premise meta b imply meta c end line line ell h because 1rule 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 because prop lemma mp2 modus ponens ell h modus ponens ell f modus ponens ell g indeed meta a imply meta c qed end math ] "
\end{list}
\subsubsection{Sv\ae{}kkelse}
Vi f\aa{}r ofte brug for det f\o{}lgende r\ae{}sonnement: Hvis formlen " [ math meta a end math ] " g\ae{}lder ubetinget, s\aa{} g\ae{}lder den ogs\aa{} under antagelse af en vilk\aa{}rlig anden formel " [ math meta b end math ] ". Lemmaet " [ math prop lemma weakening end math ] " udtrykker dette r\ae{}sonnement som f\o{}lger:
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma weakening says for all terms meta a comma meta b indeed meta b infer meta a imply meta b end lemma end math ] "
\item Vi beviser " [ math prop lemma weakening end math ] " ved hj\ae{}lp af deduktionsreglen:
\item " [ math system Q proof of prop lemma weakening reads block any term macro indent meta a comma meta b end line line ell b premise macro indent meta b end line line ell a premise macro indent meta a end line because 1rule repetition modus ponens ell b indeed macro indent meta b end line line ell c end block any term meta a comma meta b end line line ell e because 1rule deduction modus ponens ell c indeed meta b imply meta a imply meta b end line line ell d premise meta b end line because 1rule mp modus ponens ell e modus ponens ell d indeed meta a imply meta b qed end math ] "
\end{list}
\subsubsection{Modsigelse}
Det sidste lemma i dette afsnit vedr\o{}rer strengt taget ikke implikation, men derimod inferens (" [ math var x infer var y end math ] "). Lemmaet " [ math prop lemma from contradiction end math ] " udsiger, at vi kan bevise hvad som helst, hvis vi har bevist to formler, der modsiger hinanden:
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma from contradiction says for all terms meta a comma meta b indeed meta a infer not0 meta a infer meta b end lemma end math ] "
\item Beviset bruger " [ math prop lemma weakening end math ] " og slutningsreglen " [ math 1rule ad absurdum end math ] ":
\item " [ math system Q proof of prop lemma from contradiction reads any term meta a comma meta b end line line ell a premise meta a end line line ell b premise not0 meta a end line line ell c because prop lemma weakening modus ponens ell a indeed not0 meta b imply meta a end line line ell d because prop lemma weakening modus ponens ell b indeed not0 meta b imply not0 meta a end line because 1rule ad absurdum modus ponens ell c modus ponens ell d indeed meta b qed end math ] "
\end{list}
\subsection{H\aa{}ndtering af dobbeltnegationer}
De to lemmaer " [ math prop lemma remove double neg end math ] " og " [ math prop lemma add double neg end math ] " tillader os hhv.\ at fjerne og tilf\o{}je dobbeltnegationer. Jeg vil ikke kommentere beviserne:
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma remove double neg says for all terms meta a indeed not0 not0 meta a infer meta a end lemma end math ] "
\item " [ math system Q proof of prop lemma remove double neg reads any term meta a end line line ell a premise not0 not0 meta a end line line ell b because prop lemma weakening modus ponens ell a indeed not0 meta a imply not0 not0 meta a end line line ell e because prop lemma auto imply indeed not0 meta a imply not0 meta a end line because 1rule ad absurdum modus ponens ell e modus ponens ell b indeed meta a qed end math ] "
\item " [ math in theory system Q lemma prop lemma add double neg says for all terms meta a indeed meta a infer not0 not0 meta a end lemma end math ] "
\item " [ math system Q proof of prop lemma add double neg reads block any term macro indent meta a end line line ell b premise macro indent not0 not0 not0 meta a end line because prop lemma remove double neg modus ponens ell b indeed macro indent not0 meta a end line line ell d end block any term meta a end line line ell e because 1rule deduction modus ponens ell d indeed not0 not0 not0 meta a imply not0 meta a end line line ell a premise meta a end line line ell q because prop lemma weakening modus ponens ell a indeed not0 not0 not0 meta a imply meta a end line because 1rule ad absurdum modus ponens ell q modus ponens ell e indeed not0 not0 meta a qed end math ] "
\end{list}
\subsection{Modus tollens og besl\ae{}gtede lemmaer}
Hovedresultatet fra dette afsnit er slutningsreglen modus tollens, bevist som et lemma:
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma mt says for all terms meta a comma meta b indeed meta a imply meta b infer not0 meta b infer not0 meta a end lemma end math ] "
\item For at vise " [ math prop lemma mt end math ] " begynder vi med et teknisk lemma, der ikke har den store v\ae{}rdi i sig selv:
\item " [ math in theory system Q lemma prop lemma technicality says for all terms meta a comma meta b indeed meta a imply meta b infer not0 not0 meta a imply meta b end lemma end math ] "
\item " [ math system Q proof of prop lemma technicality reads 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 premise macro indent not0 not0 meta a end line line ell c because prop lemma remove double neg modus ponens ell b indeed macro indent meta a end line because 1rule mp modus ponens ell a modus ponens ell c indeed macro indent meta b end line line ell d end block any term meta a comma meta b end line line ell e because 1rule deduction modus ponens ell d indeed parenthesis meta a imply meta b end parenthesis imply not0 not0 meta a imply meta b end line line ell p premise meta a imply meta b end line because 1rule mp modus ponens ell e modus ponens ell p indeed not0 not0 meta a imply meta b qed end math ] "
\item Uafh\ae{}ngigt af " [ math prop lemma technicality end math ] " kan vi vise en version af " [ math prop lemma mt end math ] ", hvor " [ math meta a end math ] " optr\ae{}der i negeret form:
\item " [ math in theory system Q lemma prop lemma negative mt says for all terms meta a comma meta b indeed not0 meta a imply meta b infer not0 meta b infer meta a end lemma end math ] "
\item " [ math system Q proof of prop lemma negative mt reads any term meta a comma meta b end line line ell a premise not0 meta a imply meta b end line line ell b premise not0 meta b end line line ell c because prop lemma weakening modus ponens ell b indeed not0 meta a imply not0 meta b end line because 1rule ad absurdum modus ponens ell a modus ponens ell c indeed meta a qed end math ] "
\item Ud fra " [ math prop lemma technicality end math ] " og " [ math prop lemma negative mt end math ] " kan vi nu vise " [ math prop lemma mt end math ] ":
\item " [ math in theory system Q lemma prop lemma mt says for all terms meta a comma meta b indeed meta a imply meta b infer not0 meta b infer not0 meta a end lemma end math ] "
\item " [ math system Q proof of prop lemma mt 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 not0 meta b end line line ell c because prop lemma technicality indeed not0 not0 meta a imply meta b end line because prop lemma negative mt modus ponens ell c modus ponens ell b indeed not0 meta a qed end math ] "
\item Vi slutter dette underafsnit med en variant af " [ math prop lemma mt end math ] ", som erstatter en inferens med en implikation:
\item " [ math in theory system Q lemma prop lemma contrapositive says for all terms meta a comma meta b indeed meta a imply meta b infer not0 meta b imply not0 meta a end lemma end math ] "
\item N\aa{}r en inferens skal erstattes med en implikation, er det altid deduktionsreglen, der skal i spil:
\item " [ math system Q proof of prop lemma contrapositive reads 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 premise macro indent not0 meta b end line because prop lemma mt modus ponens ell a modus ponens ell b indeed macro indent not0 meta a end line line ell c end block any term meta a comma meta b end line line ell d premise meta a imply meta b end line line ell e because 1rule deduction modus ponens ell c indeed parenthesis meta a imply meta b end parenthesis imply not0 meta b imply not0 meta a end line because 1rule mp modus ponens ell e modus ponens ell d indeed not0 meta b imply not0 meta a qed end math ] "
\end{list}
\subsection{Konjunktion}
Hovedm\aa{}let med dette underafsnit er at konvertere mellem formlerne " [ math meta a end math ] " og " [ math meta b end math ] " og deres konjunktion " [ bracket meta a and0 meta b end bracket ] ".
\subsubsection{Forening af konjunkter}
Vi begynder med at sl\aa{} " [ math meta a end math ] " og " [ math meta b end math ] " sammen til " [ bracket meta a and0 meta b end bracket ] ":
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma join conjuncts says for all terms meta a comma meta b indeed meta a infer meta b infer meta a and0 meta b end lemma end math ] "
\item Beviset for " [ math prop lemma join conjuncts end math ] " er af teknisk karakter. Vi viser den makroekspanderede form " [ bracket not0 parenthesis meta a imply not0 meta b end parenthesis end bracket ] ", som vi i bevisets sidste linie konverterer til " [ bracket meta a and0 meta b end bracket ] ". Denne sidste linie er ikke n\o{}dvendig for bevischeckeren, men den g\o{}r beviset lidt nemmere at l\ae{}se:
\item " [ math system Q proof of prop lemma join conjuncts reads block any term macro indent meta a comma meta b end line line ell z premise macro indent meta a end line line ell c premise macro indent meta a imply not0 meta b end line because 1rule mp modus ponens ell c modus ponens ell z indeed macro indent not0 meta b end line line ell d end block any term meta a comma meta b end line line ell e because 1rule deduction modus ponens ell d indeed meta a imply parenthesis meta a imply not0 meta b end parenthesis imply not0 meta b end line line ell a premise meta a end line line ell b premise meta b end line line ell f because 1rule mp modus ponens ell e modus ponens ell a indeed parenthesis meta a imply not0 meta b end parenthesis imply not0 meta b end line line ell q because prop lemma add double neg modus ponens ell b indeed not0 not0 meta b end line line ell big z because prop lemma mt modus ponens ell f modus ponens ell q indeed not0 parenthesis meta a imply not0 meta b end parenthesis end line because 1rule repetition modus ponens ell big z indeed meta a and0 meta b qed end math ] "
\end{list}
\subsubsection{Udskilning af anden konjunkt}
Tautologien " [ math prop lemma second conjunct end math ] " lader os udskille den anden konjunkt fra " [ bracket meta a and0 meta b end bracket ] ". Jeg vil ikke kommentere beviset:
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma second conjunct says for all terms meta a comma meta b indeed meta a and0 meta b infer meta b end lemma end math ] "
\item " [ math system Q proof of prop lemma second conjunct reads block any term macro indent macro indent meta a comma meta b end line line ell c premise macro indent not0 meta b end line because prop lemma weakening modus ponens ell c indeed macro indent meta a imply not0 meta b end line line ell e end block any term meta a comma meta b end line line ell f because 1rule deduction modus ponens ell e indeed not0 meta b imply meta a imply not0 meta b end line line ell big z premise meta a and0 meta b end line line ell a because 1rule repetition modus ponens ell big z indeed not0 parenthesis meta a imply not0 meta b end parenthesis end line because prop lemma negative mt modus ponens ell f modus ponens ell a indeed meta b qed end math ] "
\end{list}
\subsubsection{Udskilning af f\o{}rste konjunkt}
For at udskille " [ math meta a end math ] " fra " [ bracket meta a and0 meta b end bracket ] " viser vi f\o{}rst, at " [ bracket meta a and0 meta b end bracket ] " er kommutativ. Jeg vil ikke kommentere beviset:
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma and commutativity says for all terms meta a comma meta b indeed meta a and0 meta b infer meta b and0 meta a end lemma end math ] "
\item " [ math system Q proof of prop lemma and commutativity reads block any term macro indent macro indent meta a comma meta b end line line ell b premise macro indent meta b imply not0 meta a end line line ell c premise macro indent meta a end line line ell d because prop lemma add double neg modus ponens ell c indeed macro indent not0 not0 meta a end line because prop lemma mt modus ponens ell b modus ponens ell d indeed macro indent not0 meta b end line line ell e end block any term meta a comma meta b end line line ell f because 1rule deduction modus ponens ell e indeed parenthesis meta b imply not0 meta a end parenthesis imply meta a imply not0 meta b end line line ell big z premise meta a and0 meta b end line line ell a because 1rule repetition indeed not0 parenthesis meta a imply not0 meta b end parenthesis end line line ell big y because prop lemma mt modus ponens ell f modus ponens ell a indeed not0 parenthesis meta b imply not0 meta a end parenthesis end line because 1rule repetition modus ponens ell big y indeed meta b and0 meta a qed end math ] "
\item Nu er det let at udskille den f\o{}rste konjunkt fra " [ bracket meta a and0 meta b end bracket ] ": F\o{}rst vender vi konjunktionen om til " [ bracket meta b and0 meta a end bracket ] " ved hj\ae{}lp af " [ math prop lemma and commutativity end math ] ", og s\aa{} udskiller vi " [ math meta a end math ] " ved hj\ae{}lp af " [ math prop lemma second conjunct end math ] ":
\item " [ math in theory system Q lemma prop lemma first conjunct says for all terms meta a comma meta b indeed meta a and0 meta b infer meta a end lemma end math ] "
\item " [ math system Q proof of prop lemma first conjunct reads any term meta a comma meta b end line line ell a premise meta a and0 meta b end line line ell b because prop lemma and commutativity modus ponens ell a indeed meta b and0 meta a end line because prop lemma second conjunct modus ponens ell b indeed meta a qed end math ] "
\end{list}
\subsection{Dobbeltimplikation}
I dette underafsnit viser vi tre enkle resultater vedr.\ dobbeltimplikation.
\subsubsection{Brug sammen med modus ponens}
De f\o{}lgende to tautologier g\o{}r det let at bruge anvende slutningsreglen " [ math 1rule mp end math ] " p\aa{} dobbeltimplikationer. Beviserne er enkle og kr\ae{}ver ingen kommentarer:
\begin{list}{}{
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\item " [ math in theory system Q lemma prop lemma iff first says for all terms meta a comma meta b indeed meta a iff meta b infer meta b infer meta a end lemma end math ] "
\item " [ math system Q proof of prop lemma iff first reads any term meta a comma meta b end line line ell a premise meta a iff meta b end line line ell b premise meta b end line line ell c because prop lemma second conjunct modus ponens ell a indeed meta b imply meta a end line because 1rule mp modus ponens ell c modus ponens ell b indeed meta a qed end math ] "
\item" [ math in theory system Q lemma prop lemma iff second says for all terms meta a comma meta b indeed meta a iff meta b infer meta a infer meta b end lemma end math ] "
\item " [ math system Q proof of prop lemma iff second reads any term meta a comma meta b end line line ell a premise meta a iff meta b end line line ell b premise meta a end line line ell c because prop lemma first conjunct modus ponens ell a indeed meta a imply meta b end line because 1rule mp modus ponens ell c modus ponens ell b indeed meta b qed end math ] "
\end{list}
\subsubsection{Kommutativitet}
Lemmaet " [ math prop lemma iff commutativity end math ] " f\o{}lger direkte af, at operatoren " [ bracket var x and0 var y end bracket ] "
er kommutativ:
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\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system Q lemma prop lemma iff commutativity says for all terms meta a comma meta b indeed meta a iff meta b infer meta b iff meta a end lemma end math ] "
\item " [ math system Q proof of prop lemma iff commutativity reads any term meta a comma meta b end line line ell a premise meta a iff meta b end line line ell b because 1rule repetition modus ponens ell a indeed parenthesis meta a imply meta b end parenthesis and0 parenthesis meta b imply meta a end parenthesis end line line ell c because prop lemma and commutativity modus ponens ell b indeed parenthesis meta b imply meta a end parenthesis and0 parenthesis meta a imply meta b end parenthesis end line because 1rule repetition modus ponens ell c indeed meta b iff meta a qed end math ] "
\end{list}
\subsection{Disjunktion}
Dette underafsnit indeholder tre lemmaer vedr.\ disjunktion, som vi fordeler p\aa{} to under-underafsnit.
\subsubsection{Sv\ae{}kkelse}
Givet en p\aa{}stand " [ math meta b end math ] " vil vi gerne udlede de svagere p\aa{}stande " [ bracket meta a or0 meta b end bracket ] " og " [ bracket meta b or0 meta a end bracket ] ". Den f\o{}rste slutning varetages af lemmaet " [ math prop lemma weaken or first end math ] ":
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system Q lemma prop lemma weaken or first says for all terms meta a comma meta b indeed meta b infer meta a or0 meta b end lemma end math ] "
\item Beviset best\aa{}r af en simpel anvendelse af " [ math prop lemma weakening end math ] ":
\item " [ math system Q proof of prop lemma weaken or first reads any term meta a comma meta b end line line ell a premise meta b end line line ell b because prop lemma weakening modus ponens ell a indeed not0 meta a imply meta b end line because 1rule repetition modus ponens ell b indeed meta a or0 meta b qed end math ] "
\item Slutningen fra " [ math meta a end math ] " til " [ bracket meta a or0 meta b end bracket ] " varetages af lemmaet " [ math prop lemma weaken or second end math ] ":
\item " [ math in theory system Q lemma prop lemma weaken or second says for all terms meta a comma meta b indeed meta a infer meta a or0 meta b end lemma end math ] "
\item Kernen i beviset for " [ math prop lemma weaken or second end math ] " er en anvendelse af " [ math prop lemma from contradiction end math ] ":
\item " [ math system Q proof of prop lemma weaken or second reads block any term macro indent meta a comma meta b end line line ell a premise macro indent meta a end line line ell b premise macro indent not0 meta a end line because prop lemma from contradiction modus ponens ell a modus ponens ell b indeed macro indent meta b end line line ell c end block any term meta a comma meta b end line line ell e because 1rule deduction modus ponens ell c indeed meta a imply not0 meta a imply meta b end line line ell d premise meta a end line line ell f because 1rule mp modus ponens ell e modus ponens ell d indeed not0 meta a imply meta b end line because 1rule repetition modus ponens ell f indeed meta a or0 meta b qed end math ] "
\end{list}
\subsubsection{Slutning ud fra disjunktion}
Lemmaet " [ math prop lemma from disjuncts end math ] " lader os drage slutninger ud fra en disjunktion:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math in theory system Q lemma prop lemma from disjuncts says for all terms meta a comma meta b comma meta c indeed meta a or0 meta b infer meta a imply meta c infer meta b imply meta c infer meta c end lemma end math ] "
\item Om beviset vil jeg kun sige, at det er en ret elegant \o{}velse i bevisteknik:
\item " [ math system Q proof of prop lemma from disjuncts reads any term meta a comma meta b comma meta c end line line ell a premise meta a or0 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 1rule repetition modus ponens ell a indeed not0 meta a imply meta b end line line ell e because prop lemma contrapositive modus ponens ell d indeed not0 meta b imply not0 not0 meta a end line line ell f because prop lemma technicality modus ponens ell b indeed not0 not0 meta a imply meta c end line line ell g because prop lemma imply transitivity modus ponens ell e modus ponens ell f indeed not0 meta b imply meta c end line line ell h because prop lemma contrapositive modus ponens ell g indeed not0 meta c imply not0 not0 meta b end line line ell i because prop lemma contrapositive modus ponens ell c indeed not0 meta c imply not0 meta b end line because 1rule ad absurdum modus ponens ell i modus ponens ell h indeed meta c qed end math ] "
\end{list}
(********************)
\appendix
" [ flush left math priority table preassociative priority am 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 cdots equal priority object-var equal priority ex-var equal priority ph-var equal priority vaerdi equal priority variabel equal priority op x end op equal priority op2 x comma x end op2 equal priority define-equal x comma x end equal equal priority contains-empty x end empty equal priority 1deduction x conclude x end 1deduction equal priority 1deduction zero x conclude x end 1deduction equal priority 1deduction side x conclude x condition x end 1deduction equal priority 1deduction one x conclude x condition x end 1deduction equal priority 1deduction two x conclude x condition x end 1deduction equal priority 1deduction three x conclude x condition x bound x end 1deduction equal priority 1deduction four x conclude x condition x bound x end 1deduction equal priority 1deduction four star x conclude x condition x bound x end 1deduction equal priority 1deduction five x condition x bound x end 1deduction equal priority 1deduction six x conclude x exception x bound x end 1deduction equal priority 1deduction six star x conclude x exception x bound x end 1deduction equal priority 1deduction seven x end 1deduction equal priority 1deduction eight x bound x end 1deduction equal priority 1deduction eight star x bound x end 1deduction equal priority ex1 equal priority ex2 equal priority ex3 equal priority ex10 equal priority ex20 equal priority existential var x end var equal priority x is existential var equal priority exist-sub x is x where x is x end sub equal priority exist-sub0 x is x where x is x end sub equal priority exist-sub1 x is x where x is x end sub equal priority exist-sub* x is x where x is x end sub equal priority placeholder-var1 equal priority placeholder-var2 equal priority placeholder-var3 equal priority placeholder-var x end var equal priority x is placeholder-var equal priority ph-sub x is x where x is x end sub equal priority ph-sub0 x is x where x is x end sub equal priority ph-sub1 x is x where x is x end sub equal priority ph-sub* x is x where x is x end sub equal priority var big set equal priority object big set equal priority meta big set equal priority zermelo empty set equal priority system Q equal priority 1rule mp equal priority 1rule gen equal priority 1rule repetition equal priority 1rule ad absurdum equal priority 1rule deduction equal priority 1rule exist intro equal priority axiom extensionality equal priority axiom empty set equal priority axiom pair definition equal priority axiom union definition equal priority axiom power definition equal priority axiom separation definition equal priority prop lemma add double neg equal priority prop lemma remove double neg equal priority prop lemma and commutativity equal priority prop lemma auto imply equal priority prop lemma contrapositive equal priority prop lemma first conjunct equal priority prop lemma second conjunct equal priority prop lemma from contradiction equal priority prop lemma from disjuncts equal priority prop lemma iff commutativity equal priority prop lemma iff first equal priority prop lemma iff second equal priority prop lemma imply transitivity equal priority prop lemma join conjuncts equal priority prop lemma mp2 equal priority prop lemma mp3 equal priority prop lemma mp4 equal priority prop lemma mp5 equal priority prop lemma mt equal priority prop lemma negative mt equal priority prop lemma technicality equal priority prop lemma weakening equal priority prop lemma weaken or first equal priority prop lemma weaken or second equal priority lemma formula2pair equal priority lemma pair2formula equal priority lemma formula2union equal priority lemma union2formula equal priority lemma formula2separation equal priority lemma separation2formula equal priority lemma subset in power set equal priority lemma power set is subset0 equal priority lemma power set is subset equal priority lemma power set is subset0-switch equal priority lemma power set is subset-switch equal priority lemma set equality suff condition equal priority lemma set equality suff condition(t)0 equal priority lemma set equality suff condition(t) equal priority lemma set equality skip quantifier equal priority lemma set equality nec condition equal priority lemma reflexivity0 equal priority lemma reflexivity equal priority lemma symmetry0 equal priority lemma symmetry equal priority lemma transitivity0 equal priority lemma transitivity equal priority lemma er is reflexive equal priority lemma er is symmetric equal priority lemma er is transitive equal priority lemma empty set is subset equal priority lemma member not empty0 equal priority lemma member not empty equal priority lemma unique empty set0 equal priority lemma unique empty set equal priority lemma ==Reflexivity equal priority lemma ==Symmetry equal priority lemma ==Transitivity0 equal priority lemma ==Transitivity equal priority lemma transfer ~is0 equal priority lemma transfer ~is equal priority lemma pair subset0 equal priority lemma pair subset1 equal priority lemma pair subset equal priority lemma same pair equal priority lemma same singleton equal priority lemma union subset equal priority lemma same union equal priority lemma separation subset equal priority lemma same separation equal priority lemma same binary union equal priority lemma intersection subset equal priority lemma same intersection equal priority lemma auto member equal priority lemma eq-system not empty0 equal priority lemma eq-system not empty equal priority lemma eq subset0 equal priority lemma eq subset equal priority lemma equivalence nec condition0 equal priority lemma equivalence nec condition equal priority lemma none-equivalence nec condition0 equal priority lemma none-equivalence nec condition1 equal priority lemma none-equivalence nec condition equal priority lemma equivalence class is subset equal priority lemma equivalence classes are disjoint equal priority lemma all disjoint equal priority lemma all disjoint-imply equal priority lemma bs subset union(bs/r) equal priority lemma union(bs/r) subset bs equal priority lemma union(bs/r) is bs equal priority theorem eq-system is partition equal priority var ep equal priority var fx equal priority var fy equal priority var fz equal priority var fu equal priority var fv equal priority var rx equal priority var ry equal priority var rz equal priority var ru equal priority meta ep equal priority meta fx equal priority meta fy equal priority meta fz equal priority meta fu equal priority meta fv equal priority meta rx equal priority meta ry equal priority meta rz equal priority meta ru equal priority 0 equal priority 1 equal priority (-1) equal priority 2 equal priority 1/2 equal priority 0f equal priority 1f equal priority 00 equal priority 01 equal priority axiom leqReflexivity equal priority axiom leqAntisymmetry equal priority axiom leqTransitivity equal priority axiom leqTotality equal priority axiom leqAddition equal priority axiom leqMultiplication equal priority axiom plusAssociativity equal priority axiom plusCommutativity equal priority axiom negative equal priority axiom plus0 equal priority axiom timesAssociativity equal priority axiom timesCommutativity equal priority axiom reciprocal equal priority axiom times1 equal priority axiom distribution equal priority axiom 0not1 equal priority axiom equality equal priority axiom eqLeq equal priority axiom eqAddition equal priority axiom eqMultiplication equal priority lemma set equality nec condition(1) equal priority lemma set equality nec condition(2) equal priority 1rule ifThenElse true equal priority 1rule ifThenElse false equal priority 1rule from=f equal priority 1rule to=f equal priority 1rule from
\section{Pyk definitioner} \label{sec:pyk}
\begin{flushleft}
