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\begin{isabellebody}%
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\def\isabellecontext{Examples{\isadigit{3}}}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isatagtheory
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%
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\endisatagtheory
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{\isafoldtheory}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isamarkupsubsection{Third Version: Local Interpretation%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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In the above example, the fact that \isa{{\isasymle}} is a partial
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order for the natural numbers was used in the proof of the
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second goal. In general, proofs of the equations may involve
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theorems implied by the fact the assumptions of the instantiated
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locale hold for the instantiating structure. If these theorems have
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been shown abstractly in the locale they can be made available
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conveniently in the context through an auxiliary local interpretation (keyword
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\isakeyword{interpret}). This interpretation is inside the proof of the global
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interpretation. The third revision of the example illustrates this.%
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\end{isamarkuptext}%
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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{interpretation}\isamarkupfalse%
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\ nat{\isacharcolon}\ partial{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
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\ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ {\isacharparenleft}op\ {\isasymle}{\isacharparenright}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\isacommand{proof}\isamarkupfalse%
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\ {\isacharminus}\isanewline
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\ \ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}partial{\isacharunderscore}order\ {\isacharparenleft}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isacommand{by}\isamarkupfalse%
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\ unfold{\isacharunderscore}locales\ auto\isanewline
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\ \ \isacommand{then}\isamarkupfalse%
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\ \isacommand{interpret}\isamarkupfalse%
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\ nat{\isacharcolon}\ partial{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ {\isacharbrackleft}nat{\isacharcomma}\ nat{\isacharbrackright}\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\ \isacommand{{\isachardot}}\isamarkupfalse%
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\isanewline
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\ \ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ {\isacharparenleft}op\ {\isasymle}{\isacharparenright}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
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\ nat{\isachardot}less{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
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\ auto\isanewline
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\isacommand{qed}\isamarkupfalse%
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%
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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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The inner interpretation does not require an
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elaborate new proof, it is immediate from the preceeding fact and
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proved with ``.''.
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This interpretation enriches the local proof context by
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the very theorems also obtained in the interpretation from
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Section~\ref{sec:po-first}, and \isa{nat{\isachardot}less{\isacharunderscore}def} may directly be
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used to unfold the definition. Theorems from the local
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interpretation disappear after leaving the proof context --- that
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is, after the closing \isakeyword{qed} --- and are
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then replaced by those with the desired substitutions of the strict
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order.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Further Interpretations%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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Further interpretations are necessary to reuse theorems from
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the other locales. In \isa{lattice} the operations \isa{{\isasymsqinter}} and
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\isa{{\isasymsqunion}} are substituted by \isa{min} and
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\isa{max}. The entire proof for the
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interpretation is reproduced in order to give an example of a more
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elaborate interpretation proof.%
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\end{isamarkuptext}%
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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{interpretation}\isamarkupfalse%
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\ nat{\isacharcolon}\ lattice\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
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\ \ \isakeyword{where}\ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ min\ x\ y{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ max\ x\ y{\isachardoublequoteclose}\isanewline
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\isacommand{proof}\isamarkupfalse%
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\ {\isacharminus}\isanewline
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\ \ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}lattice\ {\isacharparenleft}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}%
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\begin{isamarkuptxt}%
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We have already shown that this is a partial order,%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\ \ \ \ \isacommand{apply}\isamarkupfalse%
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\ unfold{\isacharunderscore}locales%
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\begin{isamarkuptxt}%
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hence only the lattice axioms remain to be shown: \begin{isabelle}%
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\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}inf{\isachardot}\ partial{\isacharunderscore}order{\isachardot}is{\isacharunderscore}inf\ op\ {\isasymle}\ x\ y\ inf\isanewline
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\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}sup{\isachardot}\ partial{\isacharunderscore}order{\isachardot}is{\isacharunderscore}sup\ op\ {\isasymle}\ x\ y\ sup%
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\end{isabelle} After unfolding \isa{is{\isacharunderscore}inf} and \isa{is{\isacharunderscore}sup},%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\ \ \ \ \isacommand{apply}\isamarkupfalse%
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\ {\isacharparenleft}unfold\ nat{\isachardot}is{\isacharunderscore}inf{\isacharunderscore}def\ nat{\isachardot}is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}%
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\begin{isamarkuptxt}%
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the goals become \begin{isabelle}%
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\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}inf{\isasymle}x{\isachardot}\ inf\ {\isasymle}\ y\ {\isasymand}\ {\isacharparenleft}{\isasymforall}z{\isachardot}\ z\ {\isasymle}\ x\ {\isasymand}\ z\ {\isasymle}\ y\ {\isasymlongrightarrow}\ z\ {\isasymle}\ inf{\isacharparenright}\isanewline
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\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}sup{\isasymge}x{\isachardot}\ y\ {\isasymle}\ sup\ {\isasymand}\ {\isacharparenleft}{\isasymforall}z{\isachardot}\ x\ {\isasymle}\ z\ {\isasymand}\ y\ {\isasymle}\ z\ {\isasymlongrightarrow}\ sup\ {\isasymle}\ z{\isacharparenright}%
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\end{isabelle} which can be solved
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by Presburger arithmetic.%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\ \ \ \ \isacommand{by}\isamarkupfalse%
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\ arith{\isacharplus}%
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\begin{isamarkuptxt}%
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In order to show the equations, we put ourselves in a
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situation where the lattice theorems can be used in a convenient way.%
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\end{isamarkuptxt}%
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\isamarkuptrue%
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\ \ \isacommand{then}\isamarkupfalse%
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\ \isacommand{interpret}\isamarkupfalse%
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\ nat{\isacharcolon}\ lattice\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\ \isacommand{{\isachardot}}\isamarkupfalse%
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\isanewline
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\ \ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ min\ x\ y{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isacommand{by}\isamarkupfalse%
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\ {\isacharparenleft}bestsimp\ simp{\isacharcolon}\ nat{\isachardot}meet{\isacharunderscore}def\ nat{\isachardot}is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\isanewline
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\ \ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ max\ x\ y{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isacommand{by}\isamarkupfalse%
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\ {\isacharparenleft}bestsimp\ simp{\isacharcolon}\ nat{\isachardot}join{\isacharunderscore}def\ nat{\isachardot}is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\isanewline
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\isacommand{qed}\isamarkupfalse%
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%
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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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That the relation \isa{{\isasymle}} is a total order completes this
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sequence of interpretations.%
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\end{isamarkuptext}%
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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{interpretation}\isamarkupfalse%
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\ nat{\isacharcolon}\ total{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
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\ \ \isacommand{by}\isamarkupfalse%
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\ unfold{\isacharunderscore}locales\ arith%
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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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Theorems that are available in the theory at this point are shown in
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Table~\ref{tab:nat-lattice}.
