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\begin {document}
\title {The Logiweb sequent calculus}
\author {Klaus Grue\thanks{Department of Computer Science, University of Copenhagen (DIKU)}}
\date {\today}
\maketitle
\tableofcontents
" [ ragged right ] "
\begin{abstract}
The `Logiweb sequent calculus' is suited for terse, human readable formulation of axioms, inference rules, theories, lemmas, and proofs. It supports different styles of theories like equational theories and FOL based theories. It permits to express arbitrary side conditions in the `Logiweb programming language'. As an example of use, the calculus allows to express the deduction rule as an inference rule which makes it easy to get started using a theory. The calculus has operations for `dereferencing' and `referencing' which allow to convert e.g.\ the name of a lemma into the contents of the lemma and vice versa. The calculus interacts smoothly with fully automatic tactics as well as proofs in which part of the work is done by a human author and part is done by proof tactics.
\end{abstract}
\section{Introduction}\label{sec:Introduction}
Logiweb \cite{grue04,grue05,base} is a system for electronic publication of logic. It allows authors different places in the world to define theories, state and prove lemmas, and to publish {\em Logiweb pages} that contain the results. The present paper is an example of a Logiweb page, and the present paper is correct in the sense that it has been verified by Logiweb.
Core Logiweb allows users to program proof systems in the {\em Logiweb programming language} and to publish the proof system. Users who use the same proof system and the same axiomatic theory may benefit from the results of each other in that a proof written by one author may make references across the internet to lemmas proved by another author. Cross theory cooperation (e.g.\ use of ZFC results in NBG) could be more cumbersome and cross proof system cooperation even more so, depending on the theories and systems involved.
The present paper introduces {\em Logiweb sequent calculus} which allows users to express arbitrary axiomatic theories on the same footing. That calculus supports arbitrary styles of logic (e.g.\ FOL or equational) equally well. The calculus is machine friendly in that proofs are easy to generate, verify, store, and transmit and it is general so that users of Logiweb are not tempted to make a proof system each. User friendliness in the sense that theories, lemmas, and proofs should be easy to read and write is taken care of by Logiwebs rendering and Turing complete macro expansion facilities combined with the support for proof tactics in the implementation of the calculus in \cite{base}.
The present paper gives an overview. Details are in a web appendix \cite{appendix}.
\section{Logiweb sequent calculus}
Let " [ math metavar var v end metavar end math ] " and " [ math metavar var t end metavar end math ] " denote the syntax classes of metavariables and object terms, respectively. The format " [ math metavar var s end metavar end math ] " of Logiweb statements is " [ math metavar var s end metavar [ "::=" ] metavar var v end metavar alternative metavar var t end metavar alternative metavar var s end metavar infer metavar var s end metavar alternative metavar var t end metavar endorse metavar var s end metavar alternative all metavar var v end metavar indeed metavar var s end metavar alternative absurdity alternative metavar var s end metavar rule plus metavar var s end metavar end math ] ". As an example of use, consider the following:
\[ " [ array left , center , left , left is "Let" tab sequent example axiom tab "denote" tab all metavar var a end metavar indeed metavar var a end metavar plus zero equal metavar var a end metavar , "." safe row "Let" tab sequent example rule tab "denote" tab all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar equal metavar var b end metavar infer metavar var a end metavar equal metavar var c end metavar infer metavar var b end metavar equal metavar var c end metavar , "." safe row "Let" tab sequent example contradiction tab "denote" tab zero equal one infer absurdity , "." safe row "Let" tab sequent example theory tab "denote" tab sequent example rule rule plus sequent example contradiction rule plus sequent example axiom , "." safe row "Let" tab sequent example lemma tab "denote" tab sequent example theory infer all metavar var a end metavar indeed metavar var a end metavar equal metavar var a end metavar , "." end array ] " \]
\noindent The construct " [ math all metavar var x end metavar indeed metavar var a end metavar end math ] " states that " [ math metavar var a end metavar end math ] " is provable for all object terms " [ math metavar var x end metavar end math ] ". Hence, " [ math sequent example axiom end math ] " above is an axiom scheme which says that " [ math metavar var a end metavar plus zero equal metavar var a end metavar end math ] " for all object terms " [ math metavar var a end metavar end math ] ".
The construct " [ math metavar var a end metavar infer metavar var b end metavar end math ] " states that if " [ math metavar var a end metavar end math ] " is provable then " [ math metavar var b end metavar end math ] " is provable. " [ math metavar var a end metavar infer metavar var b end metavar end math ] " is right associative and has higher priority than " [ math all metavar var x end metavar indeed metavar var a end metavar end math ] ", so " [ math sequent example rule end math ] " above means " [ math 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 equal metavar var b end metavar infer metavar var a end metavar equal metavar var c end metavar infer metavar var b end metavar equal metavar var c end metavar end math ] ". Hence, " [ math sequent example rule end math ] " is the inference rule which states that " [ math metavar var a end metavar equal metavar var b end metavar end math ] " and " [ math metavar var a end metavar equal metavar var c end metavar end math ] " infer " [ math metavar var b end metavar equal metavar var c end metavar end math ] ". The macro that expands " [ math all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var d end metavar end math ] " into " [ math all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var d end metavar end math ] " is defined in \cite{appendix}.
The construct " [ math absurdity end math ] " denotes absurdity, i.e. meta-falsehood (we reserve " [ math false end math ] " and " [ math bottom end math ] " to denote falsehood and infinite looping, respectively, at the object level). Hence, " [ math sequent example contradiction end math ] " above states that " [ math zero equal one end math ] " is an absurdity.
The construct " [ math metavar var a end metavar rule plus metavar var b end metavar end math ] " states that both " [ math metavar var a end metavar end math ] " and " [ math metavar var b end metavar end math ] " are provable. Hence, " [ math sequent example theory end math ] " above is the axiomatic theory which comprises the axiom scheme " [ math sequent example axiom end math ] ", the inference rule " [ math sequent example rule end math ] ", and the statement that " [ math zero equal one end math ] " is an absurdity.
The lemma " [ math sequent example lemma end math ] " above says that " [ math metavar var a end metavar equal metavar var a end metavar end math ] " is provable in the theory " [ math sequent example theory end math ] ".
The construct " [ math metavar var a end metavar endorse metavar var b end metavar end math ] " states that if computation of " [ math metavar var a end metavar end math ] " yields " [ math true end math ] " then " [ math metavar var b end metavar end math ] " is provable. " [ math metavar var a end metavar end math ] " must be expressed in the Logiweb programming language (c.f.\ \cite{base}). " [ math true end math ] " denotes truth in the Logiweb programming language. If " [ math metavar var a end metavar endorse metavar var b end metavar end math ] " then we shall say that " [ math metavar var a end metavar end math ] " {\em endorses} " [ math metavar var b end metavar end math ] ". The endorsement operator allows to express side conditions as we shall see later.
A {\em Logiweb sequent} is a triple " [ math var p pair var s pair var c pair true end math ] " where " [ math var c end math ] " is a Logiweb statement and " [ math var p end math ] " and " [ math var s end math ] " are finite sets of Logiweb statements. The sequent " [ math set var p sub one end sub comma "\ldots" comma var p sub var m end sub end set pair set var s sub one end sub comma "\ldots" comma var s sub var n end sub end set pair var c pair true end math ] " represents " [ math var p sub one end sub infer "\ldots" infer var p sub var m end sub infer var s sub one end sub endorse "\ldots" endorse var s sub var n end sub endorse var c end math ] ". The operations of Logiweb sequent calculus are:
\[" [ array right , left , left , left is var a init tab evaluates to tab the empty set comma the empty set comma var a infer var a pair true safe row var a infer var p pair var s pair var c pair true tab evaluates to tab var p term minus var a end minus comma var s comma var a infer var c pair true safe row var a endorse var p pair var s pair var c pair true tab evaluates to tab var p comma var s term minus var a end minus comma var a endorse var c pair true safe row all var x indeed var p pair var s pair var c pair true tab evaluates to tab var p pair var s pair all var x indeed var c pair true , "\footnotemark" safe row var p comma var s comma var a infer var c pair true modus tab evaluates to tab var p term plus var a end plus pair var s pair var c pair true safe row var p comma var s comma var a endorse var c pair true modus tab evaluates to tab var p pair var s term plus var a end plus pair var c pair true safe row var p pair var s pair all var x indeed var c pair true at var a tab evaluates to tab var p pair var s pair substitute var c set var x to var a end substitute pair true , "\footnotemark" safe row var p comma var s comma var a endorse var b pair true verify tab evaluates to tab var p pair var s pair var b pair true , "\footnotemark" safe row var p comma var s comma var a infer var b infer var c pair true curry plus tab evaluates to tab var p comma var s comma var a rule plus var b infer var c pair true safe row var p comma var s comma var a rule plus var b infer var c pair true curry minus tab evaluates to tab var p comma var s comma var a infer var b infer var c pair true safe row var p pair var s pair var n pair true dereference tab evaluates to tab var p pair var s pair var c pair true , "\footnotemark" safe row var p pair var s pair var c pair true id est var n tab evaluates to tab var p pair var s pair var n pair true , "\addtocounter{footnote}{-1}\footnotemark" safe row var p sub one end sub pair var s sub one end sub pair var c sub one end sub pair true cut var p sub two end sub pair var s sub two end sub pair var c sub two end sub pair true tab evaluates to tab var p sub one end sub term union var p sub two end sub term minus var c sub one end sub end minus pair var s sub one end sub term union var s sub two end sub pair var c sub two end sub pair true end array ] "\]%
%
\footnotetext[1]{if " [ math var x end math ] " is not free in any member of " [ math var p end math ] " or " [ math var s end math ] "}%
%
\footnotetext[2]{if " [ math var a end math ] " is free for " [ math var x end math ] " in " [ math var c end math ] "}%
%
\footnotetext[3]{if computation of " [ math var a end math ] " yields " [ math true end math ] " in the Logiweb programming language}%
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\footnotetext[4]{if " [ math var n end math ] " is defined to denote " [ math var c end math ] "}
\noindent Evaluation of a sequent operation gives a sequent or an exception. Exceptional cases are omitted above. Now let " [ math sequent example axiom prime end math ] " denote " [ math sequent example theory infer all metavar var a end metavar indeed metavar var a end metavar plus zero equal metavar var a end metavar end math ] ". We have
\[ " [ array left , left , left is sequent example theory infer sequent example theory init modus dereference cut sequent example rule rule plus sequent example contradiction infer sequent example lemma init curry plus dereference id est sequent example axiom prime tab evaluates to safe row sequent example theory infer the empty set comma the empty set comma sequent example theory infer sequent example theory pair true modus dereference cut sequent example rule rule plus sequent example contradiction infer the empty set comma the empty set comma sequent example axiom infer sequent example axiom pair true curry plus dereference id est sequent example axiom prime tab evaluates to safe row sequent example theory infer set sequent example theory end set pair the empty set pair sequent example theory pair true dereference cut the empty set comma the empty set comma sequent example rule rule plus sequent example contradiction infer sequent example axiom infer sequent example axiom pair true curry plus dereference id est sequent example axiom prime tab evaluates to safe row sequent example theory infer set sequent example theory end set comma the empty set comma sequent example rule rule plus sequent example contradiction rule plus sequent example axiom pair true cut the empty set comma the empty set comma sequent example rule rule plus sequent example contradiction rule plus sequent example axiom infer sequent example axiom pair true dereference id est sequent example axiom prime tab evaluates to safe row sequent example theory infer set sequent example theory end set comma the empty set comma sequent example rule rule plus sequent example contradiction rule plus sequent example axiom pair true cut set sequent example rule rule plus sequent example contradiction rule plus sequent example axiom end set pair the empty set pair sequent example axiom pair true dereference id est sequent example axiom prime tab evaluates to safe row sequent example theory infer set sequent example theory end set pair the empty set pair sequent example lemma pair true dereference id est sequent example axiom prime , evaluates to , sequent example theory infer set sequent example theory end set pair the empty set pair all metavar var a end metavar indeed metavar var a end metavar plus zero equal metavar var a end metavar pair true id est sequent example axiom prime tab evaluates to safe row the empty set comma the empty set comma sequent example theory infer all metavar var a end metavar indeed metavar var a end metavar plus zero equal metavar var a end metavar pair true id est sequent example axiom prime , evaluates to , the empty set pair the empty set pair sequent example axiom prime pair true end array ] " \]
\noindent which proves " [ math sequent example axiom prime end math ] ".
The syntax of {\em Logiweb sequent terms} reads " [ math metavar var q end metavar [ "::=" ] metavar var s end metavar init alternative metavar var s end metavar infer metavar var q end metavar alternative metavar var s end metavar endorse metavar var q end metavar alternative all metavar var v end metavar indeed metavar var q end metavar alternative metavar var q end metavar modus alternative metavar var q end metavar at metavar var s end metavar alternative metavar var q end metavar verify alternative metavar var q end metavar curry plus alternative metavar var q end metavar curry minus alternative metavar var q end metavar id est metavar var s end metavar alternative metavar var q end metavar dereference alternative metavar var q end metavar cut metavar var q end metavar end math ] ". A sequent term " [ math var q end math ] " is said to {\em prove} the statement " [ math var c end math ] " if " [ math var q end math ] " evaluates to " [ math the empty set pair the empty set pair var c pair true end math ] " (\cite{base} allows " [ math var q end math ] " to evaluate to " [ math var p pair the empty set pair var c pair true end math ] " provided all elements of " [ math var p end math ] " are names of previously proved lemmas).
\section{Object theories}\label{section:ObjectTheories}
Logiweb sequent calculus is like assembly language: general and machine friendly, but it needs syntactic sugar. The user friendliness of the combined system of Logiweb, Logiweb sequent calculus, and syntactic sugar has been tested during a course in logic in which ten students wrote Logiweb pages that proved " [ math object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var end math ] " from the raw axioms of Mendelsons system " [ math system s end math ] " (Peano arithmetic) \cite{mendelson}. It was found that the students could prove commutativity, get the proof verified, and write a report on that in less than three weeks, starting with little experience in Logiweb, mechanical proof checking, and logic in general. For links to the reports, consult \url{http://yoa.dk/}.
While proving " [ math object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var end math ] " from the raw axioms may be a good student exercise, it was found that the deduction theorem and parallel instantiation of object variables was badly needed. Versions of theorems that used inference " [ math unicode end of text infer unicode end of text end math ] " and meta variables were easier to use than theorems using implication " [ math unicode end of text imply unicode end of text end math ] " and object variables, but the axiom of induction forced implication and object variables upon the students. Instantiation of object variables is cumbersome because axiom A4 in \cite{mendelson} only allows to instantiate one variable at a time whereas the students needed parallel instantiation, e.g.\ when swapping two object variables. And hypothetical reasoning either required manual unfolding of the deduction theorem or programming of a proof tactic that could cost a lot of CPU-time for large developments.
