author | ballarin |
Sat, 28 Mar 2009 22:14:21 +0100 | |
changeset 30780 | c3f1e8a9e0b5 |
parent 30580 | cc5a55d7a5be |
child 30826 | a53f4872400e |
permissions | -rw-r--r-- |
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\begin{isabellebody}% |
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\def\isabellecontext{Examples}% |
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\isadelimtheory |
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\endisadelimtheory |
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\isatagtheory |
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\isacommand{theory}\isamarkupfalse% |
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\ Examples\isanewline |
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\isakeyword{imports}\ Main\ GCD\isanewline |
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\isakeyword{begin}% |
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\endisatagtheory |
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{\isafoldtheory}% |
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\isadelimtheory |
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\isanewline |
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\endisadelimtheory |
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\isadeliminvisible |
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\isanewline |
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\endisadeliminvisible |
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\isataginvisible |
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\isacommand{hide}\isamarkupfalse% |
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\ const\ Lattices{\isachardot}lattice\isanewline |
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\isacommand{pretty{\isacharunderscore}setmargin}\isamarkupfalse% |
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\ {\isadigit{6}}{\isadigit{5}}% |
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\endisataginvisible |
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{\isafoldinvisible}% |
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\isadeliminvisible |
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\endisadeliminvisible |
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\isamarkupsection{Introduction% |
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} |
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\isamarkuptrue% |
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\begin{isamarkuptext}% |
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Locales are based on contexts. A \emph{context} can be seen as a |
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formula schema |
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\[ |
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\isa{{\isasymAnd}x\isactrlsub {\isadigit{1}}{\isasymdots}x\isactrlsub n{\isachardot}\ {\isasymlbrakk}\ A\isactrlsub {\isadigit{1}}{\isacharsemicolon}\ {\isasymdots}\ {\isacharsemicolon}A\isactrlsub m\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymdots}} |
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\] |
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where variables \isa{x\isactrlsub {\isadigit{1}}}, \ldots, \isa{x\isactrlsub n} are called |
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\emph{parameters} and the premises $\isa{A\isactrlsub {\isadigit{1}}}, \ldots, |
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\isa{A\isactrlsub m}$ \emph{assumptions}. A formula \isa{C} |
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is a \emph{theorem} in the context if it is a conclusion |
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\[ |
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%\label{eq-fact-in-context} |
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\isa{{\isasymAnd}x\isactrlsub {\isadigit{1}}{\isasymdots}x\isactrlsub n{\isachardot}\ {\isasymlbrakk}\ A\isactrlsub {\isadigit{1}}{\isacharsemicolon}\ {\isasymdots}\ {\isacharsemicolon}A\isactrlsub m\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ C}. |
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\] |
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Isabelle/Isar's notion of context goes beyond this logical view. |
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Its contexts record, in a consecutive order, proved |
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conclusions along with attributes, which |
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may control proof procedures. Contexts also contain syntax information |
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for parameters and for terms depending on them.% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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% |
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\isamarkupsection{Simple Locales% |
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} |
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\isamarkuptrue% |
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% |
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\begin{isamarkuptext}% |
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Locales can be seen as persistent contexts. In its simplest form, a |
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\emph{locale declaration} consists of a sequence of context elements |
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declaring parameters (keyword \isakeyword{fixes}) and assumptions |
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(keyword \isakeyword{assumes}). The following is the specification of |
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partial orders, as locale \isa{partial{\isacharunderscore}order}.% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\ \ \isacommand{locale}\isamarkupfalse% |
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\ partial{\isacharunderscore}order\ {\isacharequal}\isanewline |
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\ \ \ \ \isakeyword{fixes}\ le\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqsubseteq}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\isanewline |
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\ \ \ \ \isakeyword{assumes}\ refl\ {\isacharbrackleft}intro{\isacharcomma}\ simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \ \ \isakeyword{and}\ anti{\isacharunderscore}sym\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}\ x\ {\isasymsqsubseteq}\ y{\isacharsemicolon}\ y\ {\isasymsqsubseteq}\ x\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \ \ \isakeyword{and}\ trans\ {\isacharbrackleft}trans{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}\ x\ {\isasymsqsubseteq}\ y{\isacharsemicolon}\ y\ {\isasymsqsubseteq}\ z\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}% |
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\begin{isamarkuptext}% |
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The parameter of this locale is \isa{le}, with infix syntax |
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\isa{{\isasymsqsubseteq}}. There is an implicit type parameter \isa{{\isacharprime}a}. It |
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is not necessary to declare parameter types: most general types will |
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be inferred from the context elements for all parameters. |
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The above declaration not only introduces the locale, it also |
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defines the \emph{locale predicate} \isa{partial{\isacharunderscore}order} with |
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definition \isa{partial{\isacharunderscore}order{\isacharunderscore}def}: |
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\begin{isabelle}% |
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\ \ partial{\isacharunderscore}order\ {\isacharquery}le\ {\isasymequiv}\isanewline |
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\isaindent{\ \ }{\isacharparenleft}{\isasymforall}x{\isachardot}\ {\isacharquery}le\ x\ x{\isacharparenright}\ {\isasymand}\isanewline |
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\isaindent{\ \ }{\isacharparenleft}{\isasymforall}x\ y{\isachardot}\ {\isacharquery}le\ x\ y\ {\isasymlongrightarrow}\ {\isacharquery}le\ y\ x\ {\isasymlongrightarrow}\ x\ {\isacharequal}\ y{\isacharparenright}\ {\isasymand}\isanewline |
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\isaindent{\ \ }{\isacharparenleft}{\isasymforall}x\ y\ z{\isachardot}\ {\isacharquery}le\ x\ y\ {\isasymlongrightarrow}\ {\isacharquery}le\ y\ z\ {\isasymlongrightarrow}\ {\isacharquery}le\ x\ z{\isacharparenright}% |
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\end{isabelle} |
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The specification of a locale is fixed, but its list of conclusions |
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may be extended through Isar commands that take a \emph{target} argument. |
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In the following, \isakeyword{definition} and |
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\isakeyword{theorem} are illustrated. |
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Table~\ref{tab:commands-with-target} lists Isar commands that accept |
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a target. There are various ways of specifying the target. A |
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target for a single command may be indicated with keyword |
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\isakeyword{in} in the following way: |
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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}{ll} |
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\isakeyword{definition} & definition through an equation \\ |
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\isakeyword{inductive} & inductive definition \\ |
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\isakeyword{fun}, \isakeyword{function} & recursive function \\ |
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\isakeyword{abbreviation} & syntactic abbreviation \\ |
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\isakeyword{theorem}, etc.\ & theorem statement with proof \\ |
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\isakeyword{theorems}, etc.\ & redeclaration of theorems |
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\end{tabular} |
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\end{center} |
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\hrule |
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\caption{Isar commands that accept a target.} |
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\label{tab:commands-with-target} |
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\end{table}% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\ \ \isacommand{definition}\isamarkupfalse% |
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\ {\isacharparenleft}\isakeyword{in}\ partial{\isacharunderscore}order{\isacharparenright}\isanewline |
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\ \ \ \ less\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqsubset}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\isanewline |
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\ \ \ \ \isakeyword{where}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqsubset}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqsubseteq}\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}{\isachardoublequoteclose}% |
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\begin{isamarkuptext}% |
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A definition in a locale depends on the locale parameters. |
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Here, a global constant \isa{partial{\isacharunderscore}order{\isachardot}less} is declared, which is lifted over the |
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locale parameter \isa{le}. Its definition is the global theorem |
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\isa{partial{\isacharunderscore}order{\isachardot}less{\isacharunderscore}def}: |
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\begin{isabelle}% |
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\ \ partial{\isacharunderscore}order\ {\isacharquery}le\ {\isasymLongrightarrow}\isanewline |
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\isaindent{\ \ }partial{\isacharunderscore}order{\isachardot}less\ {\isacharquery}le\ {\isacharquery}x\ {\isacharquery}y\ {\isacharequal}\ {\isacharparenleft}{\isacharquery}le\ {\isacharquery}x\ {\isacharquery}y\ {\isasymand}\ {\isacharquery}x\ {\isasymnoteq}\ {\isacharquery}y{\isacharparenright}% |
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\end{isabelle} |
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At the same time, the locale is extended by syntax information |