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\begin{table}
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\hrule
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\vspace{2ex}
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\begin{center}
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\begin{tabular}{l}
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\isa{nat{\isachardot}less{\isacharunderscore}def} from locale \isa{partial{\isacharunderscore}order}: \\
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\quad \isa{{\isacharparenleft}{\isacharquery}x\ {\isacharless}\ {\isacharquery}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharquery}x\ {\isasymle}\ {\isacharquery}y\ {\isasymand}\ {\isacharquery}x\ {\isasymnoteq}\ {\isacharquery}y{\isacharparenright}} \\
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\isa{nat{\isachardot}meet{\isacharunderscore}left} from locale \isa{lattice}: \\
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\quad \isa{min\ {\isacharquery}x\ {\isacharquery}y\ {\isasymle}\ {\isacharquery}x} \\
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\isa{nat{\isachardot}join{\isacharunderscore}distr} from locale \isa{distrib{\isacharunderscore}lattice}: \\
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\quad \isa{max\ {\isacharquery}x\ {\isacharparenleft}min\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ min\ {\isacharparenleft}max\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharparenleft}max\ {\isacharquery}x\ {\isacharquery}z{\isacharparenright}} \\
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\isa{nat{\isachardot}less{\isacharunderscore}total} from locale \isa{total{\isacharunderscore}order}: \\
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\quad \isa{{\isacharquery}x\ {\isacharless}\ {\isacharquery}y\ {\isasymor}\ {\isacharquery}x\ {\isacharequal}\ {\isacharquery}y\ {\isasymor}\ {\isacharquery}y\ {\isacharless}\ {\isacharquery}x}
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\end{tabular}
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\end{center}
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\hrule
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\caption{Interpreted theorems for \isa{{\isasymle}} on the natural numbers.}
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\label{tab:nat-lattice}
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\end{table}
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Note that since the locale hierarchy reflects that total orders are
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distributive lattices, an explicit interpretation of distributive
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lattices for the order relation on natural numbers is not neccessary.
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Why not push this idea further and just give the last interpretation
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as a single interpretation instead of the sequence of three? The
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reasons for this are twofold:
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\begin{itemize}
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\item
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Often it is easier to work in an incremental fashion, because later
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interpretations require theorems provided by earlier
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interpretations.
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\item
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Assume that a definition is made in some locale $l_1$, and that $l_2$
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imports $l_1$. Let an equation for the definition be
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proved in an interpretation of $l_2$. The equation will be unfolded
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in interpretations of theorems added to $l_2$ or below in the import
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hierarchy, but not for theorems added above $l_2$.
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Hence, an equation interpreting a definition should always be given in
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an interpretation of the locale where the definition is made, not in
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an interpretation of a locale further down the hierarchy.
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\end{itemize}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Lattice \isa{dvd} on \isa{nat}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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Divisibility on the natural numbers is a distributive lattice
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but not a total order. Interpretation again proceeds
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incrementally.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{interpretation}\isamarkupfalse%
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\ nat{\isacharunderscore}dvd{\isacharcolon}\ partial{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
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\ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ dvd\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}{\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{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}partial{\isacharunderscore}order\ {\isacharparenleft}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isacommand{by}\isamarkupfalse%
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\ unfold{\isacharunderscore}locales\ {\isacharparenleft}auto\ simp{\isacharcolon}\ dvd{\isacharunderscore}def{\isacharparenright}\isanewline
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\ \ \isacommand{then}\isamarkupfalse%
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\ \isacommand{interpret}\isamarkupfalse%
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\ nat{\isacharunderscore}dvd{\isacharcolon}\ partial{\isacharunderscore}order\ {\isacharbrackleft}{\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\ \isacommand{{\isachardot}}\isamarkupfalse%
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\isanewline
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\ \ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ dvd\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isacommand{apply}\isamarkupfalse%
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\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}less{\isacharunderscore}def{\isacharparenright}\isanewline
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\ \ \ \ \isacommand{apply}\isamarkupfalse%
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265 |
\ auto\isanewline
|
|
266 |
\ \ \ \ \isacommand{done}\isamarkupfalse%
|
|
267 |
\isanewline
|
|
268 |
\isacommand{qed}\isamarkupfalse%
|
|
269 |
%
|
|
270 |
\endisatagproof
|
|
271 |
{\isafoldproof}%
|
|
272 |
%
|
|
273 |
\isadelimproof
|
|
274 |
%
|
|
275 |
\endisadelimproof
|
|
276 |
%
|
|
277 |
\begin{isamarkuptext}%
|
|
278 |
Note that there is no symbol for strict divisibility. Instead,
|
|
279 |
interpretation substitutes \isa{x\ dvd\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y}.%
|
|
280 |
\end{isamarkuptext}%
|
|
281 |
\isamarkuptrue%
|
|
282 |
\isacommand{interpretation}\isamarkupfalse%
|
|
283 |
\ nat{\isacharunderscore}dvd{\isacharcolon}\ lattice\ {\isacharbrackleft}{\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
|
|
284 |
\ \ \isakeyword{where}\ nat{\isacharunderscore}dvd{\isacharunderscore}meet{\isacharunderscore}eq{\isacharcolon}\isanewline
|
|
285 |
\ \ \ \ \ \ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ gcd\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
286 |
\ \ \ \ \isakeyword{and}\ nat{\isacharunderscore}dvd{\isacharunderscore}join{\isacharunderscore}eq{\isacharcolon}\isanewline
|
|
287 |
\ \ \ \ \ \ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ lcm\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