An {\em inference rule of deduction} would eliminate these problems. If, however, " [ math for all objects object var var x end var indeed for all objects object var var y end var indeed object var var x end var equal object var var y end var imply object var var y end var equal object var var x end var end math ] " is an object statement and " [ math all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar equal metavar var b end metavar infer metavar var b end metavar equal metavar var a end metavar end math ] " is a meta statement, then the deduction rule is a meta meta statement. We do not want to introduce a meta meta level above the Logiweb sequent calculus just to express deduction so we shall express deduction using the side condition machinery of the sequent calculus.
\subsection{Peano arithmetic}
A modified version of Mendelsons system " [ math system s end math ] " (Peano arithmetic) \cite{mendelson} may be formulated thus:
\begin{list}{}{
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\item \makebox[0.45\textwidth][l]{" [ math define statement of system s as all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc rule plus 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 rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar equal metavar var b end metavar infer metavar var a end metavar suc equal metavar var b end metavar suc rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar suc equal metavar var b end metavar suc infer metavar var a end metavar equal metavar var b end metavar rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed lambda var x dot deduction zero quote metavar var a end metavar end quote conclude quote metavar var b end metavar end quote end deduction endorse metavar var a end metavar infer metavar var b end metavar rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar times metavar var b end metavar suc equal metavar var a end metavar times metavar var b end metavar plus metavar var a end metavar rule plus all metavar var a end metavar indeed metavar var a end metavar plus zero equal metavar var a end metavar rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed not metavar var b end metavar imply not metavar var a end metavar infer not metavar var b end metavar imply metavar var a end metavar infer metavar var b end metavar rule plus 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 equal metavar var b end metavar infer metavar var a end metavar equal metavar var c end metavar infer metavar var b end metavar equal metavar var c end metavar rule plus all metavar var x end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed sub zero quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var x end metavar end quote is quote zero end quote end sub endorse sub zero quote metavar var c end metavar end quote is quote metavar var a end metavar end quote where quote metavar var x end metavar end quote is quote metavar var x end metavar suc end quote end sub endorse metavar var b end metavar infer metavar var a end metavar imply metavar var c end metavar infer metavar var a end metavar rule plus all metavar var a end metavar indeed not zero equal metavar var a end metavar suc rule plus 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 rule plus all metavar var a end metavar indeed metavar var a end metavar times zero equal zero end define end math ] "}" [ math define statement of rule mp as system s 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 rule mp as rule tactic end define end math ] "
\item \makebox[0.45\textwidth][l]{" [ math define statement of rule gen as system s 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 rule gen as rule tactic end define end math ] "}" [ math define statement of deduction as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed lambda var x dot deduction zero quote metavar var a end metavar end quote conclude quote metavar var b end metavar end quote end deduction endorse metavar var a end metavar infer metavar var b end metavar end define , define proof of deduction as rule tactic end define end math ] "
\item \makebox[0.45\textwidth][l]{}" [ math define statement of axiom s two as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar equal metavar var b end metavar infer metavar var a end metavar suc equal metavar var b end metavar suc end define , define proof of axiom s two as rule tactic end define end math ] "
\item \makebox[0.45\textwidth][l]{" [ math define statement of axiom s three as system s infer all metavar var a end metavar indeed not zero equal metavar var a end metavar suc end define , define proof of axiom s three as rule tactic end define end math ] "}" [ math define statement of axiom s four as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar suc equal metavar var b end metavar suc infer metavar var a end metavar equal metavar var b end metavar end define , define proof of axiom s four as rule tactic end define end math ] "
\item \makebox[0.45\textwidth][l]{" [ math define statement of axiom s five as system s infer all metavar var a end metavar indeed metavar var a end metavar plus zero equal metavar var a end metavar end define , define proof of axiom s five as rule tactic end define end math ] "}" [ math define statement of axiom s six as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc end define , define proof of axiom s six as rule tactic end define end math ] "
\item \makebox[0.45\textwidth][l]{" [ math define statement of axiom s seven as system s infer all metavar var a end metavar indeed metavar var a end metavar times zero equal zero end define , define proof of axiom s seven as rule tactic end define end math ] "}" [ math define statement of axiom s eight as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar times metavar var b end metavar suc equal metavar var a end metavar times metavar var b end metavar plus metavar var a end metavar end define , define proof of axiom s eight as rule tactic end define end math ] "
\item " [ math define statement of double negation as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed not metavar var b end metavar imply not metavar var a end metavar infer not metavar var b end metavar imply metavar var a end metavar infer metavar var b end metavar end define , define proof of double negation as rule tactic end define end math ] "
\item " [ math define statement of axiom s one as system s 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 equal metavar var b end metavar infer metavar var a end metavar equal metavar var c end metavar infer metavar var b end metavar equal metavar var c end metavar end define , define proof of axiom s one as rule tactic end define end math ] "
\item " [ math define statement of axiom s nine as system s infer all metavar var x end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed sub zero quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var x end metavar end quote is quote zero end quote end sub endorse sub zero quote metavar var c end metavar end quote is quote metavar var a end metavar end quote where quote metavar var x end metavar end quote is quote metavar var x end metavar suc end quote end sub endorse metavar var b end metavar infer metavar var a end metavar imply metavar var c end metavar infer metavar var a end metavar end define , define proof of axiom s nine as rule tactic end define end math ] "\footnote{" [ math sub zero 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 c end metavar end quote end sub end math ] " says `the object term " [ math metavar var b end metavar end math ] " where the object variable " [ math metavar var x end metavar end math ] " is replaced by the object term " [ math metavar var c end metavar end math ] " is alpha equivalent to the object term " [ math metavar var a end metavar end math ] ", c.f.\ \cite{appendix}}
\end{list}
\noindent As defined in \cite{base}, " [ math define statement of axiom s five as system s infer all metavar var a end metavar indeed metavar var a end metavar plus zero equal metavar var a end metavar end define , define proof of axiom s five as rule tactic end define end math ] " macro expands into a lemma and a proof where the lemma says " [ math system s infer all metavar var a end metavar indeed metavar var a end metavar plus zero equal metavar var a end metavar end math ] " and the proof is a proof of that lemma. " [ math define statement of system s as all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc rule plus 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 rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar equal metavar var b end metavar infer metavar var a end metavar suc equal metavar var b end metavar suc rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar suc equal metavar var b end metavar suc infer metavar var a end metavar equal metavar var b end metavar rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed lambda var x dot deduction zero quote metavar var a end metavar end quote conclude quote metavar var b end metavar end quote end deduction endorse metavar var a end metavar infer metavar var b end metavar rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar times metavar var b end metavar suc equal metavar var a end metavar times metavar var b end metavar plus metavar var a end metavar rule plus all metavar var a end metavar indeed metavar var a end metavar plus zero equal metavar var a end metavar rule plus all metavar var a end metavar indeed all metavar var b end metavar indeed not metavar var b end metavar imply not metavar var a end metavar infer not metavar var b end metavar imply metavar var a end metavar infer metavar var b end metavar rule plus 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 equal metavar var b end metavar infer metavar var a end metavar equal metavar var c end metavar infer metavar var b end metavar equal metavar var c end metavar rule plus all metavar var x end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed sub zero quote metavar var b end metavar end quote is quote metavar var a end metavar end quote where quote metavar var x end metavar end quote is quote zero end quote end sub endorse sub zero quote metavar var c end metavar end quote is quote metavar var a end metavar end quote where quote metavar var x end metavar end quote is quote metavar var x end metavar suc end quote end sub endorse metavar var b end metavar infer metavar var a end metavar imply metavar var c end metavar infer metavar var a end metavar rule plus all metavar var a end metavar indeed not zero equal metavar var a end metavar suc rule plus 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 rule plus all metavar var a end metavar indeed metavar var a end metavar times zero equal zero end define end math ] " macro expands into a definition which defines " [ math system s end math ] " as the conjunction of " [ math all metavar var a end metavar indeed metavar var a end metavar plus zero equal metavar var a end metavar end math ] " and all the other rules attributed to " [ math system s end math ] ". The " [ math define statement of meta x as var x end define end math ] " macro is somewhat complex since it has to scan the entire page to find all rules related to the theory being defined. The benefit of collecting rules from the entire page is that it gives authors the freedom to introduce axioms one by one.
The deduction rule " [ math deduction end math ] " is such that e.g.\ " [ math all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar equal metavar var b end metavar infer metavar var b end metavar equal metavar var a end metavar end math ] " allows to conclude " [ math metavar var a end metavar equal metavar var b end metavar imply metavar var b end metavar equal metavar var a end metavar end math ] " and " [ math object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var end math ] " allows to conclude " [ math metavar var a end metavar plus metavar var b end metavar equal metavar var b end metavar plus metavar var a end metavar end math ] " so " [ math deduction end math ] " implements both deduction and parallel instantiation. All complexity is hidden in the side condition which is defined in \cite{appendix}.
Having the deduction rule, axioms A1, A2, A4, and A5 in \cite{mendelson} become superfluous. For proofs of those axioms based on deduction, see \cite{appendix}.
As an example of a development, \cite{appendix} proves the following lemmas from \cite{mendelson} in system " [ math system s end math ] ":
\begin{list}{}{
\setlength{\leftmargin}{5em}
\setlength{\itemindent}{-5em}}
\item " [ math define statement of prop three two a as system s infer all metavar var a end metavar indeed metavar var a end metavar equal metavar var a end metavar end define end math ] "
\item " [ math define statement of prop three two b as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar equal metavar var b end metavar infer metavar var b end metavar equal metavar var a end metavar end define end math ] "
\item " [ math define statement of prop three two c as system s 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 equal metavar var b end metavar infer metavar var b end metavar equal metavar var c end metavar infer metavar var a end metavar equal metavar var c end metavar end define end math ] "
\item " [ math define statement of prop three two d as system s 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 equal metavar var c end metavar infer metavar var b end metavar equal metavar var c end metavar infer metavar var a end metavar equal metavar var b end metavar end define end math ] "
\item " [ math define statement of prop three two e as system s 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 equal metavar var b end metavar infer metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar end define end math ] "
\item " [ math define statement of prop three two f as system s infer all metavar var a end metavar indeed metavar var a end metavar equal zero plus metavar var a end metavar end define end math ] "
\item " [ math define statement of prop three two g as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar suc plus metavar var b end metavar equal metavar var a end metavar plus metavar var b end metavar suc end define end math ] "
\item " [ math define statement of prop three two h as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar plus metavar var b end metavar equal metavar var b end metavar plus metavar var a end metavar end define end math ] "
\item " [ math define statement of prop three two i as system s 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 equal metavar var b end metavar infer metavar var c end metavar plus metavar var a end metavar equal metavar var c end metavar plus metavar var b end metavar end define end math ] "
\end{list}
\noindent In the present paper we merely show the proof of an auxiliary lemma which proves the induction step of " [ math prop three two f end math ] ":
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define statement of prop three two f two as system s infer all metavar var a end metavar indeed metavar var a end metavar equal zero plus metavar var a end metavar imply metavar var a end metavar suc equal zero plus metavar var a end metavar suc end define end math ] "
\item " [ math define proof of prop three two f two as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed all metavar var a end metavar indeed metavar var a end metavar equal zero plus metavar var a end metavar infer axiom s two modus ponens metavar var a end metavar equal zero plus metavar var a end metavar conclude metavar var a end metavar suc equal zero plus metavar var a end metavar suc cut axiom s six conclude zero plus metavar var a end metavar suc equal zero plus metavar var a end metavar suc cut prop three two d modus ponens metavar var a end metavar suc equal zero plus metavar var a end metavar suc modus ponens zero plus metavar var a end metavar suc equal zero plus metavar var a end metavar suc conclude metavar var a end metavar suc equal zero plus metavar var a end metavar suc cut deduction modus ponens all metavar var a end metavar indeed metavar var a end metavar equal zero plus metavar var a end metavar infer metavar var a end metavar suc equal zero plus metavar var a end metavar suc conclude metavar var a end metavar equal zero plus metavar var a end metavar imply metavar var a end metavar suc equal zero plus metavar var a end metavar suc end quote state proof state cache var c end expand end define end math ] "
\end{list}
\noindent As defined in \cite{base}, " [ math all metavar var a end metavar indeed metavar var b end metavar end math ] " macro expands into " [ math all metavar var a end metavar indeed metavar var b end metavar end math ] ". " [ math make root visible axiom s two modus ponens "L04" conclude metavar var a end metavar suc equal zero plus metavar var a end metavar suc cut metavar var b end metavar end visible end math ] " macro expands into a local macro definition and a call to a proof tactic. The local macro definition defines " [ math "L05" end math ] " as shorthand for " [ math metavar var a end metavar suc equal zero plus metavar var a end metavar suc end math ] " and the proof tactic expands the line into " [ math axiom s two init modus dereference modus at metavar var a end metavar at zero plus metavar var a end metavar cut metavar var b end metavar end math ] " using unification. For more details see \cite{appendix}.
\section{Conclusion and further work}
Logiweb sequent calculus with examples from traditional Peano arithmetic has been presented. For further examples, click `Map Theory' at http://yoa.dk/. Map theory is an equational theory. In that theory, all definitions present on a lemmas home page become axioms according to an `axiom of definition' which is expressible by a side condition in the Logiweb sequent calculus.
The traditional deduction theorem requires that no application of Gen is made to variables free in the premise. The side condition of the deduction rule above is quite different. Proving the consistency of the deduction rule is work of the future but is expected to be straightforward since, in the standard model, a statement holds for all natural numbers if and only if it holds for all terms.
%------------------------------------------------------------------------------
\bibliographystyle{alpha}
\bibliography{./page}
%------------------------------------------------------------------------------
\everymath{}
\everydisplay{}
\end{document}
End of file
File page.bib
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End of file
latex page
bibtex page
latex page
latex page
dvipdfm page
File appendix.tex
\documentclass [fleqn]{article}
\everymath{\rm}
\usepackage{latexsym}
\setlength {\overfullrule }{0mm}
\input{lgwinclude}
\usepackage{url}
\usepackage[dvipdfm=true]{hyperref}
\hypersetup{pdfpagemode=none}
\hypersetup{pdfstartpage=1}
\hypersetup{pdfstartview=FitBH}
\hypersetup{pdfpagescrop={120 130 490 730}}
\hypersetup{pdftitle=Logiweb sequent calculus - Appendix}
\hypersetup{colorlinks=true}
\begin {document}
\title {Logiweb sequent calculus, Appendix}
\author {Klaus Grue}
\date {\today}
\maketitle
\tableofcontents
\section{Introduction}
The present appendix is part of the {\em Logiweb page} available at
\begin{raggedright}
\begin{sloppypar}
\noindent \url{\lgwCheckUrl body/tex/page.pdf}.
