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\begin{isabellebody}%
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\def\isabellecontext{First{\isacharunderscore}Order{\isacharunderscore}Logic}%
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%
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\isamarkupheader{Example: First-Order Logic%
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}
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\isamarkuptrue%
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%
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\isadelimvisible
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%
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\endisadelimvisible
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%
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\isatagvisible
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\isacommand{theory}\isamarkupfalse%
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\ First{\isacharunderscore}Order{\isacharunderscore}Logic\isanewline
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\isakeyword{imports}\ Pure\isanewline
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\isakeyword{begin}%
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\endisatagvisible
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{\isafoldvisible}%
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%
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\isadelimvisible
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%
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\endisadelimvisible
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%
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\begin{isamarkuptext}%
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\noindent In order to commence a new object-logic within
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Isabelle/Pure we introduce abstract syntactic categories \isa{{\isachardoublequote}i{\isachardoublequote}}
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for individuals and \isa{{\isachardoublequote}o{\isachardoublequote}} for object-propositions. The latter
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is embedded into the language of Pure propositions by means of a
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separate judgment.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{typedecl}\isamarkupfalse%
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\ i\isanewline
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\isacommand{typedecl}\isamarkupfalse%
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\ o\isanewline
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\isanewline
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\isacommand{judgment}\isamarkupfalse%
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\isanewline
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\ \ Trueprop\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}o\ {\isasymRightarrow}\ prop{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}{\isachardoublequoteopen}{\isacharunderscore}{\isachardoublequoteclose}\ {\isadigit{5}}{\isacharparenright}%
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\begin{isamarkuptext}%
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\noindent Note that the object-logic judgement is implicit in the
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syntax: writing \isa{A} produces \isa{{\isachardoublequote}Trueprop\ A{\isachardoublequote}} internally.
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From the Pure perspective this means ``\isa{A} is derivable in the
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object-logic''.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Equational reasoning \label{sec:framework-ex-equal}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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Equality is axiomatized as a binary predicate on individuals, with
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reflexivity as introduction, and substitution as elimination
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principle. Note that the latter is particularly convenient in a
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framework like Isabelle, because syntactic congruences are
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implicitly produced by unification of \isa{{\isachardoublequote}B\ x{\isachardoublequote}} against
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expressions containing occurrences of \isa{x}.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{axiomatization}\isamarkupfalse%
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\isanewline
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\ \ equal\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}i\ {\isasymRightarrow}\ i\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{infix}\ {\isachardoublequoteopen}{\isacharequal}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\isanewline
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\isakeyword{where}\isanewline
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\ \ refl\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
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\ \ subst\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isacharequal}\ y\ {\isasymLongrightarrow}\ B\ x\ {\isasymLongrightarrow}\ B\ y{\isachardoublequoteclose}%
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\begin{isamarkuptext}%
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\noindent Substitution is very powerful, but also hard to control in
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full generality. We derive some common symmetry~/ transitivity
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schemes of as particular consequences.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{theorem}\isamarkupfalse%
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\ sym\ {\isacharbrackleft}sym{\isacharbrackright}{\isacharcolon}\isanewline
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\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}x\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
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\ \ \isakeyword{shows}\ {\isachardoublequoteopen}y\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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\isacommand{proof}\isamarkupfalse%
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\ {\isacharminus}\isanewline
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\ \ \isacommand{have}\isamarkupfalse%
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\ {\isachardoublequoteopen}x\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
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\isanewline
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\ \ \isacommand{with}\isamarkupfalse%
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\ {\isacharbackquoteopen}x\ {\isacharequal}\ y{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}y\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
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\isanewline
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\isacommand{qed}\isamarkupfalse%
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%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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\isanewline
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%
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\endisadelimproof
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\isanewline
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\isacommand{theorem}\isamarkupfalse%
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\ forw{\isacharunderscore}subst\ {\isacharbrackleft}trans{\isacharbrackright}{\isacharcolon}\isanewline
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\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}y\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}B\ x{\isachardoublequoteclose}\isanewline
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\ \ \isakeyword{shows}\ {\isachardoublequoteopen}B\ y{\isachardoublequoteclose}\isanewline
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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\isacommand{proof}\isamarkupfalse%
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\ {\isacharminus}\isanewline
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\ \ \isacommand{from}\isamarkupfalse%
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\ {\isacharbackquoteopen}y\ {\isacharequal}\ x{\isacharbackquoteclose}\ \isacommand{have}\isamarkupfalse%
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\ {\isachardoublequoteopen}x\ {\isacharequal}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
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\isanewline
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\ \ \isacommand{from}\isamarkupfalse%
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\ this\ \isakeyword{and}\ {\isacharbackquoteopen}B\ x{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}B\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
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\isanewline
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\isacommand{qed}\isamarkupfalse%
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%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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\isanewline
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%
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\endisadelimproof
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\isanewline
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\isacommand{theorem}\isamarkupfalse%
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\ back{\isacharunderscore}subst\ {\isacharbrackleft}trans{\isacharbrackright}{\isacharcolon}\isanewline
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\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}B\ x{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}x\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
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\ \ \isakeyword{shows}\ {\isachardoublequoteopen}B\ y{\isachardoublequoteclose}\isanewline
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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\isacommand{proof}\isamarkupfalse%
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\ {\isacharminus}\isanewline
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\ \ \isacommand{from}\isamarkupfalse%
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\ {\isacharbackquoteopen}x\ {\isacharequal}\ y{\isacharbackquoteclose}\ \isakeyword{and}\ {\isacharbackquoteopen}B\ x{\isacharbackquoteclose}\isanewline
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\ \ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}B\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
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\isanewline
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\isacommand{qed}\isamarkupfalse%
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%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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\isanewline
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%
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\endisadelimproof
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\isanewline
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\isacommand{theorem}\isamarkupfalse%
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\ trans\ {\isacharbrackleft}trans{\isacharbrackright}{\isacharcolon}\isanewline
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\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}x\ {\isacharequal}\ y{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}y\ {\isacharequal}\ z{\isachardoublequoteclose}\isanewline
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\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isacharequal}\ z{\isachardoublequoteclose}\isanewline
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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\isacommand{proof}\isamarkupfalse%
