In interpretation: equations are not propagated through the hierarchy automatically.

--- a/doc-src/Locales/Locales/Examples3.thy	Sun Mar 29 17:22:17 2009 +0200
+++ b/doc-src/Locales/Locales/Examples3.thy	Sun Mar 29 17:25:06 2009 +0200
@@ -15,7 +15,7 @@
   interpretation.  The third revision of the example illustrates this.  *}
 
 interpretation %visible nat: partial_order "op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"
-  where "partial_order.less op \<le> (x::nat) y = (x < y)"
+  where nat_less_eq: "partial_order.less op \<le> (x::nat) y = (x < y)"
 proof -
   show "partial_order (op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool)"
     by unfold_locales auto
@@ -49,10 +49,11 @@
   elaborate interpretation proof.  *}
 
 interpretation %visible nat: lattice "op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"
-  where "lattice.meet op \<le> (x::nat) y = min x y"
-    and "lattice.join op \<le> (x::nat) y = max x y"
+  where "partial_order.less op \<le> (x::nat) y = (x < y)"
+    and nat_meet_eq: "lattice.meet op \<le> (x::nat) y = min x y"
+    and nat_join_eq: "lattice.join op \<le> (x::nat) y = max x y"
 proof -
-  show "lattice (op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool)"
+  show lattice: "lattice (op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool)"
     txt {* We have already shown that this is a partial order, *}
     apply unfold_locales
     txt {* hence only the lattice axioms remain to be shown: @{subgoals
@@ -61,20 +62,41 @@
     txt {* the goals become @{subgoals [display]} which can be solved
       by Presburger arithmetic. *}
     by arith+
-  txt {* In order to show the equations, we put ourselves in a
+  txt {* For the first of the equations, we refer to the theorem
+  generated in the previous interpretation. *}
+  show "partial_order.less op \<le> (x::nat) y = (x < y)"
+    by (rule nat_less_eq)
+  txt {* In order to show the remaining equations, we put ourselves in a
     situation where the lattice theorems can be used in a convenient way. *}
-  then interpret nat: lattice "op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool" .
+  from lattice interpret nat: lattice "op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool" .
   show "lattice.meet op \<le> (x::nat) y = min x y"
     by (bestsimp simp: nat.meet_def nat.is_inf_def)
   show "lattice.join op \<le> (x::nat) y = max x y"
     by (bestsimp simp: nat.join_def nat.is_sup_def)
 qed
 
-text {* That the relation @{text \<le>} is a total order completes this
-  sequence of interpretations. *}
+text {* The interpretation that the relation @{text \<le>} is a total
+  order follows next. *}
 
 interpretation %visible nat: total_order "op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"
-  by unfold_locales arith
+  where "partial_order.less op \<le> (x::nat) y = (x < y)"
+    and "lattice.meet op \<le> (x::nat) y = min x y"
+    and "lattice.join op \<le> (x::nat) y = max x y"
+proof -
+  show "total_order (op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool)" by unfold_locales arith
+qed (rule nat_less_eq nat_meet_eq nat_join_eq)+
+
+text {* Note that since the locale hierarchy reflects that total
+  orders are distributive lattices, an explicit interpretation of
+  distributive lattices for the order relation on natural numbers is
+  only necessary for mapping the definitions to the right operators on
+  @{typ nat}. *}
+
+interpretation %visible nat: distrib_lattice "op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"
+  where "partial_order.less op \<le> (x::nat) y = (x < y)"
+    and "lattice.meet op \<le> (x::nat) y = min x y"
+    and "lattice.join op \<le> (x::nat) y = max x y"
+  by unfold_locales [1] (rule nat_less_eq nat_meet_eq nat_join_eq)+
 
 text {* Theorems that are available in the theory at this point are shown in
   Table~\ref{tab:nat-lattice}.
@@ -98,29 +120,6 @@
 \caption{Interpreted theorems for @{text \<le>} on the natural numbers.}
 \label{tab:nat-lattice}
 \end{table}
-
-  Note that since the locale hierarchy reflects that total orders are
-  distributive lattices, an explicit interpretation of distributive
-  lattices for the order relation on natural numbers is not neccessary.
-
-  Why not push this idea further and just give the last interpretation
-  as a single interpretation instead of the sequence of three?  The
-  reasons for this are twofold:
-\begin{itemize}
-\item
-  Often it is easier to work in an incremental fashion, because later
-  interpretations require theorems provided by earlier
-  interpretations.
-\item
-  Assume that a definition is made in some locale $l_1$, and that $l_2$
-  imports $l_1$.  Let an equation for the definition be
-  proved in an interpretation of $l_2$.  The equation will be unfolded
-  in interpretations of theorems added to $l_2$ or below in the import
-  hierarchy, but not for theorems added above $l_2$.
-  Hence, an equation interpreting a definition should always be given in
-  an interpretation of the locale where the definition is made, not in
-  an interpretation of a locale further down the hierarchy.
-\end{itemize}
   *}
 