\end{sloppypar}
\end{raggedright}
The hex number which contains a RIPEMD-160 \cite{dobbertin96} hash key and a time stamp identifies the present paper uniquely on Logiweb. The part before the hex number is the address of a {\em Logiweb relay} which redirects the users browser to the paper. That part can be replaced by the address of any Logiweb relay. The part after the hex number indicates the address of the body of the paper relative to the official, binary representation (2/ is shorthand for ../../).
Actually, the present page is rendered as three pdf files present at \href{\lgwCheckUrl body/tex/page.pdf}{2\linebreak [0]/\linebreak [0]body\linebreak [0]/\linebreak [0]tex\linebreak [0]/\linebreak [0]page.pdf}, \href{\lgwCheckUrl body/tex/appendix.pdf}{2\linebreak [0]/\linebreak [0]body\linebreak [0]/\linebreak [0]tex\linebreak [0]/\linebreak [0]appendix.pdf}, and \href{\lgwCheckUrl body/tex/chores.pdf}{2\linebreak [0]/\linebreak [0]body\linebreak [0]/\linebreak [0]tex\linebreak [0]/\linebreak [0]chores.pdf} where appendix.pdf is the present appendix. Many proofs and definitions are deferred to the present appendix to spare the reader of the main paper in the page.pdf file. The chores.pdf file defines how to render each construct in \TeX. A good place to start browsing is \href{\lgwCheckUrl index.html}{2\linebreak [0]/\linebreak [0]index.html}.
Each Logiweb page has a Logiweb bibliography which lists previously published pages which the present page relies on. The present Logiweb page references the \href{\lgwBaseUrl index.html}{base} page at
\begin{raggedright}
\begin{sloppypar}
\noindent \url{\lgwBaseUrl body/tex/page.pdf}.
\end{sloppypar}
\end{raggedright}
\noindent which defines the proof system used to verify the present page (c.f. \href{\lgwCheckUrl bibliography/tex/page.pdf}{2\linebreak [0]/\linebreak [0]bibliography\linebreak [0]/\linebreak [0]tex\linebreak [0]/\linebreak [0]page.pdf}).
As mentioned in the main paper, the user friendliness of the Logiweb sequent calculus is achieved by using the Turing complete macro expansion facility of Logiweb. To see the proofs present in the main paper after macro expansion, consult \href{\lgwCheckUrl expansion/tex/page.pdf}{2\linebreak [0]/\linebreak [0]expansion\linebreak [0]/\linebreak [0]tex\linebreak [0]/\linebreak [0]page.pdf}. To see the proofs of the present appendix after macro expansion, consult \href{\lgwCheckUrl expansion/tex/appendix.pdf}{2\linebreak [0]/\linebreak [0]expansion\linebreak [0]/\linebreak [0]tex\linebreak [0]/\linebreak [0]appendix.pdf}.
\section{Notation}
\subsection{Hiding construct}
" [ math proclaim var x hide as text "hide" end text end proclaim end math ] " proclaims " [ math var x hide end math ] " to denote hiding. We introduce it here to have a visible hiding construct instead of the invisible `unicode start of text' operator normally used for hiding, c.f.\ the \href{\lgwBaseUrl index.html}{base} page. The use of hiding in " [ 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 var y end define end quote end define end define hide end math ] " later in this appendix is play for the gallery since the " [ math var x end math ] " is introduced on the \href{\lgwBaseUrl index.html}{base} page and since macro definitions only have effect when they occur on the home page of the construct being macro defined.
\subsection{Metaquantification}\label{section:Metaquantification}
The following macro definition makes e.g.\ " [ math all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var d end metavar end math ] " expand into " [ math all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var d end metavar end math ] ". Note that " [ math all var x indeed var y end math ] " is meta quantification as defined on the \href{\lgwBaseUrl index.html}{base} page which, unfortunately, is rendered exactly like object quantification in the present document. Such notational clashes are unavoidable in mathematics in general but becomes extremely visible in a system like Logiweb where referenced papers are only a click away. To avoid confusion between the two, identically looking quantifiers, we only use the meta quantifier from the base page in the macro definition below and otherwise use " [ math for all objects var x indeed var y end math ] " to denote object quantification. The " [ math var x end math ] " operator used in the definition protects against macro expansion. The left hand side of the macro definition is protected against macro expansion as is the `recursive' call of the macro that occurs in the right hand side.
\vspace{1ex}
\noindent " [ math define macro of for all terms var x indeed var y as lambda var t dot lambda var s dot lambda var c dot state expand tagged if not var t first term root equal quote var x comma var y end quote then quote expand var t term quote all var x indeed var y end quote stack quote var x end quote pair var t first pair quote var y end quote pair var t second pair true end expand else quote expand var t term quote all var x indeed for all terms var y indeed var z end quote stack quote var x end quote pair var t first first pair quote var y end quote pair var t first second pair quote var z end quote pair var t second pair true end expand end if state var s cache var c end expand end define end math ] "
\vspace{1ex}
\noindent The definition above is a general macro definition as opposed to more restricted macro definitions like " [ math define macro of any term var i end line var p 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 any term var i end line var p as for all terms var i indeed var p end define end quote end define end define end math ] " which just define a left hand side to expand into a right hand side. The " [ 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 var y end define end quote end define end define hide end math ] " construct itself is defined by a general macro definition and happens to expand into a general macro definition, so the " [ 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 var y end define end quote end define end define hide end math ] " construct is just syntactic sugar.
A general macro definition assigns a `macro' aspect to the principal operator of the right hand side. The definition ignores the parameters of the principal operator. Instead, the right hand side of the general macro definition must take three arguments, " [ math var t end math ] ", " [ math var s end math ] ", and " [ math var c end math ] ". When a macro aspect is invoked, it is applied to the term " [ math var t end math ] " to be expanded, a `macro state' " [ math var s end math ] ", and a `Logiweb cache' " [ math var c end math ] ".
The macro state should itself be a function of three parameters " [ math var t end math ] ", " [ math var s end math ] ", and " [ math var c end math ] " and it is the function to be invoked when continuing the macro expansion. A macro like the macro protection macro " [ math var x end math ] " protects its argument " [ math var x end math ] " against macro expansion by returning it instead of calling " [ math var s end math ] " on it. The construct above calls the general macro expansion invokation function " [ math state expand var t state var s cache var c end expand end math ] " which simply applies " [ math var s end math ] " to " [ math var t end math ] ", " [ math var s end math ] ", and " [ math var c end math ] ". So the definition above does something to " [ math var t end math ] " and then continues macro expansion.
The Logiweb cache " [ math var c end math ] " is the cache of the home page of " [ math var t end math ] ", i.e.\ the page on which " [ math var t end math ] " occurs. The cache contains, among other, all definitions present on the home page and all pages transitively referenced by the home page. It also contains other things like compiled versions of all value defined constructs. This is badly needed for efficiency reasons because macro expansion is done not by the Logiweb computing engine itself but by a self-interpretter which is run by the Logiweb engine and simulates the Logiweb engine. This self-interpretter, which resides inside the macro state " [ math var s end math ] ", fetches compiled versions of constructs from the cache " [ math var c end math ] " for efficiency reasons.
The term " [ math var t end math ] " has the structure described in Section \ref{section:LogiwebTerms}. " [ math var t second end math ] " is the second subtree of " [ math var t end math ] " and " [ math var t first second end math ] " is the second subtree of the first subtree of " [ math var t end math ] ". " [ math var t term root equal var t prime end math ] " is true when " [ math var t end math ] " and " [ math var t prime end math ] " have the same principal operator.
The " [ math quote expand var t term var p stack var a end expand end math ] " construct expands the pattern " [ math var p end math ] " according to the association list " [ math var a end math ] ". Some day, when the author has some spare time, a new version of " [ math quote expand var t term var p stack var a end expand end math ] " will be implemented in which the association list is replaced by a facility like the `comma' in Lisp backquotes.
The construct " [ math quote expand var t term var p stack var a end expand end math ] " not only expands the pattern " [ math var p end math ] " according to the association list " [ math var a end math ] ", it also extracts the `debugging information' of the principal operator of " [ math var t end math ] " and installs that debugging information in all operators that come from the pattern " [ math var p end math ] ". The debugging information indicates the exact location of the construct before macro expansion.
The " [ math quote var t end quote end math ] " construct evaluates to the Logiweb representation of the term " [ math var t end math ] ".
\subsection{The `Arbitrary' proof constructor}
The `Arbitrary' proof constructor defined on the \href{\lgwBaseUrl index.html}{base} page does not take advantage of the meta quantification macro above. For that reason we introduce a new construct for introducing arbitrary meta variables:
" [ math define macro of any term var i end line var p 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 any term var i end line var p as for all terms var i indeed var p end define end quote end define end define end math ] "
Actually, the definition above was used as an example of a simple macro in Section \ref{section:Metaquantification}. When the same aspect of the same construct is defined more than once (e.g.\ the macro aspect of " [ math all var i indeed var p end math ] "), only the first definition has effect. But the Logiweb compiler prints a warning if the same aspect of the same construct have definitions that contradict each other. The repeated definition of " [ math all var i indeed var p end math ] " above gives no warnings since the two definitions have identical right hand sides.
\subsection{Macro indentation}
The following construct increases the left margin. Should be used in inline math mode only. Should be replaced by automatic indentation inside blocks as soon as possible.
" [ math define macro of macro indent 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 macro indent var x as var x end define end quote end define end define end math ] "
\subsection{Protected macro expansion}
The following construct is a general macro definition construct which both protects its left hand side against macro expansion and renders its left hand side using the tex name aspect. To see the definitions of the tex and tex name aspects of " [ math define macro of var x as var y end define hide end math ] " consult the \href{\lgwCheckUrl body/tex/chores.pdf}{chores} part of the present Logiweb page.
" [ math define macro of general macro define var x as var y end define 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 general macro define var x as var y end define as define macro of protect var x end protect as var y end define end define end quote end define end define end math ] "
\subsection{Object quantification}
The following is a repetion of Section \ref{section:Metaquantification} but concerns object quantification. It allows to write " [ math for all objects var x indeed for all objects var y indeed for all objects var z indeed var u end math ] " for " [ math for all objects var x indeed for all objects var y indeed for all objects var z indeed var u end math ] ".
\vspace{1ex}
\noindent " [ math define macro of for all var x indeed var y as lambda var t dot lambda var s dot lambda var c dot state expand tagged if not var t first term root equal quote var x comma var y end quote then quote expand var t term quote for all objects var x indeed var y end quote stack quote var x end quote pair var t first pair quote var y end quote pair var t second pair true end expand else quote expand var t term quote for all objects var x indeed for all var y indeed var z end quote stack quote var x end quote pair var t first first pair quote var y end quote pair var t first second pair quote var z end quote pair var t second pair true end expand end if state var s cache var c end expand end define end math ] "
\subsection{Proof constructs}
The " [ math var b cut var p end math ] " construct puts a parenthesis around the block " [ math var b end math ] " and places a cut operator between " [ math var b end math ] " and " [ math var p end math ] ". The construct is complicated by the line number " [ math var l end math ] " which should be macro defined locally to denote the conclusion of the block " [ math var b end math ] ". This is non-trivial since the conclusion is the last line prefixed by all premises, side-conditions, and meta-quantifiers introduced in earlier lines.
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define macro of block var b line var l end block var p as lambda var t dot lambda var s dot lambda var c dot block one var t state var s cache var c end block end define end math ] "
\item " [ math define value of block one var t state var s cache var c end block as var t tagged guard var s tagged guard var c tagged guard let one lambda var b dot let one lambda var x dot let one lambda var q dot let one lambda var q dot quote expand var t term quote var b cut var q end quote stack quote var b end quote pair var b pair quote var q end quote pair var q pair true end expand apply state expand var q state var s cache var c end expand end let apply quote expand var t term quote let var l abbreviate var x in var p end quote stack quote var l end quote pair var t second pair quote var p end quote pair var t third pair quote var x end quote pair var x pair true end expand end let apply block two var b end block end let apply state expand var t first state var s cache var c end expand end let end define end math ] "
\item " [ math define value of block two var b end block as tagged if var b term root equal quote var x infer var y end quote then quote expand var b term quote var x infer var y end quote stack quote var x end quote pair var b first pair quote var y end quote pair block two var b second end block pair true end expand else tagged if var b term root equal quote var x endorse var y end quote then quote expand var b term quote var x endorse var y end quote stack quote var x end quote pair var b first pair quote var y end quote pair block two var b second end block pair true end expand else tagged if var b term root equal quote all var x indeed var y end quote then quote expand var b term quote all var x indeed var y end quote stack quote var x end quote pair var b first pair quote var y end quote pair block two var b second end block pair true end expand else tagged if var b term root equal quote var x cut var y end quote then block two var b second end block else tagged if var b term root equal quote var x conclude var y end quote then var b second else absurdity end if end if end if end if end if end define end math ] "
\item " [ math define macro of because var a indeed var i end line 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 because var a indeed var i end line as parenthesis var a conclude var i end parenthesis end define end quote end define end define end math ] "
\end{list}
\subsection{Modus ponens at the object level}
We shall apply modus ponens to two arguments so often that this idiom deserves its own macro:
" [ math define macro of var x object modus ponens 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 object modus ponens var y as rule mp modus ponens var x modus ponens var y end define end quote end define end define end math ] "
\section{Side conditions}
\subsection{Logiweb terms}\label{section:LogiwebTerms}
Metavariables, object terms, statements, and sequent terms are all implemented as Logiweb terms which essentially have the following syntax:
\newcommand{\literal}[1]{\mbox{``#1''}}
\[\begin{array}{lll}
pdigit & ::= & \literal{1} \mathrel{|} \literal{2} \mathrel{|} \literal{3} \mathrel{|} \literal{4} \mathrel{|} \literal{5} \mathrel{|} \literal{6} \mathrel{|} \literal{7} \mathrel{|} \literal{8} \mathrel{|} \literal{9} \\
digit & ::= & \literal{0} \mathrel{|} pdigit \\
positive & ::= & pdigit \mathrel{|} positive \ digit \\
cardinal & ::= & \literal{0} \mathrel{|} positive \\
reference & ::= & cardinal \\
identifier & ::= & cardinal \\
symbol & ::= & \literal{(} \ reference \literal{\mbox {\tt \char32 }} \ identifier \ \literal{)} \\
arglist & ::= & \literal{)} \mathrel{|} \literal{\mbox {\tt \char32 }} \ term \ arglist \\
term & ::= & \literal{(} \ symbol \ arglist \\
\end{array}\]
\noindent The only thing omitted from the syntax is a third element of symbols which contains debugging information. For details on debugging information, consult the \href{\lgwBaseUrl index.html}{base} page.
