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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) + (simp add: subgroup.subset subgroup_self) + 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