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hiding this construction in the context of the locale. That is, |
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\isa{partial{\isacharunderscore}order{\isachardot}less\ le} is printed and parsed as infix |
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\isa{{\isasymsqsubset}}.% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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% |
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\begin{isamarkuptext}% |
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Finally, the conclusion of the definition |
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is added to the locale, \isa{less{\isacharunderscore}def}: |
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\begin{isabelle}% |
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\ \ {\isacharparenleft}{\isacharquery}x\ {\isasymsqsubset}\ {\isacharquery}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharquery}x\ {\isasymsqsubseteq}\ {\isacharquery}y\ {\isasymand}\ {\isacharquery}x\ {\isasymnoteq}\ {\isacharquery}y{\isacharparenright}% |
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\end{isabelle}% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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% |
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\begin{isamarkuptext}% |
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As an example of a theorem statement in the locale, here is the |
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derivation of a transitivity law.% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\ \ \isacommand{lemma}\isamarkupfalse% |
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\ {\isacharparenleft}\isakeyword{in}\ partial{\isacharunderscore}order{\isacharparenright}\ less{\isacharunderscore}le{\isacharunderscore}trans\ {\isacharbrackleft}trans{\isacharbrackright}{\isacharcolon}\isanewline |
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\ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}\ x\ {\isasymsqsubset}\ y{\isacharsemicolon}\ y\ {\isasymsqsubseteq}\ z\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isasymsqsubset}\ z{\isachardoublequoteclose}\isanewline |
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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{unfolding}\isamarkupfalse% |
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\ less{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse% |
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\ {\isacharparenleft}blast\ intro{\isacharcolon}\ trans{\isacharparenright}% |
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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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In the context of the proof, assumptions and theorems of the |
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locale may be used. Attributes are effective: \isa{anti{\isacharunderscore}sym} was |
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declared as introduction rule, hence it is in the context's set of |
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rules used by the classical reasoner by default.% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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% |
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\begin{isamarkuptext}% |
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When working with locales, sequences of commands with the same |
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target are frequent. A block of commands, delimited by |
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\isakeyword{begin} and \isakeyword{end}, makes a theory-like style |
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of working possible. All commands inside the block refer to the |
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same target. A block may immediately follow a locale |
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declaration, which makes that locale the target. Alternatively the |
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target for a block may be given with the \isakeyword{context} |
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command. |
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This style of working is illustrated in the block below, where |
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notions of infimum and supremum for partial orders are introduced, |
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together with theorems.% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\ \ \isacommand{context}\isamarkupfalse% |
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\ partial{\isacharunderscore}order\ \isakeyword{begin}\isanewline |
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\isanewline |
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\ \ \isacommand{definition}\isamarkupfalse% |
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\isanewline |
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\ \ \ \ is{\isacharunderscore}inf\ \isakeyword{where}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i\ {\isacharequal}\isanewline |
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\ \ \ \ \ \ {\isacharparenleft}i\ {\isasymsqsubseteq}\ x\ {\isasymand}\ i\ {\isasymsqsubseteq}\ y\ {\isasymand}\ {\isacharparenleft}{\isasymforall}z{\isachardot}\ z\ {\isasymsqsubseteq}\ x\ {\isasymand}\ z\ {\isasymsqsubseteq}\ y\ {\isasymlongrightarrow}\ z\ {\isasymsqsubseteq}\ i{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline |
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\isanewline |
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\ \ \isacommand{definition}\isamarkupfalse% |
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\isanewline |
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\ \ \ \ is{\isacharunderscore}sup\ \isakeyword{where}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s\ {\isacharequal}\isanewline |
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\ \ \ \ \ \ {\isacharparenleft}x\ {\isasymsqsubseteq}\ s\ {\isasymand}\ y\ {\isasymsqsubseteq}\ s\ {\isasymand}\ {\isacharparenleft}{\isasymforall}z{\isachardot}\ x\ {\isasymsqsubseteq}\ z\ {\isasymand}\ y\ {\isasymsqsubseteq}\ z\ {\isasymlongrightarrow}\ s\ {\isasymsqsubseteq}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline |
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% |
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\isadeliminvisible |
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\isanewline |
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\ \ % |
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\endisadeliminvisible |
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% |
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\isataginvisible |
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\isacommand{lemma}\isamarkupfalse% |
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\ is{\isacharunderscore}infI\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}i\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ i\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\isanewline |
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\ \ \ \ \ \ {\isacharparenleft}{\isasymAnd}z{\isachardot}\ z\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ i{\isacharparenright}\ {\isasymLongrightarrow}\ is{\isacharunderscore}inf\ x\ y\ i{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \isacommand{by}\isamarkupfalse% |
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\ {\isacharparenleft}unfold\ is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\ blast\isanewline |
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\isanewline |
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\ \ \isacommand{lemma}\isamarkupfalse% |
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\ is{\isacharunderscore}inf{\isacharunderscore}lower\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline |
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\ \ \ \ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i\ {\isasymLongrightarrow}\ {\isacharparenleft}i\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ i\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \isacommand{by}\isamarkupfalse% |
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\ {\isacharparenleft}unfold\ is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\ blast\isanewline |
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\isanewline |
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\ \ \isacommand{lemma}\isamarkupfalse% |
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\ is{\isacharunderscore}inf{\isacharunderscore}greatest\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline |
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\ \ \ \ \ \ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ i{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \isacommand{by}\isamarkupfalse% |
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\ {\isacharparenleft}unfold\ is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\ blast% |
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\endisataginvisible |
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{\isafoldinvisible}% |
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% |
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\isadeliminvisible |
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\isanewline |
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% |
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\endisadeliminvisible |
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\isanewline |
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\ \ \isacommand{theorem}\isamarkupfalse% |
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\ is{\isacharunderscore}inf{\isacharunderscore}uniq{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}is{\isacharunderscore}inf\ x\ y\ i{\isacharsemicolon}\ is{\isacharunderscore}inf\ x\ y\ i{\isacharprime}{\isasymrbrakk}\ {\isasymLongrightarrow}\ i\ {\isacharequal}\ i{\isacharprime}{\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{assume}\isamarkupfalse% |
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\ inf{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \isacommand{assume}\isamarkupfalse% |
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\ inf{\isacharprime}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isacharprime}{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \isacommand{show}\isamarkupfalse% |
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\ {\isacharquery}thesis\isanewline |
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\ \ \ \ \isacommand{proof}\isamarkupfalse% |
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\ {\isacharparenleft}rule\ anti{\isacharunderscore}sym{\isacharparenright}\isanewline |
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\ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
266 |
\ inf{\isacharprime}\ \isacommand{show}\isamarkupfalse% |
|
267 |
\ {\isachardoublequoteopen}i\ {\isasymsqsubseteq}\ i{\isacharprime}{\isachardoublequoteclose}\isanewline |
|
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\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
269 |
\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}greatest{\isacharparenright}\isanewline |
|
270 |
\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
271 |
\ inf\ \isacommand{show}\isamarkupfalse% |
|
272 |
\ {\isachardoublequoteopen}i\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
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\isanewline |
|
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\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
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\ inf\ \isacommand{show}\isamarkupfalse% |
|
276 |
\ {\isachardoublequoteopen}i\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
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\isanewline |
|
278 |
\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
279 |
\isanewline |
|
280 |
\ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
281 |
\ inf\ \isacommand{show}\isamarkupfalse% |
|
282 |
\ {\isachardoublequoteopen}i{\isacharprime}\ {\isasymsqsubseteq}\ i{\isachardoublequoteclose}\isanewline |
|
283 |
\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
284 |
\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}greatest{\isacharparenright}\isanewline |
|
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\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
286 |
\ inf{\isacharprime}\ \isacommand{show}\isamarkupfalse% |
|
287 |
\ {\isachardoublequoteopen}i{\isacharprime}\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
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\isanewline |
|
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\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