Each Logiweb page introduces one or more new constructs. As an example, the present page introduces the binary operation " [ math var x imply var y end math ] ".
Each Logiweb page has a unique reference, and each construct introduced by a page has an identifier which is unique within the page. Hence, reference and identifier together determine a construct uniquely.
\subsection{Deduction}
\begin{list}{}{
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\item " [ math define macro of deduction var p conclude var c end deduction 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 deduction var p conclude var c end deduction as lambda var x dot deduction zero quote var p end quote conclude quote var c end quote end deduction end define end quote end define end define end math ] " \newline True if the premise " [ math var p end math ] " implies the conclusion " [ math var c end math ] " according to the deduction rule. The construct quotes its arguments and pass them on to " [ math deduction zero var p conclude var c end deduction end math ] ". The quoting and the " [ math lambda var x dot "\cdots" end math ] " makes " [ math lambda var x dot deduction zero quote var p end quote conclude quote var c end quote end deduction end math ] " suited as a side condition. The present side condition ignores " [ math var x end math ] ". But when a side condition is invoked, it is applied to the cache of the home page of the side condition (the page on which the side condition occurs). That way the side condition gets access to all definitions present on the home page and all pages transitively referenced by the home page. As an example, that has allowed to define an `axiom of definition' in another theory on another Logiweb page in which all value definitions automatically become axiom schemes.
\item " [ math define value of deduction zero var p conclude var c end deduction as var c tagged guard tagged if deduction eight var p bound true end deduction then deduction one deduction seven var p end deduction conclude var c condition true end deduction else false end if end define end math ] " \newline True if the premise " [ math var p end math ] " implies the conclusion " [ math var c end math ] " according to the deduction rule. The function checks the premise for meta-closedness, strips meta-quantifiers from " [ math var p end math ] " and then calls " [ math deduction one var p conclude var c condition var s end deduction end math ] " with an empty list " [ math var s end math ] " of side conditions. The " [ math var x tagged guard var y end math ] " construct evaluates and discards " [ math var x end math ] " and then returns " [ math var y end math ] ". It is included to ensure strictness of " [ math deduction zero var p conclude var c end deduction end math ] " which in turn increases the efficiency. Some day the present author will define a `strict value definition' operator so that users don't have to add " [ math var x tagged guard var y end math ] " operators manually. The " [ math var x tagged guard var y end math ] " construct takes zero CPU-time when used in functions that are `fit' for optimization. See the \href{\lgwBaseUrl index.html}{base} page for details on `fitness' analysis.
\item " [ math define value of deduction one var p conclude var c condition var s end deduction as tagged if var c term root equal quote var x endorse var y end quote then deduction one var p conclude var c second condition var c first pair var s end deduction else deduction two var p conclude var c condition var s end deduction end if end define end math ] " \newline Move side conditions from the conclusion " [ math var c end math ] " to the list " [ math var s end math ] ". " [ math var x term root equal var y end math ] " is true if the terms " [ math var x end math ] " and " [ math var y end math ] " have the same root (i.e.\ the same principal operator) so " [ math var c term root equal quote var x endorse var y end quote end math ] " is true if the principal operator of " [ math var c end math ] " is an endorsement. " [ math var c first end math ] " and " [ math var c second end math ] " are the first and second argument, respectively, of the endorsement.
\item " [ math define value of deduction two var p conclude var c condition var s end deduction 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 deduction three var p first conclude var c first condition var s bound true end deduction and deduction two var p second conclude var c second condition var s end deduction else deduction four var p conclude var c condition var s bound deduction six var p conclude var c exception true bound true end deduction end deduction end select end define end math ] " \newline This function is true if the premise " [ math var p end math ] " is equal to the conclusion " [ math var c end math ] " except that inferences are replaced by implications. " [ math tagged if var x then var y else var z end if end math ] " and " [ math var x select var y else var z end select end math ] " express the same conditional construct. Choosing between the two is a purely stylistic choice.
\item " [ math define value of deduction three var p conclude var c condition var s bound var b end deduction as tagged if not var c term root equal quote for all objects var x indeed var y end quote then deduction four var p conclude var c condition var s bound var b end deduction 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 deduction four var p conclude var c condition var s bound var b end deduction else deduction three var p conclude var c second condition var s bound var c first pair var c first pair var b end deduction end if end if end define end math ] " \newline " [ math deduction three var p conclude var c condition var s bound var b end deduction end math ] " moves quantified variables from the premise " [ math var p end math ] " to the association list " [ math var b end math ] " of `bindings'. Each quantified variable " [ math var x end math ] " is added to " [ math var b end math ] " as the pair " [ math var x pair var x end math ] " indicating the " [ math var x end math ] " should be replaced by " [ math var x end math ] " (which is a way of saying that the variable cannot be instantiated). Moving of bound variables stops when there are no more bound variables to move or when the premise and conclusion happen to quantify the same variable. In both cases we have reached a point where premise and conclusion should be identical except for instantaition of variables. " [ math var x term equal var y end math ] " is true if the terms " [ math var x end math ] " and " [ math var y end math ] " are identical. Logiweb terms contain debugging information which indicates the precise location each operator before macro expansion. " [ math var x term equal var y end math ] " ignores this debugging information when it tests two terms for identity.
\item " [ math define value of deduction four var p conclude var c condition var s bound var b end deduction 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 deduction four var p second conclude var c second condition var s bound var p first pair var p first pair var b end deduction else tagged if not var p term root equal quote metavar var x end metavar end quote then deduction four star var p tail conclude var c tail condition var s bound var b end deduction else var p first term equal var c first and deduction five var p condition var s bound var b end deduction end if end if end if end if end define end math ] " \newline Test that the premise " [ math var p end math ] " is identical to the conclusion " [ math var c end math ] " except for instantiation of object variables as specified by the association list " [ math var b end math ] ". A term is an object variable if the root of the term is the `object variable operator'. For that reason, one can test whether or not a variable is an object variable by comparing its root to the root of an arbitrary object variable. For each meta variable, check that there are side conditions which ensure that the meta variables denote terms whose free object variables do not get caught in the context. Meta variables are recognized the same way as object variables. The object variable " [ math object var var x end var end math ] " and the meta variable " [ math metavar var x end metavar end math ] " macro expand into " [ math object var var x end var end math ] " and " [ math metavar var x end metavar end math ] ", respectively, adding the object and meta variable operator, respectively, to the term " [ math var x end math ] ". The term " [ math var x end math ] " is neither an object nor a meta variable. " [ math var x end math ] " is a Logiweb programming language variable because no Logiweb definition assigns a value to it.
\item " [ math define value of deduction four star var p conclude var c condition var s bound var b end deduction as var c tagged guard var s tagged guard var b tagged guard tagged if var p then true else deduction four var p head conclude var c head condition var s bound var b end deduction and deduction four star var p tail conclude var c tail condition var s bound var b end deduction end if end define end math ] " \newline Elementwise application of " [ math deduction four var p conclude var c condition var s bound var b end deduction end math ] ". " [ math var x head end math ] " and " [ math var x tail end math ] " denote the head and tail, respectively, of the pair " [ math var x end math ] ". Terms are represented as lists comprising the principal operator followed by arguments just like Lisp S-expressions. The principal operator in turn is a list whose first two elements are cardinals that identify the operator followed by debugging information.
\item " [ math define value of deduction five var p condition var s bound var b end deduction 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 deduction five var p condition var s bound var b tail end deduction end if end define end math ] " \newline For each variable that is bound in the context according to the association list " [ math var b end math ] ", check that the list " [ math var s end math ] " of side conditions ensures that the meta variables " [ math var p end math ] " denotes a term whose free variables do not get caught in the context. " [ math var x term in var y end math ] " is true if the term " [ math var x end math ] " belongs to the list " [ math var y end math ] " of terms.
\item " [ math define value of deduction six var p conclude var c exception var e bound var b end deduction 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 deduction six var p second conclude var c second exception var c first pair var e bound var b end deduction else deduction six star var p tail conclude var c tail exception var e bound var b end deduction end if end if end if end if end define end math ] "
\item " [ math define value of deduction six star var p conclude var c exception var e bound var b end deduction 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 deduction six star var p tail conclude var c tail exception var e bound deduction six var p head conclude var c head exception var e bound var b end deduction end deduction end if end define end math ] " \newline Elementwise application of " [ math deduction six var p conclude var c exception var e bound var b end deduction end math ] ".
\item " [ math define value of deduction seven var p end deduction as var p term root equal quote all var x indeed var y end quote select deduction seven var p second end deduction else var p end select end define end math ] " \newline Removal of all metaquantifiers.
\item " [ math define value of deduction eight var p bound var b end deduction as tagged if var p term root equal quote all var x indeed var y end quote then deduction eight var p second bound var p first pair var b end deduction else tagged if var p term root equal quote metavar var a end metavar end quote then var p term in var b else deduction eight star var p tail bound var b end deduction end if end if end define end math ] " \newline True if all free meta variables of " [ math var p end math ] " occur in the list " [ math var b end math ] " of meta variables.
\item " [ math define value of deduction eight star var p bound var b end deduction as var b tagged guard tagged if var p then true else tagged if deduction eight var p head bound var b end deduction then deduction eight star var p tail bound var b end deduction else false end if end if end define end math ] " \newline Elementwise application of " [ math deduction eight var p bound var b end deduction end math ] ".
\end{list}
\subsection{Avoidance}
\begin{list}{}{
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\item " [ math define value of var x is object var as var x term root equal quote object var var x end var end quote end define end math ] "
\item " [ math define macro of var x avoid 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 avoid var y as quote var x end quote avoid zero quote var y end quote end define end quote end define end define end math ] "
\item " [ math define value of var x avoid zero var y as lambda var c dot var x is object var and var y is metaclosed and var x avoid one var y end define end math ] "
\item " [ math define value of var x avoid one var y as tagged if var y is object var then not var x term equal var y else newline tagged if not var y term root equal quote for all objects var x indeed var y end quote then var x avoid star var y tail else newline tagged if var x term equal var y first then true else var x avoid one var y second end if end if end if end define end math ] "
\item " [ math define value of var x avoid star var y as var x tagged guard tagged if var y then true else tagged if var x avoid one var y head then var x avoid star var y tail else false end if end if end define end math ] "
\end{list}
\subsection{Substition}
\begin{list}{}{
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\item " [ math define macro of 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 sub var a is var b where var x is var t end sub as sub zero 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 sub zero var a is var b where var x is var t end sub as lambda var c dot var x is object var and sub one var a is var b where var x is var t end sub end define end math ] "
\item " [ math define value of sub one 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 tagged if var b term root equal quote for all objects var u indeed var v end quote then var b first term equal var x else false end if then var a term equal var b else newline tagged if var b is object 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 sub star 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 sub star 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 sub one var a head is var b head where var x is var t end sub then sub star 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}
\section{Proofs}
\subsection{Proofs of FOL axioms using the inference of deduction}
We now prove axiom schemes A1 and A2 from \cite{mendelson}. Furthermore, we prove two, particular instances of A4 and A5 in a way that generalizes to arbitrary instances of those axiom schemes.
Furthermore, we prove a " [ math repetition end math ] " lemma which says " [ math all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar end math ] " by a `hand made proof', i.e.\ by giving a proof metatactic explicitly which generates the sequent proof.
A page is verified by a top level verifier. The top level verifier of the present page is defined on the \href{\lgwBaseUrl index.html}{base} page. That verifier invokes, among other, a proof verifier. The proof verifier invokes each proof (where each proof is supposed to be a proof metatactic). When it does so, it applies the proof metatactic to a Logiweb cache " [ math var c end math ] " and a pair " [ math var n pair var l end math ] ". The cache " [ math var c end math ] " is the Logiweb cache of the home page of the proof, i.e.\ a structure which contains all definitions present on the home page plus all transitively referenced Logiweb pages. " [ math var n end math ] " is the identifier of the lemma (a cardinal which identifies the lemma uniquely within the home page) plus the contents of the lemma. The proof metatactic given below for proving " [ math repetition end math ] " ignores both arguments and just constructs a sequent term that proves " [ math all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar end math ] ". All other proofs in the present paper macro expand into proofs that are proved by a metatactic which scans the proof for object tactics and invokes those object tactics. The only object tactic used is named $\gg$ and does unification.
