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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 ] "
(*** 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 define macro of var x and0 var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x and0 var y as not0 parenthesis var x imply not0 var y end parenthesis end define end quote end define end define end math ] "
" [ math define macro of var x or0 var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x or0 var y as not0 var x imply var y end define end quote end define end define end math ] "
" [ math define macro of var x iff var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define end define end math ] "
}
\subsection{Negerede formler}
Det er ganske enkelt at definere negeret lighed (" [ math not0 var x zermelo is var y end math ] ") og negeret medlemskab (" [ math not0 var x in0 var y end math ] "):
\display{
" [ math define macro of var x zermelo ~is var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x zermelo ~is var y as not0 var x zermelo is var y end define end quote end define end define end math ] "
" [ math define macro of var x zermelo ~in var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x zermelo ~in var y as not0 var x in0 var y end define end quote end define end define end math ] "\footnote{H\o{}jresiderne i disse definitioner skal l\ae{}ses som hhv.\ " [ bracket not0 var x zermelo is var y end bracket ] " og " [ bracket not0 var x in0 var y 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 define macro of var x is subset of var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define end define end math ] "
}
\subsection{Singleton-m\ae{}ngde} \label{sec:single}
" [ bracket zermelo pair var x comma var x end pair end bracket ] " er m\ae{}ngden, der indeholder " [ math var x end math ] " som sit eneste
element. Vi definerer " [ bracket zermelo pair var x comma var x end pair end bracket ] " ved at parre " [ math var x end math ] " med sig selv:
\display{" [ math define macro of zermelo singleton var x end singleton as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define zermelo singleton var x end singleton as zermelo pair var x comma var x end pair end define end quote end define 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 define macro of binary-union var x comma var y end union as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define 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 define macro of intersection var x comma var y end intersection as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define end define end math ] "
}
\subsection{Relation} \label{sec:relation}
Det ordnede par " [ math zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var x comma var y end pair 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 pair zermelo pair var x comma var x end pair comma zermelo pair var x comma var y end pair 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 define macro of zermelo ordered pair var x comma var y end pair as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define 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 define macro of var x is related to var y under var r as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define end define end math ] "}
Vi kommer faktisk ikke til at bruge disse to definitioner i rapporten; beviserne vil behandle " [ bracket zermelo pair zermelo pair var x comma var x end pair comma zermelo pair var x comma var y end pair end pair in0 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 define macro of var r is reflexive relation in var x as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define 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 define macro of var r is symmetric relation in var x as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define 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 define macro of var r is transitive relation in var x as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define end define end math ] "}
Endelig er en \ae{}kvivalensrelation det samme som en relation, der er refleksiv, symmetrisk og transitiv:
\verb| |
\display{" [ math define macro of var r is equivalence relation in var x as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define 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 metavar var big set end metavar end math ] " og objektvariablen " [ math object var var big set end var end math ] ":
\display{
" [ math define macro of meta big set as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta big set as metavar var big set end metavar end define end quote end define end define end math ] "
" [ math define macro of object big set as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define object big set as object var var big set end var end define end quote end define end define end math ] "\footnote{Navnene ``" [ math metavar var big set end metavar end math ] "'' og ``" [ math object var var big set end var 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 the set of ph in var big set such that zermelo pair zermelo pair placeholder-var1 comma placeholder-var1 end pair comma zermelo pair placeholder-var1 comma var x end pair end pair in0 var r end set 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 define macro of equivalence class of var x in var big set modulo var r as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define end define end math ] "
}
\AE{}kvivalenssystemet " [ math the set of ph in power var big set end power such that not0 existential var var t end var in0 var big set imply not0 the set of ph in var big set such that zermelo pair zermelo pair placeholder-var1 comma placeholder-var1 end pair comma zermelo pair placeholder-var1 comma existential var var t end var end pair end pair in0 var r end set zermelo is placeholder-var2 end set 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 the set of ph in power var big set end power such that not0 existential var var t end var in0 var big set imply not0 the set of ph in var big set such that zermelo pair zermelo pair placeholder-var1 comma placeholder-var1 end pair comma zermelo pair placeholder-var1 comma existential var var t end var end pair end pair in0 var r end set zermelo is placeholder-var2 end set end math ] " som en delm\ae{}ngde af potensm\ae{}ngden " [ math power var big set end power end math ] ":
\verb| |
\display{
" [ math define macro of eq-system of var big set modulo var r as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define 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 define macro of var p is partition of var big set as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define 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 lambda var x dot 1deduction zero quote var p end quote conclude quote var c end quote end 1deduction end bracket ] " er en kopi af " [ bracket lambda var x dot deduction zero quote var p end quote conclude quote var c end quote end deduction end bracket ] " fra \cite{kn:check}:
\begin{list}{}{
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\item " [ math define macro of 1deduction var p conclude var c end 1deduction as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define 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 define value of 1deduction zero var p conclude var c end 1deduction as var c tagged guard tagged if 1deduction eight var p bound true end 1deduction then 1deduction side 1deduction seven var p end 1deduction conclude var c condition true end 1deduction else false end if 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 define value of 1deduction side var p conclude var c condition var s end 1deduction as tagged if var p term root equal quote var x endorse var y end quote then 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 1deduction one var p conclude var c condition var s end 1deduction end if 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 define value of 1deduction one var p conclude var c condition var s end 1deduction as tagged 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 if end define end math ] "
\item " [ math define value of 1deduction two var p conclude var c condition var s end 1deduction as var s tagged guard 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 define value of 1deduction three var p conclude var c condition var s bound var b end 1deduction as tagged if not var c term root equal quote for all objects var x indeed var y end quote then 1deduction four var p conclude var c condition var s bound var b end 1deduction else tagged if var p term root equal quote for all objects 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 1deduction three var p conclude var c second condition var s bound var c first pair var c first pair var b end 1deduction end if end if end define end math ] "
\item " [ math define value of 1deduction four var p conclude var c condition var s bound var b end 1deduction as var s tagged guard var b tagged guard tagged if var p term root equal quote object var var x end var end quote then lookup var p stack var b default true end lookup term equal var c else tagged if not var p term root equal var c then false else tagged if var p term root equal quote for all objects 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 var p first pair var p first pair var b end 1deduction else tagged if not var p term root equal quote metavar var x end metavar end quote then 1deduction four star var p tail conclude var c tail condition var s bound var b end 1deduction else var p first term equal var c first and 1deduction five var p condition var s bound var b end 1deduction end if end if end if end if end define end math ] "
\item " [ math define value of 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 tagged 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 if end define end math ] "
\item " [ math define value of 1deduction five var p condition var s bound var b end 1deduction as var p tagged guard var s tagged guard tagged if var b then true else quote quote var x end quote avoid zero quote var y end quote end quote head pair quote quote x end quote end quote head pair var b head head pair true pair quote quote var x end quote end quote head pair var p pair true pair true term in var s and 1deduction five var p condition var s bound var b tail end 1deduction end if end define end math ] "
\item " [ math define value of 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 tagged if var p term root equal quote object var var x end var end quote then var p term in var e select var b else var p pair var c pair var b end select else tagged if not var p term root equal var c then true else tagged if var p term root equal quote metavar var a end metavar end quote then var b else tagged if var p term root equal quote for all objects 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 1deduction six star var p tail conclude var c tail exception var e bound var b end 1deduction end if end if end if end if end define end math ] "
\item " [ math define value of 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 tagged 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 if end define end math ] "
\item " [ math define value of 1deduction seven var p end 1deduction as var p term root equal quote all 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 define value of 1deduction eight var p bound var b end 1deduction as tagged if var p term root equal quote all var x indeed var y end quote then 1deduction eight var p second bound var p first pair var b end 1deduction else tagged if var p term root equal quote metavar var a end metavar end quote then var p term in var b else 1deduction eight star var p tail bound var b end 1deduction end if end if end define end math ] "
\item " [ math define value of 1deduction eight star var p bound var b end 1deduction as var b tagged guard tagged if var p then true else tagged if 1deduction eight var p head bound var b end 1deduction then 1deduction eight star var p tail bound var b end 1deduction else false end if end if end define end math ] "
\end{list}
(*** EKSISTENS-VARIABLE ***)
\display{" [ math define value of 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 define macro of ex1 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define ex1 as existential var var a end var end define end quote end define end define end math ] "
}
" [ math define macro of ex2 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define ex2 as existential var var b end var end define end quote end define end define end math ] "
" [ math define macro of ex10 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define ex10 as existential var var j end var end define end quote end define end define end math ] "
" [ math define macro of ex20 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define ex20 as existential var var t end var end define end quote end define end define end math ] "
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\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define macro of exist-sub var a is var b where var x is var t end sub as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define end define end math ] "
\item " [ math define value of 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 define value of 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 tagged if var b term root equal quote for all objects var u indeed var v end quote then false else newline tagged if var b is existential var and var b term equal var x then var a term equal var t else tagged if newline var a term root equal var b then exist-sub* var a tail is var b tail where var x is var t end sub else false end if end if end if end define end math ] "
\item " [ math define value of 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 tagged if exist-sub1 var a head is var b head where var x is var t end sub then exist-sub* var a tail is var b tail where var x is var t end sub else false end if end if end define end math ] "
\end{list}
(*** AKSIOMATISK SYSTEM ***)
" [ math define statement of system Q as all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar * metavar var z end metavar rule plus all metavar var fx end metavar indeed all metavar var fy end metavar indeed R( metavar var fx end metavar ) << R( metavar var fy end metavar ) infer metavar var fx end metavar
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of 1rule mp as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule mp as rule tactic end define end math ] "
\item " [ math define statement of 1rule gen as system Q infer all metavar var x end metavar indeed all metavar var a end metavar indeed metavar var a end metavar infer for all objects metavar var x end metavar indeed metavar var a end metavar end define , define proof of 1rule gen as rule tactic end define end math ] "
\item " [ math define statement of 1rule repetition as system Q infer all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar end define , define proof of 1rule repetition as rule tactic end define end math ] "
\item " [ math define statement of 1rule ad absurdum as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var b end metavar imply metavar var a end metavar infer not0 metavar var b end metavar imply not0 metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule ad absurdum as rule tactic end define end math ] "
\item " [ math define statement of 1rule deduction as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed lambda var x dot 1deduction zero quote metavar var a end metavar end quote conclude quote metavar var b end metavar end quote end 1deduction endorse metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule deduction as rule tactic end define end math ] "
\item " [ math define statement of 1rule exist intro as system Q infer all metavar var x end metavar indeed all metavar var t end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed exist-sub0 quote metavar var a end metavar end quote is quote metavar var b end metavar end quote where quote metavar var x end metavar end quote is quote metavar var t end metavar end quote end sub endorse metavar var a end metavar infer metavar var b end metavar end define , define proof of 1rule exist intro as rule tactic end define end math ] "
\item " [ math define statement of axiom extensionality as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar zermelo is metavar var y end metavar imply for all objects object var var s end var indeed not0 object var var s end var in0 metavar var x end metavar imply object var var s end var in0 metavar var y end metavar imply not0 object var var s end var in0 metavar var y end metavar imply object var var s end var in0 metavar var x end metavar imply not0 for all objects object var var s end var indeed not0 object var var s end var in0 metavar var x end metavar imply object var var s end var in0 metavar var y end metavar imply not0 object var var s end var in0 metavar var y end metavar imply object var var s end var in0 metavar var x end metavar imply metavar var x end metavar zermelo is metavar var y end metavar end define , define proof of axiom extensionality as rule tactic end define end math ] "
\item " [ math define statement of axiom empty set as system Q infer all metavar var s end metavar indeed not0 metavar var s end metavar in0 zermelo empty set end define , define proof of axiom empty set as rule tactic end define end math ] "
\item " [ math define statement of axiom pair definition as system Q infer all metavar var s end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var s end metavar in0 zermelo pair metavar var x end metavar comma metavar var y end metavar end pair imply not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply not0 not0 metavar var s end metavar zermelo is metavar var x end metavar imply metavar var s end metavar zermelo is metavar var y end metavar imply metavar var s end metavar in0 zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end define , define proof of axiom pair definition as rule tactic end define end math ] "
\item " [ math define statement of axiom union definition as system Q infer all metavar var s end metavar indeed all metavar var x end metavar indeed not0 metavar var s end metavar in0 union metavar var x end metavar end union imply not0 metavar var s end metavar in0 existential var var j end var imply not0 existential var var j end var in0 metavar var x end metavar imply not0 not0 metavar var s end metavar in0 existential var var j end var imply not0 existential var var j end var in0 metavar var x end metavar imply metavar var s end metavar in0 union metavar var x end metavar end union end define , define proof of axiom union definition as rule tactic end define end math ] "
\item " [ math define statement of axiom power definition as system Q infer all metavar var s end metavar indeed all metavar var x end metavar indeed not0 metavar var s end metavar in0 power metavar var x end metavar end power imply for all objects object var var s end var indeed object var var s end var in0 metavar var s end metavar imply object var var s end var in0 metavar var x end metavar imply not0 for all objects object var var s end var indeed object var var s end var in0 metavar var s end metavar imply object var var s end var in0 metavar var x end metavar imply metavar var s end metavar in0 power metavar var x end metavar end power end define , define proof of axiom power definition as rule tactic end define end math ] "
\item " [ math define statement of axiom separation definition as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var p end metavar indeed all metavar var x end metavar indeed all metavar var z end metavar indeed metavar var p end metavar is placeholder-var and ph-sub metavar var b end metavar is metavar var a end metavar where metavar var p end metavar is metavar var z end metavar end sub endorse not0 metavar var z end metavar in0 the set of ph in metavar var x end metavar such that metavar var a end metavar end set imply not0 metavar var z end metavar in0 metavar var x end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var z end metavar in0 metavar var x end metavar imply not0 metavar var b end metavar imply metavar var z end metavar in0 the set of ph in metavar var x end metavar such that metavar var a end metavar end set end define , define proof of axiom separation definition as rule tactic end define 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}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma mp2 as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar end define end math ] "
\item " [ math define statement of prop lemma mp3 as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar end define end math ] "
\item " [ math define statement of prop lemma mp4 as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer metavar var e end metavar end define end math ] "
\item " [ math define statement of prop lemma mp5 as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed all metavar var f end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer metavar var e end metavar infer metavar var f end metavar end define end math ] "
\end{list}
\subsubsection{Det f\o{}rste bevis}
Vi begynder med at bevise " [ math prop lemma mp2 end math ] ":
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma mp2 as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar end define end math ] "
\item " [ math define proof of prop lemma mp2 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar infer metavar var b end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar modus ponens metavar var a end metavar conclude metavar var b end metavar imply metavar var c end metavar cut 1rule mp modus ponens metavar var b end metavar imply metavar var c end metavar modus ponens metavar var b end metavar conclude metavar var c end metavar end quote state proof state cache var c end expand end define 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 metavar var b end metavar imply metavar var c end metavar 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 metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar 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 metavar var a end metavar comma metavar var b end metavar comma metavar var c end metavar 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 all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar 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}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma mp3 as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar end define end math ] "
\item
" [ math define proof of prop lemma mp3 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer prop lemma mp2 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar modus ponens metavar var a end metavar modus ponens metavar var b end metavar conclude metavar var c end metavar imply metavar var d end metavar cut 1rule mp modus ponens metavar var c end metavar imply metavar var d end metavar modus ponens metavar var c end metavar conclude metavar var d end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of prop lemma mp4 as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer metavar var e end metavar end define end math ] "
" [ math define proof of prop lemma mp4 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer prop lemma mp2 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar modus ponens metavar var a end metavar modus ponens metavar var b end metavar conclude metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar cut prop lemma mp2 modus ponens metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar modus ponens metavar var c end metavar modus ponens metavar var d end metavar conclude metavar var e end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of prop lemma mp5 as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed all metavar var f end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer metavar var e end metavar infer metavar var f end metavar end define end math ] "
" [ math define proof of prop lemma mp5 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed all metavar var f end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar infer metavar var a end metavar infer metavar var b end metavar infer metavar var c end metavar infer metavar var d end metavar infer metavar var e end metavar infer prop lemma mp3 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar modus ponens metavar var a end metavar modus ponens metavar var b end metavar modus ponens metavar var c end metavar conclude metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar cut prop lemma mp2 modus ponens metavar var d end metavar imply metavar var e end metavar imply metavar var f end metavar modus ponens metavar var d end metavar modus ponens metavar var e end metavar conclude metavar var f end metavar end quote state proof state cache var c end expand end define 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}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma auto imply as system Q infer all metavar var a end metavar indeed metavar var a end metavar imply metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma auto imply as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed metavar var a end metavar infer 1rule repetition modus ponens metavar var a end metavar conclude metavar var a end metavar cut all metavar var a end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar conclude metavar var a end metavar imply metavar var a end metavar end quote state proof state cache var c end expand end define 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 all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar 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 all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar end bracket ] " til implikationen " [ bracket metavar var a end metavar imply metavar var a end metavar 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}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma imply transitivity as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar imply metavar var c end metavar end define 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 define proof of prop lemma imply transitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar conclude metavar var b end metavar cut 1rule mp modus ponens metavar var b end metavar imply metavar var c end metavar modus ponens metavar var b end metavar conclude metavar var c end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var b end metavar imply metavar var c end metavar infer 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar infer metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar infer metavar var c end metavar conclude metavar var a end metavar imply metavar var b end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply metavar var c end metavar cut prop lemma mp2 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply metavar var c end metavar modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var b end metavar imply metavar var c end metavar conclude metavar var a end metavar imply metavar var c end metavar end quote state proof state cache var c end expand end define 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 metavar var a end metavar end math ] " g\ae{}lder ubetinget, s\aa{} g\ae{}lder den ogs\aa{} under antagelse af en vilk\aa{}rlig anden formel " [ math metavar var b end metavar end math ] ". Lemmaet " [ math prop lemma weakening end math ] " udtrykker dette r\ae{}sonnement som f\o{}lger:
\begin{list}{}{
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\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma weakening as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar infer metavar var a end metavar imply metavar var b end metavar end define end math ] "
\item Vi beviser " [ math prop lemma weakening end math ] " ved hj\ae{}lp af deduktionsreglen:
\item " [ math define proof of prop lemma weakening as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar infer metavar var a end metavar infer 1rule repetition modus ponens metavar var b end metavar conclude metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar infer metavar var a end metavar infer metavar var b end metavar conclude metavar var b end metavar imply metavar var a end metavar imply metavar var b end metavar cut metavar var b end metavar infer 1rule mp modus ponens metavar var b end metavar imply metavar var a end metavar imply metavar var b end metavar modus ponens metavar var b end metavar conclude metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define 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}{}{
\setlength{\leftmargin}{0em}
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\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma from contradiction as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var a end metavar infer metavar var b end metavar end define end math ] "
\item Beviset bruger " [ math prop lemma weakening end math ] " og slutningsreglen " [ math 1rule ad absurdum end math ] ":
\item " [ math define proof of prop lemma from contradiction as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var a end metavar infer prop lemma weakening modus ponens metavar var a end metavar conclude not0 metavar var b end metavar imply metavar var a end metavar cut prop lemma weakening modus ponens not0 metavar var a end metavar conclude not0 metavar var b end metavar imply not0 metavar var a end metavar cut 1rule ad absurdum modus ponens not0 metavar var b end metavar imply metavar var a end metavar modus ponens not0 metavar var b end metavar imply not0 metavar var a end metavar conclude metavar var b end metavar end quote state proof state cache var c end expand end define 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}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma remove double neg as system Q infer all metavar var a end metavar indeed not0 not0 metavar var a end metavar infer metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma remove double neg as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed not0 not0 metavar var a end metavar infer prop lemma weakening modus ponens not0 not0 metavar var a end metavar conclude not0 metavar var a end metavar imply not0 not0 metavar var a end metavar cut prop lemma auto imply conclude not0 metavar var a end metavar imply not0 metavar var a end metavar cut 1rule ad absurdum modus ponens not0 metavar var a end metavar imply not0 metavar var a end metavar modus ponens not0 metavar var a end metavar imply not0 not0 metavar var a end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of prop lemma add double neg as system Q infer all metavar var a end metavar indeed metavar var a end metavar infer not0 not0 metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma add double neg as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed not0 not0 not0 metavar var a end metavar infer prop lemma remove double neg modus ponens not0 not0 not0 metavar var a end metavar conclude not0 metavar var a end metavar cut all metavar var a end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed not0 not0 not0 metavar var a end metavar infer not0 metavar var a end metavar conclude not0 not0 not0 metavar var a end metavar imply not0 metavar var a end metavar cut metavar var a end metavar infer prop lemma weakening modus ponens metavar var a end metavar conclude not0 not0 not0 metavar var a end metavar imply metavar var a end metavar cut 1rule ad absurdum modus ponens not0 not0 not0 metavar var a end metavar imply metavar var a end metavar modus ponens not0 not0 not0 metavar var a end metavar imply not0 metavar var a end metavar conclude not0 not0 metavar var a end metavar end quote state proof state cache var c end expand end define 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 define statement of prop lemma mt as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer not0 metavar var a end metavar end define 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 define statement of prop lemma technicality as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 not0 metavar var a end metavar imply metavar var b end metavar end define end math ] "
\item " [ math define proof of prop lemma technicality as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 not0 metavar var a end metavar infer prop lemma remove double neg modus ponens not0 not0 metavar var a end metavar conclude metavar var a end metavar cut 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar conclude metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 not0 metavar var a end metavar infer metavar var b end metavar conclude metavar var a end metavar imply metavar var b end metavar imply not0 not0 metavar var a end metavar imply metavar var b end metavar cut metavar var a end metavar imply metavar var b end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar imply not0 not0 metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar imply metavar var b end metavar conclude not0 not0 metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define 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 metavar var a end metavar end math ] " optr\ae{}der i negeret form:
\item " [ math define statement of prop lemma negative mt as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma negative mt as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer prop lemma weakening modus ponens not0 metavar var b end metavar conclude not0 metavar var a end metavar imply not0 metavar var b end metavar cut 1rule ad absurdum modus ponens not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define 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 define statement of prop lemma mt as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer not0 metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma mt as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer prop lemma technicality conclude not0 not0 metavar var a end metavar imply metavar var b end metavar cut prop lemma negative mt modus ponens not0 not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var b end metavar conclude not0 metavar var a end metavar end quote state proof state cache var c end expand end define 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 define statement of prop lemma contrapositive as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar imply not0 metavar var a end metavar end define end math ] "
\item N\aa{}r en inferens skal erstattes med en implikation, er det altid deduktionsreglen, der skal i spil:
\item " [ math define proof of prop lemma contrapositive as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer prop lemma mt modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var b end metavar conclude not0 metavar var a end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer not0 metavar var a end metavar conclude metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply not0 metavar var a end metavar cut 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply not0 metavar var a end metavar modus ponens metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var b end metavar imply not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{Konjunktion}
Hovedm\aa{}let med dette underafsnit er at konvertere mellem formlerne " [ math metavar var a end metavar end math ] " og " [ math metavar var b end metavar end math ] " og deres konjunktion " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] ".
\subsubsection{Forening af konjunkter}
Vi begynder med at sl\aa{} " [ math metavar var a end metavar end math ] " og " [ math metavar var b end metavar end math ] " sammen til " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] ":
\begin{list}{}{
\setlength{\leftmargin}{0em}
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\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma join conjuncts as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer metavar var b end metavar infer not0 metavar var a end metavar imply not0 metavar var b end metavar end define end math ] "
\item Beviset for " [ math prop lemma join conjuncts end math ] " er af teknisk karakter. Vi viser den makroekspanderede form " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] ", som vi i bevisets sidste linie konverterer til " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar 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 define proof of prop lemma join conjuncts as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer metavar var a end metavar imply not0 metavar var b end metavar infer 1rule mp modus ponens metavar var a end metavar imply not0 metavar var b end metavar modus ponens metavar var a end metavar conclude not0 metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer metavar var a end metavar imply not0 metavar var b end metavar infer not0 metavar var b end metavar conclude metavar var a end metavar imply metavar var a end metavar imply not0 metavar var b end metavar imply not0 metavar var b end metavar cut metavar var a end metavar infer metavar var b end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var a end metavar imply not0 metavar var b end metavar imply not0 metavar var b end metavar modus ponens metavar var a end metavar conclude metavar var a end metavar imply not0 metavar var b end metavar imply not0 metavar var b end metavar cut prop lemma add double neg modus ponens metavar var b end metavar conclude not0 not0 metavar var b end metavar cut prop lemma mt modus ponens metavar var a end metavar imply not0 metavar var b end metavar imply not0 metavar var b end metavar modus ponens not0 not0 metavar var b end metavar conclude not0 metavar var a end metavar imply not0 metavar var b end metavar cut 1rule repetition modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude not0 metavar var a end metavar imply not0 metavar var b end metavar end quote state proof state cache var c end expand end define 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 not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] ". Jeg vil ikke kommentere beviset:
\begin{list}{}{
\setlength{\leftmargin}{0em}
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\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma second conjunct as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar infer metavar var b end metavar end define end math ] "
\item " [ math define proof of prop lemma second conjunct as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var b end metavar infer prop lemma weakening modus ponens not0 metavar var b end metavar conclude metavar var a end metavar imply not0 metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var b end metavar infer metavar var a end metavar imply not0 metavar var b end metavar conclude not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var b end metavar cut not0 metavar var a end metavar imply not0 metavar var b end metavar infer 1rule repetition modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude not0 metavar var a end metavar imply not0 metavar var b end metavar cut prop lemma negative mt modus ponens not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var b end metavar modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Udskilning af f\o{}rste konjunkt}
For at udskille " [ math metavar var a end metavar end math ] " fra " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] " viser vi f\o{}rst, at " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] " er kommutativ. Jeg vil ikke kommentere beviset:
\begin{list}{}{
\setlength{\leftmargin}{0em}
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\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma and commutativity as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar infer not0 metavar var b end metavar imply not0 metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma and commutativity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar imply not0 metavar var a end metavar infer metavar var a end metavar infer prop lemma add double neg modus ponens metavar var a end metavar conclude not0 not0 metavar var a end metavar cut prop lemma mt modus ponens metavar var b end metavar imply not0 metavar var a end metavar modus ponens not0 not0 metavar var a end metavar conclude not0 metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar imply not0 metavar var a end metavar infer metavar var a end metavar infer not0 metavar var b end metavar conclude metavar var b end metavar imply not0 metavar var a end metavar imply metavar var a end metavar imply not0 metavar var b end metavar cut not0 metavar var a end metavar imply not0 metavar var b end metavar infer 1rule repetition conclude not0 metavar var a end metavar imply not0 metavar var b end metavar cut prop lemma mt modus ponens metavar var b end metavar imply not0 metavar var a end metavar imply metavar var a end metavar imply not0 metavar var b end metavar modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude not0 metavar var b end metavar imply not0 metavar var a end metavar cut 1rule repetition modus ponens not0 metavar var b end metavar imply not0 metavar var a end metavar conclude not0 metavar var b end metavar imply not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item Nu er det let at udskille den f\o{}rste konjunkt fra " [ bracket not0 metavar var a end metavar imply not0 metavar var b end metavar end bracket ] ": F\o{}rst vender vi konjunktionen om til " [ bracket not0 metavar var b end metavar imply not0 metavar var a end metavar end bracket ] " ved hj\ae{}lp af " [ math prop lemma and commutativity end math ] ", og s\aa{} udskiller vi " [ math metavar var a end metavar end math ] " ved hj\ae{}lp af " [ math prop lemma second conjunct end math ] ":
\item " [ math define statement of prop lemma first conjunct as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar infer metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma first conjunct as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar infer prop lemma and commutativity modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar conclude not0 metavar var b end metavar imply not0 metavar var a end metavar cut prop lemma second conjunct modus ponens not0 metavar var b end metavar imply not0 metavar var a end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define 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}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma iff first as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer metavar var b end metavar infer metavar var a end metavar end define end math ] "
\item " [ math define proof of prop lemma iff first as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer metavar var b end metavar infer prop lemma second conjunct modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar conclude metavar var b end metavar imply metavar var a end metavar cut 1rule mp modus ponens metavar var b end metavar imply metavar var a end metavar modus ponens metavar var b end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item" [ math define statement of prop lemma iff second as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer metavar var a end metavar infer metavar var b end metavar end define end math ] "
\item " [ math define proof of prop lemma iff second as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer metavar var a end metavar infer prop lemma first conjunct modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar conclude metavar var a end metavar imply metavar var b end metavar cut 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar conclude metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsubsection{Kommutativitet}
Lemmaet " [ math prop lemma iff commutativity end math ] " f\o{}lger direkte af, at operatoren " [ bracket not0 var x imply not0 var y end bracket ] "
er kommutativ:
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop lemma iff commutativity as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar end define end math ] "
\item " [ math define proof of prop lemma iff commutativity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar infer 1rule repetition modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar cut prop lemma and commutativity modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var b end metavar imply metavar var a end metavar conclude not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar cut 1rule repetition modus ponens not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var b end metavar imply metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define 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 metavar var b end metavar end math ] " vil vi gerne udlede de svagere p\aa{}stande " [ bracket not0 metavar var a end metavar imply metavar var b end metavar end bracket ] " og " [ bracket not0 metavar var b end metavar imply metavar var a end metavar 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 define statement of prop lemma weaken or first as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar infer not0 metavar var a end metavar imply metavar var b end metavar end define end math ] "
\item Beviset best\aa{}r af en simpel anvendelse af " [ math prop lemma weakening end math ] ":
\item " [ math define proof of prop lemma weaken or first as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var b end metavar infer prop lemma weakening modus ponens metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar cut 1rule repetition modus ponens not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\item Slutningen fra " [ math metavar var a end metavar end math ] " til " [ bracket not0 metavar var a end metavar imply metavar var b end metavar end bracket ] " varetages af lemmaet " [ math prop lemma weaken or second end math ] ":
\item " [ math define statement of prop lemma weaken or second as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var a end metavar imply metavar var b end metavar end define 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 define proof of prop lemma weaken or second as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var a end metavar infer prop lemma from contradiction modus ponens metavar var a end metavar modus ponens not0 metavar var a end metavar conclude metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var a end metavar infer metavar var b end metavar conclude metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar cut metavar var a end metavar infer 1rule mp modus ponens metavar var a end metavar imply not0 metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar cut 1rule repetition modus ponens not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define 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 define statement of prop lemma from disjuncts as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer metavar var a end metavar imply metavar var c end metavar infer metavar var b end metavar imply metavar var c end metavar infer metavar var c end metavar end define end math ] "
\item Om beviset vil jeg kun sige, at det er en ret elegant \o{}velse i bevisteknik:
\item " [ math define proof of prop lemma from disjuncts as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer metavar var a end metavar imply metavar var c end metavar infer metavar var b end metavar imply metavar var c end metavar infer 1rule repetition modus ponens not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar cut prop lemma contrapositive modus ponens not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var b end metavar imply not0 not0 metavar var a end metavar cut prop lemma technicality modus ponens metavar var a end metavar imply metavar var c end metavar conclude not0 not0 metavar var a end metavar imply metavar var c end metavar cut prop lemma imply transitivity modus ponens not0 metavar var b end metavar imply not0 not0 metavar var a end metavar modus ponens not0 not0 metavar var a end metavar imply metavar var c end metavar conclude not0 metavar var b end metavar imply metavar var c end metavar cut prop lemma contrapositive modus ponens not0 metavar var b end metavar imply metavar var c end metavar conclude not0 metavar var c end metavar imply not0 not0 metavar var b end metavar cut prop lemma contrapositive modus ponens metavar var b end metavar imply metavar var c end metavar conclude not0 metavar var c end metavar imply not0 metavar var b end metavar cut 1rule ad absurdum modus ponens not0 metavar var c end metavar imply not0 metavar var b end metavar modus ponens not0 metavar var c end metavar imply not0 not0 metavar var b end metavar conclude metavar var c end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
(********************)
\appendix
" [ flush left math define priority of am as 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 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 define tex of am as text "am" end text end define end math ] "
\item " [ math define tex of cdots as text "(\cdots{})" end text end define end math ] "
\item " [ math define tex of object-var as text "\texttt{Objekt-var}" end text end define end math ] "
\item " [ math define tex of ex-var as text "\texttt{Ex-var}" end text end define end math ] "
\item " [ math define tex of ph-var as text "\texttt{Ph-var}" end text end define end math ] "
\item " [ math define tex of vaerdi as text "\texttt{V\ae{}rdi}" end text end define end math ] "
\item " [ math define tex of variabel as text "\texttt{Variabel}" end text end define end math ] "
\item " [ math define tex of op var x end op as text "Op(#1.