" [ math protect define pyk of cdots as text "cdots" end text end define linebreak define pyk of object-var as text "object-var" end text end define linebreak define pyk of ex-var as text "ex-var" end text end define linebreak define pyk of ph-var as text "ph-var" end text end define linebreak define pyk of vaerdi as text "vaerdi" end text end define linebreak define pyk of variabel as text "variabel" end text end define linebreak define pyk of op x end op as text "op "! end op" end text end define linebreak define pyk of op2 x comma x end op2 as text "op2 "! comma "! end op2" end text end define linebreak define pyk of define-equal x comma x end equal as text "define-equal "! comma "! end equal" end text end define linebreak define pyk of contains-empty x end empty as text "contains-empty "! end empty" end text end define linebreak define pyk of 1deduction x conclude x end 1deduction as text "1deduction "! conclude "! end 1deduction" end text end define linebreak define pyk of 1deduction zero x conclude x end 1deduction as text "1deduction zero "! conclude "! end 1deduction" end text end define linebreak define pyk of 1deduction side x conclude x condition x end 1deduction as text "1deduction side "! conclude "! condition "! end 1deduction" end text end define linebreak define pyk of 1deduction one x conclude x condition x end 1deduction as text "1deduction one "! conclude "! condition "! end 1deduction" end text end define linebreak define pyk of 1deduction two x conclude x condition x end 1deduction as text "1deduction two "! conclude "! condition "! end 1deduction" end text end define linebreak define pyk of 1deduction three x conclude x condition x bound x end 1deduction as text "1deduction three "! conclude "! condition "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction four x conclude x condition x bound x end 1deduction as text "1deduction four "! conclude "! condition "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction four star x conclude x condition x bound x end 1deduction as text "1deduction four star "! conclude "! condition "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction five x condition x bound x end 1deduction as text "1deduction five "! condition "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction six x conclude x exception x bound x end 1deduction as text "1deduction six "! conclude "! exception "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction six star x conclude x exception x bound x end 1deduction as text "1deduction six star "! conclude "! exception "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction seven x end 1deduction as text "1deduction seven "! end 1deduction" end text end define linebreak define pyk of 1deduction eight x bound x end 1deduction as text "1deduction eight "! bound "! end 1deduction" end text end define linebreak define pyk of 1deduction eight star x bound x end 1deduction as text "1deduction eight star "! bound "! end 1deduction" end text end define linebreak define pyk of ex1 as text "ex1" end text end define linebreak define pyk of ex2 as text "ex2" end text end define linebreak define pyk of ex3 as text "ex3" end text end define linebreak define pyk of ex10 as text "ex10" end text end define linebreak define pyk of ex20 as text "ex20" end text end define linebreak define pyk of existential var x end var as text "existential var "! end var" end text end define linebreak define pyk of x is existential var as text ""! is existential var" end text end define linebreak define pyk of exist-sub x is x where x is x end sub as text "exist-sub "! is "! where "! is "! end sub" end text end define linebreak define pyk of exist-sub0 x is x where x is x end sub as text "exist-sub0 "! is "! where "! is "! end sub" end text end define linebreak define pyk of exist-sub1 x is x where x is x end sub as text "exist-sub1 "! is "! where "! is "! end sub" end text end define linebreak define pyk of exist-sub* x is x where x is x end sub as text "exist-sub* "! is "! where "! is "! end sub" end text end define linebreak define pyk of placeholder-var1 as text "placeholder-var1" end text end define linebreak define pyk of placeholder-var2 as text "placeholder-var2" end text end define linebreak define pyk of placeholder-var3 as text "placeholder-var3" end text end define linebreak define pyk of placeholder-var x end var as text "placeholder-var "! end var" end text end define linebreak define pyk of x is placeholder-var as text ""! is placeholder-var" end text end define linebreak define pyk of ph-sub x is x where x is x end sub as text "ph-sub "! is "! where "! is "! end sub" end text end define linebreak define pyk of ph-sub0 x is x where x is x end sub as text "ph-sub0 "! is "! where "! is "! end sub" end text end define linebreak define pyk of ph-sub1 x is x where x is x end sub as text "ph-sub1 "! is "! where "! is "! end sub" end text end define linebreak define pyk of ph-sub* x is x where x is x end sub as text "ph-sub* "! is "! where "! is "! end sub" end text end define linebreak define pyk of var big set as text "var big set" end text end define linebreak define pyk of object big set as text "object big set" end text end define linebreak define pyk of meta big set as text "meta big set" end text end define linebreak define pyk of zermelo empty set as text "zermelo empty set" end text end define linebreak define pyk of system Q as text "system Q" end text end define linebreak define pyk of 1rule mp as text "1rule mp" end text end define linebreak define pyk of 1rule gen as text "1rule gen" end text end define linebreak define pyk of 1rule repetition as text "1rule repetition" end text end define linebreak define pyk of 1rule ad absurdum as text "1rule ad absurdum" end text end define linebreak define pyk of 1rule deduction as text "1rule deduction" end text end define linebreak define pyk of 1rule exist intro as text "1rule exist intro" end text end define linebreak define pyk of axiom extensionality as text "axiom extensionality" end text end define linebreak define pyk of axiom empty set as text "axiom empty set" end text end define linebreak define pyk of axiom pair definition as text "axiom pair definition" end text end define linebreak define pyk of axiom union definition as text "axiom union definition" end text end define linebreak define pyk of axiom power definition as text "axiom power definition" end text end define linebreak define pyk of axiom separation definition as text "axiom separation definition" end text end define linebreak define pyk of prop lemma add double neg as text "prop lemma add double neg" end text end define linebreak define pyk of prop lemma remove double neg as text "prop lemma remove double neg" end text end define linebreak define pyk of prop lemma and commutativity as text "prop lemma and commutativity" end text end define linebreak define pyk of prop lemma auto imply as text "prop lemma auto imply" end text end define linebreak define pyk of prop lemma contrapositive as text "prop lemma contrapositive" end text end define linebreak define pyk of prop lemma first conjunct as text "prop lemma first conjunct" end text end define linebreak define pyk of prop lemma second conjunct as text "prop lemma second conjunct" end text end define linebreak define pyk of prop lemma from contradiction as text "prop lemma from contradiction" end text end define linebreak define pyk of prop lemma from disjuncts as text "prop lemma from disjuncts" end text end define linebreak define pyk of prop lemma iff commutativity as text "prop lemma iff commutativity" end text end define linebreak define pyk of prop lemma iff first as text "prop lemma iff first" end text end define linebreak define pyk of prop lemma iff second as text "prop lemma iff second" end text end define linebreak define pyk of prop lemma imply transitivity as text "prop lemma imply transitivity" end text end define linebreak define pyk of prop lemma join conjuncts as text "prop lemma join conjuncts" end text end define linebreak define pyk of prop lemma mp2 as text "prop lemma mp2" end text end define linebreak define pyk of prop lemma mp3 as text "prop lemma mp3" end text end define linebreak define pyk of prop lemma mp4 as text "prop lemma mp4" end text end define linebreak define pyk of prop lemma mp5 as text "prop lemma mp5" end text end define linebreak define pyk of prop lemma mt as text "prop lemma mt" end text end define linebreak define pyk of prop lemma negative mt as text "prop lemma negative mt" end text end define linebreak define pyk of prop lemma technicality as text "prop lemma technicality" end text end define linebreak define pyk of prop lemma weakening as text "prop lemma weakening" end text end define linebreak define pyk of prop lemma weaken or first as text "prop lemma weaken or first" end text end define linebreak define pyk of prop lemma weaken or second as text "prop lemma weaken or second" end text end define linebreak define pyk of lemma formula2pair as text "lemma formula2pair" end text end define linebreak define pyk of lemma pair2formula as text "lemma pair2formula" end text end define linebreak define pyk of lemma formula2union as text "lemma formula2union" end text end define linebreak define pyk of lemma union2formula as text "lemma union2formula" end text end define linebreak define pyk of lemma formula2separation as text "lemma formula2separation" end text end define linebreak define pyk of lemma separation2formula as text "lemma separation2formula" end text end define linebreak define pyk of lemma subset in power set as text "lemma subset in power set" end text end define linebreak define pyk of lemma power set is subset0 as text "lemma power set is subset0" end text end define linebreak define pyk of lemma power set is subset as text "lemma power set is subset" end text end define linebreak define pyk of lemma power set is subset0-switch as text "lemma power set is subset0-switch" end text end define linebreak define pyk of lemma power set is subset-switch as text "lemma power set is subset-switch" end text end define linebreak define pyk of lemma set equality suff condition as text "lemma set equality suff condition" end text end define linebreak define pyk of lemma set equality suff condition(t)0 as text "lemma set equality suff condition(t)0" end text end define linebreak define pyk of lemma set equality suff condition(t) as text "lemma set equality suff condition(t)" end text end define linebreak define pyk of lemma set equality skip quantifier as text "lemma set equality skip quantifier" end text end define linebreak define pyk of lemma set equality nec condition as text "lemma set equality nec condition" end text end define linebreak define pyk of lemma reflexivity0 as text "lemma reflexivity0" end text end define linebreak define pyk of lemma reflexivity as text "lemma reflexivity" end text end define linebreak define pyk of lemma symmetry0 as text "lemma symmetry0" end text end define linebreak define pyk of lemma symmetry as text "lemma symmetry" end text end define linebreak define pyk of lemma transitivity0 as text "lemma transitivity0" end text end define linebreak define pyk of lemma transitivity as text "lemma transitivity" end text end define linebreak define pyk of lemma er is reflexive as text "lemma er is reflexive" end text end define linebreak define pyk of lemma er is symmetric as text "lemma er is symmetric" end text end define linebreak define pyk of lemma er is transitive as text "lemma er is transitive" end text end define linebreak define pyk of lemma empty set is subset as text "lemma empty set is subset" end text end define linebreak define pyk of lemma member not empty0 as text "lemma member not empty0" end text end define linebreak define pyk of lemma member not empty as text "lemma member not empty" end text end define linebreak define pyk of lemma unique empty set0 as text "lemma unique empty set0" end text end define linebreak define pyk of lemma unique empty set as text "lemma unique empty set" end text end define linebreak define pyk of lemma ==Reflexivity as text "lemma ==Reflexivity" end text end define linebreak define pyk of lemma ==Symmetry as text "lemma ==Symmetry" end text end define linebreak define pyk of lemma ==Transitivity0 as text "lemma ==Transitivity0" end text end define linebreak define pyk of lemma ==Transitivity as text "lemma ==Transitivity" end text end define linebreak define pyk of lemma transfer ~is0 as text "lemma transfer ~is0" end text end define linebreak define pyk of lemma transfer ~is as text "lemma transfer ~is" end text end define linebreak define pyk of lemma pair subset0 as text "lemma pair subset0" end text end define linebreak define pyk of lemma pair subset1 as text "lemma pair subset1" end text end define linebreak define pyk of lemma pair subset as text "lemma pair subset" end text end define linebreak define pyk of lemma same pair as text "lemma same pair" end text end define linebreak define pyk of lemma same singleton as text "lemma same singleton" end text end define linebreak define pyk of lemma union subset as text "lemma union subset" end text end define linebreak define pyk of lemma same union as text "lemma same union" end text end define linebreak define pyk of lemma separation subset as text "lemma separation subset" end text end define linebreak define pyk of lemma same separation as text "lemma same separation" end text end define linebreak define pyk of lemma same binary union as text "lemma same binary union" end text end define linebreak define pyk of lemma intersection subset as text "lemma intersection subset" end text end define linebreak define pyk of lemma same intersection as text "lemma same intersection" end text end define linebreak define pyk of lemma auto member as text "lemma auto member" end text end define linebreak define pyk of lemma eq-system not empty0 as text "lemma eq-system not empty0" end text end define linebreak define pyk of lemma eq-system not empty as text "lemma eq-system not empty" end text end define linebreak define pyk of lemma eq subset0 as text "lemma eq subset0" end text end define linebreak define pyk of lemma eq subset as text "lemma eq subset" end text end define linebreak define pyk of lemma equivalence nec condition0 as text "lemma equivalence nec condition0" end text end define linebreak define pyk of lemma equivalence nec condition as text "lemma equivalence nec condition" end text end define linebreak define pyk of lemma none-equivalence nec condition0 as text "lemma none-equivalence nec condition0" end text end define linebreak define pyk of lemma none-equivalence nec condition1 as text "lemma none-equivalence nec condition1" end text end define linebreak define pyk of lemma none-equivalence nec condition as text "lemma none-equivalence nec condition" end text end define linebreak define pyk of lemma equivalence class is subset as text "lemma equivalence class is subset" end text end define linebreak define pyk of lemma equivalence classes are disjoint as text "lemma equivalence classes are disjoint" end text end define linebreak define pyk of lemma all disjoint as text "lemma all disjoint" end text end define linebreak define pyk of lemma all disjoint-imply as text "lemma all disjoint-imply" end text end define linebreak define pyk of lemma bs subset union(bs/r) as text "lemma bs subset union(bs/r)" end text end define linebreak define pyk of lemma union(bs/r) subset bs as text "lemma union(bs/r) subset bs" end text end define linebreak define pyk of lemma union(bs/r) is bs as text "lemma union(bs/r) is bs" end text end define linebreak define pyk of theorem eq-system is partition as text "theorem eq-system is partition" end text end define linebreak define pyk of var ep as text "var ep" end text end define linebreak define pyk of var fx as text "var fx" end text end define linebreak define pyk of var fy as text "var fy" end text end define linebreak define pyk of var fz as text "var fz" end text end define linebreak define pyk of var fu as text "var fu" end text end define linebreak define pyk of var fv as text "var fv" end text end define linebreak define pyk of var rx as text "var rx" end text end define linebreak define pyk of var ry as text "var ry" end text end define linebreak define pyk of var rz as text "var rz" end text end define linebreak define pyk of var ru as text "var ru" end text end define linebreak define pyk of meta ep as text "meta ep" end text end define linebreak define pyk of meta fx as text "meta fx" end text end define linebreak define pyk of meta fy as text "meta fy" end text end define linebreak define pyk of meta fz as text "meta fz" end text end define linebreak define pyk of meta fu as text "meta fu" end text end define linebreak define pyk of meta fv as text "meta fv" end text end define linebreak define pyk of meta rx as text "meta rx" end text end define linebreak define pyk of meta ry as text "meta ry" end text end define linebreak define pyk of meta rz as text "meta rz" end text end define linebreak define pyk of meta ru as text "meta ru" end text end define linebreak define pyk of 0 as text "0" end text end define linebreak define pyk of 1 as text "1" end text end define linebreak define pyk of (-1) as text "(-1)" end text end define linebreak define pyk of 2 as text "2" end text end define linebreak define pyk of 1/2 as text "1/2" end text end define linebreak define pyk of 0f as text "0f" end text end define linebreak define pyk of 1f as text "1f" end text end define linebreak define pyk of 00 as text "00" end text end define linebreak define pyk of 01 as text "01" end text end define linebreak define pyk of axiom leqReflexivity as text "axiom leqReflexivity" end text end define linebreak define pyk of axiom leqAntisymmetry as text "axiom leqAntisymmetry" end text end define linebreak define pyk of axiom leqTransitivity as text "axiom leqTransitivity" end text end define linebreak define pyk of axiom leqTotality as text "axiom leqTotality" end text end define linebreak define pyk of axiom leqAddition as text "axiom leqAddition" end text end define linebreak define pyk of axiom leqMultiplication as text "axiom leqMultiplication" end text end define linebreak define pyk of axiom plusAssociativity as text "axiom plusAssociativity" end text end define linebreak define pyk of axiom plusCommutativity as text "axiom plusCommutativity" end text end define linebreak define pyk of axiom negative as text "axiom negative" end text end define linebreak define pyk of axiom plus0 as text "axiom plus0" end text end define linebreak define pyk of axiom timesAssociativity as text "axiom timesAssociativity" end text end define linebreak define pyk of axiom timesCommutativity as text "axiom timesCommutativity" end text end define linebreak define pyk of axiom reciprocal as text "axiom reciprocal" end text end define linebreak define pyk of axiom times1 as text "axiom times1" end text end define linebreak define pyk of axiom distribution as text "axiom distribution" end text end define linebreak define pyk of axiom 0not1 as text "axiom 0not1" end text end define linebreak define pyk of axiom equality as text "axiom equality" end text end define linebreak define pyk of axiom eqLeq as text "axiom eqLeq" end text end define linebreak define pyk of axiom eqAddition as text "axiom eqAddition" end text end define linebreak define pyk of axiom eqMultiplication as text "axiom eqMultiplication" end text end define linebreak define pyk of lemma set equality nec condition(1) as text "lemma set equality nec condition(1)" end text end define linebreak define pyk of lemma set equality nec condition(2) as text "lemma set equality nec condition(2)" end text end define linebreak define pyk of 1rule ifThenElse true as text "1rule ifThenElse true" end text end define linebreak define pyk of 1rule ifThenElse false as text "1rule ifThenElse false" end text end define linebreak define pyk of 1rule from=f as text "1rule from=f" end text end define linebreak define pyk of 1rule to=f as text "1rule to=f" end text end define linebreak define pyk of 1rule from
\newpage
\section{\TeX{} definitioner} \label{sec:tex}
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math tex define am as "am" end define end math ] "
\item " [ math tex define cdots as "(\cdots{})" end define end math ] "
\item " [ math tex define object-var as "\texttt{Objekt-var}" end define end math ] "
\item " [ math tex define ex-var as "\texttt{Ex-var}" end define end math ] "
\item " [ math tex define ph-var as "\texttt{Ph-var}" end define end math ] "
\item " [ math tex define vaerdi as "\texttt{V\ae{}rdi}" end define end math ] "
\item " [ math tex define variabel as "\texttt{Variabel}" end define end math ] "
\item " [ math tex define op var x end op as "Op(#1.
)" end define end math ] "
\item " [ math tex define op2 var x comma var y end op2 as "Op(#1.
,#2.
)" end define end math ] "
\item " [ math tex define define-equal var x comma var y end equal as "#1.
\mathrel {\ddot {==}} #2." end define end math ] "
\item " [ math tex define contains-empty var x end empty as "ContainsEmpty(#1.
)" end define end math ] "
" [ math tex define 1deduction var x conclude var y end 1deduction as "
Dedu(#1.
,#2.
)" end define end math ] "
" [ math tex define 1deduction zero var x conclude var y end 1deduction as "
Dedu_0(#1.
,#2.
)" end define end math ] "
" [ math tex define 1deduction side var x conclude var y condition var z end 1deduction as "Dedu_{s}(#1.
,#2.
,#3.
)" end define end math ] "
" [ math tex define 1deduction one var x conclude var y condition var z end 1deduction as "
Dedu_1(#1.
,#2.
,#3.
)" end define end math ] "
" [ math tex define 1deduction two var x conclude var y condition var z end 1deduction as "
Dedu_2(#1.
,#2.
,#3.
)" end define end math ] "
" [ math tex define 1deduction three var x conclude var y condition var z bound var u end 1deduction as "
Dedu_3(#1.
,#2.
,#3.
,#4.
)" end define end math ] "
" [ math tex define 1deduction four var x conclude var y condition var z bound var u end 1deduction as "
Dedu_4(#1.
,#2.
,#3.
,#4.
)" end define end math ] "
" [ math tex define 1deduction four star var x conclude var y condition var z bound var u end 1deduction as "
Dedu_4^*(#1.
,#2.
,#3.
,#4.
)" end define end math ] "
" [ math tex define 1deduction five var x condition var y bound var z end 1deduction as "
Dedu_5(#1.
,#2.
,#3.
)" end define end math ] "
" [ math tex define 1deduction six var p conclude var c exception var e bound var b end 1deduction as "
Dedu_6(#1.
,#2.
,#3.
,#4.
)" end define end math ] "
" [ math tex define 1deduction six star var p conclude var c exception var e bound var b end 1deduction as "
Dedu_6^*(#1.
,#2.
,#3.
,#4.
)" end define end math ] "
" [ math tex define 1deduction seven var p end 1deduction as "
Dedu_7(#1.
)" end define end math ] "
" [ math tex define 1deduction eight var p bound var b end 1deduction as "
Dedu_8(#1.
,#2.
)" end define end math ] "
" [ math tex define 1deduction eight star var p bound var b end 1deduction as "
Dedu_8^*(#1.
,#2.
)" end define end math ] "
\item " [ math tex define ex1 as "Ex_{1}" end define end math ] "
\item " [ math tex define ex2 as "Ex_{2}" end define end math ] "
\item " [ math tex define ex10 as "Ex_{10}" end define end math ] "
\item " [ math tex define ex20 as "Ex_{20}" end define end math ] "
\item " [ math tex define existential var var x end var as "#1.
_{Ex}" end define end math ] "
\item " [ math tex define var x is existential var as "#1.
^{Ex}" end define end math ] "
" [ math tex define exist-sub var x is var y where var z is var u end sub as "\langle #1.
{\equiv} #2.
| #3.
{:==} #4.
\rangle_{Ex} " end define end math ] "
" [ math tex define exist-sub0 var x is var y where var z is var u end sub as "\langle #1.
{\equiv}^0 #2.
| #3.
{:==} #4.
\rangle_{Ex} " end define end math ] "
" [ math tex define exist-sub1 var x is var y where var z is var u end sub as "\langle #1.
{\equiv}^1 #2.
| #3.
{:==} #4.
\rangle_{Ex} " end define end math ] "
" [ math tex define exist-sub* var x is var y where var z is var u end sub as "\langle #1.
{\equiv}^* #2.
| #3.
{:==} #4.
\rangle_{Ex} " end define end math ] "
\item " [ math tex define placeholder-var1 as "ph_{1}" end define end math ] "
\item " [ math tex define placeholder-var2 as "ph_{2}" end define end math ] "
\item " [ math tex define placeholder-var3 as "ph_{3}" end define end math ] "
\item " [ math tex define placeholder-var var x end var as "#1.
_{Ph} " end define end math ] "
\item " [ math tex define var x is placeholder-var as "#1.
^{Ph}" end define end math ] "
" [ math tex define ph-sub var x is var y where var z is var u end sub as "\langle #1.
{\equiv} #2.
| #3.
{:==} #4.
\rangle_{Ph} " end define end math ] "
" [ math tex define ph-sub0 var x is var y where var z is var u end sub as "\langle #1.
{\equiv}^0 #2.
| #3.
{:==} #4.
\rangle_{Ph} " end define end math ] "
" [ math tex define ph-sub1 var x is var y where var z is var u end sub as "\langle #1.
{\equiv}^1 #2.
| #3.
{:==} #4.
\rangle_{Ph} " end define end math ] "
" [ math tex define ph-sub* var x is var y where var z is var u end sub as "\langle #1.
{\equiv}^* #2.
| #3.
{:==} #4.
\rangle_{Ph} " end define end math ] "
\item " [ math tex define var big set as "\mathsf {bs}" end define end math ] "
\item " [ math tex define object big set as " \mathsf {OBS}" end define end math ] "
" [ math tex define meta big set as "{\cal BS}" end define end math ] "
\item " [ math tex define zermelo empty set as "\mathrm{\O}" end define end math ] "
\item " [ math tex define system Q as "ZFsub" end define end math ] "
\item " [ math tex define 1rule mp as "MP" end define end math ] "
\item " [ math tex define 1rule gen as "Gen" end define end math ] "
\item " [ math tex define 1rule repetition as "Repetition" end define end math ] "
\item " [ math tex define 1rule ad absurdum as "Neg" end define end math ] "
\item " [ math tex define 1rule deduction as "Ded" end define end math ] "
\item " [ math tex define 1rule exist intro as "ExistIntro" end define end math ] "
\item " [ math tex define axiom extensionality as "Extensionality" end define end math ] "
\item " [ math tex define axiom empty set as "\O{}def" end define end math ] "
\item " [ math tex define axiom pair definition as "PairDef" end define end math ] "
\item " [ math tex define axiom union definition as "UnionDef" end define end math ] "
\item " [ math tex define axiom power definition as "PowerDef" end define end math ] "
\item " [ math tex define axiom separation definition as "SeparationDef" end define end math ] "
\item " [ math tex define prop lemma add double neg as "AddDoubleNeg" end define end math ] "
\item " [ math tex define prop lemma remove double neg as "RemoveDoubleNeg" end define end math ] "
\item " [ math tex define prop lemma and commutativity as "AndCommutativity" end define end math ] "
\item " [ math tex define prop lemma auto imply as "AutoImply" end define end math ] "
\item " [ math tex define prop lemma contrapositive as "Contrapositive" end define end math ] "
\item " [ math tex define prop lemma first conjunct as "FirstConjunct" end define end math ] "
\item " [ math tex define prop lemma second conjunct as "SecondConjunct" end define end math ] "
\item " [ math tex define prop lemma from contradiction as "FromContradiction" end define end math ] "
\item " [ math tex define prop lemma from disjuncts as "FromDisjuncts" end define end math ] "
\item " [ math tex define prop lemma iff commutativity as "IffCommutativity" end define end math ] "
\item " [ math tex define prop lemma iff first as "IffFirst" end define end math ] "
\item " [ math tex define prop lemma iff second as "IffSecond" end define end math ] "
\item " [ math tex define prop lemma imply transitivity as "ImplyTransitivity" end define end math ] "
\item " [ math tex define prop lemma join conjuncts as "JoinConjuncts" end define end math ] "
\item " [ math tex define prop lemma mp2 as "MP2" end define end math ] "
\item " [ math tex define prop lemma mp3 as "MP3" end define end math ] "
\item " [ math tex define prop lemma mp4 as "MP4" end define end math ] "
\item " [ math tex define prop lemma mp5 as "MP5" end define end math ] "
\item " [ math tex define prop lemma mt as "MT" end define end math ] "
\item " [ math tex define prop lemma negative mt as "NegativeMT" end define end math ] "
\item " [ math tex define prop lemma technicality as "Technicality" end define end math ] "
\item " [ math tex define prop lemma weakening as "Weakening" end define end math ] "
\item " [ math tex define prop lemma weaken or first as "WeakenOr1" end define end math ] "
\item " [ math tex define prop lemma weaken or second as "WeakenOr2" end define end math ] "
\item " [ math tex define lemma pair2formula as "Pair2Formula" end define end math ] "
\item " [ math tex define lemma formula2pair as "Formula2Pair" end define end math ] "
\item " [ math tex define lemma union2formula as "Union2Formula" end define end math ] "
\item " [ math tex define lemma formula2union as "Formula2Union" end define end math ] "
\item " [ math tex define lemma separation2formula as "Sep2Formula" end define end math ] "
\item " [ math tex define lemma formula2separation as "Formula2Sep" end define end math ] "
\item " [ math tex define lemma subset in power set as "SubsetInPower" end define end math ] "
\item " [ math tex define lemma power set is subset0 as "HelperPowerIsSub" end define end math ] "
\item " [ math tex define lemma power set is subset as "PowerIsSub" end define end math ] "
\item " [ math tex define lemma power set is subset0-switch as "(Switch)HelperPowerIsSub" end define end math ] "
\item " [ math tex define lemma power set is subset-switch as "(Switch)PowerIsSub" end define end math ] "
\item " [ math tex define lemma set equality suff condition as "ToSetEquality" end define end math ] "
\item " [ math tex define lemma set equality suff condition(t)0 as "HelperToSetEquality(t)" end define end math ] "
\item " [ math tex define lemma set equality suff condition(t) as "ToSetEquality(t)" end define end math ] "
\item " [ math tex define lemma set equality skip quantifier as "HelperFromSetEquality" end define end math ] "
\item " [ math tex define lemma set equality nec condition as "FromSetEquality" end define end math ] "
\item " [ math tex define lemma reflexivity0 as "HelperReflexivity" end define end math ] "
\item " [ math tex define lemma reflexivity as "Reflexivity" end define end math ] "
\item " [ math tex define lemma symmetry0 as "HelperSymmetry" end define end math ] "
\item " [ math tex define lemma symmetry as "Symmetry" end define end math ] "
\item " [ math tex define lemma transitivity0 as "HelperTransitivity" end define end math ] "
\item " [ math tex define lemma transitivity as "Transitivity" end define end math ] ",
\item " [ math tex define lemma er is reflexive as "ERisReflexive" end define end math ] "
\item " [ math tex define lemma er is symmetric as "ERisSymmetric" end define end math ] "
\item " [ math tex define lemma er is transitive as "ERisTransitive" end define end math ] "
\item " [ math tex define lemma empty set is subset as "\O{}isSubset" end define end math ] "
\item " [ math tex define lemma member not empty0 as "HelperMemberNot\O{}" end define end math ] "
\item " [ math tex define lemma member not empty as "MemberNot\O{}" end define end math ] "
\item " [ math tex define lemma unique empty set0 as "HelperUnique\O{}" end define end math ] "
\item " [ math tex define lemma unique empty set as "Unique\O{}" end define end math ] "
\item " [ math tex define lemma ==Reflexivity as "==\!{}Reflexivity" end define end math ] "
\item " [ math tex define lemma ==Symmetry as "==\!{}Symmetry" end define end math ] "
\item " [ math tex define lemma ==Transitivity0 as "Helper\!{}==\!{}Transitivity" end define end math ] "
\item " [ math tex define lemma ==Transitivity as "\!{}==\!{}Transitivity" end define end math ] "
\item " [ math tex define lemma transfer ~is0 as "HelperTransferNotEq" end define end math ] "
\item " [ math tex define lemma transfer ~is as "TransferNotEq" end define end math ] "
\item " [ math tex define lemma pair subset0 as "HelperPairSubset" end define end math ] "
\item " [ math tex define lemma pair subset1 as "Helper(2)PairSubset" end define end math ] "
\item " [ math tex define lemma pair subset as "PairSubset" end define end math ] "
\item " [ math tex define lemma same pair as "SamePair" end define end math ] "
\item " [ math tex define lemma same singleton as "SameSingleton" end define end math ] "
\item " [ math tex define lemma union subset as "UnionSubset" end define end math ] "
\item " [ math tex define lemma same union as "SameUnion" end define end math ] "
\item " [ math tex define lemma separation subset as "SeparationSubset" end define end math ] "
\item " [ math tex define lemma same separation as "SameSeparation" end define end math ] "
\item " [ math tex define lemma same binary union as "SameBinaryUnion" end define end math ] "
\item " [ math tex define lemma intersection subset as "IntersectionSubset" end define end math ] "
\item " [ math tex define lemma same intersection as "SameIntersection" end define end math ] "
\item " [ math tex define lemma auto member as "AutoMember" end define end math ] "
\item " [ math tex define lemma eq-system not empty0 as "HelperEqSysNot\O{}" end define end math ] "
\item " [ math tex define lemma eq-system not empty as "EqSysNot\O{}" end define end math ] "
\item " [ math tex define lemma eq subset0 as "HelperEqSubset" end define end math ] "
\item " [ math tex define lemma eq subset as "EqSubset" end define end math ] "
\item " [ math tex define lemma equivalence nec condition as "EqNecessary" end define end math ] "
\item " [ math tex define lemma equivalence nec condition0 as "HelperEqNecessary" end define end math ] "
\item " [ math tex define lemma none-equivalence nec condition0 as "HelperNoneEqNecessary" end define end math ] "
\item " [ math tex define lemma none-equivalence nec condition1 as "Helper(2)NoneEqNecessary" end define end math ] "
\item " [ math tex define lemma none-equivalence nec condition as "NoneEqNecessary" end define end math ] "
\item " [ math tex define lemma equivalence class is subset as "EqClassIsSubset" end define end math ] "
\item " [ math tex define lemma equivalence classes are disjoint as "EqClassesAreDisjoint" end define end math ] "
\item " [ math tex define lemma all disjoint as "AllDisjoint" end define end math ] "
\item " [ math tex define lemma all disjoint-imply as "AllDisjointImply" end define end math ] "
\item " [ math tex define lemma bs subset union(bs/r) as "BSsubset" end define end math ] "
\item " [ math tex define lemma union(bs/r) subset bs as "Union(BS/R)subset" end define end math ] "
\item " [ math tex define lemma union(bs/r) is bs as "UnionIdentity" end define end math ] "
\item " [ math tex define theorem eq-system is partition as "EqSysIsPartition" end define end math ] "
\item " [ math tex define eq-system of var x modulo var y as "#1.
/ #2." end define end math ] "
\item " [ math tex define intersection var x comma var y end intersection as "#1.
\cap #2." end define end math ] "
\item " [ math tex define union var x end union as "\cup #1." end define end math ] "
\item " [ math tex define binary-union var x comma var y end union as "#1.
\mathrel{\cup} #2." end define end math ] "
\item " [ math tex define power var x end power as "P(#1.
)" end define end math ] "
\item " [ math tex define zermelo singleton var x end singleton as "\{#1.
\}" end define end math ] "
\item " [ math tex define zermelo pair var x comma var y end pair as "\{#1.
,#2.
\}" end define end math ] "
\item " [ math tex define zermelo ordered pair var x comma var y end pair as "\langle #1.
,#2.
\rangle" end define end math ] ",
\item " [ math tex define var x in0 var y as "#1.
\mathrel{\in} #2." end define end math ] "
\item " [ math tex define var x is related to var y under var z as "#3.
(#1.
,#2.
)" end define end math ] "
\item " [ math tex define var r is reflexive relation in var x as "ReflRel(#1.
,#2.
)" end define end math ] "
\item " [ math tex define var r is symmetric relation in var x as "SymRel(#1.
,#2.
)" end define end math ] "
\item " [ math tex define var r is transitive relation in var x as "TransRel(#1.
,#2.
)" end define end math ] "
\item " [ math tex define var r is equivalence relation in var x as "EqRel(#1.
,#2.
)" end define end math ] "
\item " [ math tex define equivalence class of var x in var big set modulo var r as "[#1.
\mathrel{\in} #2.
]_{#3.
}" end define end math ] "
\item " [ math tex define var x is partition of var y as "Partition(#1.
,#2.
)" end define end math ] "
\item " [ math tex define var x zermelo is var y as "#1.
\!\mathrel{==}\! #2." end define end math ] "
\item " [ math tex define var x is subset of var y as "#1.
\mathrel{\subseteq} #2." end define end math ] "
\item " [ math tex define not0 var x as "\dot{\neg}\, #1." end define end math ] "
\item " [ math tex define var x zermelo ~in var y as "#1.
\mathrel{\notin} #2." end define end math ] "
\item " [ math tex define var x zermelo ~is var y as "#1.
\mathrel{\neq} #2." end define end math ] "
\item " [ math tex define var x and0 var y as "#1.
\mathrel{\dot{\wedge}} #2." end define end math ] "
\item " [ math tex define var x or0 var y as "#1.
\mathrel{\dot{\vee}} #2." end define end math ] "
" [ math tex define var x iff var y as "#1.
\mathrel{\dot{\Leftrightarrow}} #2." end define end math ] "
\item " [ math tex define the set of ph in var x such that var a end set as " \{ ph \mathrel{\in} #1.
\mid #2.