\begin{list}{}{
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\item " [ math define statement of repetition as system s infer all metavar var a end metavar indeed metavar var a end metavar infer metavar var a end metavar end define end math ] "
\item " [ math define statement of lemma a one as system s 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 imply metavar var a end metavar end define end math ] "
\item " [ math define statement of lemma a two as system s 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 imply metavar var a end metavar imply metavar var b end metavar imply metavar var a end metavar imply metavar var c end metavar end define end math ] "
\item " [ math define statement of lemma a four as system s infer for all objects object var var x end var indeed for all objects object var var y end var indeed object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var imply two plus three equal three plus two end define end math ] "
\item " [ math define statement of lemma a five as system s infer for all objects object var var x end var indeed two plus three equal five imply two plus three plus object var var x end var equal five plus object var var x end var imply two plus three equal five imply for all objects object var var x end var indeed two plus three plus object var var x end var equal five plus object var var x end var end define end math ] "
\end{list}
\begin{list}{}{
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\setlength{\itemsep}{1ex}}
\item " [ math define proof of repetition as lambda var c dot lambda var x dot quote system s infer all metavar var a end metavar indeed metavar var a end metavar init end quote end define end math ] "
\item " [ math define proof of lemma a one as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed 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 repetition modus ponens metavar var a end metavar conclude metavar var a end metavar cut 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 b end metavar infer metavar var a end metavar conclude metavar var a end metavar imply metavar var b end metavar imply metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define proof of lemma a two as lambda var c dot lambda var x dot proof expand quote system s 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 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 imply metavar var b end metavar infer metavar var a end metavar infer rule 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 rule 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 rule 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 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 imply metavar var c end metavar infer metavar var a end metavar imply metavar var b 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 c end metavar imply metavar var a end metavar imply metavar var b end metavar imply 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 ] "
\item " [ math define proof of lemma a four as lambda var c dot lambda var x dot proof expand quote system s infer object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var infer repetition modus ponens object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var conclude object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var cut deduction modus ponens object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var infer object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var conclude for all objects object var var x end var indeed for all objects object var var y end var indeed object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var imply two plus three equal three plus two end quote state proof state cache var c end expand end define end math ] "
\item " [ math define proof of lemma a five as lambda var c dot lambda var x dot proof expand quote system s infer two plus three equal five imply two plus three plus object var var x end var equal five plus object var var x end var infer two plus three equal five infer rule mp modus ponens two plus three equal five imply two plus three plus object var var x end var equal five plus object var var x end var modus ponens two plus three equal five conclude two plus three plus object var var x end var equal five plus object var var x end var cut rule gen modus ponens two plus three plus object var var x end var equal five plus object var var x end var conclude for all objects object var var x end var indeed two plus three plus object var var x end var equal five plus object var var x end var cut deduction modus ponens two plus three equal five imply two plus three plus object var var x end var equal five plus object var var x end var infer two plus three equal five infer for all objects object var var x end var indeed two plus three plus object var var x end var equal five plus object var var x end var conclude for all objects object var var x end var indeed two plus three equal five imply two plus three plus object var var x end var equal five plus object var var x end var imply two plus three equal five imply for all objects object var var x end var indeed two plus three plus object var var x end var equal five plus object var var x end var end quote state proof state cache var c end expand end define end math ] "
\end{list}
\subsection{A proof of " [ math var x plus var y equal var y plus var x end math ] "}
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\item " [ math define statement of prop three two a as system s infer all metavar var a end metavar indeed metavar var a end metavar equal metavar var a end metavar end define end math ] "
\item " [ math define statement of prop three two b as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar equal metavar var b end metavar infer metavar var b end metavar equal metavar var a end metavar end define end math ] "
\item " [ math define statement of prop three two c as system s 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 equal metavar var b end metavar infer metavar var b end metavar equal metavar var c end metavar infer metavar var a end metavar equal metavar var c end metavar end define end math ] "
\item " [ math define statement of prop three two d as system s 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 equal metavar var c end metavar infer metavar var b end metavar equal metavar var c end metavar infer metavar var a end metavar equal metavar var b end metavar end define end math ] "
\item " [ math define statement of prop three two e as system s 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 equal metavar var b end metavar infer metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar end define end math ] "
\item " [ math define statement of prop three two f as system s infer all metavar var a end metavar indeed metavar var a end metavar equal zero plus metavar var a end metavar end define end math ] "
\item " [ math define statement of prop three two g as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar suc plus metavar var b end metavar equal metavar var a end metavar plus metavar var b end metavar suc end define end math ] "
\item " [ math define statement of prop three two h as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar plus metavar var b end metavar equal metavar var b end metavar plus metavar var a end metavar end define end math ] "
\end{list}
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\item " [ math define proof of prop three two a as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed axiom s five conclude metavar var a end metavar plus zero equal metavar var a end metavar cut axiom s one modus ponens metavar var a end metavar plus zero equal metavar var a end metavar modus ponens metavar var a end metavar plus zero equal metavar var a end metavar conclude metavar var a end metavar equal metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define proof of prop three two b as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar equal metavar var b end metavar infer prop three two a conclude metavar var a end metavar equal metavar var a end metavar cut axiom s one modus ponens metavar var a end metavar equal metavar var b end metavar modus ponens metavar var a end metavar equal metavar var a end metavar conclude metavar var b end metavar equal metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define proof of prop three two c as lambda var c dot lambda var x dot proof expand quote system s 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 equal metavar var b end metavar infer metavar var b end metavar equal metavar var c end metavar infer prop three two b modus ponens metavar var a end metavar equal metavar var b end metavar conclude metavar var b end metavar equal metavar var a end metavar cut axiom s one modus ponens metavar var b end metavar equal metavar var a end metavar modus ponens metavar var b end metavar equal metavar var c end metavar conclude metavar var a end metavar equal metavar var c end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define proof of prop three two d as lambda var c dot lambda var x dot proof expand quote system s 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 equal metavar var c end metavar infer metavar var b end metavar equal metavar var c end metavar infer prop three two b modus ponens metavar var b end metavar equal metavar var c end metavar conclude metavar var c end metavar equal metavar var b end metavar cut prop three two c modus ponens metavar var a end metavar equal metavar var c end metavar modus ponens metavar var c end metavar equal metavar var b end metavar conclude metavar var a end metavar equal metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of prop three two e one as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar equal metavar var b end metavar imply metavar var a end metavar plus zero equal metavar var b end metavar plus zero end define end math ] "
\item " [ math define proof of prop three two e one as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar equal metavar var b end metavar infer axiom s five conclude metavar var a end metavar plus zero equal metavar var a end metavar cut prop three two c modus ponens metavar var a end metavar plus zero equal metavar var a end metavar modus ponens metavar var a end metavar equal metavar var b end metavar conclude metavar var a end metavar plus zero equal metavar var b end metavar cut axiom s five conclude metavar var b end metavar plus zero equal metavar var b end metavar cut prop three two d modus ponens metavar var a end metavar plus zero equal metavar var b end metavar modus ponens metavar var b end metavar plus zero equal metavar var b end metavar conclude metavar var a end metavar plus zero equal metavar var b end metavar plus zero cut deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar equal metavar var b end metavar infer metavar var a end metavar plus zero equal metavar var b end metavar plus zero conclude metavar var a end metavar equal metavar var b end metavar imply metavar var a end metavar plus zero equal metavar var b end metavar plus zero end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of prop three two e two as system s 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 equal metavar var b end metavar imply metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar imply metavar var a end metavar equal metavar var b end metavar imply metavar var a end metavar plus metavar var c end metavar suc equal metavar var b end metavar plus metavar var c end metavar suc end define end math ] "
\item " [ math define proof of prop three two e two as lambda var c dot lambda var x dot proof expand quote system s 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 a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar equal metavar var b end metavar imply metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar infer metavar var a end metavar equal metavar var b end metavar infer rule mp modus ponens metavar var a end metavar equal metavar var b end metavar imply metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar modus ponens metavar var a end metavar equal metavar var b end metavar conclude metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar cut axiom s two modus ponens metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar conclude metavar var a end metavar plus metavar var c end metavar suc equal metavar var b end metavar plus metavar var c end metavar suc cut axiom s six conclude metavar var a end metavar plus metavar var c end metavar suc equal metavar var a end metavar plus metavar var c end metavar suc cut prop three two c modus ponens metavar var a end metavar plus metavar var c end metavar suc equal metavar var a end metavar plus metavar var c end metavar suc modus ponens metavar var a end metavar plus metavar var c end metavar suc equal metavar var b end metavar plus metavar var c end metavar suc conclude metavar var a end metavar plus metavar var c end metavar suc equal metavar var b end metavar plus metavar var c end metavar suc cut axiom s six conclude metavar var b end metavar plus metavar var c end metavar suc equal metavar var b end metavar plus metavar var c end metavar suc cut prop three two d modus ponens metavar var a end metavar plus metavar var c end metavar suc equal metavar var b end metavar plus metavar var c end metavar suc modus ponens metavar var b end metavar plus metavar var c end metavar suc equal metavar var b end metavar plus metavar var c end metavar suc conclude metavar var a end metavar plus metavar var c end metavar suc equal metavar var b end metavar plus metavar var c end metavar suc cut 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 equal metavar var b end metavar imply metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar infer metavar var a end metavar equal metavar var b end metavar infer metavar var a end metavar plus metavar var c end metavar suc equal metavar var b end metavar plus metavar var c end metavar suc conclude metavar var a end metavar equal metavar var b end metavar imply metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar imply metavar var a end metavar equal metavar var b end metavar imply metavar var a end metavar plus metavar var c end metavar suc equal metavar var b end metavar plus metavar var c end metavar suc end quote state proof state cache var c end expand end define end math ] "
\item " [ math define proof of prop three two e as lambda var c dot lambda var x dot proof expand quote system s 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 equal metavar var b end metavar infer prop three two e one conclude object var var x end var equal object var var y end var imply object var var x end var plus zero equal object var var y end var plus zero cut prop three two e two conclude object var var x end var equal object var var y end var imply object var var x end var plus object var var z end var equal object var var y end var plus object var var z end var imply object var var x end var equal object var var y end var imply object var var x end var plus object var var z end var suc equal object var var y end var plus object var var z end var suc cut axiom s nine at object var var z end var modus ponens object var var x end var equal object var var y end var imply object var var x end var plus zero equal object var var y end var plus zero modus ponens object var var x end var equal object var var y end var imply object var var x end var plus object var var z end var equal object var var y end var plus object var var z end var imply object var var x end var equal object var var y end var imply object var var x end var plus object var var z end var suc equal object var var y end var plus object var var z end var suc conclude object var var x end var equal object var var y end var imply object var var x end var plus object var var z end var equal object var var y end var plus object var var z end var cut deduction modus ponens object var var x end var equal object var var y end var imply object var var x end var plus object var var z end var equal object var var y end var plus object var var z end var conclude metavar var a end metavar equal metavar var b end metavar imply metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar cut rule mp modus ponens metavar var a end metavar equal metavar var b end metavar imply metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar modus ponens metavar var a end metavar equal metavar var b end metavar conclude metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of prop three two f one as system s infer zero equal zero plus zero end define end math ] "
\item " [ math define proof of prop three two f one as lambda var c dot lambda var x dot proof expand quote system s infer axiom s five conclude zero plus zero equal zero cut prop three two b modus ponens zero plus zero equal zero conclude zero equal zero plus zero end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of prop three two f two as system s infer all metavar var a end metavar indeed metavar var a end metavar equal zero plus metavar var a end metavar imply metavar var a end metavar suc equal zero plus metavar var a end metavar suc end define end math ] "
\item " [ math define proof of prop three two f two as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed all metavar var a end metavar indeed metavar var a end metavar equal zero plus metavar var a end metavar infer axiom s two modus ponens metavar var a end metavar equal zero plus metavar var a end metavar conclude metavar var a end metavar suc equal zero plus metavar var a end metavar suc cut axiom s six conclude zero plus metavar var a end metavar suc equal zero plus metavar var a end metavar suc cut prop three two d modus ponens metavar var a end metavar suc equal zero plus metavar var a end metavar suc modus ponens zero plus metavar var a end metavar suc equal zero plus metavar var a end metavar suc conclude metavar var a end metavar suc equal zero plus metavar var a end metavar suc cut deduction modus ponens all metavar var a end metavar indeed metavar var a end metavar equal zero plus metavar var a end metavar infer metavar var a end metavar suc equal zero plus metavar var a end metavar suc conclude metavar var a end metavar equal zero plus metavar var a end metavar imply metavar var a end metavar suc equal zero plus metavar var a end metavar suc end quote state proof state cache var c end expand end define end math ] "
\item " [ math define proof of prop three two f as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed prop three two f one conclude zero equal zero plus zero cut prop three two f two conclude object var var x end var equal zero plus object var var x end var imply object var var x end var suc equal zero plus object var var x end var suc cut axiom s nine at object var var x end var modus ponens zero equal zero plus zero modus ponens object var var x end var equal zero plus object var var x end var imply object var var x end var suc equal zero plus object var var x end var suc conclude object var var x end var equal zero plus object var var x end var cut deduction modus ponens object var var x end var equal zero plus object var var x end var conclude metavar var a end metavar equal zero plus 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 three two g one as system s infer all metavar var a end metavar indeed metavar var a end metavar suc plus zero equal metavar var a end metavar plus zero suc end define end math ] "
\item " [ math define proof of prop three two g one as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed axiom s five conclude metavar var a end metavar suc plus zero equal metavar var a end metavar suc cut axiom s five conclude metavar var a end metavar plus zero equal metavar var a end metavar cut axiom s two modus ponens metavar var a end metavar plus zero equal metavar var a end metavar conclude metavar var a end metavar plus zero suc equal metavar var a end metavar suc cut prop three two d modus ponens metavar var a end metavar suc plus zero equal metavar var a end metavar suc modus ponens metavar var a end metavar plus zero suc equal metavar var a end metavar suc conclude metavar var a end metavar suc plus zero equal metavar var a end metavar plus zero suc end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of prop three two g two as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar suc plus metavar var b end metavar equal metavar var a end metavar plus metavar var b end metavar suc imply metavar var a end metavar suc plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc suc end define end math ] "