)" end text end define end math ] "
\item " [ math define tex of op2 var x comma var y end op2 as text "Op(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of define-equal var x comma var y end equal as text "#1.
\mathrel {\ddot {==}} #2." end text end define end math ] "
\item " [ math define tex of contains-empty var x end empty as text "ContainsEmpty(#1.
)" end text end define end math ] "
" [ math define tex of 1deduction var x conclude var y end 1deduction as text "
Dedu(#1.
,#2.
)" end text end define end math ] "
" [ math define tex of 1deduction zero var x conclude var y end 1deduction as text "
Dedu_0(#1.
,#2.
)" end text end define end math ] "
" [ math define tex of 1deduction side var x conclude var y condition var z end 1deduction as text "Dedu_{s}(#1.
,#2.
,#3.
)" end text end define end math ] "
" [ math define tex of 1deduction one var x conclude var y condition var z end 1deduction as text "
Dedu_1(#1.
,#2.
,#3.
)" end text end define end math ] "
" [ math define tex of 1deduction two var x conclude var y condition var z end 1deduction as text "
Dedu_2(#1.
,#2.
,#3.
)" end text end define end math ] "
" [ math define tex of 1deduction three var x conclude var y condition var z bound var u end 1deduction as text "
Dedu_3(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
" [ math define tex of 1deduction four var x conclude var y condition var z bound var u end 1deduction as text "
Dedu_4(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
" [ math define tex of 1deduction four star var x conclude var y condition var z bound var u end 1deduction as text "
Dedu_4^*(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
" [ math define tex of 1deduction five var x condition var y bound var z end 1deduction as text "
Dedu_5(#1.
,#2.
,#3.
)" end text end define end math ] "
" [ math define tex of 1deduction six var p conclude var c exception var e bound var b end 1deduction as text "
Dedu_6(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
" [ math define tex of 1deduction six star var p conclude var c exception var e bound var b end 1deduction as text "
Dedu_6^*(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
" [ math define tex of 1deduction seven var p end 1deduction as text "
Dedu_7(#1.
)" end text end define end math ] "
" [ math define tex of 1deduction eight var p bound var b end 1deduction as text "
Dedu_8(#1.
,#2.
)" end text end define end math ] "
" [ math define tex of 1deduction eight star var p bound var b end 1deduction as text "
Dedu_8^*(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of ex1 as text "Ex_{1}" end text end define end math ] "
\item " [ math define tex of ex2 as text "Ex_{2}" end text end define end math ] "
\item " [ math define tex of ex10 as text "Ex_{10}" end text end define end math ] "
\item " [ math define tex of ex20 as text "Ex_{20}" end text end define end math ] "
\item " [ math define tex of existential var var x end var as text "#1.
_{Ex}" end text end define end math ] "
\item " [ math define tex of var x is existential var as text "#1.
^{Ex}" end text end define end math ] "
" [ math define tex of exist-sub var x is var y where var z is var u end sub as text "\langle #1.
{\equiv} #2.
| #3.
{:==} #4.
\rangle_{Ex} " end text end define end math ] "
" [ math define tex of exist-sub0 var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^0 #2.
| #3.
{:==} #4.
\rangle_{Ex} " end text end define end math ] "
" [ math define tex of exist-sub1 var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^1 #2.
| #3.
{:==} #4.
\rangle_{Ex} " end text end define end math ] "
" [ math define tex of exist-sub* var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^* #2.
| #3.
{:==} #4.
\rangle_{Ex} " end text end define end math ] "
\item " [ math define tex of placeholder-var1 as text "ph_{1}" end text end define end math ] "
\item " [ math define tex of placeholder-var2 as text "ph_{2}" end text end define end math ] "
\item " [ math define tex of placeholder-var3 as text "ph_{3}" end text end define end math ] "
\item " [ math define tex of placeholder-var var x end var as text "#1.
_{Ph} " end text end define end math ] "
\item " [ math define tex of var x is placeholder-var as text "#1.
^{Ph}" end text end define end math ] "
" [ math define tex of ph-sub var x is var y where var z is var u end sub as text "\langle #1.
{\equiv} #2.
| #3.
{:==} #4.
\rangle_{Ph} " end text end define end math ] "
" [ math define tex of ph-sub0 var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^0 #2.
| #3.
{:==} #4.
\rangle_{Ph} " end text end define end math ] "
" [ math define tex of ph-sub1 var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^1 #2.
| #3.
{:==} #4.
\rangle_{Ph} " end text end define end math ] "
" [ math define tex of ph-sub* var x is var y where var z is var u end sub as text "\langle #1.
{\equiv}^* #2.
| #3.
{:==} #4.
\rangle_{Ph} " end text end define end math ] "
\item " [ math define tex of var big set as text "\mathsf {bs}" end text end define end math ] "
\item " [ math define tex of object big set as text " \mathsf {OBS}" end text end define end math ] "
" [ math define tex of meta big set as text "{\cal BS}" end text end define end math ] "
\item " [ math define tex of zermelo empty set as text "\mathrm{\O}" end text end define end math ] "
\item " [ math define tex of system Q as text "ZFsub" end text end define end math ] "
\item " [ math define tex of 1rule mp as text "MP" end text end define end math ] "
\item " [ math define tex of 1rule gen as text "Gen" end text end define end math ] "
\item " [ math define tex of 1rule repetition as text "Repetition" end text end define end math ] "
\item " [ math define tex of 1rule ad absurdum as text "Neg" end text end define end math ] "
\item " [ math define tex of 1rule deduction as text "Ded" end text end define end math ] "
\item " [ math define tex of 1rule exist intro as text "ExistIntro" end text end define end math ] "
\item " [ math define tex of axiom extensionality as text "Extensionality" end text end define end math ] "
\item " [ math define tex of axiom empty set as text "\O{}def" end text end define end math ] "
\item " [ math define tex of axiom pair definition as text "PairDef" end text end define end math ] "
\item " [ math define tex of axiom union definition as text "UnionDef" end text end define end math ] "
\item " [ math define tex of axiom power definition as text "PowerDef" end text end define end math ] "
\item " [ math define tex of axiom separation definition as text "SeparationDef" end text end define end math ] "
\item " [ math define tex of prop lemma add double neg as text "AddDoubleNeg" end text end define end math ] "
\item " [ math define tex of prop lemma remove double neg as text "RemoveDoubleNeg" end text end define end math ] "
\item " [ math define tex of prop lemma and commutativity as text "AndCommutativity" end text end define end math ] "
\item " [ math define tex of prop lemma auto imply as text "AutoImply" end text end define end math ] "
\item " [ math define tex of prop lemma contrapositive as text "Contrapositive" end text end define end math ] "
\item " [ math define tex of prop lemma first conjunct as text "FirstConjunct" end text end define end math ] "
\item " [ math define tex of prop lemma second conjunct as text "SecondConjunct" end text end define end math ] "
\item " [ math define tex of prop lemma from contradiction as text "FromContradiction" end text end define end math ] "
\item " [ math define tex of prop lemma from disjuncts as text "FromDisjuncts" end text end define end math ] "
\item " [ math define tex of prop lemma iff commutativity as text "IffCommutativity" end text end define end math ] "
\item " [ math define tex of prop lemma iff first as text "IffFirst" end text end define end math ] "
\item " [ math define tex of prop lemma iff second as text "IffSecond" end text end define end math ] "
\item " [ math define tex of prop lemma imply transitivity as text "ImplyTransitivity" end text end define end math ] "
\item " [ math define tex of prop lemma join conjuncts as text "JoinConjuncts" end text end define end math ] "
\item " [ math define tex of prop lemma mp2 as text "MP2" end text end define end math ] "
\item " [ math define tex of prop lemma mp3 as text "MP3" end text end define end math ] "
\item " [ math define tex of prop lemma mp4 as text "MP4" end text end define end math ] "
\item " [ math define tex of prop lemma mp5 as text "MP5" end text end define end math ] "
\item " [ math define tex of prop lemma mt as text "MT" end text end define end math ] "
\item " [ math define tex of prop lemma negative mt as text "NegativeMT" end text end define end math ] "
\item " [ math define tex of prop lemma technicality as text "Technicality" end text end define end math ] "
\item " [ math define tex of prop lemma weakening as text "Weakening" end text end define end math ] "
\item " [ math define tex of prop lemma weaken or first as text "WeakenOr1" end text end define end math ] "
\item " [ math define tex of prop lemma weaken or second as text "WeakenOr2" end text end define end math ] "
\item " [ math define tex of lemma pair2formula as text "Pair2Formula" end text end define end math ] "
\item " [ math define tex of lemma formula2pair as text "Formula2Pair" end text end define end math ] "
\item " [ math define tex of lemma union2formula as text "Union2Formula" end text end define end math ] "
\item " [ math define tex of lemma formula2union as text "Formula2Union" end text end define end math ] "
\item " [ math define tex of lemma separation2formula as text "Sep2Formula" end text end define end math ] "
\item " [ math define tex of lemma formula2separation as text "Formula2Sep" end text end define end math ] "
\item " [ math define tex of lemma subset in power set as text "SubsetInPower" end text end define end math ] "
\item " [ math define tex of lemma power set is subset0 as text "HelperPowerIsSub" end text end define end math ] "
\item " [ math define tex of lemma power set is subset as text "PowerIsSub" end text end define end math ] "
\item " [ math define tex of lemma power set is subset0-switch as text "(Switch)HelperPowerIsSub" end text end define end math ] "
\item " [ math define tex of lemma power set is subset-switch as text "(Switch)PowerIsSub" end text end define end math ] "
\item " [ math define tex of lemma set equality suff condition as text "ToSetEquality" end text end define end math ] "
\item " [ math define tex of lemma set equality suff condition(t)0 as text "HelperToSetEquality(t)" end text end define end math ] "
\item " [ math define tex of lemma set equality suff condition(t) as text "ToSetEquality(t)" end text end define end math ] "
\item " [ math define tex of lemma set equality skip quantifier as text "HelperFromSetEquality" end text end define end math ] "
\item " [ math define tex of lemma set equality nec condition as text "FromSetEquality" end text end define end math ] "
\item " [ math define tex of lemma reflexivity0 as text "HelperReflexivity" end text end define end math ] "
\item " [ math define tex of lemma reflexivity as text "Reflexivity" end text end define end math ] "
\item " [ math define tex of lemma symmetry0 as text "HelperSymmetry" end text end define end math ] "
\item " [ math define tex of lemma symmetry as text "Symmetry" end text end define end math ] "
\item " [ math define tex of lemma transitivity0 as text "HelperTransitivity" end text end define end math ] "
\item " [ math define tex of lemma transitivity as text "Transitivity" end text end define end math ] ",
\item " [ math define tex of lemma er is reflexive as text "ERisReflexive" end text end define end math ] "
\item " [ math define tex of lemma er is symmetric as text "ERisSymmetric" end text end define end math ] "
\item " [ math define tex of lemma er is transitive as text "ERisTransitive" end text end define end math ] "
\item " [ math define tex of lemma empty set is subset as text "\O{}isSubset" end text end define end math ] "
\item " [ math define tex of lemma member not empty0 as text "HelperMemberNot\O{}" end text end define end math ] "
\item " [ math define tex of lemma member not empty as text "MemberNot\O{}" end text end define end math ] "
\item " [ math define tex of lemma unique empty set0 as text "HelperUnique\O{}" end text end define end math ] "
\item " [ math define tex of lemma unique empty set as text "Unique\O{}" end text end define end math ] "
\item " [ math define tex of lemma ==Reflexivity as text "==\!{}Reflexivity" end text end define end math ] "
\item " [ math define tex of lemma ==Symmetry as text "==\!{}Symmetry" end text end define end math ] "
\item " [ math define tex of lemma ==Transitivity0 as text "Helper\!{}==\!{}Transitivity" end text end define end math ] "
\item " [ math define tex of lemma ==Transitivity as text "\!{}==\!{}Transitivity" end text end define end math ] "
\item " [ math define tex of lemma transfer ~is0 as text "HelperTransferNotEq" end text end define end math ] "
\item " [ math define tex of lemma transfer ~is as text "TransferNotEq" end text end define end math ] "
\item " [ math define tex of lemma pair subset0 as text "HelperPairSubset" end text end define end math ] "
\item " [ math define tex of lemma pair subset1 as text "Helper(2)PairSubset" end text end define end math ] "
\item " [ math define tex of lemma pair subset as text "PairSubset" end text end define end math ] "
\item " [ math define tex of lemma same pair as text "SamePair" end text end define end math ] "
\item " [ math define tex of lemma same singleton as text "SameSingleton" end text end define end math ] "
\item " [ math define tex of lemma union subset as text "UnionSubset" end text end define end math ] "
\item " [ math define tex of lemma same union as text "SameUnion" end text end define end math ] "
\item " [ math define tex of lemma separation subset as text "SeparationSubset" end text end define end math ] "
\item " [ math define tex of lemma same separation as text "SameSeparation" end text end define end math ] "
\item " [ math define tex of lemma same binary union as text "SameBinaryUnion" end text end define end math ] "
\item " [ math define tex of lemma intersection subset as text "IntersectionSubset" end text end define end math ] "
\item " [ math define tex of lemma same intersection as text "SameIntersection" end text end define end math ] "
\item " [ math define tex of lemma auto member as text "AutoMember" end text end define end math ] "
\item " [ math define tex of lemma eq-system not empty0 as text "HelperEqSysNot\O{}" end text end define end math ] "
\item " [ math define tex of lemma eq-system not empty as text "EqSysNot\O{}" end text end define end math ] "
\item " [ math define tex of lemma eq subset0 as text "HelperEqSubset" end text end define end math ] "
\item " [ math define tex of lemma eq subset as text "EqSubset" end text end define end math ] "
\item " [ math define tex of lemma equivalence nec condition as text "EqNecessary" end text end define end math ] "
\item " [ math define tex of lemma equivalence nec condition0 as text "HelperEqNecessary" end text end define end math ] "
\item " [ math define tex of lemma none-equivalence nec condition0 as text "HelperNoneEqNecessary" end text end define end math ] "
\item " [ math define tex of lemma none-equivalence nec condition1 as text "Helper(2)NoneEqNecessary" end text end define end math ] "
\item " [ math define tex of lemma none-equivalence nec condition as text "NoneEqNecessary" end text end define end math ] "
\item " [ math define tex of lemma equivalence class is subset as text "EqClassIsSubset" end text end define end math ] "
\item " [ math define tex of lemma equivalence classes are disjoint as text "EqClassesAreDisjoint" end text end define end math ] "
\item " [ math define tex of lemma all disjoint as text "AllDisjoint" end text end define end math ] "
\item " [ math define tex of lemma all disjoint-imply as text "AllDisjointImply" end text end define end math ] "
\item " [ math define tex of lemma bs subset union(bs/r) as text "BSsubset" end text end define end math ] "
\item " [ math define tex of lemma union(bs/r) subset bs as text "Union(BS/R)subset" end text end define end math ] "
\item " [ math define tex of lemma union(bs/r) is bs as text "UnionIdentity" end text end define end math ] "
\item " [ math define tex of theorem eq-system is partition as text "EqSysIsPartition" end text end define end math ] "
\item " [ math define tex of eq-system of var x modulo var y as text "#1.
/ #2." end text end define end math ] "
\item " [ math define tex of intersection var x comma var y end intersection as text "#1.
\cap #2." end text end define end math ] "
\item " [ math define tex of union var x end union as text "\cup #1." end text end define end math ] "
\item " [ math define tex of binary-union var x comma var y end union as text "#1.
\mathrel{\cup} #2." end text end define end math ] "
\item " [ math define tex of power var x end power as text "P(#1.
)" end text end define end math ] "
\item " [ math define tex of zermelo singleton var x end singleton as text "\{#1.
\}" end text end define end math ] "
\item " [ math define tex of zermelo pair var x comma var y end pair as text "\{#1.
,#2.
\}" end text end define end math ] "
\item " [ math define tex of zermelo ordered pair var x comma var y end pair as text "\langle #1.
,#2.
\rangle" end text end define end math ] ",
\item " [ math define tex of var x in0 var y as text "#1.
\mathrel{\in} #2." end text end define end math ] "
\item " [ math define tex of var x is related to var y under var z as text "#3.
(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of var r is reflexive relation in var x as text "ReflRel(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of var r is symmetric relation in var x as text "SymRel(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of var r is transitive relation in var x as text "TransRel(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of var r is equivalence relation in var x as text "EqRel(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of equivalence class of var x in var big set modulo var r as text "[#1.
\mathrel{\in} #2.
]_{#3.
}" end text end define end math ] "
\item " [ math define tex of var x is partition of var y as text "Partition(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of var x zermelo is var y as text "#1.
\!\mathrel{==}\! #2." end text end define end math ] "
\item " [ math define tex of var x is subset of var y as text "#1.
\mathrel{\subseteq} #2." end text end define end math ] "
\item " [ math define tex of not0 var x as text "\dot{\neg}\, #1." end text end define end math ] "
\item " [ math define tex of var x zermelo ~in var y as text "#1.
\mathrel{\notin} #2." end text end define end math ] "
\item " [ math define tex of var x zermelo ~is var y as text "#1.
\mathrel{\neq} #2." end text end define end math ] "
\item " [ math define tex of var x and0 var y as text "#1.
\mathrel{\dot{\wedge}} #2." end text end define end math ] "
\item " [ math define tex of var x or0 var y as text "#1.
\mathrel{\dot{\vee}} #2." end text end define end math ] "
" [ math define tex of var x iff var y as text "#1.
\mathrel{\dot{\Leftrightarrow}} #2." end text end define end math ] "
\item " [ math define tex of the set of ph in var x such that var a end set as text " \{ ph \mathrel{\in} #1.
\mid #2.
\}" end text end define end math ] "
\end{list}
------------- RRRRRRRRRRRRRRR -------------
(*** aksiomer ***)
" [ math define statement of axiom leqReflexivity as system Q infer all metavar var x end metavar indeed metavar var x end metavar <= metavar var x end metavar end define , define proof of axiom leqReflexivity as rule tactic end define end math ] "
" [ math define statement of axiom leqAntisymmetry as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar end define , define proof of axiom leqAntisymmetry as rule tactic end define end math ] "
" [ math define statement of axiom leqTransitivity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var z end metavar imply metavar var x end metavar <= metavar var z end metavar end define , define proof of axiom leqTransitivity as rule tactic end define end math ] "
" [ math define statement of axiom leqTotality as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar end define , define proof of axiom leqTotality as rule tactic end define end math ] "
" [ math define statement of axiom leqAddition as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar end define , define proof of axiom leqAddition as rule tactic end define end math ] "
" [ math define statement of axiom leqMultiplication as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var z end metavar imply metavar var x end metavar <= metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar end define , define proof of axiom leqMultiplication as rule tactic end define end math ] "
" [ math define statement of axiom plusAssociativity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var y end metavar + metavar var z end metavar end define , define proof of axiom plusAssociativity as rule tactic end define end math ] "
" [ math define statement of axiom plusCommutativity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var y end metavar + metavar var x end metavar end define , define proof of axiom plusCommutativity as rule tactic end define end math ] "
" [ math define statement of axiom negative as system Q infer all metavar var x end metavar indeed metavar var x end metavar + - metavar var x end metavar = 0 end define , define proof of axiom negative as rule tactic end define end math ] "
" [ math define statement of axiom plus0 as system Q infer all metavar var x end metavar indeed metavar var x end metavar + 0 = metavar var x end metavar end define , define proof of axiom plus0 as rule tactic end define end math ] "
" [ math define statement of axiom timesAssociativity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar * metavar var z end metavar end define , define proof of axiom timesAssociativity as rule tactic end define end math ] "
" [ math define statement of axiom timesCommutativity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar * metavar var y end metavar = metavar var y end metavar * metavar var x end metavar end define , define proof of axiom timesCommutativity as rule tactic end define end math ] "
" [ math define statement of axiom reciprocal as system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar = 0 imply metavar var x end metavar * 1/ metavar var x end metavar = 1 end define , define proof of axiom reciprocal as rule tactic end define end math ] "
" [ math define statement of axiom times1 as system Q infer all metavar var x end metavar indeed metavar var x end metavar * 1 = metavar var x end metavar end define , define proof of axiom times1 as rule tactic end define end math ] "
" [ math define statement of axiom distribution as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar + metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar end define , define proof of axiom distribution as rule tactic end define end math ] "
" [ math define statement of axiom 0not1 as system Q infer not0 0 = 1 end define , define proof of axiom 0not1 as rule tactic end define end math ] "
" [ math define statement of axiom equality as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var z end metavar imply metavar var y end metavar = metavar var z end metavar end define , define proof of axiom equality as rule tactic end define end math ] "
" [ math define statement of axiom eqLeq as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar <= metavar var y end metavar end define , define proof of axiom eqLeq as rule tactic end define end math ] "
" [ math define statement of axiom eqAddition as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end define , define proof of axiom eqAddition as rule tactic end define end math ] "
" [ math define statement of axiom eqMultiplication as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end define , define proof of axiom eqMultiplication as rule tactic end define end math ] "
(*** XX snydeaksiomer ***)
" [ math define statement of lemma ==Reflexivity as system Q infer all metavar var rx end metavar indeed metavar var rx end metavar == metavar var rx end metavar end define , define proof of lemma ==Reflexivity as rule tactic end define end math ] "
" [ math define statement of lemma ==Symmetry as system Q infer all metavar var rx end metavar indeed all metavar var ry end metavar indeed metavar var rx end metavar == metavar var ry end metavar infer metavar var ry end metavar == metavar var rx end metavar end define , define proof of lemma ==Symmetry as rule tactic end define end math ] "
" [ math define statement of lemma ==Transitivity as system Q infer all metavar var rx end metavar indeed all metavar var ry end metavar indeed all metavar var rz end metavar indeed metavar var rx end metavar == metavar var ry end metavar infer metavar var ry end metavar == metavar var rz end metavar infer metavar var rx end metavar == metavar var rz end metavar end define , define proof of lemma ==Transitivity as rule tactic end define end math ] "
XX ikke 100procent identisk med originalen fra equivalence-relations
" [ math define statement of lemma set equality nec condition(1) as system Q infer all metavar var fx end metavar indeed all metavar var rx end metavar indeed all metavar var ry end metavar indeed metavar var rx end metavar == metavar var ry end metavar infer metavar var fx end metavar in0 metavar var rx end metavar infer metavar var fx end metavar in0 metavar var ry end metavar end define , define proof of lemma set equality nec condition(1) as rule tactic end define end math ] "
XX boer bevises ud fra nummer 1
" [ math define statement of lemma set equality nec condition(2) as system Q infer all metavar var fx end metavar indeed all metavar var rx end metavar indeed all metavar var ry end metavar indeed metavar var rx end metavar == metavar var ry end metavar infer metavar var fx end metavar in0 metavar var ry end metavar infer metavar var fx end metavar in0 metavar var rx end metavar end define , define proof of lemma set equality nec condition(2) as rule tactic end define end math ] "
" [ math define statement of 1rule ifThenElse true as system Q infer all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var a end metavar infer if( metavar var a end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar end define , define proof of 1rule ifThenElse true as rule tactic end define end math ] "
" [ math define statement of 1rule ifThenElse false as system Q infer all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var a end metavar infer if( metavar var a end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar end define , define proof of 1rule ifThenElse false as rule tactic end define end math ] "
" [ math define statement of 1rule fromSameF as system Q infer all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar sameF metavar var fy end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar end define , define proof of 1rule fromSameF as rule tactic end define end math ] "
" [ math define statement of 1rule toSameF as system Q infer all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar infer metavar var fx end metavar sameF metavar var fy end metavar end define , define proof of 1rule toSameF as rule tactic end define end math ] "
" [ math define statement of 1rule from=f as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar =f metavar var fy end metavar infer [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] end define , define proof of 1rule from=f as rule tactic end define end math ] "
XX hm... det er nok med bare 1 meta m
XX loesning: objektkvantor
" [ math define statement of 1rule to=f as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer metavar var fx end metavar =f metavar var fy end metavar end define , define proof of 1rule to=f as rule tactic end define end math ] "
" [ math define statement of 1rule from
" [ math define statement of 1rule to
" [ math define statement of axiom plusF as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end define , define proof of axiom plusF as rule tactic end define end math ] "
" [ math define statement of axiom minusF as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed [ -f metavar var fx end metavar ; metavar var m end metavar ] = - [ metavar var fx end metavar ; metavar var m end metavar ] end define , define proof of axiom minusF as rule tactic end define end math ] "
" [ math define statement of axiom timesF as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end define , define proof of axiom timesF as rule tactic end define end math ] "
" [ math define statement of axiom 0f as system Q infer all metavar var m end metavar indeed [ 0f ; metavar var m end metavar ] = 0 end define , define proof of axiom 0f as rule tactic end define end math ] "
" [ math define statement of axiom 1f as system Q infer all metavar var m end metavar indeed [ 1f ; metavar var m end metavar ] = 1 end define , define proof of axiom 1f as rule tactic end define end math ] "
" [ math define statement of 1rule to==XX as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var rx end metavar indeed all metavar var ry end metavar indeed metavar var fx end metavar in0 metavar var rx end metavar imply metavar var fy end metavar in0 metavar var ry end metavar imply metavar var fx end metavar sameF metavar var fy end metavar infer metavar var rx end metavar == metavar var ry end metavar end define , define proof of 1rule to==XX as rule tactic end define end math ] "
" [ math define statement of 1rule from== as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed R( metavar var fx end metavar ) == R( metavar var fy end metavar ) infer metavar var fx end metavar sameF metavar var fy end metavar end define , define proof of 1rule from== as rule tactic end define end math ] "
" [ math define statement of 1rule to== as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar sameF metavar var fy end metavar infer R( metavar var fx end metavar ) == R( metavar var fy end metavar ) end define , define proof of 1rule to== as rule tactic end define end math ] "
" [ math define statement of 1rule from<
" [ math define statement of 1rule from<
" [ math define statement of 1rule from<
" [ math define statement of 1rule to<
" [ math define statement of 1rule from<< as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed R( metavar var fx end metavar ) << R( metavar var fy end metavar ) infer metavar var fx end metavar
" [ math define statement of 1rule to<< as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar
" [ math define statement of 1rule fromInR as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar in0 R( metavar var fy end metavar ) infer metavar var fx end metavar sameF metavar var fy end metavar end define , define proof of 1rule fromInR as rule tactic end define end math ] "
" [ math define statement of axiom plusR as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed R( var fx +f var fy ) == R( metavar var fx end metavar +f metavar var fy end metavar ) end define , define proof of axiom plusR as rule tactic end define end math ] "
" [ math define statement of axiom timesR as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar ) end define , define proof of axiom timesR as rule tactic end define end math ] "
(*** makroer ***)
" [ math define macro of meta ep as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta ep as metavar var ep end metavar end define end quote end define end define end math ] "
" [ math define macro of meta fx as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta fx as metavar var fx end metavar end define end quote end define end define end math ] "
" [ math define macro of meta fy as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta fy as metavar var fy end metavar end define end quote end define end define end math ] "
" [ math define macro of meta fz as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta fz as metavar var fz end metavar end define end quote end define end define end math ] "
" [ math define macro of meta fu as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta fu as metavar var fu end metavar end define end quote end define end define end math ] "
" [ math define macro of meta fv as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta fv as metavar var fv end metavar end define end quote end define end define end math ] "
" [ math define macro of meta rx as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta rx as metavar var rx end metavar end define end quote end define end define end math ] "
" [ math define macro of meta ry as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta ry as metavar var ry end metavar end define end quote end define end define end math ] "
" [ math define macro of meta rz as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta rz as metavar var rz end metavar end define end quote end define end define end math ] "
" [ math define macro of meta ru as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define meta ru as metavar var ru end metavar end define end quote end define end define end math ] "
" [ math define macro of ex3 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define ex3 as existential var var c end var end define end quote end define end define end math ] "
" [ math define macro of var x <<== var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x <<== var y as var x << var y or0 var x == var y end define end quote end define end define end math ] "
" [ math define macro of (-1) as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define (-1) as - 1 end define end quote end define end define end math ] "
" [ math define macro of 2 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define 2 as parenthesis 1 + 1 end parenthesis end define end quote end define end define end math ] "
" [ math define macro of 1/2 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define 1/2 as 1/ 2 end define end quote end define end define end math ] "
" [ math define macro of var x < var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x < var y as var x <= var y and0 var x != var y end define end quote end define end define end math ] "
" [ math define macro of var x != var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x != var y as not0 var x = var y end define end quote end define end define end math ] "
" [ math define macro of var x - var y as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define var x - var y as var x + parenthesis - var y end parenthesis end define end quote end define end define end math ] "
" [ math define macro of | var x | as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define | var x | as if( 0 <= var x , var x , - var x ) end define end quote end define end define end math ] "
" [ math define macro of 00 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define 00 as R( 0f ) end define end quote end define end define end math ] "
" [ math define macro of 01 as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define 01 as R( 1f ) end define end quote end define end define end math ] "
" [ math define macro of R( var fx ) ++ R( var fy ) as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define R( var fx ) ++ R( var fy ) as R( var fx +f var fy ) end define end quote end define end define end math ] "
" [ math define macro of --R( var fx ) as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define --R( var fx ) as R( -f var fx ) end define end quote end define end define end math ] " XX noedvendig?
