author  ballarin 
Sat, 17 Oct 2009 22:58:18 +0200  
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parent 32982  40810d98f4c9 
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permissions  rwrr 
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theory Examples3 
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imports Examples 
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begin 
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text {* \vspace{5ex} *} 
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subsection {* Third Version: Local Interpretation 

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\label{sec:localinterpretation} *} 

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text {* In the above example, the fact that @{term "op \<le>"} is a partial 
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order for the integers was used in the second goal to 
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discharge the premise in the definition of @{text "op \<sqsubset>"}. In 
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general, proofs of the equations not only may involve definitions 

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from the interpreted locale but arbitrarily complex arguments in the 
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context of the locale. Therefore is would be convenient to have the 
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interpreted locale conclusions temporary available in the proof. 

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This can be achieved by a locale interpretation in the proof body. 

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The command for local interpretations is \isakeyword{interpret}. We 

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repeat the example from the previous section to illustrate this. *} 

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interpretation %visible int: partial_order "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool" 
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where "partial_order.less op \<le> (x::int) y = (x < y)" 

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proof  
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show "partial_order (op \<le> :: int \<Rightarrow> int \<Rightarrow> bool)" 
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by unfold_locales auto 
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then interpret int: partial_order "op \<le> :: [int, int] \<Rightarrow> bool" . 
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show "partial_order.less op \<le> (x::int) y = (x < y)" 

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unfolding int.less_def by auto 

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qed 
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text {* The inner interpretation is immediate from the preceding fact 
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and proved by assumption (Isar short hand ``.''). It enriches the 

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local proof context by the theorems 

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also obtained in the interpretation from Section~\ref{sec:pofirst}, 
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and @{text int.less_def} may directly be used to unfold the 
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definition. Theorems from the local interpretation disappear after 
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leaving the proof context  that is, after the succeeding 
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\isakeyword{next} or \isakeyword{qed} statement. *} 
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subsection {* Further Interpretations *} 

40 

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text {* Further interpretations are necessary for 
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the other locales. In @{text lattice} the operations~@{text \<sqinter>} 
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and~@{text \<squnion>} are substituted by @{term "min :: int \<Rightarrow> int \<Rightarrow> int"} 

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and @{term "max :: int \<Rightarrow> int \<Rightarrow> int"}. The entire proof for the 

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interpretation is reproduced to give an example of a more 
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elaborate interpretation proof. Note that the equations are named 
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so they can be used in a later example. *} 

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interpretation %visible int: lattice "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool" 
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where int_min_eq: "lattice.meet op \<le> (x::int) y = min x y" 
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and int_max_eq: "lattice.join op \<le> (x::int) y = max x y" 

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proof  
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show "lattice (op \<le> :: int \<Rightarrow> int \<Rightarrow> bool)" 
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txt {* \normalsize We have already shown that this is a partial 
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order, *} 

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apply unfold_locales 

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txt {* \normalsize hence only the lattice axioms remain to be 

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shown. 
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@{subgoals [display]} 

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By @{text is_inf} and @{text is_sup}, *} 

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apply (unfold int.is_inf_def int.is_sup_def) 
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txt {* \normalsize the goals are transformed to these 
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statements: 

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@{subgoals [display]} 

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This is Presburger arithmetic, which can be solved by the 

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method @{text arith}. *} 

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by arith+ 
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txt {* \normalsize In order to show the equations, we put ourselves 

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in a situation where the lattice theorems can be used in a 

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convenient way. *} 

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then interpret int: lattice "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool" . 
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show "lattice.meet op \<le> (x::int) y = min x y" 

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by (bestsimp simp: int.meet_def int.is_inf_def) 

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show "lattice.join op \<le> (x::int) y = max x y" 

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by (bestsimp simp: int.join_def int.is_sup_def) 

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qed 
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text {* Next follows that @{text "op \<le>"} is a total order, again for 
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the integers. *} 
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interpretation %visible int: total_order "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool" 
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by unfold_locales arith 
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text {* Theorems that are available in the theory at this point are shown in 