\}" end define end math ] "
\end{list}
------------- RRRRRRRRRRRRRRR -------------
(*** aksiomer ***)
" [ math in theory system Q rule axiom leqReflexivity says for all terms meta x indeed meta x <= meta x end rule end math ] "
" [ math in theory system Q rule axiom leqAntisymmetry says for all terms meta x comma meta y indeed meta x <= meta y imply meta y <= meta x imply meta x = meta y end rule end math ] "
" [ math in theory system Q rule axiom leqTransitivity says for all terms meta x comma meta y comma meta z indeed meta x <= meta y imply meta y <= meta z imply meta x <= meta z end rule end math ] "
" [ math in theory system Q rule axiom leqTotality says for all terms meta x comma meta y indeed meta x <= meta y or0 meta y <= meta x end rule end math ] "
" [ math in theory system Q rule axiom leqAddition says for all terms meta x comma meta y comma meta z indeed meta x <= meta y imply meta x + meta z <= meta y + meta z end rule end math ] "
" [ math in theory system Q rule axiom leqMultiplication says for all terms meta x comma meta y comma meta z indeed 0 <= meta z imply meta x <= meta y imply meta x * meta z <= meta y * meta z end rule end math ] "
" [ math in theory system Q rule axiom plusAssociativity says for all terms meta x comma meta y comma meta z indeed parenthesis meta x + meta y end parenthesis + meta z = meta x + parenthesis meta y + meta z end parenthesis end rule end math ] "
" [ math in theory system Q rule axiom plusCommutativity says for all terms meta x comma meta y indeed meta x + meta y = meta y + meta x end rule end math ] "
" [ math in theory system Q rule axiom negative says for all terms meta x indeed meta x + parenthesis - meta x end parenthesis = 0 end rule end math ] "
" [ math in theory system Q rule axiom plus0 says for all terms meta x indeed meta x + 0 = meta x end rule end math ] "
" [ math in theory system Q rule axiom timesAssociativity says for all terms meta x comma meta y comma meta z indeed parenthesis meta x * meta y end parenthesis * meta z = meta x * parenthesis meta y * meta z end parenthesis end rule end math ] "
" [ math in theory system Q rule axiom timesCommutativity says for all terms meta x comma meta y indeed meta x * meta y = meta y * meta x end rule end math ] "
" [ math in theory system Q rule axiom reciprocal says for all terms meta x indeed meta x != 0 imply meta x * 1/ meta x = 1 end rule end math ] "
" [ math in theory system Q rule axiom times1 says for all terms meta x indeed meta x * 1 = meta x end rule end math ] "
" [ math in theory system Q rule axiom distribution says for all terms meta x comma meta y comma meta z indeed meta x * parenthesis meta y + meta z end parenthesis = parenthesis meta x * meta y end parenthesis + parenthesis meta x * meta z end parenthesis end rule end math ] "
" [ math in theory system Q rule axiom 0not1 says 0 != 1 end rule end math ] "
" [ math in theory system Q rule axiom equality says for all terms meta x comma meta y comma meta z indeed meta x = meta y imply meta x = meta z imply meta y = meta z end rule end math ] "
" [ math in theory system Q rule axiom eqLeq says for all terms meta x comma meta y indeed meta x = meta y imply meta x <= meta y end rule end math ] "
" [ math in theory system Q rule axiom eqAddition says for all terms meta x comma meta y comma meta z indeed meta x = meta y imply meta x + meta z = meta y + meta z end rule end math ] "
" [ math in theory system Q rule axiom eqMultiplication says for all terms meta x comma meta y comma meta z indeed meta x = meta y imply meta x * meta z = meta y * meta z end rule end math ] "
(*** XX snydeaksiomer ***)
" [ math in theory system Q rule lemma ==Reflexivity says for all terms meta rx indeed meta rx == meta rx end rule end math ] "
" [ math in theory system Q rule lemma ==Symmetry says for all terms meta rx comma meta ry indeed meta rx == meta ry infer meta ry == meta rx end rule end math ] "
" [ math in theory system Q rule lemma ==Transitivity says for all terms meta rx comma meta ry comma meta rz indeed meta rx == meta ry infer meta ry == meta rz infer meta rx == meta rz end rule end math ] "
XX ikke 100procent identisk med originalen fra equivalence-relations
" [ math in theory system Q rule lemma set equality nec condition(1) says for all terms meta fx comma meta rx comma meta ry indeed meta rx == meta ry infer meta fx in0 meta rx infer meta fx in0 meta ry end rule end math ] "
XX boer bevises ud fra nummer 1
" [ math in theory system Q rule lemma set equality nec condition(2) says for all terms meta fx comma meta rx comma meta ry indeed meta rx == meta ry infer meta fx in0 meta ry infer meta fx in0 meta rx end rule end math ] "
" [ math in theory system Q rule 1rule ifThenElse true says for all terms meta a comma meta x comma meta y indeed meta a infer if( meta a , meta x , meta y ) = meta x end rule end math ] "
" [ math in theory system Q rule 1rule ifThenElse false says for all terms meta a comma meta x comma meta y indeed not0 meta a infer if( meta a , meta x , meta y ) = meta y end rule end math ] "
" [ math in theory system Q rule 1rule fromSameF says for all terms meta m comma meta ep comma meta fx comma meta fy indeed meta fx sameF meta fy infer 0 < meta ep infer ex3 <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end rule end math ] "
" [ math in theory system Q rule 1rule toSameF says for all terms meta m comma meta ep comma meta fx comma meta fy indeed 0 < meta ep imply ex3 <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep infer meta fx sameF meta fy end rule end math ] "
" [ math in theory system Q rule 1rule from=f says for all terms meta m comma meta fx comma meta fy indeed meta fx =f meta fy infer [ meta fx ; meta m ] = [ meta fy ; meta m ] end rule end math ] "
XX hm... det er nok med bare 1 meta m
XX loesning: objektkvantor
" [ math in theory system Q rule 1rule to=f says for all terms meta m comma meta fx comma meta fy indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] infer meta fx =f meta fy end rule end math ] "
" [ math in theory system Q rule 1rule from
" [ math in theory system Q rule 1rule to
" [ math in theory system Q rule axiom plusF says for all terms meta m comma meta fx comma meta fy indeed [ meta fx +f meta fy ; meta m ] = [ meta fx ; meta m ] + [ meta fy ; meta m ] end rule end math ] "
" [ math in theory system Q rule axiom minusF says for all terms meta m comma meta fx indeed [ -f meta fx ; meta m ] = - [ meta fx ; meta m ] end rule end math ] "
" [ math in theory system Q rule axiom timesF says for all terms meta m comma meta fx comma meta fy indeed [ meta fx *f meta fy ; meta m ] = [ meta fx ; meta m ] * [ meta fy ; meta m ] end rule end math ] "
" [ math in theory system Q rule axiom 0f says for all terms meta m indeed [ 0f ; meta m ] = 0 end rule end math ] "
" [ math in theory system Q rule axiom 1f says for all terms meta m indeed [ 1f ; meta m ] = 1 end rule end math ] "
" [ math in theory system Q rule 1rule to==XX says for all terms meta fx comma meta fy comma meta rx comma meta ry indeed meta fx in0 meta rx imply meta fy in0 meta ry imply meta fx sameF meta fy infer meta rx == meta ry end rule end math ] "
" [ math in theory system Q rule 1rule from== says for all terms meta fx comma meta fy indeed R( meta fx ) == R( meta fy ) infer meta fx sameF meta fy end rule end math ] "
" [ math in theory system Q rule 1rule to== says for all terms meta fx comma meta fy indeed meta fx sameF meta fy infer R( meta fx ) == R( meta fy ) end rule end math ] "
" [ math in theory system Q rule 1rule from<
" [ math in theory system Q rule 1rule from<
" [ math in theory system Q rule 1rule from<
" [ math in theory system Q rule 1rule to<
" [ math in theory system Q rule 1rule from<< says for all terms meta fx comma meta fy indeed R( meta fx ) << R( meta fy ) infer meta fx
" [ math in theory system Q rule 1rule to<< says for all terms meta fx comma meta fy indeed meta fx
" [ math in theory system Q rule 1rule fromInR says for all terms meta fx comma meta fy indeed meta fx in0 R( meta fy ) infer meta fx sameF meta fy end rule end math ] "
" [ math in theory system Q rule axiom plusR says for all terms meta fx comma meta fy indeed R( meta fx ) ++ R( meta fy ) == R( meta fx +f meta fy ) end rule end math ] "
" [ math in theory system Q rule axiom timesR says for all terms meta fx comma meta fy indeed R( meta fx ) ** R( meta fy ) == R( meta fx *f meta fy ) end rule end math ] "
(*** makroer ***)
" [ math macro define meta ep as metavar var ep end metavar end define end math ] "
" [ math macro define meta fx as metavar var fx end metavar end define end math ] "
" [ math macro define meta fy as metavar var fy end metavar end define end math ] "
" [ math macro define meta fz as metavar var fz end metavar end define end math ] "
" [ math macro define meta fu as metavar var fu end metavar end define end math ] "
" [ math macro define meta fv as metavar var fv end metavar end define end math ] "
" [ math macro define meta rx as metavar var rx end metavar end define end math ] "
" [ math macro define meta ry as metavar var ry end metavar end define end math ] "
" [ math macro define meta rz as metavar var rz end metavar end define end math ] "
" [ math macro define meta ru as metavar var ru end metavar end define end math ] "
" [ math macro define ex3 as existential var var c end var end define end math ] "
" [ math macro define var x <<== var y as var x << var y or0 var x == var y end define end math ] "
" [ math macro define (-1) as - 1 end define end math ] "
" [ math macro define 2 as parenthesis 1 + 1 end parenthesis end define end math ] "
" [ math macro define 1/2 as 1/ 2 end define end math ] "
" [ math macro define var x < var y as var x <= var y and0 var x != var y end define end math ] "
" [ math macro define var x != var y as not0 var x = var y end define end math ] "
" [ math macro define var x - var y as var x + parenthesis - var y end parenthesis end define end math ] "
" [ math macro define | var x | as if( 0 <= var x , var x , - var x ) end define end math ] "
" [ math macro define 00 as R( 0f ) end define end math ] "
" [ math macro define 01 as R( 1f ) end define end math ] "
" [ math macro define R( var fx ) ++ R( var fy ) as R( var fx +f var fy ) end define end math ] "
" [ math macro define --R( var fx ) as R( -f var fx ) end define end math ] " XX noedvendig?
" [ math macro define R( var fx ) -- R( var fy ) as R( var fx ) ++ R( -f var fy ) end define end math ] "
XX noedvendigt med [R( ) -- R( )] konstruktionen?
(*** REGELLEMMAER ***)
" [ math in theory system Q lemma lemma leqTransitivity says for all terms meta x comma meta y comma meta z indeed meta x <= meta y infer meta y <= meta z infer meta x <= meta z end lemma end math ] "
" [ math system Q proof of lemma leqTransitivity reads any term meta x comma meta y comma meta z end line line ell a premise meta x <= meta y end line line ell b premise meta y <= meta z end line line ell c because axiom leqTransitivity indeed meta x <= meta y imply meta y <= meta z imply meta x <= meta z end line because prop lemma mp2 modus ponens ell c modus ponens ell a modus ponens ell b indeed meta x <= meta z qed end math ] "
" [ math in theory system Q lemma lemma leqAntisymmetry says for all terms meta x comma meta y indeed meta x <= meta y infer meta y <= meta x infer meta x = meta y end lemma end math ] "
" [ math system Q proof of lemma leqAntisymmetry reads any term meta x comma meta y end line line ell a premise meta x <= meta y end line line ell b premise meta y <= meta x end line line ell c because axiom leqAntisymmetry indeed meta x <= meta y imply meta y <= meta x imply meta x = meta y end line because prop lemma mp2 modus ponens ell c modus ponens ell a modus ponens ell b indeed meta x = meta y qed end math ] "
" [ math in theory system Q lemma lemma leqAddition says for all terms meta x comma meta y comma meta z indeed meta x <= meta y infer meta x + meta z <= meta y + meta z end lemma end math ] "
" [ math system Q proof of lemma leqAddition reads any term meta x comma meta y comma meta z end line line ell a premise meta x <= meta y end line line ell b because axiom leqAddition indeed meta x <= meta y imply meta x + meta z <= meta y + meta z end line because 1rule mp modus ponens ell b modus ponens ell a indeed meta x + meta z <= meta y + meta z qed end math ] "
" [ math in theory system Q lemma lemma leqMultiplication says for all terms meta x comma meta y comma meta z indeed 0 <= meta z infer meta x <= meta y infer meta x * meta z <= meta y * meta z end lemma end math ] "
" [ math system Q proof of lemma leqMultiplication reads any term meta x comma meta y comma meta z end line line ell b premise 0 <= meta z end line line ell a premise meta x <= meta y end line line ell c because axiom leqMultiplication indeed 0 <= meta z imply meta x <= meta y imply meta x * meta z <= meta y * meta z end line because prop lemma mp2 modus ponens ell c modus ponens ell b modus ponens ell a indeed meta x * meta z <= meta y * meta z qed end math ] "
" [ math in theory system Q lemma lemma reciprocal says for all terms meta x indeed meta x != 0 infer meta x * 1/ meta x = 1 end lemma end math ] "
" [ math system Q proof of lemma reciprocal reads any term meta x end line line ell a premise meta x != 0 end line line ell b because axiom reciprocal indeed meta x != 0 imply meta x * 1/ meta x = 1 end line because 1rule mp modus ponens ell b modus ponens ell a indeed meta x * 1/ meta x = 1 qed end math ] "
" [ math in theory system Q lemma lemma equality says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta x = meta z infer meta y = meta z end lemma end math ] "
" [ math system Q proof of lemma equality reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell b premise meta x = meta z end line line ell c because axiom equality indeed meta x = meta y imply meta x = meta z imply meta y = meta z end line because prop lemma mp2 modus ponens ell c modus ponens ell a modus ponens ell b indeed meta y = meta z qed end math ] "
" [ math in theory system Q lemma lemma eqLeq says for all terms meta x comma meta y indeed meta x = meta y infer meta x <= meta y end lemma end math ] "
" [ math system Q proof of lemma eqLeq reads any term meta x comma meta y end line line ell a premise meta x = meta y end line line ell b because axiom eqLeq indeed meta x = meta y imply meta x <= meta y end line because 1rule mp modus ponens ell b modus ponens ell a indeed meta x <= meta y qed end math ] "
" [ math in theory system Q lemma lemma eqAddition says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta x + meta z = meta y + meta z end lemma end math ] "
" [ math system Q proof of lemma eqAddition reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell b because axiom eqAddition indeed meta x = meta y imply meta x + meta z = meta y + meta z end line because 1rule mp modus ponens ell b modus ponens ell a indeed meta x + meta z = meta y + meta z qed end math ] "
" [ math in theory system Q lemma lemma eqMultiplication says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta x * meta z = meta y * meta z end lemma end math ] "
" [ math system Q proof of lemma eqMultiplication reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell c because axiom eqMultiplication indeed meta x = meta y imply meta x * meta z = meta y * meta z end line because 1rule mp modus ponens ell c modus ponens ell a indeed meta x * meta z = meta y * meta z qed end math ] "
(*** UDSAGNSLOGIK ***)
" [ math in theory system Q lemma prop lemma to negated imply says for all terms meta a comma meta b indeed meta a infer not0 meta b infer not0 parenthesis meta a imply meta b end parenthesis end lemma end math ] "
" [ math system Q proof of prop lemma to negated imply reads block any term meta a comma meta b end line line ell a premise meta a end line line ell b premise not0 meta b end line line ell c premise not0 not0 parenthesis meta a imply meta b end parenthesis end line line ell d because prop lemma remove double neg modus ponens ell c indeed meta a imply meta b end line line ell e because 1rule mp modus ponens ell d modus ponens ell a indeed meta b end line because prop lemma from contradiction modus ponens ell e modus ponens ell b indeed not0 parenthesis meta a imply meta b end parenthesis end line line ell f end block any term meta a comma meta b end line line ell g because 1rule deduction modus ponens ell f indeed meta a imply not0 meta b imply not0 not0 parenthesis meta a imply meta b end parenthesis imply not0 parenthesis meta a imply meta b end parenthesis end line line ell h premise meta a end line line ell i premise not0 meta b end line line ell j because prop lemma mp2 modus ponens ell g modus ponens ell h modus ponens ell i indeed not0 not0 parenthesis meta a imply meta b end parenthesis imply not0 parenthesis meta a imply meta b end parenthesis end line line ell k because prop lemma auto imply indeed not0 not0 parenthesis meta a imply meta b end parenthesis imply not0 not0 parenthesis meta a imply meta b end parenthesis end line because 1rule ad absurdum modus ponens ell j modus ponens ell k indeed not0 parenthesis meta a imply meta b end parenthesis qed end math ] "
" [ math in theory system Q lemma prop lemma tertium non datur says for all terms meta a indeed meta a or0 not0 meta a end lemma end math ] "
" [ math system Q proof of prop lemma tertium non datur reads any term meta a end line line ell a because prop lemma auto imply indeed not0 meta a imply not0 meta a end line because 1rule repetition modus ponens ell a indeed meta a or0 not0 meta a qed end math ] "
" [ math in theory system Q lemma prop lemma from negations says for all terms meta a comma meta b indeed meta a imply meta b infer not0 meta a imply meta b infer meta b end lemma end math ] "
" [ math system Q proof of prop lemma from negations 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 not0 meta a imply meta b end line line ell c because prop lemma tertium non datur indeed meta a or0 not0 meta a end line because prop lemma from disjuncts modus ponens ell c modus ponens ell a modus ponens ell b indeed meta b qed end math ] "
" [ math in theory system Q lemma prop lemma imply negation says for all terms meta a indeed meta a imply not0 meta a infer not0 meta a end lemma end math ] "
" [ math system Q proof of prop lemma imply negation reads any term meta a end line line ell a premise meta a imply not0 meta a end line line ell b because prop lemma auto imply indeed not0 meta a imply not0 meta a end line line ell c because prop lemma tertium non datur indeed meta a or0 not0 meta a end line because prop lemma from disjuncts modus ponens ell c modus ponens ell a modus ponens ell b indeed not0 meta a qed end math ] "
" [ math in theory system Q lemma prop lemma from three disjuncts says for all terms meta a comma meta b comma meta c comma meta d indeed meta a or0 meta b or0 meta c infer meta a imply meta d infer meta b imply meta d infer meta c imply meta d infer meta d end lemma end math ] "
" [ math system Q proof of prop lemma from three disjuncts reads block any term meta a comma meta b comma meta c comma meta d end line line ell big a premise meta a or0 meta b or0 meta c end line line ell big b premise meta b imply meta d end line line ell big c premise meta c imply meta d end line line ell big d premise not0 meta a end line line ell big e because 1rule repetition modus ponens ell big a indeed not0 meta a imply parenthesis meta b or0 meta c end parenthesis end line line ell big f because 1rule mp modus ponens ell big e modus ponens ell big d indeed meta b or0 meta c end line because prop lemma from disjuncts modus ponens ell big f modus ponens ell big b modus ponens ell big c indeed meta d end line line ell big g end block any term meta a comma meta b comma meta c comma meta d end line line ell a because 1rule deduction modus ponens ell big g indeed meta a or0 meta b or0 meta c imply parenthesis meta b imply meta d end parenthesis imply parenthesis meta c imply meta d end parenthesis imply not0 meta a imply meta d end line line ell b because prop lemma auto imply indeed parenthesis meta a imply meta d end parenthesis imply meta a imply meta d end line line ell c premise meta a or0 meta b or0 meta c end line line ell d premise meta a imply meta d end line line ell e premise meta b imply meta d end line line ell f premise meta c imply meta d end line line ell g because prop lemma mp3 modus ponens ell a modus ponens ell c modus ponens ell e modus ponens ell f indeed not0 meta a imply meta d end line line ell h because 1rule mp modus ponens ell b modus ponens ell d indeed meta a imply meta d end line because prop lemma from negations modus ponens ell h modus ponens ell g indeed meta d qed end math ] "
" [ math in theory system Q lemma prop lemma negate first disjunct says for all terms meta a comma meta b indeed meta a or0 meta b infer not0 meta a infer meta b end lemma end math ] "
" [ math system Q proof of prop lemma negate first disjunct reads any term meta a comma meta b end line line ell a premise meta a or0 meta b end line line ell b premise not0 meta a end line line ell c because 1rule repetition modus ponens ell a indeed not0 meta a imply meta b end line because 1rule mp modus ponens ell c modus ponens ell b indeed meta b qed end math ] "
" [ math in theory system Q lemma prop lemma negate second disjunct says for all terms meta a comma meta b indeed meta a or0 meta b infer not0 meta b infer meta a end lemma end math ] "
" [ math system Q proof of prop lemma negate second disjunct reads any term meta a comma meta b end line line ell a premise meta a or0 meta b end line line ell b premise not0 meta b end line line ell c because 1rule repetition modus ponens ell a indeed not0 meta a imply meta b end line because prop lemma negative mt modus ponens ell c modus ponens ell b indeed meta a qed end math ] "
(***)
" [ math in theory system Q lemma prop lemma expand disjuncts says for all terms meta a comma meta b comma meta c comma meta d indeed meta a or0 meta b infer meta c or0 meta d infer meta b or0 meta d or0 parenthesis meta a and0 meta c end parenthesis end lemma end math ] "
" [ math system Q proof of prop lemma expand disjuncts reads block any term meta a comma meta b comma meta c comma meta d end line line ell a premise meta a or0 meta b end line line ell b premise meta c or0 meta d end line line ell c premise not0 meta b end line line ell d premise not0 meta d end line line ell e because prop lemma negate second disjunct modus ponens ell a modus ponens ell c indeed meta a end line line ell f because prop lemma negate second disjunct modus ponens ell b modus ponens ell d indeed meta c end line because prop lemma join conjuncts modus ponens ell e modus ponens ell f indeed meta a and0 meta c end line line ell g end block any term meta a comma meta b comma meta c comma meta d end line line ell h because 1rule deduction modus ponens ell g indeed meta a or0 meta b imply meta c or0 meta d imply not0 meta b imply not0 meta d imply meta a and0 meta c end line line ell i premise meta a or0 meta b end line line ell j premise meta c or0 meta d end line line ell k because prop lemma mp2 modus ponens ell h modus ponens ell i modus ponens ell j indeed not0 meta b imply not0 meta d imply meta a and0 meta c end line because 1rule repetition modus ponens ell k indeed meta b or0 meta d or0 parenthesis meta a and0 meta c end parenthesis qed end math ] "
" [ math in theory system Q lemma prop lemma from two times two disjuncts says for all terms meta a comma meta b comma meta c comma meta d comma meta e indeed meta a or0 meta b infer meta c or0 meta d infer meta a imply meta c imply meta e infer meta a imply meta d imply meta e infer meta b imply meta c imply meta e infer meta b imply meta d imply meta e infer meta e end lemma end math ] "
" [ math system Q proof of prop lemma from two times two disjuncts reads block any term meta a comma meta b comma meta c comma meta d comma meta e end line line ell a premise meta c or0 meta d end line line ell b premise meta a imply meta c imply meta e end line line ell c premise meta a imply meta d imply meta e end line line ell d premise meta a end line line ell e because 1rule mp modus ponens ell b modus ponens ell d indeed meta c imply meta e end line line ell f because 1rule mp modus ponens ell c modus ponens ell d indeed meta d imply meta e end line because prop lemma from disjuncts modus ponens ell a modus ponens ell e modus ponens ell f indeed meta e end line line ell g end block block any term meta a comma meta b comma meta c comma meta d comma meta e end line line ell big a premise meta a or0 meta b end line line ell big b premise meta c or0 meta d end line line ell big c premise meta b imply meta c imply meta e end line line ell big d premise meta b imply meta d imply meta e end line line ell big e premise not0 meta a end line line ell big f because prop lemma negate first disjunct modus ponens ell big a modus ponens ell big e indeed meta b end line line ell big g because 1rule mp modus ponens ell big c modus ponens ell big f indeed meta c imply meta e end line line ell big h because 1rule mp modus ponens ell big d modus ponens ell big f indeed meta d imply meta e end line because prop lemma from disjuncts modus ponens ell big b modus ponens ell big g modus ponens ell big h indeed meta e end line line ell big i end block any term meta a comma meta b comma meta c comma meta d comma meta e end line line ell h because 1rule deduction modus ponens ell g indeed meta c or0 meta d imply parenthesis meta a imply meta c imply meta e end parenthesis imply parenthesis meta a imply meta d imply meta e end parenthesis imply meta a imply meta e end line line ell i because 1rule deduction modus ponens ell big i indeed meta a or0 meta b imply meta c or0 meta d imply parenthesis meta b imply meta c imply meta e end parenthesis imply parenthesis meta b imply meta d imply meta e end parenthesis imply not0 meta a imply meta e end line line ell j premise meta a or0 meta b end line line ell k premise meta c or0 meta d end line line ell l premise meta a imply meta c imply meta e end line line ell m premise meta a imply meta d imply meta e end line line ell n premise meta b imply meta c imply meta e end line line ell o premise meta b imply meta d imply meta e end line line ell p because prop lemma mp3 modus ponens ell h modus ponens ell k modus ponens ell l modus ponens ell m indeed meta a imply meta e end line line ell q because prop lemma mp4 modus ponens ell i modus ponens ell j modus ponens ell k modus ponens ell n modus ponens ell o indeed not0 meta a imply meta e end line because prop lemma from negations modus ponens ell p modus ponens ell q indeed meta e qed end math ] "
(*** EQUALITY ***)
" [ math in theory system Q lemma lemma eqReflexivity says for all terms meta x indeed meta x = meta x end lemma end math ] "
" [ math system Q proof of lemma eqReflexivity reads any term meta x end line line ell a because axiom leqReflexivity indeed meta x <= meta x end line because lemma leqAntisymmetry modus ponens ell a modus ponens ell a indeed meta x = meta x qed end math ] "
" [ math in theory system Q lemma lemma eqSymmetry says for all terms meta x comma meta y indeed meta x = meta y infer meta y = meta x end lemma end math ] "
" [ math system Q proof of lemma eqSymmetry reads any term meta x comma meta y end line line ell a premise meta x = meta y end line line ell b because lemma eqReflexivity indeed meta x = meta x end line because lemma equality modus ponens ell a modus ponens ell b indeed meta y = meta x qed end math ] "
" [ math in theory system Q lemma lemma eqTransitivity says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta y = meta z infer meta x = meta z end lemma end math ] "
" [ math system Q proof of lemma eqTransitivity reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell b premise meta y = meta z end line line ell c because lemma eqSymmetry modus ponens ell a indeed meta y = meta x end line because lemma equality modus ponens ell c modus ponens ell b indeed meta x = meta z qed end math ] "
" [ math in theory system Q lemma lemma eqTransitivity4 says for all terms meta x comma meta y comma meta z comma meta u indeed meta x = meta y infer meta y = meta z infer meta z = meta u infer meta x = meta u end lemma end math ] "
" [ math system Q proof of lemma eqTransitivity4 reads any term meta x comma meta y comma meta z comma meta u end line line ell a premise meta x = meta y end line line ell b premise meta y = meta z end line line ell c premise meta z = meta u end line line ell d because lemma eqTransitivity modus ponens ell a modus ponens ell b indeed meta x = meta z end line because lemma eqTransitivity modus ponens ell d modus ponens ell c indeed meta x = meta u qed end math ] "
" [ math in theory system Q lemma lemma eqTransitivity5 says for all terms meta x comma meta y comma meta z comma meta u comma meta v indeed meta x = meta y infer meta y = meta z infer meta z = meta u infer meta u = meta v infer meta x = meta v end lemma end math ] "
" [ math system Q proof of lemma eqTransitivity5 reads any term meta x comma meta y comma meta z comma meta u comma meta v end line line ell a premise meta x = meta y end line line ell b premise meta y = meta z end line line ell c premise meta z = meta u end line line ell d premise meta u = meta v end line line ell e because lemma eqTransitivity4 modus ponens ell a modus ponens ell b modus ponens ell c indeed meta x = meta u end line because lemma eqTransitivity modus ponens ell e modus ponens ell d indeed meta x = meta v qed end math ] "
" [ math in theory system Q lemma lemma eqTransitivity6 says for all terms meta x comma meta y comma meta z comma meta u comma meta v comma meta w indeed meta x = meta y infer meta y = meta z infer meta z = meta u infer meta u = meta v infer meta v = meta w infer meta x = meta w end lemma end math ] "
" [ math system Q proof of lemma eqTransitivity6 reads any term meta x comma meta y comma meta z comma meta u comma meta v comma meta w end line line ell a premise meta x = meta y end line line ell b premise meta y = meta z end line line ell c premise meta z = meta u end line line ell d premise meta u = meta v end line line ell e premise meta v = meta w end line line ell f because lemma eqTransitivity5 modus ponens ell a modus ponens ell b modus ponens ell c modus ponens ell d indeed meta x = meta v end line because lemma eqTransitivity modus ponens ell f modus ponens ell e indeed meta x = meta w qed end math ] "
" [ math in theory system Q lemma lemma plus0Left says for all terms meta x indeed 0 + meta x = meta x end lemma end math ] "
" [ math system Q proof of lemma plus0Left reads any term meta x end line line ell a because axiom plus0 indeed meta x + 0 = meta x end line line ell b because axiom plusCommutativity indeed 0 + meta x = meta x + 0 end line because lemma eqTransitivity modus ponens ell b modus ponens ell a indeed 0 + meta x = meta x qed end math ] "
" [ math in theory system Q lemma lemma times1Left says for all terms meta x indeed 1 * meta x = meta x end lemma end math ] "
" [ math system Q proof of lemma times1Left reads any term meta x end line line ell a because axiom times1 indeed meta x * 1 = meta x end line line ell b because axiom timesCommutativity indeed 1 * meta x = meta x * 1 end line because lemma eqTransitivity modus ponens ell b modus ponens ell a indeed 1 * meta x = meta x qed end math ] "
" [ math in theory system Q lemma lemma eqAdditionLeft says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta z + meta x = meta z + meta y end lemma end math ] "
" [ math system Q proof of lemma eqAdditionLeft reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell b because lemma eqAddition modus ponens ell a indeed meta x + meta z = meta y + meta z end line line ell c because axiom plusCommutativity indeed meta z + meta x = meta x + meta z end line line ell d because axiom plusCommutativity indeed meta y + meta z = meta z + meta y end line because lemma eqTransitivity4 modus ponens ell c modus ponens ell b modus ponens ell d indeed meta z + meta x = meta z + meta y qed end math ] "
" [ math in theory system Q lemma lemma eqMultiplicationLeft says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta z * meta x = meta z * meta y end lemma end math ] "
" [ math system Q proof of lemma eqMultiplicationLeft reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell b because lemma eqMultiplication modus ponens ell a indeed meta x * meta z = meta y * meta z end line line ell c because axiom timesCommutativity indeed meta z * meta x = meta x * meta z end line line ell d because axiom timesCommutativity indeed meta y * meta z = meta z * meta y end line because lemma eqTransitivity4 modus ponens ell c modus ponens ell b modus ponens ell d indeed meta z * meta x = meta z * meta y qed end math ] "