" [ math define macro of R( var fx ) -- R( var fy ) as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote macro define R( var fx ) -- R( var fy ) as R( var fx ) ++ R( -f var fy ) end define end quote end define end define end math ] "
XX noedvendigt med [R( ) -- R( )] konstruktionen?
(*** REGELLEMMAER ***)
" [ math define statement of lemma leqTransitivity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer metavar var x end metavar <= metavar var z end metavar end define end math ] "
" [ math define proof of lemma leqTransitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer axiom leqTransitivity conclude metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var z end metavar imply metavar var x end metavar <= metavar var z end metavar cut prop lemma mp2 modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var z end metavar imply metavar var x end metavar <= metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude metavar var x end metavar <= metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma leqAntisymmetry as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar end define end math ] "
" [ math define proof of lemma leqAntisymmetry as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar infer axiom leqAntisymmetry conclude metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar cut prop lemma mp2 modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var x end metavar conclude metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma leqAddition as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar end define end math ] "
" [ math define proof of lemma leqAddition as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer axiom leqAddition conclude metavar var x end metavar <= metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar cut 1rule mp modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma leqMultiplication as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar infer metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar end define end math ] "
" [ math define proof of lemma leqMultiplication as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar infer axiom leqMultiplication conclude 0 <= metavar var z end metavar imply metavar var x end metavar <= metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar cut prop lemma mp2 modus ponens 0 <= metavar var z end metavar imply metavar var x end metavar <= metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar modus ponens 0 <= metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma reciprocal as system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar = 0 infer metavar var x end metavar * 1/ metavar var x end metavar = 1 end define end math ] "
" [ math define proof of lemma reciprocal as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar = 0 infer axiom reciprocal conclude not0 metavar var x end metavar = 0 imply metavar var x end metavar * 1/ metavar var x end metavar = 1 cut 1rule mp modus ponens not0 metavar var x end metavar = 0 imply metavar var x end metavar * 1/ metavar var x end metavar = 1 modus ponens not0 metavar var x end metavar = 0 conclude metavar var x end metavar * 1/ metavar var x end metavar = 1 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma equality as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar = metavar var z end metavar infer metavar var y end metavar = metavar var z end metavar end define end math ] "
" [ math define proof of lemma equality as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar = metavar var z end metavar infer axiom equality conclude metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var z end metavar imply metavar var y end metavar = metavar var z end metavar cut prop lemma mp2 modus ponens metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var z end metavar imply metavar var y end metavar = metavar var z end metavar modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar = metavar var z end metavar conclude metavar var y end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma eqLeq as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var y end metavar end define end math ] "
" [ math define proof of lemma eqLeq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer axiom eqLeq conclude metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar <= metavar var y end metavar cut 1rule mp modus ponens metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar <= metavar var y end metavar modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma eqAddition as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end define end math ] "
" [ math define proof of lemma eqAddition as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer axiom eqAddition conclude metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut 1rule mp modus ponens metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma eqMultiplication as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end define end math ] "
" [ math define proof of lemma eqMultiplication as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer axiom eqMultiplication conclude metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut 1rule mp modus ponens metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
(*** UDSAGNSLOGIK ***)
" [ math define statement of prop lemma to negated imply as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var b end metavar infer not0 metavar var a end metavar imply metavar var b end metavar end define end math ] "
" [ math define proof of prop lemma to negated imply as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var b end metavar infer not0 not0 metavar var a end metavar imply metavar var b end metavar infer prop lemma remove double neg modus ponens not0 not0 metavar var a end metavar imply metavar var b end metavar conclude metavar var a end metavar imply metavar var b end metavar cut 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar conclude metavar var b end metavar cut prop lemma from contradiction modus ponens metavar var b end metavar modus ponens not0 metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar infer not0 metavar var b end metavar infer not0 not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var a end metavar imply metavar var b end metavar conclude metavar var a end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var a end metavar imply metavar var b end metavar cut metavar var a end metavar infer not0 metavar var b end metavar infer prop lemma mp2 modus ponens metavar var a end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar modus ponens not0 metavar var b end metavar conclude not0 not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var a end metavar imply metavar var b end metavar cut prop lemma auto imply conclude not0 not0 metavar var a end metavar imply metavar var b end metavar imply not0 not0 metavar var a end metavar imply metavar var b end metavar cut 1rule ad absurdum modus ponens not0 not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 not0 metavar var a end metavar imply metavar var b end metavar imply not0 not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of prop lemma tertium non datur as system Q infer all metavar var a end metavar indeed not0 metavar var a end metavar imply not0 metavar var a end metavar end define end math ] "
" [ math define proof of prop lemma tertium non datur as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed prop lemma auto imply conclude not0 metavar var a end metavar imply not0 metavar var a end metavar cut 1rule repetition modus ponens not0 metavar var a end metavar imply not0 metavar var a end metavar conclude not0 metavar var a end metavar imply not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of prop lemma from negations as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var a end metavar imply metavar var b end metavar infer metavar var b end metavar end define end math ] "
" [ math define proof of prop lemma from negations as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 metavar var a end metavar imply metavar var b end metavar infer prop lemma tertium non datur conclude not0 metavar var a end metavar imply not0 metavar var a end metavar cut prop lemma from disjuncts modus ponens not0 metavar var a end metavar imply not0 metavar var a end metavar modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var a end metavar imply metavar var b end metavar conclude metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of prop lemma imply negation as system Q infer all metavar var a end metavar indeed metavar var a end metavar imply not0 metavar var a end metavar infer not0 metavar var a end metavar end define end math ] "
" [ math define proof of prop lemma imply negation as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed metavar var a end metavar imply not0 metavar var a end metavar infer prop lemma auto imply conclude not0 metavar var a end metavar imply not0 metavar var a end metavar cut prop lemma tertium non datur conclude not0 metavar var a end metavar imply not0 metavar var a end metavar cut prop lemma from disjuncts modus ponens not0 metavar var a end metavar imply not0 metavar var a end metavar modus ponens metavar var a end metavar imply not0 metavar var a end metavar modus ponens not0 metavar var a end metavar imply not0 metavar var a end metavar conclude not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of prop lemma from three disjuncts as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar imply metavar var d end metavar infer metavar var b end metavar imply metavar var d end metavar infer metavar var c end metavar imply metavar var d end metavar infer metavar var d end metavar end define end math ] "
" [ math define proof of prop lemma from three disjuncts as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var c end metavar infer metavar var b end metavar imply metavar var d end metavar infer metavar var c end metavar imply metavar var d end metavar infer not0 metavar var a end metavar infer 1rule repetition modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var c end metavar conclude not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var c end metavar cut 1rule mp modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var c end metavar modus ponens not0 metavar var a end metavar conclude not0 metavar var b end metavar imply metavar var c end metavar cut prop lemma from disjuncts modus ponens not0 metavar var b end metavar imply metavar var c end metavar modus ponens metavar var b end metavar imply metavar var d end metavar modus ponens metavar var c end metavar imply metavar var d end metavar conclude metavar var d end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var c end metavar infer metavar var b end metavar imply metavar var d end metavar infer metavar var c end metavar imply metavar var d end metavar infer not0 metavar var a end metavar infer metavar var d end metavar conclude not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var c end metavar imply metavar var b end metavar imply metavar var d end metavar imply metavar var c end metavar imply metavar var d end metavar imply not0 metavar var a end metavar imply metavar var d end metavar cut prop lemma auto imply conclude metavar var a end metavar imply metavar var d end metavar imply metavar var a end metavar imply metavar var d end metavar cut not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var c end metavar infer metavar var a end metavar imply metavar var d end metavar infer metavar var b end metavar imply metavar var d end metavar infer metavar var c end metavar imply metavar var d end metavar infer prop lemma mp3 modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var c end metavar imply metavar var b end metavar imply metavar var d end metavar imply metavar var c end metavar imply metavar var d end metavar imply not0 metavar var a end metavar imply metavar var d end metavar modus ponens not0 metavar var a end metavar imply not0 metavar var b end metavar imply metavar var c end metavar modus ponens metavar var b end metavar imply metavar var d end metavar modus ponens metavar var c end metavar imply metavar var d end metavar conclude not0 metavar var a end metavar imply metavar var d end metavar cut 1rule mp modus ponens metavar var a end metavar imply metavar var d end metavar imply metavar var a end metavar imply metavar var d end metavar modus ponens metavar var a end metavar imply metavar var d end metavar conclude metavar var a end metavar imply metavar var d end metavar cut prop lemma from negations modus ponens metavar var a end metavar imply metavar var d end metavar modus ponens not0 metavar var a end metavar imply metavar var d end metavar conclude metavar var d end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of prop lemma negate first disjunct as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var a end metavar infer metavar var b end metavar end define end math ] "
" [ math define proof of prop lemma negate first disjunct as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var a end metavar infer 1rule repetition modus ponens not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar cut 1rule mp modus ponens not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var a end metavar conclude metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of prop lemma negate second disjunct as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer metavar var a end metavar end define end math ] "
" [ math define proof of prop lemma negate second disjunct as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var b end metavar infer 1rule repetition modus ponens not0 metavar var a end metavar imply metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar cut prop lemma negative mt modus ponens not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var b end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
(***)
" [ math define statement of prop lemma expand disjuncts as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var c end metavar imply metavar var d end metavar infer not0 metavar var b end metavar imply not0 metavar var d end metavar imply not0 metavar var a end metavar imply not0 metavar var c end metavar end define end math ] "
" [ math define proof of prop lemma expand disjuncts as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var c end metavar imply metavar var d end metavar infer not0 metavar var b end metavar infer not0 metavar var d end metavar infer prop lemma negate second disjunct modus ponens not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var b end metavar conclude metavar var a end metavar cut prop lemma negate second disjunct modus ponens not0 metavar var c end metavar imply metavar var d end metavar modus ponens not0 metavar var d end metavar conclude metavar var c end metavar cut prop lemma join conjuncts modus ponens metavar var a end metavar modus ponens metavar var c end metavar conclude not0 metavar var a end metavar imply not0 metavar var c end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var c end metavar imply metavar var d end metavar infer not0 metavar var b end metavar infer not0 metavar var d end metavar infer not0 metavar var a end metavar imply not0 metavar var c end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var c end metavar imply metavar var d end metavar imply not0 metavar var b end metavar imply not0 metavar var d end metavar imply not0 metavar var a end metavar imply not0 metavar var c end metavar cut not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var c end metavar imply metavar var d end metavar infer prop lemma mp2 modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var c end metavar imply metavar var d end metavar imply not0 metavar var b end metavar imply not0 metavar var d end metavar imply not0 metavar var a end metavar imply not0 metavar var c end metavar modus ponens not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var c end metavar imply metavar var d end metavar conclude not0 metavar var b end metavar imply not0 metavar var d end metavar imply not0 metavar var a end metavar imply not0 metavar var c end metavar cut 1rule repetition modus ponens not0 metavar var b end metavar imply not0 metavar var d end metavar imply not0 metavar var a end metavar imply not0 metavar var c end metavar conclude not0 metavar var b end metavar imply not0 metavar var d end metavar imply not0 metavar var a end metavar imply not0 metavar var c end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of prop lemma from two times two disjuncts as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var c end metavar imply metavar var d end metavar infer metavar var a end metavar imply metavar var c end metavar imply metavar var e end metavar infer metavar var a end metavar imply metavar var d end metavar imply metavar var e end metavar infer metavar var b end metavar imply metavar var c end metavar imply metavar var e end metavar infer metavar var b end metavar imply metavar var d end metavar imply metavar var e end metavar infer metavar var e end metavar end define end math ] "
" [ math define proof of prop lemma from two times two disjuncts as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed not0 metavar var c end metavar imply metavar var d end metavar infer metavar var a end metavar imply metavar var c end metavar imply metavar var e end metavar infer metavar var a end metavar imply metavar var d end metavar imply metavar var e end metavar infer metavar var a end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var c end metavar imply metavar var e end metavar modus ponens metavar var a end metavar conclude metavar var c end metavar imply metavar var e end metavar cut 1rule mp modus ponens metavar var a end metavar imply metavar var d end metavar imply metavar var e end metavar modus ponens metavar var a end metavar conclude metavar var d end metavar imply metavar var e end metavar cut prop lemma from disjuncts modus ponens not0 metavar var c end metavar imply metavar var d end metavar modus ponens metavar var c end metavar imply metavar var e end metavar modus ponens metavar var d end metavar imply metavar var e end metavar conclude metavar var e end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var c end metavar imply metavar var d end metavar infer metavar var b end metavar imply metavar var c end metavar imply metavar var e end metavar infer metavar var b end metavar imply metavar var d end metavar imply metavar var e end metavar infer not0 metavar var a end metavar infer prop lemma negate first disjunct modus ponens not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var a end metavar conclude metavar var b end metavar cut 1rule mp modus ponens metavar var b end metavar imply metavar var c end metavar imply metavar var e end metavar modus ponens metavar var b end metavar conclude metavar var c end metavar imply metavar var e end metavar cut 1rule mp modus ponens metavar var b end metavar imply metavar var d end metavar imply metavar var e end metavar modus ponens metavar var b end metavar conclude metavar var d end metavar imply metavar var e end metavar cut prop lemma from disjuncts modus ponens not0 metavar var c end metavar imply metavar var d end metavar modus ponens metavar var c end metavar imply metavar var e end metavar modus ponens metavar var d end metavar imply metavar var e end metavar conclude metavar var e end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed not0 metavar var c end metavar imply metavar var d end metavar infer metavar var a end metavar imply metavar var c end metavar imply metavar var e end metavar infer metavar var a end metavar imply metavar var d end metavar imply metavar var e end metavar infer metavar var a end metavar infer metavar var e end metavar conclude not0 metavar var c end metavar imply metavar var d end metavar imply metavar var a end metavar imply metavar var c end metavar imply metavar var e end metavar imply metavar var a end metavar imply metavar var d end metavar imply metavar var e end metavar imply metavar var a end metavar imply metavar var e end metavar cut 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed all metavar var e end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var c end metavar imply metavar var d end metavar infer metavar var b end metavar imply metavar var c end metavar imply metavar var e end metavar infer metavar var b end metavar imply metavar var d end metavar imply metavar var e end metavar infer not0 metavar var a end metavar infer metavar var e end metavar conclude not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var c end metavar imply metavar var d end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var e end metavar imply metavar var b end metavar imply metavar var d end metavar imply metavar var e end metavar imply not0 metavar var a end metavar imply metavar var e end metavar cut not0 metavar var a end metavar imply metavar var b end metavar infer not0 metavar var c end metavar imply metavar var d end metavar infer metavar var a end metavar imply metavar var c end metavar imply metavar var e end metavar infer metavar var a end metavar imply metavar var d end metavar imply metavar var e end metavar infer metavar var b end metavar imply metavar var c end metavar imply metavar var e end metavar infer metavar var b end metavar imply metavar var d end metavar imply metavar var e end metavar infer prop lemma mp3 modus ponens not0 metavar var c end metavar imply metavar var d end metavar imply metavar var a end metavar imply metavar var c end metavar imply metavar var e end metavar imply metavar var a end metavar imply metavar var d end metavar imply metavar var e end metavar imply metavar var a end metavar imply metavar var e end metavar modus ponens not0 metavar var c end metavar imply metavar var d end metavar modus ponens metavar var a end metavar imply metavar var c end metavar imply metavar var e end metavar modus ponens metavar var a end metavar imply metavar var d end metavar imply metavar var e end metavar conclude metavar var a end metavar imply metavar var e end metavar cut prop lemma mp4 modus ponens not0 metavar var a end metavar imply metavar var b end metavar imply not0 metavar var c end metavar imply metavar var d end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var e end metavar imply metavar var b end metavar imply metavar var d end metavar imply metavar var e end metavar imply not0 metavar var a end metavar imply metavar var e end metavar modus ponens not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var c end metavar imply metavar var d end metavar modus ponens metavar var b end metavar imply metavar var c end metavar imply metavar var e end metavar modus ponens metavar var b end metavar imply metavar var d end metavar imply metavar var e end metavar conclude not0 metavar var a end metavar imply metavar var e end metavar cut prop lemma from negations modus ponens metavar var a end metavar imply metavar var e end metavar modus ponens not0 metavar var a end metavar imply metavar var e end metavar conclude metavar var e end metavar end quote state proof state cache var c end expand end define end math ] "
(*** EQUALITY ***)
" [ math define statement of lemma eqReflexivity as system Q infer all metavar var x end metavar indeed metavar var x end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma eqReflexivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom leqReflexivity conclude metavar var x end metavar <= metavar var x end metavar cut lemma leqAntisymmetry modus ponens metavar var x end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var x end metavar conclude metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma eqSymmetry as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma eqSymmetry as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer lemma eqReflexivity conclude metavar var x end metavar = metavar var x end metavar cut lemma equality modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar = metavar var x end metavar conclude metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma eqTransitivity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var x end metavar = metavar var z end metavar end define end math ] "
" [ math define proof of lemma eqTransitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer lemma eqSymmetry modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar cut lemma equality modus ponens metavar var y end metavar = metavar var x end metavar modus ponens metavar var y end metavar = metavar var z end metavar conclude metavar var x end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma eqTransitivity4 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var x end metavar = metavar var u end metavar end define end math ] "
" [ math define proof of lemma eqTransitivity4 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer lemma eqTransitivity modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar = metavar var z end metavar conclude metavar var x end metavar = metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var z end metavar modus ponens metavar var z end metavar = metavar var u end metavar conclude metavar var x end metavar = metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma eqTransitivity5 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var u end metavar = metavar var v end metavar infer metavar var x end metavar = metavar var v end metavar end define end math ] "
" [ math define proof of lemma eqTransitivity5 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var u end metavar = metavar var v end metavar infer lemma eqTransitivity4 modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar = metavar var z end metavar modus ponens metavar var z end metavar = metavar var u end metavar conclude metavar var x end metavar = metavar var u end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var u end metavar modus ponens metavar var u end metavar = metavar var v end metavar conclude metavar var x end metavar = metavar var v end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma eqTransitivity6 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var u end metavar = metavar var v end metavar infer metavar var v end metavar = metavar var w end metavar infer metavar var x end metavar = metavar var w end metavar end define end math ] "
" [ math define proof of lemma eqTransitivity6 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var u end metavar = metavar var v end metavar infer metavar var v end metavar = metavar var w end metavar infer lemma eqTransitivity5 modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar = metavar var z end metavar modus ponens metavar var z end metavar = metavar var u end metavar modus ponens metavar var u end metavar = metavar var v end metavar conclude metavar var x end metavar = metavar var v end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var v end metavar modus ponens metavar var v end metavar = metavar var w end metavar conclude metavar var x end metavar = metavar var w end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma plus0Left as system Q infer all metavar var x end metavar indeed 0 + metavar var x end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma plus0Left as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom plus0 conclude metavar var x end metavar + 0 = metavar var x end metavar cut axiom plusCommutativity conclude 0 + metavar var x end metavar = metavar var x end metavar + 0 cut lemma eqTransitivity modus ponens 0 + metavar var x end metavar = metavar var x end metavar + 0 modus ponens metavar var x end metavar + 0 = metavar var x end metavar conclude 0 + metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma times1Left as system Q infer all metavar var x end metavar indeed 1 * metavar var x end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma times1Left as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom times1 conclude metavar var x end metavar * 1 = metavar var x end metavar cut axiom timesCommutativity conclude 1 * metavar var x end metavar = metavar var x end metavar * 1 cut lemma eqTransitivity modus ponens 1 * metavar var x end metavar = metavar var x end metavar * 1 modus ponens metavar var x end metavar * 1 = metavar var x end metavar conclude 1 * metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma eqAdditionLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar + metavar var x end metavar = metavar var z end metavar + metavar var y end metavar end define end math ] "
" [ math define proof of lemma eqAdditionLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer lemma eqAddition modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar conclude metavar var z end metavar + metavar var x end metavar = metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma eqMultiplicationLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar * metavar var x end metavar = metavar var z end metavar * metavar var y end metavar end define end math ] "
" [ math define proof of lemma eqMultiplicationLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer lemma eqMultiplication modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut axiom timesCommutativity conclude metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var z end metavar cut axiom timesCommutativity conclude metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var z end metavar modus ponens metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar modus ponens metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar conclude metavar var z end metavar * metavar var x end metavar = metavar var z end metavar * metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma distributionOut as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var z end metavar end define end math ] "
" [ math define proof of lemma distributionOut as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed axiom distribution conclude metavar var x end metavar * metavar var y end metavar + metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar * metavar var y end metavar + metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar conclude metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma three2twoTerms as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var y end metavar + metavar var z end metavar = metavar var u end metavar infer metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var u end metavar end define end math ] "
" [ math define proof of lemma three2twoTerms as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var y end metavar + metavar var z end metavar = metavar var u end metavar infer lemma eqAdditionLeft modus ponens metavar var y end metavar + metavar var z end metavar = metavar var u end metavar conclude metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var u end metavar cut axiom plusAssociativity conclude metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var y end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var y end metavar + metavar var z end metavar modus ponens metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var u end metavar conclude metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma three2threeTerms as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar end define end math ] "
" [ math define proof of lemma three2threeTerms as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed axiom plusCommutativity conclude metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar cut lemma three2twoTerms modus ponens metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar cut axiom plusAssociativity conclude metavar var x end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar modus ponens metavar var x end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma three2twoFactors as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var y end metavar * metavar var z end metavar = metavar var u end metavar infer metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var u end metavar end define end math ] "
" [ math define proof of lemma three2twoFactors as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var y end metavar * metavar var z end metavar = metavar var u end metavar infer lemma eqMultiplicationLeft modus ponens metavar var y end metavar * metavar var z end metavar = metavar var u end metavar conclude metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var u end metavar cut axiom timesAssociativity conclude metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar * metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar * metavar var z end metavar modus ponens metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var u end metavar conclude metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma addEquations as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar end define end math ] "
" [ math define proof of lemma addEquations as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar = metavar var u end metavar infer lemma eqAddition modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut lemma eqAdditionLeft modus ponens metavar var z end metavar = metavar var u end metavar conclude metavar var y end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma subtractEquations as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var x end metavar = metavar var y end metavar end define end math ] "
" [ math define proof of lemma subtractEquations as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar infer metavar var z end metavar = metavar var u end metavar infer lemma eqAddition modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar conclude metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar cut lemma plus0Left conclude 0 + metavar var z end metavar = metavar var z end metavar cut lemma eqTransitivity modus ponens 0 + metavar var z end metavar = metavar var z end metavar modus ponens metavar var z end metavar = metavar var u end metavar conclude 0 + metavar var z end metavar = metavar var u end metavar cut lemma positiveToRight(Eq) modus ponens 0 + metavar var z end metavar = metavar var u end metavar conclude 0 = metavar var u end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens 0 = metavar var u end metavar + - metavar var z end metavar conclude metavar var u end metavar + - metavar var z end metavar = 0 cut lemma eqAdditionLeft modus ponens metavar var u end metavar + - metavar var z end metavar = 0 conclude metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar + 0 cut axiom plusAssociativity conclude metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar cut axiom plus0 conclude metavar var y end metavar + 0 = metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar modus ponens metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar + 0 modus ponens metavar var y end metavar + 0 = metavar var y end metavar conclude metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar modus ponens metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar conclude metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma subtractEquationsLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar infer metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar = metavar var u end metavar end define end math ] "
" [ math define proof of lemma subtractEquationsLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar infer metavar var x end metavar = metavar var y end metavar infer axiom plusCommutativity conclude metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var y end metavar + metavar var u end metavar = metavar var u end metavar + metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar modus ponens metavar var y end metavar + metavar var u end metavar = metavar var u end metavar + metavar var y end metavar conclude metavar var z end metavar + metavar var x end metavar = metavar var u end metavar + metavar var y end metavar cut lemma subtractEquations modus ponens metavar var z end metavar + metavar var x end metavar = metavar var u end metavar + metavar var y end metavar modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var z end metavar = metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma eqNegated as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer - metavar var x end metavar = - metavar var y end metavar end define end math ] "
" [ math define proof of lemma eqNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer axiom negative conclude metavar var x end metavar + - metavar var x end metavar = 0 cut axiom negative conclude metavar var y end metavar + - metavar var y end metavar = 0 cut lemma eqSymmetry modus ponens metavar var y end metavar + - metavar var y end metavar = 0 conclude 0 = metavar var y end metavar + - metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + - metavar var x end metavar = 0 modus ponens 0 = metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar + - metavar var x end metavar = metavar var y end metavar + - metavar var y end metavar cut lemma subtractEquationsLeft modus ponens metavar var x end metavar + - metavar var x end metavar = metavar var y end metavar + - metavar var y end metavar modus ponens metavar var x end metavar = metavar var y end metavar conclude - metavar var x end metavar = - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma positiveToRight(Eq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var z end metavar infer metavar var x end metavar = metavar var z end metavar + - metavar var y end metavar end define end math ] "
" [ math define proof of lemma positiveToRight(Eq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var z end metavar infer lemma eqAddition modus ponens metavar var x end metavar + metavar var y end metavar = metavar var z end metavar conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var z end metavar + - metavar var y end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var z end metavar + - metavar var y end metavar conclude metavar var x end metavar = metavar var z end metavar + - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma positiveToLeft(Eq)(1 term) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar + - metavar var y end metavar = 0 end define end math ] "
" [ math define proof of lemma positiveToLeft(Eq)(1 term) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer lemma eqAddition modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var y end metavar cut axiom negative conclude metavar var y end metavar + - metavar var y end metavar = 0 cut lemma eqTransitivity modus ponens metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var y end metavar modus ponens metavar var y end metavar + - metavar var y end metavar = 0 conclude metavar var x end metavar + - metavar var y end metavar = 0 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma positiveToRight(Leq)(1 term) as system Q infer all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var y end metavar <= metavar var z end metavar infer 0 <= metavar var z end metavar + - metavar var y end metavar end define end math ] "
" [ math define proof of lemma positiveToRight(Leq)(1 term) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var y end metavar <= metavar var z end metavar infer lemma plus0Left conclude 0 + metavar var y end metavar = metavar var y end metavar cut lemma eqSymmetry modus ponens 0 + metavar var y end metavar = metavar var y end metavar conclude metavar var y end metavar = 0 + metavar var y end metavar cut lemma subLeqLeft modus ponens metavar var y end metavar = 0 + metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude 0 + metavar var y end metavar <= metavar var z end metavar cut lemma positiveToRight(Leq) modus ponens 0 + metavar var y end metavar <= metavar var z end metavar conclude 0 <= metavar var z end metavar + - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma negativeToLeft(Eq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar + - metavar var z end metavar infer metavar var x end metavar + metavar var z end metavar = metavar var y end metavar end define end math ] "
" [ math define proof of lemma negativeToLeft(Eq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar + - metavar var z end metavar infer lemma eqAddition modus ponens metavar var x end metavar = metavar var y end metavar + - metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma three2threeTerms conclude metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma x=x+y-y conclude metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar modus ponens metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
(*** NO EQUALITY ***)
" [ math define statement of lemma lessNeq as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar = metavar var y end metavar end define end math ] "
" [ math define proof of lemma lessNeq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer 1rule repetition modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut prop lemma second conjunct modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma neqSymmetry as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var y end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma neqSymmetry as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar = metavar var x end metavar infer lemma eqSymmetry modus ponens metavar var y end metavar = metavar var x end metavar conclude metavar var x end metavar = metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar = metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar cut not0 metavar var x end metavar = metavar var y end metavar infer prop lemma mt modus ponens metavar var y end metavar = metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma neqNegated as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer not0 - metavar var x end metavar = - metavar var y end metavar end define end math ] "