 
@@ -131,7 +130,7 @@
   incrementally. *}
 
 interpretation nat_dvd: partial_order "op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"
-  where "partial_order.less op dvd (x::nat) y = (x dvd y \<and> x \<noteq> y)"
+  where nat_dvd_less_eq: "partial_order.less op dvd (x::nat) y = (x dvd y \<and> x \<noteq> y)"
 proof -
   show "partial_order (op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool)"
     by unfold_locales (auto simp: dvd_def)
@@ -147,10 +146,9 @@
   x \<noteq> y"}.  *}
 
 interpretation nat_dvd: lattice "op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"
-  where nat_dvd_meet_eq:
-      "lattice.meet op dvd = gcd"
-    and nat_dvd_join_eq:
-      "lattice.join op dvd = lcm"
+  where "partial_order.less op dvd (x::nat) y = (x dvd y \<and> x \<noteq> y)"
+    and nat_dvd_meet_eq: "lattice.meet op dvd = gcd"
+    and nat_dvd_join_eq: "lattice.join op dvd = lcm"
 proof -
   show "lattice (op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool)"
     apply unfold_locales
@@ -161,6 +159,8 @@
     apply (auto intro: lcm_dvd1 lcm_dvd2 lcm_least)
     done
   then interpret nat_dvd: lattice "op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool" .
+  show "partial_order.less op dvd (x::nat) y = (x dvd y \<and> x \<noteq> y)"
+    by (rule nat_dvd_less_eq)
   show "lattice.meet op dvd = gcd"
     apply (auto simp add: expand_fun_eq)
     apply (unfold nat_dvd.meet_def)
@@ -201,11 +201,18 @@
 
 interpretation %visible nat_dvd:
   distrib_lattice "op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"
-  apply unfold_locales
-  txt {* @{subgoals [display]} *}
-  apply (unfold nat_dvd_meet_eq nat_dvd_join_eq)
-  txt {* @{subgoals [display]} *}
-  apply (rule gcd_lcm_distr) done
+  where "partial_order.less op dvd (x::nat) y = (x dvd y \<and> x \<noteq> y)"
+    and "lattice.meet op dvd = gcd"
+    and "lattice.join op dvd = lcm"
+proof -
+  show "distrib_lattice (op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool)"
+    apply unfold_locales
+    txt {* @{subgoals [display]} *}
+    apply (unfold nat_dvd_meet_eq nat_dvd_join_eq)
+    txt {* @{subgoals [display]} *}
+    apply (rule gcd_lcm_distr)
+    done
+qed (rule nat_dvd_less_eq nat_dvd_meet_eq nat_dvd_join_eq)+
 