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Table~\ref{tab:intlattice}. Two points are worth noting: 
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\begin{table} 

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\hrule 

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\vspace{2ex} 

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\begin{center} 

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\begin{tabular}{l} 

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@{thm [source] int.less_def} from locale @{text partial_order}: \\ 
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\quad @{thm int.less_def} \\ 

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@{thm [source] int.meet_left} from locale @{text lattice}: \\ 

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\quad @{thm int.meet_left} \\ 

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@{thm [source] int.join_distr} from locale @{text distrib_lattice}: \\ 

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\quad @{thm int.join_distr} \\ 

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@{thm [source] int.less_total} from locale @{text total_order}: \\ 

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\quad @{thm int.less_total} 

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\end{tabular} 
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\end{center} 

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\hrule 

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\caption{Interpreted theorems for~@{text \<le>} on the integers.} 
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\label{tab:intlattice} 
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\end{table} 
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\begin{itemize} 
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\item 
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Locale @{text distrib_lattice} was also interpreted. Since the 
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locale hierarchy reflects that total orders are distributive 
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lattices, the interpretation of the latter was inserted 
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automatically with the interpretation of the former. In general, 

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interpretation traverses the locale hierarchy upwards and interprets 
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all encountered locales, regardless whether imported or proved via 

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the \isakeyword{sublocale} command. Existing interpretations are 

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skipped avoiding duplicate work. 

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\item 
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The predicate @{term "op <"} appears in theorem @{thm [source] 
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int.less_total} 

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although an equation for the replacement of @{text "op \<sqsubset>"} was only 
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given in the interpretation of @{text partial_order}. The 
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interpretation equations are pushed downwards the hierarchy for 

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related interpretations  that is, for interpretations that share 

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the instances of parameters they have in common. 

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\end{itemize} 
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*} 
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text {* The interpretations for a locale $n$ within the current 
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theory may be inspected with \isakeyword{print\_interps}~$n$. This 

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prints the list of instances of $n$, for which interpretations exist. 

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For example, \isakeyword{print\_interps} @{term partial_order} 

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outputs the following: 

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\begin{small} 
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\begin{alltt} 
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int! : partial_order "op \(\le\)" 
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\end{alltt} 
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\end{small} 
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Of course, there is only one interpretation. 

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The interpretation qualifier on the left is decorated with an 

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exclamation point. This means that it is mandatory. Qualifiers 

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can either be \emph{mandatory} or \emph{optional}, designated by 
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``!'' or ``?'' respectively. Mandatory qualifiers must occur in a 

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name reference while optional ones need not. Mandatory qualifiers 

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prevent accidental hiding of names, while optional qualifiers can be 
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more convenient to use. For \isakeyword{interpretation}, the 
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default is ``!''. 
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*} 
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section {* Locale Expressions \label{sec:expressions} *} 
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text {* 

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A map~@{term \<phi>} between partial orders~@{text \<sqsubseteq>} and~@{text \<preceq>} 
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is called order preserving if @{text "x \<sqsubseteq> y"} implies @{text "\<phi> x \<preceq> 
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\<phi> y"}. This situation is more complex than those encountered so 

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far: it involves two partial orders, and it is desirable to use the 

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existing locale for both. 

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A locale for order preserving maps requires three parameters: @{text 
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le}~(\isakeyword{infixl}~@{text \<sqsubseteq>}) and @{text 
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le'}~(\isakeyword{infixl}~@{text \<preceq>}) for the orders and~@{text \<phi>} 

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for the map. 

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In order to reuse the existing locale for partial orders, which has 

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the single parameter~@{text le}, it must be imported twice, once 
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mapping its parameter to~@{text le} from the new locale and once 

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to~@{text le'}. This can be achieved with a compound locale 

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expression. 
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In general, a locale expression is a sequence of \emph{locale instances} 
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separated by~``$\textbf{+}$'' and followed by a \isakeyword{for} 

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clause. 
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An instance has the following format: 
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\begin{quote} 
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\textit{qualifier} \textbf{:} \textit{localename} 

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\textit{parameterinstantiation} 

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\end{quote} 

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We have already seen locale instances as arguments to 

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\isakeyword{interpretation} in Section~\ref{sec:interpretation}. 