" [ math define proof of lemma neqNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer - metavar var x end metavar = - metavar var y end metavar infer lemma eqNegated modus ponens - metavar var x end metavar = - metavar var y end metavar conclude - - metavar var x end metavar = - - metavar var y end metavar cut lemma doubleMinus conclude - - metavar var x end metavar = metavar var x end metavar cut lemma eqSymmetry modus ponens - - metavar var x end metavar = metavar var x end metavar conclude metavar var x end metavar = - - metavar var x end metavar cut lemma doubleMinus conclude - - metavar var y end metavar = metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar = - - metavar var x end metavar modus ponens - - metavar var x end metavar = - - metavar var y end metavar modus ponens - - metavar var y end metavar = metavar var y end metavar conclude metavar var x end metavar = metavar var y end metavar cut prop lemma from contradiction modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 - metavar var x end metavar = - metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer - metavar var x end metavar = - metavar var y end metavar infer not0 - metavar var x end metavar = - metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar imply - metavar var x end metavar = - metavar var y end metavar imply not0 - metavar var x end metavar = - metavar var y end metavar cut not0 metavar var x end metavar = metavar var y end metavar infer 1rule mp modus ponens not0 metavar var x end metavar = metavar var y end metavar imply - metavar var x end metavar = - metavar var y end metavar imply not0 - metavar var x end metavar = - metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude - metavar var x end metavar = - metavar var y end metavar imply not0 - metavar var x end metavar = - metavar var y end metavar cut prop lemma imply negation modus ponens - metavar var x end metavar = - metavar var y end metavar imply not0 - metavar var x end metavar = - metavar var y end metavar conclude not0 - metavar var x end metavar = - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma subNeqRight as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar = metavar var x end metavar infer not0 metavar var z end metavar = metavar var y end metavar end define end math ] "
" [ math define proof of lemma subNeqRight as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar = metavar var x end metavar infer lemma neqSymmetry modus ponens not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var x end metavar = metavar var z end metavar cut lemma subNeqLeft modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var y end metavar = metavar var z end metavar cut lemma neqSymmetry modus ponens not0 metavar var y end metavar = metavar var z end metavar conclude not0 metavar var z end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma subNeqLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar = metavar var z end metavar infer not0 metavar var y end metavar = metavar var z end metavar end define end math ] "
" [ math define proof of lemma subNeqLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar = metavar var z end metavar infer axiom equality conclude metavar var y end metavar = metavar var x end metavar imply metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar cut 1rule mp modus ponens metavar var y end metavar = metavar var x end metavar imply metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar modus ponens metavar var y end metavar = metavar var x end metavar conclude metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar cut prop lemma contrapositive modus ponens metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar = metavar var z end metavar imply not0 metavar var y end metavar = metavar var z end metavar cut 1rule mp modus ponens not0 metavar var x end metavar = metavar var z end metavar imply not0 metavar var y end metavar = metavar var z end metavar modus ponens not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var y end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma neqAddition as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end define end math ] "
" [ math define proof of lemma neqAddition as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar infer lemma eqReflexivity conclude metavar var z end metavar = metavar var z end metavar cut lemma subtractEquations modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var z end metavar = metavar var z end metavar conclude metavar var x end metavar = metavar var y end metavar cut prop lemma from contradiction modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar infer not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar imply not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut not0 metavar var x end metavar = metavar var y end metavar infer 1rule mp modus ponens not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar imply not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar imply not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut prop lemma imply negation modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar imply not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma neqMultiplication as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var z end metavar = 0 infer not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end define end math ] "
" [ math define proof of lemma neqMultiplication as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var z end metavar = 0 infer not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar infer lemma x=x*y*(1/y) modus ponens not0 metavar var z end metavar = 0 conclude metavar var x end metavar = metavar var x end metavar * metavar var z end metavar * 1/ metavar var z end metavar cut lemma eqMultiplication modus ponens metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude metavar var x end metavar * metavar var z end metavar * 1/ metavar var z end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar cut lemma x=x*y*(1/y) modus ponens not0 metavar var z end metavar = 0 conclude metavar var y end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar conclude metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar = metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar = metavar var x end metavar * metavar var z end metavar * 1/ metavar var z end metavar modus ponens metavar var x end metavar * metavar var z end metavar * 1/ metavar var z end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar modus ponens metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar = metavar var y end metavar conclude metavar var x end metavar = metavar var y end metavar cut prop lemma from contradiction modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var z end metavar = 0 infer not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar infer not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var z end metavar = 0 imply not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar imply not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut not0 metavar var z end metavar = 0 infer not0 metavar var x end metavar = metavar var y end metavar infer prop lemma mp2 modus ponens not0 metavar var z end metavar = 0 imply not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar imply not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar modus ponens not0 metavar var z end metavar = 0 modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar imply not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut prop lemma imply negation modus ponens metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar imply not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
(*** NEGATIVE ***)
" [ math define statement of lemma uniqueNegative as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = 0 infer metavar var x end metavar + metavar var z end metavar = 0 infer metavar var y end metavar = metavar var z end metavar end define end math ] "
" [ math define proof of lemma uniqueNegative as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = 0 infer metavar var x end metavar + metavar var z end metavar = 0 infer axiom plusCommutativity conclude metavar var y end metavar + metavar var x end metavar = metavar var x end metavar + metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var y end metavar + metavar var x end metavar = metavar var x end metavar + metavar var y end metavar modus ponens metavar var x end metavar + metavar var y end metavar = 0 conclude metavar var y end metavar + metavar var x end metavar = 0 cut lemma positiveToRight(Eq) modus ponens metavar var y end metavar + metavar var x end metavar = 0 conclude metavar var y end metavar = 0 + - metavar var x end metavar cut axiom plusCommutativity conclude metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar = 0 conclude metavar var z end metavar + metavar var x end metavar = 0 cut lemma positiveToRight(Eq) modus ponens metavar var z end metavar + metavar var x end metavar = 0 conclude metavar var z end metavar = 0 + - metavar var x end metavar cut lemma eqSymmetry modus ponens metavar var z end metavar = 0 + - metavar var x end metavar conclude 0 + - metavar var x end metavar = metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var y end metavar = 0 + - metavar var x end metavar modus ponens 0 + - metavar var x end metavar = metavar var z end metavar conclude metavar var y end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma doubleMinus as system Q infer all metavar var x end metavar indeed - - metavar var x end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma doubleMinus as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom negative conclude - metavar var x end metavar + - - metavar var x end metavar = 0 cut lemma x+y=zBackwards modus ponens - metavar var x end metavar + - - metavar var x end metavar = 0 conclude 0 = - - metavar var x end metavar + - metavar var x end metavar cut lemma negativeToLeft(Eq) modus ponens 0 = - - metavar var x end metavar + - metavar var x end metavar conclude 0 + metavar var x end metavar = - - metavar var x end metavar cut lemma plus0Left conclude 0 + metavar var x end metavar = metavar var x end metavar cut lemma equality modus ponens 0 + metavar var x end metavar = - - metavar var x end metavar modus ponens 0 + metavar var x end metavar = metavar var x end metavar conclude - - metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
(*** LEQ, nummer 1 af 2 ***)
" [ math define statement of lemma leqLessEq as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar end define end math ] "
" [ math define proof of lemma leqLessEq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma fromNotLess modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar <= metavar var x end metavar cut lemma leqAntisymmetry modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var x end metavar conclude metavar var x end metavar = metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar cut metavar var x end metavar <= metavar var y end metavar infer 1rule mp modus ponens metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar cut 1rule repetition modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma lessLeq as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var y end metavar end define end math ] "
" [ math define proof of lemma lessLeq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer 1rule repetition modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut prop lemma first conjunct modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma from leqGeq as system Q infer all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar imply metavar var a end metavar infer metavar var y end metavar <= metavar var x end metavar imply metavar var a end metavar infer metavar var a end metavar end define end math ] "
" [ math define proof of lemma from leqGeq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar imply metavar var a end metavar infer metavar var y end metavar <= metavar var x end metavar imply metavar var a end metavar infer axiom leqTotality conclude not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar cut prop lemma from disjuncts modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var a end metavar modus ponens metavar var y end metavar <= metavar var x end metavar imply metavar var a end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma subLeqRight as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar <= metavar var x end metavar infer metavar var z end metavar <= metavar var y end metavar end define end math ] "
" [ math define proof of lemma subLeqRight as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar <= metavar var x end metavar infer lemma eqLeq modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut lemma leqTransitivity modus ponens metavar var z end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var z end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma subLeqLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var z end metavar infer metavar var y end metavar <= metavar var z end metavar end define end math ] "
" [ math define proof of lemma subLeqLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var z end metavar infer lemma eqSymmetry modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar cut lemma eqLeq modus ponens metavar var y end metavar = metavar var x end metavar conclude metavar var y end metavar <= metavar var x end metavar cut lemma leqTransitivity modus ponens metavar var y end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var z end metavar conclude metavar var y end metavar <= metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma leqPlus1 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var x end metavar <= metavar var y end metavar + 1 imply not0 not0 metavar var x end metavar = metavar var y end metavar + 1 end define end math ] "
" [ math define proof of lemma leqPlus1 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer lemma 0<1 conclude not0 0 <= 1 imply not0 not0 0 = 1 cut lemma lessAdditionLeft modus ponens not0 0 <= 1 imply not0 not0 0 = 1 conclude not0 metavar var y end metavar + 0 <= metavar var y end metavar + 1 imply not0 not0 metavar var y end metavar + 0 = metavar var y end metavar + 1 cut axiom plus0 conclude metavar var y end metavar + 0 = metavar var y end metavar cut lemma subLessLeft modus ponens metavar var y end metavar + 0 = metavar var y end metavar modus ponens not0 metavar var y end metavar + 0 <= metavar var y end metavar + 1 imply not0 not0 metavar var y end metavar + 0 = metavar var y end metavar + 1 conclude not0 metavar var y end metavar <= metavar var y end metavar + 1 imply not0 not0 metavar var y end metavar = metavar var y end metavar + 1 cut lemma leqLessTransitivity modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens not0 metavar var y end metavar <= metavar var y end metavar + 1 imply not0 not0 metavar var y end metavar = metavar var y end metavar + 1 conclude not0 metavar var x end metavar <= metavar var y end metavar + 1 imply not0 not0 metavar var x end metavar = metavar var y end metavar + 1 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma positiveToRight(Leq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar <= metavar var z end metavar infer metavar var x end metavar <= metavar var z end metavar + - metavar var y end metavar end define end math ] "
" [ math define proof of lemma positiveToRight(Leq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar <= metavar var z end metavar infer lemma leqAddition modus ponens metavar var x end metavar + metavar var y end metavar <= metavar var z end metavar conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar <= metavar var z end metavar + - metavar var y end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar <= metavar var z end metavar + - metavar var y end metavar conclude metavar var x end metavar <= metavar var z end metavar + - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma leqAdditionLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar end define end math ] "
" [ math define proof of lemma leqAdditionLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var x end metavar + metavar var z end metavar = metavar var z end metavar + metavar var x end metavar cut axiom plusCommutativity conclude metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var z end metavar = metavar var z end metavar + metavar var x end metavar modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar conclude metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar modus ponens metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar conclude metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma leqSubtraction as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar end define end math ] "
" [ math define proof of lemma leqSubtraction as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar infer lemma leqAddition modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var x end metavar cut lemma x=x+y-y conclude metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var x end metavar modus ponens metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar modus ponens metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var x end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma leqSubtractionLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar infer metavar var x end metavar <= metavar var y end metavar end define end math ] "
" [ math define proof of lemma leqSubtractionLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar infer axiom plusCommutativity conclude metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var z end metavar + metavar var y end metavar = metavar var y end metavar + metavar var z end metavar cut lemma subLeqLeft modus ponens metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar modus ponens metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var z end metavar + metavar var y end metavar cut lemma subLeqRight modus ponens metavar var z end metavar + metavar var y end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar cut lemma leqSubtraction modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar conclude metavar var x end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma thirdGeq as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var end define end math ] "
" [ math define proof of lemma thirdGeq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer axiom leqReflexivity conclude metavar var y end metavar <= metavar var y end metavar cut prop lemma join conjuncts modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 metavar var y end metavar <= metavar var y end metavar cut 1rule exist intro at existential var var c end var at metavar var y end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 metavar var y end metavar <= metavar var y end metavar conclude not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer axiom leqReflexivity conclude metavar var x end metavar <= metavar var x end metavar cut prop lemma join conjuncts modus ponens metavar var x end metavar <= metavar var x end metavar modus ponens metavar var y end metavar <= metavar var x end metavar conclude not0 metavar var x end metavar <= metavar var x end metavar imply not0 metavar var y end metavar <= metavar var x end metavar cut 1rule exist intro at existential var var c end var at metavar var x end metavar modus ponens not0 metavar var x end metavar <= metavar var x end metavar imply not0 metavar var y end metavar <= metavar var x end metavar conclude not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var conclude metavar var x end metavar <= metavar var y end metavar imply not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var conclude metavar var y end metavar <= metavar var x end metavar imply not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var cut axiom leqTotality conclude not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar cut prop lemma from disjuncts modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar imply not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var modus ponens metavar var y end metavar <= metavar var x end metavar imply not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var conclude not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma leqNegated as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer - metavar var y end metavar <= - metavar var x end metavar end define end math ] "
" [ math define proof of lemma leqNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + - metavar var x end metavar <= metavar var y end metavar + - metavar var x end metavar cut axiom negative conclude metavar var x end metavar + - metavar var x end metavar = 0 cut lemma subLeqLeft modus ponens metavar var x end metavar + - metavar var x end metavar = 0 modus ponens metavar var x end metavar + - metavar var x end metavar <= metavar var y end metavar + - metavar var x end metavar conclude 0 <= metavar var y end metavar + - metavar var x end metavar cut axiom plusCommutativity conclude metavar var y end metavar + - metavar var x end metavar = - metavar var x end metavar + metavar var y end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + - metavar var x end metavar = - metavar var x end metavar + metavar var y end metavar modus ponens 0 <= metavar var y end metavar + - metavar var x end metavar conclude 0 <= - metavar var x end metavar + metavar var y end metavar cut lemma leqAddition modus ponens 0 <= - metavar var x end metavar + metavar var y end metavar conclude 0 + - metavar var y end metavar <= - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma plus0Left conclude 0 + - metavar var y end metavar = - metavar var y end metavar cut lemma x=x+y-y conclude - metavar var x end metavar = - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens - metavar var x end metavar = - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = - metavar var x end metavar cut lemma subLeqLeft modus ponens 0 + - metavar var y end metavar = - metavar var y end metavar modus ponens 0 + - metavar var y end metavar <= - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude - metavar var y end metavar <= - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma subLeqRight modus ponens - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = - metavar var x end metavar modus ponens - metavar var y end metavar <= - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude - metavar var y end metavar <= - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma addEquations(Leq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var z end metavar <= metavar var u end metavar infer metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar end define end math ] "
" [ math define proof of lemma addEquations(Leq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var z end metavar <= metavar var u end metavar infer lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar cut lemma leqAdditionLeft modus ponens metavar var z end metavar <= metavar var u end metavar conclude metavar var y end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar cut lemma leqTransitivity modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "
(*** LESS ***)
" [ math define statement of lemma leqNeqLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar end define end math ] "
" [ math define proof of lemma leqNeqLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var x end metavar = metavar var y end metavar infer prop lemma join conjuncts modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut 1rule repetition modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma fromLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var y end metavar <= metavar var x end metavar end define end math ] "
" [ math define proof of lemma fromLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer lemma toNotLess modus ponens metavar var y end metavar <= metavar var x end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer prop lemma add double neg modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut prop lemma mt modus ponens metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens not0 not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma toLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar infer not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma toLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar infer lemma fromNotLess modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude metavar var x end metavar <= metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar infer metavar var x end metavar <= metavar var y end metavar conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar imply metavar var x end metavar <= metavar var y end metavar cut not0 metavar var x end metavar <= metavar var y end metavar infer prop lemma negative mt modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar imply metavar var x end metavar <= metavar var y end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma fromNotLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar end define end math ] "
" [ math define proof of lemma fromNotLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var y end metavar infer 1rule repetition modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut prop lemma remove double neg modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut 1rule mp modus ponens metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude not0 not0 metavar var x end metavar = metavar var y end metavar cut prop lemma remove double neg modus ponens not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar = metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar cut lemma eqLeq modus ponens metavar var y end metavar = metavar var x end metavar conclude metavar var y end metavar <= metavar var x end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar cut not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer 1rule mp modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar cut prop lemma auto imply conclude metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar <= metavar var x end metavar cut axiom leqTotality conclude not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar cut prop lemma from disjuncts modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar modus ponens metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar <= metavar var x end metavar conclude metavar var y end metavar <= metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma toNotLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma toNotLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar infer lemma leqAntisymmetry modus ponens metavar var y end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar cut prop lemma add double neg modus ponens metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var y end metavar = metavar var x end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar infer not0 not0 metavar var y end metavar = metavar var x end metavar conclude metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut metavar var x end metavar <= metavar var y end metavar infer 1rule mp modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut prop lemma add double neg modus ponens metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut 1rule repetition modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut 1rule repetition modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma lessAddition as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end define end math ] "
" [ math define proof of lemma lessAddition as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma lessLeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar cut lemma lessNeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar cut lemma neqAddition modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut prop lemma join conjuncts modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar modus ponens not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma lessAdditionLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var z end metavar + metavar var x end metavar = metavar var z end metavar + metavar var y end metavar end define end math ] "
" [ math define proof of lemma lessAdditionLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma lessAddition modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var x end metavar + metavar var z end metavar = metavar var z end metavar + metavar var x end metavar cut lemma subLessLeft modus ponens metavar var x end metavar + metavar var z end metavar = metavar var z end metavar + metavar var x end metavar modus ponens not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var z end metavar + metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar cut lemma subLessRight modus ponens metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar modus ponens not0 metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var z end metavar + metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var z end metavar + metavar var x end metavar = metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma lessMultiplication as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end define end math ] "
" [ math define proof of lemma lessMultiplication as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma lessLeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut lemma lessLeq modus ponens not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar conclude 0 <= metavar var z end metavar cut lemma leqMultiplication modus ponens 0 <= metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar cut lemma lessNeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar cut lemma lessNeq modus ponens not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar conclude not0 0 = metavar var z end metavar cut lemma neqSymmetry modus ponens not0 0 = metavar var z end metavar conclude not0 metavar var z end metavar = 0 cut lemma neqMultiplication modus ponens not0 metavar var z end metavar = 0 modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut lemma leqNeqLess modus ponens metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar modus ponens not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma lessMultiplicationLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar * metavar var x end metavar <= metavar var z end metavar * metavar var y end metavar imply not0 not0 metavar var z end metavar * metavar var x end metavar = metavar var z end metavar * metavar var y end metavar end define end math ] "
" [ math define proof of lemma lessMultiplicationLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma lessMultiplication modus ponens not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut axiom timesCommutativity conclude metavar var x end metavar * metavar var z end metavar = metavar var z end metavar * metavar var x end metavar cut axiom timesCommutativity conclude metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar cut lemma subLessLeft modus ponens metavar var x end metavar * metavar var z end metavar = metavar var z end metavar * metavar var x end metavar modus ponens not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var z end metavar * metavar var x end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var z end metavar * metavar var x end metavar = metavar var y end metavar * metavar var z end metavar cut lemma subLessRight modus ponens metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar modus ponens not0 metavar var z end metavar * metavar var x end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var z end metavar * metavar var x end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var z end metavar * metavar var x end metavar <= metavar var z end metavar * metavar var y end metavar imply not0 not0 metavar var z end metavar * metavar var x end metavar = metavar var z end metavar * metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma lessDivision as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var z end metavar infer not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar end define end math ] "
" [ math define proof of lemma lessDivision as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var z end metavar infer not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar infer lemma fromLess modus ponens not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar cut axiom leqMultiplication conclude 0 <= metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar cut 1rule mp modus ponens 0 <= metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar modus ponens 0 <= metavar var z end metavar conclude metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar cut prop lemma contrapositive modus ponens metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar conclude not0 metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar imply not0 metavar var y end metavar <= metavar var x end metavar cut 1rule mp modus ponens not0 metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar imply not0 metavar var y end metavar <= metavar var x end metavar modus ponens not0 metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar cut lemma toLess modus ponens not0 metavar var y end metavar <= metavar var x end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma addEquations(Less) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar <= metavar var u end metavar imply not0 not0 metavar var z end metavar = metavar var u end metavar infer not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar end define end math ] "
" [ math define proof of lemma addEquations(Less) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar <= metavar var u end metavar imply not0 not0 metavar var z end metavar = metavar var u end metavar infer lemma lessAddition modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut lemma lessAdditionLeft modus ponens not0 metavar var z end metavar <= metavar var u end metavar imply not0 not0 metavar var z end metavar = metavar var u end metavar conclude not0 metavar var y end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var y end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar cut lemma lessTransitivity modus ponens not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens not0 metavar var y end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var y end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar conclude not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma leqLessTransitivity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end define end math ] "
" [ math define proof of lemma leqLessTransitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer metavar var x end metavar = metavar var z end metavar infer prop lemma first conjunct modus ponens not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar conclude metavar var y end metavar <= metavar var z end metavar cut prop lemma second conjunct modus ponens not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar conclude not0 metavar var y end metavar = metavar var z end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar = metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var z end metavar <= metavar var y end metavar cut lemma leqAntisymmetry modus ponens metavar var y end metavar <= metavar var z end metavar modus ponens metavar var z end metavar <= metavar var y end metavar conclude metavar var y end metavar = metavar var z end metavar cut prop lemma from contradiction modus ponens metavar var y end metavar = metavar var z end metavar modus ponens not0 metavar var y end metavar = metavar var z end metavar conclude not0 metavar var x end metavar = metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer metavar var x end metavar = metavar var z end metavar infer not0 metavar var x end metavar = metavar var z end metavar conclude metavar var x end metavar <= metavar var y end metavar imply not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar imply not0 metavar var x end metavar = metavar var z end metavar cut metavar var x end metavar <= metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer prop lemma mp2 modus ponens metavar var x end metavar <= metavar var y end metavar imply not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar imply not0 metavar var x end metavar = metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar conclude metavar var x end metavar = metavar var z end metavar imply not0 metavar var x end metavar = metavar var z end metavar cut prop lemma imply negation modus ponens metavar var x end metavar = metavar var z end metavar imply not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar = metavar var z end metavar cut prop lemma first conjunct modus ponens not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar conclude metavar var y end metavar <= metavar var z end metavar cut lemma leqTransitivity modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude metavar var x end metavar <= metavar var z end metavar cut prop lemma join conjuncts modus ponens metavar var x end metavar <= metavar var z end metavar modus ponens not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma lessLeqTransitivity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end define end math ] "
" [ math define proof of lemma lessLeqTransitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer metavar var z end metavar = metavar var x end metavar infer prop lemma first conjunct modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut prop lemma second conjunct modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar cut lemma subLeqRight modus ponens metavar var z end metavar = metavar var x end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude metavar var y end metavar <= metavar var x end metavar cut lemma leqAntisymmetry modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var x end metavar conclude metavar var x end metavar = metavar var y end metavar cut prop lemma from contradiction modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var z end metavar = metavar var x end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer metavar var z end metavar = metavar var x end metavar infer not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var y end metavar <= metavar var z end metavar imply metavar var z end metavar = metavar var x end metavar imply not0 metavar var z end metavar = metavar var x end metavar cut not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer prop lemma mp2 modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var y end metavar <= metavar var z end metavar imply metavar var z end metavar = metavar var x end metavar imply not0 metavar var z end metavar = metavar var x end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude metavar var z end metavar = metavar var x end metavar imply not0 metavar var z end metavar = metavar var x end metavar cut prop lemma imply negation modus ponens metavar var z end metavar = metavar var x end metavar imply not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var z end metavar = metavar var x end metavar cut lemma neqSymmetry modus ponens not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var x end metavar = metavar var z end metavar cut prop lemma first conjunct modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut lemma leqTransitivity modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude metavar var x end metavar <= metavar var z end metavar cut prop lemma join conjuncts modus ponens metavar var x end metavar <= metavar var z end metavar modus ponens not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma lessTransitivity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end define end math ] "
" [ math define proof of lemma lessTransitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer prop lemma first conjunct modus ponens not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar conclude metavar var y end metavar <= metavar var z end metavar cut lemma lessLeqTransitivity modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma lessTotality as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma lessTotality as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar = metavar var y end metavar infer lemma fromNotLess modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar <= metavar var x end metavar cut lemma neqSymmetry modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var y end metavar = metavar var x end metavar cut lemma leqNeqLess modus ponens metavar var y end metavar <= metavar var x end metavar modus ponens not0 metavar var y end metavar = metavar var x end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut 1rule repetition modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma subLessRight as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar infer not0 metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var z end metavar = metavar var y end metavar end define end math ] "
" [ math define proof of lemma subLessRight as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar infer 1rule repetition modus ponens not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar cut prop lemma first conjunct modus ponens not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar conclude metavar var z end metavar <= metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var z end metavar <= metavar var x end metavar conclude metavar var z end metavar <= metavar var y end metavar cut prop lemma second conjunct modus ponens not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var z end metavar = metavar var x end metavar cut lemma subNeqRight modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var z end metavar = metavar var y end metavar cut prop lemma join conjuncts modus ponens metavar var z end metavar <= metavar var y end metavar modus ponens not0 metavar var z end metavar = metavar var y end metavar conclude not0 metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var z end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma subLessLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar end define end math ] "
" [ math define proof of lemma subLessLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar infer 1rule repetition modus ponens not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar cut prop lemma first conjunct modus ponens not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar conclude metavar var x end metavar <= metavar var z end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar <= metavar var z end metavar conclude metavar var y end metavar <= metavar var z end metavar cut prop lemma second conjunct modus ponens not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar = metavar var z end metavar cut lemma subNeqLeft modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var y end metavar = metavar var z end metavar cut prop lemma join conjuncts modus ponens metavar var y end metavar <= metavar var z end metavar modus ponens not0 metavar var y end metavar = metavar var z end metavar conclude not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma negativeLessPositive as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 - metavar var x end metavar <= metavar var x end metavar imply not0 not0 - metavar var x end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma negativeLessPositive as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer prop lemma first conjunct modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude 0 <= metavar var x end metavar cut lemma leqAddition modus ponens 0 <= metavar var x end metavar conclude 0 + - metavar var x end metavar <= metavar var x end metavar + - metavar var x end metavar cut lemma plus0Left conclude 0 + - metavar var x end metavar = - metavar var x end metavar cut axiom negative conclude metavar var x end metavar + - metavar var x end metavar = 0 cut lemma subLeqLeft modus ponens 0 + - metavar var x end metavar = - metavar var x end metavar modus ponens 0 + - metavar var x end metavar <= metavar var x end metavar + - metavar var x end metavar conclude - metavar var x end metavar <= metavar var x end metavar + - metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar + - metavar var x end metavar = 0 modus ponens - metavar var x end metavar <= metavar var x end metavar + - metavar var x end metavar conclude - metavar var x end metavar <= 0 cut lemma leqLessTransitivity modus ponens - metavar var x end metavar <= 0 modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 - metavar var x end metavar <= metavar var x end metavar imply not0 not0 - metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma lessNegated as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 - metavar var y end metavar <= - metavar var x end metavar imply not0 not0 - metavar var y end metavar = - metavar var x end metavar end define end math ] "