 
 text {* Theorems that are available in the theory after these

--- a/doc-src/Locales/Locales/document/Examples3.tex	Sun Mar 29 17:22:17 2009 +0200
+++ b/doc-src/Locales/Locales/document/Examples3.tex	Sun Mar 29 17:25:06 2009 +0200
@@ -42,7 +42,7 @@
 \isatagvisible
 \isacommand{interpretation}\isamarkupfalse%
 \ nat{\isacharcolon}\ partial{\isacharunderscore}order\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{where}\ nat{\isacharunderscore}less{\isacharunderscore}eq{\isacharcolon}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
 \isacommand{proof}\isamarkupfalse%
 \ {\isacharminus}\isanewline
 \ \ \isacommand{show}\isamarkupfalse%
@@ -105,12 +105,13 @@
 \isatagvisible
 \isacommand{interpretation}\isamarkupfalse%
 \ nat{\isacharcolon}\ lattice\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ min\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ max\ x\ y{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ nat{\isacharunderscore}meet{\isacharunderscore}eq{\isacharcolon}\ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ min\ x\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ nat{\isacharunderscore}join{\isacharunderscore}eq{\isacharcolon}\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ max\ x\ y{\isachardoublequoteclose}\isanewline
 \isacommand{proof}\isamarkupfalse%
 \ {\isacharminus}\isanewline
 \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}lattice\ {\isacharparenleft}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}%
+\ lattice{\isacharcolon}\ {\isachardoublequoteopen}lattice\ {\isacharparenleft}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}%
 \begin{isamarkuptxt}%
 We have already shown that this is a partial order,%
 \end{isamarkuptxt}%
@@ -137,12 +138,21 @@
 \ \ \ \ \isacommand{by}\isamarkupfalse%
 \ arith{\isacharplus}%
 \begin{isamarkuptxt}%
-In order to show the equations, we put ourselves in a
+For the first of the equations, we refer to the theorem
+  generated in the previous interpretation.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ nat{\isacharunderscore}less{\isacharunderscore}eq{\isacharparenright}%
+\begin{isamarkuptxt}%
+In order to show the remaining equations, we put ourselves in a
     situation where the lattice theorems can be used in a convenient way.%
 \end{isamarkuptxt}%
 \isamarkuptrue%
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{interpret}\isamarkupfalse%
+\ \ \isacommand{from}\isamarkupfalse%
+\ lattice\ \isacommand{interpret}\isamarkupfalse%
 \ nat{\isacharcolon}\ lattice\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
 \isanewline
 \ \ \isacommand{show}\isamarkupfalse%
@@ -163,8 +173,8 @@
 \endisadelimvisible
 %
 \begin{isamarkuptext}%
-That the relation \isa{{\isasymle}} is a total order completes this
-  sequence of interpretations.%
+The interpretation that the relation \isa{{\isasymle}} is a total
+  order follows next.%
 \end{isamarkuptext}%
 \isamarkuptrue%
 %
@@ -175,8 +185,44 @@
 \isatagvisible
 \isacommand{interpretation}\isamarkupfalse%
 \ nat{\isacharcolon}\ total{\isacharunderscore}order\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ min\ x\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ max\ x\ y{\isachardoublequoteclose}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}total{\isacharunderscore}order\ {\isacharparenleft}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
+\ unfold{\isacharunderscore}locales\ arith\isanewline
+\isacommand{qed}\isamarkupfalse%
+\ {\isacharparenleft}rule\ nat{\isacharunderscore}less{\isacharunderscore}eq\ nat{\isacharunderscore}meet{\isacharunderscore}eq\ nat{\isacharunderscore}join{\isacharunderscore}eq{\isacharparenright}{\isacharplus}%
+\endisatagvisible
+{\isafoldvisible}%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\begin{isamarkuptext}%
+Note that since the locale hierarchy reflects that total
+  orders are distributive lattices, an explicit interpretation of
+  distributive lattices for the order relation on natural numbers is
+  only necessary for mapping the definitions to the right operators on
+  \isa{nat}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimvisible
+%
+\endisadelimvisible
+%
+\isatagvisible
+\isacommand{interpretation}\isamarkupfalse%
+\ nat{\isacharcolon}\ distrib{\isacharunderscore}lattice\ {\isachardoublequoteopen}op\ {\isasymle}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharless}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ min\ x\ y{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ max\ x\ y{\isachardoublequoteclose}\isanewline
 \ \ \isacommand{by}\isamarkupfalse%