290 |
\ inf{\isacharprime}\ \isacommand{show}\isamarkupfalse% |
|
291 |
\ {\isachardoublequoteopen}i{\isacharprime}\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
292 |
\isanewline |
|
293 |
\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
294 |
\isanewline |
|
295 |
\ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
296 |
\isanewline |
|
297 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
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% |
|
299 |
\endisatagproof |
|
300 |
{\isafoldproof}% |
|
301 |
% |
|
302 |
\isadelimproof |
|
303 |
\isanewline |
|
304 |
% |
|
305 |
\endisadelimproof |
|
306 |
% |
|
307 |
\isadeliminvisible |
|
308 |
\isanewline |
|
309 |
\ \ % |
|
310 |
\endisadeliminvisible |
|
311 |
% |
|
312 |
\isataginvisible |
|
313 |
\isacommand{theorem}\isamarkupfalse% |
|
314 |
\ is{\isacharunderscore}inf{\isacharunderscore}related\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ is{\isacharunderscore}inf\ x\ y\ x{\isachardoublequoteclose}\isanewline |
|
315 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
316 |
\ {\isacharminus}\isanewline |
|
317 |
\ \ \ \ \isacommand{assume}\isamarkupfalse% |
|
318 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline |
|
319 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
320 |
\ {\isacharquery}thesis\isanewline |
|
321 |
\ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
322 |
\isanewline |
|
323 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
324 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
325 |
\isanewline |
|
326 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
327 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
328 |
\ fact\isanewline |
|
329 |
\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse% |
|
330 |
\ z\ \isacommand{assume}\isamarkupfalse% |
|
331 |
\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{show}\isamarkupfalse% |
|
332 |
\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
333 |
\ fact\isanewline |
|
334 |
\ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
335 |
\isanewline |
|
336 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
337 |
\isanewline |
|
338 |
\isanewline |
|
339 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
340 |
\ is{\isacharunderscore}supI\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\isanewline |
|
341 |
\ \ \ \ \ \ {\isacharparenleft}{\isasymAnd}z{\isachardot}\ x\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ s\ {\isasymsqsubseteq}\ z{\isacharparenright}\ {\isasymLongrightarrow}\ is{\isacharunderscore}sup\ x\ y\ s{\isachardoublequoteclose}\isanewline |
|
342 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
343 |
\ {\isacharparenleft}unfold\ is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\ blast\isanewline |
|
344 |
\isanewline |
|
345 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
346 |
\ is{\isacharunderscore}sup{\isacharunderscore}least\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline |
|
347 |
\ \ \ \ \ \ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s\ {\isasymLongrightarrow}\ x\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ s\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline |
|
348 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
349 |
\ {\isacharparenleft}unfold\ is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\ blast\isanewline |
|
350 |
\isanewline |
|
351 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
352 |
\ is{\isacharunderscore}sup{\isacharunderscore}upper\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline |
|
353 |
\ \ \ \ \ \ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s\ {\isasymLongrightarrow}\ {\isacharparenleft}x\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequoteclose}\isanewline |
|
354 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
355 |
\ {\isacharparenleft}unfold\ is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\ blast% |
|
356 |
\endisataginvisible |
|
357 |
{\isafoldinvisible}% |
|
358 |
% |
|
359 |
\isadeliminvisible |
|
360 |
\isanewline |
|
361 |
% |
|
362 |
\endisadeliminvisible |
|
363 |
\isanewline |
|
364 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
365 |
\ is{\isacharunderscore}sup{\isacharunderscore}uniq{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}is{\isacharunderscore}sup\ x\ y\ s{\isacharsemicolon}\ is{\isacharunderscore}sup\ x\ y\ s{\isacharprime}{\isasymrbrakk}\ {\isasymLongrightarrow}\ s\ {\isacharequal}\ s{\isacharprime}{\isachardoublequoteclose}\isanewline |
|
366 |
% |
|
367 |
\isadelimproof |
|
368 |
\ \ % |
|
369 |
\endisadelimproof |
|
370 |
% |
|
371 |
\isatagproof |
|
372 |
\isacommand{proof}\isamarkupfalse% |
|
373 |
\ {\isacharminus}\isanewline |
|
374 |
\ \ \ \ \isacommand{assume}\isamarkupfalse% |
|
375 |
\ sup{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isachardoublequoteclose}\isanewline |
|
376 |
\ \ \ \ \isacommand{assume}\isamarkupfalse% |
|
377 |
\ sup{\isacharprime}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isacharprime}{\isachardoublequoteclose}\isanewline |
|
378 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
379 |
\ {\isacharquery}thesis\isanewline |
|
380 |
\ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
381 |
\ {\isacharparenleft}rule\ anti{\isacharunderscore}sym{\isacharparenright}\isanewline |
|
382 |
\ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
383 |
\ sup\ \isacommand{show}\isamarkupfalse% |
|
384 |
\ {\isachardoublequoteopen}s\ {\isasymsqsubseteq}\ s{\isacharprime}{\isachardoublequoteclose}\isanewline |
|
385 |
\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
386 |
\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}least{\isacharparenright}\isanewline |
|
387 |
\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
388 |
\ sup{\isacharprime}\ \isacommand{show}\isamarkupfalse% |
|
389 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ s{\isacharprime}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
390 |
\isanewline |
|
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\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
392 |
\ sup{\isacharprime}\ \isacommand{show}\isamarkupfalse% |
|
393 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ s{\isacharprime}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
394 |
\isanewline |
|
395 |
\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
396 |
\isanewline |
|
397 |
\ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
398 |
\ sup{\isacharprime}\ \isacommand{show}\isamarkupfalse% |
|
399 |
\ {\isachardoublequoteopen}s{\isacharprime}\ {\isasymsqsubseteq}\ s{\isachardoublequoteclose}\isanewline |
|
400 |
\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
401 |
\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}least{\isacharparenright}\isanewline |
|
402 |
\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
403 |
\ sup\ \isacommand{show}\isamarkupfalse% |
|
404 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ s{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
405 |
\isanewline |
|
406 |
\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
|
407 |
\ sup\ \isacommand{show}\isamarkupfalse% |
|
408 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ s{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
409 |
\isanewline |
|
410 |
\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
411 |
\isanewline |
|
412 |
\ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
413 |
\isanewline |
|
414 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
415 |
% |
|
416 |
\endisatagproof |
|
417 |
{\isafoldproof}% |
|
418 |
% |
|
419 |
\isadelimproof |
|
420 |
\isanewline |
|
421 |
% |
|
422 |
\endisadelimproof |
|
423 |
% |
|
424 |
\isadeliminvisible |
|
425 |
\isanewline |
|
426 |
\ \ % |
|
427 |
\endisadeliminvisible |
|
428 |
% |
|
429 |
\isataginvisible |
|
430 |
\isacommand{theorem}\isamarkupfalse% |
|
431 |
\ is{\isacharunderscore}sup{\isacharunderscore}related\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ is{\isacharunderscore}sup\ x\ y\ y{\isachardoublequoteclose}\isanewline |
|
432 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
433 |
\ {\isacharminus}\isanewline |
|
434 |
\ \ \ \ \isacommand{assume}\isamarkupfalse% |
|
435 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline |
|
436 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
437 |
\ {\isacharquery}thesis\isanewline |
|
438 |
\ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
439 |
\isanewline |
|
440 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
441 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
442 |
\ fact\isanewline |
|
443 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
444 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
445 |
\isanewline |
|
446 |
\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse% |
|
447 |
\ z\ \isacommand{assume}\isamarkupfalse% |
|
448 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline |
|
449 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
450 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
451 |
\ fact\isanewline |
|
452 |
\ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
453 |
\isanewline |
|
454 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
455 |
% |
|
456 |
\endisataginvisible |
|
457 |
{\isafoldinvisible}% |
|
458 |
% |
|
459 |
\isadeliminvisible |
|
460 |
\isanewline |
|
461 |
% |
|
462 |
\endisadeliminvisible |
|
463 |
\isanewline |
|
464 |
\ \ \isacommand{end}\isamarkupfalse% |
|
465 |
% |
|
466 |
\begin{isamarkuptext}% |
|
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|
467 |
The definitions of \isa{is{\isacharunderscore}inf} and \isa{is{\isacharunderscore}sup} look |
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|
468 |
like ordinary definitions in theories. Despite, what is going on |
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|
469 |
behind the scenes is more complex. The definition of \isa{is{\isacharunderscore}inf}, say, creates a constant \isa{partial{\isacharunderscore}order{\isachardot}is{\isacharunderscore}inf} where |
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|
470 |
the locale parameters that occur in the definition of \isa{is{\isacharunderscore}inf} |
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|
471 |
are additional arguments. Writing \isa{is{\isacharunderscore}inf\ x\ y\ inf} is just |
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|
472 |
a notational convenience for \isa{partial{\isacharunderscore}order{\isachardot}is{\isacharunderscore}inf\ op\ {\isasymsqsubseteq}\ x\ y\ inf}.% |
27063 | 473 |
\end{isamarkuptext}% |
474 |
\isamarkuptrue% |
|
475 |
% |
|
476 |
\begin{isamarkuptext}% |
|
477 |
Two commands are provided to inspect locales: |
|
478 |
\isakeyword{print\_locales} lists the names of all locales of the |
|
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|
479 |
current theory; \isakeyword{print\_locale}~$n$ prints the parameters |
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|
480 |
and assumptions of locale $n$; \isakeyword{print\_locale!}~$n$ |
27063 | 481 |
additionally outputs the conclusions. |
482 |
||
483 |
The syntax of the locale commands discussed in this tutorial is |
|
30780 | 484 |
shown in Table~\ref{tab:commands}. The grammer is complete with the |
485 |
exception of additional context elements not discussed here. See the |
|
27063 | 486 |
Isabelle/Isar Reference Manual~\cite{IsarRef} |
487 |
for full documentation.% |
|
488 |
\end{isamarkuptext}% |
|
489 |
\isamarkuptrue% |
|
490 |
% |
|
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|
491 |
\isamarkupsection{Import \label{sec:import}% |
27063 | 492 |
} |
493 |
\isamarkuptrue% |
|
494 |
% |
|
495 |
\begin{isamarkuptext}% |
|
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|
496 |
Algebraic structures are commonly defined by adding operations and |
27063 | 497 |
properties to existing structures. For example, partial orders |
498 |
are extended to lattices and total orders. Lattices are extended to |
|
499 |
distributive lattices. |
|
500 |
||
501 |
With locales, this inheritance is achieved through \emph{import} of a |
|
502 |
locale. The import comes before the context elements.% |
|
503 |
\end{isamarkuptext}% |
|
504 |
\isamarkuptrue% |
|
505 |
\ \ \isacommand{locale}\isamarkupfalse% |
|
506 |
\ lattice\ {\isacharequal}\ partial{\isacharunderscore}order\ {\isacharplus}\isanewline |
|
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|
507 |
\ \ \ \ \isakeyword{assumes}\ ex{\isacharunderscore}inf{\isacharcolon}\ {\isachardoublequoteopen}{\isasymexists}inf{\isachardot}\ is{\isacharunderscore}inf\ x\ y\ inf{\isachardoublequoteclose}\isanewline |
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|
508 |
\ \ \ \ \ \ \isakeyword{and}\ ex{\isacharunderscore}sup{\isacharcolon}\ {\isachardoublequoteopen}{\isasymexists}sup{\isachardot}\ is{\isacharunderscore}sup\ x\ y\ sup{\isachardoublequoteclose}\isanewline |
27063 | 509 |
\ \ \isakeyword{begin}% |
510 |
\begin{isamarkuptext}% |
|
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|
511 |