" [ math define proof of lemma lessNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma lessLeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut lemma leqNegated modus ponens metavar var x end metavar <= metavar var y end metavar conclude - metavar var y end metavar <= - metavar var x end metavar cut lemma lessNeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar cut lemma neqNegated modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 - metavar var x end metavar = - metavar var y end metavar cut lemma neqSymmetry modus ponens not0 - metavar var x end metavar = - metavar var y end metavar conclude not0 - metavar var y end metavar = - metavar var x end metavar cut lemma leqNeqLess modus ponens - metavar var y end metavar <= - metavar var x end metavar modus ponens not0 - metavar var y end metavar = - metavar var x end metavar conclude not0 - metavar var y end metavar <= - metavar var x end metavar imply not0 not0 - metavar var y end metavar = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma positiveNegated as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 - metavar var x end metavar <= 0 imply not0 not0 - metavar var x end metavar = 0 end define end math ] "
" [ math define proof of lemma positiveNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma lessNegated modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 - metavar var x end metavar <= - 0 imply not0 not0 - metavar var x end metavar = - 0 cut lemma -0=0 conclude - 0 = 0 cut lemma subLessRight modus ponens - 0 = 0 modus ponens not0 - metavar var x end metavar <= - 0 imply not0 not0 - metavar var x end metavar = - 0 conclude not0 - metavar var x end metavar <= 0 imply not0 not0 - metavar var x end metavar = 0 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma nonpositiveNegated as system Q infer all metavar var x end metavar indeed metavar var x end metavar <= 0 infer 0 <= - metavar var x end metavar end define end math ] "
" [ math define proof of lemma nonpositiveNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed metavar var x end metavar <= 0 infer lemma leqNegated modus ponens metavar var x end metavar <= 0 conclude - 0 <= - metavar var x end metavar cut lemma -0=0 conclude - 0 = 0 cut lemma subLeqLeft modus ponens - 0 = 0 modus ponens - 0 <= - metavar var x end metavar conclude 0 <= - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma negativeNegated as system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar end define end math ] "
" [ math define proof of lemma negativeNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer lemma lessNegated modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 - 0 <= - metavar var x end metavar imply not0 not0 - 0 = - metavar var x end metavar cut lemma -0=0 conclude - 0 = 0 cut lemma subLessLeft modus ponens - 0 = 0 modus ponens not0 - 0 <= - metavar var x end metavar imply not0 not0 - 0 = - metavar var x end metavar conclude not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma nonnegativeNegated as system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer - metavar var x end metavar <= 0 end define end math ] "
" [ math define proof of lemma nonnegativeNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer lemma leqNegated modus ponens 0 <= metavar var x end metavar conclude - metavar var x end metavar <= - 0 cut lemma -0=0 conclude - 0 = 0 cut lemma subLeqRight modus ponens - 0 = 0 modus ponens - metavar var x end metavar <= - 0 conclude - metavar var x end metavar <= 0 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma positiveHalved as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 0 <= 1/ 1 + 1 * metavar var x end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var x end metavar end define end math ] "
" [ math define proof of lemma positiveHalved as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma 0<1/2 conclude not0 0 <= 1/ 1 + 1 imply not0 not0 0 = 1/ 1 + 1 cut lemma lessMultiplicationLeft modus ponens not0 0 <= 1/ 1 + 1 imply not0 not0 0 = 1/ 1 + 1 modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 1/ 1 + 1 * 0 <= 1/ 1 + 1 * metavar var x end metavar imply not0 not0 1/ 1 + 1 * 0 = 1/ 1 + 1 * metavar var x end metavar cut lemma x*0=0 conclude 1/ 1 + 1 * 0 = 0 cut lemma subLessLeft modus ponens 1/ 1 + 1 * 0 = 0 modus ponens not0 1/ 1 + 1 * 0 <= 1/ 1 + 1 * metavar var x end metavar imply not0 not0 1/ 1 + 1 * 0 = 1/ 1 + 1 * metavar var x end metavar conclude not0 0 <= 1/ 1 + 1 * metavar var x end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
(*** NUMERISK ***)
" [ math define statement of lemma nonnegativeNumerical as system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar end define end math ] "
" [ math define proof of lemma nonnegativeNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer 1rule ifThenElse true modus ponens 0 <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar cut 1rule repetition modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma positiveNumerical as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar end define end math ] "
" [ math define proof of lemma positiveNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma lessLeq modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude 0 <= metavar var x end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma negativeNumerical as system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar end define end math ] "
" [ math define proof of lemma negativeNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer lemma fromLess modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 0 <= metavar var x end metavar cut 1rule ifThenElse false modus ponens not0 0 <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar cut 1rule repetition modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma nonpositiveNumerical as system Q infer all metavar var x end metavar indeed metavar var x end metavar <= 0 infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar end define end math ] "
" [ math define proof of lemma nonpositiveNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer lemma negativeNumerical modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar cut all metavar var x end metavar indeed metavar var x end metavar = 0 infer lemma eqSymmetry modus ponens metavar var x end metavar = 0 conclude 0 = metavar var x end metavar cut lemma eqLeq modus ponens 0 = metavar var x end metavar conclude 0 <= metavar var x end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar cut lemma -0=0 conclude - 0 = 0 cut lemma eqSymmetry modus ponens - 0 = 0 conclude 0 = - 0 cut lemma eqNegated modus ponens 0 = metavar var x end metavar conclude - 0 = - metavar var x end metavar cut lemma eqTransitivity5 modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar modus ponens metavar var x end metavar = 0 modus ponens 0 = - 0 modus ponens - 0 = - metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar cut 1rule deduction modus ponens all metavar var x end metavar indeed metavar var x end metavar = 0 infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar conclude metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar cut metavar var x end metavar <= 0 infer lemma leqLessEq modus ponens metavar var x end metavar <= 0 conclude not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply metavar var x end metavar = 0 cut prop lemma from disjuncts modus ponens not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply metavar var x end metavar = 0 modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar modus ponens metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma |0|=0 as system Q infer if( 0 <= 0 , 0 , - 0 ) = 0 end define end math ] "
" [ math define proof of lemma |0|=0 as lambda var c dot lambda var x dot proof expand quote system Q infer axiom leqReflexivity conclude 0 <= 0 cut lemma nonnegativeNumerical modus ponens 0 <= 0 conclude if( 0 <= 0 , 0 , - 0 ) = 0 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma 0<=|x| as system Q infer all metavar var x end metavar indeed 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) end define end math ] "
" [ math define proof of lemma 0<=|x| as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar conclude metavar var x end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma subLeqRight modus ponens metavar var x end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) modus ponens 0 <= metavar var x end metavar conclude 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut all metavar var x end metavar indeed not0 0 <= metavar var x end metavar infer lemma toLess modus ponens not0 0 <= metavar var x end metavar conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 cut lemma negativeNumerical modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar conclude - metavar var x end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma negativeNegated modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar cut lemma lessLeq modus ponens not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar conclude 0 <= - metavar var x end metavar cut lemma subLeqRight modus ponens - metavar var x end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) modus ponens 0 <= - metavar var x end metavar conclude 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed 0 <= metavar var x end metavar infer 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude 0 <= metavar var x end metavar imply 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut 1rule deduction modus ponens all metavar var x end metavar indeed not0 0 <= metavar var x end metavar infer 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude not0 0 <= metavar var x end metavar imply 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut prop lemma from negations modus ponens 0 <= metavar var x end metavar imply 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) modus ponens not0 0 <= metavar var x end metavar imply 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma sameNumerical as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end define end math ] "
" [ math define proof of lemma sameNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar infer lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar = metavar var y end metavar modus ponens 0 <= metavar var x end metavar conclude 0 <= metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var y end metavar conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var y end metavar cut lemma eqSymmetry modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var y end metavar conclude metavar var y end metavar = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma eqTransitivity4 modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed not0 0 <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar infer lemma toLess modus ponens not0 0 <= metavar var x end metavar conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 cut lemma negativeNumerical modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar cut lemma subLessLeft modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 metavar var y end metavar <= 0 imply not0 not0 metavar var y end metavar = 0 cut lemma negativeNumerical modus ponens not0 metavar var y end metavar <= 0 imply not0 not0 metavar var y end metavar = 0 conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = - metavar var y end metavar cut lemma eqSymmetry modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = - metavar var y end metavar conclude - metavar var y end metavar = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma eqNegated modus ponens metavar var x end metavar = metavar var y end metavar conclude - metavar var x end metavar = - metavar var y end metavar cut lemma eqTransitivity4 modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar modus ponens - metavar var x end metavar = - metavar var y end metavar modus ponens - metavar var y end metavar = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude 0 <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 0 <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude not0 0 <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut prop lemma from negations modus ponens 0 <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens not0 0 <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude metavar var x end metavar = metavar var y end metavar imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut 1rule mp modus ponens metavar var x end metavar = metavar var y end metavar imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens metavar var x end metavar = metavar var y end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma signNumerical(+) as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) end define end math ] "
" [ math define proof of lemma signNumerical(+) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma positiveNumerical modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar cut lemma positiveNegated modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 - metavar var x end metavar <= 0 imply not0 not0 - metavar var x end metavar = 0 cut lemma negativeNumerical modus ponens not0 - metavar var x end metavar <= 0 imply not0 not0 - metavar var x end metavar = 0 conclude if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = - - metavar var x end metavar cut lemma doubleMinus conclude - - metavar var x end metavar = metavar var x end metavar cut lemma eqTransitivity modus ponens if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = - - metavar var x end metavar modus ponens - - metavar var x end metavar = metavar var x end metavar conclude if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = metavar var x end metavar cut lemma eqSymmetry modus ponens if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = metavar var x end metavar conclude metavar var x end metavar = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) cut lemma eqTransitivity modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar modus ponens metavar var x end metavar = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma signNumerical as system Q infer all metavar var x end metavar indeed if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) end define end math ] "
" [ math define proof of lemma signNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer lemma negativeNegated modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar cut lemma signNumerical(+) modus ponens not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar conclude if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = if( 0 <= - - metavar var x end metavar , - - metavar var x end metavar , - - - metavar var x end metavar ) cut lemma doubleMinus conclude - - metavar var x end metavar = metavar var x end metavar cut lemma sameNumerical modus ponens - - metavar var x end metavar = metavar var x end metavar conclude if( 0 <= - - metavar var x end metavar , - - metavar var x end metavar , - - - metavar var x end metavar ) = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma eqTransitivity modus ponens if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = if( 0 <= - - metavar var x end metavar , - - metavar var x end metavar , - - - metavar var x end metavar ) modus ponens if( 0 <= - - metavar var x end metavar , - - metavar var x end metavar , - - - metavar var x end metavar ) = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma eqSymmetry modus ponens if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) cut all metavar var x end metavar indeed metavar var x end metavar = 0 infer lemma eqNegated modus ponens metavar var x end metavar = 0 conclude - metavar var x end metavar = - 0 cut lemma -0=0 conclude - 0 = 0 cut lemma eqSymmetry modus ponens metavar var x end metavar = 0 conclude 0 = metavar var x end metavar cut lemma eqTransitivity4 modus ponens - metavar var x end metavar = - 0 modus ponens - 0 = 0 modus ponens 0 = metavar var x end metavar conclude - metavar var x end metavar = metavar var x end metavar cut lemma eqSymmetry modus ponens - metavar var x end metavar = metavar var x end metavar conclude metavar var x end metavar = - metavar var x end metavar cut lemma sameNumerical modus ponens metavar var x end metavar = - metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) cut all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma signNumerical(+) modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) cut 1rule deduction modus ponens all metavar var x end metavar indeed metavar var x end metavar = 0 infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) conclude metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) cut 1rule deduction modus ponens all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) conclude not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) cut lemma lessTotality conclude not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply not0 metavar var x end metavar = 0 imply not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar cut prop lemma from three disjuncts modus ponens not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply not0 metavar var x end metavar = 0 imply not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) modus ponens metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma numericalDifference as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed if( 0 <= metavar var x end metavar + - metavar var y end metavar , metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar ) = if( 0 <= metavar var y end metavar + - metavar var x end metavar , metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar ) end define end math ] "
" [ math define proof of lemma numericalDifference as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed lemma signNumerical conclude if( 0 <= metavar var x end metavar + - metavar var y end metavar , metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar ) = if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) cut lemma minusNegated conclude - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar cut lemma sameNumerical modus ponens - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar conclude if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) = if( 0 <= metavar var y end metavar + - metavar var x end metavar , metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar ) cut lemma eqTransitivity modus ponens if( 0 <= metavar var x end metavar + - metavar var y end metavar , metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar ) = if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) modus ponens if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) = if( 0 <= metavar var y end metavar + - metavar var x end metavar , metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar ) conclude if( 0 <= metavar var x end metavar + - metavar var y end metavar , metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar ) = if( 0 <= metavar var y end metavar + - metavar var x end metavar , metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma splitNumericalSumHelper as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) <= if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) + if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end define end math ] "
" [ math define proof of lemma splitNumericalSumHelper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) <= if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) + if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) infer lemma signNumerical conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) cut lemma signNumerical conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma addEquations modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) + if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) + if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) conclude if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) + if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma -x-y=-(x+y) conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar cut lemma sameNumerical modus ponens - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar conclude if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) = if( 0 <= - metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar , - - metavar var x end metavar + metavar var y end metavar ) cut lemma signNumerical conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= - metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar , - - metavar var x end metavar + metavar var y end metavar ) cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= - metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar , - - metavar var x end metavar + metavar var y end metavar ) conclude if( 0 <= - metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar , - - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) cut lemma eqTransitivity modus ponens if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) = if( 0 <= - metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar , - - metavar var x end metavar + metavar var y end metavar ) modus ponens if( 0 <= - metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar , - - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) conclude if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) cut lemma subLeqRight modus ponens if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) + if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) <= if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) + if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) conclude if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma subLeqLeft modus ponens if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) modus ponens if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma splitNumericalSum(++) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end define end math ] "
" [ math define proof of lemma splitNumericalSum(++) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer lemma addEquations(Leq) modus ponens 0 <= metavar var x end metavar modus ponens 0 <= metavar var y end metavar conclude 0 + 0 <= metavar var x end metavar + metavar var y end metavar cut axiom plus0 conclude 0 + 0 = 0 cut lemma subLeqLeft modus ponens 0 + 0 = 0 modus ponens 0 + 0 <= metavar var x end metavar + metavar var y end metavar conclude 0 <= metavar var x end metavar + metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar + metavar var y end metavar conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var y end metavar conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var y end metavar cut lemma addEquations modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var y end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma eqTransitivity modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar modus ponens metavar var x end metavar + metavar var y end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma eqLeq modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma splitNumericalSum(--) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end define end math ] "
" [ math define proof of lemma splitNumericalSum(--) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer lemma nonpositiveNegated modus ponens metavar var x end metavar <= 0 conclude 0 <= - metavar var x end metavar cut lemma nonpositiveNegated modus ponens metavar var y end metavar <= 0 conclude 0 <= - metavar var y end metavar cut lemma splitNumericalSum(++) modus ponens 0 <= - metavar var x end metavar modus ponens 0 <= - metavar var y end metavar conclude if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) <= if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) + if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma splitNumericalSumHelper modus ponens if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) <= if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) + if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma splitNumericalSum(+-, smallNegative) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) end define end math ] "
" [ math define proof of lemma splitNumericalSum(+-, smallNegative) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer lemma leqAdditionLeft modus ponens metavar var y end metavar <= 0 conclude metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar + 0 cut axiom plus0 conclude metavar var x end metavar + 0 = metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar + 0 = metavar var x end metavar modus ponens metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar + 0 conclude metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar cut lemma positiveToRight(Leq)(1 term) modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma nonpositiveNumerical modus ponens metavar var y end metavar <= 0 conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = - metavar var y end metavar cut lemma eqNegated modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = - metavar var y end metavar conclude - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = - - metavar var y end metavar cut lemma doubleMinus conclude - - metavar var y end metavar = metavar var y end metavar cut lemma eqTransitivity modus ponens - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = - - metavar var y end metavar modus ponens - - metavar var y end metavar = metavar var y end metavar conclude - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar cut lemma addEquations modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar modus ponens - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var y end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar cut lemma subLeqRight modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar modus ponens 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude 0 <= metavar var x end metavar + metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar + metavar var y end metavar conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar conclude metavar var x end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var y end metavar = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) modus ponens metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma splitNumericalSum(+-, bigNegative) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) imply not0 not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end define end math ] "
" [ math define proof of lemma splitNumericalSum(+-, bigNegative) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) imply not0 not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) infer lemma nonnegativeNegated modus ponens 0 <= metavar var x end metavar conclude - metavar var x end metavar <= 0 cut lemma nonpositiveNegated modus ponens metavar var y end metavar <= 0 conclude 0 <= - metavar var y end metavar cut lemma signNumerical conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) cut lemma subLessLeft modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) modus ponens not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) imply not0 not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude not0 if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) imply not0 not0 if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma signNumerical conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma subLessRight modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) modus ponens not0 if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) imply not0 not0 if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude not0 if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) <= if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) imply not0 not0 if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma lessLeq modus ponens not0 if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) <= if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) imply not0 not0 if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) = if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) conclude if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) <= if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma splitNumericalSum(+-, smallNegative) modus ponens 0 <= - metavar var y end metavar modus ponens - metavar var x end metavar <= 0 modus ponens if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) <= if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) conclude if( 0 <= - metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar , - - metavar var y end metavar + - metavar var x end metavar ) <= if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma signNumerical conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= - metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar , - - metavar var x end metavar + metavar var y end metavar ) cut lemma -x-y=-(x+y) conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar cut axiom plusCommutativity conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var y end metavar + - metavar var x end metavar cut lemma equality modus ponens - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar modus ponens - metavar var x end metavar + - metavar var y end metavar = - metavar var y end metavar + - metavar var x end metavar conclude - metavar var x end metavar + metavar var y end metavar = - metavar var y end metavar + - metavar var x end metavar cut lemma sameNumerical modus ponens - metavar var x end metavar + metavar var y end metavar = - metavar var y end metavar + - metavar var x end metavar conclude if( 0 <= - metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar , - - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= - metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar , - - metavar var y end metavar + - metavar var x end metavar ) cut lemma eqTransitivity modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= - metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar , - - metavar var x end metavar + metavar var y end metavar ) modus ponens if( 0 <= - metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar , - - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= - metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar , - - metavar var y end metavar + - metavar var x end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= - metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar , - - metavar var y end metavar + - metavar var x end metavar ) cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= - metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar , - - metavar var y end metavar + - metavar var x end metavar ) conclude if( 0 <= - metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar , - - metavar var y end metavar + - metavar var x end metavar ) = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) cut lemma eqSymmetry modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) conclude if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma subLeqLeft modus ponens if( 0 <= - metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar , - - metavar var y end metavar + - metavar var x end metavar ) = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) modus ponens if( 0 <= - metavar var y end metavar + - metavar var x end metavar , - metavar var y end metavar + - metavar var x end metavar , - - metavar var y end metavar + - metavar var x end metavar ) <= if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma subLeqRight modus ponens if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma splitNumericalSum(+-) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end define end math ] "
" [ math define proof of lemma splitNumericalSum(+-) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma splitNumericalSum(+-, smallNegative) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma 0<=|x| conclude 0 <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma leqAdditionLeft modus ponens 0 <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut axiom plus0 conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + 0 = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma subLeqLeft modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + 0 = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma leqTransitivity modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed not0 if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma toLess modus ponens not0 if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) imply not0 not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma splitNumericalSum(+-, bigNegative) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 modus ponens not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) imply not0 not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma 0<=|x| conclude 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma leqAddition modus ponens 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude 0 + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma plus0Left conclude 0 + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma subLeqLeft modus ponens 0 + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens 0 + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma leqTransitivity modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude not0 if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer prop lemma from negations modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens not0 if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut prop lemma mp2 modus ponens 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma splitNumericalSum(-+) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end define end math ] "
" [ math define proof of lemma splitNumericalSum(-+) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer lemma nonpositiveNegated modus ponens metavar var x end metavar <= 0 conclude 0 <= - metavar var x end metavar cut lemma nonnegativeNegated modus ponens 0 <= metavar var y end metavar conclude - metavar var y end metavar <= 0 cut lemma splitNumericalSum(+-) modus ponens 0 <= - metavar var x end metavar modus ponens - metavar var y end metavar <= 0 conclude if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) <= if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) + if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma splitNumericalSumHelper modus ponens if( 0 <= - metavar var x end metavar + - metavar var y end metavar , - metavar var x end metavar + - metavar var y end metavar , - - metavar var x end metavar + - metavar var y end metavar ) <= if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) + if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma splitNumericalSum as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end define end math ] "
" [ math define proof of lemma splitNumericalSum as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer lemma splitNumericalSum(++) modus ponens 0 <= metavar var x end metavar modus ponens 0 <= metavar var y end metavar conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma splitNumericalSum(+-) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer lemma splitNumericalSum(-+) modus ponens metavar var x end metavar <= 0 modus ponens 0 <= metavar var y end metavar conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer lemma splitNumericalSum(--) modus ponens metavar var x end metavar <= 0 modus ponens metavar var y end metavar <= 0 conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude 0 <= metavar var x end metavar imply 0 <= metavar var y end metavar imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude metavar var x end metavar <= 0 imply 0 <= metavar var y end metavar imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude metavar var x end metavar <= 0 imply metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply 0 <= metavar var y end metavar imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens metavar var x end metavar <= 0 imply 0 <= metavar var y end metavar imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude 0 <= metavar var y end metavar imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens metavar var x end metavar <= 0 imply metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma from leqGeq modus ponens 0 <= metavar var y end metavar imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens metavar var y end metavar <= 0 imply if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma insertMiddleTerm(Numerical) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar + - metavar var z end metavar , metavar var x end metavar + - metavar var z end metavar , - metavar var x end metavar + - metavar var z end metavar ) + if( 0 <= metavar var z end metavar + metavar var y end metavar , metavar var z end metavar + metavar var y end metavar , - metavar var z end metavar + metavar var y end metavar ) end define end math ] "
" [ math define proof of lemma insertMiddleTerm(Numerical) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed lemma splitNumericalSum conclude if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar + - metavar var z end metavar , metavar var x end metavar + - metavar var z end metavar , - metavar var x end metavar + - metavar var z end metavar ) + if( 0 <= metavar var z end metavar + metavar var y end metavar , metavar var z end metavar + metavar var y end metavar , - metavar var z end metavar + metavar var y end metavar ) cut lemma insertMiddleTerm(Sum) conclude metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar cut lemma sameNumerical modus ponens metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) cut lemma subLeqLeft modus ponens if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) modus ponens if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar + - metavar var z end metavar , metavar var x end metavar + - metavar var z end metavar , - metavar var x end metavar + - metavar var z end metavar ) + if( 0 <= metavar var z end metavar + metavar var y end metavar , metavar var z end metavar + metavar var y end metavar , - metavar var z end metavar + metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar + - metavar var z end metavar , metavar var x end metavar + - metavar var z end metavar , - metavar var x end metavar + - metavar var z end metavar ) + if( 0 <= metavar var z end metavar + metavar var y end metavar , metavar var z end metavar + metavar var y end metavar , - metavar var z end metavar + metavar var y end metavar ) end quote state proof state cache var c end expand end define end math ] "
(*** REGNESTYKKER ***)
" [ math define statement of lemma x+y=zBackwards as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var y end metavar + metavar var x end metavar end define end math ] "
" [ math define proof of lemma x+y=zBackwards as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var z end metavar infer axiom plusCommutativity conclude metavar var x end metavar + metavar var y end metavar = metavar var y end metavar + metavar var x end metavar cut lemma equality modus ponens metavar var x end metavar + metavar var y end metavar = metavar var z end metavar conclude metavar var z end metavar = metavar var y end metavar + metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma x*y=zBackwards as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var y end metavar * metavar var x end metavar end define end math ] "
" [ math define proof of lemma x*y=zBackwards as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar = metavar var z end metavar infer axiom timesCommutativity conclude metavar var x end metavar * metavar var y end metavar = metavar var y end metavar * metavar var x end metavar cut lemma equality modus ponens metavar var x end metavar * metavar var y end metavar = metavar var z end metavar conclude metavar var z end metavar = metavar var y end metavar * metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma x=x+(y-y) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar end define end math ] "
" [ math define proof of lemma x=x+(y-y) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed axiom plus0 conclude metavar var x end metavar + 0 = metavar var x end metavar cut axiom negative conclude metavar var y end metavar + - metavar var y end metavar = 0 cut lemma eqSymmetry modus ponens metavar var y end metavar + - metavar var y end metavar = 0 conclude 0 = metavar var y end metavar + - metavar var y end metavar cut lemma eqAdditionLeft modus ponens 0 = metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar + 0 = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma equality modus ponens metavar var x end metavar + 0 = metavar var x end metavar modus ponens metavar var x end metavar + 0 = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma x=x+y-y as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar end define end math ] "
" [ math define proof of lemma x=x+y-y as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed lemma x=x+(y-y) conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut axiom plusAssociativity conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma x=x*y*(1/y) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var y end metavar = 0 infer metavar var x end metavar = metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar end define end math ] "
" [ math define proof of lemma x=x*y*(1/y) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var y end metavar = 0 infer axiom times1 conclude metavar var x end metavar * 1 = metavar var x end metavar cut lemma reciprocal modus ponens not0 metavar var y end metavar = 0 conclude metavar var y end metavar * 1/ metavar var y end metavar = 1 cut lemma three2twoFactors modus ponens metavar var y end metavar * 1/ metavar var y end metavar = 1 conclude metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = metavar var x end metavar * 1 cut lemma eqTransitivity modus ponens metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = metavar var x end metavar * 1 modus ponens metavar var x end metavar * 1 = metavar var x end metavar conclude metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = metavar var x end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = metavar var x end metavar conclude metavar var x end metavar = metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma insertMiddleTerm(Sum) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar end define end math ] "