-\ unfold{\isacharunderscore}locales\ arith%
+\ unfold{\isacharunderscore}locales\ {\isacharbrackleft}{\isadigit{1}}{\isacharbrackright}\ {\isacharparenleft}rule\ nat{\isacharunderscore}less{\isacharunderscore}eq\ nat{\isacharunderscore}meet{\isacharunderscore}eq\ nat{\isacharunderscore}join{\isacharunderscore}eq{\isacharparenright}{\isacharplus}%
 \endisatagvisible
 {\isafoldvisible}%
 %
@@ -198,38 +244,15 @@
   \isa{nat{\isachardot}meet{\isacharunderscore}left} from locale \isa{lattice}: \\
   \quad \isa{min\ {\isacharquery}x\ {\isacharquery}y\ {\isasymle}\ {\isacharquery}x} \\
   \isa{nat{\isachardot}join{\isacharunderscore}distr} from locale \isa{distrib{\isacharunderscore}lattice}: \\
-  \quad \isa{lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharquery}x\ {\isacharparenleft}lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ lattice{\isachardot}meet\ op\ {\isasymle}\ {\isacharparenleft}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharparenleft}lattice{\isachardot}join\ op\ {\isasymle}\ {\isacharquery}x\ {\isacharquery}z{\isacharparenright}} \\
+  \quad \isa{max\ {\isacharquery}x\ {\isacharparenleft}min\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ min\ {\isacharparenleft}max\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharparenleft}max\ {\isacharquery}x\ {\isacharquery}z{\isacharparenright}} \\
   \isa{nat{\isachardot}less{\isacharunderscore}total} from locale \isa{total{\isacharunderscore}order}: \\
-  \quad \isa{partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharquery}x\ {\isacharquery}y\ {\isasymor}\ {\isacharquery}x\ {\isacharequal}\ {\isacharquery}y\ {\isasymor}\ partial{\isacharunderscore}order{\isachardot}less\ op\ {\isasymle}\ {\isacharquery}y\ {\isacharquery}x}
+  \quad \isa{{\isacharquery}x\ {\isacharless}\ {\isacharquery}y\ {\isasymor}\ {\isacharquery}x\ {\isacharequal}\ {\isacharquery}y\ {\isasymor}\ {\isacharquery}y\ {\isacharless}\ {\isacharquery}x}
 \end{tabular}
 \end{center}
 \hrule
 \caption{Interpreted theorems for \isa{{\isasymle}} on the natural numbers.}
 \label{tab:nat-lattice}
-\end{table}
-
-  Note that since the locale hierarchy reflects that total orders are
-  distributive lattices, an explicit interpretation of distributive
-  lattices for the order relation on natural numbers is not neccessary.
-
-  Why not push this idea further and just give the last interpretation
-  as a single interpretation instead of the sequence of three?  The
-  reasons for this are twofold:
-\begin{itemize}
-\item
-  Often it is easier to work in an incremental fashion, because later
-  interpretations require theorems provided by earlier
-  interpretations.
-\item
-  Assume that a definition is made in some locale $l_1$, and that $l_2$
-  imports $l_1$.  Let an equation for the definition be
-  proved in an interpretation of $l_2$.  The equation will be unfolded
-  in interpretations of theorems added to $l_2$ or below in the import
-  hierarchy, but not for theorems added above $l_2$.
-  Hence, an equation interpreting a definition should always be given in
-  an interpretation of the locale where the definition is made, not in
-  an interpretation of a locale further down the hierarchy.
-\end{itemize}%
+\end{table}%
 \end{isamarkuptext}%
 \isamarkuptrue%
 %
@@ -245,7 +268,7 @@
 \isamarkuptrue%
 \isacommand{interpretation}\isamarkupfalse%
 \ nat{\isacharunderscore}dvd{\isacharcolon}\ partial{\isacharunderscore}order\ {\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ dvd\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{where}\ nat{\isacharunderscore}dvd{\isacharunderscore}less{\isacharunderscore}eq{\isacharcolon}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ dvd\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
 %
 \isadelimproof
 %
@@ -286,10 +309,9 @@
 \isamarkuptrue%
 \isacommand{interpretation}\isamarkupfalse%
 \ nat{\isacharunderscore}dvd{\isacharcolon}\ lattice\ {\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\ nat{\isacharunderscore}dvd{\isacharunderscore}meet{\isacharunderscore}eq{\isacharcolon}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ dvd\ {\isacharequal}\ gcd{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ nat{\isacharunderscore}dvd{\isacharunderscore}join{\isacharunderscore}eq{\isacharcolon}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ dvd\ {\isacharequal}\ lcm{\isachardoublequoteclose}\isanewline