" [ math in theory system Q lemma lemma distributionOut says for all terms meta x comma meta y comma meta z indeed meta x * meta y + meta x * meta z = meta x * parenthesis meta y + meta z end parenthesis end lemma end math ] "
" [ math system Q proof of lemma distributionOut reads any term meta x comma meta y comma meta z end line line ell a because axiom distribution indeed meta x * parenthesis meta y + meta z end parenthesis = meta x * meta y + meta x * meta z end line because lemma eqSymmetry modus ponens ell a indeed meta x * meta y + meta x * meta z = meta x * parenthesis meta y + meta z end parenthesis qed end math ] "
" [ math in theory system Q lemma lemma three2twoTerms says for all terms meta x comma meta y comma meta z comma meta u indeed meta y + meta z = meta u infer meta x + meta y + meta z = meta x + meta u end lemma end math ] "
" [ math system Q proof of lemma three2twoTerms reads any term meta x comma meta y comma meta z comma meta u end line line ell a premise meta y + meta z = meta u end line line ell b because lemma eqAdditionLeft modus ponens ell a indeed meta x + parenthesis meta y + meta z end parenthesis = meta x + meta u end line line ell c because axiom plusAssociativity indeed meta x + meta y + meta z = meta x + parenthesis meta y + meta z end parenthesis end line because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed meta x + meta y + meta z = meta x + meta u qed end math ] "
" [ math in theory system Q lemma lemma three2threeTerms says for all terms meta x comma meta y comma meta z indeed meta x + meta y + meta z = meta x + meta z + meta y end lemma end math ] "
" [ math system Q proof of lemma three2threeTerms reads any term meta x comma meta y comma meta z end line line ell a because axiom plusCommutativity indeed meta y + meta z = meta z + meta y end line line ell b because lemma three2twoTerms modus ponens ell a indeed meta x + meta y + meta z = meta x + parenthesis meta z + meta y end parenthesis end line line ell c because axiom plusAssociativity indeed meta x + meta z + meta y = meta x + parenthesis meta z + meta y end parenthesis end line line ell d because lemma eqSymmetry modus ponens ell c indeed meta x + parenthesis meta z + meta y end parenthesis = meta x + meta z + meta y end line because lemma eqTransitivity modus ponens ell b modus ponens ell d indeed meta x + meta y + meta z = meta x + meta z + meta y qed end math ] "
" [ math in theory system Q lemma lemma three2twoFactors says for all terms meta x comma meta y comma meta z comma meta u indeed meta y * meta z = meta u infer meta x * meta y * meta z = meta x * meta u end lemma end math ] "
" [ math system Q proof of lemma three2twoFactors reads any term meta x comma meta y comma meta z comma meta u end line line ell a premise meta y * meta z = meta u end line line ell b because lemma eqMultiplicationLeft modus ponens ell a indeed meta x * parenthesis meta y * meta z end parenthesis = meta x * meta u end line line ell c because axiom timesAssociativity indeed meta x * meta y * meta z = meta x * parenthesis meta y * meta z end parenthesis end line because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed meta x * meta y * meta z = meta x * meta u qed end math ] "
" [ math in theory system Q lemma lemma addEquations says for all terms meta x comma meta y comma meta z comma meta u indeed meta x = meta y infer meta z = meta u infer meta x + meta z = meta y + meta u end lemma end math ] "
" [ math system Q proof of lemma addEquations reads any term meta x comma meta y comma meta z comma meta u end line line ell a premise meta x = meta y end line line ell b premise meta z = meta u end line line ell c because lemma eqAddition modus ponens ell a indeed meta x + meta z = meta y + meta z end line line ell d because lemma eqAdditionLeft modus ponens ell b indeed meta y + meta z = meta y + meta u end line because lemma eqTransitivity modus ponens ell c modus ponens ell d indeed meta x + meta z = meta y + meta u qed end math ] "
" [ math in theory system Q lemma lemma subtractEquations says for all terms meta x comma meta y comma meta z comma meta u indeed meta x + meta z = meta y + meta u infer meta z = meta u infer meta x = meta y end lemma end math ] "
" [ math system Q proof of lemma subtractEquations reads any term meta x comma meta y comma meta z comma meta u end line line ell b premise meta x + meta z = meta y + meta u end line line ell a premise meta z = meta u end line line ell c because lemma eqAddition modus ponens ell b indeed meta x + meta z - meta z = meta y + meta u - meta z end line line ell d because lemma plus0Left indeed 0 + meta z = meta z end line line ell e because lemma eqTransitivity modus ponens ell d modus ponens ell a indeed 0 + meta z = meta u end line line ell f because lemma positiveToRight(Eq) modus ponens ell e indeed 0 = meta u - meta z end line line ell g because lemma eqSymmetry modus ponens ell f indeed meta u - meta z = 0 end line line ell h because lemma eqAdditionLeft modus ponens ell g indeed meta y + parenthesis meta u - meta z end parenthesis = meta y + 0 end line line ell i because axiom plusAssociativity indeed meta y + meta u - meta z = meta y + parenthesis meta u - meta z end parenthesis end line line ell j because axiom plus0 indeed meta y + 0 = meta y end line line ell k because lemma eqTransitivity4 modus ponens ell i modus ponens ell h modus ponens ell j indeed meta y + meta u - meta z = meta y end line line ell m because lemma x=x+y-y indeed meta x = meta x + meta z - meta z end line because lemma eqTransitivity4 modus ponens ell m modus ponens ell c modus ponens ell k indeed meta x = meta y qed end math ] "
" [ math in theory system Q lemma lemma subtractEquationsLeft says for all terms meta x comma meta y comma meta z comma meta u indeed meta x + meta z = meta y + meta u infer meta x = meta y infer meta z = meta u end lemma end math ] "
" [ math system Q proof of lemma subtractEquationsLeft reads any term meta x comma meta y comma meta z comma meta u end line line ell b premise meta x + meta z = meta y + meta u end line line ell a premise meta x = meta y end line line ell c because axiom plusCommutativity indeed meta z + meta x = meta x + meta z end line line ell d because axiom plusCommutativity indeed meta y + meta u = meta u + meta y end line line ell e because lemma eqTransitivity4 modus ponens ell c modus ponens ell b modus ponens ell d indeed meta z + meta x = meta u + meta y end line because lemma subtractEquations modus ponens ell e modus ponens ell a indeed meta z = meta u qed end math ] "
" [ math in theory system Q lemma lemma eqNegated says for all terms meta x comma meta y indeed meta x = meta y infer - meta x = - meta y end lemma end math ] "
" [ math system Q proof of lemma eqNegated reads any term meta x comma meta y end line line ell a premise meta x = meta y end line line ell b because axiom negative indeed meta x - meta x = 0 end line line ell c because axiom negative indeed meta y - meta y = 0 end line line ell d because lemma eqSymmetry modus ponens ell c indeed 0 = meta y - meta y end line line ell e because lemma eqTransitivity modus ponens ell b modus ponens ell d indeed meta x - meta x = meta y - meta y end line because lemma subtractEquationsLeft modus ponens ell e modus ponens ell a indeed - meta x = - meta y qed end math ] "
" [ math in theory system Q lemma lemma positiveToRight(Eq) says for all terms meta x comma meta y comma meta z indeed meta x + meta y = meta z infer meta x = meta z - meta y end lemma end math ] "
" [ math system Q proof of lemma positiveToRight(Eq) reads any term meta x comma meta y comma meta z end line line ell a premise meta x + meta y = meta z end line line ell b because lemma eqAddition modus ponens ell a indeed meta x + meta y - meta y = meta z - meta y end line line ell c because lemma x=x+y-y indeed meta x = meta x + meta y - meta y end line because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed meta x = meta z - meta y qed end math ] "
" [ math in theory system Q lemma lemma positiveToLeft(Eq)(1 term) says for all terms meta x comma meta y indeed meta x = meta y infer meta x - meta y = 0 end lemma end math ] "
" [ math system Q proof of lemma positiveToLeft(Eq)(1 term) reads any term meta x comma meta y end line line ell a premise meta x = meta y end line line ell b because lemma eqAddition modus ponens ell a indeed meta x - meta y = meta y - meta y end line line ell c because axiom negative indeed meta y - meta y = 0 end line because lemma eqTransitivity modus ponens ell b modus ponens ell c indeed meta x - meta y = 0 qed end math ] "
" [ math in theory system Q lemma lemma positiveToRight(Leq)(1 term) says for all terms meta y comma meta z indeed meta y <= meta z infer 0 <= meta z - meta y end lemma end math ] "
" [ math system Q proof of lemma positiveToRight(Leq)(1 term) reads any term meta y comma meta z end line line ell a premise meta y <= meta z end line line ell b because lemma plus0Left indeed 0 + meta y = meta y end line line ell c because lemma eqSymmetry modus ponens ell b indeed meta y = 0 + meta y end line line ell d because lemma subLeqLeft modus ponens ell c modus ponens ell a indeed 0 + meta y <= meta z end line because lemma positiveToRight(Leq) modus ponens ell d indeed 0 <= meta z - meta y qed end math ] "
" [ math in theory system Q lemma lemma negativeToLeft(Eq) says for all terms meta x comma meta y comma meta z indeed meta x = meta y - meta z infer meta x + meta z = meta y end lemma end math ] "
" [ math system Q proof of lemma negativeToLeft(Eq) reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y - meta z end line line ell b because lemma eqAddition modus ponens ell a indeed meta x + meta z = meta y - meta z + meta z end line line ell c because lemma three2threeTerms indeed meta y - meta z + meta z = meta y + meta z - meta z end line line ell d because lemma x=x+y-y indeed meta y = meta y + meta z - meta z end line line ell e because lemma eqSymmetry modus ponens ell d indeed meta y + meta z - meta z = meta y end line because lemma eqTransitivity4 modus ponens ell b modus ponens ell c modus ponens ell e indeed meta x + meta z = meta y qed end math ] "
(*** NO EQUALITY ***)
" [ math in theory system Q lemma lemma lessNeq says for all terms meta x comma meta y indeed meta x < meta y infer meta x != meta y end lemma end math ] "
" [ math system Q proof of lemma lessNeq reads any term meta x comma meta y end line line ell a premise meta x < meta y end line line ell b because 1rule repetition modus ponens ell a indeed meta x <= meta y and0 not0 parenthesis meta x = meta y end parenthesis end line because prop lemma second conjunct modus ponens ell b indeed meta x != meta y qed end math ] "
" [ math in theory system Q lemma lemma neqSymmetry says for all terms meta x comma meta y indeed meta x != meta y infer meta y != meta x end lemma end math ] "
" [ math system Q proof of lemma neqSymmetry reads block any term meta x comma meta y end line line ell a premise meta y = meta x end line because lemma eqSymmetry modus ponens ell a indeed meta x = meta y end line line ell b end block any term meta x comma meta y end line line ell c because 1rule deduction modus ponens ell b indeed meta y = meta x imply meta x = meta y end line line ell d premise meta x != meta y end line because prop lemma mt modus ponens ell c modus ponens ell d indeed meta y != meta x qed end math ] "
" [ math in theory system Q lemma lemma neqNegated says for all terms meta x comma meta y indeed meta x != meta y infer - meta x != - meta y end lemma end math ] "
" [ math system Q proof of lemma neqNegated reads block any term meta x comma meta y end line line ell big a premise meta x != meta y end line line ell big b premise - meta x = - meta y end line line ell big c because lemma eqNegated modus ponens ell big b indeed - - meta x = - - meta y end line line ell big d because lemma doubleMinus indeed - - meta x = meta x end line line ell big e because lemma eqSymmetry modus ponens ell big d indeed meta x = - - meta x end line line ell big f because lemma doubleMinus indeed - - meta y = meta y end line line ell big g because lemma eqTransitivity4 modus ponens ell big e modus ponens ell big c modus ponens ell big f indeed meta x = meta y end line because prop lemma from contradiction modus ponens ell big g modus ponens ell big a indeed - meta x != - meta y end line line ell big h end block any term meta x comma meta y end line line ell a because 1rule deduction modus ponens ell big h indeed meta x != meta y imply - meta x = - meta y imply not0 - meta x = - meta y end line line ell b premise meta x != meta y end line line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed - meta x = - meta y imply not0 - meta x = - meta y end line because prop lemma imply negation modus ponens ell c indeed not0 - meta x = - meta y qed end math ] "
" [ math in theory system Q lemma lemma subNeqRight says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta z != meta x infer meta z != meta y end lemma end math ] "
" [ math system Q proof of lemma subNeqRight reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell b premise meta z != meta x end line line ell c because lemma neqSymmetry modus ponens ell b indeed meta x != meta z end line line ell d because lemma subNeqLeft modus ponens ell a modus ponens ell c indeed meta y != meta z end line because lemma neqSymmetry modus ponens ell d indeed meta z != meta y qed end math ] "
" [ math in theory system Q lemma lemma subNeqLeft says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta x != meta z infer meta y != meta z end lemma end math ] "
" [ math system Q proof of lemma subNeqLeft reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell b premise meta x != meta z end line line ell c because axiom equality indeed meta y = meta x imply meta y = meta z imply meta x = meta z end line line ell d because lemma eqSymmetry modus ponens ell a indeed meta y = meta x end line line ell e because 1rule mp modus ponens ell c modus ponens ell d indeed meta y = meta z imply meta x = meta z end line line ell f because prop lemma contrapositive modus ponens ell e indeed meta x != meta z imply meta y != meta z end line because 1rule mp modus ponens ell f modus ponens ell b indeed meta y != meta z qed end math ] "
" [ math in theory system Q lemma lemma neqAddition says for all terms meta x comma meta y comma meta z indeed meta x != meta y infer meta x + meta z != meta y + meta z end lemma end math ] "
" [ math system Q proof of lemma neqAddition reads block any term meta x comma meta y comma meta z end line line ell big a premise meta x != meta y end line line ell big b premise meta x + meta z = meta y + meta z end line line ell big c because lemma eqReflexivity indeed meta z = meta z end line line ell big d because lemma subtractEquations modus ponens ell big b modus ponens ell big c indeed meta x = meta y end line because prop lemma from contradiction modus ponens ell big d modus ponens ell big a indeed meta x + meta z != meta y + meta z end line line ell big e end block any term meta x comma meta y comma meta z end line line ell a because 1rule deduction modus ponens ell big e indeed meta x != meta y imply meta x + meta z = meta y + meta z imply meta x + meta z != meta y + meta z end line line ell b premise meta x != meta y end line line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed meta x + meta z = meta y + meta z imply meta x + meta z != meta y + meta z end line because prop lemma imply negation modus ponens ell c indeed meta x + meta z != meta y + meta z qed end math ] "
" [ math in theory system Q lemma lemma neqMultiplication says for all terms meta x comma meta y comma meta z indeed meta z != 0 infer meta x != meta y infer meta x * meta z != meta y * meta z end lemma end math ] "
" [ math system Q proof of lemma neqMultiplication reads block any term meta x comma meta y comma meta z end line line ell a premise meta z != 0 end line line ell b premise meta x != meta y end line line ell c premise meta x * meta z = meta y * meta z end line line ell d because lemma x=x*y*(1/y) modus ponens ell a indeed meta x = meta x * meta z * 1/ meta z end line line ell e because lemma eqMultiplication modus ponens ell c indeed meta x * meta z * 1/ meta z = meta y * meta z * 1/ meta z end line line ell f because lemma x=x*y*(1/y) modus ponens ell a indeed meta y = meta y * meta z * 1/ meta z end line line ell g because lemma eqSymmetry modus ponens ell f indeed meta y * meta z * 1/ meta z = meta y end line line ell h because lemma eqTransitivity4 modus ponens ell d modus ponens ell e modus ponens ell g indeed meta x = meta y end line because prop lemma from contradiction modus ponens ell h modus ponens ell b indeed meta x * meta z != meta y * meta z end line line ell i end block any term meta x comma meta y comma meta z end line line ell j because 1rule deduction modus ponens ell i indeed meta z != 0 imply meta x != meta y imply meta x * meta z = meta y * meta z imply meta x * meta z != meta y * meta z end line line ell k premise meta z != 0 end line line ell l premise meta x != meta y end line line ell m because prop lemma mp2 modus ponens ell j modus ponens ell k modus ponens ell l indeed meta x * meta z = meta y * meta z imply meta x * meta z != meta y * meta z end line because prop lemma imply negation modus ponens ell m indeed meta x * meta z != meta y * meta z qed end math ] "
(*** NEGATIVE ***)
" [ math in theory system Q lemma lemma uniqueNegative says for all terms meta x comma meta y comma meta z indeed meta x + meta y = 0 infer meta x + meta z = 0 infer meta y = meta z end lemma end math ] "
" [ math system Q proof of lemma uniqueNegative reads any term meta x comma meta y comma meta z end line line ell a premise meta x + meta y = 0 end line line ell b premise meta x + meta z = 0 end line line ell c because axiom plusCommutativity indeed meta y + meta x = meta x + meta y end line line ell d because lemma eqTransitivity modus ponens ell c modus ponens ell a indeed meta y + meta x = 0 end line line ell e because lemma positiveToRight(Eq) modus ponens ell d indeed meta y = 0 - meta x end line line ell f because axiom plusCommutativity indeed meta z + meta x = meta x + meta z end line line ell g because lemma eqTransitivity modus ponens ell f modus ponens ell b indeed meta z + meta x = 0 end line line ell h because lemma positiveToRight(Eq) modus ponens ell g indeed meta z = 0 - meta x end line line ell i because lemma eqSymmetry modus ponens ell h indeed 0 - meta x = meta z end line because lemma eqTransitivity modus ponens ell e modus ponens ell i indeed meta y = meta z qed end math ] "
" [ math in theory system Q lemma lemma doubleMinus says for all terms meta x indeed - - meta x = meta x end lemma end math ] "
" [ math system Q proof of lemma doubleMinus reads any term meta x end line line ell a because axiom negative indeed - meta x - - meta x = 0 end line line ell b because lemma x+y=zBackwards modus ponens ell a indeed 0 = - - meta x - meta x end line line ell c because lemma negativeToLeft(Eq) modus ponens ell b indeed 0 + meta x = - - meta x end line line ell d because lemma plus0Left indeed 0 + meta x = meta x end line because lemma equality modus ponens ell c modus ponens ell d indeed - - meta x = meta x qed end math ] "
(*** LEQ, nummer 1 af 2 ***)
" [ math in theory system Q lemma lemma leqLessEq says for all terms meta x comma meta y indeed meta x <= meta y infer meta x < meta y or0 meta x = meta y end lemma end math ] "
" [ math system Q proof of lemma leqLessEq reads block any term meta x comma meta y end line line ell a premise meta x <= meta y end line line ell b premise not0 meta x < meta y end line line ell c because lemma fromNotLess modus ponens ell b indeed meta y <= meta x end line because lemma leqAntisymmetry modus ponens ell a modus ponens ell c indeed meta x = meta y end line line ell d end block any term meta x comma meta y end line line ell big a because 1rule deduction modus ponens ell d indeed meta x <= meta y imply not0 meta x < meta y imply meta x = meta y end line line ell big b premise meta x <= meta y end line line ell big c because 1rule mp modus ponens ell big a modus ponens ell big b indeed not0 meta x < meta y imply meta x = meta y end line because 1rule repetition modus ponens ell big c indeed meta x < meta y or0 meta x = meta y qed end math ] "
" [ math in theory system Q lemma lemma lessLeq says for all terms meta x comma meta y indeed meta x < meta y infer meta x <= meta y end lemma end math ] "
" [ math system Q proof of lemma lessLeq reads any term meta x comma meta y end line line ell a premise meta x < meta y end line line ell b because 1rule repetition modus ponens ell a indeed meta x <= meta y and0 not0 parenthesis meta x = meta y end parenthesis end line because prop lemma first conjunct modus ponens ell b indeed meta x <= meta y qed end math ] "
" [ math in theory system Q lemma lemma from leqGeq says for all terms meta a comma meta x comma meta y indeed meta x <= meta y imply meta a infer meta y <= meta x imply meta a infer meta a end lemma end math ] "
" [ math system Q proof of lemma from leqGeq reads any term meta a comma meta x comma meta y end line line ell a premise meta x <= meta y imply meta a end line line ell b premise meta y <= meta x imply meta a end line line ell c because axiom leqTotality indeed meta x <= meta y or0 meta y <= meta x end line because prop lemma from disjuncts modus ponens ell c modus ponens ell a modus ponens ell b indeed meta a qed end math ] "
" [ math in theory system Q lemma lemma subLeqRight says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta z <= meta x infer meta z <= meta y end lemma end math ] "
" [ math system Q proof of lemma subLeqRight reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell b premise meta z <= meta x end line line ell c because lemma eqLeq modus ponens ell a indeed meta x <= meta y end line because lemma leqTransitivity modus ponens ell b modus ponens ell c indeed meta z <= meta y qed end math ] "
" [ math in theory system Q lemma lemma subLeqLeft says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta x <= meta z infer meta y <= meta z end lemma end math ] "
" [ math system Q proof of lemma subLeqLeft reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell b premise meta x <= meta z end line line ell c because lemma eqSymmetry modus ponens ell a indeed meta y = meta x end line line ell d because lemma eqLeq modus ponens ell c indeed meta y <= meta x end line because lemma leqTransitivity modus ponens ell d modus ponens ell b indeed meta y <= meta z qed end math ] "
" [ math in theory system Q lemma lemma leqPlus1 says for all terms meta x comma meta y indeed meta x <= meta y infer meta x < meta y + 1 end lemma end math ] "
" [ math system Q proof of lemma leqPlus1 reads any term meta x comma meta y end line line ell a premise meta x <= meta y end line line ell b because lemma 0<1 indeed 0 < 1 end line line ell c because lemma lessAdditionLeft modus ponens ell b indeed meta y + 0 < meta y + 1 end line line ell d because axiom plus0 indeed meta y + 0 = meta y end line line ell e because lemma subLessLeft modus ponens ell d modus ponens ell c indeed meta y < meta y + 1 end line because lemma leqLessTransitivity modus ponens ell a modus ponens ell e indeed meta x < meta y + 1 qed end math ] "
" [ math in theory system Q lemma lemma positiveToRight(Leq) says for all terms meta x comma meta y comma meta z indeed meta x + meta y <= meta z infer meta x <= meta z - meta y end lemma end math ] "
" [ math system Q proof of lemma positiveToRight(Leq) reads any term meta x comma meta y comma meta z end line line ell a premise meta x + meta y <= meta z end line line ell b because lemma leqAddition modus ponens ell a indeed meta x + meta y - meta y <= meta z - meta y end line line ell c because lemma x=x+y-y indeed meta x = meta x + meta y - meta y end line line ell d because lemma eqSymmetry modus ponens ell c indeed meta x + meta y - meta y = meta x end line because lemma subLeqLeft modus ponens ell d modus ponens ell b indeed meta x <= meta z - meta y qed end math ] "
" [ math in theory system Q lemma lemma leqAdditionLeft says for all terms meta x comma meta y comma meta z indeed meta x <= meta y infer meta z + meta x <= meta z + meta y end lemma end math ] "
" [ math system Q proof of lemma leqAdditionLeft reads any term meta x comma meta y comma meta z end line line ell a premise meta x <= meta y end line line ell b because lemma leqAddition modus ponens ell a indeed meta x + meta z <= meta y + meta z end line line ell c because axiom plusCommutativity indeed meta x + meta z = meta z + meta x end line line ell d because axiom plusCommutativity indeed meta y + meta z = meta z + meta y end line line ell e because lemma subLeqLeft modus ponens ell c modus ponens ell b indeed meta z + meta x <= meta y + meta z end line because lemma subLeqRight modus ponens ell d modus ponens ell e indeed meta z + meta x <= meta z + meta y qed end math ] "
" [ math in theory system Q lemma lemma leqSubtraction says for all terms meta x comma meta y comma meta z indeed meta x + meta z <= meta y + meta z infer meta x <= meta y end lemma end math ] "
" [ math system Q proof of lemma leqSubtraction reads any term meta x comma meta y comma meta z end line line ell a premise meta x + meta z <= meta y + meta z end line line ell b because lemma leqAddition modus ponens ell a indeed meta x + meta z - meta z <= meta y + meta z - meta z end line line ell c because lemma x=x+y-y indeed meta x = meta x + meta z - meta z end line line ell d because lemma eqSymmetry modus ponens ell c indeed meta x + meta z - meta z = meta x end line line ell e because lemma x=x+y-y indeed meta y = meta y + meta z - meta z end line line ell f because lemma eqSymmetry modus ponens ell e indeed meta y + meta z - meta z = meta y end line line ell g because lemma subLeqLeft modus ponens ell d modus ponens ell b indeed meta x <= meta y + meta z - meta z end line because lemma subLeqRight modus ponens ell f modus ponens ell g indeed meta x <= meta y qed end math ] "
" [ math in theory system Q lemma lemma leqSubtractionLeft says for all terms meta x comma meta y comma meta z indeed meta z + meta x <= meta z + meta y infer meta x <= meta y end lemma end math ] "
" [ math system Q proof of lemma leqSubtractionLeft reads any term meta x comma meta y comma meta z end line line ell a premise meta z + meta x <= meta z + meta y end line line ell b because axiom plusCommutativity indeed meta z + meta x = meta x + meta z end line line ell c because axiom plusCommutativity indeed meta z + meta y = meta y + meta z end line line ell d because lemma subLeqLeft modus ponens ell b modus ponens ell a indeed meta x + meta z <= meta z + meta y end line line ell e because lemma subLeqRight modus ponens ell c modus ponens ell d indeed meta x + meta z <= meta y + meta z end line because lemma leqSubtraction modus ponens ell e indeed meta x <= meta y qed end math ] "
" [ math in theory system Q lemma lemma thirdGeq says for all terms meta x comma meta y indeed meta x <= ex3 and0 meta y <= ex3 end lemma end math ] "
" [ math system Q proof of lemma thirdGeq reads block any term meta x comma meta y end line line ell a premise meta x <= meta y end line line ell b because axiom leqReflexivity indeed meta y <= meta y end line line ell c because prop lemma join conjuncts modus ponens ell a modus ponens ell b indeed meta x <= meta y and0 meta y <= meta y end line because 1rule exist intro at ex3 at meta y modus ponens ell c indeed meta x <= ex3 and0 meta y <= ex3 end line line ell d end block block any term meta x comma meta y end line line ell e premise meta y <= meta x end line line ell f because axiom leqReflexivity indeed meta x <= meta x end line line ell g because prop lemma join conjuncts modus ponens ell f modus ponens ell e indeed meta x <= meta x and0 meta y <= meta x end line because 1rule exist intro at ex3 at meta x modus ponens ell g indeed meta x <= ex3 and0 meta y <= ex3 end line line ell h end block any term meta x comma meta y end line line ell i because 1rule deduction modus ponens ell d indeed meta x <= meta y imply meta x <= ex3 and0 meta y <= ex3 end line line ell j because 1rule deduction modus ponens ell h indeed meta y <= meta x imply meta x <= ex3 and0 meta y <= ex3 end line line ell k because axiom leqTotality indeed meta x <= meta y or0 meta y <= meta x end line because prop lemma from disjuncts modus ponens ell k modus ponens ell i modus ponens ell j indeed meta x <= ex3 and0 meta y <= ex3 qed end math ] "
" [ math in theory system Q lemma lemma leqNegated says for all terms meta x comma meta y indeed meta x <= meta y infer - meta y <= - meta x end lemma end math ] "
" [ math system Q proof of lemma leqNegated reads any term meta x comma meta y end line line ell a premise meta x <= meta y end line line ell b because lemma leqAddition modus ponens ell a indeed meta x - meta x <= meta y - meta x end line line ell c because axiom negative indeed meta x - meta x = 0 end line line ell d because lemma subLeqLeft modus ponens ell c modus ponens ell b indeed 0 <= meta y - meta x end line line ell e because axiom plusCommutativity indeed meta y - meta x = - meta x + meta y end line line ell f because lemma subLeqRight modus ponens ell e modus ponens ell d indeed 0 <= - meta x + meta y end line line ell g because lemma leqAddition modus ponens ell f indeed 0 - meta y <= - meta x + meta y - meta y end line line ell h because lemma plus0Left indeed 0 - meta y = - meta y end line line ell i because lemma x=x+y-y indeed - meta x = - meta x + meta y - meta y end line line ell j because lemma eqSymmetry modus ponens ell i indeed - meta x + meta y - meta y = - meta x end line line ell k because lemma subLeqLeft modus ponens ell h modus ponens ell g indeed - meta y <= - meta x + meta y - meta y end line because lemma subLeqRight modus ponens ell j modus ponens ell k indeed - meta y <= - meta x qed end math ] "
" [ math in theory system Q lemma lemma addEquations(Leq) says for all terms meta x comma meta y comma meta z comma meta u indeed meta x <= meta y infer meta z <= meta u infer meta x + meta z <= meta y + meta u end lemma end math ] "
" [ math system Q proof of lemma addEquations(Leq) reads any term meta x comma meta y comma meta z comma meta u end line line ell a premise meta x <= meta y end line line ell b premise meta z <= meta u end line line ell c because lemma leqAddition modus ponens ell a indeed meta x + meta z <= meta y + meta z end line line ell d because lemma leqAdditionLeft modus ponens ell b indeed meta y + meta z <= meta y + meta u end line because lemma leqTransitivity modus ponens ell c modus ponens ell d indeed meta x + meta z <= meta y + meta u qed end math ] "
(*** LESS ***)
" [ math in theory system Q lemma lemma leqNeqLess says for all terms meta x comma meta y indeed meta x <= meta y infer meta x != meta y infer meta x < meta y end lemma end math ] "
" [ math system Q proof of lemma leqNeqLess reads any term meta x comma meta y end line line ell a premise meta x <= meta y end line line ell b premise meta x != meta y end line line ell c because prop lemma join conjuncts modus ponens ell a modus ponens ell b indeed meta x <= meta y and0 meta x != meta y end line because 1rule repetition modus ponens ell c indeed meta x < meta y qed end math ] "
" [ math in theory system Q lemma lemma fromLess says for all terms meta x comma meta y indeed meta x < meta y infer not0 meta y <= meta x end lemma end math ] "
" [ math system Q proof of lemma fromLess reads block any term meta x comma meta y end line line ell a premise meta y <= meta x end line because lemma toNotLess modus ponens ell a indeed not0 meta x < meta y end line line ell b end block any term meta x comma meta y end line line ell c because 1rule deduction modus ponens ell b indeed meta y <= meta x imply not0 meta x < meta y end line line ell d premise meta x < meta y end line line ell e because prop lemma add double neg modus ponens ell d indeed not0 not0 meta x < meta y end line because prop lemma mt modus ponens ell c modus ponens ell e indeed not0 meta y <= meta x qed end math ] "
" [ math in theory system Q lemma lemma toLess says for all terms meta x comma meta y indeed not0 meta x <= meta y infer meta y < meta x end lemma end math ] "
" [ math system Q proof of lemma toLess reads block any term meta x comma meta y end line line ell a premise not0 meta y < meta x end line because lemma fromNotLess modus ponens ell a indeed meta x <= meta y end line line ell b end block any term meta x comma meta y end line line ell c because 1rule deduction modus ponens ell b indeed not0 meta y < meta x imply meta x <= meta y end line line ell d premise not0 meta x <= meta y end line because prop lemma negative mt modus ponens ell c modus ponens ell d indeed meta y < meta x qed end math ] "