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\ {\isacharminus}\isanewline
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\ \ \isacommand{from}\isamarkupfalse%
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\ {\isacharbackquoteopen}y\ {\isacharequal}\ z{\isacharbackquoteclose}\ \isakeyword{and}\ {\isacharbackquoteopen}x\ {\isacharequal}\ y{\isacharbackquoteclose}\isanewline
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\ \ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}x\ {\isacharequal}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
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\isanewline
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\isacommand{qed}\isamarkupfalse%
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%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isamarkupsubsection{Basic group theory%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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As an example for equational reasoning we consider some bits of
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group theory. The subsequent locale definition postulates group
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operations and axioms; we also derive some consequences of this
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specification.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{locale}\isamarkupfalse%
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\ group\ {\isacharequal}\isanewline
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\ \ \isakeyword{fixes}\ prod\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}i\ {\isasymRightarrow}\ i\ {\isasymRightarrow}\ i{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{infix}\ {\isachardoublequoteopen}{\isasymcirc}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
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\ \ \ \ \isakeyword{and}\ inv\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}i\ {\isasymRightarrow}\ i{\isachardoublequoteclose}\ \ {\isacharparenleft}{\isachardoublequoteopen}{\isacharparenleft}{\isacharunderscore}{\isasyminverse}{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbrackleft}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isacharparenright}\isanewline
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\ \ \ \ \isakeyword{and}\ unit\ {\isacharcolon}{\isacharcolon}\ i\ \ {\isacharparenleft}{\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}\isanewline
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\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymcirc}\ y{\isacharparenright}\ {\isasymcirc}\ z\ {\isacharequal}\ x\ {\isasymcirc}\ {\isacharparenleft}y\ {\isasymcirc}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isakeyword{and}\ left{\isacharunderscore}unit{\isacharcolon}\ \ {\isachardoublequoteopen}{\isadigit{1}}\ {\isasymcirc}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isakeyword{and}\ left{\isacharunderscore}inv{\isacharcolon}\ {\isachardoublequoteopen}x{\isasyminverse}\ {\isasymcirc}\ x\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
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\isakeyword{begin}\isanewline
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\isanewline
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\isacommand{theorem}\isamarkupfalse%
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\ right{\isacharunderscore}inv{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isatagproof
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\isacommand{proof}\isamarkupfalse%
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\ {\isacharminus}\isanewline
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\ \ \isacommand{have}\isamarkupfalse%
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\ {\isachardoublequoteopen}x\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ {\isadigit{1}}\ {\isasymcirc}\ {\isacharparenleft}x\ {\isasymcirc}\ x{\isasyminverse}{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
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\ {\isacharparenleft}rule\ left{\isacharunderscore}unit\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline
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\ \ \isacommand{also}\isamarkupfalse%
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\ \isacommand{have}\isamarkupfalse%
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\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{1}}\ {\isasymcirc}\ x{\isacharparenright}\ {\isasymcirc}\ x{\isasyminverse}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
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\ {\isacharparenleft}rule\ assoc\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline
|
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|
224 |
\ \ \isacommand{also}\isamarkupfalse%
|
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|
225 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
226 |
\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminverse}{\isacharparenright}{\isasyminverse}\ {\isasymcirc}\ x{\isasyminverse}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30057
|
227 |
\ {\isacharparenleft}rule\ left{\isacharunderscore}inv\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline
|
wenzelm@30057
|
228 |
\ \ \isacommand{also}\isamarkupfalse%
|
wenzelm@30057
|
229 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
230 |
\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymcirc}\ x\ {\isacharequal}\ {\isacharparenleft}x{\isasyminverse}{\isacharparenright}{\isasyminverse}\ {\isasymcirc}\ {\isacharparenleft}x{\isasyminverse}\ {\isasymcirc}\ x{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30057
|
231 |
\ {\isacharparenleft}rule\ assoc{\isacharparenright}\isanewline
|
wenzelm@30057
|
232 |
\ \ \isacommand{also}\isamarkupfalse%
|
wenzelm@30057
|
233 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
234 |
\ {\isachardoublequoteopen}x{\isasyminverse}\ {\isasymcirc}\ x\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30057
|
235 |
\ {\isacharparenleft}rule\ left{\isacharunderscore}inv{\isacharparenright}\isanewline
|
wenzelm@30057
|
236 |
\ \ \isacommand{also}\isamarkupfalse%
|
wenzelm@30057
|
237 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
238 |
\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharparenleft}x{\isasyminverse}{\isacharparenright}{\isasyminverse}\ {\isasymcirc}\ {\isasymdots}{\isacharparenright}\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminverse}{\isacharparenright}{\isasyminverse}\ {\isasymcirc}\ {\isacharparenleft}{\isadigit{1}}\ {\isasymcirc}\ x{\isasyminverse}{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30057
|
239 |
\ {\isacharparenleft}rule\ assoc{\isacharparenright}\isanewline
|
wenzelm@30057
|
240 |
\ \ \isacommand{also}\isamarkupfalse%
|
wenzelm@30057
|
241 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
242 |
\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ x{\isasyminverse}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30057
|
243 |
\ {\isacharparenleft}rule\ left{\isacharunderscore}unit{\isacharparenright}\isanewline
|
wenzelm@30057
|
244 |
\ \ \isacommand{also}\isamarkupfalse%
|
wenzelm@30057
|
245 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
246 |
\ {\isachardoublequoteopen}{\isacharparenleft}x{\isasyminverse}{\isacharparenright}{\isasyminverse}\ {\isasymcirc}\ {\isasymdots}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30057
|
247 |
\ {\isacharparenleft}rule\ left{\isacharunderscore}inv{\isacharparenright}\isanewline
|
wenzelm@30057
|
248 |
\ \ \isacommand{finally}\isamarkupfalse%
|
wenzelm@30057
|
249 |
\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30057
|
250 |
\ {\isachardoublequoteopen}x\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
251 |
\isanewline
|
wenzelm@30057
|
252 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30057
|
253 |
%
|
wenzelm@30057
|
254 |
\endisatagproof
|
wenzelm@30057
|
255 |
{\isafoldproof}%
|
wenzelm@30057
|
256 |
%
|
wenzelm@30057
|
257 |
\isadelimproof
|
wenzelm@30057
|
258 |
\isanewline
|
wenzelm@30057
|
259 |
%
|
wenzelm@30057
|
260 |
\endisadelimproof
|
wenzelm@30057
|
261 |
\isanewline
|
wenzelm@30057
|
262 |
\isacommand{theorem}\isamarkupfalse%
|
wenzelm@30057
|
263 |
\ right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymcirc}\ {\isadigit{1}}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
264 |
%
|
wenzelm@30057
|
265 |
\isadelimproof
|
wenzelm@30057
|
266 |
%
|
wenzelm@30057
|
267 |
\endisadelimproof
|
wenzelm@30057
|
268 |
%
|
wenzelm@30057
|
269 |
\isatagproof
|
wenzelm@30057
|
270 |
\isacommand{proof}\isamarkupfalse%
|
wenzelm@30057
|
271 |
\ {\isacharminus}\isanewline
|
wenzelm@30057
|
272 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
273 |
\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharequal}\ x{\isasyminverse}\ {\isasymcirc}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30057
|
274 |
\ {\isacharparenleft}rule\ left{\isacharunderscore}inv\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline
|
wenzelm@30057
|
275 |
\ \ \isacommand{also}\isamarkupfalse%
|
wenzelm@30057
|
276 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
277 |
\ {\isachardoublequoteopen}x\ {\isasymcirc}\ {\isasymdots}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymcirc}\ x{\isasyminverse}{\isacharparenright}\ {\isasymcirc}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30057
|
278 |
\ {\isacharparenleft}rule\ assoc\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline
|
wenzelm@30057
|
279 |
\ \ \isacommand{also}\isamarkupfalse%
|
wenzelm@30057
|
280 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
281 |
\ {\isachardoublequoteopen}x\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30057
|
282 |
\ {\isacharparenleft}rule\ right{\isacharunderscore}inv{\isacharparenright}\isanewline
|
wenzelm@30057
|
283 |
\ \ \isacommand{also}\isamarkupfalse%
|
wenzelm@30057
|
284 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
285 |
\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymcirc}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30057
|
286 |
\ {\isacharparenleft}rule\ left{\isacharunderscore}unit{\isacharparenright}\isanewline
|
wenzelm@30057
|
287 |
\ \ \isacommand{finally}\isamarkupfalse%
|
wenzelm@30057
|
288 |
\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30057
|
289 |
\ {\isachardoublequoteopen}x\ {\isasymcirc}\ {\isadigit{1}}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
290 |
\isanewline
|
wenzelm@30057
|
291 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30057
|
292 |
%
|
wenzelm@30057
|
293 |
\endisatagproof
|
wenzelm@30057
|
294 |
{\isafoldproof}%
|
wenzelm@30057
|
295 |
%
|
wenzelm@30057
|
296 |
\isadelimproof
|
wenzelm@30057
|
297 |
%
|
wenzelm@30057
|
298 |
\endisadelimproof
|
wenzelm@30057
|
299 |
%
|
wenzelm@30057
|
300 |
\begin{isamarkuptext}%
|
wenzelm@30066
|
301 |
\noindent Reasoning from basic axioms is often tedious. Our proofs
|
wenzelm@30066
|
302 |
work by producing various instances of the given rules (potentially
|
wenzelm@30066
|
303 |
the symmetric form) using the pattern ``\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{eq}~\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isachardoublequote}{\isacharparenleft}rule\ r{\isacharparenright}{\isachardoublequote}}'' and composing the chain of
|
wenzelm@30057
|
304 |
results via \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}/\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}. These steps may
|
wenzelm@30057
|
305 |
involve any of the transitivity rules declared in
|
wenzelm@30057
|
306 |
\secref{sec:framework-ex-equal}, namely \isa{trans} in combining
|
wenzelm@30057
|
307 |
the first two results in \isa{right{\isacharunderscore}inv} and in the final steps of
|
wenzelm@30057
|
308 |
both proofs, \isa{forw{\isacharunderscore}subst} in the first combination of \isa{right{\isacharunderscore}unit}, and \isa{back{\isacharunderscore}subst} in all other calculational steps.
|
wenzelm@30057
|
309 |
|
wenzelm@30057
|
310 |
Occasional substitutions in calculations are adequate, but should
|
wenzelm@30057
|
311 |
not be over-emphasized. The other extreme is to compose a chain by
|
wenzelm@30057
|
312 |
plain transitivity only, with replacements occurring always in
|
wenzelm@30057
|
313 |
topmost position. For example:%
|