\item " [ math define proof of prop three two g two as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar suc plus metavar var b end metavar equal metavar var a end metavar plus metavar var b end metavar suc infer axiom s two modus ponens metavar var a end metavar suc plus metavar var b end metavar equal metavar var a end metavar plus metavar var b end metavar suc conclude metavar var a end metavar suc plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc suc cut axiom s six conclude metavar var a end metavar suc plus metavar var b end metavar suc equal metavar var a end metavar suc plus metavar var b end metavar suc cut prop three two c modus ponens metavar var a end metavar suc plus metavar var b end metavar suc equal metavar var a end metavar suc plus metavar var b end metavar suc modus ponens metavar var a end metavar suc plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc suc conclude metavar var a end metavar suc plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc suc cut axiom s six conclude metavar var a end metavar plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc cut axiom s two modus ponens metavar var a end metavar plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc conclude metavar var a end metavar plus metavar var b end metavar suc suc equal metavar var a end metavar plus metavar var b end metavar suc suc cut prop three two d modus ponens metavar var a end metavar suc plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc suc modus ponens metavar var a end metavar plus metavar var b end metavar suc suc equal metavar var a end metavar plus metavar var b end metavar suc suc conclude metavar var a end metavar suc plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc suc cut deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar suc plus metavar var b end metavar equal metavar var a end metavar plus metavar var b end metavar suc infer metavar var a end metavar suc plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc suc conclude metavar var a end metavar suc plus metavar var b end metavar equal metavar var a end metavar plus metavar var b end metavar suc imply metavar var a end metavar suc plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc suc end quote state proof state cache var c end expand end define end math ] "
\item " [ math define proof of prop three two g as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed prop three two g one conclude object var var x end var suc plus zero equal object var var x end var plus zero suc cut prop three two g two conclude object var var x end var suc plus object var var y end var equal object var var x end var plus object var var y end var suc imply object var var x end var suc plus object var var y end var suc equal object var var x end var plus object var var y end var suc suc cut axiom s nine at object var var y end var modus ponens object var var x end var suc plus zero equal object var var x end var plus zero suc modus ponens object var var x end var suc plus object var var y end var equal object var var x end var plus object var var y end var suc imply object var var x end var suc plus object var var y end var suc equal object var var x end var plus object var var y end var suc suc conclude object var var x end var suc plus object var var y end var equal object var var x end var plus object var var y end var suc cut deduction modus ponens object var var x end var suc plus object var var y end var equal object var var x end var plus object var var y end var suc conclude metavar var a end metavar suc plus metavar var b end metavar equal metavar var a end metavar plus metavar var b end metavar suc end quote state proof state cache var c end expand end define end math ] "
\item " [ math define statement of prop three two h one as system s infer all metavar var a end metavar indeed metavar var a end metavar plus zero equal zero plus metavar var a end metavar end define end math ] "
\item " [ math define proof of prop three two h one as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed axiom s five conclude metavar var a end metavar plus zero equal metavar var a end metavar cut prop three two f conclude metavar var a end metavar equal zero plus metavar var a end metavar cut prop three two c modus ponens metavar var a end metavar plus zero equal metavar var a end metavar modus ponens metavar var a end metavar equal zero plus metavar var a end metavar conclude metavar var a end metavar plus zero equal zero plus 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 three two h two as system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar plus metavar var b end metavar equal metavar var b end metavar plus metavar var a end metavar imply metavar var a end metavar plus metavar var b end metavar suc equal metavar var b end metavar suc plus metavar var a end metavar end define end math ] "
\item " [ math define proof of prop three two h two as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar plus metavar var b end metavar equal metavar var b end metavar plus metavar var a end metavar infer axiom s two modus ponens metavar var a end metavar plus metavar var b end metavar equal metavar var b end metavar plus metavar var a end metavar conclude metavar var a end metavar plus metavar var b end metavar suc equal metavar var b end metavar plus metavar var a end metavar suc cut axiom s six conclude metavar var a end metavar plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc cut prop three two c modus ponens metavar var a end metavar plus metavar var b end metavar suc equal metavar var a end metavar plus metavar var b end metavar suc modus ponens metavar var a end metavar plus metavar var b end metavar suc equal metavar var b end metavar plus metavar var a end metavar suc conclude metavar var a end metavar plus metavar var b end metavar suc equal metavar var b end metavar plus metavar var a end metavar suc cut prop three two g conclude metavar var b end metavar suc plus metavar var a end metavar equal metavar var b end metavar plus metavar var a end metavar suc cut prop three two d modus ponens metavar var a end metavar plus metavar var b end metavar suc equal metavar var b end metavar plus metavar var a end metavar suc modus ponens metavar var b end metavar suc plus metavar var a end metavar equal metavar var b end metavar plus metavar var a end metavar suc conclude metavar var a end metavar plus metavar var b end metavar suc equal metavar var b end metavar suc plus metavar var a end metavar cut deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar plus metavar var b end metavar equal metavar var b end metavar plus metavar var a end metavar infer metavar var a end metavar plus metavar var b end metavar suc equal metavar var b end metavar suc plus metavar var a end metavar conclude metavar var a end metavar plus metavar var b end metavar equal metavar var b end metavar plus metavar var a end metavar imply metavar var a end metavar plus metavar var b end metavar suc equal metavar var b end metavar suc plus metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\item " [ math define proof of prop three two h as lambda var c dot lambda var x dot proof expand quote system s infer all metavar var a end metavar indeed all metavar var b end metavar indeed prop three two h one conclude object var var x end var plus zero equal zero plus object var var x end var cut prop three two h two conclude object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var imply object var var x end var plus object var var y end var suc equal object var var y end var suc plus object var var x end var cut axiom s nine at object var var y end var modus ponens object var var x end var plus zero equal zero plus object var var x end var modus ponens object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var imply object var var x end var plus object var var y end var suc equal object var var y end var suc plus object var var x end var conclude object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var cut deduction modus ponens object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var conclude metavar var a end metavar plus metavar var b end metavar equal metavar var b end metavar plus metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
we will start by doing " [ math prop three two i end math ] "
\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}
\item " [ math define proof of prop three two i as lambda var c dot lambda var x dot proof expand quote system s 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 a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar equal metavar var b end metavar infer prop three two e modus ponens metavar var a end metavar equal metavar var b end metavar conclude metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar cut prop three two h conclude metavar var a end metavar plus metavar var c end metavar equal metavar var c end metavar plus metavar var a end metavar cut prop three two h conclude metavar var b end metavar plus metavar var c end metavar equal metavar var c end metavar plus metavar var b end metavar cut axiom s one modus ponens metavar var a end metavar plus metavar var c end metavar equal metavar var b end metavar plus metavar var c end metavar modus ponens metavar var a end metavar plus metavar var c end metavar equal metavar var c end metavar plus metavar var a end metavar conclude metavar var b end metavar plus metavar var c end metavar equal metavar var c end metavar plus metavar var a end metavar cut prop three two b modus ponens metavar var b end metavar plus metavar var c end metavar equal metavar var c end metavar plus metavar var a end metavar conclude metavar var c end metavar plus metavar var a end metavar equal metavar var b end metavar plus metavar var c end metavar cut prop three two c modus ponens metavar var c end metavar plus metavar var a end metavar equal metavar var b end metavar plus metavar var c end metavar modus ponens metavar var b end metavar plus metavar var c end metavar equal metavar var c end metavar plus metavar var b end metavar conclude metavar var c end metavar plus metavar var a end metavar equal metavar var c end metavar plus metavar var b end metavar cut 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 equal metavar var b end metavar infer metavar var c end metavar plus metavar var a end metavar equal metavar var c end metavar plus metavar var b end metavar conclude metavar var a end metavar equal metavar var b end metavar infer metavar var c end metavar plus metavar var a end metavar equal metavar var c end metavar plus metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "
\end{list}
%------------------------------------------------------------------------------
\bibliographystyle{alpha}
\bibliography{./page}
%------------------------------------------------------------------------------
\everymath{}
\everydisplay{}
\end{document}
End of file
latex appendix
bibtex appendix
latex appendix
latex appendix
dvipdfm appendix
File chores.tex
\documentclass [fleqn]{article}
\everymath{\rm}
\usepackage{latexsym}
\setlength {\overfullrule }{0mm}
\input{lgwinclude}
\usepackage{url}
\usepackage[dvipdfm=true]{hyperref}
\hypersetup{pdfpagemode=none}
\hypersetup{pdfstartpage=1}
\hypersetup{pdfstartview=FitBH}
\hypersetup{pdfpagescrop={120 130 490 730}}
\hypersetup{pdftitle=Logiweb sequent calculus - Chores}
\hypersetup{colorlinks=true}
\begin {document}
\title {Logiweb sequent calculus, Chores}
\author {Klaus Grue}
\date {\today}
\maketitle
\tableofcontents
\section{Test cases}
\begin{list}{}{
\setlength{\leftmargin}{5em}
\setlength{\itemindent}{-5em}}
\item " [ math test quote object var var x end var end quote avoid zero quote object var var y end var equal object var var z end var imply for all objects object var var x end var indeed object var var x end var equal object var var y end var end quote apply check end test end math ] "
\item " [ math false test quote object var var x end var end quote avoid zero quote object var var x end var equal object var var z end var imply for all objects object var var x end var indeed object var var x end var equal object var var y end var end quote apply check end test end math ] "
\item " [ math false test quote object var var x end var end quote avoid zero quote object var var y end var equal object var var x end var imply for all objects object var var x end var indeed object var var x end var equal object var var y end var end quote apply check end test end math ] "
\item " [ math false test quote object var var x end var end quote avoid zero quote object var var y end var equal object var var z end var imply for all objects object var var y end var indeed object var var x end var equal object var var y end var end quote apply check end test end math ] "
\item " [ math test sub zero quote object var var a end var end quote is quote object var var a end var end quote where quote object var var b end var end quote is quote object var var c end var end quote end sub apply check end test end math ] "
\item " [ math false test sub zero quote object var var b end var end quote is quote object var var a end var end quote where quote object var var b end var end quote is quote object var var c end var end quote end sub apply check end test end math ] "
\item " [ math false test sub zero quote object var var c end var end quote is quote object var var a end var end quote where quote object var var b end var end quote is quote object var var c end var end quote end sub apply check end test end math ] "
\item " [ math false test sub zero quote object var var a end var end quote is quote object var var b end var end quote where quote object var var b end var end quote is quote object var var c end var end quote end sub apply check end test end math ] "
\item " [ math false test sub zero quote object var var b end var end quote is quote object var var b end var end quote where quote object var var b end var end quote is quote object var var c end var end quote end sub apply check end test end math ] "
\item " [ math test sub zero quote object var var c end var end quote is quote object var var b end var end quote where quote object var var b end var end quote is quote object var var c end var end quote end sub apply check end test end math ] "
\item " [ math test sub zero quote for all objects object var var a end var indeed object var var a end var equal object var var b end var end quote is quote for all objects object var var a end var indeed object var var a end var equal object var var b end var end quote where quote object var var a end var end quote is quote object var var c end var end quote end sub apply check end test end math ] "
\item " [ math test sub zero quote for all objects object var var a end var indeed object var var a end var equal object var var c end var end quote is quote for all objects object var var a end var indeed object var var a end var equal object var var b end var end quote where quote object var var b end var end quote is quote object var var c end var end quote end sub apply check end test end math ] "
\item " [ math test sub zero quote for all objects object var var a end var indeed object var var a end var equal zero plus object var var a end var imply object var var c end var times object var var d end var equal zero plus object var var c end var times object var var d end var end quote is quote for all objects object var var a end var indeed object var var a end var equal zero plus object var var a end var imply object var var b end var equal zero plus object var var b end var end quote where quote object var var b end var end quote is quote object var var c end var times object var var d end var end quote end sub apply check end test end math ] "
\item " [ math test sub zero quote for all objects object var var a end var indeed object var var a end var equal zero plus object var var a end var imply object var var b end var equal zero plus object var var b end var end quote is quote for all objects object var var a end var indeed object var var a end var equal zero plus object var var a end var imply object var var b end var equal zero plus object var var b end var end quote where quote object var var a end var end quote is quote object var var c end var end quote end sub apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote zero end quote conclude quote zero end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote zero end quote conclude quote one end quote end deduction apply check end test end math ] "
\item " [ math test deduction eight quote all metavar var a end metavar indeed metavar var a end metavar end quote bound true end deduction end test end math ] "
\item " [ math test deduction seven quote all metavar var a end metavar indeed metavar var a end metavar end quote end deduction term equal quote metavar var a end metavar end quote end test end math ] "
\item " [ math test lambda var x dot deduction zero quote all metavar var a end metavar indeed metavar var a end metavar end quote conclude quote metavar var a end metavar end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote metavar var a end metavar end quote conclude quote metavar var b end metavar end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote all metavar var a end metavar indeed metavar var a end metavar end quote conclude quote metavar var b end metavar end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote 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 end quote conclude quote metavar var a end metavar imply metavar var b end metavar end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote 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 end quote conclude quote metavar var a end metavar imply metavar var a end metavar end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote 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 end quote conclude quote metavar var b end metavar imply metavar var b end metavar end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote 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 end quote conclude quote zero end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote zero end quote conclude quote metavar var a end metavar imply metavar var a end metavar end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote 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 infer metavar var b end metavar infer metavar var c end metavar end quote conclude quote metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote 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 metavar var a end metavar end quote conclude quote metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote 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 infer metavar var b end metavar infer metavar var c end metavar end quote conclude quote metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote zero end quote conclude quote object var var x end var end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote object var var x end var end quote conclude quote zero end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote object var var x end var end quote conclude quote object var var x end var end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote for all objects object var var x end var indeed object var var x end var end quote conclude quote object var var x end var end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote object var var x end var end quote conclude quote for all objects object var var y end var indeed object var var z end var end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote for all objects object var var x end var indeed object var var x end var end quote conclude quote for all objects object var var x end var indeed object var var x end var end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote zero infer zero end quote conclude quote zero imply zero end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote object var var x end var infer zero end quote conclude quote zero imply zero end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote zero infer object var var x end var end quote conclude quote zero imply zero end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote object var var x end var infer object var var x end var end quote conclude quote zero imply zero end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote zero infer zero end quote conclude quote for all objects object var var x end var indeed zero imply zero end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote object var var x end var infer zero end quote conclude quote for all objects object var var x end var indeed object var var x end var imply zero end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote zero infer object var var x end var end quote conclude quote for all objects object var var x end var indeed zero imply object var var x end var end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote object var var x end var infer object var var x end var end quote conclude quote for all objects object var var x end var indeed object var var x end var imply object var var x end var end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote zero infer zero end quote conclude quote zero imply for all objects object var var x end var indeed zero end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote object var var x end var infer zero end quote conclude quote zero imply for all objects object var var x end var indeed zero end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote zero infer object var var x end var end quote conclude quote zero imply for all objects object var var y end var indeed object var var z end var end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote object var var x end var infer object var var x end var end quote conclude quote zero imply for all objects object var var x end var indeed object var var x end var end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote zero infer zero end quote conclude quote for all objects object var var x end var indeed zero imply for all objects object var var x end var indeed zero end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote object var var x end var infer zero end quote conclude quote for all objects object var var x end var indeed object var var x end var imply for all objects object var var x end var indeed zero end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote zero infer object var var x end var end quote conclude quote for all objects object var var x end var indeed zero imply two end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote object var var x end var infer object var var x end var end quote conclude quote for all objects object var var x end var indeed object var var x end var imply three end quote end deduction apply check end test end math ] "