These assumptions refer to the predicates for infimum |
27063 | 512 |
and supremum defined in \isa{partial{\isacharunderscore}order}. In the current |
513 |
implementation of locales, syntax from definitions of the imported |
|
514 |
locale is unavailable in the locale declaration, neither are their |
|
515 |
names. Hence we refer to the constants of the theory. The names |
|
516 |
and syntax is available below, in the context of the locale.% |
|
517 |
\end{isamarkuptext}% |
|
518 |
\isamarkuptrue% |
|
519 |
\ \ \isacommand{definition}\isamarkupfalse% |
|
520 |
\isanewline |
|
521 |
\ \ \ \ meet\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqinter}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isacharequal}\ {\isacharparenleft}THE\ inf{\isachardot}\ is{\isacharunderscore}inf\ x\ y\ inf{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
522 |
\isanewline |
|
523 |
\ \ \isacommand{definition}\isamarkupfalse% |
|
524 |
\isanewline |
|
525 |
\ \ \ \ join\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymsqunion}{\isachardoublequoteclose}\ {\isadigit{6}}{\isadigit{5}}{\isacharparenright}\ \isakeyword{where}\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isacharequal}\ {\isacharparenleft}THE\ sup{\isachardot}\ is{\isacharunderscore}sup\ x\ y\ sup{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
526 |
% |
|
527 |
\isadeliminvisible |
|
528 |
\isanewline |
|
529 |
\ \ % |
|
530 |
\endisadeliminvisible |
|
531 |
% |
|
532 |
\isataginvisible |
|
533 |
\isacommand{lemma}\isamarkupfalse% |
|
534 |
\ meet{\isacharunderscore}equality\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i\ {\isasymLongrightarrow}\ x\ {\isasymsqinter}\ y\ {\isacharequal}\ i{\isachardoublequoteclose}\isanewline |
|
535 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
536 |
\ {\isacharparenleft}unfold\ meet{\isacharunderscore}def{\isacharparenright}\isanewline |
|
537 |
\ \ \ \ \isacommand{assume}\isamarkupfalse% |
|
538 |
\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isachardoublequoteclose}\isanewline |
|
539 |
\ \ \ \ \isacommand{then}\isamarkupfalse% |
|
540 |
\ \isacommand{show}\isamarkupfalse% |
|
541 |
\ {\isachardoublequoteopen}{\isacharparenleft}THE\ i{\isachardot}\ is{\isacharunderscore}inf\ x\ y\ i{\isacharparenright}\ {\isacharequal}\ i{\isachardoublequoteclose}\isanewline |
|
542 |
\ \ \ \ \ \ \isacommand{by}\isamarkupfalse% |
|
543 |
\ {\isacharparenleft}rule\ the{\isacharunderscore}equality{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}uniq\ {\isacharbrackleft}OF\ {\isacharunderscore}\ {\isacharbackquoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isacharbackquoteclose}{\isacharbrackright}{\isacharparenright}\isanewline |
|
544 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
545 |
\isanewline |
|
546 |
\isanewline |
|
547 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
548 |
\ meetI\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline |
|
549 |
\ \ \ \ \ \ {\isachardoublequoteopen}i\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ i\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}z{\isachardot}\ z\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ i{\isacharparenright}\ {\isasymLongrightarrow}\ x\ {\isasymsqinter}\ y\ {\isacharequal}\ i{\isachardoublequoteclose}\isanewline |
|
550 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
551 |
\ {\isacharparenleft}rule\ meet{\isacharunderscore}equality{\isacharcomma}\ rule\ is{\isacharunderscore}infI{\isacharparenright}\ blast{\isacharplus}\isanewline |
|
552 |
\isanewline |
|
553 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
554 |
\ is{\isacharunderscore}inf{\isacharunderscore}meet\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ {\isacharparenleft}x\ {\isasymsqinter}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
555 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
556 |
\ {\isacharparenleft}unfold\ meet{\isacharunderscore}def{\isacharparenright}\isanewline |
|
557 |
\ \ \ \ \isacommand{from}\isamarkupfalse% |
|
558 |
\ ex{\isacharunderscore}inf\ \isacommand{obtain}\isamarkupfalse% |
|
559 |
\ i\ \isakeyword{where}\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
560 |
\isanewline |
|
561 |
\ \ \ \ \isacommand{then}\isamarkupfalse% |
|
562 |
\ \isacommand{show}\isamarkupfalse% |
|
563 |
\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ {\isacharparenleft}THE\ i{\isachardot}\ is{\isacharunderscore}inf\ x\ y\ i{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
564 |
\ \ \ \ \ \ \isacommand{by}\isamarkupfalse% |
|
565 |
\ {\isacharparenleft}rule\ theI{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}uniq\ {\isacharbrackleft}OF\ {\isacharunderscore}\ {\isacharbackquoteopen}is{\isacharunderscore}inf\ x\ y\ i{\isacharbackquoteclose}{\isacharbrackright}{\isacharparenright}\isanewline |
|
566 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
567 |
\isanewline |
|
568 |
\isanewline |
|
569 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
570 |
\ meet{\isacharunderscore}left\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline |
|
571 |
\ \ \ \ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline |
|
572 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
573 |
\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}lower{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}meet{\isacharparenright}\isanewline |
|
574 |
\isanewline |
|
575 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
576 |
\ meet{\isacharunderscore}right\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline |
|
577 |
\ \ \ \ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline |
|
578 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
579 |
\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}lower{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}meet{\isacharparenright}\isanewline |
|
580 |
\isanewline |
|
581 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
582 |
\ meet{\isacharunderscore}le\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline |
|
583 |
\ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}\ z\ {\isasymsqsubseteq}\ x{\isacharsemicolon}\ z\ {\isasymsqsubseteq}\ y\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ z\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ y{\isachardoublequoteclose}\isanewline |
|
584 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
585 |
\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}greatest{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}inf{\isacharunderscore}meet{\isacharparenright}\isanewline |
|
586 |
\isanewline |
|
587 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
588 |
\ join{\isacharunderscore}equality\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s\ {\isasymLongrightarrow}\ x\ {\isasymsqunion}\ y\ {\isacharequal}\ s{\isachardoublequoteclose}\isanewline |
|
589 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
590 |
\ {\isacharparenleft}unfold\ join{\isacharunderscore}def{\isacharparenright}\isanewline |
|
591 |
\ \ \ \ \isacommand{assume}\isamarkupfalse% |
|
592 |
\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isachardoublequoteclose}\isanewline |
|
593 |
\ \ \ \ \isacommand{then}\isamarkupfalse% |
|
594 |
\ \isacommand{show}\isamarkupfalse% |
|
595 |
\ {\isachardoublequoteopen}{\isacharparenleft}THE\ s{\isachardot}\ is{\isacharunderscore}sup\ x\ y\ s{\isacharparenright}\ {\isacharequal}\ s{\isachardoublequoteclose}\isanewline |
|
596 |
\ \ \ \ \ \ \isacommand{by}\isamarkupfalse% |
|
597 |
\ {\isacharparenleft}rule\ the{\isacharunderscore}equality{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}uniq\ {\isacharbrackleft}OF\ {\isacharunderscore}\ {\isacharbackquoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isacharbackquoteclose}{\isacharbrackright}{\isacharparenright}\isanewline |
|
598 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
599 |
\isanewline |
|
600 |
\isanewline |
|
601 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
602 |
\ joinI\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ s\ {\isasymLongrightarrow}\isanewline |
|
603 |
\ \ \ \ \ \ {\isacharparenleft}{\isasymAnd}z{\isachardot}\ x\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ y\ {\isasymsqsubseteq}\ z\ {\isasymLongrightarrow}\ s\ {\isasymsqsubseteq}\ z{\isacharparenright}\ {\isasymLongrightarrow}\ x\ {\isasymsqunion}\ y\ {\isacharequal}\ s{\isachardoublequoteclose}\isanewline |
|
604 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
605 |
\ {\isacharparenleft}rule\ join{\isacharunderscore}equality{\isacharcomma}\ rule\ is{\isacharunderscore}supI{\isacharparenright}\ blast{\isacharplus}\isanewline |
|
606 |
\isanewline |
|
607 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
608 |
\ is{\isacharunderscore}sup{\isacharunderscore}join\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ {\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
609 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
610 |
\ {\isacharparenleft}unfold\ join{\isacharunderscore}def{\isacharparenright}\isanewline |
|
611 |
\ \ \ \ \isacommand{from}\isamarkupfalse% |
|
612 |
\ ex{\isacharunderscore}sup\ \isacommand{obtain}\isamarkupfalse% |
|
613 |
\ s\ \isakeyword{where}\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
614 |
\isanewline |
|
615 |
\ \ \ \ \isacommand{then}\isamarkupfalse% |
|
616 |
\ \isacommand{show}\isamarkupfalse% |
|
617 |
\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ {\isacharparenleft}THE\ s{\isachardot}\ is{\isacharunderscore}sup\ x\ y\ s{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
618 |
\ \ \ \ \ \ \isacommand{by}\isamarkupfalse% |
|
619 |
\ {\isacharparenleft}rule\ theI{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}uniq\ {\isacharbrackleft}OF\ {\isacharunderscore}\ {\isacharbackquoteopen}is{\isacharunderscore}sup\ x\ y\ s{\isacharbackquoteclose}{\isacharbrackright}{\isacharparenright}\isanewline |
|
620 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
621 |
\isanewline |
|
622 |
\isanewline |
|
623 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
624 |
\ join{\isacharunderscore}left\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline |
|
625 |
\ \ \ \ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\isanewline |
|
626 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
627 |
\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}upper{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}join{\isacharparenright}\isanewline |
|
628 |
\isanewline |
|
629 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
630 |
\ join{\isacharunderscore}right\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline |
|
631 |
\ \ \ \ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\isanewline |
|
632 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
633 |
\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}upper{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}join{\isacharparenright}\isanewline |
|
634 |
\isanewline |
|
635 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
636 |
\ join{\isacharunderscore}le\ {\isacharbrackleft}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\isanewline |
|
637 |
\ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}\ x\ {\isasymsqsubseteq}\ z{\isacharsemicolon}\ y\ {\isasymsqsubseteq}\ z\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isasymsqunion}\ y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline |
|
638 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
639 |
\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}least{\isacharparenright}\ {\isacharparenleft}rule\ is{\isacharunderscore}sup{\isacharunderscore}join{\isacharparenright}\isanewline |
|
640 |
\isanewline |
|
641 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
642 |
\ meet{\isacharunderscore}assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqinter}\ y{\isacharparenright}\ {\isasymsqinter}\ z\ {\isacharequal}\ x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
643 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
644 |
\ {\isacharparenleft}rule\ meetI{\isacharparenright}\isanewline |
|
645 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
646 |
\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ y{\isachardoublequoteclose}\isanewline |
|
647 |
\ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
648 |
\isanewline |
|
649 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
650 |
\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
651 |
\isanewline |
|
652 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
653 |
\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline |
|
654 |
\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
655 |
\ {\isacharminus}\isanewline |
|
656 |
\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse% |
|
657 |
\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ y\ {\isasymsqinter}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
658 |
\isanewline |
|
659 |
\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse% |
|
660 |
\ \isacommand{have}\isamarkupfalse% |
|
661 |
\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
662 |