wenzelm@30057
|
314 |
\end{isamarkuptext}%
|
wenzelm@30057
|
315 |
\isamarkuptrue%
|
wenzelm@30057
|
316 |
%
|
wenzelm@30057
|
317 |
\isadelimproof
|
wenzelm@30057
|
318 |
%
|
wenzelm@30057
|
319 |
\endisadelimproof
|
wenzelm@30057
|
320 |
%
|
wenzelm@30057
|
321 |
\isatagproof
|
wenzelm@30057
|
322 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
323 |
\ {\isachardoublequoteopen}x\ {\isasymcirc}\ {\isadigit{1}}\ {\isacharequal}\ x\ {\isasymcirc}\ {\isacharparenleft}x{\isasyminverse}\ {\isasymcirc}\ x{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{unfolding}\isamarkupfalse%
|
wenzelm@30057
|
324 |
\ left{\isacharunderscore}inv\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
325 |
\isanewline
|
wenzelm@30057
|
326 |
\ \ \isacommand{also}\isamarkupfalse%
|
wenzelm@30057
|
327 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
328 |
\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymcirc}\ x{\isasyminverse}{\isacharparenright}\ {\isasymcirc}\ x{\isachardoublequoteclose}\ \isacommand{unfolding}\isamarkupfalse%
|
wenzelm@30057
|
329 |
\ assoc\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
330 |
\isanewline
|
wenzelm@30057
|
331 |
\ \ \isacommand{also}\isamarkupfalse%
|
wenzelm@30057
|
332 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
333 |
\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ {\isadigit{1}}\ {\isasymcirc}\ x{\isachardoublequoteclose}\ \isacommand{unfolding}\isamarkupfalse%
|
wenzelm@30057
|
334 |
\ right{\isacharunderscore}inv\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
335 |
\isanewline
|
wenzelm@30057
|
336 |
\ \ \isacommand{also}\isamarkupfalse%
|
wenzelm@30057
|
337 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
338 |
\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{unfolding}\isamarkupfalse%
|
wenzelm@30057
|
339 |
\ left{\isacharunderscore}unit\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
340 |
\isanewline
|
wenzelm@30057
|
341 |
\ \ \isacommand{finally}\isamarkupfalse%
|
wenzelm@30057
|
342 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
343 |
\ {\isachardoublequoteopen}x\ {\isasymcirc}\ {\isadigit{1}}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
344 |
%
|
wenzelm@30057
|
345 |
\endisatagproof
|
wenzelm@30057
|
346 |
{\isafoldproof}%
|
wenzelm@30057
|
347 |
%
|
wenzelm@30057
|
348 |
\isadelimproof
|
wenzelm@30057
|
349 |
%
|
wenzelm@30057
|
350 |
\endisadelimproof
|
wenzelm@30057
|
351 |
%
|
wenzelm@30057
|
352 |
\begin{isamarkuptext}%
|
wenzelm@30057
|
353 |
\noindent Here we have re-used the built-in mechanism for unfolding
|
wenzelm@30057
|
354 |
definitions in order to normalize each equational problem. A more
|
wenzelm@30057
|
355 |
realistic object-logic would include proper setup for the Simplifier
|
wenzelm@30057
|
356 |
(\secref{sec:simplifier}), the main automated tool for equational
|
wenzelm@30066
|
357 |
reasoning in Isabelle. Then ``\hyperlink{command.unfolding}{\mbox{\isa{\isacommand{unfolding}}}}~\isa{left{\isacharunderscore}inv}~\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isachardot}{\isachardot}}}}}'' would become ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isachardoublequote}{\isacharparenleft}simp\ only{\isacharcolon}\ left{\isacharunderscore}inv{\isacharparenright}{\isachardoublequote}}'' etc.%
|
wenzelm@30057
|
358 |
\end{isamarkuptext}%
|
wenzelm@30057
|
359 |
\isamarkuptrue%
|
wenzelm@30057
|
360 |
\isacommand{end}\isamarkupfalse%
|
wenzelm@30057
|
361 |
%
|
wenzelm@30066
|
362 |
\isamarkupsubsection{Propositional logic \label{sec:framework-ex-prop}%
|
wenzelm@30057
|
363 |
}
|
wenzelm@30057
|
364 |
\isamarkuptrue%
|
wenzelm@30057
|
365 |
%
|
wenzelm@30057
|
366 |
\begin{isamarkuptext}%
|
wenzelm@30057
|
367 |
We axiomatize basic connectives of propositional logic: implication,
|
wenzelm@30057
|
368 |
disjunction, and conjunction. The associated rules are modeled
|
wenzelm@30066
|
369 |
after Gentzen's system of Natural Deduction \cite{Gentzen:1935}.%
|
wenzelm@30057
|
370 |
\end{isamarkuptext}%
|
wenzelm@30057
|
371 |
\isamarkuptrue%
|
wenzelm@30057
|
372 |
\isacommand{axiomatization}\isamarkupfalse%
|
wenzelm@30057
|
373 |
\isanewline
|
wenzelm@30057
|
374 |
\ \ imp\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}o\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{infixr}\ {\isachardoublequoteopen}{\isasymlongrightarrow}{\isachardoublequoteclose}\ {\isadigit{2}}{\isadigit{5}}{\isacharparenright}\ \isakeyword{where}\isanewline
|
wenzelm@30057
|
375 |
\ \ impI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}A\ {\isasymLongrightarrow}\ B{\isacharparenright}\ {\isasymLongrightarrow}\ A\ {\isasymlongrightarrow}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
|
wenzelm@30057
|
376 |
\ \ impD\ {\isacharbrackleft}dest{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}A\ {\isasymlongrightarrow}\ B{\isacharparenright}\ {\isasymLongrightarrow}\ A\ {\isasymLongrightarrow}\ B{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
377 |
\isanewline
|
wenzelm@30057
|
378 |
\isacommand{axiomatization}\isamarkupfalse%
|
wenzelm@30057
|
379 |
\isanewline
|
wenzelm@30057
|
380 |
\ \ disj\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}o\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{infixr}\ {\isachardoublequoteopen}{\isasymor}{\isachardoublequoteclose}\ {\isadigit{3}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\isanewline
|
wenzelm@30057
|
381 |
\ \ disjI\isactrlisub {\isadigit{1}}\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}A\ {\isasymLongrightarrow}\ A\ {\isasymor}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
|
wenzelm@30066
|
382 |
\ \ disjI\isactrlisub {\isadigit{2}}\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}B\ {\isasymLongrightarrow}\ A\ {\isasymor}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
|
wenzelm@30066
|
383 |
\ \ disjE\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}A\ {\isasymor}\ B\ {\isasymLongrightarrow}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}B\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
384 |
\isanewline
|
wenzelm@30057
|
385 |
\isacommand{axiomatization}\isamarkupfalse%
|
wenzelm@30057
|
386 |
\isanewline
|
wenzelm@30057
|
387 |
\ \ conj\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}o\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{infixr}\ {\isachardoublequoteopen}{\isasymand}{\isachardoublequoteclose}\ {\isadigit{3}}{\isadigit{5}}{\isacharparenright}\ \isakeyword{where}\isanewline
|
wenzelm@30057
|
388 |
\ \ conjI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ A\ {\isasymand}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
|
wenzelm@30066
|
389 |
\ \ conjD\isactrlisub {\isadigit{1}}{\isacharcolon}\ {\isachardoublequoteopen}A\ {\isasymand}\ B\ {\isasymLongrightarrow}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
|
wenzelm@30066
|
390 |
\ \ conjD\isactrlisub {\isadigit{2}}{\isacharcolon}\ {\isachardoublequoteopen}A\ {\isasymand}\ B\ {\isasymLongrightarrow}\ B{\isachardoublequoteclose}%
|
wenzelm@30057
|
391 |
\begin{isamarkuptext}%
|
wenzelm@30057
|
392 |
\noindent The conjunctive destructions have the disadvantage that
|
wenzelm@30057
|
393 |
decomposing \isa{{\isachardoublequote}A\ {\isasymand}\ B{\isachardoublequote}} involves an immediate decision which
|
wenzelm@30057
|
394 |
component should be projected. The more convenient simultaneous
|
wenzelm@30057
|
395 |
elimination \isa{{\isachardoublequote}A\ {\isasymand}\ B\ {\isasymLongrightarrow}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequote}} can be derived as
|
wenzelm@30057
|
396 |
follows:%
|
wenzelm@30057
|
397 |
\end{isamarkuptext}%
|
wenzelm@30057
|
398 |
\isamarkuptrue%
|
wenzelm@30057
|
399 |
\isacommand{theorem}\isamarkupfalse%
|
wenzelm@30057
|
400 |
\ conjE\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\isanewline
|
wenzelm@30057
|
401 |
\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}A\ {\isasymand}\ B{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
402 |
\ \ \isakeyword{obtains}\ A\ \isakeyword{and}\ B\isanewline
|
wenzelm@30057
|
403 |
%
|
wenzelm@30057
|
404 |
\isadelimproof
|
wenzelm@30057
|
405 |
%
|
wenzelm@30057
|
406 |
\endisadelimproof
|
wenzelm@30057
|
407 |
%
|
wenzelm@30057
|
408 |
\isatagproof
|
wenzelm@30057
|
409 |
\isacommand{proof}\isamarkupfalse%
|
wenzelm@30057
|
410 |
\isanewline
|
wenzelm@30057
|
411 |
\ \ \isacommand{from}\isamarkupfalse%
|
wenzelm@30057
|
412 |
\ {\isacharbackquoteopen}A\ {\isasymand}\ B{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30066
|
413 |
\ A\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30066
|
414 |
\ {\isacharparenleft}rule\ conjD\isactrlisub {\isadigit{1}}{\isacharparenright}\isanewline
|
wenzelm@30057
|
415 |
\ \ \isacommand{from}\isamarkupfalse%
|
wenzelm@30057
|
416 |
\ {\isacharbackquoteopen}A\ {\isasymand}\ B{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30066
|
417 |
\ B\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30066
|
418 |
\ {\isacharparenleft}rule\ conjD\isactrlisub {\isadigit{2}}{\isacharparenright}\isanewline
|
wenzelm@30057
|
419 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30057
|
420 |
%
|
wenzelm@30057
|
421 |
\endisatagproof
|
wenzelm@30057
|
422 |
{\isafoldproof}%
|
wenzelm@30057
|
423 |
%
|
wenzelm@30057
|
424 |
\isadelimproof
|
wenzelm@30057
|
425 |
%
|
wenzelm@30057
|
426 |
\endisadelimproof
|
wenzelm@30057
|
427 |
%
|
wenzelm@30057
|
428 |
\begin{isamarkuptext}%
|
wenzelm@30057
|
429 |
\noindent Here is an example of swapping conjuncts with a single
|
wenzelm@30057
|
430 |
intermediate elimination step:%
|
wenzelm@30057
|
431 |
\end{isamarkuptext}%
|
wenzelm@30057
|
432 |
\isamarkuptrue%
|
wenzelm@30057
|
433 |
%
|
wenzelm@30057
|
434 |
\isadelimproof
|
wenzelm@30057
|
435 |
%
|
wenzelm@30057
|
436 |
\endisadelimproof
|
wenzelm@30057
|
437 |
%
|
wenzelm@30057
|
438 |
\isatagproof
|
wenzelm@30057
|
439 |
\ \ \isacommand{assume}\isamarkupfalse%
|
wenzelm@30057
|
440 |
\ {\isachardoublequoteopen}A\ {\isasymand}\ B{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
441 |
\ \ \isacommand{then}\isamarkupfalse%
|
wenzelm@30057
|
442 |
\ \isacommand{obtain}\isamarkupfalse%
|
wenzelm@30057
|
443 |
\ B\ \isakeyword{and}\ A\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
444 |
\isanewline
|
wenzelm@30057
|
445 |
\ \ \isacommand{then}\isamarkupfalse%
|
wenzelm@30057
|
446 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
447 |
\ {\isachardoublequoteopen}B\ {\isasymand}\ A{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
448 |
%
|
wenzelm@30057
|
449 |
\endisatagproof
|
wenzelm@30057
|
450 |
{\isafoldproof}%
|
wenzelm@30057
|
451 |
%
|
wenzelm@30057
|
452 |
\isadelimproof
|
wenzelm@30057
|
453 |
%
|
wenzelm@30057
|
454 |
\endisadelimproof
|
wenzelm@30057
|
455 |
%
|
wenzelm@30057
|
456 |
\begin{isamarkuptext}%
|
wenzelm@30066
|
457 |
\noindent Note that the analogous elimination rule for disjunction
|
wenzelm@30066
|
458 |
``\isa{{\isachardoublequote}{\isasymASSUMES}\ A\ {\isasymor}\ B\ {\isasymOBTAINS}\ A\ {\isasymBBAR}\ B{\isachardoublequote}}'' coincides with
|
wenzelm@30066
|
459 |
the original axiomatization of \isa{disjE}.
|
wenzelm@30057
|
460 |
|
wenzelm@30057
|
461 |
\medskip We continue propositional logic by introducing absurdity
|
wenzelm@30057
|
462 |
with its characteristic elimination. Plain truth may then be
|
wenzelm@30057
|
463 |
defined as a proposition that is trivially true.%
|
wenzelm@30057
|
464 |
\end{isamarkuptext}%
|
wenzelm@30057
|
465 |
\isamarkuptrue%
|
wenzelm@30057
|
466 |
\isacommand{axiomatization}\isamarkupfalse%
|
wenzelm@30057
|
467 |
\isanewline
|
wenzelm@30057
|
468 |
\ \ false\ {\isacharcolon}{\isacharcolon}\ o\ \ {\isacharparenleft}{\isachardoublequoteopen}{\isasymbottom}{\isachardoublequoteclose}{\isacharparenright}\ \isakeyword{where}\isanewline
|
wenzelm@30057
|
469 |
\ \ falseE\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymbottom}\ {\isasymLongrightarrow}\ A{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
470 |
\isanewline
|
wenzelm@30057
|
471 |
\isacommand{definition}\isamarkupfalse%
|
wenzelm@30057
|
472 |
\isanewline
|
wenzelm@30057
|
473 |
\ \ true\ {\isacharcolon}{\isacharcolon}\ o\ \ {\isacharparenleft}{\isachardoublequoteopen}{\isasymtop}{\isachardoublequoteclose}{\isacharparenright}\ \isakeyword{where}\isanewline
|
wenzelm@30057
|
474 |