" [ math in theory system Q lemma lemma fromNotLess says for all terms meta x comma meta y indeed not0 parenthesis meta x < meta y end parenthesis infer meta y <= meta x end lemma end math ] "
" [ math system Q proof of lemma fromNotLess reads block any term meta x comma meta y end line line ell a premise not0 parenthesis meta x < meta y end parenthesis end line line ell big a premise meta x <= meta y end line line ell c because 1rule repetition modus ponens ell a indeed not0 not0 parenthesis meta x <= meta y imply not0 meta x != meta y end parenthesis end line line ell d because prop lemma remove double neg modus ponens ell c indeed meta x <= meta y imply not0 meta x != meta y end line line ell e because 1rule mp modus ponens ell d modus ponens ell big a indeed not0 meta x != meta y end line line ell f because prop lemma remove double neg modus ponens ell e indeed meta x = meta y end line line ell g because lemma eqSymmetry modus ponens ell f indeed meta y = meta x end line because lemma eqLeq modus ponens ell g indeed meta y <= meta x end line line ell h end block any term meta x comma meta y end line line ell i because 1rule deduction modus ponens ell h indeed not0 meta x < meta y imply meta x <= meta y imply meta y <= meta x end line line ell j premise not0 meta x < meta y end line line ell k because 1rule mp modus ponens ell i modus ponens ell j indeed meta x <= meta y imply meta y <= meta x end line line ell l because prop lemma auto imply indeed meta y <= meta x imply meta y <= meta x end line line ell m because axiom leqTotality indeed meta x <= meta y or0 meta y <= meta x end line because prop lemma from disjuncts modus ponens ell m modus ponens ell k modus ponens ell l indeed meta y <= meta x qed end math ] "
" [ math in theory system Q lemma lemma toNotLess says for all terms meta x comma meta y indeed meta x <= meta y infer not0 meta y < meta x end lemma end math ] "
" [ math system Q proof of lemma toNotLess reads block any term meta x comma meta y end line line ell a premise meta x <= meta y end line line ell b premise meta y <= meta x end line line ell c because lemma leqAntisymmetry modus ponens ell b modus ponens ell a indeed meta y = meta x end line because prop lemma add double neg modus ponens ell c indeed not0 not0 meta y = meta x end line line ell d end block any term meta x comma meta y end line line ell e because 1rule deduction modus ponens ell d indeed meta x <= meta y imply meta y <= meta x imply not0 not0 meta y = meta x end line line ell f premise meta x <= meta y end line line ell g because 1rule mp modus ponens ell e modus ponens ell f indeed meta y <= meta x imply not0 not0 meta y = meta x end line line ell h because prop lemma add double neg modus ponens ell g indeed not0 not0 parenthesis meta y <= meta x imply not0 not0 meta y = meta x end parenthesis end line line ell i because 1rule repetition modus ponens ell h indeed not0 parenthesis meta y <= meta x and0 not0 meta y = meta x end parenthesis end line because 1rule repetition modus ponens ell i indeed not0 meta y < meta x qed end math ] "
" [ math in theory system Q lemma lemma lessAddition says for all terms meta x comma meta y comma meta z indeed meta x < meta y infer meta x + meta z < meta y + meta z end lemma end math ] "
" [ math system Q proof of lemma lessAddition reads any term meta x comma meta y comma meta z end line line ell a premise meta x < meta y end line line ell b because lemma lessLeq modus ponens ell a indeed meta x <= meta y end line line ell c because lemma leqAddition modus ponens ell b indeed meta x + meta z <= meta y + meta z end line line ell d because lemma lessNeq modus ponens ell a indeed meta x != meta y end line line ell e because lemma neqAddition modus ponens ell d indeed meta x + meta z != meta y + meta z end line because prop lemma join conjuncts modus ponens ell c modus ponens ell e indeed meta x + meta z < meta y + meta z qed end math ] "
" [ math in theory system Q lemma lemma lessAdditionLeft says for all terms meta x comma meta y comma meta z indeed meta x < meta y infer meta z + meta x < meta z + meta y end lemma end math ] "
" [ math system Q proof of lemma lessAdditionLeft reads any term meta x comma meta y comma meta z end line line ell a premise meta x < meta y end line line ell b because lemma lessAddition modus ponens ell a indeed meta x + meta z < meta y + meta z end line line ell c because axiom plusCommutativity indeed meta x + meta z = meta z + meta x end line line ell d because lemma subLessLeft modus ponens ell c modus ponens ell b indeed meta z + meta x < meta y + meta z end line line ell e because axiom plusCommutativity indeed meta y + meta z = meta z + meta y end line because lemma subLessRight modus ponens ell e modus ponens ell d indeed meta z + meta x < meta z + meta y qed end math ] "
" [ math in theory system Q lemma lemma lessMultiplication says for all terms meta x comma meta y comma meta z indeed 0 < meta z infer meta x < meta y infer meta x * meta z < meta y * meta z end lemma end math ] "
" [ math system Q proof of lemma lessMultiplication reads any term meta x comma meta y comma meta z end line line ell a premise 0 < meta z end line line ell b premise meta x < meta y end line line ell c because lemma lessLeq modus ponens ell b indeed meta x <= meta y end line line ell d because lemma lessLeq modus ponens ell a indeed 0 <= meta z end line line ell e because lemma leqMultiplication modus ponens ell d modus ponens ell c indeed meta x * meta z <= meta y * meta z end line line ell f because lemma lessNeq modus ponens ell b indeed meta x != meta y end line line ell g because lemma lessNeq modus ponens ell a indeed 0 != meta z end line line ell big a because lemma neqSymmetry modus ponens ell g indeed meta z != 0 end line line ell h because lemma neqMultiplication modus ponens ell big a modus ponens ell f indeed meta x * meta z != meta y * meta z end line because lemma leqNeqLess modus ponens ell e modus ponens ell h indeed meta x * meta z < meta y * meta z qed end math ] "
" [ math in theory system Q lemma lemma lessMultiplicationLeft says for all terms meta x comma meta y comma meta z indeed 0 < meta z infer meta x < meta y infer meta z * meta x < meta z * meta y end lemma end math ] "
" [ math system Q proof of lemma lessMultiplicationLeft reads any term meta x comma meta y comma meta z end line line ell a premise 0 < meta z end line line ell b premise meta x < meta y end line line ell c because lemma lessMultiplication modus ponens ell a modus ponens ell b indeed meta x * meta z < meta y * meta z end line line ell d because axiom timesCommutativity indeed meta x * meta z = meta z * meta x end line line ell e because axiom timesCommutativity indeed meta y * meta z = meta z * meta y end line line ell f because lemma subLessLeft modus ponens ell d modus ponens ell c indeed meta z * meta x < meta y * meta z end line because lemma subLessRight modus ponens ell e modus ponens ell f indeed meta z * meta x < meta z * meta y qed end math ] "
" [ math in theory system Q lemma lemma lessDivision says for all terms meta x comma meta y comma meta z indeed 0 <= meta z infer meta x * meta z < meta y * meta z infer meta x < meta y end lemma end math ] "
" [ math system Q proof of lemma lessDivision reads any term meta x comma meta y comma meta z end line line ell a premise 0 <= meta z end line line ell b premise meta x * meta z < meta y * meta z end line line ell c because lemma fromLess modus ponens ell b indeed not0 meta y * meta z <= meta x * meta z end line line ell d because axiom leqMultiplication indeed 0 <= meta z imply meta y <= meta x imply meta y * meta z <= meta x * meta z end line line ell e because 1rule mp modus ponens ell d modus ponens ell a indeed meta y <= meta x imply meta y * meta z <= meta x * meta z end line line ell f because prop lemma contrapositive modus ponens ell e indeed not0 meta y * meta z <= meta x * meta z imply not0 meta y <= meta x end line line ell g because 1rule mp modus ponens ell f modus ponens ell c indeed not0 meta y <= meta x end line because lemma toLess modus ponens ell g indeed meta x < meta y qed end math ] "
" [ math in theory system Q lemma lemma addEquations(Less) says for all terms meta x comma meta y comma meta z comma meta u indeed meta x < meta y infer meta z < meta u infer meta x + meta z < meta y + meta u end lemma end math ] "
" [ math system Q proof of lemma addEquations(Less) reads any term meta x comma meta y comma meta z comma meta u end line line ell a premise meta x < meta y end line line ell b premise meta z < meta u end line line ell c because lemma lessAddition modus ponens ell a indeed meta x + meta z < meta y + meta z end line line ell d because lemma lessAdditionLeft modus ponens ell b indeed meta y + meta z < meta y + meta u end line because lemma lessTransitivity modus ponens ell c modus ponens ell d indeed meta x + meta z < meta y + meta u qed end math ] "
" [ math in theory system Q lemma lemma leqLessTransitivity says for all terms meta x comma meta y comma meta z indeed meta x <= meta y infer meta y < meta z infer meta x < meta z end lemma end math ] "
" [ math system Q proof of lemma leqLessTransitivity reads block any term meta x comma meta y comma meta z end line line ell a premise meta x <= meta y end line line ell b premise meta y < meta z end line line ell big a premise meta x = meta z end line line ell c because prop lemma first conjunct modus ponens ell b indeed meta y <= meta z end line line ell d because prop lemma second conjunct modus ponens ell b indeed meta y != meta z end line line ell f because lemma subLeqLeft modus ponens ell big a modus ponens ell a indeed meta z <= meta y end line line ell g because lemma leqAntisymmetry modus ponens ell c modus ponens ell f indeed meta y = meta z end line because prop lemma from contradiction modus ponens ell g modus ponens ell d indeed meta x != meta z end line line ell h end block any term meta x comma meta y comma meta z end line line ell i because 1rule deduction modus ponens ell h indeed meta x <= meta y imply meta y < meta z imply meta x = meta z imply meta x != meta z end line line ell j premise meta x <= meta y end line line ell k premise meta y < meta z end line line ell l because prop lemma mp2 modus ponens ell i modus ponens ell j modus ponens ell k indeed meta x = meta z imply meta x != meta z end line line ell m because prop lemma imply negation modus ponens ell l indeed meta x != meta z end line line ell o because prop lemma first conjunct modus ponens ell k indeed meta y <= meta z end line line ell q because lemma leqTransitivity modus ponens ell j modus ponens ell o indeed meta x <= meta z end line because prop lemma join conjuncts modus ponens ell q modus ponens ell m indeed meta x < meta z qed end math ] "
" [ math in theory system Q lemma lemma lessLeqTransitivity says for all terms meta x comma meta y comma meta z indeed meta x < meta y infer meta y <= meta z infer meta x < meta z end lemma end math ] "
" [ math system Q proof of lemma lessLeqTransitivity reads block any term meta x comma meta y comma meta z end line line ell a premise meta x < meta y end line line ell b premise meta y <= meta z end line line ell big a premise meta z = meta x end line line ell c because prop lemma first conjunct modus ponens ell a indeed meta x <= meta y end line line ell d because prop lemma second conjunct modus ponens ell a indeed meta x != meta y end line line ell f because lemma subLeqRight modus ponens ell big a modus ponens ell b indeed meta y <= meta x end line line ell g because lemma leqAntisymmetry modus ponens ell c modus ponens ell f indeed meta x = meta y end line because prop lemma from contradiction modus ponens ell g modus ponens ell d indeed meta z != meta x end line line ell h end block any term meta x comma meta y comma meta z end line line ell i because 1rule deduction modus ponens ell h indeed meta x < meta y imply meta y <= meta z imply meta z = meta x imply meta z != meta x end line line ell j premise meta x < meta y end line line ell k premise meta y <= meta z end line line ell l because prop lemma mp2 modus ponens ell i modus ponens ell j modus ponens ell k indeed meta z = meta x imply meta z != meta x end line line ell m because prop lemma imply negation modus ponens ell l indeed meta z != meta x end line line ell n because lemma neqSymmetry modus ponens ell m indeed meta x != meta z end line line ell o because prop lemma first conjunct modus ponens ell j indeed meta x <= meta y end line line ell q because lemma leqTransitivity modus ponens ell o modus ponens ell k indeed meta x <= meta z end line because prop lemma join conjuncts modus ponens ell q modus ponens ell n indeed meta x < meta z qed end math ] "
" [ math in theory system Q lemma lemma lessTransitivity says for all terms meta x comma meta y comma meta z indeed meta x < meta y infer meta y < meta z infer meta x < meta z end lemma end math ] "
" [ math system Q proof of lemma lessTransitivity reads any term meta x comma meta y comma meta z end line line ell a premise meta x < meta y end line line ell b premise meta y < meta z end line line ell c because prop lemma first conjunct modus ponens ell b indeed meta y <= meta z end line because lemma lessLeqTransitivity modus ponens ell a modus ponens ell c indeed meta x < meta z qed end math ] "
" [ math in theory system Q lemma lemma lessTotality says for all terms meta x comma meta y indeed meta x < meta y or0 meta x = meta y or0 meta y < meta x end lemma end math ] "
" [ math system Q proof of lemma lessTotality reads block any term meta x comma meta y end line line ell a premise not0 meta x < meta y end line line ell b premise meta x != meta y end line line ell c because lemma fromNotLess modus ponens ell a indeed meta y <= meta x end line line ell d because lemma neqSymmetry modus ponens ell b indeed meta y != meta x end line because lemma leqNeqLess modus ponens ell c modus ponens ell d indeed meta y < meta x end line line ell e end block any term meta x comma meta y end line line ell f because 1rule deduction modus ponens ell e indeed not0 meta x < meta y imply meta x != meta y imply meta y < meta x end line because 1rule repetition modus ponens ell f indeed meta x < meta y or0 meta x = meta y or0 meta y < meta x qed end math ] "
" [ math in theory system Q lemma lemma subLessRight says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta z < meta x infer meta z < meta y end lemma end math ] "
" [ math system Q proof of lemma subLessRight reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell b premise meta z < meta x end line line ell c because 1rule repetition modus ponens ell b indeed meta z <= meta x and0 meta z != meta x end line line ell d because prop lemma first conjunct modus ponens ell c indeed meta z <= meta x end line line ell e because lemma subLeqRight modus ponens ell a modus ponens ell d indeed meta z <= meta y end line line ell f because prop lemma second conjunct modus ponens ell c indeed meta z != meta x end line line ell k because lemma subNeqRight modus ponens ell a modus ponens ell f indeed meta z != meta y end line because prop lemma join conjuncts modus ponens ell e modus ponens ell k indeed meta z < meta y qed end math ] "
" [ math in theory system Q lemma lemma subLessLeft says for all terms meta x comma meta y comma meta z indeed meta x = meta y infer meta x < meta z infer meta y < meta z end lemma end math ] "
" [ math system Q proof of lemma subLessLeft reads any term meta x comma meta y comma meta z end line line ell a premise meta x = meta y end line line ell b premise meta x < meta z end line line ell c because 1rule repetition modus ponens ell b indeed meta x <= meta z and0 meta x != meta z end line line ell d because prop lemma first conjunct modus ponens ell c indeed meta x <= meta z end line line ell e because lemma subLeqLeft modus ponens ell a modus ponens ell d indeed meta y <= meta z end line line ell f because prop lemma second conjunct modus ponens ell c indeed meta x != meta z end line line ell k because lemma subNeqLeft modus ponens ell a modus ponens ell f indeed meta y != meta z end line because prop lemma join conjuncts modus ponens ell e modus ponens ell k indeed meta y < meta z qed end math ] "
" [ math in theory system Q lemma lemma negativeLessPositive says for all terms meta x indeed 0 < meta x infer - meta x < meta x end lemma end math ] "
" [ math system Q proof of lemma negativeLessPositive reads any term meta x end line line ell a premise 0 < meta x end line line ell b because prop lemma first conjunct modus ponens ell a indeed 0 <= meta x end line line ell d because lemma leqAddition modus ponens ell b indeed 0 - meta x <= meta x - meta x end line line ell e because lemma plus0Left indeed 0 - meta x = - meta x end line line ell f because axiom negative indeed meta x - meta x = 0 end line line ell g because lemma subLeqLeft modus ponens ell e modus ponens ell d indeed - meta x <= meta x - meta x end line line ell h because lemma subLeqRight modus ponens ell f modus ponens ell g indeed - meta x <= 0 end line because lemma leqLessTransitivity modus ponens ell h modus ponens ell a indeed - meta x < meta x qed end math ] "
" [ math in theory system Q lemma lemma lessNegated says for all terms meta x comma meta y indeed meta x < meta y infer - meta y < - meta x end lemma end math ] "
" [ math system Q proof of lemma lessNegated reads any term meta x comma meta y end line line ell a premise meta x < meta y end line line ell b because lemma lessLeq modus ponens ell a indeed meta x <= meta y end line line ell c because lemma leqNegated modus ponens ell b indeed - meta y <= - meta x end line line ell d because lemma lessNeq modus ponens ell a indeed meta x != meta y end line line ell e because lemma neqNegated modus ponens ell d indeed not0 - meta x = - meta y end line line ell f because lemma neqSymmetry modus ponens ell e indeed not0 - meta y = - meta x end line because lemma leqNeqLess modus ponens ell c modus ponens ell f indeed - meta y < - meta x qed end math ] "
" [ math in theory system Q lemma lemma positiveNegated says for all terms meta x indeed 0 < meta x infer - meta x < 0 end lemma end math ] "
" [ math system Q proof of lemma positiveNegated reads any term meta x end line line ell a premise 0 < meta x end line line ell b because lemma lessNegated modus ponens ell a indeed - meta x < - 0 end line line ell c because lemma -0=0 indeed - 0 = 0 end line because lemma subLessRight modus ponens ell c modus ponens ell b indeed - meta x < 0 qed end math ] "
" [ math in theory system Q lemma lemma nonpositiveNegated says for all terms meta x indeed meta x <= 0 infer 0 <= - meta x end lemma end math ] "
" [ math system Q proof of lemma nonpositiveNegated reads any term meta x end line line ell a premise meta x <= 0 end line line ell b because lemma leqNegated modus ponens ell a indeed - 0 <= - meta x end line line ell c because lemma -0=0 indeed - 0 = 0 end line because lemma subLeqLeft modus ponens ell c modus ponens ell b indeed 0 <= - meta x qed end math ] "
" [ math in theory system Q lemma lemma negativeNegated says for all terms meta x indeed meta x < 0 infer 0 < - meta x end lemma end math ] "
" [ math system Q proof of lemma negativeNegated reads any term meta x end line line ell a premise meta x < 0 end line line ell b because lemma lessNegated modus ponens ell a indeed - 0 < - meta x end line line ell c because lemma -0=0 indeed - 0 = 0 end line because lemma subLessLeft modus ponens ell c modus ponens ell b indeed 0 < - meta x qed end math ] "
" [ math in theory system Q lemma lemma nonnegativeNegated says for all terms meta x indeed 0 <= meta x infer - meta x <= 0 end lemma end math ] "
" [ math system Q proof of lemma nonnegativeNegated reads any term meta x end line line ell a premise 0 <= meta x end line line ell b because lemma leqNegated modus ponens ell a indeed - meta x <= - 0 end line line ell c because lemma -0=0 indeed - 0 = 0 end line because lemma subLeqRight modus ponens ell c modus ponens ell b indeed - meta x <= 0 qed end math ] "
" [ math in theory system Q lemma lemma positiveHalved says for all terms meta x indeed 0 < meta x infer 0 < 1/2 * meta x end lemma end math ] "
" [ math system Q proof of lemma positiveHalved reads any term meta x end line line ell a premise 0 < meta x end line line ell b because lemma 0<1/2 indeed 0 < 1/2 end line line ell c because lemma lessMultiplicationLeft modus ponens ell b modus ponens ell a indeed 1/2 * 0 < 1/2 * meta x end line line ell d because lemma x*0=0 indeed 1/2 * 0 = 0 end line because lemma subLessLeft modus ponens ell d modus ponens ell c indeed 0 < 1/2 * meta x qed end math ] "
(*** NUMERISK ***)
" [ math in theory system Q lemma lemma nonnegativeNumerical says for all terms meta x indeed 0 <= meta x infer | meta x | = meta x end lemma end math ] "
" [ math system Q proof of lemma nonnegativeNumerical reads any term meta x end line line ell a premise 0 <= meta x end line line ell b because 1rule ifThenElse true modus ponens ell a indeed if( 0 <= meta x , meta x , - meta x ) = meta x end line because 1rule repetition modus ponens ell b indeed | meta x | = meta x qed end math ] "
" [ math in theory system Q lemma lemma positiveNumerical says for all terms meta x indeed 0 < meta x infer | meta x | = meta x end lemma end math ] "
" [ math system Q proof of lemma positiveNumerical reads any term meta x end line line ell a premise 0 < meta x end line line ell b because lemma lessLeq modus ponens ell a indeed 0 <= meta x end line because lemma nonnegativeNumerical modus ponens ell b indeed | meta x | = meta x qed end math ] "
" [ math in theory system Q lemma lemma negativeNumerical says for all terms meta x indeed meta x < 0 infer | meta x | = - meta x end lemma end math ] "
" [ math system Q proof of lemma negativeNumerical reads any term meta x end line line ell a premise meta x < 0 end line line ell b because lemma fromLess modus ponens ell a indeed not0 0 <= meta x end line line ell c because 1rule ifThenElse false modus ponens ell b indeed if( 0 <= meta x , meta x , - meta x ) = - meta x end line because 1rule repetition modus ponens ell c indeed | meta x | = - meta x qed end math ] "
" [ math in theory system Q lemma lemma nonpositiveNumerical says for all terms meta x indeed meta x <= 0 infer | meta x | = - meta x end lemma end math ] "
" [ math system Q proof of lemma nonpositiveNumerical reads block any term meta x end line line ell a premise meta x < 0 end line because lemma negativeNumerical modus ponens ell a indeed | meta x | = - meta x end line line ell b end block block any term meta x end line line ell c premise meta x = 0 end line line ell d because lemma eqSymmetry modus ponens ell c indeed 0 = meta x end line line ell e because lemma eqLeq modus ponens ell d indeed 0 <= meta x end line line ell f because lemma nonnegativeNumerical modus ponens ell e indeed | meta x | = meta x end line line ell g because lemma -0=0 indeed - 0 = 0 end line line ell h because lemma eqSymmetry modus ponens ell g indeed 0 = - 0 end line line ell i because lemma eqNegated modus ponens ell d indeed - 0 = - meta x end line because lemma eqTransitivity5 modus ponens ell f modus ponens ell c modus ponens ell h modus ponens ell i indeed | meta x | = - meta x end line line ell j end block any term meta x end line line ell big a because 1rule deduction modus ponens ell b indeed meta x < 0 imply | meta x | = - meta x end line line ell big b because 1rule deduction modus ponens ell j indeed meta x = 0 imply | meta x | = - meta x end line line ell big c premise meta x <= 0 end line line ell big d because lemma leqLessEq modus ponens ell big c indeed meta x < 0 or0 meta x = 0 end line because prop lemma from disjuncts modus ponens ell big d modus ponens ell big a modus ponens ell big b indeed | meta x | = - meta x qed end math ] "
" [ math in theory system Q lemma lemma |0|=0 says | 0 | = 0 end lemma end math ] "
" [ math system Q proof of lemma |0|=0 reads line ell a because axiom leqReflexivity indeed 0 <= 0 end line because lemma nonnegativeNumerical modus ponens ell a indeed | 0 | = 0 qed end math ] "
" [ math in theory system Q lemma lemma 0<=|x| says for all terms meta x indeed 0 <= | meta x | end lemma end math ] "
" [ math system Q proof of lemma 0<=|x| reads block any term meta x end line line ell a premise 0 <= meta x end line line ell b because lemma nonnegativeNumerical modus ponens ell a indeed | meta x | = meta x end line line ell c because lemma eqSymmetry modus ponens ell b indeed meta x = | meta x | end line because lemma subLeqRight modus ponens ell c modus ponens ell a indeed 0 <= | meta x | end line line ell d end block block any term meta x end line line ell e premise not0 0 <= meta x end line line ell f because lemma toLess modus ponens ell e indeed meta x < 0 end line line ell g because lemma negativeNumerical modus ponens ell f indeed | meta x | = - meta x end line line ell h because lemma eqSymmetry modus ponens ell g indeed - meta x = | meta x | end line line ell i because lemma negativeNegated modus ponens ell f indeed 0 < - meta x end line line ell j because lemma lessLeq modus ponens ell i indeed 0 <= - meta x end line because lemma subLeqRight modus ponens ell h modus ponens ell j indeed 0 <= | meta x | end line line ell k end block any term meta x end line line ell l because 1rule deduction modus ponens ell d indeed 0 <= meta x imply 0 <= | meta x | end line line ell m because 1rule deduction modus ponens ell k indeed not0 0 <= meta x imply 0 <= | meta x | end line because prop lemma from negations modus ponens ell l modus ponens ell m indeed 0 <= | meta x | qed end math ] "
" [ math in theory system Q lemma lemma sameNumerical says for all terms meta x comma meta y indeed meta x = meta y infer | meta x | = | meta y | end lemma end math ] "
" [ math system Q proof of lemma sameNumerical reads block any term meta x comma meta y end line line ell big a premise 0 <= meta x end line line ell big b premise meta x = meta y end line line ell big c because lemma nonnegativeNumerical modus ponens ell big a indeed | meta x | = meta x end line line ell big d because lemma subLeqRight modus ponens ell big b modus ponens ell big a indeed 0 <= meta y end line line ell big e because lemma nonnegativeNumerical modus ponens ell big d indeed | meta y | = meta y end line line ell big f because lemma eqSymmetry modus ponens ell big e indeed meta y = | meta y | end line because lemma eqTransitivity4 modus ponens ell big c modus ponens ell big b modus ponens ell big f indeed | meta x | = | meta y | end line line ell big g end block block any term meta x comma meta y end line line ell d premise not0 0 <= meta x end line line ell e premise meta x = meta y end line line ell f because lemma toLess modus ponens ell d indeed meta x < 0 end line line ell g because lemma negativeNumerical modus ponens ell f indeed | meta x | = - meta x end line line ell h because lemma subLessLeft modus ponens ell e modus ponens ell f indeed meta y < 0 end line line ell i because lemma negativeNumerical modus ponens ell h indeed | meta y | = - meta y end line line ell j because lemma eqSymmetry modus ponens ell i indeed - meta y = | meta y | end line line ell k because lemma eqNegated modus ponens ell e indeed - meta x = - meta y end line because lemma eqTransitivity4 modus ponens ell g modus ponens ell k modus ponens ell j indeed | meta x | = | meta y | end line line ell l end block any term meta x comma meta y end line line ell m premise meta x = meta y end line line ell n because 1rule deduction modus ponens ell big g indeed 0 <= meta x imply meta x = meta y imply | meta x | = | meta y | end line line ell o because 1rule deduction modus ponens ell l indeed not0 0 <= meta x imply meta x = meta y imply | meta x | = | meta y | end line line ell p because prop lemma from negations modus ponens ell n modus ponens ell o indeed meta x = meta y imply | meta x | = | meta y | end line because 1rule mp modus ponens ell p modus ponens ell m indeed | meta x | = | meta y | qed end math ] "
" [ math in theory system Q lemma lemma signNumerical(+) says for all terms meta x indeed 0 < meta x infer | meta x | = | - meta x | end lemma end math ] "
" [ math system Q proof of lemma signNumerical(+) reads any term meta x end line line ell a premise 0 < meta x end line line ell b because lemma positiveNumerical modus ponens ell a indeed | meta x | = meta x end line line ell c because lemma positiveNegated modus ponens ell a indeed - meta x < 0 end line line ell d because lemma negativeNumerical modus ponens ell c indeed | - meta x | = - - meta x end line line ell e because lemma doubleMinus indeed - - meta x = meta x end line line ell f because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed | - meta x | = meta x end line line ell g because lemma eqSymmetry modus ponens ell f indeed meta x = | - meta x | end line because lemma eqTransitivity modus ponens ell b modus ponens ell g indeed | meta x | = | - meta x | qed end math ] "
" [ math in theory system Q lemma lemma signNumerical says for all terms meta x indeed | meta x | = | - meta x | end lemma end math ] "
" [ math system Q proof of lemma signNumerical reads block any term meta x end line line ell a premise meta x < 0 end line line ell b because lemma negativeNegated modus ponens ell a indeed 0 < - meta x end line line ell c because lemma signNumerical(+) modus ponens ell b indeed | - meta x | = | - - meta x | end line line ell d because lemma doubleMinus indeed - - meta x = meta x end line line ell e because lemma sameNumerical modus ponens ell d indeed | - - meta x | = | meta x | end line line ell f because lemma eqTransitivity modus ponens ell c modus ponens ell e indeed | - meta x | = | meta x | end line because lemma eqSymmetry modus ponens ell f indeed | meta x | = | - meta x | end line line ell big a end block block any term meta x end line line ell a premise meta x = 0 end line line ell b because lemma eqNegated modus ponens ell a indeed - meta x = - 0 end line line ell c because lemma -0=0 indeed - 0 = 0 end line line ell d because lemma eqSymmetry modus ponens ell a indeed 0 = meta x end line line ell e because lemma eqTransitivity4 modus ponens ell b modus ponens ell c modus ponens ell d indeed - meta x = meta x end line line ell f because lemma eqSymmetry modus ponens ell e indeed meta x = - meta x end line because lemma sameNumerical modus ponens ell f indeed | meta x | = | - meta x | end line line ell big b end block block any term meta x end line line ell a premise 0 < meta x end line because lemma signNumerical(+) modus ponens ell a indeed | meta x | = | - meta x | end line line ell big c end block any term meta x end line line ell big d because 1rule deduction modus ponens ell big a indeed meta x < 0 imply | meta x | = | - meta x | end line line ell big e because 1rule deduction modus ponens ell big b indeed meta x = 0 imply | meta x | = | - meta x | end line line ell big f because 1rule deduction modus ponens ell big c indeed 0 < meta x imply | meta x | = | - meta x | end line line ell big g because lemma lessTotality indeed meta x < 0 or0 meta x = 0 or0 0 < meta x end line because prop lemma from three disjuncts modus ponens ell big g modus ponens ell big d modus ponens ell big e modus ponens ell big f indeed | meta x | = | - meta x | qed end math ] "
" [ math in theory system Q lemma lemma numericalDifference says for all terms meta x comma meta y indeed | meta x - meta y | = | meta y - meta x | end lemma end math ] "
" [ math system Q proof of lemma numericalDifference reads any term meta x comma meta y end line line ell a because lemma signNumerical indeed | meta x - meta y | = | - parenthesis meta x - meta y end parenthesis | end line line ell b because lemma minusNegated indeed - parenthesis meta x - meta y end parenthesis = meta y - meta x end line line ell c because lemma sameNumerical modus ponens ell b indeed | - parenthesis meta x - meta y end parenthesis | = | meta y - meta x | end line because lemma eqTransitivity modus ponens ell a modus ponens ell c indeed | meta x - meta y | = | meta y - meta x | qed end math ] "
" [ math in theory system Q lemma lemma splitNumericalSumHelper says for all terms meta x comma meta y indeed | - meta x - meta y | <= | - meta x | + | - meta y | infer | meta x + meta y | <= | meta x | + | meta y | end lemma end math ] "