" [ math define proof of lemma insertMiddleTerm(Sum) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma three2threeTerms conclude metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar conclude metavar var x end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma eqAddition modus ponens metavar var x end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar conclude metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar cut axiom plusAssociativity conclude metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar modus ponens metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma insertMiddleTerm(Difference) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var y end metavar + metavar var z end metavar end define end math ] "
" [ math define proof of lemma insertMiddleTerm(Difference) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed lemma insertMiddleTerm(Sum) conclude metavar var x end metavar + - metavar var y end metavar = metavar var x end metavar + - - metavar var z end metavar + - metavar var z end metavar + - metavar var y end metavar cut lemma doubleMinus conclude - - metavar var z end metavar = metavar var z end metavar cut lemma eqAdditionLeft modus ponens - - metavar var z end metavar = metavar var z end metavar conclude metavar var x end metavar + - - metavar var z end metavar = metavar var x end metavar + metavar var z end metavar cut axiom plusCommutativity conclude - metavar var z end metavar + - metavar var y end metavar = - metavar var y end metavar + - metavar var z end metavar cut lemma -x-y=-(x+y) conclude - metavar var y end metavar + - metavar var z end metavar = - metavar var y end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens - metavar var z end metavar + - metavar var y end metavar = - metavar var y end metavar + - metavar var z end metavar modus ponens - metavar var y end metavar + - metavar var z end metavar = - metavar var y end metavar + metavar var z end metavar conclude - metavar var z end metavar + - metavar var y end metavar = - metavar var y end metavar + metavar var z end metavar cut lemma addEquations modus ponens metavar var x end metavar + - - metavar var z end metavar = metavar var x end metavar + metavar var z end metavar modus ponens - metavar var z end metavar + - metavar var y end metavar = - metavar var y end metavar + metavar var z end metavar conclude metavar var x end metavar + - - metavar var z end metavar + - metavar var z end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var y end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + - metavar var y end metavar = metavar var x end metavar + - - metavar var z end metavar + - metavar var z end metavar + - metavar var y end metavar modus ponens metavar var x end metavar + - - metavar var z end metavar + - metavar var z end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var y end metavar + metavar var z end metavar conclude metavar var x end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma x*0+x=x as system Q infer all metavar var x end metavar indeed metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma x*0+x=x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom times1 conclude metavar var x end metavar * 1 = metavar var x end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar * 1 = metavar var x end metavar conclude metavar var x end metavar = metavar var x end metavar * 1 cut lemma eqAdditionLeft modus ponens metavar var x end metavar = metavar var x end metavar * 1 conclude metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar * 1 cut axiom distribution conclude metavar var x end metavar * 0 + 1 = metavar var x end metavar * 0 + metavar var x end metavar * 1 cut lemma eqSymmetry modus ponens metavar var x end metavar * 0 + 1 = metavar var x end metavar * 0 + metavar var x end metavar * 1 conclude metavar var x end metavar * 0 + metavar var x end metavar * 1 = metavar var x end metavar * 0 + 1 cut lemma plus0Left conclude 0 + 1 = 1 cut lemma eqMultiplicationLeft modus ponens 0 + 1 = 1 conclude metavar var x end metavar * 0 + 1 = metavar var x end metavar * 1 cut lemma eqTransitivity5 modus ponens metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar * 1 modus ponens metavar var x end metavar * 0 + metavar var x end metavar * 1 = metavar var x end metavar * 0 + 1 modus ponens metavar var x end metavar * 0 + 1 = metavar var x end metavar * 1 modus ponens metavar var x end metavar * 1 = metavar var x end metavar conclude metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma x*0=0 as system Q infer all metavar var x end metavar indeed metavar var x end metavar * 0 = 0 end define end math ] "
" [ math define proof of lemma x*0=0 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma x=x+(y-y) conclude metavar var x end metavar * 0 = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar cut axiom plusAssociativity conclude metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar conclude metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar cut lemma x*0+x=x conclude metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar cut lemma eqAddition modus ponens metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar conclude metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar + - metavar var x end metavar cut axiom negative conclude metavar var x end metavar + - metavar var x end metavar = 0 cut lemma eqTransitivity5 modus ponens metavar var x end metavar * 0 = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar modus ponens metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar modus ponens metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar + - metavar var x end metavar modus ponens metavar var x end metavar + - metavar var x end metavar = 0 conclude metavar var x end metavar * 0 = 0 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma (-1)*(-1)+(-1)*1=0 as system Q infer - 1 * - 1 + - 1 * 1 = 0 end define end math ] "
" [ math define proof of lemma (-1)*(-1)+(-1)*1=0 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma distributionOut conclude - 1 * - 1 + - 1 * 1 = - 1 * - 1 + 1 cut axiom negative conclude 1 + - 1 = 0 cut axiom plusCommutativity conclude - 1 + 1 = 1 + - 1 cut lemma eqTransitivity modus ponens - 1 + 1 = 1 + - 1 modus ponens 1 + - 1 = 0 conclude - 1 + 1 = 0 cut lemma eqMultiplicationLeft modus ponens - 1 + 1 = 0 conclude - 1 * - 1 + 1 = - 1 * 0 cut lemma x*0=0 conclude - 1 * 0 = 0 cut lemma eqTransitivity4 modus ponens - 1 * - 1 + - 1 * 1 = - 1 * - 1 + 1 modus ponens - 1 * - 1 + 1 = - 1 * 0 modus ponens - 1 * 0 = 0 conclude - 1 * - 1 + - 1 * 1 = 0 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma (-1)*(-1)=1 as system Q infer - 1 * - 1 = 1 end define end math ] "
" [ math define proof of lemma (-1)*(-1)=1 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma x=x+(y-y) conclude - 1 * - 1 = - 1 * - 1 + 1 + - 1 cut axiom times1 conclude - 1 * 1 = - 1 cut lemma eqSymmetry modus ponens - 1 * 1 = - 1 conclude - 1 = - 1 * 1 cut lemma eqAdditionLeft modus ponens - 1 = - 1 * 1 conclude 1 + - 1 = 1 + - 1 * 1 cut lemma eqAdditionLeft modus ponens 1 + - 1 = 1 + - 1 * 1 conclude - 1 * - 1 + 1 + - 1 = - 1 * - 1 + 1 + - 1 * 1 cut axiom plusCommutativity conclude 1 + - 1 * 1 = - 1 * 1 + 1 cut lemma eqAdditionLeft modus ponens 1 + - 1 * 1 = - 1 * 1 + 1 conclude - 1 * - 1 + 1 + - 1 * 1 = - 1 * - 1 + - 1 * 1 + 1 cut axiom plusAssociativity conclude - 1 * - 1 + - 1 * 1 + 1 = - 1 * - 1 + - 1 * 1 + 1 cut lemma eqSymmetry modus ponens - 1 * - 1 + - 1 * 1 + 1 = - 1 * - 1 + - 1 * 1 + 1 conclude - 1 * - 1 + - 1 * 1 + 1 = - 1 * - 1 + - 1 * 1 + 1 cut lemma (-1)*(-1)+(-1)*1=0 conclude - 1 * - 1 + - 1 * 1 = 0 cut lemma eqAddition modus ponens - 1 * - 1 + - 1 * 1 = 0 conclude - 1 * - 1 + - 1 * 1 + 1 = 0 + 1 cut lemma plus0Left conclude 0 + 1 = 1 cut lemma eqTransitivity5 modus ponens - 1 * - 1 = - 1 * - 1 + 1 + - 1 modus ponens - 1 * - 1 + 1 + - 1 = - 1 * - 1 + 1 + - 1 * 1 modus ponens - 1 * - 1 + 1 + - 1 * 1 = - 1 * - 1 + - 1 * 1 + 1 modus ponens - 1 * - 1 + - 1 * 1 + 1 = - 1 * - 1 + - 1 * 1 + 1 conclude - 1 * - 1 = - 1 * - 1 + - 1 * 1 + 1 cut lemma eqTransitivity4 modus ponens - 1 * - 1 = - 1 * - 1 + - 1 * 1 + 1 modus ponens - 1 * - 1 + - 1 * 1 + 1 = 0 + 1 modus ponens 0 + 1 = 1 conclude - 1 * - 1 = 1 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma 0<1Helper as system Q infer 1 <= 0 imply 0 <= 1 end define end math ] "
" [ math define proof of lemma 0<1Helper as lambda var c dot lambda var x dot proof expand quote system Q infer 1 <= 0 infer lemma leqAddition modus ponens 1 <= 0 conclude 1 + - 1 <= 0 + - 1 cut axiom negative conclude 1 + - 1 = 0 cut lemma subLeqLeft modus ponens 1 + - 1 = 0 modus ponens 1 + - 1 <= 0 + - 1 conclude 0 <= 0 + - 1 cut lemma plus0Left conclude 0 + - 1 = - 1 cut lemma subLeqRight modus ponens 0 + - 1 = - 1 modus ponens 0 <= 0 + - 1 conclude 0 <= - 1 cut lemma leqMultiplication modus ponens 0 <= - 1 modus ponens 0 <= - 1 conclude 0 * - 1 <= - 1 * - 1 cut lemma x*0=0 conclude - 1 * 0 = 0 cut axiom timesCommutativity conclude 0 * - 1 = - 1 * 0 cut lemma eqTransitivity modus ponens 0 * - 1 = - 1 * 0 modus ponens - 1 * 0 = 0 conclude 0 * - 1 = 0 cut lemma subLeqLeft modus ponens 0 * - 1 = 0 modus ponens 0 * - 1 <= - 1 * - 1 conclude 0 <= - 1 * - 1 cut lemma (-1)*(-1)=1 conclude - 1 * - 1 = 1 cut lemma subLeqRight modus ponens - 1 * - 1 = 1 modus ponens 0 <= - 1 * - 1 conclude 0 <= 1 cut 1rule deduction modus ponens 1 <= 0 infer 0 <= 1 conclude 1 <= 0 imply 0 <= 1 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma 0<1 as system Q infer not0 0 <= 1 imply not0 not0 0 = 1 end define end math ] "
" [ math define proof of lemma 0<1 as lambda var c dot lambda var x dot proof expand quote system Q infer axiom leqTotality conclude not0 0 <= 1 imply 1 <= 0 cut prop lemma auto imply conclude 0 <= 1 imply 0 <= 1 cut lemma 0<1Helper conclude 1 <= 0 imply 0 <= 1 cut prop lemma from disjuncts modus ponens not0 0 <= 1 imply 1 <= 0 modus ponens 0 <= 1 imply 0 <= 1 modus ponens 1 <= 0 imply 0 <= 1 conclude 0 <= 1 cut axiom 0not1 conclude not0 0 = 1 cut prop lemma join conjuncts modus ponens 0 <= 1 modus ponens not0 0 = 1 conclude not0 0 <= 1 imply not0 not0 0 = 1 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma 0<2 as system Q infer not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 end define end math ] "
" [ math define proof of lemma 0<2 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma 0<1 conclude not0 0 <= 1 imply not0 not0 0 = 1 cut lemma lessAddition modus ponens not0 0 <= 1 imply not0 not0 0 = 1 conclude not0 0 + 1 <= 1 + 1 imply not0 not0 0 + 1 = 1 + 1 cut lemma plus0Left conclude 0 + 1 = 1 cut lemma subLessLeft modus ponens 0 + 1 = 1 modus ponens not0 0 + 1 <= 1 + 1 imply not0 not0 0 + 1 = 1 + 1 conclude not0 1 <= 1 + 1 imply not0 not0 1 = 1 + 1 cut lemma lessTransitivity modus ponens not0 0 <= 1 imply not0 not0 0 = 1 modus ponens not0 1 <= 1 + 1 imply not0 not0 1 = 1 + 1 conclude not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma 0<1/2 as system Q infer not0 0 <= 1/ 1 + 1 imply not0 not0 0 = 1/ 1 + 1 end define end math ] "
" [ math define proof of lemma 0<1/2 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma 0<2 conclude not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 cut prop lemma first conjunct modus ponens not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 conclude 0 <= 1 + 1 cut prop lemma second conjunct modus ponens not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 conclude not0 0 = 1 + 1 cut lemma neqSymmetry modus ponens not0 0 = 1 + 1 conclude not0 1 + 1 = 0 cut lemma 0<1 conclude not0 0 <= 1 imply not0 not0 0 = 1 cut lemma x*0=0 conclude 1 + 1 * 0 = 0 cut lemma x*y=zBackwards modus ponens 1 + 1 * 0 = 0 conclude 0 = 0 * 1 + 1 cut lemma subLessLeft modus ponens 0 = 0 * 1 + 1 modus ponens not0 0 <= 1 imply not0 not0 0 = 1 conclude not0 0 * 1 + 1 <= 1 imply not0 not0 0 * 1 + 1 = 1 cut lemma reciprocal modus ponens not0 1 + 1 = 0 conclude 1 + 1 * 1/ 1 + 1 = 1 cut lemma x*y=zBackwards modus ponens 1 + 1 * 1/ 1 + 1 = 1 conclude 1 = 1/ 1 + 1 * 1 + 1 cut lemma subLessRight modus ponens 1 = 1/ 1 + 1 * 1 + 1 modus ponens not0 0 * 1 + 1 <= 1 imply not0 not0 0 * 1 + 1 = 1 conclude not0 0 * 1 + 1 <= 1/ 1 + 1 * 1 + 1 imply not0 not0 0 * 1 + 1 = 1/ 1 + 1 * 1 + 1 cut lemma lessDivision modus ponens 0 <= 1 + 1 modus ponens not0 0 * 1 + 1 <= 1/ 1 + 1 * 1 + 1 imply not0 not0 0 * 1 + 1 = 1/ 1 + 1 * 1 + 1 conclude not0 0 <= 1/ 1 + 1 imply not0 not0 0 = 1/ 1 + 1 end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma x+x=2*x as system Q infer all metavar var x end metavar indeed metavar var x end metavar + metavar var x end metavar = 1 + 1 * metavar var x end metavar end define end math ] "
" [ math define proof of lemma x+x=2*x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom times1 conclude metavar var x end metavar * 1 = metavar var x end metavar cut lemma eqSymmetry conclude metavar var x end metavar = metavar var x end metavar * 1 cut lemma eqAdditionLeft modus ponens metavar var x end metavar = metavar var x end metavar * 1 conclude metavar var x end metavar + metavar var x end metavar = metavar var x end metavar + metavar var x end metavar * 1 cut lemma eqAddition modus ponens metavar var x end metavar = metavar var x end metavar * 1 conclude metavar var x end metavar + metavar var x end metavar * 1 = metavar var x end metavar * 1 + metavar var x end metavar * 1 cut lemma eqTransitivity modus ponens metavar var x end metavar + metavar var x end metavar = metavar var x end metavar + metavar var x end metavar * 1 modus ponens metavar var x end metavar + metavar var x end metavar * 1 = metavar var x end metavar * 1 + metavar var x end metavar * 1 conclude metavar var x end metavar + metavar var x end metavar = metavar var x end metavar * 1 + metavar var x end metavar * 1 cut lemma distributionOut conclude metavar var x end metavar * 1 + metavar var x end metavar * 1 = metavar var x end metavar * 1 + 1 cut 1rule repetition modus ponens metavar var x end metavar * 1 + metavar var x end metavar * 1 = metavar var x end metavar * 1 + 1 conclude metavar var x end metavar * 1 + metavar var x end metavar * 1 = metavar var x end metavar * 1 + 1 cut axiom timesCommutativity conclude metavar var x end metavar * 1 + 1 = 1 + 1 * metavar var x end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar + metavar var x end metavar = metavar var x end metavar * 1 + metavar var x end metavar * 1 modus ponens metavar var x end metavar * 1 + metavar var x end metavar * 1 = metavar var x end metavar * 1 + 1 modus ponens metavar var x end metavar * 1 + 1 = 1 + 1 * metavar var x end metavar conclude metavar var x end metavar + metavar var x end metavar = 1 + 1 * metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma (1/2)x+(1/2)x=x as system Q infer all metavar var x end metavar indeed 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var x end metavar = metavar var x end metavar end define end math ] "
" [ math define proof of lemma (1/2)x+(1/2)x=x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma 0<2 conclude not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 cut lemma lessNeq modus ponens not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 conclude not0 0 = 1 + 1 cut lemma neqSymmetry modus ponens not0 0 = 1 + 1 conclude not0 1 + 1 = 0 cut lemma x+x=2*x conclude 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar cut axiom timesAssociativity conclude 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar cut lemma eqSymmetry modus ponens 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar conclude 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar cut lemma reciprocal modus ponens not0 1 + 1 = 0 conclude 1 + 1 * 1/ 1 + 1 = 1 cut lemma eqMultiplication modus ponens 1 + 1 * 1/ 1 + 1 = 1 conclude 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 * metavar var x end metavar cut lemma times1Left conclude 1 * metavar var x end metavar = metavar var x end metavar cut lemma eqTransitivity5 modus ponens 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar modus ponens 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar modus ponens 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 * metavar var x end metavar modus ponens 1 * metavar var x end metavar = metavar var x end metavar conclude 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma times(-1) as system Q infer all metavar var x end metavar indeed metavar var x end metavar * - 1 = - metavar var x end metavar end define , define proof of lemma times(-1) as rule tactic end define end math ] "
" [ math define proof of lemma times(-1) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom negative conclude 1 + - 1 = 0 cut axiom plusCommutativity conclude - 1 + 1 = 1 + - 1 cut lemma eqTransitivity modus ponens - 1 + 1 = 1 + - 1 modus ponens 1 + - 1 = 0 conclude - 1 + 1 = 0 cut lemma eqMultiplicationLeft modus ponens - 1 + 1 = 0 conclude metavar var x end metavar * - 1 + 1 = metavar var x end metavar * 0 cut lemma x*0=0 conclude metavar var x end metavar * 0 = 0 cut lemma eqTransitivity modus ponens metavar var x end metavar * - 1 + 1 = metavar var x end metavar * 0 modus ponens metavar var x end metavar * 0 = 0 conclude metavar var x end metavar * - 1 + 1 = 0 cut axiom distribution conclude metavar var x end metavar * - 1 + 1 = metavar var x end metavar * - 1 + metavar var x end metavar * 1 cut lemma eqSymmetry modus ponens metavar var x end metavar * - 1 + 1 = metavar var x end metavar * - 1 + metavar var x end metavar * 1 conclude metavar var x end metavar * - 1 + metavar var x end metavar * 1 = metavar var x end metavar * - 1 + 1 cut lemma eqTransitivity modus ponens metavar var x end metavar * - 1 + metavar var x end metavar * 1 = metavar var x end metavar * - 1 + 1 modus ponens metavar var x end metavar * - 1 + 1 = 0 conclude metavar var x end metavar * - 1 + metavar var x end metavar * 1 = 0 cut lemma positiveToRight(Eq) modus ponens metavar var x end metavar * - 1 + metavar var x end metavar * 1 = 0 conclude metavar var x end metavar * - 1 = 0 + - metavar var x end metavar * 1 cut lemma plus0Left conclude 0 + - metavar var x end metavar * 1 = - metavar var x end metavar * 1 cut lemma eqTransitivity modus ponens metavar var x end metavar * - 1 = 0 + - metavar var x end metavar * 1 modus ponens 0 + - metavar var x end metavar * 1 = - metavar var x end metavar * 1 conclude metavar var x end metavar * - 1 = - metavar var x end metavar * 1 cut axiom times1 conclude metavar var x end metavar * 1 = metavar var x end metavar cut lemma eqNegated modus ponens metavar var x end metavar * 1 = metavar var x end metavar conclude - metavar var x end metavar * 1 = - metavar var x end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar * - 1 = - metavar var x end metavar * 1 modus ponens - metavar var x end metavar * 1 = - metavar var x end metavar conclude metavar var x end metavar * - 1 = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma times(-1)Left as system Q infer all metavar var x end metavar indeed - 1 * metavar var x end metavar = - metavar var x end metavar end define end math ] "
" [ math define proof of lemma times(-1)Left as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma times(-1) conclude metavar var x end metavar * - 1 = - metavar var x end metavar cut axiom timesCommutativity conclude - 1 * metavar var x end metavar = metavar var x end metavar * - 1 cut lemma eqTransitivity modus ponens - 1 * metavar var x end metavar = metavar var x end metavar * - 1 modus ponens metavar var x end metavar * - 1 = - metavar var x end metavar conclude - 1 * metavar var x end metavar = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma -x-y=-(x+y) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar end define end math ] "
" [ math define proof of lemma -x-y=-(x+y) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed lemma times(-1)Left conclude - 1 * metavar var x end metavar = - metavar var x end metavar cut lemma times(-1)Left conclude - 1 * metavar var y end metavar = - metavar var y end metavar cut lemma addEquations modus ponens - 1 * metavar var x end metavar = - metavar var x end metavar modus ponens - 1 * metavar var y end metavar = - metavar var y end metavar conclude - 1 * metavar var x end metavar + - 1 * metavar var y end metavar = - metavar var x end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens - 1 * metavar var x end metavar + - 1 * metavar var y end metavar = - metavar var x end metavar + - metavar var y end metavar conclude - metavar var x end metavar + - metavar var y end metavar = - 1 * metavar var x end metavar + - 1 * metavar var y end metavar cut lemma distributionOut conclude - 1 * metavar var x end metavar + - 1 * metavar var y end metavar = - 1 * metavar var x end metavar + metavar var y end metavar cut lemma times(-1)Left conclude - 1 * metavar var x end metavar + metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar cut lemma eqTransitivity4 modus ponens - metavar var x end metavar + - metavar var y end metavar = - 1 * metavar var x end metavar + - 1 * metavar var y end metavar modus ponens - 1 * metavar var x end metavar + - 1 * metavar var y end metavar = - 1 * metavar var x end metavar + metavar var y end metavar modus ponens - 1 * metavar var x end metavar + metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma minusNegated as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar end define end math ] "
" [ math define proof of lemma minusNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed lemma doubleMinus conclude - - metavar var y end metavar = metavar var y end metavar cut lemma eqAddition modus ponens - - metavar var y end metavar = metavar var y end metavar conclude - - metavar var y end metavar + - metavar var x end metavar = metavar var y end metavar + - metavar var x end metavar cut lemma eqSymmetry modus ponens - - metavar var y end metavar + - metavar var x end metavar = metavar var y end metavar + - metavar var x end metavar conclude metavar var y end metavar + - metavar var x end metavar = - - metavar var y end metavar + - metavar var x end metavar cut lemma -x-y=-(x+y) conclude - - metavar var y end metavar + - metavar var x end metavar = - - metavar var y end metavar + metavar var x end metavar cut axiom plusCommutativity conclude - metavar var y end metavar + metavar var x end metavar = metavar var x end metavar + - metavar var y end metavar cut lemma eqNegated modus ponens - metavar var y end metavar + metavar var x end metavar = metavar var x end metavar + - metavar var y end metavar conclude - - metavar var y end metavar + metavar var x end metavar = - metavar var x end metavar + - metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var y end metavar + - metavar var x end metavar = - - metavar var y end metavar + - metavar var x end metavar modus ponens - - metavar var y end metavar + - metavar var x end metavar = - - metavar var y end metavar + metavar var x end metavar modus ponens - - metavar var y end metavar + metavar var x end metavar = - metavar var x end metavar + - metavar var y end metavar conclude metavar var y end metavar + - metavar var x end metavar = - metavar var x end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar + - metavar var x end metavar = - metavar var x end metavar + - metavar var y end metavar conclude - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma -0=0 as system Q infer - 0 = 0 end define end math ] "
" [ math define proof of lemma -0=0 as lambda var c dot lambda var x dot proof expand quote system Q infer axiom negative conclude 0 + - 0 = 0 cut axiom plus0 conclude 0 + 0 = 0 cut lemma uniqueNegative modus ponens 0 + - 0 = 0 modus ponens 0 + 0 = 0 conclude - 0 = 0 end quote state proof state cache var c end expand end define end math ] "
(*** LEQ, nummer 2 af 2 ***)
" [ math define statement of lemma negativeToLeft(Leq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar infer metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar end define end math ] "
" [ math define proof of lemma negativeToLeft(Leq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar infer lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma x=x+y-y conclude metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma three2threeTerms conclude metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar conclude metavar var y end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar conclude metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "
(*** SAME-F ***)
" [ math define statement of lemma sameFsymmetry as system Q infer all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar sameF metavar var fy end metavar infer metavar var fy end metavar sameF metavar var fx end metavar end define end math ] "
" [ math define proof of lemma sameFsymmetry as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar sameF metavar var fy end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer existential var var c end var <= metavar var m end metavar infer 1rule fromSameF modus ponens metavar var fx end metavar sameF metavar var fy end metavar modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar conclude existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut 1rule mp modus ponens existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar modus ponens existential var var c end var <= metavar var m end metavar conclude not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut lemma numericalDifference conclude if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) cut lemma subLessLeft modus ponens if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) modus ponens not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar sameF metavar var fy end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer existential var var c end var <= metavar var m end metavar infer not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude metavar var fx end metavar sameF metavar var fy end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut metavar var fx end metavar sameF metavar var fy end metavar infer 1rule mp modus ponens metavar var fx end metavar sameF metavar var fy end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar modus ponens metavar var fx end metavar sameF metavar var fy end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut 1rule toSameF modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude metavar var fy end metavar sameF metavar var fx end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma plusF(Sym) as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] end define end math ] "
" [ math define proof of lemma plusF(Sym) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed axiom plusF conclude [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma eqSymmetry modus ponens [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma timesF(Sym) as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] end define end math ] "
" [ math define proof of lemma timesF(Sym) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed axiom timesF conclude [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma eqSymmetry modus ponens [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma plus0f as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed metavar var fx end metavar +f 0f =f metavar var fx end metavar end define end math ] "
" [ math define proof of lemma plus0f as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed axiom plusF conclude [ metavar var fx end metavar +f 0f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ 0f ; metavar var m end metavar ] cut axiom 0f conclude [ 0f ; metavar var m end metavar ] = 0 cut lemma eqAdditionLeft modus ponens [ 0f ; metavar var m end metavar ] = 0 conclude [ metavar var fx end metavar ; metavar var m end metavar ] + [ 0f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + 0 cut axiom plus0 conclude [ metavar var fx end metavar ; metavar var m end metavar ] + 0 = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqTransitivity4 modus ponens [ metavar var fx end metavar +f 0f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ 0f ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + [ 0f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + 0 modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + 0 = [ metavar var fx end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar +f 0f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut 1rule to=f modus ponens [ metavar var fx end metavar +f 0f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] conclude metavar var fx end metavar +f 0f =f metavar var fx end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma sameFtransitivity as system Q infer all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed metavar var fx end metavar sameF metavar var fy end metavar infer metavar var fy end metavar sameF metavar var fz end metavar infer metavar var fx end metavar sameF metavar var fz end metavar end define end math ] "
" [ math define proof of lemma sameFtransitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed metavar var fx end metavar sameF metavar var fy end metavar infer metavar var fy end metavar sameF metavar var fz end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer existential var var c end var <= metavar var m end metavar infer lemma positiveHalved modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar conclude not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar cut 1rule fromSameF modus ponens metavar var fx end metavar sameF metavar var fy end metavar modus ponens not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar conclude existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = 1/ 1 + 1 * metavar var ep end metavar cut 1rule mp modus ponens existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = 1/ 1 + 1 * metavar var ep end metavar modus ponens existential var var c end var <= metavar var m end metavar conclude not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = 1/ 1 + 1 * metavar var ep end metavar cut 1rule fromSameF modus ponens metavar var fy end metavar sameF metavar var fz end metavar modus ponens not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar conclude existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = 1/ 1 + 1 * metavar var ep end metavar cut 1rule mp modus ponens existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = 1/ 1 + 1 * metavar var ep end metavar modus ponens existential var var c end var <= metavar var m end metavar conclude not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = 1/ 1 + 1 * metavar var ep end metavar cut lemma addEquations(Less) modus ponens not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = 1/ 1 + 1 * metavar var ep end metavar modus ponens not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = 1/ 1 + 1 * metavar var ep end metavar conclude not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) + if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= 1/ 1 + 1 * metavar var ep end metavar + 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) + if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = 1/ 1 + 1 * metavar var ep end metavar + 1/ 1 + 1 * metavar var ep end metavar cut lemma (1/2)x+(1/2)x=x conclude 1/ 1 + 1 * metavar var ep end metavar + 1/ 1 + 1 * metavar var ep end metavar = metavar var ep end metavar cut lemma subLessRight modus ponens 1/ 1 + 1 * metavar var ep end metavar + 1/ 1 + 1 * metavar var ep end metavar = metavar var ep end metavar modus ponens not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) + if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= 1/ 1 + 1 * metavar var ep end metavar + 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) + if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = 1/ 1 + 1 * metavar var ep end metavar + 1/ 1 + 1 * metavar var ep end metavar conclude not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) + if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) + if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut lemma insertMiddleTerm(Numerical) conclude if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) + if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) cut lemma leqLessTransitivity modus ponens if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) + if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) modus ponens not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) + if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) + if( 0 <= [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed metavar var fx end metavar sameF metavar var fy end metavar infer metavar var fy end metavar sameF metavar var fz end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer existential var var c end var <= metavar var m end metavar infer not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude metavar var fx end metavar sameF metavar var fy end metavar imply metavar var fy end metavar sameF metavar var fz end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut metavar var fx end metavar sameF metavar var fy end metavar infer metavar var fy end metavar sameF metavar var fz end metavar infer prop lemma mp2 modus ponens metavar var fx end metavar sameF metavar var fy end metavar imply metavar var fy end metavar sameF metavar var fz end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar modus ponens metavar var fx end metavar sameF metavar var fy end metavar modus ponens metavar var fy end metavar sameF metavar var fz end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut 1rule toSameF modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude metavar var fx end metavar sameF metavar var fz end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma =f to sameF as system Q infer all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar =f metavar var fy end metavar infer metavar var fx end metavar sameF metavar var fy end metavar end define end math ] "
" [ math define proof of lemma =f to sameF as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar =f metavar var fy end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer 1rule from=f modus ponens metavar var fx end metavar =f metavar var fy end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma positiveToLeft(Eq)(1 term) modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = 0 cut lemma sameNumerical modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = 0 conclude if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = if( 0 <= 0 , 0 , - 0 ) cut lemma |0|=0 conclude if( 0 <= 0 , 0 , - 0 ) = 0 cut lemma eqTransitivity modus ponens if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = if( 0 <= 0 , 0 , - 0 ) modus ponens if( 0 <= 0 , 0 , - 0 ) = 0 conclude if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = 0 cut lemma eqSymmetry modus ponens if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = 0 conclude 0 = if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) cut lemma subLessLeft modus ponens 0 = if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar conclude not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut prop lemma weakening modus ponens not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude 0 <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut 1rule exist intro at existential var var c end var at 0 modus ponens 0 <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar =f metavar var fy end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude metavar var fx end metavar =f metavar var fy end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut metavar var fx end metavar =f metavar var fy end metavar infer 1rule mp modus ponens metavar var fx end metavar =f metavar var fy end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar modus ponens metavar var fx end metavar =f metavar var fy end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut 1rule toSameF modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude metavar var fx end metavar sameF metavar var fy end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma f2R(Plus) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed all metavar var fv end metavar indeed metavar var fx end metavar +f metavar var fy end metavar sameF metavar var fz end metavar infer R( var fx +f var fy ) == R( metavar var fz end metavar ) end define end math ] "
" [ math define proof of lemma f2R(Plus) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed all metavar var fv end metavar indeed metavar var fx end metavar +f metavar var fy end metavar sameF metavar var fz end metavar infer metavar var fu end metavar in0 R( metavar var fx end metavar +f metavar var fy end metavar ) infer metavar var fv end metavar in0 R( metavar var fz end metavar ) infer 1rule fromInR modus ponens metavar var fu end metavar in0 R( metavar var fx end metavar +f metavar var fy end metavar ) conclude metavar var fu end metavar sameF metavar var fx end metavar +f metavar var fy end metavar cut 1rule fromInR modus ponens metavar var fv end metavar in0 R( metavar var fz end metavar ) conclude metavar var fv end metavar sameF metavar var fz end metavar cut lemma sameFsymmetry modus ponens metavar var fv end metavar sameF metavar var fz end metavar conclude metavar var fz end metavar sameF metavar var fv end metavar cut lemma sameFtransitivity modus ponens metavar var fu end metavar sameF metavar var fx end metavar +f metavar var fy end metavar modus ponens metavar var fx end metavar +f metavar var fy end metavar sameF metavar var fz end metavar conclude metavar var fu end metavar sameF metavar var fz end metavar cut lemma sameFtransitivity modus ponens metavar var fu end metavar sameF metavar var fz end metavar modus ponens metavar var fz end metavar sameF metavar var fv end metavar conclude metavar var fu end metavar sameF metavar var fv end metavar cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed all metavar var fv end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed all metavar var fv end metavar indeed metavar var fx end metavar +f metavar var fy end metavar sameF metavar var fz end metavar infer metavar var fu end metavar in0 R( metavar var fx end metavar +f metavar var fy end metavar ) infer metavar var fv end metavar in0 R( metavar var fz end metavar ) infer metavar var fu end metavar sameF metavar var fv end metavar conclude metavar var fx end metavar +f metavar var fy end metavar sameF metavar var fz end metavar imply metavar var fu end metavar in0 R( metavar var fx end metavar +f metavar var fy end metavar ) imply metavar var fv end metavar in0 R( metavar var fz end metavar ) imply metavar var fu end metavar sameF metavar var fv end metavar cut metavar var fx end metavar +f metavar var fy end metavar sameF metavar var fz end metavar infer 1rule mp modus ponens metavar var fx end metavar +f metavar var fy end metavar sameF metavar var fz end metavar imply metavar var fu end metavar in0 R( metavar var fx end metavar +f metavar var fy end metavar ) imply metavar var fv end metavar in0 R( metavar var fz end metavar ) imply metavar var fu end metavar sameF metavar var fv end metavar modus ponens metavar var fx end metavar +f metavar var fy end metavar sameF metavar var fz end metavar conclude metavar var fu end metavar in0 R( metavar var fx end metavar +f metavar var fy end metavar ) imply metavar var fv end metavar in0 R( metavar var fz end metavar ) imply metavar var fu end metavar sameF metavar var fv end metavar cut 1rule to==XX modus ponens metavar var fu end metavar in0 R( metavar var fx end metavar +f metavar var fy end metavar ) imply metavar var fv end metavar in0 R( metavar var fz end metavar ) imply metavar var fu end metavar sameF metavar var fv end metavar conclude R( metavar var fx end metavar +f metavar var fy end metavar ) == R( metavar var fz end metavar ) cut axiom plusR conclude R( var fx +f var fy ) == R( metavar var fx end metavar +f metavar var fy end metavar ) cut lemma ==Transitivity modus ponens R( var fx +f var fy ) == R( metavar var fx end metavar +f metavar var fy end metavar ) modus ponens R( metavar var fx end metavar +f metavar var fy end metavar ) == R( metavar var fz end metavar ) conclude R( var fx +f var fy ) == R( metavar var fz end metavar ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma f2R(Times) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed metavar var fx end metavar *f metavar var fy end metavar sameF metavar var fz end metavar infer R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) == R( metavar var fz end metavar ) end define end math ] "
" [ math define proof of lemma f2R(Times) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed metavar var fx end metavar *f metavar var fy end metavar sameF metavar var fz end metavar infer 1rule to== modus ponens metavar var fx end metavar *f metavar var fy end metavar sameF metavar var fz end metavar conclude R( metavar var fx end metavar *f metavar var fy end metavar ) == R( metavar var fz end metavar ) cut axiom timesR conclude R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar ) cut lemma ==Transitivity modus ponens R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar ) modus ponens R( metavar var fx end metavar *f metavar var fy end metavar ) == R( metavar var fz end metavar ) conclude R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) == R( metavar var fz end metavar ) end quote state proof state cache var c end expand end define end math ] "
(*** R-AFDELINGEN ***)
" [ math define statement of lemma plusR(Sym) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed R( metavar var fx end metavar +f metavar var fy end metavar ) == R( var fx +f var fy ) end define end math ] "
" [ math define proof of lemma plusR(Sym) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed axiom plusR conclude R( var fx +f var fy ) == R( metavar var fx end metavar +f metavar var fy end metavar ) cut lemma ==Symmetry modus ponens R( var fx +f var fy ) == R( metavar var fx end metavar +f metavar var fy end metavar ) conclude R( metavar var fx end metavar +f metavar var fy end metavar ) == R( var fx +f var fy ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma timesR(Sym) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed R( metavar var fx end metavar *f metavar var fy end metavar ) == R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) end define end math ] "
" [ math define proof of lemma timesR(Sym) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed axiom timesR conclude R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar ) cut lemma ==Symmetry modus ponens R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar ) conclude R( metavar var fx end metavar *f metavar var fy end metavar ) == R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) end quote state proof state cache var c end expand end define end math ] "
(*** LEQ-R ***)
" [ math define statement of lemma eqLeq(R) as system Q infer all metavar var rx end metavar indeed all metavar var ry end metavar indeed metavar var rx end metavar == metavar var ry end metavar infer not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar end define end math ] "
