src/HOL/GroupTheory/Bij.thy

--- a/src/HOL/GroupTheory/Bij.thy	Wed Sep 25 11:23:26 2002 +0200
+++ b/src/HOL/GroupTheory/Bij.thy	Thu Sep 26 10:40:13 2002 +0200
@@ -1,42 +1,131 @@
 (*  Title:      HOL/GroupTheory/Bij
     ID:         $Id$
     Author:     Florian Kammueller, with new proofs by L C Paulson
-    Copyright   1998-2001  University of Cambridge
-
-Bijections of a set and the group of bijections
-	Sigma version with locales
 *)
 
-Bij = Group +
+
+header{*Bijections of a Set, Permutation Groups, Automorphism Groups*} 
+
+theory Bij = Group:
 
 constdefs
   Bij :: "'a set => (('a => 'a)set)" 
-    "Bij S == {f. f \\<in> S \\<rightarrow> S & f`S = S & inj_on f S}"
+    --{*Only extensional functions, since otherwise we get too many.*}
+    "Bij S == extensional S \<inter> {f. f`S = S & inj_on f S}"
+
+   BijGroup ::  "'a set => (('a => 'a) group)"
+   "BijGroup S == (| carrier = Bij S, 
+		     sum  = %g: Bij S. %f: Bij S. compose S g f,
+		     gminus = %f: Bij S. %x: S. (Inv S f) x, 
+		     zero = %x: S. x |)"
+
+
+declare Id_compose [simp] compose_Id [simp]
+
+lemma Bij_imp_extensional: "f \<in> Bij S ==> f \<in> extensional S"
+by (simp add: Bij_def)
+
+lemma Bij_imp_funcset: "f \<in> Bij S ==> f \<in> S -> S"
+by (auto simp add: Bij_def Pi_def)
+
+lemma Bij_imp_apply: "f \<in> Bij S ==> f ` S = S"
+by (simp add: Bij_def)
+
+lemma Bij_imp_inj_on: "f \<in> Bij S ==> inj_on f S"
+by (simp add: Bij_def)
+
+lemma BijI: "[| f \<in> extensional(S); f`S = S; inj_on f S |] ==> f \<in> Bij S"
+by (simp add: Bij_def)
+
 
-constdefs 
-BijGroup ::  "'a set => (('a => 'a) grouptype)"
-"BijGroup S == (| carrier = Bij S, 
-                  bin_op  = %g: Bij S. %f: Bij S. compose S g f,
-                  inverse = %f: Bij S. %x: S. (Inv S f) x, 
-                  unit    = %x: S. x |)"
+subsection{*Bijections Form a Group*}
+
+lemma restrict_Inv_Bij: "f \<in> Bij S ==> (%x:S. (Inv S f) x) \<in> Bij S"
+apply (simp add: Bij_def)
+apply (intro conjI)
+txt{*Proving @{term "restrict (Inv S f) S ` S = S"}*}
+ apply (rule equalityI)
+  apply (force simp add: Inv_mem) --{*first inclusion*}
+ apply (rule subsetI)   --{*second inclusion*}
+ apply (rule_tac x = "f x" in image_eqI)
+  apply (force intro:  simp add: Inv_f_f)
+ apply blast
+txt{*Remaining goal: @{term "inj_on (restrict (Inv S f) S) S"}*}
+apply (rule inj_onI)
+apply (auto elim: Inv_injective)
+done
+
+lemma id_Bij: "(\<lambda>x\<in>S. x) \<in> Bij S "
+apply (rule BijI)
+apply (auto simp add: inj_on_def)
+done
+
+lemma compose_Bij: "[| x \<in> Bij S; y \<in> Bij S|] ==> compose S x y \<in> Bij S"
+apply (rule BijI)
+  apply (simp add: compose_extensional) 
+ apply (blast del: equalityI
+              intro: surj_compose dest: Bij_imp_apply Bij_imp_inj_on)
+apply (blast intro: inj_on_compose dest: Bij_imp_apply Bij_imp_inj_on)
+done
 