\ \ {\isachardoublequoteopen}{\isasymtop}\ {\isasymequiv}\ {\isasymbottom}\ {\isasymlongrightarrow}\ {\isasymbottom}{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
475 |
\isanewline
|
wenzelm@30057
|
476 |
\isacommand{theorem}\isamarkupfalse%
|
wenzelm@30057
|
477 |
\ trueI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isasymtop}\isanewline
|
wenzelm@30057
|
478 |
%
|
wenzelm@30057
|
479 |
\isadelimproof
|
wenzelm@30057
|
480 |
\ \ %
|
wenzelm@30057
|
481 |
\endisadelimproof
|
wenzelm@30057
|
482 |
%
|
wenzelm@30057
|
483 |
\isatagproof
|
wenzelm@30057
|
484 |
\isacommand{unfolding}\isamarkupfalse%
|
wenzelm@30057
|
485 |
\ true{\isacharunderscore}def\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
486 |
%
|
wenzelm@30057
|
487 |
\endisatagproof
|
wenzelm@30057
|
488 |
{\isafoldproof}%
|
wenzelm@30057
|
489 |
%
|
wenzelm@30057
|
490 |
\isadelimproof
|
wenzelm@30057
|
491 |
%
|
wenzelm@30057
|
492 |
\endisadelimproof
|
wenzelm@30057
|
493 |
%
|
wenzelm@30057
|
494 |
\begin{isamarkuptext}%
|
wenzelm@30066
|
495 |
\medskip\noindent Now negation represents an implication towards
|
wenzelm@30066
|
496 |
absurdity:%
|
wenzelm@30057
|
497 |
\end{isamarkuptext}%
|
wenzelm@30057
|
498 |
\isamarkuptrue%
|
wenzelm@30057
|
499 |
\isacommand{definition}\isamarkupfalse%
|
wenzelm@30057
|
500 |
\isanewline
|
wenzelm@30057
|
501 |
\ \ not\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}o\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}{\isachardoublequoteopen}{\isasymnot}\ {\isacharunderscore}{\isachardoublequoteclose}\ {\isacharbrackleft}{\isadigit{4}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{4}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\isanewline
|
wenzelm@30057
|
502 |
\ \ {\isachardoublequoteopen}{\isasymnot}\ A\ {\isasymequiv}\ A\ {\isasymlongrightarrow}\ {\isasymbottom}{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
503 |
\isanewline
|
wenzelm@30057
|
504 |
\isacommand{theorem}\isamarkupfalse%
|
wenzelm@30057
|
505 |
\ notI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\isanewline
|
wenzelm@30057
|
506 |
\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}A\ {\isasymLongrightarrow}\ {\isasymbottom}{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
507 |
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ A{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
508 |
%
|
wenzelm@30057
|
509 |
\isadelimproof
|
wenzelm@30057
|
510 |
%
|
wenzelm@30057
|
511 |
\endisadelimproof
|
wenzelm@30057
|
512 |
%
|
wenzelm@30057
|
513 |
\isatagproof
|
wenzelm@30057
|
514 |
\isacommand{unfolding}\isamarkupfalse%
|
wenzelm@30057
|
515 |
\ not{\isacharunderscore}def\isanewline
|
wenzelm@30057
|
516 |
\isacommand{proof}\isamarkupfalse%
|
wenzelm@30057
|
517 |
\isanewline
|
wenzelm@30057
|
518 |
\ \ \isacommand{assume}\isamarkupfalse%
|
wenzelm@30057
|
519 |
\ A\isanewline
|
wenzelm@30057
|
520 |
\ \ \isacommand{then}\isamarkupfalse%
|
wenzelm@30057
|
521 |
\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30057
|
522 |
\ {\isasymbottom}\ \isacommand{by}\isamarkupfalse%
|
wenzelm@30057
|
523 |
\ {\isacharparenleft}rule\ {\isacharbackquoteopen}A\ {\isasymLongrightarrow}\ {\isasymbottom}{\isacharbackquoteclose}{\isacharparenright}\isanewline
|
wenzelm@30057
|
524 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30057
|
525 |
%
|
wenzelm@30057
|
526 |
\endisatagproof
|
wenzelm@30057
|
527 |
{\isafoldproof}%
|
wenzelm@30057
|
528 |
%
|
wenzelm@30057
|
529 |
\isadelimproof
|
wenzelm@30057
|
530 |
\isanewline
|
wenzelm@30057
|
531 |
%
|
wenzelm@30057
|
532 |
\endisadelimproof
|
wenzelm@30057
|
533 |
\isanewline
|
wenzelm@30057
|
534 |
\isacommand{theorem}\isamarkupfalse%
|
wenzelm@30057
|
535 |
\ notE\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\isanewline
|
wenzelm@30057
|
536 |
\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnot}\ A{\isachardoublequoteclose}\ \isakeyword{and}\ A\isanewline
|
wenzelm@30057
|
537 |
\ \ \isakeyword{shows}\ B\isanewline
|
wenzelm@30057
|
538 |
%
|
wenzelm@30057
|
539 |
\isadelimproof
|
wenzelm@30057
|
540 |
%
|
wenzelm@30057
|
541 |
\endisadelimproof
|
wenzelm@30057
|
542 |
%
|
wenzelm@30057
|
543 |
\isatagproof
|
wenzelm@30057
|
544 |
\isacommand{proof}\isamarkupfalse%
|
wenzelm@30057
|
545 |
\ {\isacharminus}\isanewline
|
wenzelm@30057
|
546 |
\ \ \isacommand{from}\isamarkupfalse%
|
wenzelm@30057
|
547 |
\ {\isacharbackquoteopen}{\isasymnot}\ A{\isacharbackquoteclose}\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
548 |
\ {\isachardoublequoteopen}A\ {\isasymlongrightarrow}\ {\isasymbottom}{\isachardoublequoteclose}\ \isacommand{unfolding}\isamarkupfalse%
|
wenzelm@30057
|
549 |
\ not{\isacharunderscore}def\ \isacommand{{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
550 |
\isanewline
|
wenzelm@30057
|
551 |
\ \ \isacommand{from}\isamarkupfalse%
|
wenzelm@30057
|
552 |
\ {\isacharbackquoteopen}A\ {\isasymlongrightarrow}\ {\isasymbottom}{\isacharbackquoteclose}\ \isakeyword{and}\ {\isacharbackquoteopen}A{\isacharbackquoteclose}\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
553 |
\ {\isasymbottom}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
554 |
\isanewline
|
wenzelm@30057
|
555 |
\ \ \isacommand{then}\isamarkupfalse%
|
wenzelm@30057
|
556 |
\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30057
|
557 |
\ B\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
558 |
\isanewline
|
wenzelm@30057
|
559 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30057
|
560 |
%
|
wenzelm@30057
|
561 |
\endisatagproof
|
wenzelm@30057
|
562 |
{\isafoldproof}%
|
wenzelm@30057
|
563 |
%
|
wenzelm@30057
|
564 |
\isadelimproof
|
wenzelm@30057
|
565 |
%
|
wenzelm@30057
|
566 |
\endisadelimproof
|
wenzelm@30057
|
567 |
%
|
wenzelm@30057
|
568 |
\isamarkupsubsection{Classical logic%
|
wenzelm@30057
|
569 |
}
|
wenzelm@30057
|
570 |
\isamarkuptrue%
|
wenzelm@30057
|
571 |
%
|
wenzelm@30057
|
572 |
\begin{isamarkuptext}%
|
wenzelm@30057
|
573 |
Subsequently we state the principle of classical contradiction as a
|
wenzelm@30057
|
574 |
local assumption. Thus we refrain from forcing the object-logic
|
wenzelm@30057
|
575 |
into the classical perspective. Within that context, we may derive
|
wenzelm@30057
|
576 |
well-known consequences of the classical principle.%
|
wenzelm@30057
|
577 |
\end{isamarkuptext}%
|
wenzelm@30057
|
578 |
\isamarkuptrue%
|
wenzelm@30057
|
579 |
\isacommand{locale}\isamarkupfalse%
|
wenzelm@30057
|
580 |
\ classical\ {\isacharequal}\isanewline
|
wenzelm@30057
|
581 |
\ \ \isakeyword{assumes}\ classical{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isasymnot}\ C\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
582 |
\isakeyword{begin}\isanewline
|
wenzelm@30057
|
583 |
\isanewline
|
wenzelm@30057
|
584 |
\isacommand{theorem}\isamarkupfalse%
|
wenzelm@30057
|
585 |
\ double{\isacharunderscore}negation{\isacharcolon}\isanewline
|
wenzelm@30057
|
586 |
\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnot}\ {\isasymnot}\ C{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
587 |
\ \ \isakeyword{shows}\ C\isanewline
|
wenzelm@30057
|
588 |
%
|
wenzelm@30057
|
589 |
\isadelimproof
|
wenzelm@30057
|
590 |
%
|
wenzelm@30057
|
591 |
\endisadelimproof
|
wenzelm@30057
|
592 |
%
|
wenzelm@30057
|
593 |
\isatagproof
|
wenzelm@30057
|
594 |
\isacommand{proof}\isamarkupfalse%
|
wenzelm@30057
|
595 |
\ {\isacharparenleft}rule\ classical{\isacharparenright}\isanewline
|
wenzelm@30057
|
596 |
\ \ \isacommand{assume}\isamarkupfalse%
|
wenzelm@30057
|
597 |
\ {\isachardoublequoteopen}{\isasymnot}\ C{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
598 |
\ \ \isacommand{with}\isamarkupfalse%
|
wenzelm@30057
|
599 |
\ {\isacharbackquoteopen}{\isasymnot}\ {\isasymnot}\ C{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30057
|
600 |
\ C\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
601 |
\isanewline
|
wenzelm@30057
|
602 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30057
|
603 |
%
|
wenzelm@30057
|
604 |
\endisatagproof
|
wenzelm@30057
|
605 |
{\isafoldproof}%
|
wenzelm@30057
|
606 |
%
|
wenzelm@30057
|
607 |
\isadelimproof
|
wenzelm@30057
|
608 |
\isanewline
|
wenzelm@30057
|
609 |
%
|
wenzelm@30057
|
610 |
\endisadelimproof
|
wenzelm@30057
|
611 |
\isanewline
|
wenzelm@30057
|
612 |
\isacommand{theorem}\isamarkupfalse%
|
wenzelm@30057
|
613 |
\ tertium{\isacharunderscore}non{\isacharunderscore}datur{\isacharcolon}\ {\isachardoublequoteopen}C\ {\isasymor}\ {\isasymnot}\ C{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
614 |
%
|
wenzelm@30057
|
615 |
\isadelimproof
|
wenzelm@30057
|
616 |
%
|
wenzelm@30057
|
617 |
\endisadelimproof
|
wenzelm@30057
|
618 |
%
|
wenzelm@30057
|
619 |
\isatagproof
|
wenzelm@30057
|
620 |
\isacommand{proof}\isamarkupfalse%
|
wenzelm@30057
|
621 |
\ {\isacharparenleft}rule\ double{\isacharunderscore}negation{\isacharparenright}\isanewline
|
wenzelm@30057
|
622 |
\ \ \isacommand{show}\isamarkupfalse%
|
wenzelm@30057
|
623 |
\ {\isachardoublequoteopen}{\isasymnot}\ {\isasymnot}\ {\isacharparenleft}C\ {\isasymor}\ {\isasymnot}\ C{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
624 |
\ \ \isacommand{proof}\isamarkupfalse%
|
wenzelm@30057
|
625 |
\isanewline
|
wenzelm@30057
|
626 |
\ \ \ \ \isacommand{assume}\isamarkupfalse%
|
wenzelm@30057
|
627 |
\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}C\ {\isasymor}\ {\isasymnot}\ C{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
628 |
\ \ \ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
629 |
\ {\isachardoublequoteopen}{\isasymnot}\ C{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
630 |
\ \ \ \ \isacommand{proof}\isamarkupfalse%
|
wenzelm@30057
|
631 |
\isanewline
|
wenzelm@30057
|
632 |
\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
|
wenzelm@30057
|
633 |
\ C\ \isacommand{then}\isamarkupfalse%
|
wenzelm@30057
|
634 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
635 |
\ {\isachardoublequoteopen}C\ {\isasymor}\ {\isasymnot}\ C{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
636 |
\isanewline
|
wenzelm@30057
|
637 |
\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
|
wenzelm@30057
|
638 |
\ {\isacharbackquoteopen}{\isasymnot}\ {\isacharparenleft}C\ {\isasymor}\ {\isasymnot}\ C{\isacharparenright}{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30057
|
639 |
\ {\isasymbottom}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
640 |
\isanewline
|
wenzelm@30057
|
641 |
\ \ \ \ \isacommand{qed}\isamarkupfalse%
|
wenzelm@30057
|
642 |
\isanewline
|
wenzelm@30057
|
643 |
\ \ \ \ \isacommand{then}\isamarkupfalse%
|
wenzelm@30057
|
644 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
645 |
\ {\isachardoublequoteopen}C\ {\isasymor}\ {\isasymnot}\ C{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
646 |
\isanewline
|
wenzelm@30057
|
647 |
\ \ \ \ \isacommand{with}\isamarkupfalse%
|
wenzelm@30057
|
648 |
\ {\isacharbackquoteopen}{\isasymnot}\ {\isacharparenleft}C\ {\isasymor}\ {\isasymnot}\ C{\isacharparenright}{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30057
|
649 |
\ {\isasymbottom}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
650 |
\isanewline
|
wenzelm@30057
|
651 |
\ \ \isacommand{qed}\isamarkupfalse%
|
wenzelm@30057
|
652 |
\isanewline
|
wenzelm@30057
|
653 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30057
|
654 |
%
|
wenzelm@30057
|
655 |
\endisatagproof
|
wenzelm@30057
|
656 |
{\isafoldproof}%
|
wenzelm@30057
|
657 |
%
|
wenzelm@30057
|
658 |
\isadelimproof
|
wenzelm@30057
|
659 |
%
|
wenzelm@30057
|
660 |
\endisadelimproof
|
wenzelm@30057
|
661 |
%
|
wenzelm@30057
|
662 |
\begin{isamarkuptext}%
|
wenzelm@30066
|
663 |
\noindent These examples illustrate both classical reasoning and
|
wenzelm@30066
|
664 |
non-trivial propositional proofs in general. All three rules
|
wenzelm@30066
|
665 |
characterize classical logic independently, but the original rule is
|
wenzelm@30066
|
666 |
already the most convenient to use, because it leaves the conclusion
|
wenzelm@30066
|
667 |
unchanged. Note that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymnot}\ C\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequote}} fits again into our
|
wenzelm@30066
|
668 |
format for eliminations, despite the additional twist that the
|
wenzelm@30066
|
669 |
context refers to the main conclusion. So we may write \isa{classical} as the Isar statement ``\isa{{\isachardoublequote}{\isasymOBTAINS}\ {\isasymnot}\ thesis{\isachardoublequote}}''.