\isanewline |
|
663 |
\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
664 |
\ \isacommand{show}\isamarkupfalse% |
|
665 |
\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
666 |
\isanewline |
|
667 |
\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
668 |
\isanewline |
|
669 |
\ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
670 |
\isanewline |
|
671 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
672 |
\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline |
|
673 |
\ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
674 |
\ {\isacharminus}\isanewline |
|
675 |
\ \ \ \ \ \ \isacommand{have}\isamarkupfalse% |
|
676 |
\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ y\ {\isasymsqinter}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
677 |
\isanewline |
|
678 |
\ \ \ \ \ \ \isacommand{also}\isamarkupfalse% |
|
679 |
\ \isacommand{have}\isamarkupfalse% |
|
680 |
\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
681 |
\isanewline |
|
682 |
\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
683 |
\ \isacommand{show}\isamarkupfalse% |
|
684 |
\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
685 |
\isanewline |
|
686 |
\ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
687 |
\isanewline |
|
688 |
\ \ \ \ \isacommand{fix}\isamarkupfalse% |
|
689 |
\ w\ \isacommand{assume}\isamarkupfalse% |
|
690 |
\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ y{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline |
|
691 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
692 |
\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
693 |
\ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
694 |
\isanewline |
|
695 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
696 |
\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline |
|
697 |
\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
698 |
\ {\isacharminus}\isanewline |
|
699 |
\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse% |
|
700 |
\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
701 |
\ fact\isanewline |
|
702 |
\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse% |
|
703 |
\ \isacommand{have}\isamarkupfalse% |
|
704 |
\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
705 |
\isanewline |
|
706 |
\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
707 |
\ \isacommand{show}\isamarkupfalse% |
|
708 |
\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
709 |
\isanewline |
|
710 |
\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
711 |
\isanewline |
|
712 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
713 |
\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ y\ {\isasymsqinter}\ z{\isachardoublequoteclose}\isanewline |
|
714 |
\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
715 |
\isanewline |
|
716 |
\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
717 |
\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline |
|
718 |
\ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
719 |
\ {\isacharminus}\isanewline |
|
720 |
\ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse% |
|
721 |
\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ x\ {\isasymsqinter}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
722 |
\ fact\isanewline |
|
723 |
\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse% |
|
724 |
\ \isacommand{have}\isamarkupfalse% |
|
725 |
\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
726 |
\isanewline |
|
727 |
\ \ \ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
728 |
\ \isacommand{show}\isamarkupfalse% |
|
729 |
\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
730 |
\isanewline |
|
731 |
\ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
732 |
\isanewline |
|
733 |
\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
734 |
\ {\isachardoublequoteopen}w\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
735 |
\ fact\isanewline |
|
736 |
\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
737 |
\isanewline |
|
738 |
\ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
739 |
\isanewline |
|
740 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
741 |
\isanewline |
|
742 |
\isanewline |
|
743 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
744 |
\ meet{\isacharunderscore}commute{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isacharequal}\ y\ {\isasymsqinter}\ x{\isachardoublequoteclose}\isanewline |
|
745 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
746 |
\ {\isacharparenleft}rule\ meetI{\isacharparenright}\isanewline |
|
747 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
748 |
\ {\isachardoublequoteopen}y\ {\isasymsqinter}\ x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
749 |
\isanewline |
|
750 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
751 |
\ {\isachardoublequoteopen}y\ {\isasymsqinter}\ x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
752 |
\isanewline |
|
753 |
\ \ \ \ \isacommand{fix}\isamarkupfalse% |
|
754 |
\ z\ \isacommand{assume}\isamarkupfalse% |
|
755 |
\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline |
|
756 |
\ \ \ \ \isacommand{then}\isamarkupfalse% |
|
757 |
\ \isacommand{show}\isamarkupfalse% |
|
758 |
\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ y\ {\isasymsqinter}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
759 |
\isanewline |
|
760 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
761 |
\isanewline |
|
762 |
\isanewline |
|
763 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
764 |
\ meet{\isacharunderscore}join{\isacharunderscore}absorb{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline |
|
765 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
766 |
\ {\isacharparenleft}rule\ meetI{\isacharparenright}\isanewline |
|
767 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
768 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
769 |
\isanewline |
|
770 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
771 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
772 |
\isanewline |
|
773 |
\ \ \ \ \isacommand{fix}\isamarkupfalse% |
|
774 |
\ z\ \isacommand{assume}\isamarkupfalse% |
|
775 |
\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\isanewline |
|
776 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
777 |
\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
778 |
\ fact\isanewline |
|
779 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
780 |
\isanewline |
|
781 |
\isanewline |
|
782 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
783 |
\ join{\isacharunderscore}assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqunion}\ z\ {\isacharequal}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
784 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
785 |
\ {\isacharparenleft}rule\ joinI{\isacharparenright}\isanewline |
|
786 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
787 |
\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
788 |
\ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
789 |
\isanewline |
|
790 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
791 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
792 |
\isanewline |
|
793 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
794 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
795 |
\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
796 |
\ {\isacharminus}\isanewline |
|
797 |
\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse% |
|
798 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ y\ {\isasymsqunion}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
799 |
\isanewline |
|
800 |
\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse% |
|
801 |
\ \isacommand{have}\isamarkupfalse% |
|
802 |
\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
803 |
\isanewline |
|
804 |
\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
805 |
\ \isacommand{show}\isamarkupfalse% |
|
806 |
\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
807 |
\isanewline |
|
808 |
\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
809 |
\isanewline |
|
810 |
\ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
811 |
\isanewline |
|
812 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
813 |
\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
814 |
\ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
815 |
\ {\isacharminus}\isanewline |
|
816 |
\ \ \ \ \ \ \isacommand{have}\isamarkupfalse% |
|
817 |
\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ y\ {\isasymsqunion}\ z{\isachardoublequoteclose}\ \ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
818 |
\isanewline |
|
819 |
\ \ \ \ \ \ \isacommand{also}\isamarkupfalse% |
|
820 |
\ \isacommand{have}\isamarkupfalse% |
|
821 |
\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
822 |
\isanewline |
|
823 |
\ \ \ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
824 |
\ \isacommand{show}\isamarkupfalse% |
|
825 |
\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
826 |
\isanewline |
|
827 |
\ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
828 |
\isanewline |
|
829 |
\ \ \ \ \isacommand{fix}\isamarkupfalse% |
|
830 |
\ w\ \isacommand{assume}\isamarkupfalse% |
|
831 |
\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\isanewline |
|
832 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
833 |
\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\isanewline |
|
834 |
\ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
835 |
\isanewline |
|
836 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
837 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\isanewline |
|
838 |
\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
839 |
\ {\isacharminus}\isanewline |
|
840 |
\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse% |
|
841 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
842 |
\isanewline |
|
843 |
\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse% |
|
844 |
\ \isacommand{have}\isamarkupfalse% |
|
845 |
\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
846 |
\ fact\isanewline |
|
847 |
\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
848 |
\ \isacommand{show}\isamarkupfalse% |
|
849 |
\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
850 |
\isanewline |
|
851 |
\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
852 |
\isanewline |
|
853 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
854 |
\ {\isachardoublequoteopen}y\ {\isasymsqunion}\ z\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\isanewline |
|
855 |
\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
856 |
\isanewline |
|
857 |
\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
858 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\isanewline |
|
859 |
\ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse% |
|
860 |
\ {\isacharminus}\isanewline |
|
861 |
\ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse% |
|
862 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
863 |
\isanewline |
|
864 |
\ \ \ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse% |
|
865 |
\ \isacommand{have}\isamarkupfalse% |
|
866 |
\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
867 |
\ fact\isanewline |
|
868 |
\ \ \ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
869 |
\ \isacommand{show}\isamarkupfalse% |
|
870 |
\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
871 |
\isanewline |
|
872 |
\ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
873 |
\isanewline |
|
874 |
\ \ \ \ \ \ \ \ \isacommand{show}\isamarkupfalse% |
|
875 |
\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ w{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
876 |
\ fact\isanewline |
|
877 |
\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
878 |
\isanewline |
|
879 |
\ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
880 |
\isanewline |
|
881 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
882 |
\isanewline |
|
883 |
\isanewline |
|
884 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
885 |
\ join{\isacharunderscore}commute{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isacharequal}\ y\ {\isasymsqunion}\ x{\isachardoublequoteclose}\isanewline |
|
886 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
887 |
\ {\isacharparenleft}rule\ joinI{\isacharparenright}\isanewline |
|
888 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
889 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymsqunion}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
890 |
\isanewline |
|
891 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
892 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ y\ {\isasymsqunion}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
893 |
\isanewline |
|
894 |
\ \ \ \ \isacommand{fix}\isamarkupfalse% |
|
895 |
\ z\ \isacommand{assume}\isamarkupfalse% |
|
896 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline |
|
897 |
\ \ \ \ \isacommand{then}\isamarkupfalse% |
|
898 |
\ \isacommand{show}\isamarkupfalse% |
|
899 |
\ {\isachardoublequoteopen}y\ {\isasymsqunion}\ x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