-locale bij = 
-  fixes 
-    S :: "'a set"
-    B :: "('a => 'a)set"
-    comp :: "[('a => 'a),('a => 'a)]=>('a => 'a)" (infixr "o''" 80)
-    inv'   :: "('a => 'a)=>('a => 'a)"              
-    e'   :: "('a => 'a)"
-  defines
-    B_def    "B == Bij S"
-    o'_def   "g o' f == compose S g f"
-    inv'_def   "inv' f == Inv S f"
-    e'_def   "e'  == (%x: S. x)"
+theorem group_BijGroup: "group (BijGroup S)"
+apply (simp add: group_def semigroup_def group_axioms_def 
+                 BijGroup_def restrictI compose_Bij restrict_Inv_Bij id_Bij)
+apply (auto intro!: compose_Bij) 
+  apply (blast intro: compose_assoc [symmetric] Bij_imp_funcset)
+ apply (simp add: Bij_def compose_Inv_id) 
+apply (simp add: Id_compose Bij_imp_funcset Bij_imp_extensional) 
+done
+
+
+subsection{*Automorphisms Form a Group*}
+
+lemma Bij_Inv_mem: "[|  f \<in> Bij S;  x : S |] ==> Inv S f x : S"
+by (simp add: Bij_def Inv_mem) 
+
+lemma Bij_Inv_lemma:
+ assumes eq: "!!x y. [|x \<in> S; y \<in> S|] ==> h(g x y) = g (h x) (h y)"
+ shows "[| h \<in> Bij S;  g \<in> S \<rightarrow> S \<rightarrow> S;  x \<in> S;  y \<in> S |]        
+        ==> Inv S h (g x y) = g (Inv S h x) (Inv S h y)"
+apply (simp add: Bij_def) 
+apply (subgoal_tac "EX x':S. EX y':S. x = h x' & y = h y'")
+ apply clarify
+ apply (simp add: eq [symmetric] Inv_f_f funcset_mem [THEN funcset_mem], blast) 
+done
+
+constdefs
+ auto :: "('a,'b)group_scheme => ('a => 'a)set"
+  "auto G == hom G G Int Bij (carrier G)"
 
-locale bijgroup = bij +
-  fixes 
-    BG :: "('a => 'a) grouptype"
-  defines
-    BG_def "BG == BijGroup S"
+  AutoGroup :: "[('a,'c) group_scheme] => ('a=>'a) group"
+  "AutoGroup G == BijGroup (carrier G) (|carrier := auto G |)"
+
+lemma id_in_auto: "group G ==> (%x: carrier G. x) \<in> auto G"
+by (simp add: auto_def hom_def restrictI semigroup.sum_closed 
+              group.axioms id_Bij) 
+
+lemma restrict_Inv_hom:
+      "[|group G; h \<in> hom G G; h \<in> Bij (carrier G)|]
+       ==> restrict (Inv (carrier G) h) (carrier G) \<in> hom G G"
+by (simp add: hom_def Bij_Inv_mem restrictI semigroup.sum_closed 
+              semigroup.sum_funcset group.axioms Bij_Inv_lemma)
+
+lemma subgroup_auto:
+      "group G ==> subgroup (auto G) (BijGroup (carrier G))"
+apply (rule group.subgroupI) 
+    apply (rule group_BijGroup) 
+   apply (force simp add: auto_def BijGroup_def) 
+  apply (blast intro: dest: id_in_auto) 
+ apply (simp add: auto_def BijGroup_def restrict_Inv_Bij
+                  restrict_Inv_hom) 
+apply (simp add: auto_def BijGroup_def compose_Bij)
+apply (simp add: hom_def compose_def Pi_def group.axioms)
+done
+
+theorem AutoGroup: "group G ==> group (AutoGroup G)"
+apply (drule subgroup_auto) 
+apply (simp add: subgroup_def AutoGroup_def) 
+done
+
 end