src/HOL/Algebra/Group.thy

 Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.
 *)
 
 header {* Groups *}
 
-theory Group = FuncSet:
+theory Group = FuncSet + Lattice:
 
 section {* From Magmas to Groups *}
 
 text {*
   Definitions follow \cite{Jacobson:1985}; with the exception of
   \emph{magma} which, following Bourbaki, is a set together with a
   binary, closed operation.
 *}
 
 subsection {* Definitions *}
-
-(* Object with a carrier set. *)
-
-record 'a partial_object =
-  carrier :: "'a set"
 
 record 'a semigroup = "'a partial_object" +
   mult :: "['a, 'a] => 'a" (infixl "\<otimes>\<index>" 70)
 
 record 'a monoid = "'a semigroup" +

   assumes subset [intro, simp]: "H \<subseteq> carrier G"
     and m_closed [intro, simp]: "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"
 
 declare (in submagma) magma.intro [intro] semigroup.intro [intro]
   semigroup_axioms.intro [intro]
-(*
-alternative definition of submagma
-
-locale submagma = var H + struct G +
-  assumes subset [intro, simp]: "carrier H \<subseteq> carrier G"
-    and m_equal [simp]: "mult H = mult G"
-    and m_closed [intro, simp]:
-      "[| x \<in> carrier H; y \<in> carrier H |] ==> x \<otimes> y \<in> carrier H"
-*)
 
 lemma submagma_imp_subset:
   "submagma H G ==> H \<subseteq> carrier G"
   by (rule submagma.subset)
 

 
 lemma (in comm_group) inv_mult:
   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"
   by (simp add: m_ac inv_mult_group)
 
+subsection {* Lattice of subgroups of a group *}
+
+text_raw {* \label{sec:subgroup-lattice} *}
+
+theorem (in group) subgroups_partial_order:
+  "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
+  by (rule partial_order.intro) simp_all
+
+lemma (in group) subgroup_self:
+  "subgroup (carrier G) G"
+  by (rule subgroupI) auto
+
+lemma (in group) subgroup_imp_group:
+  "subgroup H G ==> group (G(| carrier := H |))"
+  using subgroup.groupI [OF _ group.intro] .
+
+lemma (in group) is_monoid [intro, simp]:
+  "monoid G"
+  by (rule monoid.intro)
+
+lemma (in group) subgroup_inv_equality:
+  "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"
+apply (rule_tac inv_equality [THEN sym])
+  apply (rule group.l_inv [OF subgroup_imp_group, simplified])
+   apply assumption+
+ apply (rule subsetD [OF subgroup.subset])
+  apply assumption+
+apply (rule subsetD [OF subgroup.subset])
+ apply assumption
+apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified])
+  apply assumption+
+done
+
+theorem (in group) subgroups_Inter:
+  assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"
+    and not_empty: "A ~= {}"
+  shows "subgroup (\<Inter>A) G"
+proof (rule subgroupI)
+  from subgr [THEN subgroup.subset] and not_empty
+  show "\<Inter>A \<subseteq> carrier G" by blast
+next
+  from subgr [THEN subgroup.one_closed]
+  show "\<Inter>A ~= {}" by blast
+next
+  fix x assume "x \<in> \<Inter>A"
+  with subgr [THEN subgroup.m_inv_closed]
+  show "inv x \<in> \<Inter>A" by blast
+next
+  fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"
+  with subgr [THEN subgroup.m_closed]
+  show "x \<otimes> y \<in> \<Inter>A" by blast
+qed
+
+theorem (in group) subgroups_complete_lattice:
+  "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
+    (is "complete_lattice ?L")
+proof (rule partial_order.complete_lattice_criterion1)
+  show "partial_order ?L" 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 "EX G. greatest ?L G (carrier ?L)" ..
+next
+  fix A
+  assume L: "A \<subseteq> carrier ?L" 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)
+    fix H
+    assume H: "H \<in> A"
+    with L have subgroupH: "subgroup H G" by auto
+    from subgroupH have submagmaH: "submagma H G" by (rule subgroup.axioms)
+    from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
+      by (rule subgroup_imp_group)
+    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
+  next
+    fix H
+    assume H: "H \<in> Lower ?L A"
+    with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest)
+  next
+    show "A \<subseteq> carrier ?L" by (rule L)
+  next
+    show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
+  qed
+  then show "EX I. greatest ?L I (Lower ?L A)" ..
+qed
+
 end