|
wenzelm@30066
|
670 |
This also explains nicely how classical reasoning really works:
|
wenzelm@30066
|
671 |
whatever the main \isa{thesis} might be, we may always assume its
|
wenzelm@30066
|
672 |
negation!%
|
wenzelm@30057
|
673 |
\end{isamarkuptext}%
|
wenzelm@30057
|
674 |
\isamarkuptrue%
|
wenzelm@30057
|
675 |
\isacommand{end}\isamarkupfalse%
|
wenzelm@30057
|
676 |
%
|
wenzelm@30066
|
677 |
\isamarkupsubsection{Quantifiers \label{sec:framework-ex-quant}%
|
wenzelm@30057
|
678 |
}
|
wenzelm@30057
|
679 |
\isamarkuptrue%
|
wenzelm@30057
|
680 |
%
|
wenzelm@30057
|
681 |
\begin{isamarkuptext}%
|
wenzelm@30057
|
682 |
Representing quantifiers is easy, thanks to the higher-order nature
|
wenzelm@30057
|
683 |
of the underlying framework. According to the well-known technique
|
wenzelm@30057
|
684 |
introduced by Church \cite{church40}, quantifiers are operators on
|
wenzelm@30057
|
685 |
predicates, which are syntactically represented as \isa{{\isachardoublequote}{\isasymlambda}{\isachardoublequote}}-terms
|
wenzelm@30066
|
686 |
of type \isa{{\isachardoublequote}i\ {\isasymRightarrow}\ o{\isachardoublequote}}. Binder notation turns \isa{{\isachardoublequote}All\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ B\ x{\isacharparenright}{\isachardoublequote}} into \isa{{\isachardoublequote}{\isasymforall}x{\isachardot}\ B\ x{\isachardoublequote}} etc.%
|
wenzelm@30057
|
687 |
\end{isamarkuptext}%
|
wenzelm@30057
|
688 |
\isamarkuptrue%
|
wenzelm@30057
|
689 |
\isacommand{axiomatization}\isamarkupfalse%
|
wenzelm@30057
|
690 |
\isanewline
|
wenzelm@30057
|
691 |
\ \ All\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}i\ {\isasymRightarrow}\ o{\isacharparenright}\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{binder}\ {\isachardoublequoteopen}{\isasymforall}{\isachardoublequoteclose}\ {\isadigit{1}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\isanewline
|
wenzelm@30057
|
692 |
\ \ allI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isasymAnd}x{\isachardot}\ B\ x{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymforall}x{\isachardot}\ B\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
|
wenzelm@30057
|
693 |
\ \ allD\ {\isacharbrackleft}dest{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isasymforall}x{\isachardot}\ B\ x{\isacharparenright}\ {\isasymLongrightarrow}\ B\ a{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
694 |
\isanewline
|
wenzelm@30057
|
695 |
\isacommand{axiomatization}\isamarkupfalse%
|
wenzelm@30057
|
696 |
\isanewline
|
wenzelm@30057
|
697 |
\ \ Ex\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}i\ {\isasymRightarrow}\ o{\isacharparenright}\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{binder}\ {\isachardoublequoteopen}{\isasymexists}{\isachardoublequoteclose}\ {\isadigit{1}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\isanewline
|
wenzelm@30057
|
698 |
\ \ exI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}B\ a\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymexists}x{\isachardot}\ B\ x{\isacharparenright}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
|
wenzelm@30057
|
699 |
\ \ exE\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isasymexists}x{\isachardot}\ B\ x{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ B\ x\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequoteclose}%
|
wenzelm@30057
|
700 |
\begin{isamarkuptext}%
|
wenzelm@30066
|
701 |
\noindent The statement of \isa{exE} corresponds to ``\isa{{\isachardoublequote}{\isasymASSUMES}\ {\isasymexists}x{\isachardot}\ B\ x\ {\isasymOBTAINS}\ x\ {\isasymWHERE}\ B\ x{\isachardoublequote}}'' in Isar. In the
|
wenzelm@30066
|
702 |
subsequent example we illustrate quantifier reasoning involving all
|
wenzelm@30066
|
703 |
four rules:%
|
wenzelm@30057
|
704 |
\end{isamarkuptext}%
|
wenzelm@30057
|
705 |
\isamarkuptrue%
|
wenzelm@30057
|
706 |
\isacommand{theorem}\isamarkupfalse%
|
wenzelm@30057
|
707 |
\isanewline
|
wenzelm@30057
|
708 |
\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymexists}x{\isachardot}\ {\isasymforall}y{\isachardot}\ R\ x\ y{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
709 |
\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}y{\isachardot}\ {\isasymexists}x{\isachardot}\ R\ x\ y{\isachardoublequoteclose}\isanewline
|
wenzelm@30057
|
710 |
%
|
wenzelm@30057
|
711 |
\isadelimproof
|
wenzelm@30057
|
712 |
%
|
wenzelm@30057
|
713 |
\endisadelimproof
|
wenzelm@30057
|
714 |
%
|
wenzelm@30057
|
715 |
\isatagproof
|
wenzelm@30057
|
716 |
\isacommand{proof}\isamarkupfalse%
|
wenzelm@30057
|
717 |
\ \ \ \ %
|
wenzelm@30057
|
718 |
\isamarkupcmt{\isa{{\isachardoublequote}{\isasymforall}{\isachardoublequote}} introduction%
|
wenzelm@30057
|
719 |
}
|
wenzelm@30057
|
720 |
\isanewline
|
wenzelm@30057
|
721 |
\ \ \isacommand{obtain}\isamarkupfalse%
|
wenzelm@30057
|
722 |
\ x\ \isakeyword{where}\ {\isachardoublequoteopen}{\isasymforall}y{\isachardot}\ R\ x\ y{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
|
wenzelm@30057
|
723 |
\ {\isacharbackquoteopen}{\isasymexists}x{\isachardot}\ {\isasymforall}y{\isachardot}\ R\ x\ y{\isacharbackquoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
724 |
\ \ \ \ %
|
wenzelm@30057
|
725 |
\isamarkupcmt{\isa{{\isachardoublequote}{\isasymexists}{\isachardoublequote}} elimination%
|
wenzelm@30057
|
726 |
}
|
wenzelm@30057
|
727 |
\isanewline
|
wenzelm@30057
|
728 |
\ \ \isacommand{fix}\isamarkupfalse%
|
wenzelm@30057
|
729 |
\ y\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30057
|
730 |
\ {\isachardoublequoteopen}R\ x\ y{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse%
|
wenzelm@30057
|
731 |
\ {\isacharbackquoteopen}{\isasymforall}y{\isachardot}\ R\ x\ y{\isacharbackquoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
732 |
\ \ \ \ %
|
wenzelm@30057
|
733 |
\isamarkupcmt{\isa{{\isachardoublequote}{\isasymforall}{\isachardoublequote}} destruction%
|
wenzelm@30057
|
734 |
}
|
wenzelm@30057
|
735 |
\isanewline
|
wenzelm@30057
|
736 |
\ \ \isacommand{then}\isamarkupfalse%
|
wenzelm@30057
|
737 |
\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30057
|
738 |
\ {\isachardoublequoteopen}{\isasymexists}x{\isachardot}\ R\ x\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30057
|
739 |
\ \ \ \ %
|
wenzelm@30057
|
740 |
\isamarkupcmt{\isa{{\isachardoublequote}{\isasymexists}{\isachardoublequote}} introduction%
|
wenzelm@30057
|
741 |
}
|
wenzelm@30057
|
742 |
\isanewline
|
wenzelm@30057
|
743 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30057
|
744 |
%
|
wenzelm@30057
|
745 |
\endisatagproof
|
wenzelm@30057
|
746 |
{\isafoldproof}%
|
wenzelm@30057
|
747 |
%
|
wenzelm@30057
|
748 |
\isadelimproof
|
wenzelm@30057
|
749 |
%
|
wenzelm@30057
|
750 |
\endisadelimproof
|
wenzelm@30057
|
751 |
%
|
wenzelm@30066
|
752 |
\isamarkupsubsection{Canonical reasoning patterns%
|
wenzelm@30066
|
753 |
}
|
wenzelm@30066
|
754 |
\isamarkuptrue%
|
wenzelm@30066
|
755 |
%
|
wenzelm@30066
|
756 |
\begin{isamarkuptext}%
|
wenzelm@30066
|
757 |
The main rules of first-order predicate logic from
|
wenzelm@30066
|
758 |
\secref{sec:framework-ex-prop} and \secref{sec:framework-ex-quant}
|
wenzelm@30066
|
759 |
can now be summarized as follows, using the native Isar statement
|
wenzelm@30066
|
760 |
format of \secref{sec:framework-stmt}.
|
wenzelm@30066
|
761 |
|
wenzelm@30066
|
762 |
\medskip
|
wenzelm@30066
|
763 |
\begin{tabular}{l}
|
wenzelm@30066
|
764 |
\isa{{\isachardoublequote}impI{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymLongrightarrow}\ B\ {\isasymSHOWS}\ A\ {\isasymlongrightarrow}\ B{\isachardoublequote}} \\
|
wenzelm@30066
|
765 |
\isa{{\isachardoublequote}impD{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymlongrightarrow}\ B\ {\isasymAND}\ A\ {\isasymSHOWS}\ B{\isachardoublequote}} \\[1ex]
|
wenzelm@30066
|
766 |
|
wenzelm@30066
|
767 |
\isa{{\isachardoublequote}disjI\isactrlisub {\isadigit{1}}{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymSHOWS}\ A\ {\isasymor}\ B{\isachardoublequote}} \\
|
wenzelm@30066
|
768 |
\isa{{\isachardoublequote}disjI\isactrlisub {\isadigit{2}}{\isacharcolon}\ {\isasymASSUMES}\ B\ {\isasymSHOWS}\ A\ {\isasymor}\ B{\isachardoublequote}} \\
|
wenzelm@30066
|
769 |
\isa{{\isachardoublequote}disjE{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymor}\ B\ {\isasymOBTAINS}\ A\ {\isasymBBAR}\ B{\isachardoublequote}} \\[1ex]
|
wenzelm@30066
|
770 |
|
wenzelm@30066
|
771 |
\isa{{\isachardoublequote}conjI{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymAND}\ B\ {\isasymSHOWS}\ A\ {\isasymand}\ B{\isachardoublequote}} \\
|
wenzelm@30066
|
772 |
\isa{{\isachardoublequote}conjE{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymand}\ B\ {\isasymOBTAINS}\ A\ {\isasymAND}\ B{\isachardoublequote}} \\[1ex]
|
wenzelm@30066
|
773 |
|
wenzelm@30066
|
774 |
\isa{{\isachardoublequote}falseE{\isacharcolon}\ {\isasymASSUMES}\ {\isasymbottom}\ {\isasymSHOWS}\ A{\isachardoublequote}} \\
|
wenzelm@30066
|
775 |
\isa{{\isachardoublequote}trueI{\isacharcolon}\ {\isasymSHOWS}\ {\isasymtop}{\isachardoublequote}} \\[1ex]
|
wenzelm@30066
|
776 |
|
wenzelm@30066
|
777 |
\isa{{\isachardoublequote}notI{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymLongrightarrow}\ {\isasymbottom}\ {\isasymSHOWS}\ {\isasymnot}\ A{\isachardoublequote}} \\
|