" [ math system Q proof of lemma splitNumericalSumHelper reads any term meta x comma meta y end line line ell a premise | - meta x - meta y | <= | - meta x | + | - meta y | end line line ell b because lemma signNumerical indeed | meta x | = | - meta x | end line line ell c because lemma signNumerical indeed | meta y | = | - meta y | end line line ell d because lemma addEquations modus ponens ell b modus ponens ell c indeed | meta x | + | meta y | = | - meta x | + | - meta y | end line line ell e because lemma eqSymmetry modus ponens ell d indeed | - meta x | + | - meta y | = | meta x | + | meta y | end line line ell f because lemma -x-y=-(x+y) indeed - meta x - meta y = - parenthesis meta x + meta y end parenthesis end line line ell g because lemma sameNumerical modus ponens ell f indeed | - meta x - meta y | = | - parenthesis meta x + meta y end parenthesis | end line line ell h because lemma signNumerical indeed | meta x + meta y | = | - parenthesis meta x + meta y end parenthesis | end line line ell i because lemma eqSymmetry modus ponens ell h indeed | - parenthesis meta x + meta y end parenthesis | = | meta x + meta y | end line line ell j because lemma eqTransitivity modus ponens ell g modus ponens ell i indeed | - meta x - meta y | = | meta x + meta y | end line line ell k because lemma subLeqRight modus ponens ell e modus ponens ell a indeed | - meta x - meta y | <= | meta x | + | meta y | end line because lemma subLeqLeft modus ponens ell j modus ponens ell k indeed | meta x + meta y | <= | meta x | + | meta y | qed end math ] "
" [ math in theory system Q lemma lemma splitNumericalSum(++) says for all terms meta x comma meta y indeed 0 <= meta x infer 0 <= meta y infer | meta x + meta y | <= | meta x | + | meta y | end lemma end math ] "
" [ math system Q proof of lemma splitNumericalSum(++) reads any term meta x comma meta y end line line ell a premise 0 <= meta x end line line ell b premise 0 <= meta y end line line ell c because lemma addEquations(Leq) modus ponens ell a modus ponens ell b indeed 0 + 0 <= meta x + meta y end line line ell d because axiom plus0 indeed 0 + 0 = 0 end line line ell e because lemma subLeqLeft modus ponens ell d modus ponens ell c indeed 0 <= meta x + meta y end line line ell f because lemma nonnegativeNumerical modus ponens ell e indeed | meta x + meta y | = meta x + meta y end line line ell big a because lemma nonnegativeNumerical modus ponens ell a indeed | meta x | = meta x end line line ell big b because lemma nonnegativeNumerical modus ponens ell b indeed | meta y | = meta y end line line ell big c because lemma addEquations modus ponens ell big a modus ponens ell big b indeed | meta x | + | meta y | = meta x + meta y end line line ell big d because lemma eqSymmetry modus ponens ell big c indeed meta x + meta y = | meta x | + | meta y | end line line ell big e because lemma eqTransitivity modus ponens ell f modus ponens ell big d indeed | meta x + meta y | = | meta x | + | meta y | end line because lemma eqLeq modus ponens ell big e indeed | meta x + meta y | <= | meta x | + | meta y | qed end math ] "
" [ math in theory system Q lemma lemma splitNumericalSum(--) says for all terms meta x comma meta y indeed meta x <= 0 infer meta y <= 0 infer | meta x + meta y | <= | meta x | + | meta y | end lemma end math ] "
" [ math system Q proof of lemma splitNumericalSum(--) reads any term meta x comma meta y end line line ell a premise meta x <= 0 end line line ell b premise meta y <= 0 end line line ell c because lemma nonpositiveNegated modus ponens ell a indeed 0 <= - meta x end line line ell d because lemma nonpositiveNegated modus ponens ell b indeed 0 <= - meta y end line line ell e because lemma splitNumericalSum(++) modus ponens ell c modus ponens ell d indeed | - meta x - meta y | <= | - meta x | + | - meta y | end line because lemma splitNumericalSumHelper modus ponens ell e indeed | meta x + meta y | <= | meta x | + | meta y | qed end math ] "
" [ math in theory system Q lemma lemma splitNumericalSum(+-, smallNegative) says for all terms meta x comma meta y indeed 0 <= meta x infer meta y <= 0 infer | meta y | <= | meta x | infer | meta x + meta y | <= | meta x | end lemma end math ] "
" [ math system Q proof of lemma splitNumericalSum(+-, smallNegative) reads any term meta x comma meta y end line line ell a premise 0 <= meta x end line line ell b premise meta y <= 0 end line line ell c premise | meta y | <= | meta x | end line line ell big a because lemma leqAdditionLeft modus ponens ell b indeed meta x + meta y <= meta x + 0 end line line ell big b because axiom plus0 indeed meta x + 0 = meta x end line line ell big c because lemma subLeqRight modus ponens ell big b modus ponens ell big a indeed meta x + meta y <= meta x end line line ell d because lemma positiveToRight(Leq)(1 term) modus ponens ell c indeed 0 <= | meta x | - | meta y | end line line ell f because lemma nonpositiveNumerical modus ponens ell b indeed | meta y | = - meta y end line line ell g because lemma eqNegated modus ponens ell f indeed - | meta y | = - - meta y end line line ell h because lemma doubleMinus indeed - - meta y = meta y end line line ell i because lemma eqTransitivity modus ponens ell g modus ponens ell h indeed - | meta y | = meta y end line line ell e because lemma nonnegativeNumerical modus ponens ell a indeed | meta x | = meta x end line line ell j because lemma addEquations modus ponens ell e modus ponens ell i indeed | meta x | - | meta y | = meta x + meta y end line line ell k because lemma subLeqRight modus ponens ell j modus ponens ell d indeed 0 <= meta x + meta y end line line ell l because lemma nonnegativeNumerical modus ponens ell k indeed | meta x + meta y | = meta x + meta y end line line ell m because lemma eqSymmetry modus ponens ell l indeed meta x + meta y = | meta x + meta y | end line line ell big q because lemma eqSymmetry modus ponens ell e indeed meta x = | meta x | end line line ell n because lemma subLeqLeft modus ponens ell m modus ponens ell big c indeed | meta x + meta y | <= meta x end line because lemma subLeqRight modus ponens ell big q modus ponens ell n indeed | meta x + meta y | <= | meta x | qed end math ] "
" [ math in theory system Q lemma lemma splitNumericalSum(+-, bigNegative) says for all terms meta x comma meta y indeed 0 <= meta x infer meta y <= 0 infer | meta x | < | meta y | infer | meta x + meta y | <= | meta y | end lemma end math ] "
" [ math system Q proof of lemma splitNumericalSum(+-, bigNegative) reads any term meta x comma meta y end line line ell a premise 0 <= meta x end line line ell b premise meta y <= 0 end line line ell c premise | meta x | < | meta y | end line line ell big a because lemma nonnegativeNegated modus ponens ell a indeed - meta x <= 0 end line line ell big b because lemma nonpositiveNegated modus ponens ell b indeed 0 <= - meta y end line line ell big c because lemma signNumerical indeed | meta x | = | - meta x | end line line ell big d because lemma subLessLeft modus ponens ell big c modus ponens ell c indeed | - meta x | < | meta y | end line line ell big e because lemma signNumerical indeed | meta y | = | - meta y | end line line ell big f because lemma subLessRight modus ponens ell big e modus ponens ell big d indeed | - meta x | < | - meta y | end line line ell big g because lemma lessLeq modus ponens ell big f indeed | - meta x | <= | - meta y | end line line ell big h because lemma splitNumericalSum(+-, smallNegative) modus ponens ell big b modus ponens ell big a modus ponens ell big g indeed | - meta y - meta x | <= | - meta y | end line line ell d because lemma signNumerical indeed | meta x + meta y | = | - parenthesis meta x + meta y end parenthesis | end line line ell e because lemma -x-y=-(x+y) indeed - meta x - meta y = - parenthesis meta x + meta y end parenthesis end line line ell f because axiom plusCommutativity indeed - meta x - meta y = - meta y - meta x end line line ell g because lemma equality modus ponens ell e modus ponens ell f indeed - parenthesis meta x + meta y end parenthesis = - meta y - meta x end line line ell h because lemma sameNumerical modus ponens ell g indeed | - parenthesis meta x + meta y end parenthesis | = | - meta y - meta x | end line line ell i because lemma eqTransitivity modus ponens ell d modus ponens ell h indeed | meta x + meta y | = | - meta y - meta x | end line line ell j because lemma eqSymmetry modus ponens ell i indeed | - meta y - meta x | = | meta x + meta y | end line line ell k because lemma eqSymmetry modus ponens ell big e indeed | - meta y | = | meta y | end line line ell l because lemma subLeqLeft modus ponens ell j modus ponens ell big h indeed | meta x + meta y | <= | - meta y | end line because lemma subLeqRight modus ponens ell k modus ponens ell l indeed | meta x + meta y | <= | meta y | qed end math ] "
" [ math in theory system Q lemma lemma splitNumericalSum(+-) says for all terms meta x comma meta y indeed 0 <= meta x infer meta y <= 0 infer | meta x + meta y | <= | meta x | + | meta y | end lemma end math ] "
" [ math system Q proof of lemma splitNumericalSum(+-) reads block any term meta x comma meta y end line line ell c premise | meta y | <= | meta x | end line line ell a premise 0 <= meta x end line line ell b premise meta y <= 0 end line line ell d because lemma splitNumericalSum(+-, smallNegative) modus ponens ell a modus ponens ell b modus ponens ell c indeed | meta x + meta y | <= | meta x | end line line ell e because lemma 0<=|x| indeed 0 <= | meta y | end line line ell f because lemma leqAdditionLeft modus ponens ell e indeed | meta x | + 0 <= | meta x | + | meta y | end line line ell g because axiom plus0 indeed | meta x | + 0 = | meta x | end line line ell h because lemma subLeqLeft modus ponens ell g modus ponens ell f indeed | meta x | <= | meta x | + | meta y | end line because lemma leqTransitivity modus ponens ell d modus ponens ell h indeed | meta x + meta y | <= | meta x | + | meta y | end line line ell i end block block any term meta x comma meta y end line line ell big c premise not0 | meta y | <= | meta x | end line line ell big a premise 0 <= meta x end line line ell big z premise meta y <= 0 end line line ell big d because lemma toLess modus ponens ell big c indeed | meta x | < | meta y | end line line ell big e because lemma splitNumericalSum(+-, bigNegative) modus ponens ell big a modus ponens ell big z modus ponens ell big d indeed | meta x + meta y | <= | meta y | end line line ell big f because lemma 0<=|x| indeed 0 <= | meta x | end line line ell big g because lemma leqAddition modus ponens ell big f indeed 0 + | meta y | <= | meta x | + | meta y | end line line ell big h because lemma plus0Left indeed 0 + | meta y | = | meta y | end line line ell big i because lemma subLeqLeft modus ponens ell big h modus ponens ell big g indeed | meta y | <= | meta x | + | meta y | end line because lemma leqTransitivity modus ponens ell big e modus ponens ell big i indeed | meta x + meta y | <= | meta x | + | meta y | end line line ell big j end block any term meta x comma meta y end line line ell m because 1rule deduction modus ponens ell i indeed | meta y | <= | meta x | imply 0 <= meta x imply meta y <= 0 imply | meta x + meta y | <= | meta x | + | meta y | end line line ell n because 1rule deduction modus ponens ell big j indeed not0 | meta y | <= | meta x | imply 0 <= meta x imply meta y <= 0 imply | meta x + meta y | <= | meta x | + | meta y | end line line ell o premise 0 <= meta x end line line ell p premise meta y <= 0 end line line ell q because prop lemma from negations modus ponens ell m modus ponens ell n indeed 0 <= meta x imply meta y <= 0 imply | meta x + meta y | <= | meta x | + | meta y | end line because prop lemma mp2 modus ponens ell q modus ponens ell o modus ponens ell p indeed | meta x + meta y | <= | meta x | + | meta y | qed end math ] "
" [ math in theory system Q lemma lemma splitNumericalSum(-+) says for all terms meta x comma meta y indeed meta x <= 0 infer 0 <= meta y infer | meta x + meta y | <= | meta x | + | meta y | end lemma end math ] "
" [ math system Q proof of lemma splitNumericalSum(-+) reads any term meta x comma meta y end line line ell a premise meta x <= 0 end line line ell b premise 0 <= meta y end line line ell c because lemma nonpositiveNegated modus ponens ell a indeed 0 <= - meta x end line line ell d because lemma nonnegativeNegated modus ponens ell b indeed - meta y <= 0 end line line ell e because lemma splitNumericalSum(+-) modus ponens ell c modus ponens ell d indeed | - meta x - meta y | <= | - meta x | + | - meta y | end line because lemma splitNumericalSumHelper modus ponens ell e indeed | meta x + meta y | <= | meta x | + | meta y | qed end math ] "
" [ math in theory system Q lemma lemma splitNumericalSum says for all terms meta x comma meta y indeed | meta x + meta y | <= | meta x | + | meta y | end lemma end math ] "
" [ math system Q proof of lemma splitNumericalSum reads block any term meta x comma meta y end line line ell a premise 0 <= meta x end line line ell b premise 0 <= meta y end line because lemma splitNumericalSum(++) modus ponens ell a modus ponens ell b indeed | meta x + meta y | <= | meta x | + | meta y | end line line ell c end block block any term meta x comma meta y end line line ell d premise 0 <= meta x end line line ell e premise meta y <= 0 end line because lemma splitNumericalSum(+-) modus ponens ell d modus ponens ell e indeed | meta x + meta y | <= | meta x | + | meta y | end line line ell f end block block any term meta x comma meta y end line line ell g premise meta x <= 0 end line line ell h premise 0 <= meta y end line because lemma splitNumericalSum(-+) modus ponens ell g modus ponens ell h indeed | meta x + meta y | <= | meta x | + | meta y | end line line ell i end block block any term meta x comma meta y end line line ell j premise meta x <= 0 end line line ell k premise meta y <= 0 end line because lemma splitNumericalSum(--) modus ponens ell j modus ponens ell k indeed | meta x + meta y | <= | meta x | + | meta y | end line line ell l end block any term meta x comma meta y end line line ell big a because 1rule deduction modus ponens ell c indeed 0 <= meta x imply 0 <= meta y imply | meta x + meta y | <= | meta x | + | meta y | end line line ell big b because 1rule deduction modus ponens ell f indeed 0 <= meta x imply meta y <= 0 imply | meta x + meta y | <= | meta x | + | meta y | end line line ell big c because 1rule deduction modus ponens ell i indeed meta x <= 0 imply 0 <= meta y imply | meta x + meta y | <= | meta x | + | meta y | end line line ell big d because 1rule deduction modus ponens ell l indeed meta x <= 0 imply meta y <= 0 imply | meta x + meta y | <= | meta x | + | meta y | end line line ell big e because lemma from leqGeq modus ponens ell big a modus ponens ell big c indeed 0 <= meta y imply | meta x + meta y | <= | meta x | + | meta y | end line line ell big f because lemma from leqGeq modus ponens ell big b modus ponens ell big d indeed meta y <= 0 imply | meta x + meta y | <= | meta x | + | meta y | end line because lemma from leqGeq modus ponens ell big e modus ponens ell big f indeed | meta x + meta y | <= | meta x | + | meta y | qed end math ] "
" [ math in theory system Q lemma lemma insertMiddleTerm(Numerical) says for all terms meta x comma meta y comma meta z indeed | meta x + meta y | <= | meta x - meta z | + | meta z + meta y | end lemma end math ] "
" [ math system Q proof of lemma insertMiddleTerm(Numerical) reads any term meta x comma meta y comma meta z end line line ell a because lemma splitNumericalSum indeed | parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis | <= | meta x - meta z | + | meta z + meta y | end line line ell b because lemma insertMiddleTerm(Sum) indeed meta x + meta y = parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis end line line ell c because lemma sameNumerical modus ponens ell b indeed | meta x + meta y | = | parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis | end line line ell d because lemma eqSymmetry modus ponens ell c indeed | parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis | = | meta x + meta y | end line because lemma subLeqLeft modus ponens ell d modus ponens ell a indeed | meta x + meta y | <= | meta x - meta z | + | meta z + meta y | qed end math ] "
(*** REGNESTYKKER ***)
" [ math in theory system Q lemma lemma x+y=zBackwards says for all terms meta x comma meta y comma meta z indeed meta x + meta y = meta z infer meta z = meta y + meta x end lemma end math ] "
" [ math system Q proof of lemma x+y=zBackwards reads any term meta x comma meta y comma meta z end line line ell a premise meta x + meta y = meta z end line line ell b because axiom plusCommutativity indeed meta x + meta y = meta y + meta x end line because lemma equality modus ponens ell a indeed meta z = meta y + meta x qed end math ] "
" [ math in theory system Q lemma lemma x*y=zBackwards says for all terms meta x comma meta y comma meta z indeed meta x * meta y = meta z infer meta z = meta y * meta x end lemma end math ] "
" [ math system Q proof of lemma x*y=zBackwards reads any term meta x comma meta y comma meta z end line line ell a premise meta x * meta y = meta z end line line ell b because axiom timesCommutativity indeed meta x * meta y = meta y * meta x end line because lemma equality modus ponens ell a indeed meta z = meta y * meta x qed end math ] "
" [ math in theory system Q lemma lemma x=x+(y-y) says for all terms meta x comma meta y indeed meta x = meta x + parenthesis meta y - meta y end parenthesis end lemma end math ] "
" [ math system Q proof of lemma x=x+(y-y) reads any term meta x comma meta y end line line ell a because axiom plus0 indeed meta x + 0 = meta x end line line ell b because axiom negative indeed meta y - meta y = 0 end line line ell c because lemma eqSymmetry modus ponens ell b indeed 0 = meta y - meta y end line line ell d because lemma eqAdditionLeft modus ponens ell c indeed meta x + 0 = meta x + parenthesis meta y - meta y end parenthesis end line because lemma equality modus ponens ell a modus ponens ell d indeed meta x = meta x + parenthesis meta y - meta y end parenthesis qed end math ] "
" [ math in theory system Q lemma lemma x=x+y-y says for all terms meta x comma meta y indeed meta x = meta x + meta y - meta y end lemma end math ] "
" [ math system Q proof of lemma x=x+y-y reads any term meta x comma meta y end line line ell a because lemma x=x+(y-y) indeed meta x = meta x + parenthesis meta y - meta y end parenthesis end line line ell b because axiom plusAssociativity indeed meta x + meta y - meta y = meta x + parenthesis meta y - meta y end parenthesis end line line ell c because lemma eqSymmetry modus ponens ell b indeed meta x + parenthesis meta y - meta y end parenthesis = meta x + meta y - meta y end line because lemma eqTransitivity modus ponens ell a modus ponens ell c indeed meta x = meta x + meta y - meta y qed end math ] "
" [ math in theory system Q lemma lemma x=x*y*(1/y) says for all terms meta x comma meta y indeed meta y != 0 infer meta x = meta x * meta y * 1/ meta y end lemma end math ] "
" [ math system Q proof of lemma x=x*y*(1/y) reads any term meta x comma meta y end line line ell a premise meta y != 0 end line line ell b because axiom times1 indeed meta x * 1 = meta x end line line ell c because lemma reciprocal modus ponens ell a indeed meta y * 1/ meta y = 1 end line line ell d because lemma three2twoFactors modus ponens ell c indeed meta x * meta y * 1/ meta y = meta x * 1 end line line ell e because lemma eqTransitivity modus ponens ell d modus ponens ell b indeed meta x * meta y * 1/ meta y = meta x end line because lemma eqSymmetry modus ponens ell e indeed meta x = meta x * meta y * 1/ meta y qed end math ] "
" [ math in theory system Q lemma lemma insertMiddleTerm(Sum) says for all terms meta x comma meta y comma meta z indeed meta x + meta y = parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis end lemma end math ] "
" [ math system Q proof of lemma insertMiddleTerm(Sum) reads any term meta x comma meta y comma meta z end line line ell a because lemma x=x+y-y indeed meta x = meta x + meta z - meta z end line line ell b because lemma three2threeTerms indeed meta x + meta z - meta z = meta x - meta z + meta z end line line ell c because lemma eqTransitivity modus ponens ell a modus ponens ell b indeed meta x = meta x - meta z + meta z end line line ell d because lemma eqAddition modus ponens ell c indeed meta x + meta y = parenthesis meta x - meta z end parenthesis + meta z + meta y end line line ell e because axiom plusAssociativity indeed parenthesis meta x - meta z end parenthesis + meta z + meta y = parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis end line because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed meta x + meta y = parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis qed end math ] "
" [ math in theory system Q lemma lemma insertMiddleTerm(Difference) says for all terms meta x comma meta y comma meta z indeed meta x - meta y = parenthesis meta x + meta z end parenthesis - parenthesis meta y + meta z end parenthesis end lemma end math ] "
" [ math system Q proof of lemma insertMiddleTerm(Difference) reads any term meta x comma meta y comma meta z end line line ell a because lemma insertMiddleTerm(Sum) indeed meta x - meta y = parenthesis meta x - - meta z end parenthesis + parenthesis - meta z - meta y end parenthesis end line line ell b because lemma doubleMinus indeed - - meta z = meta z end line line ell c because lemma eqAdditionLeft modus ponens ell b indeed meta x - - meta z = meta x + meta z end line line ell d because axiom plusCommutativity indeed - meta z - meta y = - meta y - meta z end line line ell e because lemma -x-y=-(x+y) indeed - meta y - meta z = - parenthesis meta y + meta z end parenthesis end line line ell f because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed - meta z - meta y = - parenthesis meta y + meta z end parenthesis end line line ell g because lemma addEquations modus ponens ell c modus ponens ell f indeed parenthesis meta x - - meta z end parenthesis + parenthesis - meta z - meta y end parenthesis = parenthesis meta x + meta z end parenthesis - parenthesis meta y + meta z end parenthesis end line because lemma eqTransitivity modus ponens ell a modus ponens ell g indeed meta x - meta y = parenthesis meta x + meta z end parenthesis - parenthesis meta y + meta z end parenthesis qed end math ] "
" [ math in theory system Q lemma lemma x*0+x=x says for all terms meta x indeed meta x * 0 + meta x = meta x end lemma end math ] "
" [ math system Q proof of lemma x*0+x=x reads any term meta x end line line ell big a because axiom times1 indeed meta x * 1 = meta x end line line ell a because lemma eqSymmetry modus ponens ell big a indeed meta x = meta x * 1 end line line ell b because lemma eqAdditionLeft modus ponens ell a indeed meta x * 0 + meta x = meta x * 0 + meta x * 1 end line line ell c because axiom distribution indeed meta x * parenthesis 0 + 1 end parenthesis = meta x * 0 + meta x * 1 end line line ell d because lemma eqSymmetry modus ponens ell c indeed meta x * 0 + meta x * 1 = meta x * parenthesis 0 + 1 end parenthesis end line line ell e because lemma plus0Left indeed 0 + 1 = 1 end line line ell f because lemma eqMultiplicationLeft modus ponens ell e indeed meta x * parenthesis 0 + 1 end parenthesis = meta x * 1 end line because lemma eqTransitivity5 modus ponens ell b modus ponens ell d modus ponens ell f modus ponens ell big a indeed meta x * 0 + meta x = meta x qed end math ] "
" [ math in theory system Q lemma lemma x*0=0 says for all terms meta x indeed meta x * 0 = 0 end lemma end math ] "
" [ math system Q proof of lemma x*0=0 reads any term meta x end line line ell a because lemma x=x+(y-y) indeed meta x * 0 = meta x * 0 + parenthesis meta x - meta x end parenthesis end line line ell b because axiom plusAssociativity indeed meta x * 0 + meta x - meta x = meta x * 0 + parenthesis meta x - meta x end parenthesis end line line ell c because lemma eqSymmetry modus ponens ell b indeed meta x * 0 + parenthesis meta x - meta x end parenthesis = meta x * 0 + meta x - meta x end line line ell d because lemma x*0+x=x indeed meta x * 0 + meta x = meta x end line line ell e because lemma eqAddition modus ponens ell d indeed meta x * 0 + meta x - meta x = meta x - meta x end line line ell f because axiom negative indeed meta x - meta x = 0 end line because lemma eqTransitivity5 modus ponens ell a modus ponens ell c modus ponens ell e modus ponens ell f indeed meta x * 0 = 0 qed end math ] "
" [ math in theory system Q lemma lemma (-1)*(-1)+(-1)*1=0 says (-1) * (-1) + (-1) * 1 = 0 end lemma end math ] "
" [ math system Q proof of lemma (-1)*(-1)+(-1)*1=0 reads line ell a because lemma distributionOut indeed (-1) * (-1) + (-1) * 1 = (-1) * parenthesis (-1) + 1 end parenthesis end line line ell b because axiom negative indeed 1 + (-1) = 0 end line line ell c because axiom plusCommutativity indeed (-1) + 1 = 1 + (-1) end line line ell d because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed (-1) + 1 = 0 end line line ell e because lemma eqMultiplicationLeft modus ponens ell d indeed (-1) * parenthesis (-1) + 1 end parenthesis = (-1) * 0 end line line ell f because lemma x*0=0 indeed (-1) * 0 = 0 end line because lemma eqTransitivity4 modus ponens ell a modus ponens ell e modus ponens ell f indeed (-1) * (-1) + (-1) * 1 = 0 qed end math ] "
" [ math in theory system Q lemma lemma (-1)*(-1)=1 says (-1) * (-1) = 1 end lemma end math ] "
" [ math system Q proof of lemma (-1)*(-1)=1 reads line ell a because lemma x=x+(y-y) indeed (-1) * (-1) = (-1) * (-1) + parenthesis 1 - 1 end parenthesis end line line ell b because axiom times1 indeed (-1) * 1 = (-1) end line line ell c because lemma eqSymmetry modus ponens ell b indeed (-1) = (-1) * 1 end line line ell d because lemma eqAdditionLeft modus ponens ell c indeed 1 - 1 = 1 + (-1) * 1 end line line ell e because lemma eqAdditionLeft modus ponens ell d indeed (-1) * (-1) + parenthesis 1 - 1 end parenthesis = (-1) * (-1) + parenthesis 1 + (-1) * 1 end parenthesis end line line ell f because axiom plusCommutativity indeed 1 + (-1) * 1 = (-1) * 1 + 1 end line line ell g because lemma eqAdditionLeft modus ponens ell f indeed (-1) * (-1) + parenthesis 1 + (-1) * 1 end parenthesis = (-1) * (-1) + parenthesis (-1) * 1 + 1 end parenthesis end line line ell big a because axiom plusAssociativity indeed (-1) * (-1) + (-1) * 1 + 1 = (-1) * (-1) + parenthesis (-1) * 1 + 1 end parenthesis end line line ell h because lemma eqSymmetry modus ponens ell big a indeed (-1) * (-1) + parenthesis (-1) * 1 + 1 end parenthesis = (-1) * (-1) + (-1) * 1 + 1 end line line ell i because lemma (-1)*(-1)+(-1)*1=0 indeed (-1) * (-1) + (-1) * 1 = 0 end line line ell j because lemma eqAddition modus ponens ell i indeed (-1) * (-1) + (-1) * 1 + 1 = 0 + 1 end line line ell k because lemma plus0Left indeed 0 + 1 = 1 end line line ell l because lemma eqTransitivity5 modus ponens ell a modus ponens ell e modus ponens ell g modus ponens ell h indeed (-1) * (-1) = (-1) * (-1) + (-1) * 1 + 1 end line because lemma eqTransitivity4 modus ponens ell l modus ponens ell j modus ponens ell k indeed (-1) * (-1) = 1 qed end math ] "
" [ math in theory system Q lemma lemma 0<1Helper says 1 <= 0 imply 0 <= 1 end lemma end math ] "
" [ math system Q proof of lemma 0<1Helper reads block line ell a premise 1 <= 0 end line line ell b because lemma leqAddition modus ponens ell a indeed 1 + (-1) <= 0 + (-1) end line line ell c because axiom negative indeed 1 + (-1) = 0 end line line ell d because lemma subLeqLeft modus ponens ell c modus ponens ell b indeed 0 <= 0 + (-1) end line line ell e because lemma plus0Left indeed 0 + (-1) = (-1) end line line ell f because lemma subLeqRight modus ponens ell e modus ponens ell d indeed 0 <= (-1) end line line ell g because lemma leqMultiplication modus ponens ell f modus ponens ell f indeed 0 * (-1) <= (-1) * (-1) end line line ell h because lemma x*0=0 indeed (-1) * 0 = 0 end line line ell i because axiom timesCommutativity indeed 0 * (-1) = (-1) * 0 end line line ell j because lemma eqTransitivity modus ponens ell i modus ponens ell h indeed 0 * (-1) = 0 end line line ell k because lemma subLeqLeft modus ponens ell j modus ponens ell g indeed 0 <= (-1) * (-1) end line line ell l because lemma (-1)*(-1)=1 indeed (-1) * (-1) = 1 end line because lemma subLeqRight modus ponens ell l modus ponens ell k indeed 0 <= 1 end line line ell m end block because 1rule deduction modus ponens ell m indeed 1 <= 0 imply 0 <= 1 qed end math ] "
" [ math in theory system Q lemma lemma 0<1 says 0 < 1 end lemma end math ] "
" [ math system Q proof of lemma 0<1 reads line ell a because axiom leqTotality indeed 0 <= 1 or0 1 <= 0 end line line ell b because prop lemma auto imply indeed 0 <= 1 imply 0 <= 1 end line line ell c because lemma 0<1Helper indeed 1 <= 0 imply 0 <= 1 end line line ell d because prop lemma from disjuncts modus ponens ell a modus ponens ell b modus ponens ell c indeed 0 <= 1 end line line ell e because axiom 0not1 indeed 0 != 1 end line because prop lemma join conjuncts modus ponens ell d modus ponens ell e indeed 0 < 1 qed end math ] "
" [ math in theory system Q lemma lemma 0<2 says 0 < 2 end lemma end math ] "
" [ math system Q proof of lemma 0<2 reads line ell a because lemma 0<1 indeed 0 < 1 end line line ell c because lemma lessAddition modus ponens ell a indeed 0 + 1 < 1 + 1 end line line ell d because lemma plus0Left indeed 0 + 1 = 1 end line line ell e because lemma subLessLeft modus ponens ell d modus ponens ell c indeed 1 < 1 + 1 end line because lemma lessTransitivity modus ponens ell a modus ponens ell e indeed 0 < 2 qed end math ] "
" [ math in theory system Q lemma lemma 0<1/2 says 0 < 1/2 end lemma end math ] "
" [ math system Q proof of lemma 0<1/2 reads line ell big a because lemma 0<2 indeed 0 < 2 end line line ell big b because prop lemma first conjunct modus ponens ell big a indeed 0 <= 2 end line line ell big c because prop lemma second conjunct modus ponens ell big a indeed 0 != 2 end line line ell big d because lemma neqSymmetry modus ponens ell big c indeed 2 != 0 end line line ell a because lemma 0<1 indeed 0 < 1 end line line ell b because lemma x*0=0 indeed 2 * 0 = 0 end line line ell d because lemma x*y=zBackwards modus ponens ell b indeed 0 = 0 * 2 end line line ell e because lemma subLessLeft modus ponens ell d modus ponens ell a indeed 0 * 2 < 1 end line line ell f because lemma reciprocal modus ponens ell big d indeed 2 * 1/2 = 1 end line line ell g because lemma x*y=zBackwards modus ponens ell f indeed 1 = 1/2 * 2 end line line ell h because lemma subLessRight modus ponens ell g modus ponens ell e indeed 0 * 2 < 1/2 * 2 end line because lemma lessDivision modus ponens ell big b modus ponens ell h indeed 0 < 1/2 qed end math ] "
" [ math in theory system Q lemma lemma x+x=2*x says for all terms meta x indeed meta x + meta x = 2 * meta x end lemma end math ] "
" [ math system Q proof of lemma x+x=2*x reads any term meta x end line line ell a because axiom times1 indeed meta x * 1 = meta x end line line ell b because lemma eqSymmetry indeed meta x = meta x * 1 end line line ell c because lemma eqAdditionLeft modus ponens ell b indeed meta x + meta x = meta x + meta x * 1 end line line ell d because lemma eqAddition modus ponens ell b indeed meta x + meta x * 1 = meta x * 1 + meta x * 1 end line line ell e because lemma eqTransitivity modus ponens ell c modus ponens ell d indeed meta x + meta x = meta x * 1 + meta x * 1 end line line ell f because lemma distributionOut indeed meta x * 1 + meta x * 1 = meta x * parenthesis 1 + 1 end parenthesis end line line ell g because 1rule repetition modus ponens ell f indeed meta x * 1 + meta x * 1 = meta x * 2 end line line ell h because axiom timesCommutativity indeed meta x * 2 = 2 * meta x end line because lemma eqTransitivity4 modus ponens ell e modus ponens ell g modus ponens ell h indeed meta x + meta x = 2 * meta x qed end math ] "
" [ math in theory system Q lemma lemma (1/2)x+(1/2)x=x says for all terms meta x indeed 1/2 * meta x + 1/2 * meta x = meta x end lemma end math ] "
" [ math system Q proof of lemma (1/2)x+(1/2)x=x reads any term meta x end line line ell a because lemma 0<2 indeed 0 < 2 end line line ell b because lemma lessNeq modus ponens ell a indeed 0 != 2 end line line ell c because lemma neqSymmetry modus ponens ell b indeed 2 != 0 end line line ell d because lemma x+x=2*x indeed 1/2 * meta x + 1/2 * meta x = 2 * parenthesis 1/2 * meta x end parenthesis end line line ell e because axiom timesAssociativity indeed 2 * 1/2 * meta x = 2 * parenthesis 1/2 * meta x end parenthesis end line line ell big a because lemma eqSymmetry modus ponens ell e indeed 2 * parenthesis 1/2 * meta x end parenthesis = 2 * 1/2 * meta x end line line ell g because lemma reciprocal modus ponens ell c indeed 2 * 1/2 = 1 end line line ell h because lemma eqMultiplication modus ponens ell g indeed 2 * 1/2 * meta x = 1 * meta x end line line ell i because lemma times1Left indeed 1 * meta x = meta x end line because lemma eqTransitivity5 modus ponens ell d modus ponens ell big a modus ponens ell h modus ponens ell i indeed 1/2 * meta x + 1/2 * meta x = meta x qed end math ] "
" [ math in theory system Q rule lemma times(-1) says for all terms meta x indeed meta x * (-1) = - meta x end rule end math ] "