288 |
%
|
|
289 |
\isadelimproof
|
|
290 |
%
|
|
291 |
\endisadelimproof
|
|
292 |
%
|
|
293 |
\isatagproof
|
|
294 |
\isacommand{proof}\isamarkupfalse%
|
|
295 |
\ {\isacharminus}\isanewline
|
|
296 |
\ \ \isacommand{show}\isamarkupfalse%
|
|
297 |
\ {\isachardoublequoteopen}lattice\ {\isacharparenleft}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
298 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
299 |
\ unfold{\isacharunderscore}locales\isanewline
|
|
300 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
301 |
\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}is{\isacharunderscore}inf{\isacharunderscore}def\ nat{\isacharunderscore}dvd{\isachardot}is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\isanewline
|
|
302 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
303 |
\ {\isacharparenleft}rule{\isacharunderscore}tac\ x\ {\isacharequal}\ {\isachardoublequoteopen}gcd\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\ \isakeyword{in}\ exI{\isacharparenright}\isanewline
|
|
304 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
305 |
\ auto\ {\isacharbrackleft}{\isadigit{1}}{\isacharbrackright}\isanewline
|
|
306 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
307 |
\ {\isacharparenleft}rule{\isacharunderscore}tac\ x\ {\isacharequal}\ {\isachardoublequoteopen}lcm\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\ \isakeyword{in}\ exI{\isacharparenright}\isanewline
|
|
308 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
309 |
\ {\isacharparenleft}auto\ intro{\isacharcolon}\ lcm{\isacharunderscore}dvd{\isadigit{1}}\ lcm{\isacharunderscore}dvd{\isadigit{2}}\ lcm{\isacharunderscore}least{\isacharparenright}\isanewline
|
|
310 |
\ \ \ \ \isacommand{done}\isamarkupfalse%
|
|
311 |
\isanewline
|
|
312 |
\ \ \isacommand{then}\isamarkupfalse%
|
|
313 |
\ \isacommand{interpret}\isamarkupfalse%
|
|
314 |
\ nat{\isacharunderscore}dvd{\isacharcolon}\ lattice\ {\isacharbrackleft}{\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\ \isacommand{{\isachardot}}\isamarkupfalse%
|
|
315 |
\isanewline
|
|
316 |
\ \ \isacommand{show}\isamarkupfalse%
|
|
317 |
\ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ gcd\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
318 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
319 |
\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}meet{\isacharunderscore}def{\isacharparenright}\isanewline
|
|
320 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
321 |
\ {\isacharparenleft}rule\ the{\isacharunderscore}equality{\isacharparenright}\isanewline
|
|
322 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
323 |
\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\isanewline
|
|
324 |
\ \ \ \ \isacommand{by}\isamarkupfalse%
|
|
325 |
\ auto\isanewline
|
|
326 |
\ \ \isacommand{show}\isamarkupfalse%
|
|
327 |
\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ lcm\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
328 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
329 |
\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}join{\isacharunderscore}def{\isacharparenright}\isanewline
|
|
330 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
331 |
\ {\isacharparenleft}rule\ the{\isacharunderscore}equality{\isacharparenright}\isanewline
|
|
332 |
\ \ \ \ \isacommand{apply}\isamarkupfalse%
|
|
333 |
\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isachardot}is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\isanewline
|
|
334 |
\ \ \ \ \isacommand{by}\isamarkupfalse%
|
|
335 |
\ {\isacharparenleft}auto\ intro{\isacharcolon}\ lcm{\isacharunderscore}dvd{\isadigit{1}}\ lcm{\isacharunderscore}dvd{\isadigit{2}}\ lcm{\isacharunderscore}least{\isacharparenright}\isanewline
|
|
336 |
\isacommand{qed}\isamarkupfalse%
|
|
337 |
%
|
|
338 |
\endisatagproof
|
|
339 |
{\isafoldproof}%
|
|
340 |
%
|
|
341 |
\isadelimproof
|
|
342 |
%
|
|
343 |
\endisadelimproof
|
|
344 |
%
|
|
345 |
\begin{isamarkuptext}%
|
|
346 |
Equations \isa{nat{\isacharunderscore}dvd{\isacharunderscore}meet{\isacharunderscore}eq} and \isa{nat{\isacharunderscore}dvd{\isacharunderscore}join{\isacharunderscore}eq} are named since they are handy in the proof of
|
|
347 |
the subsequent interpretation.%
|
|
348 |
\end{isamarkuptext}%
|
|
349 |
\isamarkuptrue%
|
|
350 |
%
|
|
351 |
\isadeliminvisible
|
|
352 |
%
|
|
353 |
\endisadeliminvisible
|
|
354 |
%
|
|
355 |
\isataginvisible
|
|
356 |
\isacommand{ML}\isamarkupfalse%
|
|
357 |
\ {\isacharverbatimopen}\ set\ quick{\isacharunderscore}and{\isacharunderscore}dirty\ {\isacharverbatimclose}\isanewline
|
|
358 |
\isanewline
|
|
359 |
\isanewline
|
|
360 |
\isanewline
|
|
361 |
\isacommand{lemma}\isamarkupfalse%
|
|
362 |
\ gcd{\isacharunderscore}lcm{\isacharunderscore}distr{\isacharcolon}\isanewline
|
|
363 |
\ \ {\isachardoublequoteopen}gcd\ {\isacharparenleft}x{\isacharcomma}\ lcm\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ lcm\ {\isacharparenleft}gcd\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isacharcomma}\ gcd\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
364 |
\ \ \isacommand{sorry}\isamarkupfalse%
|
|
365 |
\isanewline
|
|
366 |
\isanewline
|
|
367 |
\isacommand{ML}\isamarkupfalse%
|
|
368 |
\ {\isacharverbatimopen}\ reset\ quick{\isacharunderscore}and{\isacharunderscore}dirty\ {\isacharverbatimclose}%
|
|
369 |
\endisataginvisible
|
|
370 |
{\isafoldinvisible}%
|
|
371 |
%
|
|
372 |
\isadeliminvisible
|
|
373 |
%
|
|
374 |
\endisadeliminvisible
|
|
375 |
\isanewline
|
|
376 |
%
|
|
377 |
\isadelimvisible
|
|
378 |
\ \ \isanewline
|
|
379 |
%
|
|
380 |
\endisadelimvisible
|
|
381 |
%
|
|
382 |
\isatagvisible
|
|
383 |
\isacommand{interpretation}\isamarkupfalse%
|
|
384 |
\ nat{\isacharunderscore}dvd{\isacharcolon}\isanewline
|
|
385 |
\ \ distrib{\isacharunderscore}lattice\ {\isacharbrackleft}{\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}{\isacharbrackright}\isanewline
|
|
386 |
\ \ \isacommand{apply}\isamarkupfalse%
|
|
387 |
\ unfold{\isacharunderscore}locales%
|
|
388 |
\begin{isamarkuptxt}%
|
|
389 |
\begin{isabelle}%
|
|
390 |
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ y\ z{\isachardot}\isanewline
|
|
391 |
\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }lattice{\isachardot}meet\ op\ dvd\ x\ {\isacharparenleft}lattice{\isachardot}join\ op\ dvd\ y\ z{\isacharparenright}\ {\isacharequal}\isanewline
|
|
392 |
\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }lattice{\isachardot}join\ op\ dvd\ {\isacharparenleft}lattice{\isachardot}meet\ op\ dvd\ x\ y{\isacharparenright}\isanewline
|
|
393 |
\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ \ }{\isacharparenleft}lattice{\isachardot}meet\ op\ dvd\ x\ z{\isacharparenright}%
|
|
394 |
\end{isabelle}%
|
|
395 |
\end{isamarkuptxt}%
|
|
396 |
\isamarkuptrue%
|
|
397 |
\ \ \isacommand{apply}\isamarkupfalse%
|
|
398 |
\ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isacharunderscore}meet{\isacharunderscore}eq\ nat{\isacharunderscore}dvd{\isacharunderscore}join{\isacharunderscore}eq{\isacharparenright}%
|
|
399 |
\begin{isamarkuptxt}%
|
|
400 |
\begin{isabelle}%
|
|
401 |
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ y\ z{\isachardot}\ gcd\ {\isacharparenleft}x{\isacharcomma}\ lcm\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ lcm\ {\isacharparenleft}gcd\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isacharcomma}\ gcd\ {\isacharparenleft}x{\isacharcomma}\ z{\isacharparenright}{\isacharparenright}%
|
|
402 |
\end{isabelle}%
|
|
403 |
\end{isamarkuptxt}%
|
|
404 |
\isamarkuptrue%
|
|
405 |
\ \ \isacommand{apply}\isamarkupfalse%
|
|
406 |
\ {\isacharparenleft}rule\ gcd{\isacharunderscore}lcm{\isacharunderscore}distr{\isacharparenright}\ \isacommand{done}\isamarkupfalse%
|
|
407 |
%
|
|
408 |
\endisatagvisible
|
|
409 |
{\isafoldvisible}%
|
|
410 |
%
|
|
411 |
\isadelimvisible
|
|
412 |
%
|
|
413 |
\endisadelimvisible
|
|
414 |
%
|
|
415 |
\begin{isamarkuptext}%
|
|
416 |
Theorems that are available in the theory after these
|
|
417 |
interpretations are shown in Table~\ref{tab:nat-dvd-lattice}.