" [ math define proof of lemma eqLeq(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var rx end metavar indeed all metavar var ry end metavar indeed metavar var rx end metavar == metavar var ry end metavar infer prop lemma weaken or first modus ponens metavar var rx end metavar == metavar var ry end metavar conclude not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar cut 1rule repetition modus ponens not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar conclude not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma thirdGeqSeries as system Q infer all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed all metavar var rx end metavar indeed all metavar var ry end metavar indeed all metavar var rz end metavar indeed all metavar var ru end metavar indeed metavar var rx end metavar << metavar var ry end metavar infer metavar var rz end metavar << metavar var ru end metavar infer metavar var fx end metavar in0 metavar var rx end metavar infer metavar var fy end metavar in0 metavar var ry end metavar infer metavar var fz end metavar in0 metavar var rz end metavar infer metavar var fu end metavar in0 metavar var ru end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer existential var var c end var <= metavar var m end metavar infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply not0 [ metavar var fz end metavar ; metavar var m end metavar ] <= [ metavar var fu end metavar ; metavar var m end metavar ] + - metavar var ep end metavar end define end math ] "
" [ math define proof of lemma thirdGeqSeries as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var fu end metavar indeed all metavar var rx end metavar indeed all metavar var ry end metavar indeed all metavar var rz end metavar indeed all metavar var ru end metavar indeed metavar var rx end metavar << metavar var ry end metavar infer metavar var rz end metavar << metavar var ru end metavar infer metavar var fx end metavar in0 metavar var rx end metavar infer metavar var fy end metavar in0 metavar var ry end metavar infer metavar var fz end metavar in0 metavar var rz end metavar infer metavar var fu end metavar in0 metavar var ru end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer existential var var c end var <= metavar var m end metavar infer 1rule from<
(*** LESS-R ***)
XX lidt grimt med de ekstra variable
" [ math define statement of lemma subLessLeft(R) as system Q infer all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var rx end metavar indeed all metavar var ry end metavar indeed all metavar var rz end metavar indeed metavar var rx end metavar == metavar var ry end metavar infer metavar var rx end metavar << metavar var rz end metavar infer metavar var ry end metavar << metavar var rz end metavar end define end math ] "
" [ math define proof of lemma subLessLeft(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var rx end metavar indeed all metavar var ry end metavar indeed all metavar var rz end metavar indeed metavar var rx end metavar == metavar var ry end metavar infer metavar var rx end metavar << metavar var rz end metavar infer metavar var fy end metavar in0 metavar var ry end metavar infer metavar var fz end metavar in0 metavar var rz end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer lemma set equality nec condition(2) modus ponens metavar var rx end metavar == metavar var ry end metavar modus ponens metavar var fy end metavar in0 metavar var ry end metavar conclude metavar var fy end metavar in0 metavar var rx end metavar cut 1rule from<
" [ math define statement of lemma subLessRight(R) as system Q infer all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var rx end metavar indeed all metavar var ry end metavar indeed all metavar var rz end metavar indeed metavar var rx end metavar == metavar var ry end metavar infer metavar var rz end metavar << metavar var rx end metavar infer metavar var rz end metavar << metavar var ry end metavar end define end math ] "
" [ math define proof of lemma subLessRight(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var ep end metavar indeed all metavar var m end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed all metavar var rx end metavar indeed all metavar var ry end metavar indeed all metavar var rz end metavar indeed metavar var rx end metavar == metavar var ry end metavar infer metavar var rz end metavar << metavar var rx end metavar infer metavar var fz end metavar in0 metavar var rz end metavar infer metavar var fy end metavar in0 metavar var ry end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer lemma set equality nec condition(2) modus ponens metavar var rx end metavar == metavar var ry end metavar conclude metavar var fy end metavar in0 metavar var rx end metavar cut 1rule from<
" [ math define statement of lemma lessLeq(R) as system Q infer all metavar var rx end metavar indeed all metavar var ry end metavar indeed metavar var rx end metavar << metavar var ry end metavar infer not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar end define end math ] "
" [ math define proof of lemma lessLeq(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var rx end metavar indeed all metavar var ry end metavar indeed metavar var rx end metavar << metavar var ry end metavar infer prop lemma weaken or second modus ponens metavar var rx end metavar << metavar var ry end metavar conclude not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar cut 1rule repetition modus ponens not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar conclude not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma <
" [ math define proof of lemma <
(*** NUMMER 1 ***)
" [ math define statement of lemma <<==Reflexivity as system Q infer all metavar var rx end metavar indeed not0 metavar var rx end metavar << metavar var rx end metavar imply metavar var rx end metavar == metavar var rx end metavar end define end math ] "
" [ math define proof of lemma <<==Reflexivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var rx end metavar indeed lemma ==Reflexivity conclude metavar var rx end metavar == metavar var rx end metavar cut lemma eqLeq(R) modus ponens metavar var rx end metavar == metavar var rx end metavar conclude not0 metavar var rx end metavar << metavar var rx end metavar imply metavar var rx end metavar == metavar var rx end metavar end quote state proof state cache var c end expand end define end math ] "
(*** NUMMER 2 ***)
" [ math define statement of lemma <<==AntisymmetryHelper(Q) as system Q infer all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar infer metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar infer metavar var a end metavar end define end math ] "
" [ math define proof of lemma <<==AntisymmetryHelper(Q) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar infer metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar infer lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar cut axiom plusAssociativity conclude metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar cut axiom plusCommutativity conclude - metavar var z end metavar + metavar var z end metavar = metavar var z end metavar + - metavar var z end metavar cut lemma eqAdditionLeft modus ponens - metavar var z end metavar + metavar var z end metavar = metavar var z end metavar + - metavar var z end metavar conclude metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma x=x+(y-y) conclude metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar modus ponens metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar conclude metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar cut lemma leqTransitivity modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var x end metavar + - metavar var z end metavar cut lemma leqSubtractionLeft modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var x end metavar + - metavar var z end metavar conclude metavar var z end metavar <= - metavar var z end metavar cut lemma toNotLess modus ponens metavar var z end metavar <= - metavar var z end metavar conclude not0 not0 - metavar var z end metavar <= metavar var z end metavar imply not0 not0 - metavar var z end metavar = metavar var z end metavar cut lemma negativeLessPositive modus ponens not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar conclude not0 - metavar var z end metavar <= metavar var z end metavar imply not0 not0 - metavar var z end metavar = metavar var z end metavar cut prop lemma from contradiction modus ponens not0 - metavar var z end metavar <= metavar var z end metavar imply not0 not0 - metavar var z end metavar = metavar var z end metavar modus ponens not0 not0 - metavar var z end metavar <= metavar var z end metavar imply not0 not0 - metavar var z end metavar = metavar var z end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma <<==Antisymmetry as system Q infer all metavar var rx end metavar indeed all metavar var ry end metavar indeed not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar infer not0 metavar var ry end metavar << metavar var rx end metavar imply metavar var ry end metavar == metavar var rx end metavar infer metavar var rx end metavar == metavar var ry end metavar end define end math ] "
" [ math define proof of lemma <<==Antisymmetry as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var rx end metavar indeed all metavar var ry end metavar indeed not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar infer not0 metavar var ry end metavar << metavar var rx end metavar imply metavar var ry end metavar == metavar var rx end metavar infer 1rule repetition modus ponens not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar conclude not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar cut 1rule repetition modus ponens not0 metavar var ry end metavar << metavar var rx end metavar imply metavar var ry end metavar == metavar var rx end metavar conclude not0 metavar var ry end metavar << metavar var rx end metavar imply metavar var ry end metavar == metavar var rx end metavar cut prop lemma expand disjuncts modus ponens not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar modus ponens not0 metavar var ry end metavar << metavar var rx end metavar imply metavar var ry end metavar == metavar var rx end metavar conclude not0 metavar var rx end metavar == metavar var ry end metavar imply not0 metavar var ry end metavar == metavar var rx end metavar imply not0 metavar var rx end metavar << metavar var ry end metavar imply not0 metavar var ry end metavar << metavar var rx end metavar cut prop lemma auto imply conclude metavar var rx end metavar == metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar cut all metavar var rx end metavar indeed all metavar var ry end metavar indeed metavar var ry end metavar == metavar var rx end metavar infer lemma ==Symmetry modus ponens metavar var ry end metavar == metavar var rx end metavar conclude metavar var rx end metavar == metavar var ry end metavar cut 1rule deduction modus ponens all metavar var rx end metavar indeed all metavar var ry end metavar indeed metavar var ry end metavar == metavar var rx end metavar infer metavar var rx end metavar == metavar var ry end metavar conclude metavar var ry end metavar == metavar var rx end metavar imply metavar var rx end metavar == metavar var ry end metavar cut all metavar var rx end metavar indeed all metavar var ry end metavar indeed not0 metavar var rx end metavar << metavar var ry end metavar imply not0 metavar var ry end metavar << metavar var rx end metavar infer prop lemma first conjunct modus ponens not0 metavar var rx end metavar << metavar var ry end metavar imply not0 metavar var ry end metavar << metavar var rx end metavar conclude metavar var rx end metavar << metavar var ry end metavar cut 1rule from<
(*** NUMMER 3 ***)
" [ math define statement of lemma <
" [ math define proof of lemma <
" [ math define statement of lemma <<==Transitivity as system Q infer all metavar var rx end metavar indeed all metavar var ry end metavar indeed all metavar var rz end metavar indeed not0 metavar var rx end metavar << metavar var ry end metavar imply metavar var rx end metavar == metavar var ry end metavar infer not0 metavar var ry end metavar << metavar var rz end metavar imply metavar var ry end metavar == metavar var rz end metavar infer not0 metavar var rx end metavar << metavar var rz end metavar imply metavar var rx end metavar == metavar var rz end metavar end define end math ] "
" [ math define proof of lemma <<==Transitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var rx end metavar indeed all metavar var ry end metavar indeed all metavar var rz end metavar indeed metavar var rx end metavar << metavar var ry end metavar infer metavar var ry end metavar << metavar var rz end metavar infer lemma <
(*** NUMMER 5 ***)
" [ math define statement of lemma ==Addition as system Q infer all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed R( metavar var fx end metavar ) == R( metavar var fy end metavar ) infer R( var fx +f var fy ) == R( var fx +f var fy ) end define end math ] "
" [ math define proof of lemma ==Addition as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed R( metavar var fx end metavar ) == R( metavar var fy end metavar ) infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer existential var var c end var <= metavar var m end metavar infer 1rule from== modus ponens R( metavar var fx end metavar ) == R( metavar var fy end metavar ) conclude metavar var fx end metavar sameF metavar var fy end metavar cut 1rule fromSameF modus ponens metavar var fx end metavar sameF metavar var fy end metavar modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar conclude existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut 1rule mp modus ponens existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar modus ponens existential var var c end var <= metavar var m end metavar conclude not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut lemma insertMiddleTerm(Difference) conclude [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] cut axiom plusF conclude [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] cut axiom plusF conclude [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqNegated modus ponens [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] conclude - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = - [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma addEquations modus ponens [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = - [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqSymmetry modus ponens [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqTransitivity modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] cut lemma sameNumerical modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] conclude if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) cut lemma subLessLeft modus ponens if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) modus ponens not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed R( metavar var fx end metavar ) == R( metavar var fy end metavar ) infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer existential var var c end var <= metavar var m end metavar infer not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude R( metavar var fx end metavar ) == R( metavar var fy end metavar ) imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut R( metavar var fx end metavar ) == R( metavar var fy end metavar ) infer 1rule mp modus ponens R( metavar var fx end metavar ) == R( metavar var fy end metavar ) imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar modus ponens R( metavar var fx end metavar ) == R( metavar var fy end metavar ) conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar cut 1rule toSameF modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply existential var var c end var <= metavar var m end metavar imply not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) <= metavar var ep end metavar imply not0 not0 if( 0 <= [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] , - [ metavar var fx end metavar +f metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] ) = metavar var ep end metavar conclude metavar var fx end metavar +f metavar var fz end metavar sameF metavar var fy end metavar +f metavar var fz end metavar cut 1rule to== modus ponens metavar var fx end metavar +f metavar var fz end metavar sameF metavar var fy end metavar +f metavar var fz end metavar conclude R( metavar var fx end metavar +f metavar var fz end metavar ) == R( metavar var fy end metavar +f metavar var fz end metavar ) cut axiom plusR conclude R( var fx +f var fy ) == R( metavar var fx end metavar +f metavar var fz end metavar ) cut axiom plusR conclude R( var fx +f var fy ) == R( metavar var fy end metavar +f metavar var fz end metavar ) cut lemma ==Symmetry modus ponens R( var fx +f var fy ) == R( metavar var fy end metavar +f metavar var fz end metavar ) conclude R( metavar var fy end metavar +f metavar var fz end metavar ) == R( var fx +f var fy ) cut lemma ==Transitivity modus ponens R( var fx +f var fy ) == R( metavar var fx end metavar +f metavar var fz end metavar ) modus ponens R( metavar var fx end metavar +f metavar var fz end metavar ) == R( metavar var fy end metavar +f metavar var fz end metavar ) conclude R( var fx +f var fy ) == R( metavar var fy end metavar +f metavar var fz end metavar ) cut lemma ==Transitivity modus ponens R( var fx +f var fy ) == R( metavar var fy end metavar +f metavar var fz end metavar ) modus ponens R( metavar var fy end metavar +f metavar var fz end metavar ) == R( var fx +f var fy ) conclude R( var fx +f var fy ) == R( var fx +f var fy ) end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma ==AdditionLeft as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed R( metavar var fx end metavar ) == R( metavar var fy end metavar ) infer R( var fx +f var fy ) == R( var fx +f var fy ) end define end math ] "
" [ math define proof of lemma ==AdditionLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed R( metavar var fx end metavar ) == R( metavar var fy end metavar ) infer lemma ==Addition modus ponens R( metavar var fx end metavar ) == R( metavar var fy end metavar ) conclude R( var fx +f var fy ) == R( var fx +f var fy ) cut lemma plusCommutativity(R) conclude R( var fx +f var fy ) == R( var fx +f var fy ) cut lemma plusCommutativity(R) conclude R( var fx +f var fy ) == R( var fx +f var fy ) cut lemma ==Transitivity modus ponens R( var fx +f var fy ) == R( var fx +f var fy ) modus ponens R( var fx +f var fy ) == R( var fx +f var fy ) conclude R( var fx +f var fy ) == R( var fx +f var fy ) cut lemma ==Transitivity modus ponens R( var fx +f var fy ) == R( var fx +f var fy ) modus ponens R( var fx +f var fy ) == R( var fx +f var fy ) conclude R( var fx +f var fy ) == R( var fx +f var fy ) end quote state proof state cache var c end expand end define end math ] "
(***)
" [ math define statement of lemma <
" [ math define proof of lemma <
(***)
" [ math define statement of lemma <<==Addition as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 R( metavar var fx end metavar ) << R( metavar var fy end metavar ) imply R( metavar var fx end metavar ) == R( metavar var fy end metavar ) infer not0 R( var fx +f var fy ) << R( var fx +f var fy ) imply R( var fx +f var fy ) == R( var fx +f var fy ) end define end math ] "
" [ math define proof of lemma <<==Addition as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed R( metavar var fx end metavar ) << R( metavar var fy end metavar ) infer lemma <
(*** NUMMER 7 ***)
" [ math define statement of lemma plusAssociativity(F) as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar =f metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar end define end math ] "
" [ math define proof of lemma plusAssociativity(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed axiom plusF conclude [ metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] cut axiom plusF conclude [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma eqAddition modus ponens [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] cut axiom plusAssociativity conclude [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma plusF(Sym) conclude [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqAdditionLeft modus ponens [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] cut lemma plusF(Sym) conclude [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqTransitivity6 modus ponens [ metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] cut 1rule to=f modus ponens [ metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] conclude metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar =f metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma plusAssociativity(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed R( metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ) == R( metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ) end define end math ] "
" [ math define proof of lemma plusAssociativity(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed lemma plusAssociativity(F) conclude metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar =f metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar cut lemma =f to sameF modus ponens metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar =f metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar conclude metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar sameF metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar cut lemma f2R(Plus) modus ponens metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar sameF metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar conclude R( var fx +f var fy ) == R( metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ) cut lemma plusR(Sym) conclude R( metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ) == R( var fx +f var fy ) cut lemma ==Transitivity modus ponens R( metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ) == R( var fx +f var fy ) modus ponens R( var fx +f var fy ) == R( metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ) conclude R( metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ) == R( metavar var fx end metavar +f metavar var fy end metavar +f metavar var fz end metavar ) end quote state proof state cache var c end expand end define end math ] "
(*** NUMMER 8 ***)
" [ math define statement of lemma plus00 as system Q infer all metavar var fx end metavar indeed R( var fx +f var fy ) == R( metavar var fx end metavar ) end define end math ] "
" [ math define proof of lemma plus00 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed lemma plus0f conclude metavar var fx end metavar +f 0f =f metavar var fx end metavar cut lemma =f to sameF modus ponens metavar var fx end metavar +f 0f =f metavar var fx end metavar conclude metavar var fx end metavar +f 0f sameF metavar var fx end metavar cut lemma f2R(Plus) modus ponens metavar var fx end metavar +f 0f sameF metavar var fx end metavar conclude R( var fx +f var fy ) == R( metavar var fx end metavar ) cut 1rule repetition modus ponens R( var fx +f var fy ) == R( metavar var fx end metavar ) conclude R( var fx +f var fy ) == R( metavar var fx end metavar ) end quote state proof state cache var c end expand end define end math ] "
(*** NUMMER 9 ***)
" [ math define statement of lemma negative(R) as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed R( var fx +f var fy ) == R( 0f ) end define end math ] "
" [ math define proof of lemma negative(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed axiom plusF conclude [ metavar var fx end metavar +f -f metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ -f metavar var fx end metavar ; metavar var m end metavar ] cut axiom minusF conclude [ -f metavar var fx end metavar ; metavar var m end metavar ] = - [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqAdditionLeft modus ponens [ -f metavar var fx end metavar ; metavar var m end metavar ] = - [ metavar var fx end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] + [ -f metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] cut axiom negative conclude [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut axiom 0f conclude [ 0f ; metavar var m end metavar ] = 0 cut lemma eqSymmetry modus ponens [ 0f ; metavar var m end metavar ] = 0 conclude 0 = [ 0f ; metavar var m end metavar ] cut lemma eqTransitivity5 modus ponens [ metavar var fx end metavar +f -f metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ -f metavar var fx end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + [ -f metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = 0 modus ponens 0 = [ 0f ; metavar var m end metavar ] conclude [ metavar var fx end metavar +f -f metavar var fx end metavar ; metavar var m end metavar ] = [ 0f ; metavar var m end metavar ] cut 1rule to=f modus ponens [ metavar var fx end metavar +f -f metavar var fx end metavar ; metavar var m end metavar ] = [ 0f ; metavar var m end metavar ] conclude metavar var fx end metavar +f -f metavar var fx end metavar =f 0f cut lemma =f to sameF modus ponens metavar var fx end metavar +f -f metavar var fx end metavar =f 0f conclude metavar var fx end metavar +f -f metavar var fx end metavar sameF 0f cut lemma f2R(Plus) modus ponens metavar var fx end metavar +f -f metavar var fx end metavar sameF 0f conclude R( var fx +f var fy ) == R( 0f ) cut 1rule repetition modus ponens R( var fx +f var fy ) == R( 0f ) conclude R( var fx +f var fy ) == R( 0f ) end quote state proof state cache var c end expand end define end math ] "
(*** NUMMER 10 ***)
" [ math define statement of lemma plusCommutativity(F) as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar +f metavar var fy end metavar =f metavar var fy end metavar +f metavar var fx end metavar end define end math ] "
" [ math define proof of lemma plusCommutativity(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed axiom plusF conclude [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] cut axiom plusCommutativity conclude [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma plusF(Sym) conclude [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar +f metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqTransitivity4 modus ponens [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar +f metavar var fx end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar +f metavar var fx end metavar ; metavar var m end metavar ] cut 1rule to=f modus ponens [ metavar var fx end metavar +f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar +f metavar var fx end metavar ; metavar var m end metavar ] conclude metavar var fx end metavar +f metavar var fy end metavar =f metavar var fy end metavar +f metavar var fx end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma plusCommutativity(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed R( var fx +f var fy ) == R( var fx +f var fy ) end define end math ] "
" [ math define proof of lemma plusCommutativity(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed lemma plusCommutativity(F) conclude metavar var fx end metavar +f metavar var fy end metavar =f metavar var fy end metavar +f metavar var fx end metavar cut lemma =f to sameF modus ponens metavar var fx end metavar +f metavar var fy end metavar =f metavar var fy end metavar +f metavar var fx end metavar conclude metavar var fx end metavar +f metavar var fy end metavar sameF metavar var fy end metavar +f metavar var fx end metavar cut lemma f2R(Plus) modus ponens metavar var fx end metavar +f metavar var fy end metavar sameF metavar var fy end metavar +f metavar var fx end metavar conclude R( var fx +f var fy ) == R( metavar var fy end metavar +f metavar var fx end metavar ) cut lemma plusR(Sym) conclude R( metavar var fy end metavar +f metavar var fx end metavar ) == R( var fx +f var fy ) cut lemma ==Transitivity modus ponens R( var fx +f var fy ) == R( metavar var fy end metavar +f metavar var fx end metavar ) modus ponens R( metavar var fy end metavar +f metavar var fx end metavar ) == R( var fx +f var fy ) conclude R( var fx +f var fy ) == R( var fx +f var fy ) end quote state proof state cache var c end expand end define end math ] "
(*** NUMMER 11 ***)
" [ math define statement of lemma timesAssociativity(F) as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar =f metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar end define end math ] "
" [ math define proof of lemma timesAssociativity(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed axiom timesF conclude [ metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut axiom timesF conclude [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma eqMultiplication modus ponens [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut axiom timesAssociativity conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma timesF(Sym) conclude [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqMultiplicationLeft modus ponens [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] cut lemma timesF(Sym) conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqTransitivity6 modus ponens [ metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] cut 1rule to=f modus ponens [ metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ; metavar var m end metavar ] conclude metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar =f metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma timesAssociativity(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed R( metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ) end define end math ] "
" [ math define proof of lemma timesAssociativity(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed lemma timesAssociativity(F) conclude metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar =f metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar cut lemma =f to sameF modus ponens metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar =f metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar conclude metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar sameF metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar cut lemma f2R(Times) modus ponens metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar sameF metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar conclude R( metavar var fx end metavar *f metavar var fy end metavar ) ** R( metavar var fz end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ) cut lemma timesR(Sym) conclude R( metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar ) ** R( metavar var fz end metavar ) cut lemma ==Transitivity modus ponens R( metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar ) ** R( metavar var fz end metavar ) modus ponens R( metavar var fx end metavar *f metavar var fy end metavar ) ** R( metavar var fz end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ) conclude R( metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar *f metavar var fz end metavar ) end quote state proof state cache var c end expand end define end math ] "
(*** NUMMER 12 ***)
" [ math define statement of lemma times1f as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed metavar var fx end metavar *f 1f =f metavar var fx end metavar end define end math ] "
" [ math define proof of lemma times1f as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed axiom timesF conclude [ metavar var fx end metavar *f 1f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ 1f ; metavar var m end metavar ] cut axiom 1f conclude [ 1f ; metavar var m end metavar ] = 1 cut lemma eqMultiplicationLeft modus ponens [ 1f ; metavar var m end metavar ] = 1 conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ 1f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * 1 cut axiom times1 conclude [ metavar var fx end metavar ; metavar var m end metavar ] * 1 = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqTransitivity4 modus ponens [ metavar var fx end metavar *f 1f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ 1f ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ 1f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * 1 modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * 1 = [ metavar var fx end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar *f 1f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut 1rule to=f modus ponens [ metavar var fx end metavar *f 1f ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] conclude metavar var fx end metavar *f 1f =f metavar var fx end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma times01 as system Q infer all metavar var fx end metavar indeed R( metavar var fx end metavar ) ** R( 1f ) == R( metavar var fx end metavar ) end define end math ] "
" [ math define proof of lemma times01 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed lemma times1f conclude metavar var fx end metavar *f 1f =f metavar var fx end metavar cut lemma =f to sameF modus ponens metavar var fx end metavar *f 1f =f metavar var fx end metavar conclude metavar var fx end metavar *f 1f sameF metavar var fx end metavar cut lemma f2R(Times) modus ponens metavar var fx end metavar *f 1f sameF metavar var fx end metavar conclude R( metavar var fx end metavar ) ** R( 1f ) == R( metavar var fx end metavar ) cut 1rule repetition modus ponens R( metavar var fx end metavar ) ** R( 1f ) == R( metavar var fx end metavar ) conclude R( metavar var fx end metavar ) ** R( 1f ) == R( metavar var fx end metavar ) end quote state proof state cache var c end expand end define end math ] "
(*** NUMMER 14 ***)
" [ math define statement of lemma timesCommutativity(F) as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar *f metavar var fy end metavar =f metavar var fy end metavar *f metavar var fx end metavar end define end math ] "
" [ math define proof of lemma timesCommutativity(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed axiom timesF conclude [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] cut axiom timesCommutativity conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma timesF(Sym) conclude [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar *f metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqTransitivity4 modus ponens [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar *f metavar var fx end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar *f metavar var fx end metavar ; metavar var m end metavar ] cut 1rule to=f modus ponens [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar *f metavar var fx end metavar ; metavar var m end metavar ] conclude metavar var fx end metavar *f metavar var fy end metavar =f metavar var fy end metavar *f metavar var fx end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma timesCommutativity(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) == R( metavar var fy end metavar ) ** R( metavar var fx end metavar ) end define end math ] "
" [ math define proof of lemma timesCommutativity(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed lemma timesCommutativity(F) conclude metavar var fx end metavar *f metavar var fy end metavar =f metavar var fy end metavar *f metavar var fx end metavar cut lemma =f to sameF modus ponens metavar var fx end metavar *f metavar var fy end metavar =f metavar var fy end metavar *f metavar var fx end metavar conclude metavar var fx end metavar *f metavar var fy end metavar sameF metavar var fy end metavar *f metavar var fx end metavar cut lemma f2R(Times) modus ponens metavar var fx end metavar *f metavar var fy end metavar sameF metavar var fy end metavar *f metavar var fx end metavar conclude R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) == R( metavar var fy end metavar *f metavar var fx end metavar ) cut lemma timesR(Sym) conclude R( metavar var fy end metavar *f metavar var fx end metavar ) == R( metavar var fy end metavar ) ** R( metavar var fx end metavar ) cut lemma ==Transitivity modus ponens R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) == R( metavar var fy end metavar *f metavar var fx end metavar ) modus ponens R( metavar var fy end metavar *f metavar var fx end metavar ) == R( metavar var fy end metavar ) ** R( metavar var fx end metavar ) conclude R( metavar var fx end metavar ) ** R( metavar var fy end metavar ) == R( metavar var fy end metavar ) ** R( metavar var fx end metavar ) end quote state proof state cache var c end expand end define end math ] "
(*** NUMMER 15 ***)
" [ math define statement of lemma distribution(F) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed metavar var fx end metavar *f metavar var fy end metavar +f metavar var fz end metavar =f metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar end define end math ] "
" [ math define proof of lemma distribution(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed axiom timesF conclude [ metavar var fx end metavar *f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] cut axiom plusF conclude [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqMultiplicationLeft modus ponens [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] cut axiom distribution conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma timesF(Sym) conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] cut lemma timesF(Sym) conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fz end metavar ; metavar var m end metavar ] cut lemma addEquations modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fz end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar *f metavar var fz end metavar ; metavar var m end metavar ] cut lemma plusF(Sym) conclude [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar *f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqTransitivity6 modus ponens [ metavar var fx end metavar *f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar *f metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar *f metavar var fy end metavar ; metavar var m end metavar ] + [ metavar var fx end metavar *f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar *f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar ; metavar var m end metavar ] cut 1rule to=f modus ponens [ metavar var fx end metavar *f metavar var fy end metavar +f metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar ; metavar var m end metavar ] conclude metavar var fx end metavar *f metavar var fy end metavar +f metavar var fz end metavar =f metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar end quote state proof state cache var c end expand end define end math ] "
" [ math define statement of lemma distribution(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed R( metavar var fx end metavar ) ** R( metavar var fy end metavar +f metavar var fz end metavar ) == R( var fx +f var fy ) end define end math ] "
" [ math define proof of lemma distribution(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed lemma distribution(F) conclude metavar var fx end metavar *f metavar var fy end metavar +f metavar var fz end metavar =f metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar cut lemma =f to sameF modus ponens metavar var fx end metavar *f metavar var fy end metavar +f metavar var fz end metavar =f metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar conclude metavar var fx end metavar *f metavar var fy end metavar +f metavar var fz end metavar sameF metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar cut lemma f2R(Times) modus ponens metavar var fx end metavar *f metavar var fy end metavar +f metavar var fz end metavar sameF metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar conclude R( metavar var fx end metavar ) ** R( metavar var fy end metavar +f metavar var fz end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar ) cut lemma plusR(Sym) conclude R( metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar ) == R( var fx +f var fy ) cut lemma ==Transitivity modus ponens R( metavar var fx end metavar ) ** R( metavar var fy end metavar +f metavar var fz end metavar ) == R( metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar ) modus ponens R( metavar var fx end metavar *f metavar var fy end metavar +f metavar var fx end metavar *f metavar var fz end metavar ) == R( var fx +f var fy ) conclude R( metavar var fx end metavar ) ** R( metavar var fy end metavar +f metavar var fz end metavar ) == R( var fx +f var fy ) end quote state proof state cache var c end expand end define end math ] "
(*** tex-definitioner ***)
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\item " [ math define tex of var ep as text "(\epsilon)" end text end define end math ] "
\item " [ math define tex of var fx as text "(fx)" end text end define end math ] "
\item " [ math define tex of var fy as text "(fy)" end text end define end math ] "
\item " [ math define tex of var fz as text "(fz)" end text end define end math ] "
\item " [ math define tex of var fu as text "(fu)" end text end define end math ] "
\item " [ math define tex of var fu as text "(fv)" end text end define end math ] "
\item " [ math define tex of var rx as text "(rx)" end text end define end math ] "
\item " [ math define tex of var ry as text "(ry)" end text end define end math ] "
\item " [ math define tex of var rz as text "(rz)" end text end define end math ] "
\item " [ math define tex of var ru as text "(ru)" end text end define end math ] "
\item " [ math define tex of meta ep as text "\epsilon" end text end define end math ] "
\item " [ math define tex of meta fx as text "FX" end text end define end math ] "
\item " [ math define tex of meta fy as text "FY" end text end define end math ] "
\item " [ math define tex of meta fz as text "FZ" end text end define end math ] "
\item " [ math define tex of meta fu as text "FU" end text end define end math ] "
\item " [ math define tex of meta fv as text "FV" end text end define end math ] "
\item " [ math define tex of meta rx as text "RX" end text end define end math ] "
\item " [ math define tex of meta ry as text "RY" end text end define end math ] "
\item " [ math define tex of meta rz as text "RZ" end text end define end math ] "
\item " [ math define tex of meta ru as text "RU" end text end define end math ] "
\item " [ math define tex of ex3 as text "Ex3" end text end define end math ] "
\item " [ math define tex of 0 as text "0" end text end define end math ] "
\item " [ math define tex of 1 as text "1" end text end define end math ] "
\item " [ math define tex of (-1) as text "(-1)" end text end define end math ] "
\item " [ math define tex of 2 as text "2" end text end define end math ] "
\item " [ math define tex of 1/2 as text "1/2" end text end define end math ] "
\item " [ math define tex of 0f as text "0f" end text end define end math ] "
\item " [ math define tex of 00 as text "00" end text end define end math ] "
\item " [ math define tex of var x = var y as text "#1.