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As before, the qualifier serves to disambiguate names from 
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different instances of the same locale. While in 

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\isakeyword{interpretation} qualifiers default to mandatory, in 

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import and in the \isakeyword{sublocale} command, they default to 

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optional. 

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Since the parameters~@{text le} and~@{text le'} are to be partial 
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orders, our locale for order preserving maps will import the these 
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instances: 

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\begin{small} 
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\begin{alltt} 
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le: partial_order le 

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le': partial_order le' 

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\end{alltt} 

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\end{small} 
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For matter of convenience we choose to name parameter names and 

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qualifiers alike. This is an arbitrary decision. Technically, qualifiers 

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and parameters are unrelated. 
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Having determined the instances, let us turn to the \isakeyword{for} 

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clause. It serves to declare locale parameters in the same way as 

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the context element \isakeyword{fixes} does. Context elements can 

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only occur after the import section, and therefore the parameters 

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referred to in the instances must be declared in the \isakeyword{for} 
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clause. The \isakeyword{for} clause is also where the syntax of these 

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parameters is declared. 
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Two context elements for the map parameter~@{text \<phi>} and the 
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assumptions that it is order preserving complete the locale 

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declaration. *} 
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locale order_preserving = 

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le: partial_order le + le': partial_order le' 
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for le (infixl "\<sqsubseteq>" 50) and le' (infixl "\<preceq>" 50) + 
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fixes \<phi> 
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assumes hom_le: "x \<sqsubseteq> y \<Longrightarrow> \<phi> x \<preceq> \<phi> y" 
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text (in order_preserving) {* Here are examples of theorems that are 
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available in the locale: 

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\hspace*{1em}@{thm [source] hom_le}: @{thm hom_le} 
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\hspace*{1em}@{thm [source] le.less_le_trans}: @{thm le.less_le_trans} 
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\hspace*{1em}@{thm [source] le'.less_le_trans}: 
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@{thm [display, indent=4] le'.less_le_trans} 

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While there is infix syntax for the strict operation associated to 
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@{term "op \<sqsubseteq>"}, there is none for the strict version of @{term 

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"op \<preceq>"}. The abbreviation @{text less} with its infix syntax is only 

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available for the original instance it was declared for. We may 

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introduce the abbreviation @{text less'} with infix syntax~@{text \<prec>} 
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with the following declaration: *} 

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abbreviation (in order_preserving) 

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less' (infixl "\<prec>" 50) where "less' \<equiv> partial_order.less le'" 

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text (in order_preserving) {* Now the theorem is displayed nicely as 
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@{thm [source] le'.less_le_trans}: 
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@{thm [display, indent=2] le'.less_le_trans} *} 

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text {* There are short notations for locale expressions. These are 
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discussed in the following. *} 

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subsection {* Default Instantiations *} 

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text {* 
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It is possible to omit parameter instantiations. The 

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instantiation then defaults to the name of 

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the parameter itself. For example, the locale expression @{text 

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partial_order} is short for @{text "partial_order le"}, since the 
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locale's single parameter is~@{text le}. We took advantage of this 
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in the \isakeyword{sublocale} declarations of 

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Section~\ref{sec:changingthehierarchy}. *} 
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255 

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subsection {* Implicit Parameters \label{sec:implicitparameters} *} 
257 

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text {* In a locale expression that occurs within a locale 
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declaration, omitted parameters additionally extend the (possibly 

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empty) \isakeyword{for} clause. 