|
|
418 |
|
|
419 |
\begin{table}
|
|
420 |
\hrule
|
|
421 |
\vspace{2ex}
|
|
422 |
\begin{center}
|
|
423 |
\begin{tabular}{l}
|
|
424 |
\isa{nat{\isacharunderscore}dvd{\isachardot}less{\isacharunderscore}def} from locale \isa{partial{\isacharunderscore}order}: \\
|
|
425 |
\quad \isa{{\isacharparenleft}{\isacharquery}x\ dvd\ {\isacharquery}y\ {\isasymand}\ {\isacharquery}x\ {\isasymnoteq}\ {\isacharquery}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharquery}x\ dvd\ {\isacharquery}y\ {\isasymand}\ {\isacharquery}x\ {\isasymnoteq}\ {\isacharquery}y{\isacharparenright}} \\
|
|
426 |
\isa{nat{\isacharunderscore}dvd{\isachardot}meet{\isacharunderscore}left} from locale \isa{lattice}: \\
|
|
427 |
\quad \isa{gcd\ {\isacharparenleft}{\isacharquery}x{\isacharcomma}\ {\isacharquery}y{\isacharparenright}\ dvd\ {\isacharquery}x} \\
|
|
428 |
\isa{nat{\isacharunderscore}dvd{\isachardot}join{\isacharunderscore}distr} from locale \isa{distrib{\isacharunderscore}lattice}: \\
|
|
429 |
\quad \isa{lcm\ {\isacharparenleft}{\isacharquery}x{\isacharcomma}\ gcd\ {\isacharparenleft}{\isacharquery}y{\isacharcomma}\ {\isacharquery}z{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ gcd\ {\isacharparenleft}lcm\ {\isacharparenleft}{\isacharquery}x{\isacharcomma}\ {\isacharquery}y{\isacharparenright}{\isacharcomma}\ lcm\ {\isacharparenleft}{\isacharquery}x{\isacharcomma}\ {\isacharquery}z{\isacharparenright}{\isacharparenright}} \\
|
|
430 |
\end{tabular}
|
|
431 |
\end{center}
|
|
432 |
\hrule
|
|
433 |
\caption{Interpreted theorems for \isa{dvd} on the natural numbers.}
|
|
434 |
\label{tab:nat-dvd-lattice}
|
|
435 |
\end{table}%
|
|
436 |
\end{isamarkuptext}%
|
|
437 |
\isamarkuptrue%
|
|
438 |
%
|
|
439 |
\begin{isamarkuptext}%
|
|
440 |
The full syntax of the interpretation commands is shown in
|
|
441 |
Table~\ref{tab:commands}. The grammar refers to
|
|
442 |
\textit{expr}, which stands for a \emph{locale} expression. Locale
|
|
443 |
expressions are discussed in Section~\ref{sec:expressions}.%
|
|
444 |
\end{isamarkuptext}%
|
|
445 |
\isamarkuptrue%
|
|
446 |
%
|
|
447 |
\isamarkupsection{Locale Expressions%
|
|
448 |
}
|
|
449 |
\isamarkuptrue%
|
|
450 |
%
|
|
451 |
\begin{isamarkuptext}%
|
|
452 |
\label{sec:expressions}
|
|
453 |
|
|
454 |
A map \isa{{\isasymphi}} between partial orders \isa{{\isasymsqsubseteq}} and \isa{{\isasympreceq}}
|
|
455 |
is called order preserving if \isa{x\ {\isasymsqsubseteq}\ y} implies \isa{{\isasymphi}\ x\ {\isasympreceq}\ {\isasymphi}\ y}. This situation is more complex than those encountered so
|
|
456 |
far: it involves two partial orders, and it is desirable to use the
|
|
457 |
existing locale for both.
|
|
458 |
|
|
459 |
Inspecting the grammar of locale commands in
|
|
460 |
Table~\ref{tab:commands} reveals that the import of a locale can be
|
|
461 |
more than just a single locale. In general, the import is a
|
|
462 |
\emph{locale expression}. Locale expressions enable to combine locales
|
|
463 |
and rename parameters. A locale name is a locale expression. If
|
|
464 |
$e_1$ and $e_2$ are locale expressions then $e_1 + e_2$ is their
|
|
465 |
\emph{merge}. If $e$ is an expression, then $e\:q_1 \ldots q_n$ is
|
|
466 |
a \emph{renamed expression} where the parameters in $e$ are renamed
|
|
467 |
to $q_1 \ldots q_n$. Using a locale expression, a locale for order
|
|
468 |
preserving maps can be declared in the following way.%
|
|
469 |
\end{isamarkuptext}%
|
|
470 |
\isamarkuptrue%
|
|
471 |
\ \ \isacommand{locale}\isamarkupfalse%
|
|
472 |
\ order{\isacharunderscore}preserving\ {\isacharequal}\isanewline
|
|
473 |
\ \ \ \ partial{\isacharunderscore}order\ {\isacharplus}\ partial{\isacharunderscore}order\ le{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasympreceq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ {\isacharplus}\isanewline
|
|
474 |
\ \ \ \ \isakeyword{fixes}\ {\isasymphi}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isachardoublequoteclose}\isanewline
|
|
475 |
\ \ \ \ \isakeyword{assumes}\ hom{\isacharunderscore}le{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ {\isasymphi}\ x\ {\isasympreceq}\ {\isasymphi}\ y{\isachardoublequoteclose}%
|
|
476 |
\begin{isamarkuptext}%
|
|
477 |
The second line contains the expression, which is the
|
|
478 |
merge of two partial order locales. The parameter of the second one
|
|
479 |
is \isa{le{\isacharprime}} with new infix syntax \isa{{\isasympreceq}}. The
|
|
480 |
parameters of the entire locale are \isa{le}, \isa{le{\isacharprime}} and
|
|
481 |
\isa{{\isasymphi}}. This is their \emph{canonical order},
|
|
482 |
which is obtained by a left-to-right traversal of the expression,
|
|
483 |
where only the new parameters are appended to the end of the list. The
|
|
484 |
parameters introduced in the locale elements of the declaration
|
|
485 |
follow.