wenzelm@30066
|
778 |
\isa{{\isachardoublequote}notE{\isacharcolon}\ {\isasymASSUMES}\ {\isasymnot}\ A\ {\isasymAND}\ A\ {\isasymSHOWS}\ B{\isachardoublequote}} \\[1ex]
|
wenzelm@30066
|
779 |
|
wenzelm@30066
|
780 |
\isa{{\isachardoublequote}allI{\isacharcolon}\ {\isasymASSUMES}\ {\isasymAnd}x{\isachardot}\ B\ x\ {\isasymSHOWS}\ {\isasymforall}x{\isachardot}\ B\ x{\isachardoublequote}} \\
|
wenzelm@30066
|
781 |
\isa{{\isachardoublequote}allE{\isacharcolon}\ {\isasymASSUMES}\ {\isasymforall}x{\isachardot}\ B\ x\ {\isasymSHOWS}\ B\ a{\isachardoublequote}} \\[1ex]
|
wenzelm@30066
|
782 |
|
wenzelm@30066
|
783 |
\isa{{\isachardoublequote}exI{\isacharcolon}\ {\isasymASSUMES}\ B\ a\ {\isasymSHOWS}\ {\isasymexists}x{\isachardot}\ B\ x{\isachardoublequote}} \\
|
wenzelm@30066
|
784 |
\isa{{\isachardoublequote}exE{\isacharcolon}\ {\isasymASSUMES}\ {\isasymexists}x{\isachardot}\ B\ x\ {\isasymOBTAINS}\ a\ {\isasymWHERE}\ B\ a{\isachardoublequote}}
|
wenzelm@30066
|
785 |
\end{tabular}
|
wenzelm@30066
|
786 |
\medskip
|
wenzelm@30066
|
787 |
|
wenzelm@30066
|
788 |
\noindent This essentially provides a declarative reading of Pure
|
wenzelm@30066
|
789 |
rules as Isar reasoning patterns: the rule statements tells how a
|
wenzelm@30066
|
790 |
canonical proof outline shall look like. Since the above rules have
|
wenzelm@30066
|
791 |
already been declared as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}}, \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}} --- each according to its
|
wenzelm@30066
|
792 |
particular shape --- we can immediately write Isar proof texts as
|
wenzelm@30066
|
793 |
follows:%
|
wenzelm@30066
|
794 |
\end{isamarkuptext}%
|
wenzelm@30066
|
795 |
\isamarkuptrue%
|
wenzelm@30066
|
796 |
%
|
wenzelm@30066
|
797 |
\isadelimproof
|
wenzelm@30066
|
798 |
%
|
wenzelm@30066
|
799 |
\endisadelimproof
|
wenzelm@30066
|
800 |
%
|
wenzelm@30066
|
801 |
\isatagproof
|
wenzelm@30066
|
802 |
%
|
wenzelm@30066
|
803 |
\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
804 |
\isanewline
|
wenzelm@30066
|
805 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
806 |
\ {\isachardoublequoteopen}A\ {\isasymlongrightarrow}\ B{\isachardoublequoteclose}\isanewline
|
wenzelm@30066
|
807 |
\ \ \isacommand{proof}\isamarkupfalse%
|
wenzelm@30066
|
808 |
\isanewline
|
wenzelm@30066
|
809 |
\ \ \ \ \isacommand{assume}\isamarkupfalse%
|
wenzelm@30066
|
810 |
\ A\isanewline
|
wenzelm@30066
|
811 |
\ \ \ \ \isacommand{show}\isamarkupfalse%
|
wenzelm@30066
|
812 |
\ B%
|
wenzelm@30066
|
813 |
\endisatagproof
|
wenzelm@30066
|
814 |
{\isafoldproof}%
|
wenzelm@30066
|
815 |
%
|
wenzelm@30066
|
816 |
\isadelimproof
|
wenzelm@30066
|
817 |
%
|
wenzelm@30066
|
818 |
\endisadelimproof
|
wenzelm@30066
|
819 |
%
|
wenzelm@30066
|
820 |
\isadelimnoproof
|
wenzelm@30066
|
821 |
\ %
|
wenzelm@30066
|
822 |
\endisadelimnoproof
|
wenzelm@30066
|
823 |
%
|
wenzelm@30066
|
824 |
\isatagnoproof
|
wenzelm@30066
|
825 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
826 |
%
|
wenzelm@30066
|
827 |
\endisatagnoproof
|
wenzelm@30066
|
828 |
{\isafoldnoproof}%
|
wenzelm@30066
|
829 |
%
|
wenzelm@30066
|
830 |
\isadelimnoproof
|
wenzelm@30066
|
831 |
\isanewline
|
wenzelm@30066
|
832 |
%
|
wenzelm@30066
|
833 |
\endisadelimnoproof
|
wenzelm@30066
|
834 |
%
|
wenzelm@30066
|
835 |
\isadelimproof
|
wenzelm@30066
|
836 |
\ \ %
|
wenzelm@30066
|
837 |
\endisadelimproof
|
wenzelm@30066
|
838 |
%
|
wenzelm@30066
|
839 |
\isatagproof
|
wenzelm@30066
|
840 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30066
|
841 |
%
|
wenzelm@30066
|
842 |
\end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
843 |
\isanewline
|
wenzelm@30066
|
844 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
845 |
\ {\isachardoublequoteopen}A\ {\isasymlongrightarrow}\ B{\isachardoublequoteclose}\ \isakeyword{and}\ A%
|
wenzelm@30066
|
846 |
\endisatagproof
|
wenzelm@30066
|
847 |
{\isafoldproof}%
|
wenzelm@30066
|
848 |
%
|
wenzelm@30066
|
849 |
\isadelimproof
|
wenzelm@30066
|
850 |
%
|
wenzelm@30066
|
851 |
\endisadelimproof
|
wenzelm@30066
|
852 |
%
|
wenzelm@30066
|
853 |
\isadelimnoproof
|
wenzelm@30066
|
854 |
\ %
|
wenzelm@30066
|
855 |
\endisadelimnoproof
|
wenzelm@30066
|
856 |
%
|
wenzelm@30066
|
857 |
\isatagnoproof
|
wenzelm@30066
|
858 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
859 |
%
|
wenzelm@30066
|
860 |
\endisatagnoproof
|
wenzelm@30066
|
861 |
{\isafoldnoproof}%
|
wenzelm@30066
|
862 |
%
|
wenzelm@30066
|
863 |
\isadelimnoproof
|
wenzelm@30066
|
864 |
\isanewline
|
wenzelm@30066
|
865 |
%
|
wenzelm@30066
|
866 |
\endisadelimnoproof
|
wenzelm@30066
|
867 |
%
|
wenzelm@30066
|
868 |
\isadelimproof
|
wenzelm@30066
|
869 |
\ \ %
|
wenzelm@30066
|
870 |
\endisadelimproof
|
wenzelm@30066
|
871 |
%
|
wenzelm@30066
|
872 |
\isatagproof
|
wenzelm@30066
|
873 |
\isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
874 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
875 |
\ B\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30066
|
876 |
%
|
wenzelm@30066
|
877 |
\end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
878 |
\isanewline
|
wenzelm@30066
|
879 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
880 |
\ A%
|
wenzelm@30066
|
881 |
\endisatagproof
|
wenzelm@30066
|
882 |
{\isafoldproof}%
|
wenzelm@30066
|
883 |
%
|
wenzelm@30066
|
884 |
\isadelimproof
|
wenzelm@30066
|
885 |
%
|
wenzelm@30066
|
886 |
\endisadelimproof
|
wenzelm@30066
|
887 |
%
|
wenzelm@30066
|
888 |
\isadelimnoproof
|
wenzelm@30066
|
889 |
\ %
|
wenzelm@30066
|
890 |
\endisadelimnoproof
|
wenzelm@30066
|
891 |
%
|
wenzelm@30066
|
892 |
\isatagnoproof
|
wenzelm@30066
|
893 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
894 |
%
|
wenzelm@30066
|
895 |
\endisatagnoproof
|
wenzelm@30066
|
896 |
{\isafoldnoproof}%
|
wenzelm@30066
|
897 |
%
|
wenzelm@30066
|
898 |
\isadelimnoproof
|
wenzelm@30066
|
899 |
\isanewline
|
wenzelm@30066
|
900 |
%
|
wenzelm@30066
|
901 |
\endisadelimnoproof
|
wenzelm@30066
|
902 |
%
|
wenzelm@30066
|
903 |
\isadelimproof
|
wenzelm@30066
|
904 |
\ \ %
|
wenzelm@30066
|
905 |
\endisadelimproof
|
wenzelm@30066
|
906 |
%
|
wenzelm@30066
|
907 |
\isatagproof
|
wenzelm@30066
|
908 |
\isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
909 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
910 |
\ {\isachardoublequoteopen}A\ {\isasymor}\ B{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30066
|
911 |
\isanewline
|
wenzelm@30066
|
912 |
\isanewline
|
wenzelm@30066
|
913 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
914 |
\ B%
|
wenzelm@30066
|
915 |
\endisatagproof
|
wenzelm@30066
|
916 |
{\isafoldproof}%
|
wenzelm@30066
|
917 |
%
|
wenzelm@30066
|
918 |
\isadelimproof
|
wenzelm@30066
|
919 |
%
|
wenzelm@30066
|
920 |
\endisadelimproof
|
wenzelm@30066
|
921 |
%
|
wenzelm@30066
|
922 |
\isadelimnoproof
|
wenzelm@30066
|
923 |
\ %
|
wenzelm@30066
|
924 |
\endisadelimnoproof
|
wenzelm@30066
|
925 |
%
|
wenzelm@30066
|
926 |
\isatagnoproof
|
wenzelm@30066
|
927 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
928 |
%
|
wenzelm@30066
|
929 |
\endisatagnoproof
|
wenzelm@30066
|
930 |
{\isafoldnoproof}%
|
wenzelm@30066
|
931 |
%
|
wenzelm@30066
|
932 |
\isadelimnoproof
|
wenzelm@30066
|
933 |
\isanewline
|
wenzelm@30066
|
934 |
%
|
wenzelm@30066
|
935 |
\endisadelimnoproof
|
wenzelm@30066
|
936 |
%
|
wenzelm@30066
|
937 |
\isadelimproof
|
wenzelm@30066
|
938 |
\ \ %
|
wenzelm@30066
|
939 |
\endisadelimproof
|
wenzelm@30066
|
940 |
%
|
wenzelm@30066
|
941 |
\isatagproof
|
wenzelm@30066
|
942 |
\isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
943 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
944 |
\ {\isachardoublequoteopen}A\ {\isasymor}\ B{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30066
|
945 |
%
|
wenzelm@30066
|
946 |
\end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
947 |
\isanewline
|
wenzelm@30066
|
948 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
949 |
\ {\isachardoublequoteopen}A\ {\isasymor}\ B{\isachardoublequoteclose}%
|
wenzelm@30066
|
950 |
\endisatagproof
|
wenzelm@30066
|
951 |
{\isafoldproof}%
|
wenzelm@30066
|
952 |
%
|
wenzelm@30066
|
953 |
\isadelimproof
|
wenzelm@30066
|
954 |
%
|
wenzelm@30066
|
955 |
\endisadelimproof
|
wenzelm@30066
|
956 |
%
|
wenzelm@30066
|
957 |
\isadelimnoproof
|
wenzelm@30066
|
958 |
\ %
|
wenzelm@30066
|
959 |
\endisadelimnoproof
|
wenzelm@30066
|
960 |
%
|
wenzelm@30066
|
961 |
\isatagnoproof
|
wenzelm@30066
|
962 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
963 |
%
|
wenzelm@30066
|
964 |
\endisatagnoproof
|
wenzelm@30066
|
965 |
{\isafoldnoproof}%
|
wenzelm@30066
|
966 |
%
|
wenzelm@30066
|
967 |
\isadelimnoproof
|
wenzelm@30066
|
968 |
\isanewline
|
wenzelm@30066
|
969 |
%
|
wenzelm@30066
|
970 |
\endisadelimnoproof
|
wenzelm@30066
|
971 |
%
|
wenzelm@30066
|
972 |
\isadelimproof
|
wenzelm@30066
|
973 |
\ \ %
|
wenzelm@30066
|
974 |
\endisadelimproof
|
wenzelm@30066
|
975 |
%
|
wenzelm@30066
|
976 |
\isatagproof
|
wenzelm@30066
|
977 |
\isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
978 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
979 |
\ C\isanewline
|
wenzelm@30066
|
980 |
\ \ \isacommand{proof}\isamarkupfalse%
|
wenzelm@30066
|
981 |
\isanewline
|
wenzelm@30066
|
982 |
\ \ \ \ \isacommand{assume}\isamarkupfalse%