900 |
\isanewline |
|
901 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
902 |
\isanewline |
|
903 |
\isanewline |
|
904 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
905 |
\ join{\isacharunderscore}meet{\isacharunderscore}absorb{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ {\isacharparenleft}x\ {\isasymsqinter}\ y{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline |
|
906 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
907 |
\ {\isacharparenleft}rule\ joinI{\isacharparenright}\isanewline |
|
908 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
909 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
910 |
\isanewline |
|
911 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
912 |
\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
913 |
\isanewline |
|
914 |
\ \ \ \ \isacommand{fix}\isamarkupfalse% |
|
915 |
\ z\ \isacommand{assume}\isamarkupfalse% |
|
916 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline |
|
917 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
918 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
919 |
\ fact\isanewline |
|
920 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
921 |
\isanewline |
|
922 |
\isanewline |
|
923 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
924 |
\ meet{\isacharunderscore}idem{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline |
|
925 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
926 |
\ {\isacharminus}\isanewline |
|
927 |
\ \ \ \ \isacommand{have}\isamarkupfalse% |
|
928 |
\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}x\ {\isasymsqunion}\ {\isacharparenleft}x\ {\isasymsqinter}\ x{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
929 |
\ {\isacharparenleft}rule\ meet{\isacharunderscore}join{\isacharunderscore}absorb{\isacharparenright}\isanewline |
|
930 |
\ \ \ \ \isacommand{also}\isamarkupfalse% |
|
931 |
\ \isacommand{have}\isamarkupfalse% |
|
932 |
\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ {\isacharparenleft}x\ {\isasymsqinter}\ x{\isacharparenright}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
933 |
\ {\isacharparenleft}rule\ join{\isacharunderscore}meet{\isacharunderscore}absorb{\isacharparenright}\isanewline |
|
934 |
\ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
935 |
\ \isacommand{show}\isamarkupfalse% |
|
936 |
\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
937 |
\isanewline |
|
938 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
939 |
\isanewline |
|
940 |
\isanewline |
|
941 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
942 |
\ meet{\isacharunderscore}related\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ x\ {\isasymsqinter}\ y\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline |
|
943 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
944 |
\ {\isacharparenleft}rule\ meetI{\isacharparenright}\isanewline |
|
945 |
\ \ \ \ \isacommand{assume}\isamarkupfalse% |
|
946 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline |
|
947 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
948 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
949 |
\isanewline |
|
950 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
951 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
952 |
\ fact\isanewline |
|
953 |
\ \ \ \ \isacommand{fix}\isamarkupfalse% |
|
954 |
\ z\ \isacommand{assume}\isamarkupfalse% |
|
955 |
\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline |
|
956 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
957 |
\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
958 |
\ fact\isanewline |
|
959 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
960 |
\isanewline |
|
961 |
\isanewline |
|
962 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
963 |
\ meet{\isacharunderscore}related{\isadigit{2}}\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ x\ {\isasymsqinter}\ y\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline |
|
964 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
965 |
\ {\isacharparenleft}drule\ meet{\isacharunderscore}related{\isacharparenright}\ {\isacharparenleft}simp\ add{\isacharcolon}\ meet{\isacharunderscore}commute{\isacharparenright}\isanewline |
|
966 |
\isanewline |
|
967 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
968 |
\ join{\isacharunderscore}related\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymLongrightarrow}\ x\ {\isasymsqunion}\ y\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline |
|
969 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
970 |
\ {\isacharparenleft}rule\ joinI{\isacharparenright}\isanewline |
|
971 |
\ \ \ \ \isacommand{assume}\isamarkupfalse% |
|
972 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline |
|
973 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
974 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
975 |
\isanewline |
|
976 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
977 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
978 |
\ fact\isanewline |
|
979 |
\ \ \ \ \isacommand{fix}\isamarkupfalse% |
|
980 |
\ z\ \isacommand{assume}\isamarkupfalse% |
|
981 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline |
|
982 |
\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
983 |
\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
984 |
\ fact\isanewline |
|
985 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
986 |
\isanewline |
|
987 |
\isanewline |
|
988 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
989 |
\ join{\isacharunderscore}related{\isadigit{2}}\ {\isacharbrackleft}elim{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x\ {\isasymLongrightarrow}\ x\ {\isasymsqunion}\ y\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline |
|
990 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
991 |
\ {\isacharparenleft}drule\ join{\isacharunderscore}related{\isacharparenright}\ {\isacharparenleft}simp\ add{\isacharcolon}\ join{\isacharunderscore}commute{\isacharparenright}\isanewline |
|
992 |
\isanewline |
|
993 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
994 |
\ meet{\isacharunderscore}connection{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqsubseteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqinter}\ y\ {\isacharequal}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
995 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
996 |
\isanewline |
|
997 |
\ \ \ \ \isacommand{assume}\isamarkupfalse% |
|
998 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline |
|
999 |
\ \ \ \ \isacommand{then}\isamarkupfalse% |
|
1000 |
\ \isacommand{have}\isamarkupfalse% |
|
1001 |
\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
1002 |
\isanewline |
|
1003 |
\ \ \ \ \isacommand{then}\isamarkupfalse% |
|
1004 |
\ \isacommand{show}\isamarkupfalse% |
|
1005 |
\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
1006 |
\isanewline |
|
1007 |
\ \ \isacommand{next}\isamarkupfalse% |
|
1008 |
\isanewline |
|
1009 |
\ \ \ \ \isacommand{have}\isamarkupfalse% |
|
1010 |
\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
1011 |
\isanewline |
|
1012 |
\ \ \ \ \isacommand{also}\isamarkupfalse% |
|
1013 |
\ \isacommand{assume}\isamarkupfalse% |
|
1014 |
\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ y\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline |
|
1015 |
\ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
1016 |
\ \isacommand{show}\isamarkupfalse% |
|
1017 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
1018 |
\isanewline |
|
1019 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
1020 |
\isanewline |
|
1021 |
\isanewline |
|
1022 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
1023 |
\ join{\isacharunderscore}connection{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqsubseteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ y\ {\isacharequal}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
1024 |
\ \ \isacommand{proof}\isamarkupfalse% |
|
1025 |
\isanewline |
|
1026 |
\ \ \ \ \isacommand{assume}\isamarkupfalse% |
|
1027 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\isanewline |
|
1028 |
\ \ \ \ \isacommand{then}\isamarkupfalse% |
|
1029 |
\ \isacommand{have}\isamarkupfalse% |
|
1030 |
\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
1031 |
\isanewline |
|
1032 |
\ \ \ \ \isacommand{then}\isamarkupfalse% |
|
1033 |
\ \isacommand{show}\isamarkupfalse% |
|
1034 |
\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isacharequal}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
1035 |
\isanewline |
|
1036 |
\ \ \isacommand{next}\isamarkupfalse% |
|
1037 |
\isanewline |
|
1038 |
\ \ \ \ \isacommand{have}\isamarkupfalse% |
|
1039 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ x\ {\isasymsqunion}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
|
1040 |
\isanewline |
|
1041 |
\ \ \ \ \isacommand{also}\isamarkupfalse% |
|
1042 |
\ \isacommand{assume}\isamarkupfalse% |
|
1043 |
\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ y\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline |
|
1044 |
\ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
1045 |
\ \isacommand{show}\isamarkupfalse% |
|
1046 |
\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
1047 |
\isanewline |
|
1048 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
1049 |
\isanewline |
|
1050 |
\isanewline |
|
1051 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
1052 |
\ meet{\isacharunderscore}connection{\isadigit{2}}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqsubseteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymsqinter}\ x\ {\isacharequal}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
1053 |
\ \ \ \ \isacommand{using}\isamarkupfalse% |
|
1054 |
\ meet{\isacharunderscore}commute\ meet{\isacharunderscore}connection\ \isacommand{by}\isamarkupfalse% |
|
1055 |
\ simp\isanewline |
|
1056 |
\isanewline |
|
1057 |
\ \ \isacommand{theorem}\isamarkupfalse% |
|
1058 |
\ join{\isacharunderscore}connection{\isadigit{2}}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymsqsubseteq}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ y\ {\isacharequal}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
1059 |
\ \ \ \ \isacommand{using}\isamarkupfalse% |
|
1060 |
\ join{\isacharunderscore}commute\ join{\isacharunderscore}connection\ \isacommand{by}\isamarkupfalse% |
|
1061 |
\ simp% |
|
1062 |
\begin{isamarkuptext}% |
|
1063 |
Naming according to Jacobson I, p.\ 459.% |
|
1064 |
\end{isamarkuptext}% |
|
1065 |
\isamarkuptrue% |
|
1066 |
\ \ \isacommand{lemmas}\isamarkupfalse% |
|
1067 |
\ L{\isadigit{1}}\ {\isacharequal}\ join{\isacharunderscore}commute\ meet{\isacharunderscore}commute\isanewline |
|
1068 |
\ \ \isacommand{lemmas}\isamarkupfalse% |
|
1069 |
\ L{\isadigit{2}}\ {\isacharequal}\ join{\isacharunderscore}assoc\ meet{\isacharunderscore}assoc\isanewline |
|
1070 |
\ \ \isanewline |
|
1071 |
\ \ \isacommand{lemmas}\isamarkupfalse% |
|
1072 |
\ L{\isadigit{4}}\ {\isacharequal}\ join{\isacharunderscore}meet{\isacharunderscore}absorb\ meet{\isacharunderscore}join{\isacharunderscore}absorb% |
|
1073 |
\endisataginvisible |
|
1074 |
{\isafoldinvisible}% |
|
1075 |
% |
|
1076 |
\isadeliminvisible |
|
1077 |
\isanewline |
|
1078 |
% |
|
1079 |
\endisadeliminvisible |
|
1080 |
\isanewline |
|
1081 |
\ \ \isacommand{end}\isamarkupfalse% |
|
1082 |
% |
|
1083 |
\begin{isamarkuptext}% |
|
1084 |
Locales for total orders and distributive lattices follow. |
|
1085 |
Each comes with an example theorem.% |
|
1086 |
\end{isamarkuptext}% |
|
1087 |
\isamarkuptrue% |
|
1088 |
\ \ \isacommand{locale}\isamarkupfalse% |
|
1089 |
\ total{\isacharunderscore}order\ {\isacharequal}\ partial{\isacharunderscore}order\ {\isacharplus}\isanewline |
|
1090 |
\ \ \ \ \isakeyword{assumes}\ total{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymor}\ y\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline |
|
1091 |
\isanewline |
|
1092 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
1093 |
\ {\isacharparenleft}\isakeyword{in}\ total{\isacharunderscore}order{\isacharparenright}\ less{\isacharunderscore}total{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubset}\ y\ {\isasymor}\ x\ {\isacharequal}\ y\ {\isasymor}\ y\ {\isasymsqsubset}\ x{\isachardoublequoteclose}\isanewline |
|
1094 |
% |
|
1095 |
\isadelimproof |
|
1096 |
\ \ \ \ % |
|
1097 |
\endisadelimproof |
|
1098 |
% |
|
1099 |
\isatagproof |
|
1100 |
\isacommand{using}\isamarkupfalse% |
|
1101 |
\ total\isanewline |
|
1102 |
\ \ \ \ \isacommand{by}\isamarkupfalse% |
|
1103 |
\ {\isacharparenleft}unfold\ less{\isacharunderscore}def{\isacharparenright}\ blast% |
|
1104 |
\endisatagproof |
|
1105 |
{\isafoldproof}% |
|
1106 |
% |
|
1107 |
\isadelimproof |
|
1108 |
\isanewline |
|
1109 |
% |
|
1110 |
\endisadelimproof |
|
1111 |
\isanewline |
|
1112 |
\ \ \isacommand{locale}\isamarkupfalse% |
|
1113 |
\ distrib{\isacharunderscore}lattice\ {\isacharequal}\ lattice\ {\isacharplus}\isanewline |
|
30580
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Updated chapters 1-5 to locale reimplementation.