|
|
486 |
|
|
487 |
In renamings parameters are referred to by position in the canonical
|
|
488 |
order; an underscore is used to skip a parameter position, which is
|
|
489 |
then not renamed. Renaming deletes the syntax of a parameter unless
|
|
490 |
a new mixfix annotation is given.
|
|
491 |
|
|
492 |
Parameter renamings are morphisms between locales. These can be
|
|
493 |
lifted to terms and theorems and thus be applied to assumptions and
|
|
494 |
conclusions. The assumption of a merge is the conjunction of the
|
|
495 |
assumptions of the merged locale. The conclusions of a merge are
|
|
496 |
obtained by appending the conclusions of the left locale and of the
|
|
497 |
right locale.%
|
|
498 |
\end{isamarkuptext}%
|
|
499 |
\isamarkuptrue%
|
|
500 |
%
|
|
501 |
\begin{isamarkuptext}%
|
|
502 |
The locale \isa{order{\isacharunderscore}preserving} contains theorems for both
|
|
503 |
orders \isa{{\isasymsqsubseteq}} and \isa{{\isasympreceq}}. How can one refer to a theorem for
|
|
504 |
a particular order, \isa{{\isasymsqsubseteq}} or \isa{{\isasympreceq}}? Names in locales are
|
|
505 |
qualified by the locale parameters. More precisely, a name is
|
|
506 |
qualified by the parameters of the locale in which its declaration
|
|
507 |
occurs. Here are examples:%
|
|
508 |
\end{isamarkuptext}%
|
|
509 |
\isamarkuptrue%
|
|
510 |
%
|
|
511 |
\isadeliminvisible
|
|
512 |
%
|
|
513 |
\endisadeliminvisible
|
|
514 |
%
|
|
515 |
\isataginvisible
|
|
516 |
\isacommand{context}\isamarkupfalse%
|
|
517 |
\ order{\isacharunderscore}preserving\ \isakeyword{begin}%
|
|
518 |
\endisataginvisible
|
|
519 |
{\isafoldinvisible}%
|
|
520 |
%
|
|
521 |
\isadeliminvisible
|
|
522 |
%
|
|
523 |
\endisadeliminvisible
|
|
524 |
%
|
|
525 |
\begin{isamarkuptext}%
|
|
526 |
\isa{le{\isachardot}less{\isacharunderscore}le{\isacharunderscore}trans}: \isa{{\isasymlbrakk}{\isacharquery}x\ {\isasymsqsubset}\ {\isacharquery}y{\isacharsemicolon}\ {\isacharquery}y\ {\isasymsqsubseteq}\ {\isacharquery}z{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}x\ {\isasymsqsubset}\ {\isacharquery}z}
|
|
527 |
|
|
528 |
\isa{le{\isacharunderscore}le{\isacharprime}{\isacharunderscore}{\isasymphi}{\isachardot}hom{\isacharunderscore}le}: \isa{{\isacharquery}x\ {\isasymsqsubseteq}\ {\isacharquery}y\ {\isasymLongrightarrow}\ {\isasymphi}\ {\isacharquery}x\ {\isasympreceq}\ {\isasymphi}\ {\isacharquery}y}%
|
|
529 |
\end{isamarkuptext}%
|
|
530 |
\isamarkuptrue%
|
|
531 |
%
|
|
532 |
\begin{isamarkuptext}%
|
|
533 |
When renaming a locale, the morphism is also applied
|
|
534 |
to the qualifiers. Hence theorems for the partial order \isa{{\isasympreceq}}
|
|
535 |
are qualified by \isa{le{\isacharprime}}. For example, \isa{le{\isacharprime}{\isachardot}less{\isacharunderscore}le{\isacharunderscore}trans}: \begin{isabelle}%
|
|
536 |
\ \ {\isasymlbrakk}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasympreceq}\ {\isacharquery}x\ {\isacharquery}y{\isacharsemicolon}\ {\isacharquery}y\ {\isasympreceq}\ {\isacharquery}z{\isasymrbrakk}\isanewline
|
|
537 |
\isaindent{\ \ }{\isasymLongrightarrow}\ partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasympreceq}\ {\isacharquery}x\ {\isacharquery}z%
|
|
538 |
\end{isabelle}%
|
|
539 |
\end{isamarkuptext}%
|
|
540 |
\isamarkuptrue%
|
|
541 |
%
|
|
542 |
\isadeliminvisible
|
|
543 |
%
|
|
544 |
\endisadeliminvisible
|
|
545 |
%
|
|
546 |
\isataginvisible
|
|
547 |
\isacommand{end}\isamarkupfalse%
|
|
548 |
%
|
|
549 |
\endisataginvisible
|
|
550 |
{\isafoldinvisible}%
|
|
551 |
%
|
|
552 |
\isadeliminvisible
|
|
553 |
%
|
|
554 |
\endisadeliminvisible
|
|
555 |
%
|
|
556 |
\begin{isamarkuptext}%
|
|
557 |
This example reveals that there is no infix syntax for the strict
|
|
558 |
version of \isa{{\isasympreceq}}! This can, of course, not be introduced
|
|
559 |
automatically, but it can be declared manually through an abbreviation.%
|
|
560 |
\end{isamarkuptext}%
|
|
561 |
\isamarkuptrue%
|
|
562 |
\ \ \isacommand{abbreviation}\isamarkupfalse%
|
|
563 |
\ {\isacharparenleft}\isakeyword{in}\ order{\isacharunderscore}preserving{\isacharparenright}\isanewline
|
|
564 |
\ \ \ \ less{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymprec}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}less{\isacharprime}\ {\isasymequiv}\ partial{\isacharunderscore}order{\isachardot}less\ le{\isacharprime}{\isachardoublequoteclose}%
|
|
565 |
\begin{isamarkuptext}%
|
|
566 |
Now the theorem is displayed nicely as
|
|
567 |
\isa{{\isasymlbrakk}{\isacharquery}x\ {\isasymprec}\ {\isacharquery}y{\isacharsemicolon}\ {\isacharquery}y\ {\isasympreceq}\ {\isacharquery}z{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}x\ {\isasymprec}\ {\isacharquery}z}.%
|
|
568 |
\end{isamarkuptext}%
|
|
569 |
\isamarkuptrue%
|
|
570 |
%
|
|
571 |
\begin{isamarkuptext}%
|
|
572 |
Not only names of theorems are qualified. In fact, all names
|
|
573 |
are qualified, in particular names introduced by definitions and
|
|
574 |
abbreviations. The name of the strict order of \isa{{\isasymsqsubseteq}} is \isa{le{\isachardot}less} and therefore \isa{le{\isacharprime}{\isachardot}less} is the name of the strict
|
|
575 |
order of \isa{{\isasympreceq}}. Hence, the equation in the above abbreviation
|
|
576 |
could have been written as \isa{less{\isacharprime}\ {\isasymequiv}\ le{\isacharprime}{\isachardot}less}.%
|
|
577 |
\end{isamarkuptext}%
|
|
578 |
\isamarkuptrue%
|
|
579 |
%
|
|
580 |
\begin{isamarkuptext}%
|
|
581 |
Two more locales illustrate working with locale expressions.