|
wenzelm@30066
|
983 |
\ A\isanewline
|
wenzelm@30066
|
984 |
\ \ \ \ \isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
985 |
\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30066
|
986 |
\ C%
|
wenzelm@30066
|
987 |
\endisatagproof
|
wenzelm@30066
|
988 |
{\isafoldproof}%
|
wenzelm@30066
|
989 |
%
|
wenzelm@30066
|
990 |
\isadelimproof
|
wenzelm@30066
|
991 |
%
|
wenzelm@30066
|
992 |
\endisadelimproof
|
wenzelm@30066
|
993 |
%
|
wenzelm@30066
|
994 |
\isadelimnoproof
|
wenzelm@30066
|
995 |
\ %
|
wenzelm@30066
|
996 |
\endisadelimnoproof
|
wenzelm@30066
|
997 |
%
|
wenzelm@30066
|
998 |
\isatagnoproof
|
wenzelm@30066
|
999 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
1000 |
%
|
wenzelm@30066
|
1001 |
\endisatagnoproof
|
wenzelm@30066
|
1002 |
{\isafoldnoproof}%
|
wenzelm@30066
|
1003 |
%
|
wenzelm@30066
|
1004 |
\isadelimnoproof
|
wenzelm@30066
|
1005 |
\isanewline
|
wenzelm@30066
|
1006 |
%
|
wenzelm@30066
|
1007 |
\endisadelimnoproof
|
wenzelm@30066
|
1008 |
%
|
wenzelm@30066
|
1009 |
\isadelimproof
|
wenzelm@30066
|
1010 |
\ \ %
|
wenzelm@30066
|
1011 |
\endisadelimproof
|
wenzelm@30066
|
1012 |
%
|
wenzelm@30066
|
1013 |
\isatagproof
|
wenzelm@30066
|
1014 |
\isacommand{next}\isamarkupfalse%
|
wenzelm@30066
|
1015 |
\isanewline
|
wenzelm@30066
|
1016 |
\ \ \ \ \isacommand{assume}\isamarkupfalse%
|
wenzelm@30066
|
1017 |
\ B\isanewline
|
wenzelm@30066
|
1018 |
\ \ \ \ \isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
1019 |
\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30066
|
1020 |
\ C%
|
wenzelm@30066
|
1021 |
\endisatagproof
|
wenzelm@30066
|
1022 |
{\isafoldproof}%
|
wenzelm@30066
|
1023 |
%
|
wenzelm@30066
|
1024 |
\isadelimproof
|
wenzelm@30066
|
1025 |
%
|
wenzelm@30066
|
1026 |
\endisadelimproof
|
wenzelm@30066
|
1027 |
%
|
wenzelm@30066
|
1028 |
\isadelimnoproof
|
wenzelm@30066
|
1029 |
\ %
|
wenzelm@30066
|
1030 |
\endisadelimnoproof
|
wenzelm@30066
|
1031 |
%
|
wenzelm@30066
|
1032 |
\isatagnoproof
|
wenzelm@30066
|
1033 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
1034 |
%
|
wenzelm@30066
|
1035 |
\endisatagnoproof
|
wenzelm@30066
|
1036 |
{\isafoldnoproof}%
|
wenzelm@30066
|
1037 |
%
|
wenzelm@30066
|
1038 |
\isadelimnoproof
|
wenzelm@30066
|
1039 |
\isanewline
|
wenzelm@30066
|
1040 |
%
|
wenzelm@30066
|
1041 |
\endisadelimnoproof
|
wenzelm@30066
|
1042 |
%
|
wenzelm@30066
|
1043 |
\isadelimproof
|
wenzelm@30066
|
1044 |
\ \ %
|
wenzelm@30066
|
1045 |
\endisadelimproof
|
wenzelm@30066
|
1046 |
%
|
wenzelm@30066
|
1047 |
\isatagproof
|
wenzelm@30066
|
1048 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30066
|
1049 |
%
|
wenzelm@30066
|
1050 |
\end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
1051 |
\isanewline
|
wenzelm@30066
|
1052 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1053 |
\ A\ \isakeyword{and}\ B%
|
wenzelm@30066
|
1054 |
\endisatagproof
|
wenzelm@30066
|
1055 |
{\isafoldproof}%
|
wenzelm@30066
|
1056 |
%
|
wenzelm@30066
|
1057 |
\isadelimproof
|
wenzelm@30066
|
1058 |
%
|
wenzelm@30066
|
1059 |
\endisadelimproof
|
wenzelm@30066
|
1060 |
%
|
wenzelm@30066
|
1061 |
\isadelimnoproof
|
wenzelm@30066
|
1062 |
\ %
|
wenzelm@30066
|
1063 |
\endisadelimnoproof
|
wenzelm@30066
|
1064 |
%
|
wenzelm@30066
|
1065 |
\isatagnoproof
|
wenzelm@30066
|
1066 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
1067 |
%
|
wenzelm@30066
|
1068 |
\endisatagnoproof
|
wenzelm@30066
|
1069 |
{\isafoldnoproof}%
|
wenzelm@30066
|
1070 |
%
|
wenzelm@30066
|
1071 |
\isadelimnoproof
|
wenzelm@30066
|
1072 |
\isanewline
|
wenzelm@30066
|
1073 |
%
|
wenzelm@30066
|
1074 |
\endisadelimnoproof
|
wenzelm@30066
|
1075 |
%
|
wenzelm@30066
|
1076 |
\isadelimproof
|
wenzelm@30066
|
1077 |
\ \ %
|
wenzelm@30066
|
1078 |
\endisadelimproof
|
wenzelm@30066
|
1079 |
%
|
wenzelm@30066
|
1080 |
\isatagproof
|
wenzelm@30066
|
1081 |
\isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
1082 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1083 |
\ {\isachardoublequoteopen}A\ {\isasymand}\ B{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30066
|
1084 |
%
|
wenzelm@30066
|
1085 |
\end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
1086 |
\isanewline
|
wenzelm@30066
|
1087 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1088 |
\ {\isachardoublequoteopen}A\ {\isasymand}\ B{\isachardoublequoteclose}%
|
wenzelm@30066
|
1089 |
\endisatagproof
|
wenzelm@30066
|
1090 |
{\isafoldproof}%
|
wenzelm@30066
|
1091 |
%
|
wenzelm@30066
|
1092 |
\isadelimproof
|
wenzelm@30066
|
1093 |
%
|
wenzelm@30066
|
1094 |
\endisadelimproof
|
wenzelm@30066
|
1095 |
%
|
wenzelm@30066
|
1096 |
\isadelimnoproof
|
wenzelm@30066
|
1097 |
\ %
|
wenzelm@30066
|
1098 |
\endisadelimnoproof
|
wenzelm@30066
|
1099 |
%
|
wenzelm@30066
|
1100 |
\isatagnoproof
|
wenzelm@30066
|
1101 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
1102 |
%
|
wenzelm@30066
|
1103 |
\endisatagnoproof
|
wenzelm@30066
|
1104 |
{\isafoldnoproof}%
|
wenzelm@30066
|
1105 |
%
|
wenzelm@30066
|
1106 |
\isadelimnoproof
|
wenzelm@30066
|
1107 |
\isanewline
|
wenzelm@30066
|
1108 |
%
|
wenzelm@30066
|
1109 |
\endisadelimnoproof
|
wenzelm@30066
|
1110 |
%
|
wenzelm@30066
|
1111 |
\isadelimproof
|
wenzelm@30066
|
1112 |
\ \ %
|
wenzelm@30066
|
1113 |
\endisadelimproof
|
wenzelm@30066
|
1114 |
%
|
wenzelm@30066
|
1115 |
\isatagproof
|
wenzelm@30066
|
1116 |
\isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
1117 |
\ \isacommand{obtain}\isamarkupfalse%
|
wenzelm@30066
|
1118 |
\ A\ \isakeyword{and}\ B\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30066
|
1119 |
%
|
wenzelm@30066
|
1120 |
\end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
1121 |
\isanewline
|
wenzelm@30066
|
1122 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1123 |
\ {\isachardoublequoteopen}{\isasymbottom}{\isachardoublequoteclose}%
|
wenzelm@30066
|
1124 |
\endisatagproof
|
wenzelm@30066
|
1125 |
{\isafoldproof}%
|
wenzelm@30066
|
1126 |
%
|
wenzelm@30066
|
1127 |
\isadelimproof
|
wenzelm@30066
|
1128 |
%
|
wenzelm@30066
|
1129 |
\endisadelimproof
|
wenzelm@30066
|
1130 |
%
|
wenzelm@30066
|
1131 |
\isadelimnoproof
|
wenzelm@30066
|
1132 |
\ %
|
wenzelm@30066
|
1133 |
\endisadelimnoproof
|
wenzelm@30066
|
1134 |
%
|
wenzelm@30066
|
1135 |
\isatagnoproof
|
wenzelm@30066
|
1136 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
1137 |
%
|
wenzelm@30066
|
1138 |
\endisatagnoproof
|
wenzelm@30066
|
1139 |
{\isafoldnoproof}%
|
wenzelm@30066
|
1140 |
%
|
wenzelm@30066
|
1141 |
\isadelimnoproof
|
wenzelm@30066
|
1142 |
\isanewline
|
wenzelm@30066
|
1143 |
%
|
wenzelm@30066
|
1144 |
\endisadelimnoproof
|
wenzelm@30066
|
1145 |
%
|
wenzelm@30066
|
1146 |
\isadelimproof
|
wenzelm@30066
|
1147 |
\ \ %
|
wenzelm@30066
|
1148 |
\endisadelimproof
|
wenzelm@30066
|
1149 |
%
|
wenzelm@30066
|
1150 |
\isatagproof
|
wenzelm@30066
|
1151 |
\isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
1152 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1153 |
\ A\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30066
|
1154 |
%
|
wenzelm@30066
|
1155 |
\end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
1156 |
\isanewline
|
wenzelm@30066
|
1157 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1158 |
\ {\isachardoublequoteopen}{\isasymtop}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30066
|
1159 |
%
|
wenzelm@30066
|
1160 |
\end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
1161 |
\isanewline
|
wenzelm@30066
|
1162 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1163 |
\ {\isachardoublequoteopen}{\isasymnot}\ A{\isachardoublequoteclose}\isanewline
|
wenzelm@30066
|
1164 |
\ \ \isacommand{proof}\isamarkupfalse%
|
wenzelm@30066
|
1165 |
\isanewline
|
wenzelm@30066
|
1166 |
\ \ \ \ \isacommand{assume}\isamarkupfalse%
|
wenzelm@30066
|
1167 |
\ A\isanewline
|
wenzelm@30066
|
1168 |
\ \ \ \ \isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
1169 |
\ \isacommand{show}\isamarkupfalse%
|
wenzelm@30066
|
1170 |
\ {\isachardoublequoteopen}{\isasymbottom}{\isachardoublequoteclose}%
|
wenzelm@30066
|
1171 |
\endisatagproof
|
wenzelm@30066
|
1172 |
{\isafoldproof}%
|
wenzelm@30066
|
1173 |
%
|
wenzelm@30066
|
1174 |
\isadelimproof
|
wenzelm@30066
|
1175 |
%
|
wenzelm@30066
|
1176 |
\endisadelimproof
|
wenzelm@30066
|
1177 |
%
|
wenzelm@30066
|
1178 |
\isadelimnoproof
|
wenzelm@30066
|
1179 |
\ %
|
wenzelm@30066
|
1180 |
\endisadelimnoproof
|
wenzelm@30066
|
1181 |
%
|
wenzelm@30066
|
1182 |
\isatagnoproof
|
wenzelm@30066
|
1183 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
1184 |
%
|
wenzelm@30066
|
1185 |
\endisatagnoproof
|
wenzelm@30066
|
1186 |
{\isafoldnoproof}%
|
wenzelm@30066
|
1187 |
%
|
wenzelm@30066
|
1188 |
\isadelimnoproof
|
wenzelm@30066
|
1189 |
\isanewline
|
wenzelm@30066
|
1190 |
%
|
wenzelm@30066
|
1191 |
\endisadelimnoproof
|
wenzelm@30066
|
1192 |
%
|
wenzelm@30066
|
1193 |
\isadelimproof
|
wenzelm@30066
|
1194 |
\ \ %
|
wenzelm@30066
|
1195 |
\endisadelimproof
|
wenzelm@30066
|
1196 |
%
|
wenzelm@30066
|
1197 |
\isatagproof
|
wenzelm@30066
|
1198 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30066
|
1199 |
%
|
wenzelm@30066
|
1200 |
\end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
1201 |
\isanewline
|
wenzelm@30066
|
1202 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1203 |