\item " [ math test lambda var x dot deduction zero quote object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var end quote conclude quote two plus three equal three plus two end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var end quote conclude quote two plus three equal two plus three end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var end quote conclude quote two plus three equal two plus two end quote end deduction apply check end test end math ] "
\item " [ math false test lambda var x dot deduction zero quote object var var x end var plus object var var y end var equal object var var y end var plus object var var x end var end quote conclude quote two plus three equal three plus three end quote end deduction apply check end test end math ] "
\end{list}
\section{Pyk definitions}
\begin{flushleft}
" [ math define pyk of general macro define x as x end define as text "general macro define "! as "! end define" end text end define linebreak define pyk of make root visible x end visible as text "make root visible "! end visible" end text end define linebreak define pyk of sequent example axiom as text "sequent example axiom" end text end define linebreak define pyk of sequent example rule as text "sequent example rule" end text end define linebreak define pyk of sequent example contradiction as text "sequent example contradiction" end text end define linebreak define pyk of sequent example theory as text "sequent example theory" end text end define linebreak define pyk of sequent example lemma as text "sequent example lemma" end text end define linebreak define pyk of set x end set as text "set "! end set" end text end define linebreak define pyk of object var x end var as text "object var "! end var" end text end define linebreak define pyk of object a as text "object a" end text end define linebreak define pyk of object b as text "object b" end text end define linebreak define pyk of object c as text "object c" end text end define linebreak define pyk of object d as text "object d" end text end define linebreak define pyk of object e as text "object e" end text end define linebreak define pyk of object f as text "object f" end text end define linebreak define pyk of object g as text "object g" end text end define linebreak define pyk of object h as text "object h" end text end define linebreak define pyk of object i as text "object i" end text end define linebreak define pyk of object j as text "object j" end text end define linebreak define pyk of object k as text "object k" end text end define linebreak define pyk of object l as text "object l" end text end define linebreak define pyk of object m as text "object m" end text end define linebreak define pyk of object n as text "object n" end text end define linebreak define pyk of object o as text "object o" end text end define linebreak define pyk of object p as text "object p" end text end define linebreak define pyk of object q as text "object q" end text end define linebreak define pyk of object r as text "object r" end text end define linebreak define pyk of object s as text "object s" end text end define linebreak define pyk of object t as text "object t" end text end define linebreak define pyk of object u as text "object u" end text end define linebreak define pyk of object v as text "object v" end text end define linebreak define pyk of object w as text "object w" end text end define linebreak define pyk of object x as text "object x" end text end define linebreak define pyk of object y as text "object y" end text end define linebreak define pyk of object z as text "object z" end text end define linebreak define pyk of sub x is x where x is x end sub as text "sub "! is "! where "! is "! end sub" end text end define linebreak define pyk of sub zero x is x where x is x end sub as text "sub zero "! is "! where "! is "! end sub" end text end define linebreak define pyk of sub one x is x where x is x end sub as text "sub one "! is "! where "! is "! end sub" end text end define linebreak define pyk of sub star x is x where x is x end sub as text "sub star "! is "! where "! is "! end sub" end text end define linebreak define pyk of deduction x conclude x end deduction as text "deduction "! conclude "! end deduction" end text end define linebreak define pyk of deduction zero x conclude x end deduction as text "deduction zero "! conclude "! end deduction" end text end define linebreak define pyk of deduction one x conclude x condition x end deduction as text "deduction one "! conclude "! condition "! end deduction" end text end define linebreak define pyk of deduction two x conclude x condition x end deduction as text "deduction two "! conclude "! condition "! end deduction" end text end define linebreak define pyk of deduction three x conclude x condition x bound x end deduction as text "deduction three "! conclude "! condition "! bound "! end deduction" end text end define linebreak define pyk of deduction four x conclude x condition x bound x end deduction as text "deduction four "! conclude "! condition "! bound "! end deduction" end text end define linebreak define pyk of deduction four star x conclude x condition x bound x end deduction as text "deduction four star "! conclude "! condition "! bound "! end deduction" end text end define linebreak define pyk of deduction five x condition x bound x end deduction as text "deduction five "! condition "! bound "! end deduction" end text end define linebreak define pyk of deduction six x conclude x exception x bound x end deduction as text "deduction six "! conclude "! exception "! bound "! end deduction" end text end define linebreak define pyk of deduction six star x conclude x exception x bound x end deduction as text "deduction six star "! conclude "! exception "! bound "! end deduction" end text end define linebreak define pyk of deduction seven x end deduction as text "deduction seven "! end deduction" end text end define linebreak define pyk of deduction eight x bound x end deduction as text "deduction eight "! bound "! end deduction" end text end define linebreak define pyk of deduction eight star x bound x end deduction as text "deduction eight star "! bound "! end deduction" end text end define linebreak define pyk of system s as text "system s" end text end define linebreak define pyk of double negation as text "double negation" end text end define linebreak define pyk of rule mp as text "rule mp" end text end define linebreak define pyk of rule gen as text "rule gen" end text end define linebreak define pyk of deduction as text "deduction" end text end define linebreak define pyk of axiom s one as text "axiom s one" end text end define linebreak define pyk of axiom s two as text "axiom s two" end text end define linebreak define pyk of axiom s three as text "axiom s three" end text end define linebreak define pyk of axiom s four as text "axiom s four" end text end define linebreak define pyk of axiom s five as text "axiom s five" end text end define linebreak define pyk of axiom s six as text "axiom s six" end text end define linebreak define pyk of axiom s seven as text "axiom s seven" end text end define linebreak define pyk of axiom s eight as text "axiom s eight" end text end define linebreak define pyk of axiom s nine as text "axiom s nine" end text end define linebreak define pyk of repetition as text "repetition" end text end define linebreak define pyk of lemma a one as text "lemma a one" end text end define linebreak define pyk of lemma a two as text "lemma a two" end text end define linebreak define pyk of lemma a four as text "lemma a four" end text end define linebreak define pyk of lemma a five as text "lemma a five" end text end define linebreak define pyk of prop three two a as text "prop three two a" end text end define linebreak define pyk of prop three two b as text "prop three two b" end text end define linebreak define pyk of prop three two c as text "prop three two c" end text end define linebreak define pyk of prop three two d as text "prop three two d" end text end define linebreak define pyk of prop three two e one as text "prop three two e one" end text end define linebreak define pyk of prop three two e two as text "prop three two e two" end text end define linebreak define pyk of prop three two e as text "prop three two e" end text end define linebreak define pyk of prop three two f one as text "prop three two f one" end text end define linebreak define pyk of prop three two f two as text "prop three two f two" end text end define linebreak define pyk of prop three two f as text "prop three two f" end text end define linebreak define pyk of prop three two g one as text "prop three two g one" end text end define linebreak define pyk of prop three two g two as text "prop three two g two" end text end define linebreak define pyk of prop three two g as text "prop three two g" end text end define linebreak define pyk of prop three two h one as text "prop three two h one" end text end define linebreak define pyk of prop three two h two as text "prop three two h two" end text end define linebreak define pyk of prop three two h as text "prop three two h" end text end define linebreak define pyk of prop three two i as text "prop three two i" end text end define linebreak define pyk of prop three two l one as text "prop three two l one" end text end define linebreak define pyk of block one x state x cache x end block as text "block one "! state "! cache "! end block" end text end define linebreak define pyk of block two x end block as text "block two "! end block" end text end define linebreak define pyk of x hide as text ""! hide" end text end define linebreak define pyk of macro indent x as text "macro indent "!" end text end define linebreak define pyk of x suc as text ""! suc" end text end define linebreak define pyk of x equal x as text ""! equal "!" end text end define linebreak define pyk of x unequal x as text ""! unequal "!" end text end define linebreak define pyk of x is object var as text ""! is object var" end text end define linebreak define pyk of x avoid zero x as text ""! avoid zero "!" end text end define linebreak define pyk of x avoid one x as text ""! avoid one "!" end text end define linebreak define pyk of x avoid star x as text ""! avoid star "!" end text end define linebreak define pyk of exist x indeed x as text "exist "! indeed "!" end text end define linebreak define pyk of for all x indeed x as text "for all "! indeed "!" end text end define linebreak define pyk of for all objects x indeed x as text "for all objects "! indeed "!" end text end define linebreak define pyk of x imply x as text ""! imply "!" end text end define linebreak define pyk of x if and only if x as text ""! if and only if "!" end text end define linebreak define pyk of x avoid x as text ""! avoid "!" end text end define linebreak define pyk of x object modus ponens x as text ""! object modus ponens "!" end text end define linebreak define pyk of for all terms x indeed x as text "for all terms "! indeed "!" end text end define linebreak define pyk of block x line x end block x as text "block "! line "! end block "!" end text end define linebreak define pyk of because x indeed x end line as text "because "! indeed "! end line" end text end define linebreak define pyk of any term x end line x as text "any term "! end line "!" end text end define linebreak define pyk of x alternative x as text ""! alternative "!" end text end define linebreak define pyk of evaluates to as text "evaluates to" end text end define linebreak define pyk of x safe row x as text ""! safe row "!" end text end define linebreak define pyk of check as text "check" end text end define linebreak unicode end of text end math ] "
\end{flushleft}
\section{\TeX\ definitions}
\begin{list}{}{
\setlength{\leftmargin}{5em}
\setlength{\itemindent}{-5em}}
\item " [ math define tex of general macro define var x as var y end define as text "
[#1/tex name/tex.
\stackrel {\circ}{=}#2.
]" end text end define end math ] "
\item " [ math define tex of make root visible var x end visible as text "#1/tex name/tex." end text end define end math ] "
\item " [ math define tex name of make root visible var x end visible as text "
RootVisible(#1.
)" end text end define end math ] "
\item " [ math define tex of var x hide as text "#1.
{}^{hide}" end text end define end math ] "
\item " [ math define tex of var x suc as text "#1.
{}'" end text end define end math ] "
\item " [ math define tex of var x equal var y as text "#1.
= #2." end text end define end math ] "
\item " [ math define tex of var x unequal var y as text "#1.
\neq #2." end text end define end math ] "
\item " [ math define tex of var x imply var y as text "#1.
\Rightarrow #2." end text end define end math ] "
\item " [ math define tex of var x if and only if var y as text "#1.
\Leftrightarrow #2." end text end define end math ] "
\item " [ math define tex of var x alternative var y as text "#1.
\mathrel{|} #2." end text end define end math ] "
\item " [ math define tex of exist var x indeed var y as text "
\exists #1.
\colon #2." end text end define end math ] "
\item " [ math define tex of for all var x indeed var y as text "
\forall #1.
\colon #2." end text end define end math ] "
\item " [ math define tex of for all objects var x indeed var y as text "
\forall_{obj} #1.
\colon #2." end text end define end math ] "
\item " [ math define tex of for all terms var x indeed var y as text "
\Pi #1.
\colon #2." end text end define end math ] "
\item " [ math define tex of any term var i end line var p as text "
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$#1.
$\hfill \makebox [0mm][l]{\quad ;}}#2." end text end define end math ] "
\item " [ math define tex name of any term var i end line var p as text "
Arbitrary \gg #1.
;#2." end text end define end math ] "
\item " [ math define tex of var x safe row var y as text "#1.
\\{}#2." end text end define end math ] "
\item " [ math define tex name of var x safe row var y as text "#1.
\backslash \backslash #2." end text end define end math ] "
\item " [ math define tex of sequent example axiom as text "
A" end text end define end math ] "
\item " [ math define tex of sequent example rule as text "
R" end text end define end math ] "
\item " [ math define tex of sequent example contradiction as text "
C" end text end define end math ] "
\item " [ math define tex of sequent example theory as text "
T" end text end define end math ] "
\item " [ math define tex of sequent example lemma as text "
L" end text end define end math ] "
\item " [ math define tex of set var x end set as text "
\{#1.
\}" end text end define end math ] "
\item " [ math define tex of system s as text "
S" end text end define end math ] "
\item " [ math define tex of double negation as text "
Neg" end text end define end math ] "
\item " [ math define tex of axiom s one as text "
S1" end text end define end math ] "
\item " [ math define tex of axiom s two as text "
S2" end text end define end math ] "
\item " [ math define tex of axiom s three as text "
S3" end text end define end math ] "
\item " [ math define tex of axiom s four as text "
S4" end text end define end math ] "
\item " [ math define tex of axiom s five as text "
S5" end text end define end math ] "
\item " [ math define tex of axiom s six as text "
S6" end text end define end math ] "
\item " [ math define tex of axiom s seven as text "
S7" end text end define end math ] "
\item " [ math define tex of axiom s eight as text "
S8" end text end define end math ] "
\item " [ math define tex of axiom s nine as text "
S9" end text end define end math ] "
\item " [ math define tex of rule mp as text "
MP" end text end define end math ] "
\item " [ math define tex of rule gen as text "
Gen" end text end define end math ] "
\item " [ math define tex of deduction as text "
Ded" end text end define end math ] "
\item " [ math define tex of repetition as text "
Repetition" end text end define end math ] "
\item " [ math define tex of lemma a one as text "
A1'" end text end define end math ] "
\item " [ math define tex of lemma a two as text "
A2'" end text end define end math ] "
\item " [ math define tex of lemma a four as text "
A4'" end text end define end math ] "
\item " [ math define tex of lemma a five as text "
A5'" end text end define end math ] "
\item " [ math define tex of var x is object var as text "#1.
{}^{var}" end text end define end math ] "
\item " [ math define tex of var x avoid var y as text "#1.
\#.#2." end text end define end math ] "
\item " [ math define tex of var x avoid zero var y as text "#1.
\#.^0#2." end text end define end math ] "
\item " [ math define tex of var x avoid one var y as text "#1.
\#.^1#2." end text end define end math ] "
\item " [ math define tex of var x avoid star var y as text "#1.
\#.^*#2." end text end define end math ] "
\item " [ math define tex of sub var x is var y where var z is var u end sub as text "
\langle #1.
{\equiv} #2.
| #3.
{:=} #4.
\rangle " end text end define end math ] "
\item " [ math define tex of sub zero var x is var y where var z is var u end sub as text "
\langle #1.
{\equiv}^0 #2.
| #3.
{:=} #4.
\rangle " end text end define end math ] "
\item " [ math define tex of sub one var x is var y where var z is var u end sub as text "
\langle #1.
{\equiv}^1 #2.
| #3.
{:=} #4.
\rangle " end text end define end math ] "
\item " [ math define tex of sub star var x is var y where var z is var u end sub as text "
\langle #1.
{\equiv}^* #2.
| #3.
{:=} #4.
\rangle " end text end define end math ] "
\item " [ math define tex of deduction var x conclude var y end deduction as text "
Ded(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of deduction zero var x conclude var y end deduction as text "
Ded_0(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of deduction one var x conclude var y condition var z end deduction as text "
Ded_1(#1.
,#2.
,#3.
)" end text end define end math ] "
\item " [ math define tex of deduction two var x conclude var y condition var z end deduction as text "
Ded_2(#1.
,#2.
,#3.
)" end text end define end math ] "
\item " [ math define tex of deduction three var x conclude var y condition var z bound var u end deduction as text "
Ded_3(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
\item " [ math define tex of deduction four var x conclude var y condition var z bound var u end deduction as text "
Ded_4(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
\item " [ math define tex of deduction four star var x conclude var y condition var z bound var u end deduction as text "
Ded_4^*(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
\item " [ math define tex of deduction five var x condition var y bound var z end deduction as text "
Ded_5(#1.
,#2.
,#3.
)" end text end define end math ] "
\item " [ math define tex of deduction six var p conclude var c exception var e bound var b end deduction as text "
Ded_6(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
\item " [ math define tex of deduction six star var p conclude var c exception var e bound var b end deduction as text "
Ded_6^*(#1.
,#2.
,#3.
,#4.
)" end text end define end math ] "
\item " [ math define tex of deduction seven var p end deduction as text "
Ded_7(#1.
)" end text end define end math ] "
\item " [ math define tex of deduction eight var p bound var b end deduction as text "
Ded_8(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of deduction eight star var p bound var b end deduction as text "
Ded_8^*(#1.
,#2.
)" end text end define end math ] "
\item " [ math define tex of block var b line var l end block var p as text "
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}#1.
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$#2.
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}#3." end text end define end math ] "
\item " [ math define tex name of block var b line var l end block var p as text "
Begin \, #1.
; #2.
: End ; #3." end text end define end math ] "
\item " [ math define tex of because var a indeed var i end line as text "
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi
$:}$#1.