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The \isakeyword{for} clause is a general construct of Isabelle/Isar 
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to mark names occurring in the preceding declaration as ``arbitrary 

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but fixed''. This is necessary for example, if the name is already 

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bound in a surrounding context. In a locale expression, names 

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occurring in parameter instantiations should be bound by a 

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\isakeyword{for} clause whenever these names are not introduced 

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elsewhere in the context  for example, on the left hand side of a 

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\isakeyword{sublocale} declaration. 

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There is an exception to this rule in locale declarations, where the 
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\isakeyword{for} clause serves to declare locale parameters. Here, 
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locale parameters for which no parameter instantiation is given are 
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implicitly added, with their mixfix syntax, at the beginning of the 

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\isakeyword{for} clause. For example, in a locale declaration, the 

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expression @{text partial_order} is short for 

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\begin{small} 
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\begin{alltt} 
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partial_order le \isakeyword{for} le (\isakeyword{infixl} "\(\sqsubseteq\)" 50)\textrm{.} 

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\end{alltt} 

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\end{small} 
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This short hand was used in the locale declarations throughout 

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Section~\ref{sec:import}. 
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*} 

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text{* 

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The following locale declarations provide more examples. A 
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map~@{text \<phi>} is a lattice homomorphism if it preserves meet and 

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join. *} 
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locale lattice_hom = 
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le: lattice + le': lattice le' for le' (infixl "\<preceq>" 50) + 
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fixes \<phi> 
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assumes hom_meet: "\<phi> (x \<sqinter> y) = le'.meet (\<phi> x) (\<phi> y)" 
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and hom_join: "\<phi> (x \<squnion> y) = le'.join (\<phi> x) (\<phi> y)" 

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text {* The parameter instantiation in the first instance of @{term 
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lattice} is omitted. This causes the parameter~@{text le} to be 

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added to the \isakeyword{for} clause, and the locale has 

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parameters~@{text le},~@{text le'} and, of course,~@{text \<phi>}. 

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Before turning to the second example, we complete the locale by 
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providing infix syntax for the meet and join operations of the 
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second lattice. 
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*} 

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context lattice_hom begin 

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abbreviation meet' (infixl "\<sqinter>''" 50) where "meet' \<equiv> le'.meet" 

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abbreviation join' (infixl "\<squnion>''" 50) where "join' \<equiv> le'.join" 

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end 

311 

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text {* The next example makes radical use of the short hand 
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facilities. A homomorphism is an endomorphism if both orders 

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coincide. *} 

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locale lattice_end = lattice_hom _ le 
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text {* The notation~@{text _} enables to omit a parameter in a 
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positional instantiation. The omitted parameter,~@{text le} becomes 

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the parameter of the declared locale and is, in the following 
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position, used to instantiate the second parameter of @{text 
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lattice_hom}. The effect is that of identifying the first in second 
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parameter of the homomorphism locale. *} 

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text {* The inheritance diagram of the situation we have now is shown 

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in Figure~\ref{fig:hom}, where the dashed line depicts an 

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interpretation which is introduced below. Parameter instantiations 
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are indicated by $\sqsubseteq \mapsto \preceq$ etc. By looking at 

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the inheritance diagram it would seem 

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that two identical copies of each of the locales @{text 
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partial_order} and @{text lattice} are imported by @{text 
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lattice_end}. This is not the case! Inheritance paths with 

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identical morphisms are automatically detected and 

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the conclusions of the respective locales appear only once. 
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\begin{figure} 

337 
\hrule \vspace{2ex} 

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\begin{center} 

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\begin{tikzpicture} 

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\node (o) at (0,0) {@{text partial_order}}; 

341 
\node (oh) at (1.5,2) {@{text order_preserving}}; 

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\node (oh1) at (1.5,0.7) {$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$}; 

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\node (oh2) at (0,1.3) {$\scriptscriptstyle \sqsubseteq \mapsto \preceq$}; 

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\node (l) at (1.5,2) {@{text lattice}}; 

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\node (lh) at (0,4) {@{text lattice_hom}}; 

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\node (lh1) at (0,2.7) {$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$}; 