" [ math system Q proof of lemma times(-1) reads any term meta x end line line ell a because axiom negative indeed 1 + (-1) = 0 end line line ell b because axiom plusCommutativity indeed (-1) + 1 = 1 + (-1) end line line ell c because lemma eqTransitivity modus ponens ell b modus ponens ell a indeed (-1) + 1 = 0 end line line ell d because lemma eqMultiplicationLeft modus ponens ell c indeed meta x * parenthesis (-1) + 1 end parenthesis = meta x * 0 end line line ell e because lemma x*0=0 indeed meta x * 0 = 0 end line line ell f because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed meta x * parenthesis (-1) + 1 end parenthesis = 0 end line line ell g because axiom distribution indeed meta x * parenthesis (-1) + 1 end parenthesis = meta x * (-1) + meta x * 1 end line line ell h because lemma eqSymmetry modus ponens ell g indeed meta x * (-1) + meta x * 1 = meta x * parenthesis (-1) + 1 end parenthesis end line line ell i because lemma eqTransitivity modus ponens ell h modus ponens ell f indeed meta x * (-1) + meta x * 1 = 0 end line line ell j because lemma positiveToRight(Eq) modus ponens ell i indeed meta x * (-1) = 0 - parenthesis meta x * 1 end parenthesis end line line ell k because lemma plus0Left indeed 0 - parenthesis meta x * 1 end parenthesis = - parenthesis meta x * 1 end parenthesis end line line ell m because lemma eqTransitivity modus ponens ell j modus ponens ell k indeed meta x * (-1) = - parenthesis meta x * 1 end parenthesis end line line ell n because axiom times1 indeed meta x * 1 = meta x end line line ell o because lemma eqNegated modus ponens ell n indeed - parenthesis meta x * 1 end parenthesis = - meta x end line because lemma eqTransitivity modus ponens ell m modus ponens ell o indeed meta x * (-1) = - meta x qed end math ] "
" [ math in theory system Q lemma lemma times(-1)Left says for all terms meta x indeed (-1) * meta x = - meta x end lemma end math ] "
" [ math system Q proof of lemma times(-1)Left reads any term meta x end line line ell a because lemma times(-1) indeed meta x * (-1) = - meta x end line line ell b because axiom timesCommutativity indeed (-1) * meta x = meta x * (-1) end line because lemma eqTransitivity modus ponens ell b modus ponens ell a indeed (-1) * meta x = - meta x qed end math ] "
" [ math in theory system Q lemma lemma -x-y=-(x+y) says for all terms meta x comma meta y indeed - meta x - meta y = - parenthesis meta x + meta y end parenthesis end lemma end math ] "
" [ math system Q proof of lemma -x-y=-(x+y) reads any term meta x comma meta y end line line ell a because lemma times(-1)Left indeed (-1) * meta x = - meta x end line line ell b because lemma times(-1)Left indeed (-1) * meta y = - meta y end line line ell c because lemma addEquations modus ponens ell a modus ponens ell b indeed (-1) * meta x + (-1) * meta y = - meta x - meta y end line line ell d because lemma eqSymmetry modus ponens ell c indeed - meta x - meta y = (-1) * meta x + (-1) * meta y end line line ell e because lemma distributionOut indeed (-1) * meta x + (-1) * meta y = (-1) * parenthesis meta x + meta y end parenthesis end line line ell f because lemma times(-1)Left indeed (-1) * parenthesis meta x + meta y end parenthesis = - parenthesis meta x + meta y end parenthesis end line because lemma eqTransitivity4 modus ponens ell d modus ponens ell e modus ponens ell f indeed - meta x - meta y = - parenthesis meta x + meta y end parenthesis qed end math ] "
" [ math in theory system Q lemma lemma minusNegated says for all terms meta x comma meta y indeed - parenthesis meta x - meta y end parenthesis = meta y - meta x end lemma end math ] "
" [ math system Q proof of lemma minusNegated reads any term meta x comma meta y end line line ell a because lemma doubleMinus indeed - - meta y = meta y end line line ell b because lemma eqAddition modus ponens ell a indeed - - meta y - meta x = meta y - meta x end line line ell c because lemma eqSymmetry modus ponens ell b indeed meta y - meta x = - - meta y - meta x end line line ell d because lemma -x-y=-(x+y) indeed - - meta y - meta x = - parenthesis - meta y + meta x end parenthesis end line line ell e because axiom plusCommutativity indeed - meta y + meta x = meta x - meta y end line line ell f because lemma eqNegated modus ponens ell e indeed - parenthesis - meta y + meta x end parenthesis = - parenthesis meta x - meta y end parenthesis end line line ell g because lemma eqTransitivity4 modus ponens ell c modus ponens ell d modus ponens ell f indeed meta y - meta x = - parenthesis meta x - meta y end parenthesis end line because lemma eqSymmetry modus ponens ell g indeed - parenthesis meta x - meta y end parenthesis = meta y - meta x qed end math ] "
" [ math in theory system Q lemma lemma -0=0 says - 0 = 0 end lemma end math ] "
" [ math system Q proof of lemma -0=0 reads line ell a because axiom negative indeed 0 - 0 = 0 end line line ell b because axiom plus0 indeed 0 + 0 = 0 end line because lemma uniqueNegative modus ponens ell a modus ponens ell b indeed - 0 = 0 qed end math ] "
(*** LEQ, nummer 2 af 2 ***)
" [ math in theory system Q lemma lemma negativeToLeft(Leq) says for all terms meta x comma meta y comma meta z indeed meta x <= meta y - meta z infer meta x + meta z <= meta y end lemma end math ] "
" [ math system Q proof of lemma negativeToLeft(Leq) reads any term meta x comma meta y comma meta z end line line ell a premise meta x <= meta y - meta z end line line ell b because lemma leqAddition modus ponens ell a indeed meta x + meta z <= meta y - meta z + meta z end line line ell c because lemma x=x+y-y indeed meta y = meta y + meta z - meta z end line line ell e because lemma three2threeTerms indeed meta y + meta z - meta z = meta y - meta z + meta z end line line ell f because lemma eqTransitivity modus ponens ell c modus ponens ell e indeed meta y = meta y - meta z + meta z end line line ell g because lemma eqSymmetry modus ponens ell f indeed meta y - meta z + meta z = meta y end line because lemma subLeqRight modus ponens ell g modus ponens ell b indeed meta x + meta z <= meta y qed end math ] "
(*** SAME-F ***)
" [ math in theory system Q lemma lemma sameFsymmetry says for all terms meta ep comma meta m comma meta fx comma meta fy indeed meta fx sameF meta fy infer meta fy sameF meta fx end lemma end math ] "
" [ math system Q proof of lemma sameFsymmetry reads block any term meta ep comma meta m comma meta fx comma meta fy end line line ell a premise meta fx sameF meta fy end line line ell b premise 0 < meta ep end line line ell c premise ex3 <= meta m end line line ell d because 1rule fromSameF modus ponens ell a modus ponens ell b indeed ex3 <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line line ell big d because 1rule mp modus ponens ell d modus ponens ell c indeed | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line line ell e because lemma numericalDifference indeed | [ meta fx ; meta m ] - [ meta fy ; meta m ] | = | [ meta fy ; meta m ] - [ meta fx ; meta m ] | end line because lemma subLessLeft modus ponens ell e modus ponens ell big d indeed | [ meta fy ; meta m ] - [ meta fx ; meta m ] | < meta ep end line line ell big f end block any term meta ep comma meta m comma meta fx comma meta fy end line line ell a because 1rule deduction modus ponens ell big f indeed meta fx sameF meta fy imply 0 < meta ep imply ex3 <= meta m imply | [ meta fy ; meta m ] - [ meta fx ; meta m ] | < meta ep end line line ell b premise meta fx sameF meta fy end line line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed 0 < meta ep imply ex3 <= meta m imply | [ meta fy ; meta m ] - [ meta fx ; meta m ] | < meta ep end line because 1rule toSameF modus ponens ell c indeed meta fy sameF meta fx qed end math ] "
" [ math in theory system Q lemma lemma plusF(Sym) says for all terms meta m comma meta fx comma meta fy indeed [ meta fx ; meta m ] + [ meta fy ; meta m ] = [ meta fx +f meta fy ; meta m ] end lemma end math ] "
" [ math system Q proof of lemma plusF(Sym) reads any term meta m comma meta fx comma meta fy end line line ell a because axiom plusF indeed [ meta fx +f meta fy ; meta m ] = [ meta fx ; meta m ] + [ meta fy ; meta m ] end line because lemma eqSymmetry modus ponens ell a indeed [ meta fx ; meta m ] + [ meta fy ; meta m ] = [ meta fx +f meta fy ; meta m ] qed end math ] "
" [ math in theory system Q lemma lemma timesF(Sym) says for all terms meta m comma meta fx comma meta fy indeed [ meta fx ; meta m ] * [ meta fy ; meta m ] = [ meta fx *f meta fy ; meta m ] end lemma end math ] "
" [ math system Q proof of lemma timesF(Sym) reads any term meta m comma meta fx comma meta fy end line line ell a because axiom timesF indeed [ meta fx *f meta fy ; meta m ] = [ meta fx ; meta m ] * [ meta fy ; meta m ] end line because lemma eqSymmetry modus ponens ell a indeed [ meta fx ; meta m ] * [ meta fy ; meta m ] = [ meta fx *f meta fy ; meta m ] qed end math ] "
" [ math in theory system Q lemma lemma plus0f says for all terms meta m comma meta fx indeed meta fx +f 0f =f meta fx end lemma end math ] "
" [ math system Q proof of lemma plus0f reads any term meta m comma meta fx end line line ell a because axiom plusF indeed [ meta fx +f 0f ; meta m ] = [ meta fx ; meta m ] + [ 0f ; meta m ] end line line ell b because axiom 0f indeed [ 0f ; meta m ] = 0 end line line ell c because lemma eqAdditionLeft modus ponens ell b indeed [ meta fx ; meta m ] + [ 0f ; meta m ] = [ meta fx ; meta m ] + 0 end line line ell d because axiom plus0 indeed [ meta fx ; meta m ] + 0 = [ meta fx ; meta m ] end line line ell e because lemma eqTransitivity4 modus ponens ell a modus ponens ell c modus ponens ell d indeed [ meta fx +f 0f ; meta m ] = [ meta fx ; meta m ] end line because 1rule to=f modus ponens ell e indeed meta fx +f 0f =f meta fx qed end math ] "
" [ math in theory system Q lemma lemma sameFtransitivity says for all terms meta m comma meta ep comma meta fx comma meta fy comma meta fz indeed meta fx sameF meta fy infer meta fy sameF meta fz infer meta fx sameF meta fz end lemma end math ] "
" [ math system Q proof of lemma sameFtransitivity reads block any term meta m comma meta ep comma meta fx comma meta fy comma meta fz end line line ell a premise meta fx sameF meta fy end line line ell b premise meta fy sameF meta fz end line line ell c premise 0 < meta ep end line line ell d premise ex3 <= meta m end line line ell e because lemma positiveHalved modus ponens ell c indeed 0 < 1/2 * meta ep end line line ell f because 1rule fromSameF modus ponens ell a modus ponens ell e indeed ex3 <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < 1/2 * meta ep end line line ell big f because 1rule mp modus ponens ell f modus ponens ell d indeed | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < 1/2 * meta ep end line line ell g because 1rule fromSameF modus ponens ell b modus ponens ell e indeed ex3 <= meta m imply | [ meta fy ; meta m ] - [ meta fz ; meta m ] | < 1/2 * meta ep end line line ell big g because 1rule mp modus ponens ell g modus ponens ell d indeed | [ meta fy ; meta m ] - [ meta fz ; meta m ] | < 1/2 * meta ep end line line ell h because lemma addEquations(Less) modus ponens ell big f modus ponens ell big g indeed | [ meta fx ; meta m ] - [ meta fy ; meta m ] | + | [ meta fy ; meta m ] - [ meta fz ; meta m ] | < 1/2 * meta ep + 1/2 * meta ep end line line ell i because lemma (1/2)x+(1/2)x=x indeed 1/2 * meta ep + 1/2 * meta ep = meta ep end line line ell j because lemma subLessRight modus ponens ell i modus ponens ell h indeed | [ meta fx ; meta m ] - [ meta fy ; meta m ] | + | [ meta fy ; meta m ] - [ meta fz ; meta m ] | < meta ep end line line ell k because lemma insertMiddleTerm(Numerical) indeed | [ meta fx ; meta m ] - [ meta fz ; meta m ] | <= | [ meta fx ; meta m ] - [ meta fy ; meta m ] | + | [ meta fy ; meta m ] - [ meta fz ; meta m ] | end line because lemma leqLessTransitivity modus ponens ell k modus ponens ell j indeed | [ meta fx ; meta m ] - [ meta fz ; meta m ] | < meta ep end line line ell big l end block any term meta m comma meta ep comma meta fx comma meta fy comma meta fz end line line ell a because 1rule deduction modus ponens ell big l indeed meta fx sameF meta fy imply meta fy sameF meta fz imply 0 < meta ep imply ex3 <= meta m imply | [ meta fx ; meta m ] - [ meta fz ; meta m ] | < meta ep end line line ell b premise meta fx sameF meta fy end line line ell c premise meta fy sameF meta fz end line line ell d because prop lemma mp2 modus ponens ell a modus ponens ell b modus ponens ell c indeed 0 < meta ep imply ex3 <= meta m imply | [ meta fx ; meta m ] - [ meta fz ; meta m ] | < meta ep end line because 1rule toSameF modus ponens ell d indeed meta fx sameF meta fz qed end math ] "
" [ math in theory system Q lemma lemma =f to sameF says for all terms meta ep comma meta m comma meta fx comma meta fy indeed meta fx =f meta fy infer meta fx sameF meta fy end lemma end math ] "
" [ math system Q proof of lemma =f to sameF reads block any term meta ep comma meta m comma meta fx comma meta fy end line line ell a premise meta fx =f meta fy end line line ell b premise 0 < meta ep end line line ell d because 1rule from=f modus ponens ell a indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] end line line ell e because lemma positiveToLeft(Eq)(1 term) modus ponens ell d indeed [ meta fx ; meta m ] - [ meta fy ; meta m ] = 0 end line line ell f because lemma sameNumerical modus ponens ell e indeed | [ meta fx ; meta m ] - [ meta fy ; meta m ] | = | 0 | end line line ell g because lemma |0|=0 indeed | 0 | = 0 end line line ell h because lemma eqTransitivity modus ponens ell f modus ponens ell g indeed | [ meta fx ; meta m ] - [ meta fy ; meta m ] | = 0 end line line ell i because lemma eqSymmetry modus ponens ell h indeed 0 = | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end line line ell j because lemma subLessLeft modus ponens ell i modus ponens ell b indeed | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line line ell k because prop lemma weakening modus ponens ell j indeed 0 <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line because 1rule exist intro at ex3 at 0 modus ponens ell k indeed ex3 <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line line ell big l end block any term meta ep comma meta m comma meta fx comma meta fy end line line ell big a because 1rule deduction modus ponens ell big l indeed meta fx =f meta fy imply 0 < meta ep imply ex3 <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line line ell a premise meta fx =f meta fy end line line ell b because 1rule mp modus ponens ell big a modus ponens ell a indeed 0 < meta ep imply ex3 <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line because 1rule toSameF modus ponens ell b indeed meta fx sameF meta fy qed end math ] "
" [ math in theory system Q lemma lemma f2R(Plus) says for all terms meta fx comma meta fy comma meta fz comma meta fu comma meta fv indeed meta fx +f meta fy sameF meta fz infer R( meta fx ) ++ R( meta fy ) == R( meta fz ) end lemma end math ] "
" [ math system Q proof of lemma f2R(Plus) reads block any term meta fx comma meta fy comma meta fz comma meta fu comma meta fv end line line ell d premise meta fx +f meta fy sameF meta fz end line line ell e premise meta fu in0 R( meta fx +f meta fy ) end line line ell f premise meta fv in0 R( meta fz ) end line line ell g because 1rule fromInR modus ponens ell e indeed meta fu sameF meta fx +f meta fy end line line ell h because 1rule fromInR modus ponens ell f indeed meta fv sameF meta fz end line line ell i because lemma sameFsymmetry modus ponens ell h indeed meta fz sameF meta fv end line line ell j because lemma sameFtransitivity modus ponens ell g modus ponens ell d indeed meta fu sameF meta fz end line because lemma sameFtransitivity modus ponens ell j modus ponens ell i indeed meta fu sameF meta fv end line line ell big a end block any term meta fx comma meta fy comma meta fz comma meta fu comma meta fv end line line ell c because 1rule deduction modus ponens ell big a indeed meta fx +f meta fy sameF meta fz imply meta fu in0 R( meta fx +f meta fy ) imply meta fv in0 R( meta fz ) imply meta fu sameF meta fv end line line ell d premise meta fx +f meta fy sameF meta fz end line line ell e because 1rule mp modus ponens ell c modus ponens ell d indeed meta fu in0 R( meta fx +f meta fy ) imply meta fv in0 R( meta fz ) imply meta fu sameF meta fv end line line ell f because 1rule to==XX modus ponens ell e indeed R( meta fx +f meta fy ) == R( meta fz ) end line line ell g because axiom plusR indeed R( meta fx ) ++ R( meta fy ) == R( meta fx +f meta fy ) end line because lemma ==Transitivity modus ponens ell g modus ponens ell f indeed R( meta fx ) ++ R( meta fy ) == R( meta fz ) qed end math ] "
" [ math in theory system Q lemma lemma f2R(Times) says for all terms meta fx comma meta fy comma meta fz indeed meta fx *f meta fy sameF meta fz infer R( meta fx ) ** R( meta fy ) == R( meta fz ) end lemma end math ] "
" [ math system Q proof of lemma f2R(Times) reads any term meta fx comma meta fy comma meta fz end line line ell a premise meta fx *f meta fy sameF meta fz end line line ell b because 1rule to== modus ponens ell a indeed R( meta fx *f meta fy ) == R( meta fz ) end line line ell c because axiom timesR indeed R( meta fx ) ** R( meta fy ) == R( meta fx *f meta fy ) end line because lemma ==Transitivity modus ponens ell c modus ponens ell b indeed R( meta fx ) ** R( meta fy ) == R( meta fz ) qed end math ] "
(*** R-AFDELINGEN ***)
" [ math in theory system Q lemma lemma plusR(Sym) says for all terms meta fx comma meta fy indeed R( meta fx +f meta fy ) == R( meta fx ) ++ R( meta fy ) end lemma end math ] "
" [ math system Q proof of lemma plusR(Sym) reads any term meta fx comma meta fy end line line ell a because axiom plusR indeed R( meta fx ) ++ R( meta fy ) == R( meta fx +f meta fy ) end line because lemma ==Symmetry modus ponens ell a indeed R( meta fx +f meta fy ) == R( meta fx ) ++ R( meta fy ) qed end math ] "
" [ math in theory system Q lemma lemma timesR(Sym) says for all terms meta fx comma meta fy indeed R( meta fx *f meta fy ) == R( meta fx ) ** R( meta fy ) end lemma end math ] "
" [ math system Q proof of lemma timesR(Sym) reads any term meta fx comma meta fy end line line ell a because axiom timesR indeed R( meta fx ) ** R( meta fy ) == R( meta fx *f meta fy ) end line because lemma ==Symmetry modus ponens ell a indeed R( meta fx *f meta fy ) == R( meta fx ) ** R( meta fy ) qed end math ] "
(*** LEQ-R ***)
" [ math in theory system Q lemma lemma eqLeq(R) says for all terms meta rx comma meta ry indeed meta rx == meta ry infer meta rx <<== meta ry end lemma end math ] "
" [ math system Q proof of lemma eqLeq(R) reads any term meta rx comma meta ry end line line ell a premise meta rx == meta ry end line line ell b because prop lemma weaken or first modus ponens ell a indeed meta rx << meta ry or0 meta rx == meta ry end line because 1rule repetition modus ponens ell b indeed meta rx <<== meta ry qed end math ] "
" [ math in theory system Q lemma lemma thirdGeqSeries says for all terms meta m comma meta ep comma meta fx comma meta fy comma meta fz comma meta fu comma meta rx comma meta ry comma meta rz comma meta ru indeed meta rx << meta ry infer meta rz << meta ru infer meta fx in0 meta rx infer meta fy in0 meta ry infer meta fz in0 meta rz infer meta fu in0 meta ru infer 0 < meta ep infer ex3 <= meta m infer [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep and0 [ meta fz ; meta m ] <= [ meta fu ; meta m ] - meta ep end lemma end math ] "
" [ math system Q proof of lemma thirdGeqSeries reads any term meta m comma meta ep comma meta fx comma meta fy comma meta fz comma meta fu comma meta rx comma meta ry comma meta rz comma meta ru end line line ell a premise meta rx << meta ry end line line ell b premise meta rz << meta ru end line line ell c premise meta fx in0 meta rx end line line ell d premise meta fy in0 meta ry end line line ell e premise meta fz in0 meta rz end line line ell f premise meta fu in0 meta ru end line line ell g premise 0 < meta ep end line line ell big a premise ex3 <= meta m end line line ell h because 1rule from<
(*** LESS-R ***)
XX lidt grimt med de ekstra variable
" [ math in theory system Q lemma lemma subLessLeft(R) says for all terms meta ep comma meta m comma meta fy comma meta fz comma meta rx comma meta ry comma meta rz indeed meta rx == meta ry infer meta rx << meta rz infer meta ry << meta rz end lemma end math ] "
" [ math system Q proof of lemma subLessLeft(R) reads block any term meta ep comma meta m comma meta fy comma meta fz comma meta rx comma meta ry comma meta rz end line line ell a premise meta rx == meta ry end line line ell b premise meta rx << meta rz end line line ell c premise meta fy in0 meta ry end line line ell d premise meta fz in0 meta rz end line line ell e premise 0 < meta ep end line line ell f because lemma set equality nec condition(2) modus ponens ell a modus ponens ell c indeed meta fy in0 meta rx end line because 1rule from<
" [ math in theory system Q lemma lemma subLessRight(R) says for all terms meta ep comma meta m comma meta fy comma meta fz comma meta rx comma meta ry comma meta rz indeed meta rx == meta ry infer meta rz << meta rx infer meta rz << meta ry end lemma end math ] "
" [ math system Q proof of lemma subLessRight(R) reads block any term meta ep comma meta m comma meta fy comma meta fz comma meta rx comma meta ry comma meta rz end line line ell a premise meta rx == meta ry end line line ell b premise meta rz << meta rx end line line ell c premise meta fz in0 meta rz end line line ell d premise meta fy in0 meta ry end line line ell e premise 0 < meta ep end line line ell f because lemma set equality nec condition(2) modus ponens ell a indeed meta fy in0 meta rx end line because 1rule from<
" [ math in theory system Q lemma lemma lessLeq(R) says for all terms meta rx comma meta ry indeed meta rx << meta ry infer meta rx <<== meta ry end lemma end math ] "
" [ math system Q proof of lemma lessLeq(R) reads any term meta rx comma meta ry end line line ell a premise meta rx << meta ry end line line ell b because prop lemma weaken or second modus ponens ell a indeed meta rx << meta ry or0 meta rx == meta ry end line because 1rule repetition modus ponens ell b indeed meta rx <<== meta ry qed end math ] "
" [ math in theory system Q lemma lemma <
" [ math system Q proof of lemma <
(*** NUMMER 1 ***)
" [ math in theory system Q lemma lemma <<==Reflexivity says for all terms meta rx indeed meta rx <<== meta rx end lemma end math ] "
" [ math system Q proof of lemma <<==Reflexivity reads any term meta rx end line line ell a because lemma ==Reflexivity indeed meta rx == meta rx end line because lemma eqLeq(R) modus ponens ell a indeed meta rx <<== meta rx qed end math ] "
(*** NUMMER 2 ***)
" [ math in theory system Q lemma lemma <<==AntisymmetryHelper(Q) says for all terms meta a comma meta x comma meta y comma meta z indeed 0 < meta z infer meta x <= meta y - meta z infer meta y <= meta x - meta z infer meta a end lemma end math ] "
" [ math system Q proof of lemma <<==AntisymmetryHelper(Q) reads any term meta a comma meta x comma meta y comma meta z end line line ell big a premise 0 < meta z end line line ell a premise meta x <= meta y - meta z end line line ell b premise meta y <= meta x - meta z end line line ell c because lemma leqAddition modus ponens ell a indeed meta x + meta z <= meta y - meta z + meta z end line line ell d because axiom plusAssociativity indeed meta y - meta z + meta z = meta y + parenthesis - meta z + meta z end parenthesis end line line ell e because axiom plusCommutativity indeed - meta z + meta z = meta z - meta z end line line ell f because lemma eqAdditionLeft modus ponens ell e indeed meta y + parenthesis - meta z + meta z end parenthesis = meta y + parenthesis meta z - meta z end parenthesis end line line ell g because lemma x=x+(y-y) indeed meta y = meta y + parenthesis meta z - meta z end parenthesis end line line ell h because lemma eqSymmetry modus ponens ell g indeed meta y + parenthesis meta z - meta z end parenthesis = meta y end line line ell i because lemma eqTransitivity4 modus ponens ell d modus ponens ell f modus ponens ell h indeed meta y - meta z + meta z = meta y end line line ell j because lemma subLeqRight modus ponens ell i modus ponens ell c indeed meta x + meta z <= meta y end line line ell k because lemma leqTransitivity modus ponens ell j modus ponens ell b indeed meta x + meta z <= meta x - meta z end line line ell l because lemma leqSubtractionLeft modus ponens ell k indeed meta z <= - meta z end line line ell m because lemma toNotLess modus ponens ell l indeed not0 - meta z < meta z end line line ell n because lemma negativeLessPositive modus ponens ell big a indeed - meta z < meta z end line because prop lemma from contradiction modus ponens ell n modus ponens ell m indeed meta a qed end math ] "
" [ math in theory system Q lemma lemma <<==Antisymmetry says for all terms meta rx comma meta ry indeed meta rx <<== meta ry infer meta ry <<== meta rx infer meta rx == meta ry end lemma end math ] "
" [ math system Q proof of lemma <<==Antisymmetry reads any term meta rx comma meta ry end line line ell a premise meta rx <<== meta ry end line line ell b premise meta ry <<== meta rx end line line ell c because 1rule repetition modus ponens ell a indeed meta rx << meta ry or0 meta rx == meta ry end line line ell d because 1rule repetition modus ponens ell b indeed meta ry << meta rx or0 meta ry == meta rx end line line ell e because prop lemma expand disjuncts modus ponens ell c modus ponens ell d indeed meta rx == meta ry or0 meta ry == meta rx or0 parenthesis meta rx << meta ry and0 meta ry << meta rx end parenthesis end line line ell f because prop lemma auto imply indeed meta rx == meta ry imply meta rx == meta ry end line block any term meta rx comma meta ry end line line ell g premise meta ry == meta rx end line because lemma ==Symmetry modus ponens ell g indeed meta rx == meta ry end line line ell h end block line ell i because 1rule deduction modus ponens ell h indeed meta ry == meta rx imply meta rx == meta ry end line block any term meta rx comma meta ry end line line ell j premise meta rx << meta ry and0 meta ry << meta rx end line line ell k because prop lemma first conjunct modus ponens ell j indeed meta rx << meta ry end line line ell l because 1rule from<
(*** NUMMER 3 ***)
" [ math in theory system Q lemma lemma <
" [ math system Q proof of lemma <
" [ math in theory system Q lemma lemma <<==Transitivity says for all terms meta rx comma meta ry comma meta rz indeed meta rx <<== meta ry infer meta ry <<== meta rz infer meta rx <<== meta rz end lemma end math ] "
" [ math system Q proof of lemma <<==Transitivity reads block any term meta rx comma meta ry comma meta rz end line line ell e premise meta rx << meta ry end line line ell f premise meta ry << meta rz end line line ell g because lemma <
(*** NUMMER 5 ***)
" [ math in theory system Q lemma lemma ==Addition says for all terms meta m comma meta ep comma meta fx comma meta fy comma meta fz indeed R( meta fx ) == R( meta fy ) infer R( meta fx ) ++ R( meta fz ) == R( meta fy ) ++ R( meta fz ) end lemma end math ] "
" [ math system Q proof of lemma ==Addition reads block any term meta m comma meta ep comma meta fx comma meta fy comma meta fz end line line ell a premise R( meta fx ) == R( meta fy ) end line line ell b premise 0 < meta ep end line line ell c premise ex3 <= meta m end line line ell d because 1rule from== modus ponens ell a indeed meta fx sameF meta fy end line line ell e because 1rule fromSameF modus ponens ell d modus ponens ell b indeed ex3 <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line line ell f because 1rule mp modus ponens ell e modus ponens ell c indeed | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line line ell g because lemma insertMiddleTerm(Difference) indeed [ meta fx ; meta m ] - [ meta fy ; meta m ] = parenthesis [ meta fx ; meta m ] + [ meta fz ; meta m ] end parenthesis - parenthesis [ meta fy ; meta m ] + [ meta fz ; meta m ] end parenthesis end line line ell h because axiom plusF indeed [ meta fx +f meta fz ; meta m ] = [ meta fx ; meta m ] + [ meta fz ; meta m ] end line line ell i because axiom plusF indeed [ meta fy +f meta fz ; meta m ] = [ meta fy ; meta m ] + [ meta fz ; meta m ] end line line ell j because lemma eqNegated modus ponens ell i indeed - [ meta fy +f meta fz ; meta m ] = - parenthesis [ meta fy ; meta m ] + [ meta fz ; meta m ] end parenthesis end line line ell k because lemma addEquations modus ponens ell h modus ponens ell j indeed [ meta fx +f meta fz ; meta m ] - [ meta fy +f meta fz ; meta m ] = parenthesis [ meta fx ; meta m ] + [ meta fz ; meta m ] end parenthesis - parenthesis [ meta fy ; meta m ] + [ meta fz ; meta m ] end parenthesis end line line ell l because lemma eqSymmetry modus ponens ell k indeed parenthesis [ meta fx ; meta m ] + [ meta fz ; meta m ] end parenthesis - parenthesis [ meta fy ; meta m ] + [ meta fz ; meta m ] end parenthesis = [ meta fx +f meta fz ; meta m ] - [ meta fy +f meta fz ; meta m ] end line line ell m because lemma eqTransitivity modus ponens ell g modus ponens ell l indeed [ meta fx ; meta m ] - [ meta fy ; meta m ] = [ meta fx +f meta fz ; meta m ] - [ meta fy +f meta fz ; meta m ] end line line ell n because lemma sameNumerical modus ponens ell m indeed | [ meta fx ; meta m ] - [ meta fy ; meta m ] | = | [ meta fx +f meta fz ; meta m ] - [ meta fy +f meta fz ; meta m ] | end line because lemma subLessLeft modus ponens ell n modus ponens ell f indeed | [ meta fx +f meta fz ; meta m ] - [ meta fy +f meta fz ; meta m ] | < meta ep end line line ell big o end block any term meta m comma meta ep comma meta fx comma meta fy comma meta fz end line line ell a because 1rule deduction modus ponens ell big o indeed R( meta fx ) == R( meta fy ) imply 0 < meta ep imply ex3 <= meta m imply | [ meta fx +f meta fz ; meta m ] - [ meta fy +f meta fz ; meta m ] | < meta ep end line line ell b premise R( meta fx ) == R( meta fy ) end line line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed 0 < meta ep imply ex3 <= meta m imply | [ meta fx +f meta fz ; meta m ] - [ meta fy +f meta fz ; meta m ] | < meta ep end line line ell d because 1rule toSameF modus ponens ell c indeed meta fx +f meta fz sameF meta fy +f meta fz end line line ell e because 1rule to== modus ponens ell d indeed R( meta fx +f meta fz ) == R( meta fy +f meta fz ) end line line ell f because axiom plusR indeed R( meta fx ) ++ R( meta fz ) == R( meta fx +f meta fz ) end line line ell g because axiom plusR indeed R( meta fy ) ++ R( meta fz ) == R( meta fy +f meta fz ) end line line ell h because lemma ==Symmetry modus ponens ell g indeed R( meta fy +f meta fz ) == R( meta fy ) ++ R( meta fz ) end line line ell i because lemma ==Transitivity modus ponens ell f modus ponens ell e indeed R( meta fx ) ++ R( meta fz ) == R( meta fy +f meta fz ) end line because lemma ==Transitivity modus ponens ell i modus ponens ell h indeed R( meta fx ) ++ R( meta fz ) == R( meta fy ) ++ R( meta fz ) qed end math ] "
" [ math in theory system Q lemma lemma ==AdditionLeft says for all terms meta fx comma meta fy comma meta fz indeed R( meta fx ) == R( meta fy ) infer R( meta fz ) ++ R( meta fx ) == R( meta fz ) ++ R( meta fy ) end lemma end math ] "
" [ math system Q proof of lemma ==AdditionLeft reads any term meta fx comma meta fy comma meta fz end line line ell a premise R( meta fx ) == R( meta fy ) end line line ell b because lemma ==Addition modus ponens ell a indeed R( meta fx ) ++ R( meta fz ) == R( meta fy ) ++ R( meta fz ) end line line ell c because lemma plusCommutativity(R) indeed R( meta fz ) ++ R( meta fx ) == R( meta fx ) ++ R( meta fz ) end line line ell d because lemma plusCommutativity(R) indeed R( meta fy ) ++ R( meta fz ) == R( meta fz ) ++ R( meta fy ) end line line ell e because lemma ==Transitivity modus ponens ell c modus ponens ell b indeed R( meta fz ) ++ R( meta fx ) == R( meta fy ) ++ R( meta fz ) end line because lemma ==Transitivity modus ponens ell e modus ponens ell d indeed R( meta fz ) ++ R( meta fx ) == R( meta fz ) ++ R( meta fy ) qed end math ] "
(***)
" [ math in theory system Q lemma lemma <
" [ math system Q proof of lemma <
(***)
" [ math in theory system Q lemma lemma <<==Addition says for all terms meta fx comma meta fy comma meta fz indeed R( meta fx ) <<== R( meta fy ) infer R( meta fx ) ++ R( meta fz ) <<== R( meta fy ) ++ R( meta fz ) end lemma end math ] "
" [ math system Q proof of lemma <<==Addition reads block any term meta fx comma meta fy comma meta fz end line line ell a premise R( meta fx ) << R( meta fy ) end line line ell b because lemma <
(*** NUMMER 7 ***)