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$#2.
$\hfill \makebox [0mm][l]{\quad ;}}" end text end define end math ] "
" [ math define tex name of because var a indeed var i end line as text "
Last\ block\ line \, #1.
\gg #2.
\,;" end text end define end math ] "
\item " [ math define tex of var x object modus ponens var y as text "#1.
\unrhd #2." end text end define end math ] "
\item " [ math define tex of prop three two a as text "
Prop\ 3.2a" end text end define end math ] "
\item " [ math define tex of prop three two b as text "
Prop\ 3.2b" end text end define end math ] "
\item " [ math define tex of prop three two c as text "
Prop\ 3.2c" end text end define end math ] "
\item " [ math define tex of prop three two d as text "
Prop\ 3.2d" end text end define end math ] "
\item " [ math define tex of prop three two e one as text "
Prop\ 3.2e_1" end text end define end math ] "
\item " [ math define tex of prop three two e two as text "
Prop\ 3.2e_2" end text end define end math ] "
\item " [ math define tex of prop three two e as text "
Prop\ 3.2e" end text end define end math ] "
\item " [ math define tex of prop three two f one as text "
Prop\ 3.2f_1" end text end define end math ] "
\item " [ math define tex of prop three two f two as text "
Prop\ 3.2f_2" end text end define end math ] "
\item " [ math define tex of prop three two f as text "
Prop\ 3.2f" end text end define end math ] "
\item " [ math define tex of prop three two g one as text "
Prop\ 3.2g_1" end text end define end math ] "
\item " [ math define tex of prop three two g two as text "
Prop\ 3.2g_2" end text end define end math ] "
\item " [ math define tex of prop three two g as text "
Prop\ 3.2g" end text end define end math ] "
\item " [ math define tex of prop three two h one as text "
Prop\ 3.2h_1" end text end define end math ] "
\item " [ math define tex of prop three two h two as text "
Prop\ 3.2h_2" end text end define end math ] "
\item " [ math define tex of prop three two h as text "
Prop\ 3.2h" end text end define end math ] "
\item " [ math define tex of prop three two i as text "
Prop\ 3.2i" end text end define end math ] "
\item " [ math define tex of prop three two l one as text "
Prop\ 3.2l_1" end text end define end math ] "
\item " [ math define tex of macro indent var x as text "
$%
\leftskip=1em%
$#1." end text end define end math ] "
\item " [ math define tex name of macro indent var x as text "
MacroIndent(#1.
)" end text end define end math ] "
\item " [ math define tex of block one var t state var s cache var c end block as text "
Block_1(#1.
,#2.
,#3.
)" end text end define end math ] "
\item " [ math define tex of block two var b end block as text "
Block_2(#1.
)" end text end define end math ] "
\item " [ math define tex of evaluates to as text "
\rightarrow " end text end define end math ] "
\end{list}
\subsection{Variables}
" [ math define tex of object var var x end var as text "\overline{#1.}" end text end define end math ] "
" [ math define macro of object a 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 a as object var var a end var end define end quote end define end define end math ] "
" [ math define macro of object b 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 b as object var var b end var end define end quote end define end define end math ] "
" [ math define macro of object c 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 c as object var var c end var end define end quote end define end define end math ] "
" [ math define macro of object d 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 d as object var var d end var end define end quote end define end define end math ] "
" [ math define macro of object e 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 e as object var var e end var end define end quote end define end define end math ] "
" [ math define macro of object f 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 f as object var var f end var end define end quote end define end define end math ] "
" [ math define macro of object g 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 g as object var var g end var end define end quote end define end define end math ] "
" [ math define macro of object h 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 h as object var var h end var end define end quote end define end define end math ] "
" [ math define macro of object i 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 i as object var var i end var end define end quote end define end define end math ] "
" [ math define macro of object j 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 j as object var var j end var end define end quote end define end define end math ] "
" [ math define macro of object k 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 k as object var var k end var end define end quote end define end define end math ] "
" [ math define macro of object l 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 l as object var var l end var end define end quote end define end define end math ] "
" [ math define macro of object m 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 m as object var var m end var end define end quote end define end define end math ] "
" [ math define macro of object n 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 n as object var var n end var end define end quote end define end define end math ] "
" [ math define macro of object o 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 o as object var var o end var end define end quote end define end define end math ] "
" [ math define macro of object p 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 p as object var var p end var end define end quote end define end define end math ] "
" [ math define macro of object q 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 q as object var var q end var end define end quote end define end define end math ] "
" [ math define macro of object 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 object r as object var var r end var end define end quote end define end define end math ] "
" [ math define macro of object s 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 s as object var var s end var end define end quote end define end define end math ] "
" [ math define macro of object t 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 t as object var var t end var end define end quote end define end define end math ] "
" [ math define macro of object u 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 u as object var var u end var end define end quote end define end define end math ] "
" [ math define macro of object v 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 v as object var var v end var end define end quote end define end define end math ] "
" [ math define macro of object w 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 w as object var var w end var end define end quote end define end define end math ] "
" [ math define macro of object 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 object x as object var var x end var end define end quote end define end define end math ] "
" [ math define macro of object 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 object y as object var var y end var end define end quote end define end define end math ] "
" [ math define macro of object z 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 z as object var var z end var end define end quote end define end define end math ] "
" [ math define tex of object a as text "
\mathit{a}" end text end define end math ] "
" [ math define tex of object b as text "
\mathit{b}" end text end define end math ] "
" [ math define tex of object c as text "
\mathit{c}" end text end define end math ] "
" [ math define tex of object d as text "
\mathit{d}" end text end define end math ] "
" [ math define tex of object e as text "
\mathit{e}" end text end define end math ] "
" [ math define tex of object f as text "
\mathit{f}" end text end define end math ] "
" [ math define tex of object g as text "
\mathit{g}" end text end define end math ] "
" [ math define tex of object h as text "
\mathit{h}" end text end define end math ] "
" [ math define tex of object i as text "
\mathit{i}" end text end define end math ] "
" [ math define tex of object j as text "
\mathit{j}" end text end define end math ] "
" [ math define tex of object k as text "
\mathit{k}" end text end define end math ] "
" [ math define tex of object l as text "
\mathit{l}" end text end define end math ] "
" [ math define tex of object m as text "
\mathit{m}" end text end define end math ] "
" [ math define tex of object n as text "
\mathit{n}" end text end define end math ] "
" [ math define tex of object o as text "
\mathit{o}" end text end define end math ] "
" [ math define tex of object p as text "
\mathit{p}" end text end define end math ] "
" [ math define tex of object q as text "
\mathit{q}" end text end define end math ] "
" [ math define tex of object r as text "
\mathit{r}" end text end define end math ] "
" [ math define tex of object s as text "
\mathit{s}" end text end define end math ] "
" [ math define tex of object t as text "
\mathit{t}" end text end define end math ] "
" [ math define tex of object u as text "
\mathit{u}" end text end define end math ] "
" [ math define tex of object v as text "
\mathit{v}" end text end define end math ] "
" [ math define tex of object w as text "
\mathit{w}" end text end define end math ] "
" [ math define tex of object x as text "
\mathit{x}" end text end define end math ] "
" [ math define tex of object y as text "
\mathit{y}" end text end define end math ] "
" [ math define tex of object z as text "
\mathit{z}" end text end define end math ] "
\section{Priority table}
" [ flush left math define priority of check as preassociative priority check 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 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 prop three two i equal priority prop three two l one equal priority block one x state x cache x end block equal priority block two x end block end priority greater than preassociative priority x sub x end sub equal priority x intro x index x pyk x tex x end intro equal priority x intro x pyk x tex x end intro equal priority x intro x index x pyk x tex x name x end intro equal priority x intro x pyk x tex x name x end intro equal priority x prime equal priority x assoc x end assoc equal priority x set x to x end set equal priority x set multi x to x end set equal priority x bit nil equal priority x bit one equal priority binary equal priority x color x end color equal priority x color star x end color equal priority x raw head equal priority x raw tail equal priority x cardinal untag equal priority x head equal priority x tail equal priority x is singular equal priority x is cardinal equal priority x is data equal priority x is atomic equal priority x cardinal retract equal priority x tagged retract equal priority x boolean retract equal priority x ref equal priority x id equal priority x debug equal priority x root equal priority x zeroth equal priority x first equal priority x second equal priority x third equal priority x fourth equal priority x fifth equal priority x sixth equal priority x seventh equal priority x eighth equal priority x ninth equal priority x is error equal priority x is metavar equal priority x is metaclosed equal priority x is metaclosed star equal priority x hide end priority greater than preassociative priority unicode start of text x end unicode text equal priority unicode end of text equal priority text x end text equal priority text x plus x equal priority text x plus indent x equal priority unicode newline x equal priority unicode space x equal priority unicode exclamation mark x equal priority unicode quotation mark x equal priority unicode number sign x equal priority unicode dollar sign x equal priority unicode percent x equal priority unicode ampersand x equal priority unicode apostrophe x equal priority unicode left parenthesis x equal priority unicode right parenthesis x equal priority unicode asterisk x equal priority unicode plus sign x equal priority unicode comma x equal priority unicode hyphen x equal priority unicode period x equal priority unicode slash x equal priority unicode zero x equal priority unicode one x equal priority unicode two x equal priority unicode three x equal priority unicode four x equal priority unicode five x equal priority unicode six x equal priority unicode seven x equal priority unicode eight x equal priority unicode nine x equal priority unicode colon x equal priority unicode semicolon x equal priority unicode less than x equal priority unicode equal sign x equal priority unicode greater than x equal priority unicode question mark x equal priority unicode commercial at x equal priority unicode capital a x equal priority unicode capital b x equal priority unicode capital c x equal priority unicode capital d x equal priority unicode capital e x equal priority unicode capital f x equal priority unicode capital g x equal priority unicode capital h x equal priority unicode capital i x equal priority unicode capital j x equal priority unicode capital k x equal priority unicode capital l x equal priority unicode capital m x equal priority unicode capital n x equal priority unicode capital o x equal priority unicode capital p x equal priority unicode capital q x equal priority unicode capital r x equal priority unicode capital s x equal priority unicode capital t x equal priority unicode capital u x equal priority unicode capital v x equal priority unicode capital w x equal priority unicode capital x x equal priority unicode capital y x equal priority unicode capital z x equal priority unicode left bracket x equal priority unicode backslash x equal priority unicode right bracket x equal priority unicode circumflex x equal priority unicode underscore x equal priority unicode grave accent x equal priority unicode small a x equal priority unicode small b x equal priority unicode small c x equal priority unicode small d x equal priority unicode small e x equal priority unicode small f x equal priority unicode small g x equal priority unicode small h x equal priority unicode small i x equal priority unicode small j x equal priority unicode small k x equal priority unicode small l x equal priority unicode small m x equal priority unicode small n x equal priority unicode small o x equal priority unicode small p x equal priority unicode small q x equal priority unicode small r x equal priority unicode small s x equal priority unicode small t x equal priority unicode small u x equal priority unicode small v x equal priority unicode small w x equal priority unicode small x x equal priority unicode small y x equal priority unicode small z x equal priority unicode left brace x equal priority unicode vertical line x equal priority unicode right brace x equal priority unicode tilde x equal priority preassociative x greater than x equal priority postassociative x greater than x equal priority priority x equal x equal priority priority x end priority equal priority newline x equal priority macro newline x equal priority macro indent x end priority greater than preassociative priority x apply x equal priority x tagged apply x end priority greater than preassociative priority x suc end priority greater than preassociative priority x times x equal priority x times zero x end priority greater than preassociative priority x plus x equal priority x plus zero x equal priority x plus one x equal priority x minus x equal priority x minus zero x equal priority x minus one x end priority greater than preassociative priority x term plus x end plus equal priority x term union x equal priority x term minus x end minus end priority greater than postassociative priority x raw pair x equal priority x eager pair x equal priority x tagged pair x equal priority x untagged double x equal priority x pair x equal priority x double x end priority greater than postassociative priority x comma x end priority greater than preassociative priority x boolean equal x equal priority x data equal x equal priority x cardinal equal x equal priority x peano equal x equal priority x tagged equal x equal priority x math equal x equal priority x reduce to x equal priority x term equal x equal priority x term list equal x equal priority x term root equal x equal priority x term in x equal priority x term subset x equal priority x term set equal x equal priority x sequent equal x equal priority x free in x equal priority x free in star x equal priority x free for x in x equal priority x free for star x in x equal priority x claim in x equal priority x less x equal priority x less zero x equal priority x less one x equal priority x equal x equal priority x unequal x equal priority x is object var equal priority x avoid zero x equal priority x avoid one x equal priority x avoid star x end priority greater than preassociative priority not x end priority greater than preassociative priority x and x equal priority x macro and x equal priority x simple and x equal priority x claim and x end priority greater than preassociative priority x or x equal priority x parallel x equal priority x macro or x end priority greater than preassociative priority exist x indeed x equal priority for all x indeed x equal priority for all objects x indeed x end priority greater than postassociative priority x macro imply x equal priority x imply x equal priority x if and only if x end priority greater than postassociative priority x guard x equal priority x spy x equal priority x tagged guard x end priority greater than preassociative priority x select x else x end select end priority greater than preassociative priority lambda x dot x equal priority tagged lambda x dot x equal priority tagging x equal priority open if x then x else x equal priority let x be x in x equal priority let x abbreviate x in x end priority greater than preassociative priority x avoid x end priority greater than preassociative priority x init equal priority x modus equal priority x verify equal priority x curry plus equal priority x curry minus equal priority x dereference end priority greater than preassociative priority x at x equal priority x modus ponens x equal priority x modus probans x equal priority x conclude x equal priority x object modus ponens x end priority greater than postassociative priority x infer x equal priority x endorse x equal priority x id est x end priority greater than preassociative priority all x indeed x equal priority for all terms x indeed x end priority greater than postassociative priority x rule plus x end priority greater than postassociative priority x cut x end priority greater than preassociative priority x proves x end priority greater than preassociative priority x proof of x reads x equal priority line x because x indeed x end line x equal priority because x indeed x qed equal priority line x premise x end line x equal priority line x side condition x end line x equal priority arbitrary x end line x equal priority locally define x as x end line x equal priority block x line x end block x equal priority because x indeed x end line equal priority any term x end line x end priority greater than postassociative priority x alternative x end priority greater than postassociative priority x , x equal priority x [ x ] x end priority greater than preassociative priority x tab x equal priority evaluates to end priority greater than preassociative priority x row x equal priority x linebreak x equal priority x safe row x end priority greater than unicode end of text end define end math end left ] "
\end{document}
End of file
latex chores
latex chores
dvipdfm chores"
The pyk compiler, version 0.grue.20060417+ by Klaus Grue,