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\node (lh2) at (1.5,3.3) {$\scriptscriptstyle \sqsubseteq \mapsto \preceq$}; 

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\node (le) at (0,6) {@{text lattice_end}}; 

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\node (le1) at (0,4.8) 

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[anchor=west]{$\scriptscriptstyle \sqsubseteq \mapsto \sqsubseteq$}; 

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\node (le2) at (0,5.2) 

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[anchor=west]{$\scriptscriptstyle \preceq \mapsto \sqsubseteq$}; 

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\draw (o)  (l); 

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\draw[dashed] (oh)  (lh); 

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\draw (lh)  (le); 

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\draw (o) .. controls (oh1.south west) .. (oh); 

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\draw (o) .. controls (oh2.north east) .. (oh); 

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\draw (l) .. controls (lh1.south west) .. (lh); 

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\draw (l) .. controls (lh2.north east) .. (lh); 

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\end{tikzpicture} 

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\end{center} 

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\hrule 

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\caption{Hierarchy of Homomorphism Locales.} 

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\label{fig:hom} 

365 
\end{figure} 

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*} 

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text {* It can be shown easily that a lattice homomorphism is order 

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preserving. As the final example of this section, a locale 

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interpretation is used to assert this: *} 
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sublocale lattice_hom \<subseteq> order_preserving proof unfold_locales 
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fix x y 
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assume "x \<sqsubseteq> y" 

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then have "y = (x \<squnion> y)" by (simp add: le.join_connection) 

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then have "\<phi> y = (\<phi> x \<squnion>' \<phi> y)" by (simp add: hom_join [symmetric]) 

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then show "\<phi> x \<preceq> \<phi> y" by (simp add: le'.join_connection) 

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qed 

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text (in lattice_hom) {* 
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Theorems and other declarations  syntax, in particular  from 
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the locale @{text order_preserving} are now active in @{text 
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lattice_hom}, for example 
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@{thm [source] hom_le}: 
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@{thm [display, indent=2] hom_le} 

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This theorem will be useful in the following section. 
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*} 
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section {* Conditional Interpretation *} 
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text {* There are situations where an interpretation is not possible 
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in the general case since the desired property is only valid if 

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certain conditions are fulfilled. Take, for example, the function 

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@{text "\<lambda>i. n * i"} that scales its argument by a constant factor. 

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This function is order preserving (and even a lattice endomorphism) 

397 
with respect to @{term "op \<le>"} provided @{text "n \<ge> 0"}. 

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It is not possible to express this using a global interpretation, 

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because it is in general unspecified whether~@{term n} is 

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nonnegative, but one may make an interpretation in an inner context 

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of a proof where full information is available. 

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This is not fully satisfactory either, since potentially 

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interpretations may be required to make interpretations in many 

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contexts. What is 

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required is an interpretation that depends on the condition  and 

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this can be done with the \isakeyword{sublocale} command. For this 

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purpose, we introduce a locale for the condition. *} 

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locale non_negative = 
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fixes n :: int 

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assumes non_neg: "0 \<le> n" 
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text {* It is again convenient to make the interpretation in an 

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incremental fashion, first for order preserving maps, the for 

416 
lattice endomorphisms. *} 

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sublocale non_negative \<subseteq> 
419 
order_preserving "op \<le>" "op \<le>" "\<lambda>i. n * i" 

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using non_neg by unfold_locales (rule mult_left_mono) 

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text {* While the proof of the previous interpretation 

423 
is straightforward from monotonicity lemmas for~@{term "op *"}, the 

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second proof follows a useful pattern. *} 

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sublocale %visible non_negative \<subseteq> lattice_end "op \<le>" "\<lambda>i. n * i" 
427 
proof (unfold_locales, unfold int_min_eq int_max_eq) 

428 
txt {* \normalsize Unfolding the locale predicates \emph{and} the 

429 
interpretation equations immediately yields two subgoals that 

430 
reflect the core conjecture. 