" [ math in theory system Q lemma lemma plusAssociativity(F) says for all terms meta m comma meta fx comma meta fy comma meta fz indeed meta fx +f meta fy +f meta fz =f meta fx +f parenthesis meta fy +f meta fz end parenthesis end lemma end math ] "
" [ math system Q proof of lemma plusAssociativity(F) reads any term meta m comma meta fx comma meta fy comma meta fz end line line ell a because axiom plusF indeed [ meta fx +f meta fy +f meta fz ; meta m ] = [ meta fx +f meta fy ; meta m ] + [ meta fz ; meta m ] end line line ell b because axiom plusF indeed [ meta fx +f meta fy ; meta m ] = [ meta fx ; meta m ] + [ meta fy ; meta m ] end line line ell c because lemma eqAddition modus ponens ell b indeed [ meta fx +f meta fy ; meta m ] + [ meta fz ; meta m ] = [ meta fx ; meta m ] + [ meta fy ; meta m ] + [ meta fz ; meta m ] end line line ell d because axiom plusAssociativity indeed [ meta fx ; meta m ] + [ meta fy ; meta m ] + [ meta fz ; meta m ] = [ meta fx ; meta m ] + parenthesis [ meta fy ; meta m ] + [ meta fz ; meta m ] end parenthesis end line line ell e because lemma plusF(Sym) indeed [ meta fy ; meta m ] + [ meta fz ; meta m ] = [ meta fy +f meta fz ; meta m ] end line line ell f because lemma eqAdditionLeft modus ponens ell e indeed [ meta fx ; meta m ] + parenthesis [ meta fy ; meta m ] + [ meta fz ; meta m ] end parenthesis = [ meta fx ; meta m ] + [ meta fy +f meta fz ; meta m ] end line line ell g because lemma plusF(Sym) indeed [ meta fx ; meta m ] + [ meta fy +f meta fz ; meta m ] = [ meta fx +f parenthesis meta fy +f meta fz end parenthesis ; meta m ] end line line ell h because lemma eqTransitivity6 modus ponens ell a modus ponens ell c modus ponens ell d modus ponens ell f modus ponens ell g indeed [ meta fx +f meta fy +f meta fz ; meta m ] = [ meta fx +f parenthesis meta fy +f meta fz end parenthesis ; meta m ] end line because 1rule to=f modus ponens ell h indeed meta fx +f meta fy +f meta fz =f meta fx +f parenthesis meta fy +f meta fz end parenthesis qed end math ] "
" [ math in theory system Q lemma lemma plusAssociativity(R) says for all terms meta fx comma meta fy comma meta fz indeed R( meta fx +f meta fy +f meta fz ) == R( meta fx +f parenthesis meta fy +f meta fz end parenthesis ) end lemma end math ] "
" [ math system Q proof of lemma plusAssociativity(R) reads any term meta fx comma meta fy comma meta fz end line line ell a because lemma plusAssociativity(F) indeed meta fx +f meta fy +f meta fz =f meta fx +f parenthesis meta fy +f meta fz end parenthesis end line line ell b because lemma =f to sameF modus ponens ell a indeed meta fx +f meta fy +f meta fz sameF meta fx +f parenthesis meta fy +f meta fz end parenthesis end line line ell c because lemma f2R(Plus) modus ponens ell b indeed R( meta fx +f meta fy ) ++ R( meta fz ) == R( meta fx +f parenthesis meta fy +f meta fz end parenthesis ) end line line ell d because lemma plusR(Sym) indeed R( meta fx +f meta fy +f meta fz ) == R( meta fx +f meta fy ) ++ R( meta fz ) end line because lemma ==Transitivity modus ponens ell d modus ponens ell c indeed R( meta fx +f meta fy +f meta fz ) == R( meta fx +f parenthesis meta fy +f meta fz end parenthesis ) qed end math ] "
(*** NUMMER 8 ***)
" [ math in theory system Q lemma lemma plus00 says for all terms meta fx indeed R( meta fx ) ++ 00 == R( meta fx ) end lemma end math ] "
" [ math system Q proof of lemma plus00 reads any term meta fx end line line ell a because lemma plus0f indeed meta fx +f 0f =f meta fx end line line ell b because lemma =f to sameF modus ponens ell a indeed meta fx +f 0f sameF meta fx end line line ell c because lemma f2R(Plus) modus ponens ell b indeed R( meta fx ) ++ R( 0f ) == R( meta fx ) end line because 1rule repetition modus ponens ell c indeed R( meta fx ) ++ R( 0f ) == R( meta fx ) qed end math ] "
(*** NUMMER 9 ***)
" [ math in theory system Q lemma lemma negative(R) says for all terms meta m comma meta fx indeed R( meta fx ) -- R( meta fx ) == 00 end lemma end math ] "
" [ math system Q proof of lemma negative(R) reads any term meta m comma meta fx end line line ell a because axiom plusF indeed [ meta fx +f -f meta fx ; meta m ] = [ meta fx ; meta m ] + [ -f meta fx ; meta m ] end line line ell b because axiom minusF indeed [ -f meta fx ; meta m ] = - [ meta fx ; meta m ] end line line ell c because lemma eqAdditionLeft modus ponens ell b indeed [ meta fx ; meta m ] + [ -f meta fx ; meta m ] = [ meta fx ; meta m ] - [ meta fx ; meta m ] end line line ell d because axiom negative indeed [ meta fx ; meta m ] - [ meta fx ; meta m ] = 0 end line line ell e because axiom 0f indeed [ 0f ; meta m ] = 0 end line line ell f because lemma eqSymmetry modus ponens ell e indeed 0 = [ 0f ; meta m ] end line line ell g because lemma eqTransitivity5 modus ponens ell a modus ponens ell c modus ponens ell d modus ponens ell f indeed [ meta fx +f -f meta fx ; meta m ] = [ 0f ; meta m ] end line line ell h because 1rule to=f modus ponens ell g indeed meta fx +f -f meta fx =f 0f end line line ell i because lemma =f to sameF modus ponens ell h indeed meta fx +f -f meta fx sameF 0f end line line ell j because lemma f2R(Plus) modus ponens ell i indeed R( meta fx ) ++ R( -f meta fx ) == R( 0f ) end line because 1rule repetition modus ponens ell j indeed R( meta fx ) -- R( meta fx ) == 00 qed end math ] "
(*** NUMMER 10 ***)
" [ math in theory system Q lemma lemma plusCommutativity(F) says for all terms meta m comma meta fx comma meta fy indeed meta fx +f meta fy =f meta fy +f meta fx end lemma end math ] "
" [ math system Q proof of lemma plusCommutativity(F) reads any term meta m comma meta fx comma meta fy end line line ell a because axiom plusF indeed [ meta fx +f meta fy ; meta m ] = [ meta fx ; meta m ] + [ meta fy ; meta m ] end line line ell b because axiom plusCommutativity indeed [ meta fx ; meta m ] + [ meta fy ; meta m ] = [ meta fy ; meta m ] + [ meta fx ; meta m ] end line line ell c because lemma plusF(Sym) indeed [ meta fy ; meta m ] + [ meta fx ; meta m ] = [ meta fy +f meta fx ; meta m ] end line line ell d because lemma eqTransitivity4 modus ponens ell a modus ponens ell b modus ponens ell c indeed [ meta fx +f meta fy ; meta m ] = [ meta fy +f meta fx ; meta m ] end line because 1rule to=f modus ponens ell d indeed meta fx +f meta fy =f meta fy +f meta fx qed end math ] "
" [ math in theory system Q lemma lemma plusCommutativity(R) says for all terms meta fx comma meta fy indeed R( meta fx ) ++ R( meta fy ) == R( meta fy ) ++ R( meta fx ) end lemma end math ] "
" [ math system Q proof of lemma plusCommutativity(R) reads any term meta fx comma meta fy end line line ell a because lemma plusCommutativity(F) indeed meta fx +f meta fy =f meta fy +f meta fx end line line ell b because lemma =f to sameF modus ponens ell a indeed meta fx +f meta fy sameF meta fy +f meta fx end line line ell c because lemma f2R(Plus) modus ponens ell b indeed R( meta fx ) ++ R( meta fy ) == R( meta fy +f meta fx ) end line line ell d because lemma plusR(Sym) indeed R( meta fy +f meta fx ) == R( meta fy ) ++ R( meta fx ) end line because lemma ==Transitivity modus ponens ell c modus ponens ell d indeed R( meta fx ) ++ R( meta fy ) == R( meta fy ) ++ R( meta fx ) qed end math ] "
(*** NUMMER 11 ***)
" [ math in theory system Q lemma lemma timesAssociativity(F) says for all terms meta m comma meta fx comma meta fy comma meta fz indeed meta fx *f meta fy *f meta fz =f meta fx *f parenthesis meta fy *f meta fz end parenthesis end lemma end math ] "
" [ math system Q proof of lemma timesAssociativity(F) reads any term meta m comma meta fx comma meta fy comma meta fz end line line ell a because axiom timesF indeed [ meta fx *f meta fy *f meta fz ; meta m ] = [ meta fx *f meta fy ; meta m ] * [ meta fz ; meta m ] end line line ell b because axiom timesF indeed [ meta fx *f meta fy ; meta m ] = [ meta fx ; meta m ] * [ meta fy ; meta m ] end line line ell c because lemma eqMultiplication modus ponens ell b indeed [ meta fx *f meta fy ; meta m ] * [ meta fz ; meta m ] = [ meta fx ; meta m ] * [ meta fy ; meta m ] * [ meta fz ; meta m ] end line line ell d because axiom timesAssociativity indeed [ meta fx ; meta m ] * [ meta fy ; meta m ] * [ meta fz ; meta m ] = [ meta fx ; meta m ] * parenthesis [ meta fy ; meta m ] * [ meta fz ; meta m ] end parenthesis end line line ell e because lemma timesF(Sym) indeed [ meta fy ; meta m ] * [ meta fz ; meta m ] = [ meta fy *f meta fz ; meta m ] end line line ell f because lemma eqMultiplicationLeft modus ponens ell e indeed [ meta fx ; meta m ] * parenthesis [ meta fy ; meta m ] * [ meta fz ; meta m ] end parenthesis = [ meta fx ; meta m ] * [ meta fy *f meta fz ; meta m ] end line line ell g because lemma timesF(Sym) indeed [ meta fx ; meta m ] * [ meta fy *f meta fz ; meta m ] = [ meta fx *f parenthesis meta fy *f meta fz end parenthesis ; meta m ] end line line ell h because lemma eqTransitivity6 modus ponens ell a modus ponens ell c modus ponens ell d modus ponens ell f modus ponens ell g indeed [ meta fx *f meta fy *f meta fz ; meta m ] = [ meta fx *f parenthesis meta fy *f meta fz end parenthesis ; meta m ] end line because 1rule to=f modus ponens ell h indeed meta fx *f meta fy *f meta fz =f meta fx *f parenthesis meta fy *f meta fz end parenthesis qed end math ] "
" [ math in theory system Q lemma lemma timesAssociativity(R) says for all terms meta fx comma meta fy comma meta fz indeed R( meta fx *f meta fy *f meta fz ) == R( meta fx *f parenthesis meta fy *f meta fz end parenthesis ) end lemma end math ] "
" [ math system Q proof of lemma timesAssociativity(R) reads any term meta fx comma meta fy comma meta fz end line line ell a because lemma timesAssociativity(F) indeed meta fx *f meta fy *f meta fz =f meta fx *f parenthesis meta fy *f meta fz end parenthesis end line line ell b because lemma =f to sameF modus ponens ell a indeed meta fx *f meta fy *f meta fz sameF meta fx *f parenthesis meta fy *f meta fz end parenthesis end line line ell c because lemma f2R(Times) modus ponens ell b indeed R( meta fx *f meta fy ) ** R( meta fz ) == R( meta fx *f parenthesis meta fy *f meta fz end parenthesis ) end line line ell d because lemma timesR(Sym) indeed R( meta fx *f meta fy *f meta fz ) == R( meta fx *f meta fy ) ** R( meta fz ) end line because lemma ==Transitivity modus ponens ell d modus ponens ell c indeed R( meta fx *f meta fy *f meta fz ) == R( meta fx *f parenthesis meta fy *f meta fz end parenthesis ) qed end math ] "
(*** NUMMER 12 ***)
" [ math in theory system Q lemma lemma times1f says for all terms meta m comma meta fx indeed meta fx *f 1f =f meta fx end lemma end math ] "
" [ math system Q proof of lemma times1f reads any term meta m comma meta fx end line line ell a because axiom timesF indeed [ meta fx *f 1f ; meta m ] = [ meta fx ; meta m ] * [ 1f ; meta m ] end line line ell b because axiom 1f indeed [ 1f ; meta m ] = 1 end line line ell c because lemma eqMultiplicationLeft modus ponens ell b indeed [ meta fx ; meta m ] * [ 1f ; meta m ] = [ meta fx ; meta m ] * 1 end line line ell d because axiom times1 indeed [ meta fx ; meta m ] * 1 = [ meta fx ; meta m ] end line line ell e because lemma eqTransitivity4 modus ponens ell a modus ponens ell c modus ponens ell d indeed [ meta fx *f 1f ; meta m ] = [ meta fx ; meta m ] end line because 1rule to=f modus ponens ell e indeed meta fx *f 1f =f meta fx qed end math ] "
" [ math in theory system Q lemma lemma times01 says for all terms meta fx indeed R( meta fx ) ** 01 == R( meta fx ) end lemma end math ] "
" [ math system Q proof of lemma times01 reads any term meta fx end line line ell a because lemma times1f indeed meta fx *f 1f =f meta fx end line line ell b because lemma =f to sameF modus ponens ell a indeed meta fx *f 1f sameF meta fx end line line ell c because lemma f2R(Times) modus ponens ell b indeed R( meta fx ) ** R( 1f ) == R( meta fx ) end line because 1rule repetition modus ponens ell c indeed R( meta fx ) ** 01 == R( meta fx ) qed end math ] "
(*** NUMMER 14 ***)
" [ math in theory system Q lemma lemma timesCommutativity(F) says for all terms meta m comma meta fx comma meta fy indeed meta fx *f meta fy =f meta fy *f meta fx end lemma end math ] "
" [ math system Q proof of lemma timesCommutativity(F) reads any term meta m comma meta fx comma meta fy end line line ell a because axiom timesF indeed [ meta fx *f meta fy ; meta m ] = [ meta fx ; meta m ] * [ meta fy ; meta m ] end line line ell b because axiom timesCommutativity indeed [ meta fx ; meta m ] * [ meta fy ; meta m ] = [ meta fy ; meta m ] * [ meta fx ; meta m ] end line line ell c because lemma timesF(Sym) indeed [ meta fy ; meta m ] * [ meta fx ; meta m ] = [ meta fy *f meta fx ; meta m ] end line line ell d because lemma eqTransitivity4 modus ponens ell a modus ponens ell b modus ponens ell c indeed [ meta fx *f meta fy ; meta m ] = [ meta fy *f meta fx ; meta m ] end line because 1rule to=f modus ponens ell d indeed meta fx *f meta fy =f meta fy *f meta fx qed end math ] "
" [ math in theory system Q lemma lemma timesCommutativity(R) says for all terms meta fx comma meta fy indeed R( meta fx ) ** R( meta fy ) == R( meta fy ) ** R( meta fx ) end lemma end math ] "
" [ math system Q proof of lemma timesCommutativity(R) reads any term meta fx comma meta fy end line line ell a because lemma timesCommutativity(F) indeed meta fx *f meta fy =f meta fy *f meta fx end line line ell b because lemma =f to sameF modus ponens ell a indeed meta fx *f meta fy sameF meta fy *f meta fx end line line ell c because lemma f2R(Times) modus ponens ell b indeed R( meta fx ) ** R( meta fy ) == R( meta fy *f meta fx ) end line line ell d because lemma timesR(Sym) indeed R( meta fy *f meta fx ) == R( meta fy ) ** R( meta fx ) end line because lemma ==Transitivity modus ponens ell c modus ponens ell d indeed R( meta fx ) ** R( meta fy ) == R( meta fy ) ** R( meta fx ) qed end math ] "
(*** NUMMER 15 ***)
" [ math in theory system Q lemma lemma distribution(F) says for all terms meta fx comma meta fy comma meta fz indeed meta fx *f parenthesis meta fy +f meta fz end parenthesis =f meta fx *f meta fy +f meta fx *f meta fz end lemma end math ] "
" [ math system Q proof of lemma distribution(F) reads any term meta fx comma meta fy comma meta fz end line line ell a because axiom timesF indeed [ meta fx *f parenthesis meta fy +f meta fz end parenthesis ; meta m ] = [ meta fx ; meta m ] * [ meta fy +f meta fz ; meta m ] end line line ell b because axiom plusF indeed [ meta fy +f meta fz ; meta m ] = [ meta fy ; meta m ] + [ meta fz ; meta m ] end line line ell c because lemma eqMultiplicationLeft modus ponens ell b indeed [ meta fx ; meta m ] * [ meta fy +f meta fz ; meta m ] = [ meta fx ; meta m ] * parenthesis [ meta fy ; meta m ] + [ meta fz ; meta m ] end parenthesis end line line ell d because axiom distribution indeed [ meta fx ; meta m ] * parenthesis [ meta fy ; meta m ] + [ meta fz ; meta m ] end parenthesis = [ meta fx ; meta m ] * [ meta fy ; meta m ] + [ meta fx ; meta m ] * [ meta fz ; meta m ] end line line ell e because lemma timesF(Sym) indeed [ meta fx ; meta m ] * [ meta fy ; meta m ] = [ meta fx *f meta fy ; meta m ] end line line ell f because lemma timesF(Sym) indeed [ meta fx ; meta m ] * [ meta fz ; meta m ] = [ meta fx *f meta fz ; meta m ] end line line ell g because lemma addEquations modus ponens ell e modus ponens ell f indeed [ meta fx ; meta m ] * [ meta fy ; meta m ] + [ meta fx ; meta m ] * [ meta fz ; meta m ] = [ meta fx *f meta fy ; meta m ] + [ meta fx *f meta fz ; meta m ] end line line ell h because lemma plusF(Sym) indeed [ meta fx *f meta fy ; meta m ] + [ meta fx *f meta fz ; meta m ] = [ meta fx *f meta fy +f meta fx *f meta fz ; meta m ] end line line ell i because lemma eqTransitivity6 modus ponens ell a modus ponens ell c modus ponens ell d modus ponens ell g modus ponens ell h indeed [ meta fx *f parenthesis meta fy +f meta fz end parenthesis ; meta m ] = [ meta fx *f meta fy +f meta fx *f meta fz ; meta m ] end line because 1rule to=f modus ponens ell i indeed meta fx *f parenthesis meta fy +f meta fz end parenthesis =f meta fx *f meta fy +f meta fx *f meta fz qed end math ] "
" [ math in theory system Q lemma lemma distribution(R) says for all terms meta fx comma meta fy comma meta fz indeed R( meta fx ) ** R( meta fy +f meta fz ) == R( meta fx *f meta fy ) ++ R( meta fx *f meta fz ) end lemma end math ] "
" [ math system Q proof of lemma distribution(R) reads any term meta fx comma meta fy comma meta fz end line line ell big b because lemma distribution(F) indeed meta fx *f parenthesis meta fy +f meta fz end parenthesis =f meta fx *f meta fy +f meta fx *f meta fz end line line ell big c because lemma =f to sameF modus ponens ell big b indeed meta fx *f parenthesis meta fy +f meta fz end parenthesis sameF meta fx *f meta fy +f meta fx *f meta fz end line line ell big d because lemma f2R(Times) modus ponens ell big c indeed R( meta fx ) ** R( meta fy +f meta fz ) == R( meta fx *f meta fy +f meta fx *f meta fz ) end line line ell big e because lemma plusR(Sym) indeed R( meta fx *f meta fy +f meta fx *f meta fz ) == R( meta fx *f meta fy ) ++ R( meta fx *f meta fz ) end line because lemma ==Transitivity modus ponens ell big d modus ponens ell big e indeed R( meta fx ) ** R( meta fy +f meta fz ) == R( meta fx *f meta fy ) ++ R( meta fx *f meta fz ) qed end math ] "
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\item " [ math tex define 1rule from== as "From==" end define end math ] "
\item " [ math tex define 1rule to== as "To==" end define end math ] "
\item " [ math tex define 1rule from<
\item " [ math tex define 1rule from<
\item " [ math tex define 1rule from<
\item " [ math tex define 1rule to<
\item " [ math tex define 1rule from<< as "From<<" end define end math ] "
\item " [ math tex define 1rule to<< as "To<<" end define end math ] "
\item " [ math tex define 1rule fromInR as "FromInR" end define end math ] "
\item " [ math tex define axiom plusR as "PlusR" end define end math ] "
\item " [ math tex define axiom timesR as "TimesR" end define end math ] "
\item " [ math tex define lemma leqAntisymmetry as "leqAntisymmetry" end define end math ] "
\item " [ math tex define lemma leqTransitivity as "leqTransitivity" end define end math ] "
\item " [ math tex define lemma leqAddition as "leqAddition" end define end math ] "
\item " [ math tex define lemma reciprocal as "Reciprocal" end define end math ] "
\item " [ math tex define lemma equality as "Equality" end define end math ] "
\item " [ math tex define lemma eqLeq as "eqLeq" end define end math ] "
\item " [ math tex define lemma eqAddition as "eqAddition" end define end math ] "
\item " [ math tex define lemma eqMultiplication as "eqMultiplication" end define end math ] "
\item " [ math tex define prop lemma to negated imply as "ToNegatedImply" end define end math ] "
\item " [ math tex define prop lemma tertium non datur as "TND" end define end math ] "
\item " [ math tex define prop lemma imply negation as "ImplyNegation" end define end math ] "
\item " [ math tex define prop lemma from negations as "FromNegations" end define end math ] "
\item " [ math tex define prop lemma from three disjuncts as "From3Disjuncts" end define end math ] "
\item " [ math tex define prop lemma negate first disjunct as "NegateDisjunct1" end define end math ] "
\item " [ math tex define prop lemma negate second disjunct as "NegateDisjunct2" end define end math ] "
\item " [ math tex define prop lemma expand disjuncts as "ExpandDisjuncts" end define end math ] "
\item " [ math tex define prop lemma from two times two disjuncts as "From2*2Disjuncts" end define end math ] "
\item " [ math tex define lemma eqReflexivity as "eqReflexivity" end define end math ] "
\item " [ math tex define lemma eqSymmetry as "eqSymmetry" end define end math ] "
\item " [ math tex define lemma eqTransitivity as "eqTransitivity" end define end math ] "
\item " [ math tex define lemma eqTransitivity4 as "eqTransitivity4" end define end math ] "
\item " [ math tex define lemma eqTransitivity5 as "eqTransitivity5" end define end math ] "
\item " [ math tex define lemma eqTransitivity6 as "eqTransitivity6" end define end math ] "
\item " [ math tex define lemma plus0Left as "plus0Left" end define end math ] "
\item " [ math tex define lemma times1Left as "times1Left" end define end math ] "
\item " [ math tex define lemma eqMultiplicationLeft as "EqMultiplicationLeft" end define end math ] "
\item " [ math tex define lemma distributionOut as "DistributionOut" end define end math ] "
\item " [ math tex define lemma three2twoTerms as "Three2twoTerms" end define end math ] "
\item " [ math tex define lemma three2threeTerms as "Three2threeTerms" end define end math ] "
\item " [ math tex define lemma three2twoFactors as "Three2threeFactors" end define end math ] "
\item " [ math tex define lemma addEquations as "AddEquations" end define end math ] "
\item " [ math tex define lemma subtractEquations as "SubtractEquations" end define end math ] "
\item " [ math tex define lemma subtractEquationsLeft as "SubtractEquationsLeft" end define end math ] "
\item " [ math tex define lemma eqNegated as "EqNegated" end define end math ] "
\item " [ math tex define lemma positiveToRight(Eq) as "PositiveToRight(Eq)" end define end math ] "
\item " [ math tex define lemma positiveToLeft(Eq)(1 term) as "PositiveToLeft(Eq)(1 term)" end define end math ] "
\item " [ math tex define lemma negativeToLeft(Eq) as "NegativeToLeft(Eq)" end define end math ] "
\item " [ math tex define lemma uniqueNegative as "UniqueNegative" end define end math ] "
\item " [ math tex define lemma doubleMinus as "DoubleMinus" end define end math ] "
\item " [ math tex define lemma lessNeq as "LessNeq" end define end math ] "
\item " [ math tex define lemma neqSymmetry as "NeqSymmetry" end define end math ] "
\item " [ math tex define lemma neqNegated as "NeqNegated" end define end math ] "
\item " [ math tex define lemma subNeqRight as "SubNeqRight" end define end math ] "
\item " [ math tex define lemma subNeqLeft as "SubNeqLeft" end define end math ] "
\item " [ math tex define lemma neqAddition as "NeqAddition" end define end math ] "
\item " [ math tex define lemma neqMultiplication as "NeqMultiplication" end define end math ] "
\item " [ math tex define lemma leqLessEq as "LeqLessEq" end define end math ] "
\item " [ math tex define lemma lessLeq as "LessLeq" end define end math ] "
\item " [ math tex define lemma from leqGeq as "FromLeqGeq" end define end math ] "
\item " [ math tex define lemma subLeqRight as "subLeqRight" end define end math ] "
\item " [ math tex define lemma subLeqLeft as "subLeqLeft" end define end math ] "
\item " [ math tex define lemma leqPlus1 as "Leq+1" end define end math ] "
\item " [ math tex define lemma positiveToRight(Leq) as "PositiveToRight(Leq)" end define end math ] "
\item " [ math tex define lemma positiveToRight(Leq)(1 term) as "PositiveToRight(Leq)(1 term)" end define end math ] "
\item " [ math tex define lemma leqAdditionLeft as "LeqAdditionLeft" end define end math ] "
\item " [ math tex define lemma leqSubtraction as "leqSubtraction" end define end math ] "
\item " [ math tex define lemma leqSubtractionLeft as "leqSubtractionLeft" end define end math ] "
\item " [ math tex define lemma leqMultiplication as "leqMultiplication" end define end math ] "
\item " [ math tex define lemma thirdGeq as "thirdGeq" end define end math ] "
\item " [ math tex define lemma leqNegated as "LeqNegated" end define end math ] "
\item " [ math tex define lemma addEquations(Leq) as "AddEquations(Leq)" end define end math ] "
\item " [ math tex define lemma leqNeqLess as "LeqNeqLess" end define end math ] "
\item " [ math tex define lemma fromLess as "FromLess" end define end math ] "
\item " [ math tex define lemma toLess as "ToLess" end define end math ] "
\item " [ math tex define lemma fromNotLess as "fromNotLess" end define end math ] "
\item " [ math tex define lemma toNotLess as "toNotLess" end define end math ] "
\item " [ math tex define lemma lessAddition as "LessAddition" end define end math ] "
\item " [ math tex define lemma lessAdditionLeft as "LessAdditionLeft" end define end math ] "
\item " [ math tex define lemma lessMultiplication as "LessMultiplication" end define end math ] "
\item " [ math tex define lemma lessMultiplicationLeft as "LessMultiplicationLeft" end define end math ] "
\item " [ math tex define lemma lessDivision as "LessDivision" end define end math ] "
\item " [ math tex define lemma addEquations(Less) as "AddEquations(Less)" end define end math ] "
\item " [ math tex define lemma negativeLessPositive as "NegativeLessPositive" end define end math ] "
\item " [ math tex define lemma leqLessTransitivity as "leqLessTransitivity" end define end math ] "
\item " [ math tex define lemma lessLeqTransitivity as "LessLeqTransitivity" end define end math ] "
\item " [ math tex define lemma lessTransitivity as "LessTransitivity" end define end math ] "
\item " [ math tex define lemma lessTotality as "LessTotality" end define end math ] "
\item " [ math tex define lemma subLessRight as "SubLessRight" end define end math ] "
\item " [ math tex define lemma subLessLeft as "SubLessLeft" end define end math ] "
\item " [ math tex define lemma lessNegated as "LessNegated" end define end math ] "
\item " [ math tex define lemma positiveNegated as "PositiveNegated" end define end math ] "
\item " [ math tex define lemma nonpositiveNegated as "NonpositiveNegated" end define end math ] "
\item " [ math tex define lemma negativeNegated as "NegativeNegated" end define end math ] "
\item " [ math tex define lemma nonnegativeNegated as "NonnegativeNegated" end define end math ] "
\item " [ math tex define lemma positiveHalved as "PositiveHalved" end define end math ] "
\item " [ math tex define lemma nonnegativeNumerical as "NonnegativeNumerical" end define end math ] "
\item " [ math tex define lemma negativeNumerical as "NegativeNumerical" end define end math ] "
\item " [ math tex define lemma positiveNumerical as "PositiveNumerical" end define end math ] "
\item " [ math tex define lemma |0|=0 as "|0|=0" end define end math ] "
\item " [ math tex define lemma 0<=|x| as "0<=|x|" end define end math ] "
\item " [ math tex define lemma sameNumerical as "SameNumerical" end define end math ] "
\item " [ math tex define lemma signNumerical(+) as "SignNumerical(+)" end define end math ] "
\item " [ math tex define lemma signNumerical as "SignNumerical" end define end math ] "
\item " [ math tex define lemma numericalDifference as "NumericalDifference" end define end math ] "
\item " [ math tex define lemma splitNumericalSumHelper as "SplitNumericalSumHelper" end define end math ] "
\item " [ math tex define lemma splitNumericalSum(++) as "splitNumericalSum(++)" end define end math ] "
\item " [ math tex define lemma splitNumericalSum(--) as "splitNumericalSum(--)" end define end math ] "
\item " [ math tex define lemma splitNumericalSum(+-, smallNegative) as "splitNumericalSum(+-small)" end define end math ] "
\item " [ math tex define lemma splitNumericalSum(+-, bigNegative) as "splitNumericalSum(+-big)" end define end math ] "
\item " [ math tex define lemma splitNumericalSum(+-) as "splitNumericalSum(+-)" end define end math ] "
\item " [ math tex define lemma splitNumericalSum(-+) as "splitNumericalSum(-+)" end define end math ] "
\item " [ math tex define lemma splitNumericalSum as "splitNumericalSum" end define end math ] "
\item " [ math tex define lemma insertMiddleTerm(Numerical) as "insertMiddleTerm(Numerical)" end define end math ] "
\item " [ math tex define lemma x+y=zBackwards as "x+y=zBackwards" end define end math ] "
\item " [ math tex define lemma x*y=zBackwards as "x*y=zBackwards" end define end math ] "
\item " [ math tex define lemma x=x+(y-y) as "x=x+(y-y)" end define end math ] "
\item " [ math tex define lemma x=x+y-y as "x=x+y-y" end define end math ] "
\item " [ math tex define lemma x=x*y*(1/y) as " " end define end math ] "
\item " [ math tex define lemma insertMiddleTerm(Sum) as "insertMiddleTerm(Sum)" end define end math ] "
\item " [ math tex define lemma insertMiddleTerm(Difference) as "insertMiddleTerm(Difference)" end define end math ] "
\item " [ math tex define lemma x*0+x=x as "x*0+x=x" end define end math ] "
\item " [ math tex define lemma x*0=0 as "x*0=0" end define end math ] "
\item " [ math tex define lemma (-1)*(-1)+(-1)*1=0 as "(-1)*(-1)+(-1)*1=0" end define end math ] "
\item " [ math tex define lemma (-1)*(-1)=1 as "(-1)*(-1)=1" end define end math ] "
\item " [ math tex define lemma 0<1Helper as "0<1Helper" end define end math ] "
\item " [ math tex define lemma 0<1 as "0<1" end define end math ] "
\item " [ math tex define lemma 0<2 as "0<2" end define end math ] "
\item " [ math tex define lemma 0<1/2 as "0<1/2" end define end math ] "
\item " [ math tex define lemma x+x=2*x as "TwoWholes" end define end math ] "
\item " [ math tex define lemma (1/2)x+(1/2)x=x as "TwoHalves" end define end math ] "
\item " [ math tex define lemma -x-y=-(x+y) as "-x-y=-(x+y)" end define end math ] "
\item " [ math tex define lemma minusNegated as "MinusNegated" end define end math ] "
\item " [ math tex define lemma times(-1) as "Times(-1)" end define end math ] "
\item " [ math tex define lemma times(-1)Left as "Times(-1)Left" end define end math ] "
\item " [ math tex define lemma -0=0 as "-0=0" end define end math ] "
\item " [ math tex define lemma negativeToLeft(Leq) as "negativeToLeft(Leq)" end define end math ] "
\item " [ math tex define lemma sameFsymmetry as "SFsymmetry" end define end math ] "
\item " [ math tex define lemma sameFtransitivity as "SFtransitivity" end define end math ] "
\item " [ math tex define lemma =f to sameF as "=fToSameF " end define end math ] "
\item " [ math tex define lemma plusF(Sym) as "PlusF(Sym)" end define end math ] "
\item " [ math tex define lemma timesF(Sym) as "TimesF(Sym)" end define end math ] "
\item " [ math tex define lemma f2R(Plus) as "f2R(Plus)" end define end math ] "
\item " [ math tex define lemma f2R(Times) as "f2R(Times)" end define end math ] "
\item " [ math tex define lemma plusR(Sym) as "PlusR(Sym)" end define end math ] "
\item " [ math tex define lemma timesR(Sym) as "TimesR(Sym)" end define end math ] "
\item " [ math tex define lemma lessLeq(R) as "LessLeq(R)" end define end math ] "
\item " [ math tex define lemma eqLeq(R) as "eqLeq(R)" end define end math ] "
\item " [ math tex define lemma thirdGeqSeries as "ThirdGeqSeries" end define end math ] "
\item " [ math tex define lemma subLessRight(R) as "SubLessRight(R)" end define end math ] "
\item " [ math tex define lemma subLessLeft(R) as "SubLessLeft(R)" end define end math ] "
\item " [ math tex define lemma <
\item " [ math tex define lemma <
\item " [ math tex define lemma <<==Reflexivity as "<<==Reflexivity" end define end math ] "
\item " [ math tex define lemma <<==AntisymmetryHelper(Q) as "<<==AntisymmetryHelper(Q)" end define end math ] "
\item " [ math tex define lemma <<==Antisymmetry as "<<==Antisymmetry" end define end math ] "
\item " [ math tex define lemma <<==Transitivity as "<<==Transitivity" end define end math ] "
\item " [ math tex define lemma plus0f as "Plus0f" end define end math ] "
\item " [ math tex define lemma plus00 as "Plus00" end define end math ] "
\item " [ math tex define lemma ==Addition as "==Addition" end define end math ] "
\item " [ math tex define lemma ==AdditionLeft as "==AdditionLeft" end define end math ] "
\item " [ math tex define lemma <
\item " [ math tex define lemma <<==Addition as "<<==Addition" end define end math ] "
\item " [ math tex define lemma plusAssociativity(F) as "PlusAssociativity(F)" end define end math ] "
\item " [ math tex define lemma plusAssociativity(R) as "PlusAssociativity(R)" end define end math ] "
\item " [ math tex define lemma negative(R) as "Negative(R)" end define end math ] "
\item " [ math tex define lemma plusCommutativity(F) as "PlusCommutativity(F)" end define end math ] "
\item " [ math tex define lemma plusCommutativity(R) as "PlusCommutativity(R)" end define end math ] "
\item " [ math tex define lemma times1f as "Times1f" end define end math ] "
\item " [ math tex define lemma times01 as "Times01" end define end math ] "
\item " [ math tex define lemma timesAssociativity(F) as "TimesAssociativity(F)" end define end math ] "
\item " [ math tex define lemma timesAssociativity(R) as "TimesAssociativity(R)" end define end math ] "
\item " [ math tex define lemma timesCommutativity(F) as "TimesCommutativity(F)" end define end math ] "
\item " [ math tex define lemma timesCommutativity(R) as "TimesCommutativity(R)" end define end math ] "
\item " [ math tex define lemma distribution(F) as "Distribution(F)" end define end math ] "
\item " [ math tex define lemma distribution(R) as "Distribution(R)" end define end math ] "
\end{list}
\end{document}
End of file
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