|
|
582 |
A map \isa{{\isasymphi}} is a lattice homomorphism if it preserves meet and join.%
|
|
583 |
\end{isamarkuptext}%
|
|
584 |
\isamarkuptrue%
|
|
585 |
\ \ \isacommand{locale}\isamarkupfalse%
|
|
586 |
\ lattice{\isacharunderscore}hom\ {\isacharequal}\ lattice\ {\isacharplus}\ lattice\ le{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasympreceq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ {\isacharplus}\isanewline
|
|
587 |
\ \ \ \ \isakeyword{fixes}\ {\isasymphi}\isanewline
|
|
588 |
\ \ \ \ \isakeyword{assumes}\ hom{\isacharunderscore}meet{\isacharcolon}\isanewline
|
|
589 |
\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymphi}\ {\isacharparenleft}lattice{\isachardot}meet\ le\ x\ y{\isacharparenright}\ {\isacharequal}\ lattice{\isachardot}meet\ le{\isacharprime}\ {\isacharparenleft}{\isasymphi}\ x{\isacharparenright}\ {\isacharparenleft}{\isasymphi}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
590 |
\ \ \ \ \ \ \isakeyword{and}\ hom{\isacharunderscore}join{\isacharcolon}\isanewline
|
|
591 |
\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymphi}\ {\isacharparenleft}lattice{\isachardot}join\ le\ x\ y{\isacharparenright}\ {\isacharequal}\ lattice{\isachardot}join\ le{\isacharprime}\ {\isacharparenleft}{\isasymphi}\ x{\isacharparenright}\ {\isacharparenleft}{\isasymphi}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
592 |
\isanewline
|
|
593 |
\ \ \isacommand{abbreviation}\isamarkupfalse%
|
|
594 |
\ {\isacharparenleft}\isakeyword{in}\ lattice{\isacharunderscore}hom{\isacharparenright}\isanewline
|
|
595 |
\ \ \ \ meet{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqinter}{\isacharprime}{\isacharprime}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}meet{\isacharprime}\ {\isasymequiv}\ le{\isacharprime}{\isachardot}meet{\isachardoublequoteclose}\isanewline
|
|
596 |
\ \ \isacommand{abbreviation}\isamarkupfalse%
|
|
597 |
\ {\isacharparenleft}\isakeyword{in}\ lattice{\isacharunderscore}hom{\isacharparenright}\isanewline
|
|
598 |
\ \ \ \ join{\isacharprime}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqunion}{\isacharprime}{\isacharprime}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}join{\isacharprime}\ {\isasymequiv}\ le{\isacharprime}{\isachardot}join{\isachardoublequoteclose}%
|
|
599 |
\begin{isamarkuptext}%
|
|
600 |
A homomorphism is an endomorphism if both orders coincide.%
|
|
601 |
\end{isamarkuptext}%
|
|
602 |
\isamarkuptrue%
|
|
603 |
\ \ \isacommand{locale}\isamarkupfalse%
|
|
604 |
\ lattice{\isacharunderscore}end\ {\isacharequal}\isanewline
|
|
605 |
\ \ \ \ lattice{\isacharunderscore}hom\ le\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqsubseteq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\ le\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqsubseteq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}%
|
|
606 |
\begin{isamarkuptext}%
|
|
607 |
The inheritance diagram of the situation we have now is shown
|
|
608 |
in Figure~\ref{fig:hom}, where the dashed line depicts an
|
|
609 |
interpretation which is introduced below. Renamings are
|
|
610 |
indicated by $\sqsubseteq \mapsto \preceq$ etc. The expression
|
|
611 |
imported by \isa{lattice{\isacharunderscore}end} identifies the first and second
|
|
612 |
parameter of \isa{lattice{\isacharunderscore}hom}. By looking at the inheritance diagram it would seem
|
|
613 |
that two identical copies of each of the locales \isa{partial{\isacharunderscore}order} and \isa{lattice} are imported. This is not the
|
|
614 |
case! Inheritance paths with identical morphisms are detected and
|
|
615 |
the conclusions of the respecitve locales appear only once.