+\ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ dvd\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ nat{\isacharunderscore}dvd{\isacharunderscore}meet{\isacharunderscore}eq{\isacharcolon}\ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ dvd\ {\isacharequal}\ gcd{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ nat{\isacharunderscore}dvd{\isacharunderscore}join{\isacharunderscore}eq{\isacharcolon}\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ dvd\ {\isacharequal}\ lcm{\isachardoublequoteclose}\isanewline
 %
 \isadelimproof
 %
@@ -319,6 +341,10 @@
 \ nat{\isacharunderscore}dvd{\isacharcolon}\ lattice\ {\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
 \isanewline
 \ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ dvd\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+\ {\isacharparenleft}rule\ nat{\isacharunderscore}dvd{\isacharunderscore}less{\isacharunderscore}eq{\isacharparenright}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
 \ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ dvd\ {\isacharequal}\ gcd{\isachardoublequoteclose}\isanewline
 \ \ \ \ \isacommand{apply}\isamarkupfalse%
 \ {\isacharparenleft}auto\ simp\ add{\isacharcolon}\ expand{\isacharunderscore}fun{\isacharunderscore}eq{\isacharparenright}\isanewline
@@ -383,7 +409,14 @@
 \isacommand{interpretation}\isamarkupfalse%
 \ nat{\isacharunderscore}dvd{\isacharcolon}\isanewline
 \ \ distrib{\isacharunderscore}lattice\ {\isachardoublequoteopen}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
+\ \ \isakeyword{where}\ {\isachardoublequoteopen}partial{\isacharunderscore}order{\isachardot}less\ op\ dvd\ {\isacharparenleft}x{\isacharcolon}{\isacharcolon}nat{\isacharparenright}\ y\ {\isacharequal}\ {\isacharparenleft}x\ dvd\ y\ {\isasymand}\ x\ {\isasymnoteq}\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}lattice{\isachardot}meet\ op\ dvd\ {\isacharequal}\ gcd{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isakeyword{and}\ {\isachardoublequoteopen}lattice{\isachardot}join\ op\ dvd\ {\isacharequal}\ lcm{\isachardoublequoteclose}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\ {\isacharminus}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isachardoublequoteopen}distrib{\isacharunderscore}lattice\ {\isacharparenleft}op\ dvd\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
 \ unfold{\isacharunderscore}locales%
 \begin{isamarkuptxt}%
 \begin{isabelle}%
@@ -394,7 +427,7 @@
 \end{isabelle}%
 \end{isamarkuptxt}%
 \isamarkuptrue%
-\ \ \isacommand{apply}\isamarkupfalse%
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
 \ {\isacharparenleft}unfold\ nat{\isacharunderscore}dvd{\isacharunderscore}meet{\isacharunderscore}eq\ nat{\isacharunderscore}dvd{\isacharunderscore}join{\isacharunderscore}eq{\isacharparenright}%
 \begin{isamarkuptxt}%
 \begin{isabelle}%
@@ -402,9 +435,12 @@
 \end{isabelle}%
 \end{isamarkuptxt}%
 \isamarkuptrue%
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}rule\ gcd{\isacharunderscore}lcm{\isacharunderscore}distr{\isacharparenright}\ \isacommand{done}\isamarkupfalse%
-%
+\ \ \ \ \isacommand{apply}\isamarkupfalse%
+\ {\isacharparenleft}rule\ gcd{\isacharunderscore}lcm{\isacharunderscore}distr{\isacharparenright}\isanewline
+\ \ \ \ \isacommand{done}\isamarkupfalse%
+\isanewline
+\isacommand{qed}\isamarkupfalse%
+\ {\isacharparenleft}rule\ nat{\isacharunderscore}dvd{\isacharunderscore}less{\isacharunderscore}eq\ nat{\isacharunderscore}dvd{\isacharunderscore}meet{\isacharunderscore}eq\ nat{\isacharunderscore}dvd{\isacharunderscore}join{\isacharunderscore}eq{\isacharparenright}{\isacharplus}%
 \endisatagvisible
 {\isafoldvisible}%
 %
@@ -426,7 +462,7 @@
   \isa{nat{\isacharunderscore}dvd{\isachardot}meet{\isacharunderscore}left} from locale \isa{lattice}: \\
   \quad \isa{gcd\ {\isacharquery}x\ {\isacharquery}y\ dvd\ {\isacharquery}x} \\
   \isa{nat{\isacharunderscore}dvd{\isachardot}join{\isacharunderscore}distr} from locale \isa{distrib{\isacharunderscore}lattice}: \\
-  \quad \isa{lattice{\isachardot}join\ op\ dvd\ {\isacharquery}x\ {\isacharparenleft}lattice{\isachardot}meet\ op\ dvd\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ lattice{\isachardot}meet\ op\ dvd\ {\isacharparenleft}lattice{\isachardot}join\ op\ dvd\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharparenleft}lattice{\isachardot}join\ op\ dvd\ {\isacharquery}x\ {\isacharquery}z{\isacharparenright}} \\
+  \quad \isa{lcm\ {\isacharquery}x\ {\isacharparenleft}gcd\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ gcd\ {\isacharparenleft}lcm\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharparenleft}lcm\ {\isacharquery}x\ {\isacharquery}z{\isacharparenright}} \\
 \end{tabular}
 \end{center}
 \hrule