ballarin
parents:
30393
diff
changeset
|
1114 |
\ \ \ \ \isakeyword{assumes}\ meet{\isacharunderscore}distr{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}\ {\isacharequal}\ x\ {\isasymsqinter}\ y\ {\isasymsqunion}\ x\ {\isasymsqinter}\ z{\isachardoublequoteclose}\isanewline |
27063 | 1115 |
\isanewline |
1116 |
\ \ \isacommand{lemma}\isamarkupfalse% |
|
1117 |
\ {\isacharparenleft}\isakeyword{in}\ distrib{\isacharunderscore}lattice{\isacharparenright}\ join{\isacharunderscore}distr{\isacharcolon}\isanewline |
|
1118 |
\ \ \ \ {\isachardoublequoteopen}x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqinter}\ {\isacharparenleft}x\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\ \ \isanewline |
|
1119 |
% |
|
1120 |
\isadelimproof |
|
1121 |
\ \ \ \ % |
|
1122 |
\endisadelimproof |
|
1123 |
% |
|
1124 |
\isatagproof |
|
1125 |
\isacommand{proof}\isamarkupfalse% |
|
1126 |
\ {\isacharminus}\isanewline |
|
1127 |
\ \ \ \ \isacommand{have}\isamarkupfalse% |
|
1128 |
\ {\isachardoublequoteopen}x\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ {\isacharparenleft}x\ {\isasymsqinter}\ z{\isacharparenright}{\isacharparenright}\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
1129 |
\ {\isacharparenleft}simp\ add{\isacharcolon}\ L{\isadigit{4}}{\isacharparenright}\isanewline |
|
1130 |
\ \ \ \ \isacommand{also}\isamarkupfalse% |
|
1131 |
\ \isacommand{have}\isamarkupfalse% |
|
1132 |
\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x\ {\isasymsqunion}\ {\isacharparenleft}{\isacharparenleft}x\ {\isasymsqinter}\ z{\isacharparenright}\ {\isasymsqunion}\ {\isacharparenleft}y\ {\isasymsqinter}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
1133 |
\ {\isacharparenleft}simp\ add{\isacharcolon}\ L{\isadigit{2}}{\isacharparenright}\isanewline |
|
1134 |
\ \ \ \ \isacommand{also}\isamarkupfalse% |
|
1135 |
\ \isacommand{have}\isamarkupfalse% |
|
1136 |
\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ x\ {\isasymsqunion}\ {\isacharparenleft}{\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqinter}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
1137 |
\ {\isacharparenleft}simp\ add{\isacharcolon}\ L{\isadigit{1}}\ meet{\isacharunderscore}distr{\isacharparenright}\isanewline |
|
1138 |
\ \ \ \ \isacommand{also}\isamarkupfalse% |
|
1139 |
\ \isacommand{have}\isamarkupfalse% |
|
1140 |
\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqinter}\ x{\isacharparenright}\ {\isasymsqunion}\ {\isacharparenleft}{\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqinter}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
1141 |
\ {\isacharparenleft}simp\ add{\isacharcolon}\ L{\isadigit{1}}\ L{\isadigit{4}}{\isacharparenright}\isanewline |
|
1142 |
\ \ \ \ \isacommand{also}\isamarkupfalse% |
|
1143 |
\ \isacommand{have}\isamarkupfalse% |
|
1144 |
\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymsqunion}\ y{\isacharparenright}\ {\isasymsqinter}\ {\isacharparenleft}x\ {\isasymsqunion}\ z{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
|
1145 |
\ {\isacharparenleft}simp\ add{\isacharcolon}\ meet{\isacharunderscore}distr{\isacharparenright}\isanewline |
|
1146 |
\ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
1147 |
\ \isacommand{show}\isamarkupfalse% |
|
1148 |
\ {\isacharquery}thesis\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
1149 |
\isanewline |
|
1150 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
1151 |
% |
|
1152 |
\endisatagproof |
|
1153 |
{\isafoldproof}% |
|
1154 |
% |
|
1155 |
\isadelimproof |
|
1156 |
% |
|
1157 |
\endisadelimproof |
|
1158 |
% |
|
1159 |
\begin{isamarkuptext}% |
|
1160 |
The locale hierachy obtained through these declarations is shown in Figure~\ref{fig:lattices}(a). |
|
1161 |
||
1162 |
\begin{figure} |
|
1163 |
\hrule \vspace{2ex} |
|
1164 |
\begin{center} |
|
1165 |
\subfigure[Declared hierachy]{ |
|
1166 |
\begin{tikzpicture} |
|
1167 |
\node (po) at (0,0) {\isa{partial{\isacharunderscore}order}}; |
|
1168 |
\node (lat) at (-1.5,-1) {\isa{lattice}}; |
|
1169 |
\node (dlat) at (-1.5,-2) {\isa{distrib{\isacharunderscore}lattice}}; |
|
1170 |
\node (to) at (1.5,-1) {\isa{total{\isacharunderscore}order}}; |
|
1171 |
\draw (po) -- (lat); |
|
1172 |
\draw (lat) -- (dlat); |
|
1173 |
\draw (po) -- (to); |
|
1174 |
% \draw[->, dashed] (lat) -- (to); |
|
1175 |
\end{tikzpicture} |
|
1176 |
} \\ |
|
1177 |
\subfigure[Total orders are lattices]{ |
|
1178 |
\begin{tikzpicture} |
|
1179 |
\node (po) at (0,0) {\isa{partial{\isacharunderscore}order}}; |
|
1180 |
\node (lat) at (0,-1) {\isa{lattice}}; |
|
1181 |
\node (dlat) at (-1.5,-2) {\isa{distrib{\isacharunderscore}lattice}}; |
|
1182 |
\node (to) at (1.5,-2) {\isa{total{\isacharunderscore}order}}; |
|
1183 |
\draw (po) -- (lat); |
|
1184 |
\draw (lat) -- (dlat); |
|
1185 |
\draw (lat) -- (to); |
|
1186 |
% \draw[->, dashed] (dlat) -- (to); |
|
1187 |
\end{tikzpicture} |
|
1188 |
} \quad |
|
1189 |
\subfigure[Total orders are distributive lattices]{ |
|
1190 |
\begin{tikzpicture} |
|
1191 |
\node (po) at (0,0) {\isa{partial{\isacharunderscore}order}}; |
|
1192 |
\node (lat) at (0,-1) {\isa{lattice}}; |
|
1193 |
\node (dlat) at (0,-2) {\isa{distrib{\isacharunderscore}lattice}}; |
|
1194 |
\node (to) at (0,-3) {\isa{total{\isacharunderscore}order}}; |
|
1195 |
\draw (po) -- (lat); |
|
1196 |
\draw (lat) -- (dlat); |
|
1197 |
\draw (dlat) -- (to); |
|
1198 |
\end{tikzpicture} |
|
1199 |
} |
|
1200 |
\end{center} |
|
1201 |
\hrule |
|
1202 |
\caption{Hierarchy of Lattice Locales.} |
|
1203 |
\label{fig:lattices} |
|
1204 |
\end{figure}% |
|
1205 |
\end{isamarkuptext}% |
|
1206 |
\isamarkuptrue% |
|
1207 |
% |
|
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changeset
|
1208 |
\isamarkupsection{Changing the Locale Hierarchy |
cc5a55d7a5be
Updated chapters 1-5 to locale reimplementation.
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\label{sec:changing-the-hierarchy}% |
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} |
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\isamarkuptrue% |
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% |
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\begin{isamarkuptext}% |
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cc5a55d7a5be
Updated chapters 1-5 to locale reimplementation.
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diff
changeset
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Total orders are lattices. Hence, by deriving the lattice |
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axioms for total orders, the hierarchy may be changed |
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and \isa{lattice} be placed between \isa{partial{\isacharunderscore}order} |
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and \isa{total{\isacharunderscore}order}, as shown in Figure~\ref{fig:lattices}(b). |
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Changes to the locale hierarchy may be declared |
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Fixed tutorial to compile with new locales; grammar of new locale commands.
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changeset
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with the \isakeyword{sublocale} command.% |
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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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937baa077df2
Fixed tutorial to compile with new locales; grammar of new locale commands.