|
|
616 |
|
|
617 |
\begin{figure}
|
|
618 |
\hrule \vspace{2ex}
|
|
619 |
\begin{center}
|
|
620 |
\begin{tikzpicture}
|
|
621 |
\node (o) at (0,0) {\isa{partial{\isacharunderscore}order}};
|
|
622 |
\node (oh) at (1.5,-2) {\isa{order{\isacharunderscore}preserving}};
|
|
623 |
\node (oh1) at (1.5,-0.7) {$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$};
|
|
624 |
\node (oh2) at (0,-1.3) {$\scriptscriptstyle \sqsubseteq \mapsto \preceq$};
|
|
625 |
\node (l) at (-1.5,-2) {\isa{lattice}};
|
|
626 |
\node (lh) at (0,-4) {\isa{lattice{\isacharunderscore}hom}};
|
|
627 |
\node (lh1) at (0,-2.7) {$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$};
|
|
628 |
\node (lh2) at (-1.5,-3.3) {$\scriptscriptstyle \sqsubseteq \mapsto \preceq$};
|
|
629 |
\node (le) at (0,-6) {\isa{lattice{\isacharunderscore}end}};
|
|
630 |
\node (le1) at (0,-4.8)
|
|
631 |
[anchor=west]{$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$};
|
|
632 |
\node (le2) at (0,-5.2)
|
|
633 |
[anchor=west]{$\scriptscriptstyle \preceq \mapsto \sqsubseteq$};
|
|
634 |
\draw (o) -- (l);
|
|
635 |
\draw[dashed] (oh) -- (lh);
|
|
636 |
\draw (lh) -- (le);
|
|
637 |
\draw (o) .. controls (oh1.south west) .. (oh);
|
|
638 |
\draw (o) .. controls (oh2.north east) .. (oh);
|
|
639 |
\draw (l) .. controls (lh1.south west) .. (lh);
|
|
640 |
\draw (l) .. controls (lh2.north east) .. (lh);
|
|
641 |
\end{tikzpicture}
|
|
642 |
\end{center}
|
|
643 |
\hrule
|
|
644 |
\caption{Hierarchy of Homomorphism Locales.}
|
|
645 |
\label{fig:hom}
|
|
646 |
\end{figure}%
|
|
647 |
\end{isamarkuptext}%
|
|
648 |
\isamarkuptrue%
|
|
649 |
%
|
|
650 |
\begin{isamarkuptext}%
|
|
651 |
It can be shown easily that a lattice homomorphism is order
|
|
652 |
preserving. As the final example of this section, a locale
|
|
653 |
interpretation is used to assert this.%
|
|
654 |
\end{isamarkuptext}%
|
|
655 |
\isamarkuptrue%
|
|
656 |
\ \ \isacommand{interpretation}\isamarkupfalse%
|
|
657 |
\ lattice{\isacharunderscore}hom\ {\isasymsubseteq}\ order{\isacharunderscore}preserving%
|
|
658 |
\isadelimproof
|
|
659 |
\ %
|
|
660 |
\endisadelimproof
|
|
661 |
%
|
|
662 |
\isatagproof
|
|
663 |
\isacommand{proof}\isamarkupfalse%
|
|
664 |
\ unfold{\isacharunderscore}locales\isanewline
|
|
665 |
\ \ \ \ \isacommand{fix}\isamarkupfalse%
|
|
666 |
\ x\ y\isanewline
|
|
667 |
\ \ \ \ \isacommand{assume}\isamarkupfalse%
|
|
668 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline
|
|
669 |
\ \ \ \ \isacommand{then}\isamarkupfalse%
|
|
670 |
\ \isacommand{have}\isamarkupfalse%
|
|
671 |
\ {\isachardoublequoteopen}y\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
|
672 |
\ {\isacharparenleft}simp\ add{\isacharcolon}\ le{\isachardot}join{\isacharunderscore}connection{\isacharparenright}\isanewline
|
|
673 |
\ \ \ \ \isacommand{then}\isamarkupfalse%
|
|
674 |
\ \isacommand{have}\isamarkupfalse%
|
|
675 |
\ {\isachardoublequoteopen}{\isasymphi}\ y\ {\isacharequal}\ {\isacharparenleft}{\isasymphi}\ x\ {\isasymsqunion}{\isacharprime}\ {\isasymphi}\ y{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
|
676 |
\ {\isacharparenleft}simp\ add{\isacharcolon}\ hom{\isacharunderscore}join\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline
|
|
677 |
\ \ \ \ \isacommand{then}\isamarkupfalse%
|
|
678 |
\ \isacommand{show}\isamarkupfalse%
|
|
679 |
\ {\isachardoublequoteopen}{\isasymphi}\ x\ {\isasympreceq}\ {\isasymphi}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
|
680 |
\ {\isacharparenleft}simp\ add{\isacharcolon}\ le{\isacharprime}{\isachardot}join{\isacharunderscore}connection{\isacharparenright}\isanewline
|
|
681 |
\ \ \isacommand{qed}\isamarkupfalse%
|
|
682 |
%
|
|
683 |
\endisatagproof
|
|
684 |
{\isafoldproof}%
|
|
685 |
%
|
|
686 |
\isadelimproof
|
|
687 |
%
|
|
688 |
\endisadelimproof
|
|
689 |
%
|
|
690 |
\begin{isamarkuptext}%
|
|
691 |
Theorems and other declarations --- syntax, in particular ---
|
|
692 |
from the locale \isa{order{\isacharunderscore}preserving} are now active in \isa{lattice{\isacharunderscore}hom}, for example
|
|
693 |
|
|
694 |
\isa{le{\isacharprime}{\isachardot}less{\isacharunderscore}le{\isacharunderscore}trans}:
|
|
695 |
\isa{{\isasymlbrakk}{\isacharquery}x\ {\isasymprec}\ {\isacharquery}y{\isacharsemicolon}\ {\isacharquery}y\ {\isasympreceq}\ {\isacharquery}z{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}x\ {\isasymprec}\ {\isacharquery}z}%
|
|
696 |
\end{isamarkuptext}%
|
|
697 |
\isamarkuptrue%
|
|
698 |
%
|
|
699 |
\isamarkupsection{Further Reading%
|
|
700 |
}
|
|
701 |
\isamarkuptrue%
|
|
702 |
%
|
|
703 |
\begin{isamarkuptext}%
|
|
704 |
More information on locales and their interpretation is
|
|
705 |
available. For the locale hierarchy of import and interpretation
|
|
706 |
dependencies see \cite{Ballarin2006a}; interpretations in theories
|
|
707 |
and proofs are covered in \cite{Ballarin2006b}. In the latter, we
|
|
708 |
show how interpretation in proofs enables to reason about families
|
|
709 |
of algebraic structures, which cannot be expressed with locales
|
|
710 |
directly.
|
|
711 |
|
|
712 |
Haftmann and Wenzel \cite{HaftmannWenzel2007} overcome a restriction
|
|
713 |
of axiomatic type classes through a combination with locale
|
|
714 |
interpretation. The result is a Haskell-style class system with a
|
|
715 |
facility to generate Haskell code. Classes are sufficient for
|
|
716 |
simple specifications with a single type parameter. The locales for
|
|
717 |
orders and lattices presented in this tutorial fall into this
|
|
718 |
category. Order preserving maps, homomorphisms and vector spaces,
|
|
719 |
on the other hand, do not.