431 
@{subgoals [display]} 

432 
It is now necessary to show, in the context of @{term 

433 
non_negative}, that multiplication by~@{term n} commutes with 

434 
@{term min} and @{term max}. *} 

435 
qed (auto simp: hom_le) 

32981  436 

32983  437 
text (in order_preserving) {* The lemma @{thm [source] hom_le} 
438 
simplifies a proof that would have otherwise been lengthy and we may 

439 
consider making it a default rule for the simplifier: *} 

32981  440 

32983  441 
lemmas (in order_preserving) hom_le [simp] 
32981  442 

443 

32983  444 
subsection {* Avoiding Infinite Chains of Interpretations 
445 
\label{sec:infinitechains} *} 

446 

447 
text {* Similar situations arise frequently in formalisations of 

448 
abstract algebra where it is desirable to express that certain 

449 
constructions preserve certain properties. For example, polynomials 

450 
over rings are rings, or  an example from the domain where the 

451 
illustrations of this tutorial are taken from  a partial order 

452 
may be obtained for a function space by pointwise lifting of the 

453 
partial order of the codomain. This corresponds to the following 

454 
interpretation: *} 

455 

456 
sublocale %visible partial_order \<subseteq> f: partial_order "\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x" 

457 
oops 

458 

459 
text {* Unfortunately this is a cyclic interpretation that leads to an 

460 
infinite chain, namely 

461 
@{text [display, indent=2] "partial_order \<subseteq> partial_order (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) \<subseteq> 

462 
partial_order (\<lambda>f g. \<forall>x y. f x y \<sqsubseteq> g x y) \<subseteq> \<dots>"} 

463 
and the interpretation is rejected. 

464 

465 
Instead it is necessary to declare a locale that is logically 

466 
equivalent to @{term partial_order} but serves to collect facts 

467 
about functions spaces where the codomain is a partial order, and 

468 
to make the interpretation in its context: *} 

32981  469 

32983  470 
locale fun_partial_order = partial_order 
471 

472 
sublocale fun_partial_order \<subseteq> 

473 
f: partial_order "\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x" 

474 
by unfold_locales (fast,rule,fast,blast intro: trans) 

475 

476 
text {* It is quite common in abstract algebra that such a construction 

477 
maps a hierarchy of algebraic structures (or specifications) to a 

478 
related hierarchy. By means of the same lifting, a function space 

479 
is a lattice if its codomain is a lattice: *} 

480 

481 
locale fun_lattice = fun_partial_order + lattice 

32981  482 

32983  483 
sublocale fun_lattice \<subseteq> f: lattice "\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x" 
484 
proof unfold_locales 

485 
fix f g 

486 
have "partial_order.is_inf (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g (\<lambda>x. f x \<sqinter> g x)" 

487 
apply (rule is_infI) apply rule+ apply (drule spec, assumption)+ done 

488 
then show "\<exists>inf. partial_order.is_inf (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g inf" 

489 
by fast 

490 
next 

491 
fix f g 

492 
have "partial_order.is_sup (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g (\<lambda>x. f x \<squnion> g x)" 

493 
apply (rule is_supI) apply rule+ apply (drule spec, assumption)+ done 

494 
then show "\<exists>sup. partial_order.is_sup (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x) f g sup" 

495 
by fast 

496 
qed 

497 

27063  498 

499 
section {* Further Reading *} 

500 

501 
text {* More information on locales and their interpretation is 

502 
available. For the locale hierarchy of import and interpretation 

32983  503 
dependencies see~\cite{Ballarin2006a}; interpretations in theories 
504 
and proofs are covered in~\cite{Ballarin2006b}. In the latter, I 

27063  505 
show how interpretation in proofs enables to reason about families 
506 
of algebraic structures, which cannot be expressed with locales 

507 
directly. 