\ {\isachardoublequoteopen}{\isasymnot}\ A{\isachardoublequoteclose}\ \isakeyword{and}\ A%
|
wenzelm@30066
|
1204 |
\endisatagproof
|
wenzelm@30066
|
1205 |
{\isafoldproof}%
|
wenzelm@30066
|
1206 |
%
|
wenzelm@30066
|
1207 |
\isadelimproof
|
wenzelm@30066
|
1208 |
%
|
wenzelm@30066
|
1209 |
\endisadelimproof
|
wenzelm@30066
|
1210 |
%
|
wenzelm@30066
|
1211 |
\isadelimnoproof
|
wenzelm@30066
|
1212 |
\ %
|
wenzelm@30066
|
1213 |
\endisadelimnoproof
|
wenzelm@30066
|
1214 |
%
|
wenzelm@30066
|
1215 |
\isatagnoproof
|
wenzelm@30066
|
1216 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
1217 |
%
|
wenzelm@30066
|
1218 |
\endisatagnoproof
|
wenzelm@30066
|
1219 |
{\isafoldnoproof}%
|
wenzelm@30066
|
1220 |
%
|
wenzelm@30066
|
1221 |
\isadelimnoproof
|
wenzelm@30066
|
1222 |
\isanewline
|
wenzelm@30066
|
1223 |
%
|
wenzelm@30066
|
1224 |
\endisadelimnoproof
|
wenzelm@30066
|
1225 |
%
|
wenzelm@30066
|
1226 |
\isadelimproof
|
wenzelm@30066
|
1227 |
\ \ %
|
wenzelm@30066
|
1228 |
\endisadelimproof
|
wenzelm@30066
|
1229 |
%
|
wenzelm@30066
|
1230 |
\isatagproof
|
wenzelm@30066
|
1231 |
\isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
1232 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1233 |
\ B\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30066
|
1234 |
%
|
wenzelm@30066
|
1235 |
\end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
1236 |
\isanewline
|
wenzelm@30066
|
1237 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1238 |
\ {\isachardoublequoteopen}{\isasymforall}x{\isachardot}\ B\ x{\isachardoublequoteclose}\isanewline
|
wenzelm@30066
|
1239 |
\ \ \isacommand{proof}\isamarkupfalse%
|
wenzelm@30066
|
1240 |
\isanewline
|
wenzelm@30066
|
1241 |
\ \ \ \ \isacommand{fix}\isamarkupfalse%
|
wenzelm@30066
|
1242 |
\ x\isanewline
|
wenzelm@30066
|
1243 |
\ \ \ \ \isacommand{show}\isamarkupfalse%
|
wenzelm@30066
|
1244 |
\ {\isachardoublequoteopen}B\ x{\isachardoublequoteclose}%
|
wenzelm@30066
|
1245 |
\endisatagproof
|
wenzelm@30066
|
1246 |
{\isafoldproof}%
|
wenzelm@30066
|
1247 |
%
|
wenzelm@30066
|
1248 |
\isadelimproof
|
wenzelm@30066
|
1249 |
%
|
wenzelm@30066
|
1250 |
\endisadelimproof
|
wenzelm@30066
|
1251 |
%
|
wenzelm@30066
|
1252 |
\isadelimnoproof
|
wenzelm@30066
|
1253 |
\ %
|
wenzelm@30066
|
1254 |
\endisadelimnoproof
|
wenzelm@30066
|
1255 |
%
|
wenzelm@30066
|
1256 |
\isatagnoproof
|
wenzelm@30066
|
1257 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
1258 |
%
|
wenzelm@30066
|
1259 |
\endisatagnoproof
|
wenzelm@30066
|
1260 |
{\isafoldnoproof}%
|
wenzelm@30066
|
1261 |
%
|
wenzelm@30066
|
1262 |
\isadelimnoproof
|
wenzelm@30066
|
1263 |
\isanewline
|
wenzelm@30066
|
1264 |
%
|
wenzelm@30066
|
1265 |
\endisadelimnoproof
|
wenzelm@30066
|
1266 |
%
|
wenzelm@30066
|
1267 |
\isadelimproof
|
wenzelm@30066
|
1268 |
\ \ %
|
wenzelm@30066
|
1269 |
\endisadelimproof
|
wenzelm@30066
|
1270 |
%
|
wenzelm@30066
|
1271 |
\isatagproof
|
wenzelm@30066
|
1272 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30066
|
1273 |
%
|
wenzelm@30066
|
1274 |
\end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
1275 |
\isanewline
|
wenzelm@30066
|
1276 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1277 |
\ {\isachardoublequoteopen}{\isasymforall}x{\isachardot}\ B\ x{\isachardoublequoteclose}%
|
wenzelm@30066
|
1278 |
\endisatagproof
|
wenzelm@30066
|
1279 |
{\isafoldproof}%
|
wenzelm@30066
|
1280 |
%
|
wenzelm@30066
|
1281 |
\isadelimproof
|
wenzelm@30066
|
1282 |
%
|
wenzelm@30066
|
1283 |
\endisadelimproof
|
wenzelm@30066
|
1284 |
%
|
wenzelm@30066
|
1285 |
\isadelimnoproof
|
wenzelm@30066
|
1286 |
\ %
|
wenzelm@30066
|
1287 |
\endisadelimnoproof
|
wenzelm@30066
|
1288 |
%
|
wenzelm@30066
|
1289 |
\isatagnoproof
|
wenzelm@30066
|
1290 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
1291 |
%
|
wenzelm@30066
|
1292 |
\endisatagnoproof
|
wenzelm@30066
|
1293 |
{\isafoldnoproof}%
|
wenzelm@30066
|
1294 |
%
|
wenzelm@30066
|
1295 |
\isadelimnoproof
|
wenzelm@30066
|
1296 |
\isanewline
|
wenzelm@30066
|
1297 |
%
|
wenzelm@30066
|
1298 |
\endisadelimnoproof
|
wenzelm@30066
|
1299 |
%
|
wenzelm@30066
|
1300 |
\isadelimproof
|
wenzelm@30066
|
1301 |
\ \ %
|
wenzelm@30066
|
1302 |
\endisadelimproof
|
wenzelm@30066
|
1303 |
%
|
wenzelm@30066
|
1304 |
\isatagproof
|
wenzelm@30066
|
1305 |
\isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
1306 |
\ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1307 |
\ {\isachardoublequoteopen}B\ a{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30066
|
1308 |
%
|
wenzelm@30066
|
1309 |
\end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
1310 |
\isanewline
|
wenzelm@30066
|
1311 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1312 |
\ {\isachardoublequoteopen}{\isasymexists}x{\isachardot}\ B\ x{\isachardoublequoteclose}\isanewline
|
wenzelm@30066
|
1313 |
\ \ \isacommand{proof}\isamarkupfalse%
|
wenzelm@30066
|
1314 |
\isanewline
|
wenzelm@30066
|
1315 |
\ \ \ \ \isacommand{show}\isamarkupfalse%
|
wenzelm@30066
|
1316 |
\ {\isachardoublequoteopen}B\ a{\isachardoublequoteclose}%
|
wenzelm@30066
|
1317 |
\endisatagproof
|
wenzelm@30066
|
1318 |
{\isafoldproof}%
|
wenzelm@30066
|
1319 |
%
|
wenzelm@30066
|
1320 |
\isadelimproof
|
wenzelm@30066
|
1321 |
%
|
wenzelm@30066
|
1322 |
\endisadelimproof
|
wenzelm@30066
|
1323 |
%
|
wenzelm@30066
|
1324 |
\isadelimnoproof
|
wenzelm@30066
|
1325 |
\ %
|
wenzelm@30066
|
1326 |
\endisadelimnoproof
|
wenzelm@30066
|
1327 |
%
|
wenzelm@30066
|
1328 |
\isatagnoproof
|
wenzelm@30066
|
1329 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
1330 |
%
|
wenzelm@30066
|
1331 |
\endisatagnoproof
|
wenzelm@30066
|
1332 |
{\isafoldnoproof}%
|
wenzelm@30066
|
1333 |
%
|
wenzelm@30066
|
1334 |
\isadelimnoproof
|
wenzelm@30066
|
1335 |
\isanewline
|
wenzelm@30066
|
1336 |
%
|
wenzelm@30066
|
1337 |
\endisadelimnoproof
|
wenzelm@30066
|
1338 |
%
|
wenzelm@30066
|
1339 |
\isadelimproof
|
wenzelm@30066
|
1340 |
\ \ %
|
wenzelm@30066
|
1341 |
\endisadelimproof
|
wenzelm@30066
|
1342 |
%
|
wenzelm@30066
|
1343 |
\isatagproof
|
wenzelm@30066
|
1344 |
\isacommand{qed}\isamarkupfalse%
|
wenzelm@30066
|
1345 |
%
|
wenzelm@30066
|
1346 |
\end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30066
|
1347 |
\isanewline
|
wenzelm@30066
|
1348 |
\ \ \isacommand{have}\isamarkupfalse%
|
wenzelm@30066
|
1349 |
\ {\isachardoublequoteopen}{\isasymexists}x{\isachardot}\ B\ x{\isachardoublequoteclose}%
|
wenzelm@30066
|
1350 |
\endisatagproof
|
wenzelm@30066
|
1351 |
{\isafoldproof}%
|
wenzelm@30066
|
1352 |
%
|
wenzelm@30066
|
1353 |
\isadelimproof
|
wenzelm@30066
|
1354 |
%
|
wenzelm@30066
|
1355 |
\endisadelimproof
|
wenzelm@30066
|
1356 |
%
|
wenzelm@30066
|
1357 |
\isadelimnoproof
|
wenzelm@30066
|
1358 |
\ %
|
wenzelm@30066
|
1359 |
\endisadelimnoproof
|
wenzelm@30066
|
1360 |
%
|
wenzelm@30066
|
1361 |
\isatagnoproof
|
wenzelm@30066
|
1362 |
\isacommand{sorry}\isamarkupfalse%
|
wenzelm@30066
|
1363 |
%
|
wenzelm@30066
|
1364 |
\endisatagnoproof
|
wenzelm@30066
|
1365 |
{\isafoldnoproof}%
|
wenzelm@30066
|
1366 |
%
|
wenzelm@30066
|
1367 |
\isadelimnoproof
|
wenzelm@30066
|
1368 |
\isanewline
|
wenzelm@30066
|
1369 |
%
|
wenzelm@30066
|
1370 |
\endisadelimnoproof
|
wenzelm@30066
|
1371 |
%
|
wenzelm@30066
|
1372 |
\isadelimproof
|
wenzelm@30066
|
1373 |
\ \ %
|
wenzelm@30066
|
1374 |
\endisadelimproof
|
wenzelm@30066
|
1375 |
%
|
wenzelm@30066
|
1376 |
\isatagproof
|
wenzelm@30066
|
1377 |
\isacommand{then}\isamarkupfalse%
|
wenzelm@30066
|
1378 |
\ \isacommand{obtain}\isamarkupfalse%
|
wenzelm@30066
|
1379 |
\ a\ \isakeyword{where}\ {\isachardoublequoteopen}B\ a{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
|
wenzelm@30066
|
1380 |
%
|
wenzelm@30066
|
1381 |
\end{minipage}
|
wenzelm@30066
|
1382 |
%
|
wenzelm@30066
|
1383 |
\endisatagproof
|
wenzelm@30066
|
1384 |
{\isafoldproof}%
|
wenzelm@30066
|
1385 |
%
|
wenzelm@30066
|
1386 |
\isadelimproof
|
wenzelm@30066
|
1387 |
%
|
wenzelm@30066
|
1388 |
\endisadelimproof
|
wenzelm@30066
|
1389 |
%
|
wenzelm@30066
|
1390 |
\begin{isamarkuptext}%
|
wenzelm@30066
|
1391 |
\bigskip\noindent Of course, these proofs are merely examples. As
|
wenzelm@30066
|
1392 |
sketched in \secref{sec:framework-subproof}, there is a fair amount
|
wenzelm@30066
|
1393 |
of flexibility in expressing Pure deductions in Isar. Here the user
|
wenzelm@30066
|
1394 |
is asked to express himself adequately, aiming at proof texts of
|
wenzelm@30066
|
1395 |
literary quality.%
|
wenzelm@30066
|
1396 |
\end{isamarkuptext}%
|
wenzelm@30066
|
1397 |
\isamarkuptrue%
|
wenzelm@30066
|
1398 |
%
|
wenzelm@30057
|
1399 |
\isadelimvisible
|
wenzelm@30057
|
1400 |
%
|
wenzelm@30057
|
1401 |
\endisadelimvisible
|
wenzelm@30057
|
1402 |
%
|
wenzelm@30057
|
1403 |
\isatagvisible
|
wenzelm@30057
|
1404 |
\isacommand{end}\isamarkupfalse%
|
wenzelm@30057
|
1405 |
%
|
wenzelm@30057
|
1406 |
\endisatagvisible
|
wenzelm@30057
|
1407 |
{\isafoldvisible}%
|
wenzelm@30057
|
1408 |
%
|
wenzelm@30057
|
1409 |
\isadelimvisible
|
wenzelm@30057
|
1410 |
%
|
wenzelm@30057
|
1411 |
\endisadelimvisible
|
wenzelm@30066
|
1412 |
\isanewline
|
wenzelm@30057
|
1413 |
\end{isabellebody}%
|
wenzelm@30057
|
1414 |
%%% Local Variables:
|
wenzelm@30057
|
1415 |
%%% mode: latex
|
wenzelm@30057
|
1416 |
%%% TeX-master: "root"
|
wenzelm@30057
|
1417 |
%%% End:
|