= #2." end text end define end math ] "
\item " [ math define tex of var x != var y as text "#1.
\neq #2." end text end define end math ] "
\item " [ math define tex of var x < var y as text "#1.
< #2." end text end define end math ] "
\item " [ math define tex of var x <= var y as text "#1.
<= #2." end text end define end math ] "
\item " [ math define tex of var x =f var y as text "#1.
=_{f}#2." end text end define end math ] "
\item " [ math define tex of var x
\item " [ math define tex of var x sameF var y as text "SF(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of var x == var y as text "#1.
== #2." end text end define end math ] "
\item " [ math define tex of var x << var y as text "#1.
<< #2." end text end define end math ] "
\item " [ math define tex of var x <<== var y as text "#1.
<<== #2." end text end define end math ] "
\item " [ math define tex of [ var x ; var y ] as text "#1.
[#2.
]" end text end define end math ] "
\item " [ math define tex of - var x as text "-#1." end text end define end math ] "
\item " [ math define tex of -f var x as text "-_{f}#1." end text end define end math ] "
\item " [ math define tex of var x + var y as text "#1.
+#2." end text end define end math ] "
\item " [ math define tex of var x - var y as text "#1.
-#2." end text end define end math ] "
\item " [ math define tex of var fx +f var fy as text "#1.
+_{f}#2." end text end define end math ] "
\item " [ math define tex of var fx -f var fy as text "#1.
-_{f}#2." end text end define end math ] "
\item " [ math define tex of var fx *f var fy as text "#1.
*_{f}#2." end text end define end math ] "
\item " [ math define tex of var x ++ var y as text "#1.
++#2." end text end define end math ] "
\item " [ math define tex of R( var fx ) -- R( var fy ) as text "R(#1.
) -- R(#2.
)" end text end define end math ] "
\item " [ math define tex of var x * var y as text "#1.
*#2." end text end define end math ] "
\item " [ math define tex of var x ** var y as text "#1.
**#2." end text end define end math ] "
\item " [ math define tex of axiom leqReflexivity as text "leqReflexivity" end text end define end math ] "
\item " [ math define tex of 1/ var x as text "rec#1." end text end define end math ] "
\item " [ math define tex of | var x | as text "|#1.|" end text end define end math ] "
\item " [ math define tex of if( var x , var y , var z ) as text "if(#1.
,#2.
,#3.
)" end text end define end math ] "
\item " [ math define tex of R( var x ) as text "R(#1.
)" end text end define end math ] "
\item " [ math define tex of --R( var x ) as text "--R(#1.
)" end text end define end math ] "
\item " [ math define tex of axiom leqAntisymmetry as text "leqAntisymmetryAxiom" end text end define end math ] "
\item " [ math define tex of axiom leqTransitivity as text "leqTransitivityAxiom" end text end define end math ] "
\item " [ math define tex of axiom leqTotality as text "leqTotality" end text end define end math ] "
\item " [ math define tex of axiom leqAddition as text "leqAdditionAxiom" end text end define end math ] "
\item " [ math define tex of axiom leqMultiplication as text "leqMultiplicationAxiom" end text end define end math ] "
\item " [ math define tex of axiom plusAssociativity as text "plusAssociativity" end text end define end math ] "
\item " [ math define tex of axiom plusCommutativity as text "plusCommutativity" end text end define end math ] "
\item " [ math define tex of axiom negative as text "Negative" end text end define end math ] "
\item " [ math define tex of axiom plus0 as text "plus0" end text end define end math ] "
\item " [ math define tex of axiom timesAssociativity as text "timesAssociativity" end text end define end math ] "
\item " [ math define tex of axiom timesCommutativity as text "timesCommutativity" end text end define end math ] "
\item " [ math define tex of axiom reciprocal as text "ReciprocalAxiom" end text end define end math ] "
\item " [ math define tex of axiom times1 as text "times1" end text end define end math ] "
\item " [ math define tex of axiom plusAssociativity as text "plusAssociativity" end text end define end math ] "
\item " [ math define tex of axiom plusCommutativity as text "plusCommutativity" end text end define end math ] "
\item " [ math define tex of axiom negative as text "Negative" end text end define end math ] "
\item " [ math define tex of axiom distribution as text "Distribution" end text end define end math ] "
\item " [ math define tex of axiom 0not1 as text "0not1" end text end define end math ] "
\item " [ math define tex of axiom equality as text "equalityAxiom" end text end define end math ] "
\item " [ math define tex of axiom eqLeq as text "eqLeqAxiom" end text end define end math ] "
\item " [ math define tex of axiom eqAddition as text "eqAdditionAxiom" end text end define end math ] "
\item " [ math define tex of axiom eqMultiplication as text "eqMultiplicationAxiom" end text end define end math ] "
\item " [ math define tex of lemma set equality nec condition(1) as text "SENC1" end text end define end math ] "
\item " [ math define tex of lemma set equality nec condition(2) as text "SENC2" end text end define end math ] "
\item " [ math define tex of 1rule ifThenElse true as text "IfThenElse(T)" end text end define end math ] "
\item " [ math define tex of 1rule ifThenElse false as text "IfThenElse(F)" end text end define end math ] "
\item " [ math define tex of 1rule from=f as text "From=f" end text end define end math ] "
\item " [ math define tex of 1rule to=f as text "To=f" end text end define end math ] "
\item " [ math define tex of 1rule from
\item " [ math define tex of 1rule to
\item " [ math define tex of axiom plusF as text "PlusF" end text end define end math ] "
\item " [ math define tex of axiom minusF as text "MinusF" end text end define end math ] "
\item " [ math define tex of axiom timesF as text "TimesF" end text end define end math ] "
\item " [ math define tex of axiom 0f as text "0f" end text end define end math ] "
\item " [ math define tex of axiom 1f as text "1f" end text end define end math ] "
\item " [ math define tex of 1rule fromSameF as text "FromSF" end text end define end math ] "
\item " [ math define tex of 1rule toSameF as text "ToSF" end text end define end math ] "
\item " [ math define tex of 1rule to==XX as text "To==XX" end text end define end math ] "
\item " [ math define tex of 1rule from== as text "From==" end text end define end math ] "
\item " [ math define tex of 1rule to== as text "To==" end text end define end math ] "
\item " [ math define tex of 1rule from<
\item " [ math define tex of 1rule from<
\item " [ math define tex of 1rule from<
\item " [ math define tex of 1rule to<
\item " [ math define tex of 1rule from<< as text "From<<" end text end define end math ] "
\item " [ math define tex of 1rule to<< as text "To<<" end text end define end math ] "
\item " [ math define tex of 1rule fromInR as text "FromInR" end text end define end math ] "
\item " [ math define tex of axiom plusR as text "PlusR" end text end define end math ] "
\item " [ math define tex of axiom timesR as text "TimesR" end text end define end math ] "
\item " [ math define tex of lemma leqAntisymmetry as text "leqAntisymmetry" end text end define end math ] "
\item " [ math define tex of lemma leqTransitivity as text "leqTransitivity" end text end define end math ] "
\item " [ math define tex of lemma leqAddition as text "leqAddition" end text end define end math ] "
\item " [ math define tex of lemma reciprocal as text "Reciprocal" end text end define end math ] "
\item " [ math define tex of lemma equality as text "Equality" end text end define end math ] "
\item " [ math define tex of lemma eqLeq as text "eqLeq" end text end define end math ] "
\item " [ math define tex of lemma eqAddition as text "eqAddition" end text end define end math ] "
\item " [ math define tex of lemma eqMultiplication as text "eqMultiplication" end text end define end math ] "
\item " [ math define tex of prop lemma to negated imply as text "ToNegatedImply" end text end define end math ] "
\item " [ math define tex of prop lemma tertium non datur as text "TND" end text end define end math ] "
\item " [ math define tex of prop lemma imply negation as text "ImplyNegation" end text end define end math ] "
\item " [ math define tex of prop lemma from negations as text "FromNegations" end text end define end math ] "
\item " [ math define tex of prop lemma from three disjuncts as text "From3Disjuncts" end text end define end math ] "
\item " [ math define tex of prop lemma negate first disjunct as text "NegateDisjunct1" end text end define end math ] "
\item " [ math define tex of prop lemma negate second disjunct as text "NegateDisjunct2" end text end define end math ] "
\item " [ math define tex of prop lemma expand disjuncts as text "ExpandDisjuncts" end text end define end math ] "
\item " [ math define tex of prop lemma from two times two disjuncts as text "From2*2Disjuncts" end text end define end math ] "
\item " [ math define tex of lemma eqReflexivity as text "eqReflexivity" end text end define end math ] "
\item " [ math define tex of lemma eqSymmetry as text "eqSymmetry" end text end define end math ] "
\item " [ math define tex of lemma eqTransitivity as text "eqTransitivity" end text end define end math ] "
\item " [ math define tex of lemma eqTransitivity4 as text "eqTransitivity4" end text end define end math ] "
\item " [ math define tex of lemma eqTransitivity5 as text "eqTransitivity5" end text end define end math ] "
\item " [ math define tex of lemma eqTransitivity6 as text "eqTransitivity6" end text end define end math ] "
\item " [ math define tex of lemma plus0Left as text "plus0Left" end text end define end math ] "
\item " [ math define tex of lemma times1Left as text "times1Left" end text end define end math ] "
\item " [ math define tex of lemma eqMultiplicationLeft as text "EqMultiplicationLeft" end text end define end math ] "
\item " [ math define tex of lemma distributionOut as text "DistributionOut" end text end define end math ] "
\item " [ math define tex of lemma three2twoTerms as text "Three2twoTerms" end text end define end math ] "
\item " [ math define tex of lemma three2threeTerms as text "Three2threeTerms" end text end define end math ] "
\item " [ math define tex of lemma three2twoFactors as text "Three2threeFactors" end text end define end math ] "
\item " [ math define tex of lemma addEquations as text "AddEquations" end text end define end math ] "
\item " [ math define tex of lemma subtractEquations as text "SubtractEquations" end text end define end math ] "
\item " [ math define tex of lemma subtractEquationsLeft as text "SubtractEquationsLeft" end text end define end math ] "
\item " [ math define tex of lemma eqNegated as text "EqNegated" end text end define end math ] "
\item " [ math define tex of lemma positiveToRight(Eq) as text "PositiveToRight(Eq)" end text end define end math ] "
\item " [ math define tex of lemma positiveToLeft(Eq)(1 term) as text "PositiveToLeft(Eq)(1 term)" end text end define end math ] "
\item " [ math define tex of lemma negativeToLeft(Eq) as text "NegativeToLeft(Eq)" end text end define end math ] "
\item " [ math define tex of lemma uniqueNegative as text "UniqueNegative" end text end define end math ] "
\item " [ math define tex of lemma doubleMinus as text "DoubleMinus" end text end define end math ] "
\item " [ math define tex of lemma lessNeq as text "LessNeq" end text end define end math ] "
\item " [ math define tex of lemma neqSymmetry as text "NeqSymmetry" end text end define end math ] "
\item " [ math define tex of lemma neqNegated as text "NeqNegated" end text end define end math ] "
\item " [ math define tex of lemma subNeqRight as text "SubNeqRight" end text end define end math ] "
\item " [ math define tex of lemma subNeqLeft as text "SubNeqLeft" end text end define end math ] "
\item " [ math define tex of lemma neqAddition as text "NeqAddition" end text end define end math ] "
\item " [ math define tex of lemma neqMultiplication as text "NeqMultiplication" end text end define end math ] "
\item " [ math define tex of lemma leqLessEq as text "LeqLessEq" end text end define end math ] "
\item " [ math define tex of lemma lessLeq as text "LessLeq" end text end define end math ] "
\item " [ math define tex of lemma from leqGeq as text "FromLeqGeq" end text end define end math ] "
\item " [ math define tex of lemma subLeqRight as text "subLeqRight" end text end define end math ] "
\item " [ math define tex of lemma subLeqLeft as text "subLeqLeft" end text end define end math ] "
\item " [ math define tex of lemma leqPlus1 as text "Leq+1" end text end define end math ] "
\item " [ math define tex of lemma positiveToRight(Leq) as text "PositiveToRight(Leq)" end text end define end math ] "
\item " [ math define tex of lemma positiveToRight(Leq)(1 term) as text "PositiveToRight(Leq)(1 term)" end text end define end math ] "
\item " [ math define tex of lemma leqAdditionLeft as text "LeqAdditionLeft" end text end define end math ] "
\item " [ math define tex of lemma leqSubtraction as text "leqSubtraction" end text end define end math ] "
\item " [ math define tex of lemma leqSubtractionLeft as text "leqSubtractionLeft" end text end define end math ] "
\item " [ math define tex of lemma leqMultiplication as text "leqMultiplication" end text end define end math ] "
\item " [ math define tex of lemma thirdGeq as text "thirdGeq" end text end define end math ] "
\item " [ math define tex of lemma leqNegated as text "LeqNegated" end text end define end math ] "
\item " [ math define tex of lemma addEquations(Leq) as text "AddEquations(Leq)" end text end define end math ] "
\item " [ math define tex of lemma leqNeqLess as text "LeqNeqLess" end text end define end math ] "
\item " [ math define tex of lemma fromLess as text "FromLess" end text end define end math ] "
\item " [ math define tex of lemma toLess as text "ToLess" end text end define end math ] "
\item " [ math define tex of lemma fromNotLess as text "fromNotLess" end text end define end math ] "
\item " [ math define tex of lemma toNotLess as text "toNotLess" end text end define end math ] "
\item " [ math define tex of lemma lessAddition as text "LessAddition" end text end define end math ] "
\item " [ math define tex of lemma lessAdditionLeft as text "LessAdditionLeft" end text end define end math ] "
\item " [ math define tex of lemma lessMultiplication as text "LessMultiplication" end text end define end math ] "
\item " [ math define tex of lemma lessMultiplicationLeft as text "LessMultiplicationLeft" end text end define end math ] "
\item " [ math define tex of lemma lessDivision as text "LessDivision" end text end define end math ] "
\item " [ math define tex of lemma addEquations(Less) as text "AddEquations(Less)" end text end define end math ] "
\item " [ math define tex of lemma negativeLessPositive as text "NegativeLessPositive" end text end define end math ] "
\item " [ math define tex of lemma leqLessTransitivity as text "leqLessTransitivity" end text end define end math ] "
\item " [ math define tex of lemma lessLeqTransitivity as text "LessLeqTransitivity" end text end define end math ] "
\item " [ math define tex of lemma lessTransitivity as text "LessTransitivity" end text end define end math ] "
\item " [ math define tex of lemma lessTotality as text "LessTotality" end text end define end math ] "
\item " [ math define tex of lemma subLessRight as text "SubLessRight" end text end define end math ] "
\item " [ math define tex of lemma subLessLeft as text "SubLessLeft" end text end define end math ] "
\item " [ math define tex of lemma lessNegated as text "LessNegated" end text end define end math ] "
\item " [ math define tex of lemma positiveNegated as text "PositiveNegated" end text end define end math ] "
\item " [ math define tex of lemma nonpositiveNegated as text "NonpositiveNegated" end text end define end math ] "
\item " [ math define tex of lemma negativeNegated as text "NegativeNegated" end text end define end math ] "
\item " [ math define tex of lemma nonnegativeNegated as text "NonnegativeNegated" end text end define end math ] "
\item " [ math define tex of lemma positiveHalved as text "PositiveHalved" end text end define end math ] "
\item " [ math define tex of lemma nonnegativeNumerical as text "NonnegativeNumerical" end text end define end math ] "
\item " [ math define tex of lemma negativeNumerical as text "NegativeNumerical" end text end define end math ] "
\item " [ math define tex of lemma positiveNumerical as text "PositiveNumerical" end text end define end math ] "
\item " [ math define tex of lemma |0|=0 as text "|0|=0" end text end define end math ] "
\item " [ math define tex of lemma 0<=|x| as text "0<=|x|" end text end define end math ] "
\item " [ math define tex of lemma sameNumerical as text "SameNumerical" end text end define end math ] "
\item " [ math define tex of lemma signNumerical(+) as text "SignNumerical(+)" end text end define end math ] "
\item " [ math define tex of lemma signNumerical as text "SignNumerical" end text end define end math ] "
\item " [ math define tex of lemma numericalDifference as text "NumericalDifference" end text end define end math ] "
\item " [ math define tex of lemma splitNumericalSumHelper as text "SplitNumericalSumHelper" end text end define end math ] "
\item " [ math define tex of lemma splitNumericalSum(++) as text "splitNumericalSum(++)" end text end define end math ] "
\item " [ math define tex of lemma splitNumericalSum(--) as text "splitNumericalSum(--)" end text end define end math ] "
\item " [ math define tex of lemma splitNumericalSum(+-, smallNegative) as text "splitNumericalSum(+-small)" end text end define end math ] "
\item " [ math define tex of lemma splitNumericalSum(+-, bigNegative) as text "splitNumericalSum(+-big)" end text end define end math ] "
\item " [ math define tex of lemma splitNumericalSum(+-) as text "splitNumericalSum(+-)" end text end define end math ] "
\item " [ math define tex of lemma splitNumericalSum(-+) as text "splitNumericalSum(-+)" end text end define end math ] "
\item " [ math define tex of lemma splitNumericalSum as text "splitNumericalSum" end text end define end math ] "
\item " [ math define tex of lemma insertMiddleTerm(Numerical) as text "insertMiddleTerm(Numerical)" end text end define end math ] "
\item " [ math define tex of lemma x+y=zBackwards as text "x+y=zBackwards" end text end define end math ] "
\item " [ math define tex of lemma x*y=zBackwards as text "x*y=zBackwards" end text end define end math ] "
\item " [ math define tex of lemma x=x+(y-y) as text "x=x+(y-y)" end text end define end math ] "
\item " [ math define tex of lemma x=x+y-y as text "x=x+y-y" end text end define end math ] "
\item " [ math define tex of lemma x=x*y*(1/y) as text " " end text end define end math ] "
\item " [ math define tex of lemma insertMiddleTerm(Sum) as text "insertMiddleTerm(Sum)" end text end define end math ] "
\item " [ math define tex of lemma insertMiddleTerm(Difference) as text "insertMiddleTerm(Difference)" end text end define end math ] "
\item " [ math define tex of lemma x*0+x=x as text "x*0+x=x" end text end define end math ] "
\item " [ math define tex of lemma x*0=0 as text "x*0=0" end text end define end math ] "
\item " [ math define tex of lemma (-1)*(-1)+(-1)*1=0 as text "(-1)*(-1)+(-1)*1=0" end text end define end math ] "
\item " [ math define tex of lemma (-1)*(-1)=1 as text "(-1)*(-1)=1" end text end define end math ] "
\item " [ math define tex of lemma 0<1Helper as text "0<1Helper" end text end define end math ] "
\item " [ math define tex of lemma 0<1 as text "0<1" end text end define end math ] "
\item " [ math define tex of lemma 0<2 as text "0<2" end text end define end math ] "
\item " [ math define tex of lemma 0<1/2 as text "0<1/2" end text end define end math ] "
\item " [ math define tex of lemma x+x=2*x as text "TwoWholes" end text end define end math ] "
\item " [ math define tex of lemma (1/2)x+(1/2)x=x as text "TwoHalves" end text end define end math ] "
\item " [ math define tex of lemma -x-y=-(x+y) as text "-x-y=-(x+y)" end text end define end math ] "
\item " [ math define tex of lemma minusNegated as text "MinusNegated" end text end define end math ] "
\item " [ math define tex of lemma times(-1) as text "Times(-1)" end text end define end math ] "
\item " [ math define tex of lemma times(-1)Left as text "Times(-1)Left" end text end define end math ] "
\item " [ math define tex of lemma -0=0 as text "-0=0" end text end define end math ] "
\item " [ math define tex of lemma negativeToLeft(Leq) as text "negativeToLeft(Leq)" end text end define end math ] "
\item " [ math define tex of lemma sameFsymmetry as text "SFsymmetry" end text end define end math ] "
\item " [ math define tex of lemma sameFtransitivity as text "SFtransitivity" end text end define end math ] "
\item " [ math define tex of lemma =f to sameF as text "=fToSameF " end text end define end math ] "
\item " [ math define tex of lemma plusF(Sym) as text "PlusF(Sym)" end text end define end math ] "
\item " [ math define tex of lemma timesF(Sym) as text "TimesF(Sym)" end text end define end math ] "
\item " [ math define tex of lemma f2R(Plus) as text "f2R(Plus)" end text end define end math ] "
\item " [ math define tex of lemma f2R(Times) as text "f2R(Times)" end text end define end math ] "
\item " [ math define tex of lemma plusR(Sym) as text "PlusR(Sym)" end text end define end math ] "
\item " [ math define tex of lemma timesR(Sym) as text "TimesR(Sym)" end text end define end math ] "
\item " [ math define tex of lemma lessLeq(R) as text "LessLeq(R)" end text end define end math ] "
\item " [ math define tex of lemma eqLeq(R) as text "eqLeq(R)" end text end define end math ] "
\item " [ math define tex of lemma thirdGeqSeries as text "ThirdGeqSeries" end text end define end math ] "
\item " [ math define tex of lemma subLessRight(R) as text "SubLessRight(R)" end text end define end math ] "
\item " [ math define tex of lemma subLessLeft(R) as text "SubLessLeft(R)" end text end define end math ] "
\item " [ math define tex of lemma <
\item " [ math define tex of lemma <
\item " [ math define tex of lemma <<==Reflexivity as text "<<==Reflexivity" end text end define end math ] "
\item " [ math define tex of lemma <<==AntisymmetryHelper(Q) as text "<<==AntisymmetryHelper(Q)" end text end define end math ] "
\item " [ math define tex of lemma <<==Antisymmetry as text "<<==Antisymmetry" end text end define end math ] "
\item " [ math define tex of lemma <<==Transitivity as text "<<==Transitivity" end text end define end math ] "
\item " [ math define tex of lemma plus0f as text "Plus0f" end text end define end math ] "
\item " [ math define tex of lemma plus00 as text "Plus00" end text end define end math ] "
\item " [ math define tex of lemma ==Addition as text "==Addition" end text end define end math ] "
\item " [ math define tex of lemma ==AdditionLeft as text "==AdditionLeft" end text end define end math ] "
\item " [ math define tex of lemma <
\item " [ math define tex of lemma <<==Addition as text "<<==Addition" end text end define end math ] "
\item " [ math define tex of lemma plusAssociativity(F) as text "PlusAssociativity(F)" end text end define end math ] "
\item " [ math define tex of lemma plusAssociativity(R) as text "PlusAssociativity(R)" end text end define end math ] "
\item " [ math define tex of lemma negative(R) as text "Negative(R)" end text end define end math ] "
\item " [ math define tex of lemma plusCommutativity(F) as text "PlusCommutativity(F)" end text end define end math ] "
\item " [ math define tex of lemma plusCommutativity(R) as text "PlusCommutativity(R)" end text end define end math ] "
\item " [ math define tex of lemma times1f as text "Times1f" end text end define end math ] "
\item " [ math define tex of lemma times01 as text "Times01" end text end define end math ] "
\item " [ math define tex of lemma timesAssociativity(F) as text "TimesAssociativity(F)" end text end define end math ] "
\item " [ math define tex of lemma timesAssociativity(R) as text "TimesAssociativity(R)" end text end define end math ] "
\item " [ math define tex of lemma timesCommutativity(F) as text "TimesCommutativity(F)" end text end define end math ] "
\item " [ math define tex of lemma timesCommutativity(R) as text "TimesCommutativity(R)" end text end define end math ] "
\item " [ math define tex of lemma distribution(F) as text "Distribution(F)" end text end define end math ] "
\item " [ math define tex of lemma distribution(R) as text "Distribution(R)" end text end define end math ] "
\end{list}
\end{document}
End of file
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The pyk compiler, version 0.grue.20060417+ by Klaus Grue,