ballarin
parents:
27375
diff
changeset
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\isacommand{sublocale}\isamarkupfalse% |
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\ total{\isacharunderscore}order\ {\isasymsubseteq}\ lattice% |
1230 |
\begin{isamarkuptxt}% |
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This enters the context of locale \isa{total{\isacharunderscore}order}, in |
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which the goal \begin{isabelle}% |
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\ {\isadigit{1}}{\isachardot}\ lattice\ op\ {\isasymsqsubseteq}% |
|
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\end{isabelle} must be shown. First, the |
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locale predicate needs to be unfolded --- for example using its |
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definition or by introduction rules |
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provided by the locale package. The methods \isa{intro{\isacharunderscore}locales} |
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and \isa{unfold{\isacharunderscore}locales} automate this. They are aware of the |
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current context and dependencies between locales and automatically |
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discharge goals implied by these. While \isa{unfold{\isacharunderscore}locales} |
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always unfolds locale predicates to assumptions, \isa{intro{\isacharunderscore}locales} only unfolds definitions along the locale |
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hierarchy, leaving a goal consisting of predicates defined by the |
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locale package. Occasionally the latter is of advantage since the goal |
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is smaller. |
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||
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For the current goal, we would like to get hold of |
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the assumptions of \isa{lattice}, hence \isa{unfold{\isacharunderscore}locales} |
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is appropriate.% |
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\end{isamarkuptxt}% |
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\isamarkuptrue% |
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\ \ \isacommand{proof}\isamarkupfalse% |
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\ unfold{\isacharunderscore}locales% |
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\begin{isamarkuptxt}% |
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Since both \isa{lattice} and \isa{total{\isacharunderscore}order} |
|
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inherit \isa{partial{\isacharunderscore}order}, the assumptions of the latter are |
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discharged, and the only subgoals that remain are the assumptions |
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introduced in \isa{lattice} \begin{isabelle}% |
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\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}inf{\isachardot}\ is{\isacharunderscore}inf\ x\ y\ inf\isanewline |
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\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}x\ y{\isachardot}\ {\isasymexists}sup{\isachardot}\ is{\isacharunderscore}sup\ x\ y\ sup% |
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\end{isabelle} |
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The proof for the first subgoal is% |
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\end{isamarkuptxt}% |
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\isamarkuptrue% |
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\ \ \ \ \isacommand{fix}\isamarkupfalse% |
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\ x\ y\isanewline |
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\ \ \ \ \isacommand{from}\isamarkupfalse% |
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\ total\ \isacommand{have}\isamarkupfalse% |
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\ {\isachardoublequoteopen}is{\isacharunderscore}inf\ x\ y\ {\isacharparenleft}if\ x\ {\isasymsqsubseteq}\ y\ then\ x\ else\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \ \ \isacommand{by}\isamarkupfalse% |
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\ {\isacharparenleft}auto\ simp{\isacharcolon}\ is{\isacharunderscore}inf{\isacharunderscore}def{\isacharparenright}\isanewline |
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\ \ \ \ \isacommand{then}\isamarkupfalse% |
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\ \isacommand{show}\isamarkupfalse% |
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\ {\isachardoublequoteopen}{\isasymexists}inf{\isachardot}\ is{\isacharunderscore}inf\ x\ y\ inf{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
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% |
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\begin{isamarkuptxt}% |
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The proof for the second subgoal is analogous and not |
|
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reproduced here.% |
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\end{isamarkuptxt}% |
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\isamarkuptrue% |
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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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\isadeliminvisible |
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\ \ % |
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\endisadeliminvisible |
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% |
|
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\isataginvisible |
|
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\isacommand{next}\isamarkupfalse% |
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\isanewline |
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\ \ \ \ \isacommand{fix}\isamarkupfalse% |
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\ x\ y\isanewline |
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\ \ \ \ \isacommand{from}\isamarkupfalse% |
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\ total\ \isacommand{have}\isamarkupfalse% |
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\ {\isachardoublequoteopen}is{\isacharunderscore}sup\ x\ y\ {\isacharparenleft}if\ x\ {\isasymsqsubseteq}\ y\ then\ y\ else\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \ \ \isacommand{by}\isamarkupfalse% |
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\ {\isacharparenleft}auto\ simp{\isacharcolon}\ is{\isacharunderscore}sup{\isacharunderscore}def{\isacharparenright}\isanewline |
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\ \ \ \ \isacommand{then}\isamarkupfalse% |
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\ \isacommand{show}\isamarkupfalse% |
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\ {\isachardoublequoteopen}{\isasymexists}sup{\isachardot}\ is{\isacharunderscore}sup\ x\ y\ sup{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% |
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% |
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\endisataginvisible |
|
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{\isafoldinvisible}% |
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% |
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\isadeliminvisible |
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% |
|
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\endisadeliminvisible |
|
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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{qed}\isamarkupfalse% |
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% |
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\endisatagvisible |
|
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{\isafoldvisible}% |
|
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% |
|
1323 |
\isadelimvisible |
|
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% |
|
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\endisadelimvisible |
|
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% |
|
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\begin{isamarkuptext}% |
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Similarly, total orders are distributive lattices.% |
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\end{isamarkuptext}% |
|
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\isamarkuptrue% |
|
29566
937baa077df2
Fixed tutorial to compile with new locales; grammar of new locale commands.
ballarin
parents:
27375
diff
changeset
|
1331 |
\ \ \isacommand{sublocale}\isamarkupfalse% |
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\ total{\isacharunderscore}order\ {\isasymsubseteq}\ distrib{\isacharunderscore}lattice\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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\ unfold{\isacharunderscore}locales\isanewline |
|
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\ \ \ \ \isacommand{fix}\isamarkupfalse% |
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\ x\ y\ z\isanewline |
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\ \ \ \ \isacommand{show}\isamarkupfalse% |
|
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\ {\isachardoublequoteopen}x\ {\isasymsqinter}\ {\isacharparenleft}y\ {\isasymsqunion}\ z{\isacharparenright}\ {\isacharequal}\ x\ {\isasymsqinter}\ y\ {\isasymsqunion}\ x\ {\isasymsqinter}\ z{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{is}\ {\isachardoublequoteopen}{\isacharquery}l\ {\isacharequal}\ {\isacharquery}r{\isachardoublequoteclose}{\isacharparenright}% |
|
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\begin{isamarkuptxt}% |
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Jacobson I, p.\ 462% |
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\end{isamarkuptxt}% |
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\isamarkuptrue% |
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\ \ \ \ \isacommand{proof}\isamarkupfalse% |
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\ {\isacharminus}\isanewline |
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\ \ \ \ \ \ \isacommand{{\isacharbraceleft}}\isamarkupfalse% |
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\ \isacommand{assume}\isamarkupfalse% |
|
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\ c{\isacharcolon}\ {\isachardoublequoteopen}y\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\ {\isachardoublequoteopen}z\ {\isasymsqsubseteq}\ x{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
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\ c\ \isacommand{have}\isamarkupfalse% |
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\ {\isachardoublequoteopen}{\isacharquery}l\ {\isacharequal}\ y\ {\isasymsqunion}\ z{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse% |
|
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\ {\isacharparenleft}metis\ c\ join{\isacharunderscore}connection{\isadigit{2}}\ join{\isacharunderscore}related{\isadigit{2}}\ meet{\isacharunderscore}related{\isadigit{2}}\ total{\isacharparenright}\isanewline |
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\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse% |
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\ \isacommand{from}\isamarkupfalse% |
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\ c\ \isacommand{have}\isamarkupfalse% |
|
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\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharquery}r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% |
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\ {\isacharparenleft}metis\ meet{\isacharunderscore}related{\isadigit{2}}{\isacharparenright}\isanewline |
|
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\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse% |
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\ \isacommand{have}\isamarkupfalse% |
|
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\ {\isachardoublequoteopen}{\isacharquery}l\ {\isacharequal}\ {\isacharquery}r{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse% |
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\ \isacommand{{\isacharbraceright}}\isamarkupfalse% |
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\isanewline |
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\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse% |
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\isanewline |
|
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\ \ \ \ \ \ \isacommand{{\isacharbraceleft}}\isamarkupfalse% |
|
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\ \isacommand{assume}\isamarkupfalse% |
|
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\ c{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymsqsubseteq}\ y\ {\isasymor}\ x\ {\isasymsqsubseteq}\ z{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse% |
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\ c\ \isacommand{have}\isamarkupfalse% |
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\ {\isachardoublequoteopen}{\isacharquery}l\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline |
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\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse% |
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\ {\isacharparenleft}metis\ join{\isacharunderscore}connection{\isadigit{2}}\ join{\isacharunderscore}related{\isadigit{2}}\ meet{\isacharunderscore}connection\ total\ trans{\isacharparenright}\isanewline |
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\ \ \ \ \ \ \ \ \isacommand{also}\isamarkupfalse% |
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\ \isacommand{from}\isamarkupfalse% |
|
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\ c\ \isacommand{have}\isamarkupfalse% |
|
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\ {\isachardoublequoteopen}{\isachardot}{\isachardot}{\isachardot}\ {\isacharequal}\ {\isacharquery}r{\isachardoublequoteclose}\isanewline |
|
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\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse% |
|
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\ {\isacharparenleft}metis\ join{\isacharunderscore}commute\ join{\isacharunderscore}related{\isadigit{2}}\ meet{\isacharunderscore}connection\ meet{\isacharunderscore}related{\isadigit{2}}\ total{\isacharparenright}\isanewline |
|
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\ \ \ \ \ \ \ \ \isacommand{finally}\isamarkupfalse% |
|
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\ \isacommand{have}\isamarkupfalse% |
|
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\ {\isachardoublequoteopen}{\isacharquery}l\ {\isacharequal}\ {\isacharquery}r{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse% |
|
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\ \isacommand{{\isacharbraceright}}\isamarkupfalse% |
|
1389 |
\isanewline |
|
1390 |
\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse% |
|
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\ \isacommand{note}\isamarkupfalse% |
|
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\ total\isanewline |
|
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\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse% |
|
1394 |
\ \isacommand{show}\isamarkupfalse% |
|
1395 |
\ {\isacharquery}thesis\ \isacommand{by}\isamarkupfalse% |
|
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\ blast\isanewline |
|
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\ \ \ \ \isacommand{qed}\isamarkupfalse% |
|
1398 |
\isanewline |
|
1399 |
\ \ \isacommand{qed}\isamarkupfalse% |
|
1400 |
% |
|
1401 |
\endisatagproof |
|
1402 |
{\isafoldproof}% |
|
1403 |
% |
|
1404 |
\isadelimproof |
|
1405 |
% |
|
1406 |
\endisadelimproof |
|
1407 |
% |
|
1408 |
\begin{isamarkuptext}% |
|
1409 |
The locale hierarchy is now as shown in Figure~\ref{fig:lattices}(c).% |
|
1410 |
\end{isamarkuptext}% |
|
1411 |
\isamarkuptrue% |
|
1412 |
% |
|
1413 |
\isadelimtheory |
|
1414 |
% |
|
1415 |
\endisadelimtheory |
|
1416 |
% |
|
1417 |
\isatagtheory |
|
27080 | 1418 |
\isacommand{end}\isamarkupfalse% |
27063 | 1419 |
% |
1420 |
\endisatagtheory |
|
1421 |
{\isafoldtheory}% |
|
1422 |
% |
|
1423 |
\isadelimtheory |
|
1424 |
% |
|
1425 |
\endisadelimtheory |
|
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\isanewline |
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\end{isabellebody}% |
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%%% Local Variables: |
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%%% mode: latex |
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%%% TeX-master: "root" |
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%%% End: |