|
|
720 |
|
|
721 |
The original work of Kamm\"uller on locales \cite{KammullerEtAl1999}
|
|
722 |
may be of interest from a historical perspective. The mathematical
|
|
723 |
background on orders and lattices is taken from Jacobson's textbook
|
|
724 |
on algebra \cite[Chapter~8]{Jacobson1985}.%
|
|
725 |
\end{isamarkuptext}%
|
|
726 |
\isamarkuptrue%
|
|
727 |
%
|
|
728 |
\begin{isamarkuptext}%
|
|
729 |
\begin{table}
|
|
730 |
\hrule
|
|
731 |
\vspace{2ex}
|
|
732 |
\begin{center}
|
|
733 |
\begin{tabular}{l>$c<$l}
|
|
734 |
\multicolumn{3}{l}{Miscellaneous} \\
|
|
735 |
|
|
736 |
\textit{attr-name} & ::=
|
|
737 |
& \textit{name} $|$ \textit{attribute} $|$
|
|
738 |
\textit{name} \textit{attribute} \\[2ex]
|
|
739 |
|
|
740 |
\multicolumn{3}{l}{Context Elements} \\
|
|
741 |
|
|
742 |
\textit{fixes} & ::=
|
|
743 |
& \textit{name} [ ``\textbf{::}'' \textit{type} ]
|
|
744 |
[ ``\textbf{(}'' \textbf{structure} ``\textbf{)}'' $|$
|
|
745 |
\textit{mixfix} ] \\
|
|
746 |
\begin{comment}
|
|
747 |
\textit{constrains} & ::=
|
|
748 |
& \textit{name} ``\textbf{::}'' \textit{type} \\
|
|
749 |
\end{comment}
|
|
750 |
\textit{assumes} & ::=
|
|
751 |
& [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
|
|
752 |
\begin{comment}
|
|
753 |
\textit{defines} & ::=
|
|
754 |
& [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
|
|
755 |
\textit{notes} & ::=
|
|
756 |
& [ \textit{attr-name} ``\textbf{=}'' ]
|
|
757 |
( \textit{qualified-name} [ \textit{attribute} ] )$^+$ \\
|
|
758 |
\end{comment}
|
|
759 |
|
|
760 |
\textit{element} & ::=
|
|
761 |
& \textbf{fixes} \textit{fixes} ( \textbf{and} \textit{fixes} )$^*$ \\
|
|
762 |
\begin{comment}
|
|
763 |
& |
|
|
764 |
& \textbf{constrains} \textit{constrains}
|
|
765 |
( \textbf{and} \textit{constrains} )$^*$ \\
|
|
766 |
\end{comment}
|
|
767 |
& |
|
|
768 |
& \textbf{assumes} \textit{assumes} ( \textbf{and} \textit{assumes} )$^*$ \\[2ex]
|
|
769 |
%\begin{comment}
|
|
770 |
% & |
|
|
771 |
% & \textbf{defines} \textit{defines} ( \textbf{and} \textit{defines} )$^*$ \\
|
|
772 |
% & |
|
|
773 |
% & \textbf{notes} \textit{notes} ( \textbf{and} \textit{notes} )$^*$ \\
|
|
774 |
%\end{comment}
|
|
775 |
|
|
776 |
\multicolumn{3}{l}{Locale Expressions} \\
|
|
777 |
|
|
778 |
\textit{rename} & ::=
|
|
779 |
& \textit{name} [ \textit{mixfix} ] $|$ ``\textbf{\_}'' \\
|
|
780 |
\textit{expr} & ::=
|
|
781 |
& \textit{renamed-expr} ( ``\textbf{+}'' \textit{renamed-expr} )$^*$ \\
|
|
782 |
\textit{renamed-expr} & ::=
|
|
783 |
& ( \textit{qualified-name} $|$
|
|
784 |
``\textbf{(}'' \textit{expr} ``\textbf{)}'' ) \textit{rename}$^*$ \\[2ex]
|
|
785 |
|
|
786 |
\multicolumn{3}{l}{Declaration of Locales} \\
|
|
787 |
|
|
788 |
\textit{locale} & ::=
|
|
789 |
& \textit{element}$^+$ \\
|
|
790 |
& | & \textit{locale-expr} [ ``\textbf{+}'' \textit{element}$^+$ ] \\
|
|
791 |
\textit{toplevel} & ::=
|
|
792 |
& \textbf{locale} \textit{name} [ ``\textbf{=}''
|
|
793 |
\textit{locale} ] \\[2ex]
|
|
794 |
|
|
795 |
\multicolumn{3}{l}{Interpretation} \\
|
|
796 |
|
|
797 |
\textit{equation} & ::= & [ \textit{attr-name} ``\textbf{:}'' ]
|
|
798 |
\textit{prop} \\
|
|
799 |
\textit{insts} & ::= & [ ``\textbf{[}'' \textit{term}$^+$
|
|
800 |
``\textbf{]}'' ] \\
|
|
801 |
& & [ \textbf{where} \textit{equation} ( \textbf{and}
|
|
802 |
\textit{equation} )$^*$ ] \\
|
|
803 |
\textit{toplevel} & ::=
|
|
804 |
& \textbf{interpretation} \textit{name} ( ``$<$'' $|$
|
|
805 |
``$\subseteq$'' ) \textit{expr} \textit{proof} \\
|
|
806 |
& |
|
|
807 |
& \textbf{interpretation} [ \textit{attr-name} ``\textbf{:}'' ]
|
|
808 |
\textit{expr} \textit{insts} \textit{proof} \\
|
|
809 |
& |
|
|
810 |
& \textbf{interpret} [ \textit{attr-name} ``\textbf{:}'' ]
|
|
811 |
\textit{expr} \textit{insts} \textit{proof} \\[2ex]
|
|
812 |
|
|
813 |
\multicolumn{3}{l}{Diagnostics} \\
|
|
814 |
|
|
815 |
\textit{toplevel} & ::=
|
|
816 |
& \textbf{print\_locale} [ ``\textbf{!}'' ] \textit{locale} \\
|
|
817 |
& | & \textbf{print\_locales}
|
|
818 |
\end{tabular}
|
|
819 |
\end{center}
|
|
820 |
\hrule
|
|
821 |
\caption{Syntax of Locale Commands.}
|
|
822 |
\label{tab:commands}
|
|
823 |
\end{table}%
|
|
824 |
\end{isamarkuptext}%
|
|
825 |
\isamarkuptrue%
|
|
826 |
%
|
|
827 |
\begin{isamarkuptext}%
|
|
828 |
\textbf{Acknowledgements.} Alexander Krauss, Tobias Nipkow,
|
|
829 |
Christian Sternagel and Makarius Wenzel have made useful comments on
|
|
830 |
a draft of this document.%
|
|
831 |
\end{isamarkuptext}%
|
|
832 |
\isamarkuptrue%
|
|
833 |
%
|
|
834 |
\isadelimtheory
|
|
835 |
%
|
|
836 |
\endisadelimtheory
|
|
837 |
%
|
|
838 |
\isatagtheory
|
|
839 |
%
|
|
840 |
\endisatagtheory
|
|
841 |
{\isafoldtheory}%
|
|
842 |
%
|
|
843 |
\isadelimtheory
|
|
844 |
%
|
|
845 |
\endisadelimtheory
|
|
846 |
\end{isabellebody}%
|
|
847 |
%%% Local Variables:
|
|
848 |
%%% mode: latex
|
|
849 |
%%% TeX-master: "root"
|
|
850 |
%%% End:
|