508 

32983  509 
Haftmann and Wenzel~\cite{HaftmannWenzel2007} overcome a restriction 
27063  510 
of axiomatic type classes through a combination with locale 
511 
interpretation. The result is a Haskellstyle class system with a 

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512 
facility to generate ML and Haskell code. Classes are sufficient for 
27063  513 
simple specifications with a single type parameter. The locales for 
514 
orders and lattices presented in this tutorial fall into this 

515 
category. Order preserving maps, homomorphisms and vector spaces, 

516 
on the other hand, do not. 

517 

32981  518 
The locales reimplementation for Isabelle 2009 provides, among other 
32983  519 
improvements, a clean integration with Isabelle/Isar's local theory 
520 
mechanisms, which are described in another paper by Haftmann and 

521 
Wenzel~\cite{HaftmannWenzel2009}. 

32981  522 

32983  523 
The original work of Kamm\"uller on locales~\cite{KammullerEtAl1999} 
524 
may be of interest from a historical perspective. My previous 

525 
report on locales and locale expressions~\cite{Ballarin2004a} 

526 
describes a simpler form of expressions than available now and is 

527 
outdated. The mathematical background on orders and lattices is 

528 
taken from Jacobson's textbook on algebra~\cite[Chapter~8]{Jacobson1985}. 

32981  529 

530 
The sources of this tutorial, which include all proofs, are 

531 
available with the Isabelle distribution at 

532 
\url{http://isabelle.in.tum.de}. 

27063  533 
*} 
534 

535 
text {* 

536 
\begin{table} 

537 
\hrule 

538 
\vspace{2ex} 

539 
\begin{center} 

540 
\begin{tabular}{l>$c<$l} 

541 
\multicolumn{3}{l}{Miscellaneous} \\ 

542 

543 
\textit{attrname} & ::= 

544 
& \textit{name} $$ \textit{attribute} $$ 

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\textit{name} \textit{attribute} \\ 
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\textit{qualifier} & ::= 
30751  547 
& \textit{name} [``\textbf{?}'' $$ ``\textbf{!}''] \\[2ex] 
27063  548 

549 
\multicolumn{3}{l}{Context Elements} \\ 

550 

551 
\textit{fixes} & ::= 

552 
& \textit{name} [ ``\textbf{::}'' \textit{type} ] 

553 
[ ``\textbf{(}'' \textbf{structure} ``\textbf{)}'' $$ 

554 
\textit{mixfix} ] \\ 

555 
\begin{comment} 

556 
\textit{constrains} & ::= 

557 
& \textit{name} ``\textbf{::}'' \textit{type} \\ 

558 
\end{comment} 

559 
\textit{assumes} & ::= 

560 
& [ \textit{attrname} ``\textbf{:}'' ] \textit{proposition} \\ 

561 
\begin{comment} 

562 
\textit{defines} & ::= 

563 
& [ \textit{attrname} ``\textbf{:}'' ] \textit{proposition} \\ 

564 
\textit{notes} & ::= 

565 
& [ \textit{attrname} ``\textbf{=}'' ] 

566 
( \textit{qualifiedname} [ \textit{attribute} ] )$^+$ \\ 

567 
\end{comment} 

568 

569 
\textit{element} & ::= 

570 
& \textbf{fixes} \textit{fixes} ( \textbf{and} \textit{fixes} )$^*$ \\ 

571 
\begin{comment} 

572 
&  

573 
& \textbf{constrains} \textit{constrains} 

574 
( \textbf{and} \textit{constrains} )$^*$ \\ 

575 
\end{comment} 

576 
&  

577 
& \textbf{assumes} \textit{assumes} ( \textbf{and} \textit{assumes} )$^*$ \\[2ex] 

578 
%\begin{comment} 

579 
% &  

580 
% & \textbf{defines} \textit{defines} ( \textbf{and} \textit{defines} )$^*$ \\ 

