src/HOL/Algebra/Group.thy
 changeset 21041 60e418260b4d parent 20318 0e0ea63fe768 child 22063 717425609192
--- a/src/HOL/Algebra/Group.thy	Sun Oct 15 12:16:20 2006 +0200
+++ b/src/HOL/Algebra/Group.thy	Mon Oct 16 10:27:54 2006 +0200
@@ -685,7 +685,7 @@
text_raw {* \label{sec:subgroup-lattice} *}

theorem (in group) subgroups_partial_order:
-  "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
+  "partial_order {H. subgroup H G} (op \<subseteq>)"
by (rule partial_order.intro) simp_all

lemma (in group) subgroup_self:
@@ -730,22 +730,23 @@
qed

theorem (in group) subgroups_complete_lattice:
-  "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
-    (is "complete_lattice ?L")
+  "complete_lattice {H. subgroup H G} (op \<subseteq>)"
+    (is "complete_lattice ?car ?le")
proof (rule partial_order.complete_lattice_criterion1)
-  show "partial_order ?L" by (rule subgroups_partial_order)
+  show "partial_order ?car ?le" by (rule subgroups_partial_order)
next
-  have "greatest ?L (carrier G) (carrier ?L)"
-    by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)
-  then show "\<exists>G. greatest ?L G (carrier ?L)" ..
+  have "order_syntax.greatest ?car ?le (carrier G) ?car"
+    by (unfold order_syntax.greatest_def)
+  then show "\<exists>G. order_syntax.greatest ?car ?le G ?car" ..
next
fix A
-  assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
+  assume L: "A \<subseteq> ?car" and non_empty: "A ~= {}"
then have Int_subgroup: "subgroup (\<Inter>A) G"
by (fastsimp intro: subgroups_Inter)
-  have "greatest ?L (\<Inter>A) (Lower ?L A)"
-    (is "greatest ?L ?Int _")
-  proof (rule greatest_LowerI)
+  have "order_syntax.greatest ?car ?le (\<Inter>A) (order_syntax.Lower ?car ?le A)"
+    (is "order_syntax.greatest _ _ ?Int _")
+  proof (rule order_syntax.greatest_LowerI)
fix H
assume H: "H \<in> A"
with L have subgroupH: "subgroup H G" by auto
@@ -754,17 +755,18 @@
from groupH have monoidH: "monoid ?H"
by (rule group.is_monoid)
from H have Int_subset: "?Int \<subseteq> H" by fastsimp
-    then show "le ?L ?Int H" by simp
+    then show "?le ?Int H" by simp
next
fix H
-    assume H: "H \<in> Lower ?L A"
-    with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest)
+    assume H: "H \<in> order_syntax.Lower ?car ?le A"
+    with L Int_subgroup show "?le H ?Int"
+      by (fastsimp simp: order_syntax.Lower_def intro: Inter_greatest)
next
-    show "A \<subseteq> carrier ?L" by (rule L)
+    show "A \<subseteq> ?car" by (rule L)
next
-    show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
+    show "?Int \<in> ?car" by simp (rule Int_subgroup)
qed
-  then show "\<exists>I. greatest ?L I (Lower ?L A)" ..
+  then show "\<exists>I. order_syntax.greatest ?car ?le I (order_syntax.Lower ?car ?le A)" ..
qed

end