581 
% &  

582 
% & \textbf{notes} \textit{notes} ( \textbf{and} \textit{notes} )$^*$ \\ 

583 
%\end{comment} 

584 

585 
\multicolumn{3}{l}{Locale Expressions} \\ 

586 

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\textit{posinsts} & ::= 
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588 
& ( \textit{term} $$ ``\textbf{\_}'' )$^*$ \\ 
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589 
\textit{namedinsts} & ::= 
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& \textbf{where} \textit{name} ``\textbf{=}'' \textit{term} 
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( \textbf{and} \textit{name} ``\textbf{=}'' \textit{term} )$^*$ \\ 
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592 
\textit{instance} & ::= 
30751  593 
& [ \textit{qualifier} ``\textbf{:}'' ] 
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\textit{qualifiedname} ( \textit{posinsts} $$ \textit{namedinst} ) \\ 
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595 
\textit{expression} & ::= 
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596 
& \textit{instance} ( ``\textbf{+}'' \textit{instance} )$^*$ 
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597 
[ \textbf{for} \textit{fixes} ( \textbf{and} \textit{fixes} )$^*$ ] \\[2ex] 
27063  598 

599 
\multicolumn{3}{l}{Declaration of Locales} \\ 

600 

601 
\textit{locale} & ::= 

602 
& \textit{element}$^+$ \\ 

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603 
&  & \textit{expression} [ ``\textbf{+}'' \textit{element}$^+$ ] \\ 
27063  604 
\textit{toplevel} & ::= 
605 
& \textbf{locale} \textit{name} [ ``\textbf{=}'' 

606 
\textit{locale} ] \\[2ex] 

607 

608 
\multicolumn{3}{l}{Interpretation} \\ 

609 

610 
\textit{equation} & ::= & [ \textit{attrname} ``\textbf{:}'' ] 

611 
\textit{prop} \\ 

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612 
\textit{equations} & ::= & \textbf{where} \textit{equation} ( \textbf{and} 
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613 
\textit{equation} )$^*$ \\ 
27063  614 
\textit{toplevel} & ::= 
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615 
& \textbf{sublocale} \textit{name} ( ``$<$'' $$ 
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616 
``$\subseteq$'' ) \textit{expression} \textit{proof} \\ 
27063  617 
&  
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618 
& \textbf{interpretation} 
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619 
\textit{expression} [ \textit{equations} ] \textit{proof} \\ 
27063  620 
&  
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621 
& \textbf{interpret} 
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622 
\textit{expression} \textit{proof} \\[2ex] 
27063  623 

624 
\multicolumn{3}{l}{Diagnostics} \\ 

625 

626 
\textit{toplevel} & ::= 

32981  627 
& \textbf{print\_locales} \\ 
628 
&  & \textbf{print\_locale} [ ``\textbf{!}'' ] \textit{locale} \\ 

629 
&  & \textbf{print\_interps} \textit{locale} 

27063  630 
\end{tabular} 
631 
\end{center} 

632 
\hrule 

30826  633 
\caption{Syntax of Locale Commands.} 
27063  634 
\label{tab:commands} 
635 
\end{table} 

636 
*} 

637 

32983  638 
text {* \textbf{Revision History.} For the present third revision of 
639 
the tutorial, much of the explanatory text 

640 
was rewritten. Inheritance of interpretation equations is 

641 
available with the forthcoming release of Isabelle, which at the 

642 
time of editing these notes is expected for the end of 2009. 

643 
The second revision accommodates changes introduced by the locales 

644 
reimplementation for Isabelle 2009. Most notably locale expressions 

645 
have been generalised from renaming to instantiation. *} 

32981  646 

27063  647 
text {* \textbf{Acknowledgements.} Alexander Krauss, Tobias Nipkow, 
32981  648 
Randy Pollack, Christian Sternagel and Makarius Wenzel have made 
32983  649 
useful comments on earlier versions of this document. The section 
650 
on conditional interpretation was inspired by a number of email 

651 
enquiries the author received from locale users, and which suggested 

652 
that this use case is important enough to deserve explicit 

653 
explanation. The term \emph{conditional interpretation} is due to 

654 
Larry Paulson. *} 

27063  655 

656 
end 